Subgroups — C₃₁×#61084 #21517843

Order 7936.

2834 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#576	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3496	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3283	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3280	C₂	1	4	yes	maximal, complemented
5	D₄×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3273	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3272	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3061	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3036	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3012	C₂	1	4	yes	maximal, complemented
12	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	4	yes	complemented
15	C₃₁×#576	—	2	16	no	—
16	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
17	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
18	C₂×C₃₁×#135	C₂²	1	2	yes	complemented
19	C₂×C₃₁×#135	C₂²	1	4	yes	complemented
20	C₂×C₃₁×#135	C₂²	1	2	yes	complemented
21	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂²×D₈×C₃₁	C₂²	1	4	yes	complemented
23	C₂×C₃₁×#131	C₂²	1	2	yes	complemented
24	C₃₁×#547	—	2	2	no	—
25	C₃₁×#547	C₂²	1	4	yes	complemented
26	C₃₁×#546	—	2	4	no	—
27	C₃₁×#546	C₂²	1	8	yes	complemented
28	D₄²×C₃₁	—	2	2	no	—
29	D₄²×C₃₁	C₂²	1	4	yes	complemented
30	C₃₁×#537	C₂²	1	4	yes	complemented
31	C₃₁×#534	—	2	4	no	—
32	C₃₁×#532	C₂²	1	1	yes	characteristic
33	C₃₁×#532	C₂²	1	2	yes	complemented
34	C₂×C₃₁×#121	—	2	4	no	—
35	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
36	C₃₁×#495	C₂²	1	2	yes	complemented
37	C₃₁×#495	C₂²	1	2	yes	complemented
38	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
39	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
40	C₃₁×#492	C₂²	1	2	yes	complemented
41	C₃₁×#468	C₂²	1	8	yes	complemented
42	C₃₁×#467	C₂²	1	4	yes	complemented
43	C₃₁×#466	C₂²	1	4	yes	complemented
44	C₃₁×#463	C₂²	1	8	yes	complemented
45	C₃₁×#461	C₂²	1	4	yes	complemented
46	C₃₁×#459	C₂²	1	4	yes	complemented
47	C₃₁×#452	C₂²	1	8	yes	complemented
48	C₃₁×#450	C₂²	1	8	yes	complemented
49	C₃₁×#449	C₂²	1	8	yes	complemented
50	C₃₁×#447	C₂²	1	8	yes	complemented
51	C₄×D₈×C₃₁	C₂²	1	2	yes	complemented
52	C₄×D₈×C₃₁	C₂²	1	2	yes	complemented
53	C₄×D₈×C₃₁	C₂²	1	2	yes	complemented
54	C₄×D₈×C₃₁	C₂²	1	2	yes	complemented
55	C₃₁×#436	C₂²	1	2	yes	complemented
56	C₃₁×#436	C₂²	1	2	yes	complemented
57	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
58	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
59	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic, complemented
60	C₃₁×#423	C₂²	1	1	yes	characteristic, complemented
61	C₃₁×#417	C₂²	1	4	yes	complemented
62	C₂×C₃₁×#102	C₂²	1	2	yes	complemented
63	C₃₁×#408	C₂²	1	2	yes	complemented
64	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
65	C₃₁×H(5,𝔽₂)	—	2	16	no	—
66	C₃₁×H(5,𝔽₂)	—	4	8	no	—
67	C₂×#41×C₃₁	—	2	4	no	—
68	C₂×#41×C₃₁	C₂³	1	4	yes	—
69	C₂×#41×C₃₁	C₂³	1	8	yes	—
70	C₂×#41×C₃₁	C₂³	1	2	yes	—
71	C₂²×D₄×C₃₁	—	2	8	no	—
72	C₂²×D₄×C₃₁	—	2	4	no	—
73	C₂²×D₄×C₃₁	—	2	4	no	—
74	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
75	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
76	C₂³×C₄×C₃₁	—	2	2	no	—
77	C₃₁×AGL(1,ℤ/8)	—	2	16	no	—
78	C₃₁×AGL(1,ℤ/8)	—	2	32	no	—
79	C₃₁×#135	—	2	16	no	—
80	C₃₁×#135	—	2	8	no	—
81	C₃₁×#135	—	4	4	no	—
82	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
83	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
84	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
85	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
86	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
87	C₂×D₈×C₃₁	—	2	16	no	—
88	C₂×D₈×C₃₁	D₄	1	8	yes	complemented
89	C₂×D₈×C₃₁	C₂³	1	8	yes	—
90	C₂×D₈×C₃₁	C₂³	1	2	yes	—
91	C₂×D₈×C₃₁	C₂³	1	2	yes	—
92	C₂×D₈×C₃₁	C₂³	1	2	yes	—
93	C₂×D₈×C₃₁	C₂³	1	2	yes	—
94	C₃₁×#131	—	2	8	no	—
95	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
96	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
97	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
98	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
99	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
101	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
102	C₃₁×#127	D₄	1	1	yes	characteristic
103	C₃₁×#127	C₂³	1	2	yes	—
104	C₃₁×#124	D₄	1	2	yes	—
105	C₃₁×#124	C₂³	1	4	yes	—
106	C₃₁×#123	—	2	8	no	—
107	C₃₁×#123	—	2	8	no	—
108	C₃₁×C₂²⋊Q₈	—	2	4	no	—
109	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
110	C₃₁×#121	—	2	8	no	—
111	C₃₁×#121	—	2	4	no	—
112	C₃₁×#121	—	2	4	no	—
113	C₃₁×#121	—	2	2	no	—
114	C₃₁×#121	—	2	2	no	—
115	C₃₁×#121	—	2	4	no	—
116	C₃₁×#121	—	2	8	no	—
117	C₃₁×#121	D₄	1	2	yes	—
118	C₃₁×#121	D₄	1	2	yes	—
119	C₃₁×#121	C₂³	1	8	yes	—
120	C₃₁×#120	—	2	8	no	—
121	C₃₁×#120	—	2	8	no	—
122	C₄×D₄×C₃₁	—	2	2	no	—
123	C₄×D₄×C₃₁	—	2	2	no	—
124	C₄×D₄×C₃₁	—	2	2	no	—
125	C₄×D₄×C₃₁	—	2	2	no	—
126	C₄×D₄×C₃₁	C₂³	1	4	yes	—
127	C₄×D₄×C₃₁	C₂³	1	4	yes	—
128	C₄×D₄×C₃₁	C₂³	1	2	yes	—
129	C₄×D₄×C₃₁	C₂³	1	2	yes	—
130	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
131	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
132	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
133	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
134	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
135	C₃₁×#107	C₂³	1	2	yes	—
136	C₃₁×#107	C₂³	1	2	yes	—
137	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
138	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
139	C₃₁×#103	C₂³	1	8	yes	—
140	C₃₁×#102	C₂³	1	4	yes	—
141	C₃₁×#102	C₂³	1	4	yes	—
142	C₃₁×#102	C₂³	1	8	yes	—
143	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
144	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
145	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
146	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
147	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
148	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
149	C₂⁴×C₃₁	—	2	2	no	—
150	C₂⁴×C₃₁	—	2	8	no	—
151	#41×C₃₁	—	2	4	no	—
152	#41×C₃₁	—	2	4	no	—
153	#41×C₃₁	—	2	16	no	—
154	#41×C₃₁	—	2	8	no	—
155	#41×C₃₁	—	2	8	no	—
156	#41×C₃₁	—	4	8	no	—
157	#41×C₃₁	—	4	16	no	—
158	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
159	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
160	C₂×D₄×C₃₁	—	2	16	no	—
161	C₂×D₄×C₃₁	—	2	8	no	—
162	C₂×D₄×C₃₁	—	2	4	no	—
163	C₂×D₄×C₃₁	—	2	8	no	—
164	C₂×D₄×C₃₁	—	2	2	no	—
165	C₂×D₄×C₃₁	—	2	2	no	—
166	C₂×D₄×C₃₁	—	2	8	no	—
167	C₂×D₄×C₃₁	—	2	2	no	—
168	C₂×D₄×C₃₁	—	2	4	no	—
169	C₂×D₄×C₃₁	—	4	8	no	—
170	C₂×D₄×C₃₁	—	4	16	no	—
171	C₂×D₄×C₃₁	—	4	32	no	—
172	C₂×D₄×C₃₁	—	8	8	no	—
173	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
174	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
175	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
176	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
177	C₂×D₄×C₃₁	C₂×D₄	1	8	yes	complemented
178	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
179	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₃₁	—	2	2	no	—
181	C₂²×C₄×C₃₁	—	2	8	no	—
182	C₂²×C₄×C₃₁	—	2	4	no	—
183	C₂²×C₄×C₃₁	—	2	4	no	—
184	C₂²×C₄×C₃₁	—	2	4	no	—
185	C₂²×C₄×C₃₁	—	8	4	no	—
186	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
189	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
190	Q₁₆×C₃₁	—	2	4	no	—
191	Q₁₆×C₃₁	—	4	2	no	—
192	SD₁₆×C₃₁	—	2	16	no	—
193	SD₁₆×C₃₁	—	2	8	no	—
194	SD₁₆×C₃₁	—	4	4	no	—
195	D₈×C₃₁	—	2	16	no	—
196	D₈×C₃₁	—	2	4	no	—
197	D₈×C₃₁	—	4	2	no	—
198	D₈×C₃₁	C₂×D₄	1	8	yes	complemented
199	D₈×C₃₁	C₂×D₄	1	8	yes	complemented
200	M₁₆×C₃₁	—	2	4	no	—
201	M₁₆×C₃₁	—	2	8	no	—
202	C₂×C₈×C₃₁	—	2	2	no	—
203	C₂×C₈×C₃₁	—	2	8	no	—
204	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	complemented
205	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
206	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
207	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
208	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
209	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
210	C₄⋊C₄×C₃₁	—	2	2	no	—
211	C₄⋊C₄×C₃₁	—	2	8	no	—
212	C₄⋊C₄×C₃₁	—	2	2	no	—
213	C₄⋊C₄×C₃₁	—	2	2	no	—
214	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
215	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
216	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
217	C₂²⋊C₄×C₃₁	—	2	8	no	—
218	C₂²⋊C₄×C₃₁	—	2	4	no	—
219	C₂²⋊C₄×C₃₁	—	2	4	no	—
220	C₂²⋊C₄×C₃₁	—	2	4	no	—
221	C₂²⋊C₄×C₃₁	—	2	8	no	—
222	C₂²⋊C₄×C₃₁	—	2	4	no	—
223	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
224	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
225	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
226	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
227	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
228	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
229	#61084	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
230	C₂³×C₃₁	—	2	4	no	—
231	C₂³×C₃₁	—	2	8	no	—
232	C₂³×C₃₁	—	2	4	no	—
233	C₂³×C₃₁	—	4	16	no	—
234	C₂³×C₃₁	—	8	2	no	—
235	C₂³×C₃₁	—	8	2	no	—
236	C₂³×C₃₁	—	8	8	no	—
237	C₂³×C₃₁	—	8	8	no	—
238	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
239	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
240	Q₈×C₃₁	—	2	4	no	—
241	Q₈×C₃₁	—	2	4	no	—
242	Q₈×C₃₁	—	2	4	no	—
243	D₄×C₃₁	—	2	8	no	—
244	D₄×C₃₁	—	2	8	no	—
245	D₄×C₃₁	—	2	4	no	—
246	D₄×C₃₁	—	2	4	no	—
247	D₄×C₃₁	—	2	4	no	—
248	D₄×C₃₁	—	2	4	no	—
249	D₄×C₃₁	—	2	4	no	—
250	D₄×C₃₁	—	4	2	no	—
251	D₄×C₃₁	—	4	16	no	—
252	D₄×C₃₁	—	4	16	no	—
253	D₄×C₃₁	—	4	8	no	—
254	D₄×C₃₁	—	4	4	no	—
255	D₄×C₃₁	—	8	2	no	—
256	D₄×C₃₁	—	8	8	no	—
257	D₄×C₃₁	C₂²×D₄	1	8	yes	—
258	C₂×C₄×C₃₁	—	2	2	no	—
259	C₂×C₄×C₃₁	—	2	4	no	—
260	C₂×C₄×C₃₁	—	2	4	no	—
261	C₂×C₄×C₃₁	—	2	4	no	—
262	C₂×C₄×C₃₁	—	2	4	no	—
263	C₂×C₄×C₃₁	—	4	1	no	—
264	C₂×C₄×C₃₁	—	4	16	no	—
265	C₂×C₄×C₃₁	—	4	8	no	—
266	C₂×C₄×C₃₁	—	4	8	no	—
267	C₂×C₄×C₃₁	—	8	4	no	—
268	C₂×C₄×C₃₁	—	8	2	no	—
269	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
273	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
274	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
275	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
276	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
277	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
278	C₈×C₃₁	—	2	4	no	—
279	C₈×C₃₁	—	2	1	no	—
280	C₈×C₃₁	—	2	1	no	—
281	C₈×C₃₁	C₂²×D₄	1	2	yes	—
282	C₈×C₃₁	C₂²×D₄	1	2	yes	—
283	C₂×#576	C₂×C₃₁	1	2	yes	complemented
284	#3496	C₂×C₃₁	1	2	yes	complemented
285	#3283	C₂×C₃₁	1	2	yes	complemented
286	#3280	C₂×C₃₁	1	4	yes	complemented
287	D₄×D₈	C₂×C₃₁	1	2	yes	complemented
288	#3273	C₂×C₃₁	1	4	yes	complemented
289	#3272	C₂×C₃₁	1	4	yes	complemented
290	#3147	C₂×C₃₁	1	1	yes	characteristic, complemented
291	#3061	C₂×C₃₁	1	2	yes	complemented
292	#3036	C₂×C₃₁	1	2	yes	complemented
293	#3012	C₂×C₃₁	1	4	yes	complemented
294	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
295	#2931	C₂×C₃₁	1	1	yes	characteristic, complemented
296	C₂²×C₃₁	—	2	1	no	—
297	C₂²×C₃₁	—	2	1	no	—
298	C₂²×C₃₁	—	2	4	no	—
299	C₂²×C₃₁	—	2	4	no	—
300	C₂²×C₃₁	—	2	8	no	—
301	C₂²×C₃₁	—	2	4	no	—
302	C₂²×C₃₁	—	4	4	no	—
303	C₂²×C₃₁	—	4	4	no	—
304	C₂²×C₃₁	—	4	4	no	—
305	C₂²×C₃₁	—	4	4	no	—
306	C₂²×C₃₁	—	4	4	no	—
307	C₂²×C₃₁	—	8	8	no	—
308	C₂²×C₃₁	—	16	16	no	—
309	C₂²×C₃₁	D₄²	1	2	yes	—
310	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
311	C₄×C₃₁	—	2	1	no	—
312	C₄×C₃₁	—	2	1	no	—
313	C₄×C₃₁	—	2	4	no	—
314	C₄×C₃₁	—	2	1	no	—
315	C₄×C₃₁	—	4	4	no	—
316	C₄×C₃₁	—	4	4	no	—
317	C₄×C₃₁	D₄²	1	2	yes	—
318	C₄×C₃₁	#576	1	2	yes	complemented
319	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
320	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
321	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	4	yes	complemented
322	#576	—	2	16	no	—
323	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	complemented
324	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	8	yes	complemented
325	C₂×#135	C₂²×C₃₁	1	2	yes	complemented
326	C₂×#135	C₂²×C₃₁	1	4	yes	complemented
327	C₂×#135	C₂²×C₃₁	1	2	yes	complemented
328	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
329	C₂²×D₈	C₂²×C₃₁	1	4	yes	complemented
330	C₂×#131	C₂²×C₃₁	1	2	yes	complemented
331	#547	—	2	2	no	—
332	#547	C₂²×C₃₁	1	4	yes	complemented
333	#546	—	2	4	no	—
334	#546	C₂²×C₃₁	1	8	yes	complemented
335	D₄²	—	2	2	no	—
336	D₄²	C₂²×C₃₁	1	4	yes	complemented
337	#537	C₂²×C₃₁	1	4	yes	complemented
338	#534	—	2	4	no	—
339	#532	C₂²×C₃₁	1	1	yes	characteristic
340	#532	C₂²×C₃₁	1	2	yes	complemented
341	C₂×#121	—	2	4	no	—
342	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
343	#495	C₂²×C₃₁	1	2	yes	complemented
344	#495	C₂²×C₃₁	1	2	yes	complemented
345	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
346	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
347	#492	C₂²×C₃₁	1	2	yes	complemented
348	#468	C₂²×C₃₁	1	8	yes	complemented
349	#467	C₂²×C₃₁	1	4	yes	complemented
350	#466	C₂²×C₃₁	1	4	yes	complemented
351	#463	C₂²×C₃₁	1	8	yes	complemented
352	#461	C₂²×C₃₁	1	4	yes	complemented
353	#459	C₂²×C₃₁	1	4	yes	complemented
354	#452	C₂²×C₃₁	1	8	yes	complemented
355	#450	C₂²×C₃₁	1	8	yes	complemented
356	#449	C₂²×C₃₁	1	8	yes	complemented
357	#447	C₂²×C₃₁	1	8	yes	complemented
358	C₄×D₈	C₂²×C₃₁	1	2	yes	complemented
359	C₄×D₈	C₂²×C₃₁	1	2	yes	complemented
360	C₄×D₈	C₂²×C₃₁	1	2	yes	complemented
361	C₄×D₈	C₂²×C₃₁	1	2	yes	complemented
362	#436	C₂²×C₃₁	1	2	yes	complemented
363	#436	C₂²×C₃₁	1	2	yes	complemented
364	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
365	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
366	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic, complemented
367	#423	C₂²×C₃₁	1	1	yes	characteristic, complemented
368	#417	C₂²×C₃₁	1	4	yes	complemented
369	C₂×#102	C₂²×C₃₁	1	2	yes	complemented
370	#408	C₂²×C₃₁	1	2	yes	complemented
371	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
372	C₂×C₃₁	—	2	2	no	—
373	C₂×C₃₁	—	4	4	no	—
374	C₂×C₃₁	—	4	8	no	—
375	C₂×C₃₁	—	8	4	no	—
376	C₂×C₃₁	C₂×D₄²	1	1	yes	characteristic
377	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
378	C₂×C₃₁	#3584	1	1	yes	characteristic
379	H(5,𝔽₂)	—	2	16	no	—
380	H(5,𝔽₂)	—	4	8	no	—
381	C₂×#41	—	2	4	no	—
382	C₂×#41	C₂³×C₃₁	1	4	yes	—
383	C₂×#41	C₂³×C₃₁	1	8	yes	—
384	C₂×#41	C₂³×C₃₁	1	2	yes	—
385	C₂²×D₄	—	2	8	no	—
386	C₂²×D₄	—	2	4	no	—
387	C₂²×D₄	—	2	4	no	—
388	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
389	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
390	C₂³×C₄	—	2	2	no	—
391	AGL(1,ℤ/8)	—	2	16	no	—
392	AGL(1,ℤ/8)	—	2	32	no	—
393	#135	—	2	16	no	—
394	#135	—	2	8	no	—
395	#135	—	4	4	no	—
396	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
397	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
398	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
399	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
400	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
401	C₂×D₈	—	2	16	no	—
402	C₂×D₈	D₄×C₃₁	1	8	yes	complemented
403	C₂×D₈	C₂³×C₃₁	1	2	yes	—
404	C₂×D₈	C₂³×C₃₁	1	8	yes	—
405	C₂×D₈	C₂³×C₃₁	1	2	yes	—
406	C₂×D₈	C₂³×C₃₁	1	2	yes	—
407	C₂×D₈	C₂³×C₃₁	1	2	yes	—
408	#131	—	2	8	no	—
409	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
410	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
411	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
412	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
413	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
414	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
415	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
416	#127	D₄×C₃₁	1	1	yes	characteristic
417	#127	C₂³×C₃₁	1	2	yes	—
418	#124	D₄×C₃₁	1	2	yes	—
419	#124	C₂³×C₃₁	1	4	yes	—
420	#123	—	2	8	no	—
421	#123	—	2	8	no	—
422	C₂²⋊Q₈	—	2	4	no	—
423	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
424	#121	—	2	8	no	—
425	#121	—	2	4	no	—
426	#121	—	2	4	no	—
427	#121	—	2	2	no	—
428	#121	—	2	2	no	—
429	#121	—	2	4	no	—
430	#121	—	2	8	no	—
431	#121	D₄×C₃₁	1	2	yes	—
432	#121	D₄×C₃₁	1	2	yes	—
433	#121	C₂³×C₃₁	1	8	yes	—
434	#120	—	2	8	no	—
435	#120	—	2	8	no	—
436	C₄×D₄	—	2	2	no	—
437	C₄×D₄	—	2	2	no	—
438	C₄×D₄	—	2	2	no	—
439	C₄×D₄	—	2	2	no	—
440	C₄×D₄	C₂³×C₃₁	1	4	yes	—
441	C₄×D₄	C₂³×C₃₁	1	4	yes	—
442	C₄×D₄	C₂³×C₃₁	1	2	yes	—
443	C₄×D₄	C₂³×C₃₁	1	2	yes	—
444	C₂×C₄⋊C₄	—	2	4	no	—
445	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
446	C₂×C₂²⋊C₄	—	2	2	no	—
447	C₂×C₂²⋊C₄	—	2	8	no	—
448	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
449	#107	C₂³×C₃₁	1	2	yes	—
450	#107	C₂³×C₃₁	1	2	yes	—
451	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
452	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
453	#103	C₂³×C₃₁	1	8	yes	—
454	#102	C₂³×C₃₁	1	4	yes	—
455	#102	C₂³×C₃₁	1	4	yes	—
456	#102	C₂³×C₃₁	1	8	yes	—
457	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
458	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
459	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
460	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
461	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
462	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
463	C₃₁	#61084	1	1	yes	characteristic, direct product factor, Sylow
464	C₂⁴	—	2	2	no	—
465	C₂⁴	—	2	8	no	—
466	#41	—	2	4	no	—
467	#41	—	2	4	no	—
468	#41	—	2	16	no	—
469	#41	—	2	8	no	—
470	#41	—	2	8	no	—
471	#41	—	4	8	no	—
472	#41	—	4	16	no	—
473	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
474	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
475	C₂×D₄	—	2	16	no	—
476	C₂×D₄	—	2	8	no	—
477	C₂×D₄	—	2	4	no	—
478	C₂×D₄	—	2	8	no	—
479	C₂×D₄	—	2	2	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	2	8	no	—
482	C₂×D₄	—	2	2	no	—
483	C₂×D₄	—	2	4	no	—
484	C₂×D₄	—	4	8	no	—
485	C₂×D₄	—	4	16	no	—
486	C₂×D₄	—	4	32	no	—
487	C₂×D₄	—	8	8	no	—
488	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
489	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
490	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
491	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
492	C₂×D₄	C₂×D₄×C₃₁	1	8	yes	complemented
493	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
494	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
495	C₂²×C₄	—	2	2	no	—
496	C₂²×C₄	—	2	8	no	—
497	C₂²×C₄	—	2	4	no	—
498	C₂²×C₄	—	2	4	no	—
499	C₂²×C₄	—	2	4	no	—
500	C₂²×C₄	—	8	4	no	—
501	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
502	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
503	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
504	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
505	Q₁₆	—	2	4	no	—
506	Q₁₆	—	4	2	no	—
507	SD₁₆	—	2	16	no	—
508	SD₁₆	—	2	8	no	—
509	SD₁₆	—	4	4	no	—
510	D₈	—	2	16	no	—
511	D₈	—	2	4	no	—
512	D₈	—	4	2	no	—
513	D₈	C₂×D₄×C₃₁	1	8	yes	complemented
514	D₈	C₂×D₄×C₃₁	1	8	yes	complemented
515	M₁₆	—	2	4	no	—
516	M₁₆	—	2	8	no	—
517	C₂×C₈	—	2	2	no	—
518	C₂×C₈	—	2	8	no	—
519	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	complemented
520	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
521	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
522	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
523	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
524	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
525	C₄⋊C₄	—	2	2	no	—
526	C₄⋊C₄	—	2	8	no	—
527	C₄⋊C₄	—	2	2	no	—
528	C₄⋊C₄	—	2	2	no	—
529	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
530	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
531	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
532	C₂²⋊C₄	—	2	8	no	—
533	C₂²⋊C₄	—	2	4	no	—
534	C₂²⋊C₄	—	2	4	no	—
535	C₂²⋊C₄	—	2	4	no	—
536	C₂²⋊C₄	—	2	8	no	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
539	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
540	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
541	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
542	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
543	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
544	C₂³	—	2	4	no	—
545	C₂³	—	2	8	no	—
546	C₂³	—	2	4	no	—
547	C₂³	—	4	16	no	—
548	C₂³	—	8	2	no	—
549	C₂³	—	8	2	no	—
550	C₂³	—	8	8	no	—
551	C₂³	—	8	8	no	—
552	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
553	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
554	Q₈	—	2	4	no	—
555	Q₈	—	2	4	no	—
556	Q₈	—	2	4	no	—
557	D₄	—	2	8	no	—
558	D₄	—	2	8	no	—
559	D₄	—	2	4	no	—
560	D₄	—	2	4	no	—
561	D₄	—	2	4	no	—
562	D₄	—	2	4	no	—
563	D₄	—	2	4	no	—
564	D₄	—	4	2	no	—
565	D₄	—	4	16	no	—
566	D₄	—	4	16	no	—
567	D₄	—	4	8	no	—
568	D₄	—	4	4	no	—
569	D₄	—	8	2	no	—
570	D₄	—	8	8	no	—
571	D₄	C₂²×D₄×C₃₁	1	8	yes	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	4	no	—
574	C₂×C₄	—	2	4	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	16	no	—
579	C₂×C₄	—	4	8	no	—
580	C₂×C₄	—	4	8	no	—
581	C₂×C₄	—	8	4	no	—
582	C₂×C₄	—	8	2	no	—
583	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
584	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
585	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
586	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
587	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
588	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
589	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
590	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
591	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
592	C₈	—	2	4	no	—
593	C₈	—	2	1	no	—
594	C₈	—	2	1	no	—
595	C₈	C₂²×D₄×C₃₁	1	2	yes	—
596	C₈	C₂²×D₄×C₃₁	1	2	yes	—
597	C₂²	—	2	1	no	—
598	C₂²	—	2	1	no	—
599	C₂²	—	2	4	no	—
600	C₂²	—	2	8	no	—
601	C₂²	—	2	4	no	—
602	C₂²	—	2	4	no	—
603	C₂²	—	4	4	no	—
604	C₂²	—	4	4	no	—
605	C₂²	—	4	4	no	—
606	C₂²	—	4	4	no	—
607	C₂²	—	4	4	no	—
608	C₂²	—	8	8	no	—
609	C₂²	—	16	16	no	—
610	C₂²	D₄²×C₃₁	1	2	yes	—
611	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
612	C₄	—	2	4	no	—
613	C₄	—	2	1	no	—
614	C₄	—	2	1	no	—
615	C₄	—	2	1	no	—
616	C₄	—	4	4	no	—
617	C₄	—	4	4	no	—
618	C₄	D₄²×C₃₁	1	2	yes	—
619	C₄	C₃₁×#576	1	2	yes	complemented
620	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
621	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
622	C₂	—	2	2	no	—
623	C₂	—	4	4	no	—
624	C₂	—	4	8	no	—
625	C₂	—	8	4	no	—
626	C₂	C₂×D₄²×C₃₁	1	1	yes	characteristic
627	C₂	C₂×C₃₁×#576	1	1	yes	characteristic
628	C₂	C₃₁×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#532 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₃₁×#576	C₂	semidirect
2. C₃₁×#3496	C₂	semidirect
3. C₃₁×#3283	C₂	semidirect
4. C₃₁×#3280	C₂	semidirect
5. D₄×D₈×C₃₁	C₂	semidirect
6. C₃₁×#3273	C₂	semidirect
7. C₃₁×#3272	C₂	semidirect
8. C₃₁×#3147	C₂	semidirect
9. C₃₁×#3061	C₂	semidirect
10. C₃₁×#3036	C₂	semidirect
11. C₃₁×#3012	C₂	semidirect
12. C₂×C₄×D₈×C₃₁	C₂	semidirect
13. C₃₁×#2931	C₂	semidirect
14. C₂×C₃₁×H(5,𝔽₂)	C₂²	semidirect
16. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
17. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
18. C₂×C₃₁×#135	C₂²	semidirect
19. C₂×C₃₁×#135	C₂²	semidirect
20. C₂×C₃₁×#135	C₂²	semidirect
22. C₂²×D₈×C₃₁	C₂²	semidirect
23. C₂×C₃₁×#131	C₂²	semidirect
25. C₃₁×#547	C₂²	semidirect
27. C₃₁×#546	C₂²	semidirect
29. D₄²×C₃₁	C₂²	semidirect
30. C₃₁×#537	C₂²	semidirect
33. C₃₁×#532	C₂²	semidirect
35. C₂×C₄×D₄×C₃₁	C₂²	semidirect
36. C₃₁×#495	C₂²	semidirect
37. C₃₁×#495	C₂²	semidirect
38. C₃₁×#494	C₂²	semidirect
39. C₃₁×#493	C₂²	semidirect
40. C₃₁×#492	C₂²	semidirect
41. C₃₁×#468	C₂²	semidirect
42. C₃₁×#467	C₂²	semidirect
43. C₃₁×#466	C₂²	semidirect
44. C₃₁×#463	C₂²	semidirect
45. C₃₁×#461	C₂²	semidirect
46. C₃₁×#459	C₂²	semidirect
47. C₃₁×#452	C₂²	semidirect
48. C₃₁×#450	C₂²	semidirect
49. C₃₁×#449	C₂²	semidirect
50. C₃₁×#447	C₂²	semidirect
51. C₄×D₈×C₃₁	C₂²	semidirect
52. C₄×D₈×C₃₁	C₂²	semidirect
53. C₄×D₈×C₃₁	C₂²	semidirect
54. C₄×D₈×C₃₁	C₂²	semidirect
55. C₃₁×#436	C₂²	semidirect
56. C₃₁×#436	C₂²	semidirect
57. C₈×D₄×C₃₁	C₂²	semidirect
58. C₈×D₄×C₃₁	C₂²	semidirect
59. C₂×C₃₁×#107	C₂²	semidirect
60. C₃₁×#423	C₂²	semidirect
61. C₃₁×#417	C₂²	semidirect
62. C₂×C₃₁×#102	C₂²	semidirect
63. C₃₁×#408	C₂²	semidirect
64. C₂×C₄×C₈×C₃₁	C₂²	semidirect
88. C₂×D₈×C₃₁	D₄	semidirect
177. C₂×D₄×C₃₁	C₂×D₄	semidirect
198. D₈×C₃₁	C₂×D₄	semidirect
199. D₈×C₃₁	C₂×D₄	semidirect
204. C₂×C₈×C₃₁	C₂×D₄	semidirect
229. #61084	C₃₁	direct (N × H)
283. C₂×#576	C₂×C₃₁	semidirect
284. #3496	C₂×C₃₁	semidirect
285. #3283	C₂×C₃₁	semidirect
286. #3280	C₂×C₃₁	semidirect
287. D₄×D₈	C₂×C₃₁	semidirect
288. #3273	C₂×C₃₁	semidirect
289. #3272	C₂×C₃₁	semidirect
290. #3147	C₂×C₃₁	semidirect
291. #3061	C₂×C₃₁	semidirect
292. #3036	C₂×C₃₁	semidirect
293. #3012	C₂×C₃₁	semidirect
294. C₂×C₄×D₈	C₂×C₃₁	semidirect
295. #2931	C₂×C₃₁	semidirect
318. C₄×C₃₁	#576	semidirect
321. C₂×H(5,𝔽₂)	C₂²×C₃₁	semidirect
323. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
324. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
325. C₂×#135	C₂²×C₃₁	semidirect
326. C₂×#135	C₂²×C₃₁	semidirect
327. C₂×#135	C₂²×C₃₁	semidirect
329. C₂²×D₈	C₂²×C₃₁	semidirect
330. C₂×#131	C₂²×C₃₁	semidirect
332. #547	C₂²×C₃₁	semidirect
334. #546	C₂²×C₃₁	semidirect
336. D₄²	C₂²×C₃₁	semidirect
337. #537	C₂²×C₃₁	semidirect
340. #532	C₂²×C₃₁	semidirect
342. C₂×C₄×D₄	C₂²×C₃₁	semidirect
343. #495	C₂²×C₃₁	semidirect
344. #495	C₂²×C₃₁	semidirect
345. #494	C₂²×C₃₁	semidirect
346. #493	C₂²×C₃₁	semidirect
347. #492	C₂²×C₃₁	semidirect
348. #468	C₂²×C₃₁	semidirect
349. #467	C₂²×C₃₁	semidirect
350. #466	C₂²×C₃₁	semidirect
351. #463	C₂²×C₃₁	semidirect
352. #461	C₂²×C₃₁	semidirect
353. #459	C₂²×C₃₁	semidirect
354. #452	C₂²×C₃₁	semidirect
355. #450	C₂²×C₃₁	semidirect
356. #449	C₂²×C₃₁	semidirect
357. #447	C₂²×C₃₁	semidirect
358. C₄×D₈	C₂²×C₃₁	semidirect
359. C₄×D₈	C₂²×C₃₁	semidirect
360. C₄×D₈	C₂²×C₃₁	semidirect
361. C₄×D₈	C₂²×C₃₁	semidirect
362. #436	C₂²×C₃₁	semidirect
363. #436	C₂²×C₃₁	semidirect
364. C₈×D₄	C₂²×C₃₁	semidirect
365. C₈×D₄	C₂²×C₃₁	semidirect
366. C₂×#107	C₂²×C₃₁	semidirect
367. #423	C₂²×C₃₁	semidirect
368. #417	C₂²×C₃₁	semidirect
369. C₂×#102	C₂²×C₃₁	semidirect
370. #408	C₂²×C₃₁	semidirect
371. C₂×C₄×C₈	C₂²×C₃₁	semidirect
402. C₂×D₈	D₄×C₃₁	semidirect
463. C₃₁	#61084	direct (N × H)
492. C₂×D₄	C₂×D₄×C₃₁	semidirect
513. D₈	C₂×D₄×C₃₁	semidirect
514. D₈	C₂×D₄×C₃₁	semidirect
519. C₂×C₈	C₂×D₄×C₃₁	semidirect
619. C₄	C₃₁×#576	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#576	2
C₃₁×#3496	2
C₃₁×#3283	2
C₃₁×#3280	2
D₄×D₈×C₃₁	2
C₃₁×#3273	2
C₃₁×#3272	2
C₃₁×#3147	2
C₃₁×#3061	2
C₃₁×#3036	2
C₃₁×#3012	2
C₂×C₄×D₈×C₃₁	2
C₃₁×#2931	2
#61084	31

Subgroups — C31×#61084 #21517843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61084 #21517843