Subgroups — C₃₁×#61089 #21517848

Order 7936.

2922 conjugacy classes of subgroups.

The subgroup lattice has 506 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×#3498	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3287	C₂	1	4	yes	maximal, complemented
4	D₄×D₈×C₃₁	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3274	C₂	1	8	yes	maximal, complemented
6	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3058	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3038	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3008	C₂	1	4	yes	maximal, complemented
10	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	4	yes	complemented
13	C₃₁×#576	—	2	16	no	—
14	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
15	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
16	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#135	C₂²	1	4	yes	complemented
18	C₂²×D₈×C₃₁	C₂²	1	2	yes	complemented
19	C₂²×D₈×C₃₁	C₂²	1	4	yes	complemented
20	C₂²×D₈×C₃₁	C₂²	1	2	yes	complemented
21	C₂×C₃₁×#131	C₂²	1	2	yes	complemented
22	C₃₁×#550	C₂²	1	8	yes	complemented
23	D₄²×C₃₁	—	2	8	no	—
24	D₄²×C₃₁	C₂²	1	8	yes	complemented
25	C₃₁×#535	—	2	4	no	—
26	C₃₁×#535	C₂²	1	4	yes	complemented
27	C₃₁×#532	—	2	2	no	—
28	C₃₁×#532	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#127	—	2	2	no	—
30	C₂×C₃₁×#127	C₂²	1	2	yes	complemented
31	C₄×#41×C₃₁	C₂²	1	2	yes	complemented
32	C₃₁×#493	C₂²	1	4	yes	complemented
33	C₃₁×#493	C₂²	1	4	yes	complemented
34	C₃₁×#469	C₂²	1	8	yes	complemented
35	C₃₁×#466	C₂²	1	8	yes	complemented
36	C₃₁×#460	C₂²	1	8	yes	complemented
37	C₃₁×#459	C₂²	1	8	yes	complemented
38	C₃₁×#449	C₂²	1	16	yes	complemented
39	C₃₁×#447	C₂²	1	16	yes	complemented
40	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
41	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
42	C₄×D₈×C₃₁	C₂²	1	2	yes	complemented
43	C₃₁×#436	C₂²	1	4	yes	complemented
44	C₈×D₄×C₃₁	C₂²	1	4	yes	complemented
45	C₃₁×#427	C₂²	1	1	yes	characteristic, complemented
46	C₃₁×#423	C₂²	1	1	yes	characteristic, complemented
47	C₃₁×#418	C₂²	1	4	yes	complemented
48	C₃₁×#416	C₂²	1	2	yes	complemented
49	C₃₁×#408	C₂²	1	2	yes	complemented
50	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
51	C₃₁×H(5,𝔽₂)	—	2	16	no	—
52	C₃₁×H(5,𝔽₂)	—	4	8	no	—
53	C₂×#41×C₃₁	—	2	4	no	—
54	C₂×#41×C₃₁	C₂³	1	8	yes	—
55	C₂×#41×C₃₁	C₂³	1	2	yes	—
56	C₂×#41×C₃₁	C₂³	1	2	yes	—
57	C₂²×D₄×C₃₁	—	2	4	no	—
58	C₂²×D₄×C₃₁	—	2	8	no	—
59	C₂²×D₄×C₃₁	—	2	8	no	—
60	C₂²×D₄×C₃₁	—	2	4	no	—
61	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
62	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
63	C₃₁×AGL(1,ℤ/8)	—	2	16	no	—
64	C₃₁×AGL(1,ℤ/8)	—	2	32	no	—
65	C₃₁×#135	—	2	16	no	—
66	C₃₁×#135	D₄	1	8	yes	complemented
67	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
69	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
70	C₂×D₈×C₃₁	—	2	16	no	—
71	C₂×D₈×C₃₁	—	2	8	no	—
72	C₂×D₈×C₃₁	—	4	4	no	—
73	C₂×D₈×C₃₁	C₂³	1	8	yes	—
74	C₂×D₈×C₃₁	C₂³	1	4	yes	—
75	C₂×D₈×C₃₁	C₂³	1	4	yes	—
76	C₂×D₈×C₃₁	C₂³	1	8	yes	—
77	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₃₁×#131	—	2	8	no	—
79	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
80	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
81	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
83	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
85	C₃₁×#127	—	2	4	no	—
86	C₃₁×#127	—	2	8	no	—
87	C₃₁×#127	—	2	2	no	—
88	C₃₁×#127	—	2	2	no	—
89	C₃₁×#127	D₄	1	1	yes	characteristic
90	C₃₁×#127	C₂³	1	8	yes	—
91	C₃₁×#124	—	2	4	no	—
92	C₃₁×#124	D₄	1	2	yes	—
93	C₃₁×#121	—	2	8	no	—
94	C₃₁×#121	—	2	16	no	—
95	C₃₁×#121	D₄	1	4	yes	—
96	C₃₁×#121	C₂³	1	16	yes	—
97	C₃₁×#120	—	2	8	no	—
98	C₃₁×#120	—	2	16	no	—
99	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
100	C₄×D₄×C₃₁	—	2	4	no	—
101	C₄×D₄×C₃₁	—	2	8	no	—
102	C₄×D₄×C₃₁	C₂³	1	4	yes	—
103	C₄×D₄×C₃₁	C₂³	1	4	yes	—
104	C₃₁×#117	—	2	2	no	—
105	C₃₁×#117	C₂³	1	2	yes	—
106	C₂×C₄²×C₃₁	—	2	2	no	—
107	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
108	C₃₁×#107	C₂³	1	2	yes	—
109	C₃₁×#106	C₂³	1	2	yes	—
110	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
111	C₃₁×#103	C₂³	1	4	yes	—
112	C₃₁×#102	C₂³	1	4	yes	—
113	C₃₁×#102	C₂³	1	16	yes	—
114	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
115	C₄×C₈×C₃₁	C₂³	1	2	yes	—
116	C₄×C₈×C₃₁	C₂³	1	2	yes	—
117	C₂⁴×C₃₁	—	2	8	no	—
118	C₂⁴×C₃₁	—	2	4	no	—
119	#41×C₃₁	—	2	4	no	—
120	#41×C₃₁	—	2	4	no	—
121	#41×C₃₁	—	2	4	no	—
122	#41×C₃₁	—	2	16	no	—
123	#41×C₃₁	—	4	8	no	—
124	#41×C₃₁	—	4	16	no	—
125	#41×C₃₁	C₂×D₄	1	8	yes	complemented
126	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
127	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
128	C₂×D₄×C₃₁	—	2	2	no	—
129	C₂×D₄×C₃₁	—	2	16	no	—
130	C₂×D₄×C₃₁	—	2	8	no	—
131	C₂×D₄×C₃₁	—	2	16	no	—
132	C₂×D₄×C₃₁	—	2	8	no	—
133	C₂×D₄×C₃₁	—	2	8	no	—
134	C₂×D₄×C₃₁	—	2	8	no	—
135	C₂×D₄×C₃₁	—	2	4	no	—
136	C₂×D₄×C₃₁	—	2	8	no	—
137	C₂×D₄×C₃₁	—	2	8	no	—
138	C₂×D₄×C₃₁	—	4	8	no	—
139	C₂×D₄×C₃₁	—	4	16	no	—
140	C₂×D₄×C₃₁	—	4	32	no	—
141	C₂×D₄×C₃₁	—	8	8	no	—
142	C₂×D₄×C₃₁	—	8	16	no	—
143	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
144	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
145	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
146	C₂×D₄×C₃₁	C₂⁴	1	8	yes	—
147	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
148	C₂²×C₄×C₃₁	—	2	2	no	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	2	8	no	—
151	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
154	Q₁₆×C₃₁	C₂×D₄	1	4	yes	complemented
155	SD₁₆×C₃₁	—	2	16	no	—
156	SD₁₆×C₃₁	C₂×D₄	1	8	yes	complemented
157	D₈×C₃₁	—	2	16	no	—
158	D₈×C₃₁	—	2	8	no	—
159	D₈×C₃₁	—	2	8	no	—
160	D₈×C₃₁	—	4	8	no	—
161	D₈×C₃₁	C₂×D₄	1	4	yes	complemented
162	M₁₆×C₃₁	—	2	4	no	—
163	M₁₆×C₃₁	—	2	8	no	—
164	C₂×C₈×C₃₁	—	2	2	no	—
165	C₂×C₈×C₃₁	—	2	8	no	—
166	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	complemented
167	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
168	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
169	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
170	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
171	C₄⋊C₄×C₃₁	—	2	4	no	—
172	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
173	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
174	C₂²⋊C₄×C₃₁	—	2	2	no	—
175	C₂²⋊C₄×C₃₁	—	2	4	no	—
176	C₂²⋊C₄×C₃₁	—	2	16	no	—
177	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
178	C₄²×C₃₁	—	2	2	no	—
179	C₄²×C₃₁	—	2	4	no	—
180	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
181	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₃₁	C₂⁴	1	2	yes	—
183	#61089	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂³×C₃₁	—	2	2	no	—
185	C₂³×C₃₁	—	2	8	no	—
186	C₂³×C₃₁	—	4	16	no	—
187	C₂³×C₃₁	—	4	8	no	—
188	C₂³×C₃₁	—	8	8	no	—
189	C₂³×C₃₁	—	8	16	no	—
190	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
191	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
192	Q₈×C₃₁	—	2	4	no	—
193	Q₈×C₃₁	—	2	2	no	—
194	Q₈×C₃₁	C₂²×D₄	1	4	yes	—
195	D₄×C₃₁	—	2	8	no	—
196	D₄×C₃₁	—	2	8	no	—
197	D₄×C₃₁	—	2	4	no	—
198	D₄×C₃₁	—	2	8	no	—
199	D₄×C₃₁	—	2	8	no	—
200	D₄×C₃₁	—	2	2	no	—
201	D₄×C₃₁	—	2	8	no	—
202	D₄×C₃₁	—	4	16	no	—
203	D₄×C₃₁	—	4	16	no	—
204	D₄×C₃₁	—	4	8	no	—
205	D₄×C₃₁	—	4	16	no	—
206	D₄×C₃₁	—	8	8	no	—
207	D₄×C₃₁	—	8	16	no	—
208	D₄×C₃₁	C₂²×D₄	1	4	yes	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	8	no	—
211	C₂×C₄×C₃₁	—	2	4	no	—
212	C₂×C₄×C₃₁	—	2	4	no	—
213	C₂×C₄×C₃₁	—	2	4	no	—
214	C₂×C₄×C₃₁	—	2	4	no	—
215	C₂×C₄×C₃₁	—	4	16	no	—
216	C₂×C₄×C₃₁	—	4	8	no	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
218	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
219	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
221	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
222	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
223	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
224	C₈×C₃₁	—	2	4	no	—
225	C₈×C₃₁	—	2	2	no	—
226	C₈×C₃₁	C₂²×D₄	1	2	yes	—
227	C₈×C₃₁	C₂²×D₄	1	2	yes	—
228	C₂×#576	C₂×C₃₁	1	2	yes	complemented
229	#3498	C₂×C₃₁	1	2	yes	complemented
230	#3287	C₂×C₃₁	1	4	yes	complemented
231	D₄×D₈	C₂×C₃₁	1	4	yes	complemented
232	#3274	C₂×C₃₁	1	8	yes	complemented
233	C₂×#493	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#3058	C₂×C₃₁	1	2	yes	complemented
235	#3038	C₂×C₃₁	1	2	yes	complemented
236	#3008	C₂×C₃₁	1	4	yes	complemented
237	C₄×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#2931	C₂×C₃₁	1	1	yes	characteristic, complemented
239	C₂²×C₃₁	—	2	2	no	—
240	C₂²×C₃₁	—	2	4	no	—
241	C₂²×C₃₁	—	2	4	no	—
242	C₂²×C₃₁	—	2	8	no	—
243	C₂²×C₃₁	—	4	4	no	—
244	C₂²×C₃₁	—	4	8	no	—
245	C₂²×C₃₁	—	4	16	no	—
246	C₂²×C₃₁	—	16	8	no	—
247	C₂²×C₃₁	—	16	16	no	—
248	C₂²×C₃₁	D₄²	1	2	yes	—
249	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
250	C₄×C₃₁	—	2	4	no	—
251	C₄×C₃₁	—	2	4	no	—
252	C₄×C₃₁	—	2	2	no	—
253	C₄×C₃₁	—	4	4	no	—
254	C₄×C₃₁	D₄²	1	2	yes	—
255	C₄×C₃₁	#576	1	4	yes	complemented
256	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
257	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
258	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	4	yes	complemented
259	#576	—	2	16	no	—
260	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	complemented
261	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	8	yes	complemented
262	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×#135	C₂²×C₃₁	1	4	yes	complemented
264	C₂²×D₈	C₂²×C₃₁	1	2	yes	complemented
265	C₂²×D₈	C₂²×C₃₁	1	4	yes	complemented
266	C₂²×D₈	C₂²×C₃₁	1	2	yes	complemented
267	C₂×#131	C₂²×C₃₁	1	2	yes	complemented
268	#550	C₂²×C₃₁	1	8	yes	complemented
269	D₄²	—	2	8	no	—
270	D₄²	C₂²×C₃₁	1	8	yes	complemented
271	#535	—	2	4	no	—
272	#535	C₂²×C₃₁	1	4	yes	complemented
273	#532	—	2	2	no	—
274	#532	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×#127	—	2	2	no	—
276	C₂×#127	C₂²×C₃₁	1	2	yes	complemented
277	C₄×#41	C₂²×C₃₁	1	2	yes	complemented
278	#493	C₂²×C₃₁	1	4	yes	complemented
279	#493	C₂²×C₃₁	1	4	yes	complemented
280	#469	C₂²×C₃₁	1	8	yes	complemented
281	#466	C₂²×C₃₁	1	8	yes	complemented
282	#460	C₂²×C₃₁	1	8	yes	complemented
283	#459	C₂²×C₃₁	1	8	yes	complemented
284	#449	C₂²×C₃₁	1	16	yes	complemented
285	#447	C₂²×C₃₁	1	16	yes	complemented
286	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
287	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
288	C₄×D₈	C₂²×C₃₁	1	2	yes	complemented
289	#436	C₂²×C₃₁	1	4	yes	complemented
290	C₈×D₄	C₂²×C₃₁	1	4	yes	complemented
291	#427	C₂²×C₃₁	1	1	yes	characteristic, complemented
292	#423	C₂²×C₃₁	1	1	yes	characteristic, complemented
293	#418	C₂²×C₃₁	1	4	yes	complemented
294	#416	C₂²×C₃₁	1	2	yes	complemented
295	#408	C₂²×C₃₁	1	2	yes	complemented
296	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
297	C₂×C₃₁	—	2	2	no	—
298	C₂×C₃₁	—	4	4	no	—
299	C₂×C₃₁	—	4	4	no	—
300	C₂×C₃₁	—	8	8	no	—
301	C₂×C₃₁	C₂×D₄²	1	1	yes	characteristic
302	C₂×C₃₁	C₂×#576	1	2	yes	—
303	H(5,𝔽₂)	—	2	16	no	—
304	H(5,𝔽₂)	—	4	8	no	—
305	C₂×#41	—	2	4	no	—
306	C₂×#41	C₂³×C₃₁	1	8	yes	—
307	C₂×#41	C₂³×C₃₁	1	2	yes	—
308	C₂×#41	C₂³×C₃₁	1	2	yes	—
309	C₂²×D₄	—	2	4	no	—
310	C₂²×D₄	—	2	8	no	—
311	C₂²×D₄	—	2	8	no	—
312	C₂²×D₄	—	2	4	no	—
313	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
314	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
315	AGL(1,ℤ/8)	—	2	16	no	—
316	AGL(1,ℤ/8)	—	2	32	no	—
317	#135	—	2	16	no	—
318	#135	D₄×C₃₁	1	8	yes	complemented
319	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
320	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
321	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
322	C₂×D₈	—	2	16	no	—
323	C₂×D₈	—	2	8	no	—
324	C₂×D₈	—	4	4	no	—
325	C₂×D₈	C₂³×C₃₁	1	4	yes	—
326	C₂×D₈	C₂³×C₃₁	1	8	yes	—
327	C₂×D₈	C₂³×C₃₁	1	4	yes	—
328	C₂×D₈	C₂³×C₃₁	1	8	yes	—
329	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
330	#131	—	2	8	no	—
331	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
332	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
333	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
334	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
335	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
336	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
337	#127	—	2	4	no	—
338	#127	—	2	8	no	—
339	#127	—	2	2	no	—
340	#127	—	2	2	no	—
341	#127	D₄×C₃₁	1	1	yes	characteristic
342	#127	C₂³×C₃₁	1	8	yes	—
343	#124	—	2	4	no	—
344	#124	D₄×C₃₁	1	2	yes	—
345	#121	—	2	8	no	—
346	#121	—	2	16	no	—
347	#121	D₄×C₃₁	1	4	yes	—
348	#121	C₂³×C₃₁	1	16	yes	—
349	#120	—	2	8	no	—
350	#120	—	2	16	no	—
351	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
352	C₄×D₄	—	2	4	no	—
353	C₄×D₄	—	2	8	no	—
354	C₄×D₄	C₂³×C₃₁	1	4	yes	—
355	C₄×D₄	C₂³×C₃₁	1	4	yes	—
356	#117	—	2	2	no	—
357	#117	C₂³×C₃₁	1	2	yes	—
358	C₂×C₄²	—	2	2	no	—
359	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
360	#107	C₂³×C₃₁	1	2	yes	—
361	#106	C₂³×C₃₁	1	2	yes	—
362	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
363	#103	C₂³×C₃₁	1	4	yes	—
364	#102	C₂³×C₃₁	1	4	yes	—
365	#102	C₂³×C₃₁	1	16	yes	—
366	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
367	C₄×C₈	C₂³×C₃₁	1	2	yes	—
368	C₄×C₈	C₂³×C₃₁	1	2	yes	—
369	C₃₁	#61089	1	1	yes	characteristic, direct product factor, Sylow
370	C₂⁴	—	2	8	no	—
371	C₂⁴	—	2	4	no	—
372	#41	—	2	4	no	—
373	#41	—	2	4	no	—
374	#41	—	2	4	no	—
375	#41	—	2	16	no	—
376	#41	—	4	8	no	—
377	#41	—	4	16	no	—
378	#41	C₂×D₄×C₃₁	1	8	yes	complemented
379	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
380	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	16	no	—
383	C₂×D₄	—	2	8	no	—
384	C₂×D₄	—	2	16	no	—
385	C₂×D₄	—	2	8	no	—
386	C₂×D₄	—	2	8	no	—
387	C₂×D₄	—	2	8	no	—
388	C₂×D₄	—	2	4	no	—
389	C₂×D₄	—	2	8	no	—
390	C₂×D₄	—	2	8	no	—
391	C₂×D₄	—	4	8	no	—
392	C₂×D₄	—	4	16	no	—
393	C₂×D₄	—	4	32	no	—
394	C₂×D₄	—	8	8	no	—
395	C₂×D₄	—	8	16	no	—
396	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
397	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
398	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
399	C₂×D₄	C₂⁴×C₃₁	1	8	yes	—
400	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	4	no	—
403	C₂²×C₄	—	2	8	no	—
404	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
405	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
406	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
407	Q₁₆	C₂×D₄×C₃₁	1	4	yes	complemented
408	SD₁₆	—	2	16	no	—
409	SD₁₆	C₂×D₄×C₃₁	1	8	yes	complemented
410	D₈	—	2	16	no	—
411	D₈	—	2	8	no	—
412	D₈	—	2	8	no	—
413	D₈	—	4	8	no	—
414	D₈	C₂×D₄×C₃₁	1	4	yes	complemented
415	M₁₆	—	2	4	no	—
416	M₁₆	—	2	8	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	8	no	—
419	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	complemented
420	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
421	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
422	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
423	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
424	C₄⋊C₄	—	2	4	no	—
425	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
426	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
427	C₂²⋊C₄	—	2	2	no	—
428	C₂²⋊C₄	—	2	4	no	—
429	C₂²⋊C₄	—	2	16	no	—
430	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
431	C₄²	—	2	2	no	—
432	C₄²	—	2	4	no	—
433	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
434	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
435	C₄²	C₂⁴×C₃₁	1	2	yes	—
436	C₂³	—	2	2	no	—
437	C₂³	—	2	8	no	—
438	C₂³	—	4	16	no	—
439	C₂³	—	4	8	no	—
440	C₂³	—	8	8	no	—
441	C₂³	—	8	16	no	—
442	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
443	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
444	Q₈	—	2	4	no	—
445	Q₈	—	2	2	no	—
446	Q₈	C₂²×D₄×C₃₁	1	4	yes	—
447	D₄	—	2	8	no	—
448	D₄	—	2	8	no	—
449	D₄	—	2	4	no	—
450	D₄	—	2	8	no	—
451	D₄	—	2	8	no	—
452	D₄	—	2	2	no	—
453	D₄	—	2	8	no	—
454	D₄	—	4	16	no	—
455	D₄	—	4	16	no	—
456	D₄	—	4	8	no	—
457	D₄	—	4	16	no	—
458	D₄	—	8	8	no	—
459	D₄	—	8	16	no	—
460	D₄	C₂²×D₄×C₃₁	1	4	yes	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	8	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	4	16	no	—
468	C₂×C₄	—	4	8	no	—
469	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
470	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
471	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
472	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
473	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
474	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
475	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
476	C₈	—	2	4	no	—
477	C₈	—	2	2	no	—
478	C₈	C₂²×D₄×C₃₁	1	2	yes	—
479	C₈	C₂²×D₄×C₃₁	1	2	yes	—
480	C₂²	—	2	2	no	—
481	C₂²	—	2	4	no	—
482	C₂²	—	2	4	no	—
483	C₂²	—	2	8	no	—
484	C₂²	—	4	4	no	—
485	C₂²	—	4	8	no	—
486	C₂²	—	4	16	no	—
487	C₂²	—	16	8	no	—
488	C₂²	—	16	16	no	—
489	C₂²	D₄²×C₃₁	1	2	yes	—
490	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
491	C₄	—	2	4	no	—
492	C₄	—	2	4	no	—
493	C₄	—	2	2	no	—
494	C₄	—	4	4	no	—
495	C₄	D₄²×C₃₁	1	2	yes	—
496	C₄	C₃₁×#576	1	4	yes	complemented
497	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
498	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
499	C₂	—	2	2	no	—
500	C₂	—	4	4	no	—
501	C₂	—	4	4	no	—
502	C₂	—	8	8	no	—
503	C₂	C₂×D₄²×C₃₁	1	1	yes	characteristic
504	C₂	C₂×C₃₁×#576	1	2	yes	—

	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×#3498	C₂	semidirect
3. C₃₁×#3287	C₂	semidirect
4. D₄×D₈×C₃₁	C₂	semidirect
5. C₃₁×#3274	C₂	semidirect
6. C₂×C₃₁×#493	C₂	semidirect
7. C₃₁×#3058	C₂	semidirect
8. C₃₁×#3038	C₂	semidirect
9. C₃₁×#3008	C₂	semidirect
10. C₄×C₃₁×#135	C₂	semidirect
11. C₃₁×#2931	C₂	semidirect
12. C₂×C₃₁×H(5,𝔽₂)	C₂²	semidirect
14. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
15. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
17. C₂×C₃₁×#135	C₂²	semidirect
18. C₂²×D₈×C₃₁	C₂²	semidirect
19. C₂²×D₈×C₃₁	C₂²	semidirect
20. C₂²×D₈×C₃₁	C₂²	semidirect
21. C₂×C₃₁×#131	C₂²	semidirect
22. C₃₁×#550	C₂²	semidirect
24. D₄²×C₃₁	C₂²	semidirect
26. C₃₁×#535	C₂²	semidirect
30. C₂×C₃₁×#127	C₂²	semidirect
31. C₄×#41×C₃₁	C₂²	semidirect
32. C₃₁×#493	C₂²	semidirect
33. C₃₁×#493	C₂²	semidirect
34. C₃₁×#469	C₂²	semidirect
35. C₃₁×#466	C₂²	semidirect
36. C₃₁×#460	C₂²	semidirect
37. C₃₁×#459	C₂²	semidirect
38. C₃₁×#449	C₂²	semidirect
39. C₃₁×#447	C₂²	semidirect
40. C₄×Q₁₆×C₃₁	C₂²	semidirect
41. C₄×SD₁₆×C₃₁	C₂²	semidirect
42. C₄×D₈×C₃₁	C₂²	semidirect
43. C₃₁×#436	C₂²	semidirect
44. C₈×D₄×C₃₁	C₂²	semidirect
45. C₃₁×#427	C₂²	semidirect
46. C₃₁×#423	C₂²	semidirect
47. C₃₁×#418	C₂²	semidirect
48. C₃₁×#416	C₂²	semidirect
49. C₃₁×#408	C₂²	semidirect
50. C₂×C₄×C₈×C₃₁	C₂²	semidirect
66. C₃₁×#135	D₄	semidirect
125. #41×C₃₁	C₂×D₄	semidirect
154. Q₁₆×C₃₁	C₂×D₄	semidirect
156. SD₁₆×C₃₁	C₂×D₄	semidirect
161. D₈×C₃₁	C₂×D₄	semidirect
166. C₂×C₈×C₃₁	C₂×D₄	semidirect
183. #61089	C₃₁	direct (N × H)
228. C₂×#576	C₂×C₃₁	semidirect
229. #3498	C₂×C₃₁	semidirect
230. #3287	C₂×C₃₁	semidirect
231. D₄×D₈	C₂×C₃₁	semidirect
232. #3274	C₂×C₃₁	semidirect
233. C₂×#493	C₂×C₃₁	semidirect
234. #3058	C₂×C₃₁	semidirect
235. #3038	C₂×C₃₁	semidirect
236. #3008	C₂×C₃₁	semidirect
237. C₄×#135	C₂×C₃₁	semidirect
238. #2931	C₂×C₃₁	semidirect
255. C₄×C₃₁	#576	semidirect
258. C₂×H(5,𝔽₂)	C₂²×C₃₁	semidirect
260. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
261. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
263. C₂×#135	C₂²×C₃₁	semidirect
264. C₂²×D₈	C₂²×C₃₁	semidirect
265. C₂²×D₈	C₂²×C₃₁	semidirect
266. C₂²×D₈	C₂²×C₃₁	semidirect
267. C₂×#131	C₂²×C₃₁	semidirect
268. #550	C₂²×C₃₁	semidirect
270. D₄²	C₂²×C₃₁	semidirect
272. #535	C₂²×C₃₁	semidirect
276. C₂×#127	C₂²×C₃₁	semidirect
277. C₄×#41	C₂²×C₃₁	semidirect
278. #493	C₂²×C₃₁	semidirect
279. #493	C₂²×C₃₁	semidirect
280. #469	C₂²×C₃₁	semidirect
281. #466	C₂²×C₃₁	semidirect
282. #460	C₂²×C₃₁	semidirect
283. #459	C₂²×C₃₁	semidirect
284. #449	C₂²×C₃₁	semidirect
285. #447	C₂²×C₃₁	semidirect
286. C₄×Q₁₆	C₂²×C₃₁	semidirect
287. C₄×SD₁₆	C₂²×C₃₁	semidirect
288. C₄×D₈	C₂²×C₃₁	semidirect
289. #436	C₂²×C₃₁	semidirect
290. C₈×D₄	C₂²×C₃₁	semidirect
291. #427	C₂²×C₃₁	semidirect
292. #423	C₂²×C₃₁	semidirect
293. #418	C₂²×C₃₁	semidirect
294. #416	C₂²×C₃₁	semidirect
295. #408	C₂²×C₃₁	semidirect
296. C₂×C₄×C₈	C₂²×C₃₁	semidirect
318. #135	D₄×C₃₁	semidirect
369. C₃₁	#61089	direct (N × H)
378. #41	C₂×D₄×C₃₁	semidirect
407. Q₁₆	C₂×D₄×C₃₁	semidirect
409. SD₁₆	C₂×D₄×C₃₁	semidirect
414. D₈	C₂×D₄×C₃₁	semidirect
419. C₂×C₈	C₂×D₄×C₃₁	semidirect
496. 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×#3498	2
C₃₁×#3287	2
D₄×D₈×C₃₁	2
C₃₁×#3274	2
C₂×C₃₁×#493	2
C₃₁×#3058	2
C₃₁×#3038	2
C₃₁×#3008	2
C₄×C₃₁×#135	2
C₃₁×#2931	2
#61089	31

Subgroups — C31×#61089 #21517848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61089 #21517848