Subgroups — C₃₁×#60810 #21517569

Order 7936.

2098 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#542	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3449	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3225	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3224	C₂	1	8	yes	maximal, complemented
5	C₃₁×#3153	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3141	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3037	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3036	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3007	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3006	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2921	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×#137	C₂²	1	4	yes	complemented
14	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
15	C₃₁×#542	—	2	4	no	—
16	C₃₁×#542	C₂²	1	8	yes	complemented
17	C₃₁×#537	—	2	1	no	—
18	C₃₁×#537	—	2	1	no	—
19	C₃₁×#536	C₂²	1	4	yes	complemented
20	C₃₁×#535	C₂²	1	4	yes	complemented
21	C₃₁×#534	—	2	1	no	—
22	C₃₁×#534	—	2	1	no	—
23	C₃₁×#532	C₂²	1	2	yes	—
24	C₃₁×#532	C₂²	1	2	yes	—
25	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₃₁×#126	—	2	1	no	—
27	C₂×C₃₁×#124	—	2	1	no	—
28	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic, complemented
29	C₃₁×#525	—	2	4	no	—
30	C₂×C₃₁×#123	—	2	1	no	—
31	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
32	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
34	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
37	C₃₁×#497	C₂²	1	4	yes	complemented
38	C₃₁×#496	C₂²	1	4	yes	complemented
39	C₃₁×#495	C₂²	1	8	yes	complemented
40	C₃₁×#489	C₂²	1	8	yes	complemented
41	C₃₁×#487	C₂²	1	4	yes	complemented
42	C₃₁×#486	C₂²	1	4	yes	complemented
43	C₃₁×#464	C₂²	1	8	yes	complemented
44	C₃₁×#463	C₂²	1	8	yes	complemented
45	C₃₁×#462	C₂²	1	4	yes	complemented
46	C₃₁×#461	C₂²	1	4	yes	complemented
47	C₃₁×#460	C₂²	1	4	yes	complemented
48	C₃₁×#459	C₂²	1	4	yes	complemented
49	C₃₁×#452	C₂²	1	8	yes	complemented
50	C₃₁×#451	C₂²	1	4	yes	complemented
51	C₃₁×#450	C₂²	1	4	yes	complemented
52	C₃₁×#449	C₂²	1	8	yes	—
53	C₃₁×#448	C₂²	1	4	yes	complemented
54	C₃₁×#447	C₂²	1	4	yes	complemented
55	C₃₁×#433	C₂²	1	8	yes	complemented
56	C₃₁×#423	C₂²	1	2	yes	complemented
57	C₃₁×#417	C₂²	1	4	yes	complemented
58	C₃₁×#417	C₂²	1	4	yes	complemented
59	C₃₁×#407	C₂²	1	2	yes	—
60	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
61	C₂×#41×C₃₁	—	2	2	no	—
62	C₂×#41×C₃₁	—	2	2	no	—
63	C₂×#41×C₃₁	—	4	8	no	—
64	C₂×#41×C₃₁	C₂³	1	4	yes	—
65	C₂×#41×C₃₁	C₂³	1	4	yes	—
66	C₂²×Q₈×C₃₁	—	2	1	no	—
67	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
69	C₂²×D₄×C₃₁	—	2	2	no	—
70	C₂²×D₄×C₃₁	—	2	1	no	—
71	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
73	C₂³×C₄×C₃₁	—	2	2	no	—
74	C₂³×C₄×C₃₁	—	2	1	no	—
75	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#137	—	4	8	no	—
77	C₃₁×AGL(1,ℤ/8)	—	4	8	no	—
78	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
79	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
80	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
81	C₂×D₈×C₃₁	C₂³	1	8	yes	—
82	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
83	C₃₁×C₄⋊Q₈	—	2	2	no	—
84	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
85	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
86	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
87	C₃₁×#127	C₂³	1	2	yes	—
88	C₃₁×#127	C₂³	1	2	yes	—
89	C₃₁×#126	—	2	8	no	—
90	C₃₁×#124	—	2	8	no	—
91	C₃₁×#124	—	2	2	no	—
92	C₃₁×#124	C₂³	1	4	yes	—
93	C₃₁×#124	C₂³	1	4	yes	—
94	C₃₁×#124	C₂³	1	4	yes	—
95	C₃₁×#123	—	2	8	no	—
96	C₃₁×#123	—	2	4	no	—
97	C₃₁×#123	—	2	4	no	—
98	C₃₁×C₂²⋊Q₈	—	2	2	no	—
99	C₃₁×C₂²⋊Q₈	—	2	8	no	—
100	C₃₁×C₂²⋊Q₈	—	2	4	no	—
101	C₃₁×C₂²⋊Q₈	—	2	4	no	—
102	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
103	C₃₁×#121	—	2	2	no	—
104	C₃₁×#121	—	2	2	no	—
105	C₃₁×#121	—	2	2	no	—
106	C₃₁×#121	—	2	2	no	—
107	C₃₁×#121	—	2	2	no	—
108	C₃₁×#121	—	2	4	no	—
109	C₃₁×#121	—	2	4	no	—
110	C₃₁×#121	C₂³	1	4	yes	—
111	C₃₁×#120	—	2	4	no	—
112	C₃₁×#120	—	2	4	no	—
113	C₄×Q₈×C₃₁	—	2	2	no	—
114	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
115	C₄×D₄×C₃₁	—	2	2	no	—
116	C₄×D₄×C₃₁	—	2	2	no	—
117	C₄×D₄×C₃₁	—	2	2	no	—
118	C₄×D₄×C₃₁	—	2	2	no	—
119	C₄×D₄×C₃₁	—	2	2	no	—
120	C₄×D₄×C₃₁	C₂³	1	4	yes	—
121	C₃₁×#117	D₄	1	4	yes	—
122	C₃₁×#117	C₂³	1	4	yes	—
123	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
124	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
125	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
127	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
128	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
129	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
130	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
131	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
132	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
133	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
134	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
135	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
136	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
137	C₂×C₄²×C₃₁	—	2	1	no	—
138	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
139	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
140	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
141	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
142	C₃₁×#103	C₂³	1	8	yes	—
143	C₃₁×#103	C₂³	1	8	yes	—
144	C₃₁×#102	C₂³	1	8	yes	—
145	C₃₁×#102	C₂³	1	8	yes	—
146	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
147	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
148	C₃₁×#97	C₂³	1	4	yes	—
149	C₄×C₈×C₃₁	C₂³	1	4	yes	—
150	C₂⁴×C₃₁	—	2	1	no	—
151	C₂⁴×C₃₁	—	2	2	no	—
152	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
153	#41×C₃₁	—	2	8	no	—
154	#41×C₃₁	—	2	8	no	—
155	#41×C₃₁	—	4	4	no	—
156	#41×C₃₁	—	4	4	no	—
157	#41×C₃₁	—	4	8	no	—
158	#41×C₃₁	—	4	8	no	—
159	C₂×Q₈×C₃₁	—	2	1	no	—
160	C₂×Q₈×C₃₁	—	2	2	no	—
161	C₂×Q₈×C₃₁	—	2	4	no	—
162	C₂×Q₈×C₃₁	—	2	4	no	—
163	C₂×Q₈×C₃₁	—	4	4	no	—
164	C₂×Q₈×C₃₁	—	8	2	no	—
165	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
166	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
167	C₂×D₄×C₃₁	—	2	4	no	—
168	C₂×D₄×C₃₁	—	2	4	no	—
169	C₂×D₄×C₃₁	—	2	4	no	—
170	C₂×D₄×C₃₁	—	2	1	no	—
171	C₂×D₄×C₃₁	—	2	2	no	—
172	C₂×D₄×C₃₁	—	2	2	no	—
173	C₂×D₄×C₃₁	—	2	2	no	—
174	C₂×D₄×C₃₁	—	2	2	no	—
175	C₂×D₄×C₃₁	—	2	2	no	—
176	C₂×D₄×C₃₁	—	2	4	no	—
177	C₂×D₄×C₃₁	—	4	4	no	—
178	C₂×D₄×C₃₁	—	4	8	no	—
179	C₂×D₄×C₃₁	—	8	2	no	—
180	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
181	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₃₁	—	2	4	no	—
183	C₂²×C₄×C₃₁	—	2	1	no	—
184	C₂²×C₄×C₃₁	—	2	1	no	—
185	C₂²×C₄×C₃₁	—	2	1	no	—
186	C₂²×C₄×C₃₁	—	2	2	no	—
187	C₂²×C₄×C₃₁	—	2	4	no	—
188	C₂²×C₄×C₃₁	—	2	4	no	—
189	C₂²×C₄×C₃₁	—	2	4	no	—
190	C₂²×C₄×C₃₁	—	4	8	no	—
191	C₂²×C₄×C₃₁	—	8	2	no	—
192	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
193	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
194	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
195	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
196	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
197	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
198	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
199	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
200	Q₁₆×C₃₁	—	4	8	no	—
201	SD₁₆×C₃₁	—	4	8	no	—
202	SD₁₆×C₃₁	—	4	8	no	—
203	D₈×C₃₁	—	4	8	no	—
204	M₁₆×C₃₁	—	2	8	no	—
205	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
206	C₄⋊C₄×C₃₁	—	2	4	no	—
207	C₄⋊C₄×C₃₁	—	2	4	no	—
208	C₄⋊C₄×C₃₁	—	2	4	no	—
209	C₄⋊C₄×C₃₁	—	2	1	no	—
210	C₄⋊C₄×C₃₁	—	2	1	no	—
211	C₄⋊C₄×C₃₁	—	2	2	no	—
212	C₄⋊C₄×C₃₁	—	2	2	no	—
213	C₄⋊C₄×C₃₁	—	2	2	no	—
214	C₄⋊C₄×C₃₁	—	2	2	no	—
215	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
216	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
217	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
218	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
219	C₂²⋊C₄×C₃₁	—	2	4	no	—
220	C₂²⋊C₄×C₃₁	—	2	4	no	—
221	C₂²⋊C₄×C₃₁	—	2	4	no	—
222	C₂²⋊C₄×C₃₁	—	2	4	no	—
223	C₂²⋊C₄×C₃₁	—	2	4	no	—
224	C₂²⋊C₄×C₃₁	—	2	4	no	—
225	C₂²⋊C₄×C₃₁	—	2	4	no	—
226	C₂²⋊C₄×C₃₁	—	2	4	no	—
227	C₂²⋊C₄×C₃₁	—	2	4	no	—
228	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
229	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
230	C₄²×C₃₁	—	2	4	no	—
231	C₄²×C₃₁	C₂×D₄	1	2	yes	—
232	C₄²×C₃₁	C₂×D₄	1	2	yes	—
233	C₄²×C₃₁	C₂⁴	1	2	yes	—
234	C₄²×C₃₁	C₂⁴	1	2	yes	—
235	#60810	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
236	C₂³×C₃₁	—	2	2	no	—
237	C₂³×C₃₁	—	2	2	no	—
238	C₂³×C₃₁	—	2	4	no	—
239	C₂³×C₃₁	—	4	1	no	—
240	C₂³×C₃₁	—	4	1	no	—
241	C₂³×C₃₁	—	4	4	no	—
242	C₂³×C₃₁	—	4	2	no	—
243	C₂³×C₃₁	—	4	2	no	—
244	C₂³×C₃₁	—	4	2	no	—
245	C₂³×C₃₁	—	8	2	no	—
246	C₂³×C₃₁	—	8	2	no	—
247	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
248	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
249	Q₈×C₃₁	—	2	4	no	—
250	Q₈×C₃₁	—	2	4	no	—
251	Q₈×C₃₁	—	2	4	no	—
252	Q₈×C₃₁	—	4	4	no	—
253	Q₈×C₃₁	—	4	2	no	—
254	Q₈×C₃₁	—	8	4	no	—
255	D₄×C₃₁	—	2	4	no	—
256	D₄×C₃₁	—	2	4	no	—
257	D₄×C₃₁	—	2	4	no	—
258	D₄×C₃₁	—	4	4	no	—
259	D₄×C₃₁	—	4	8	no	—
260	D₄×C₃₁	—	4	8	no	—
261	D₄×C₃₁	—	4	2	no	—
262	D₄×C₃₁	—	8	2	no	—
263	D₄×C₃₁	—	8	2	no	—
264	D₄×C₃₁	—	8	2	no	—
265	D₄×C₃₁	—	8	2	no	—
266	D₄×C₃₁	—	8	4	no	—
267	C₂×C₄×C₃₁	—	2	2	no	—
268	C₂×C₄×C₃₁	—	2	4	no	—
269	C₂×C₄×C₃₁	—	2	4	no	—
270	C₂×C₄×C₃₁	—	2	2	no	—
271	C₂×C₄×C₃₁	—	2	2	no	—
272	C₂×C₄×C₃₁	—	2	2	no	—
273	C₂×C₄×C₃₁	—	2	4	no	—
274	C₂×C₄×C₃₁	—	4	1	no	—
275	C₂×C₄×C₃₁	—	4	1	no	—
276	C₂×C₄×C₃₁	—	4	1	no	—
277	C₂×C₄×C₃₁	—	4	1	no	—
278	C₂×C₄×C₃₁	—	4	8	no	—
279	C₂×C₄×C₃₁	—	4	4	no	—
280	C₂×C₄×C₃₁	—	4	2	no	—
281	C₂×C₄×C₃₁	—	4	2	no	—
282	C₂×C₄×C₃₁	—	4	2	no	—
283	C₂×C₄×C₃₁	—	4	8	no	—
284	C₂×C₄×C₃₁	—	8	4	no	—
285	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
286	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
287	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
288	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
289	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
290	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
291	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
292	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
293	C₈×C₃₁	—	2	8	no	—
294	C₂×#542	C₂×C₃₁	1	1	yes	characteristic, complemented
295	#3449	C₂×C₃₁	1	1	yes	characteristic, complemented
296	#3225	C₂×C₃₁	1	8	yes	complemented
297	#3224	C₂×C₃₁	1	8	yes	complemented
298	#3153	C₂×C₃₁	1	2	yes	complemented
299	#3141	C₂×C₃₁	1	2	yes	complemented
300	#3037	C₂×C₃₁	1	2	yes	complemented
301	#3036	C₂×C₃₁	1	2	yes	complemented
302	#3007	C₂×C₃₁	1	2	yes	complemented
303	#3006	C₂×C₃₁	1	2	yes	complemented
304	#2921	C₂×C₃₁	1	1	yes	characteristic, complemented
305	C₂²×C₃₁	—	2	1	no	—
306	C₂²×C₃₁	—	2	1	no	—
307	C₂²×C₃₁	—	2	2	no	—
308	C₂²×C₃₁	—	2	2	no	—
309	C₂²×C₃₁	—	2	2	no	—
310	C₂²×C₃₁	—	4	2	no	—
311	C₂²×C₃₁	—	4	4	no	—
312	C₂²×C₃₁	—	4	4	no	—
313	C₂²×C₃₁	—	4	2	no	—
314	C₂²×C₃₁	—	4	4	no	—
315	C₂²×C₃₁	—	4	2	no	—
316	C₂²×C₃₁	—	8	2	no	—
317	C₂²×C₃₁	—	8	4	no	—
318	C₂²×C₃₁	—	8	8	no	—
319	C₂²×C₃₁	#535	1	2	yes	—
320	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
321	C₄×C₃₁	—	2	2	no	—
322	C₄×C₃₁	—	2	2	no	—
323	C₄×C₃₁	—	2	2	no	—
324	C₄×C₃₁	—	2	2	no	—
325	C₄×C₃₁	—	2	2	no	—
326	C₄×C₃₁	—	4	2	no	—
327	C₄×C₃₁	—	4	2	no	—
328	C₄×C₃₁	—	4	4	no	—
329	C₄×C₃₁	—	4	2	no	—
330	C₄×C₃₁	#535	1	2	yes	—
331	C₄×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
332	C₂²×#41	C₂²×C₃₁	1	2	yes	—
333	C₂×#137	C₂²×C₃₁	1	4	yes	complemented
334	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	complemented
335	#542	—	2	4	no	—
336	#542	C₂²×C₃₁	1	8	yes	complemented
337	#537	—	2	1	no	—
338	#537	—	2	1	no	—
339	#536	C₂²×C₃₁	1	4	yes	complemented
340	#535	C₂²×C₃₁	1	4	yes	complemented
341	#534	—	2	1	no	—
342	#534	—	2	1	no	—
343	#532	C₂²×C₃₁	1	2	yes	—
344	#532	C₂²×C₃₁	1	2	yes	—
345	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
346	C₂×#126	—	2	1	no	—
347	C₂×#124	—	2	1	no	—
348	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic, complemented
349	#525	—	2	4	no	—
350	C₂×#123	—	2	1	no	—
351	C₂×C₂²⋊Q₈	—	2	1	no	—
352	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
353	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
354	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
355	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
356	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
357	#497	C₂²×C₃₁	1	4	yes	complemented
358	#496	C₂²×C₃₁	1	4	yes	complemented
359	#495	C₂²×C₃₁	1	8	yes	complemented
360	#489	C₂²×C₃₁	1	8	yes	complemented
361	#487	C₂²×C₃₁	1	4	yes	complemented
362	#486	C₂²×C₃₁	1	4	yes	complemented
363	#464	C₂²×C₃₁	1	8	yes	complemented
364	#463	C₂²×C₃₁	1	8	yes	complemented
365	#462	C₂²×C₃₁	1	4	yes	complemented
366	#461	C₂²×C₃₁	1	4	yes	complemented
367	#460	C₂²×C₃₁	1	4	yes	complemented
368	#459	C₂²×C₃₁	1	4	yes	complemented
369	#452	C₂²×C₃₁	1	8	yes	complemented
370	#451	C₂²×C₃₁	1	4	yes	complemented
371	#450	C₂²×C₃₁	1	4	yes	complemented
372	#449	C₂²×C₃₁	1	8	yes	—
373	#448	C₂²×C₃₁	1	4	yes	complemented
374	#447	C₂²×C₃₁	1	4	yes	complemented
375	#433	C₂²×C₃₁	1	8	yes	complemented
376	#423	C₂²×C₃₁	1	2	yes	complemented
377	#417	C₂²×C₃₁	1	4	yes	complemented
378	#417	C₂²×C₃₁	1	4	yes	complemented
379	#407	C₂²×C₃₁	1	2	yes	—
380	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
381	C₂×C₃₁	—	2	2	no	—
382	C₂×C₃₁	—	4	2	no	—
383	C₂×C₃₁	—	8	2	no	—
384	C₂×C₃₁	—	8	4	no	—
385	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
386	C₂×C₃₁	#3585	1	1	yes	characteristic
387	C₂×C₃₁	#3586	1	1	yes	characteristic
388	C₂×#41	—	2	2	no	—
389	C₂×#41	—	2	2	no	—
390	C₂×#41	—	4	8	no	—
391	C₂×#41	C₂³×C₃₁	1	4	yes	—
392	C₂×#41	C₂³×C₃₁	1	4	yes	—
393	C₂²×Q₈	—	2	1	no	—
394	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
395	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
396	C₂²×D₄	—	2	2	no	—
397	C₂²×D₄	—	2	1	no	—
398	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
399	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
400	C₂³×C₄	—	2	2	no	—
401	C₂³×C₄	—	2	1	no	—
402	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
403	#137	—	4	8	no	—
404	AGL(1,ℤ/8)	—	4	8	no	—
405	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
406	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
407	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
408	C₂×D₈	C₂³×C₃₁	1	8	yes	—
409	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
410	C₄⋊Q₈	—	2	2	no	—
411	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
412	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
413	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
414	#127	C₂³×C₃₁	1	2	yes	—
415	#127	C₂³×C₃₁	1	2	yes	—
416	#126	—	2	8	no	—
417	#124	—	2	8	no	—
418	#124	—	2	2	no	—
419	#124	C₂³×C₃₁	1	4	yes	—
420	#124	C₂³×C₃₁	1	4	yes	—
421	#124	C₂³×C₃₁	1	4	yes	—
422	#123	—	2	8	no	—
423	#123	—	2	4	no	—
424	#123	—	2	4	no	—
425	C₂²⋊Q₈	—	2	2	no	—
426	C₂²⋊Q₈	—	2	8	no	—
427	C₂²⋊Q₈	—	2	4	no	—
428	C₂²⋊Q₈	—	2	4	no	—
429	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
430	#121	—	2	2	no	—
431	#121	—	2	2	no	—
432	#121	—	2	2	no	—
433	#121	—	2	2	no	—
434	#121	—	2	2	no	—
435	#121	—	2	4	no	—
436	#121	—	2	4	no	—
437	#121	C₂³×C₃₁	1	4	yes	—
438	#120	—	2	4	no	—
439	#120	—	2	4	no	—
440	C₄×Q₈	—	2	2	no	—
441	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
442	C₄×D₄	—	2	2	no	—
443	C₄×D₄	—	2	2	no	—
444	C₄×D₄	—	2	2	no	—
445	C₄×D₄	—	2	2	no	—
446	C₄×D₄	—	2	2	no	—
447	C₄×D₄	C₂³×C₃₁	1	4	yes	—
448	#117	D₄×C₃₁	1	4	yes	—
449	#117	C₂³×C₃₁	1	4	yes	—
450	C₂×C₄⋊C₄	—	2	1	no	—
451	C₂×C₄⋊C₄	—	2	1	no	—
452	C₂×C₄⋊C₄	—	2	1	no	—
453	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
454	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
455	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
456	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
457	C₂×C₂²⋊C₄	—	2	1	no	—
458	C₂×C₂²⋊C₄	—	2	1	no	—
459	C₂×C₂²⋊C₄	—	2	1	no	—
460	C₂×C₂²⋊C₄	—	2	1	no	—
461	C₂×C₂²⋊C₄	—	2	1	no	—
462	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
463	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
464	C₂×C₄²	—	2	1	no	—
465	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
466	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
467	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
468	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
469	#103	C₂³×C₃₁	1	8	yes	—
470	#103	C₂³×C₃₁	1	8	yes	—
471	#102	C₂³×C₃₁	1	8	yes	—
472	#102	C₂³×C₃₁	1	8	yes	—
473	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
474	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
475	#97	C₂³×C₃₁	1	4	yes	—
476	C₄×C₈	C₂³×C₃₁	1	4	yes	—
477	C₃₁	#60810	1	1	yes	characteristic, direct product factor, Sylow
478	C₂⁴	—	2	1	no	—
479	C₂⁴	—	2	2	no	—
480	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
481	#41	—	2	8	no	—
482	#41	—	2	8	no	—
483	#41	—	4	4	no	—
484	#41	—	4	4	no	—
485	#41	—	4	8	no	—
486	#41	—	4	8	no	—
487	C₂×Q₈	—	2	1	no	—
488	C₂×Q₈	—	2	2	no	—
489	C₂×Q₈	—	2	4	no	—
490	C₂×Q₈	—	2	4	no	—
491	C₂×Q₈	—	4	4	no	—
492	C₂×Q₈	—	8	2	no	—
493	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
494	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
495	C₂×D₄	—	2	4	no	—
496	C₂×D₄	—	2	4	no	—
497	C₂×D₄	—	2	4	no	—
498	C₂×D₄	—	2	1	no	—
499	C₂×D₄	—	2	2	no	—
500	C₂×D₄	—	2	2	no	—
501	C₂×D₄	—	2	2	no	—
502	C₂×D₄	—	2	2	no	—
503	C₂×D₄	—	2	2	no	—
504	C₂×D₄	—	2	4	no	—
505	C₂×D₄	—	4	4	no	—
506	C₂×D₄	—	4	8	no	—
507	C₂×D₄	—	8	2	no	—
508	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
509	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
510	C₂²×C₄	—	2	4	no	—
511	C₂²×C₄	—	2	1	no	—
512	C₂²×C₄	—	2	1	no	—
513	C₂²×C₄	—	2	1	no	—
514	C₂²×C₄	—	2	2	no	—
515	C₂²×C₄	—	2	4	no	—
516	C₂²×C₄	—	2	4	no	—
517	C₂²×C₄	—	2	4	no	—
518	C₂²×C₄	—	4	8	no	—
519	C₂²×C₄	—	8	2	no	—
520	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
521	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
522	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
523	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
524	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
525	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
526	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
527	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
528	Q₁₆	—	4	8	no	—
529	SD₁₆	—	4	8	no	—
530	SD₁₆	—	4	8	no	—
531	D₈	—	4	8	no	—
532	M₁₆	—	2	8	no	—
533	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
534	C₄⋊C₄	—	2	4	no	—
535	C₄⋊C₄	—	2	4	no	—
536	C₄⋊C₄	—	2	4	no	—
537	C₄⋊C₄	—	2	1	no	—
538	C₄⋊C₄	—	2	1	no	—
539	C₄⋊C₄	—	2	2	no	—
540	C₄⋊C₄	—	2	2	no	—
541	C₄⋊C₄	—	2	2	no	—
542	C₄⋊C₄	—	2	2	no	—
543	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
544	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
545	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
546	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
547	C₂²⋊C₄	—	2	4	no	—
548	C₂²⋊C₄	—	2	4	no	—
549	C₂²⋊C₄	—	2	4	no	—
550	C₂²⋊C₄	—	2	4	no	—
551	C₂²⋊C₄	—	2	4	no	—
552	C₂²⋊C₄	—	2	4	no	—
553	C₂²⋊C₄	—	2	4	no	—
554	C₂²⋊C₄	—	2	4	no	—
555	C₂²⋊C₄	—	2	4	no	—
556	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
557	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
558	C₄²	—	2	4	no	—
559	C₄²	C₂×D₄×C₃₁	1	2	yes	—
560	C₄²	C₂×D₄×C₃₁	1	2	yes	—
561	C₄²	C₂⁴×C₃₁	1	2	yes	—
562	C₄²	C₂⁴×C₃₁	1	2	yes	—
563	C₂³	—	2	2	no	—
564	C₂³	—	2	2	no	—
565	C₂³	—	2	4	no	—
566	C₂³	—	4	1	no	—
567	C₂³	—	4	1	no	—
568	C₂³	—	4	4	no	—
569	C₂³	—	4	2	no	—
570	C₂³	—	4	2	no	—
571	C₂³	—	4	2	no	—
572	C₂³	—	8	2	no	—
573	C₂³	—	8	2	no	—
574	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
575	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
576	Q₈	—	2	4	no	—
577	Q₈	—	2	4	no	—
578	Q₈	—	2	4	no	—
579	Q₈	—	4	4	no	—
580	Q₈	—	4	2	no	—
581	Q₈	—	8	4	no	—
582	D₄	—	2	4	no	—
583	D₄	—	2	4	no	—
584	D₄	—	2	4	no	—
585	D₄	—	4	4	no	—
586	D₄	—	4	8	no	—
587	D₄	—	4	8	no	—
588	D₄	—	4	2	no	—
589	D₄	—	8	2	no	—
590	D₄	—	8	2	no	—
591	D₄	—	8	2	no	—
592	D₄	—	8	2	no	—
593	D₄	—	8	4	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	4	no	—
596	C₂×C₄	—	2	4	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	4	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	4	1	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	8	no	—
606	C₂×C₄	—	4	4	no	—
607	C₂×C₄	—	4	2	no	—
608	C₂×C₄	—	4	2	no	—
609	C₂×C₄	—	4	2	no	—
610	C₂×C₄	—	4	8	no	—
611	C₂×C₄	—	8	4	no	—
612	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
613	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
614	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
615	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
616	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
617	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
618	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
619	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
620	C₈	—	2	8	no	—
621	C₂²	—	2	1	no	—
622	C₂²	—	2	1	no	—
623	C₂²	—	2	2	no	—
624	C₂²	—	2	2	no	—
625	C₂²	—	2	2	no	—
626	C₂²	—	4	4	no	—
627	C₂²	—	4	2	no	—
628	C₂²	—	4	2	no	—
629	C₂²	—	4	4	no	—
630	C₂²	—	4	4	no	—
631	C₂²	—	4	2	no	—
632	C₂²	—	8	2	no	—
633	C₂²	—	8	4	no	—
634	C₂²	—	8	8	no	—
635	C₂²	C₃₁×#535	1	2	yes	—
636	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
637	C₄	—	2	2	no	—
638	C₄	—	2	2	no	—
639	C₄	—	2	2	no	—
640	C₄	—	2	2	no	—
641	C₄	—	2	2	no	—
642	C₄	—	4	4	no	—
643	C₄	—	4	2	no	—
644	C₄	—	4	2	no	—
645	C₄	—	4	2	no	—
646	C₄	C₃₁×#535	1	2	yes	—
647	C₄	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
648	C₂	—	2	2	no	—
649	C₂	—	4	2	no	—
650	C₂	—	8	4	no	—
651	C₂	—	8	2	no	—
652	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
653	C₂	C₃₁×#3585	1	1	yes	characteristic
654	C₂	C₃₁×#3586	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 ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#542	C₂	semidirect
2. C₃₁×#3449	C₂	semidirect
3. C₃₁×#3225	C₂	semidirect
4. C₃₁×#3224	C₂	semidirect
5. C₃₁×#3153	C₂	semidirect
6. C₃₁×#3141	C₂	semidirect
7. C₃₁×#3037	C₂	semidirect
8. C₃₁×#3036	C₂	semidirect
9. C₃₁×#3007	C₂	semidirect
10. C₃₁×#3006	C₂	semidirect
11. C₃₁×#2921	C₂	semidirect
13. C₂×C₃₁×#137	C₂²	semidirect
14. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
16. C₃₁×#542	C₂²	semidirect
19. C₃₁×#536	C₂²	semidirect
20. C₃₁×#535	C₂²	semidirect
25. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
28. C₂×C₃₁×#124	C₂²	semidirect
34. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
35. C₂×C₄×D₄×C₃₁	C₂²	semidirect
37. C₃₁×#497	C₂²	semidirect
38. C₃₁×#496	C₂²	semidirect
39. C₃₁×#495	C₂²	semidirect
40. C₃₁×#489	C₂²	semidirect
41. C₃₁×#487	C₂²	semidirect
42. C₃₁×#486	C₂²	semidirect
43. C₃₁×#464	C₂²	semidirect
44. C₃₁×#463	C₂²	semidirect
45. C₃₁×#462	C₂²	semidirect
46. C₃₁×#461	C₂²	semidirect
47. C₃₁×#460	C₂²	semidirect
48. C₃₁×#459	C₂²	semidirect
49. C₃₁×#452	C₂²	semidirect
50. C₃₁×#451	C₂²	semidirect
51. C₃₁×#450	C₂²	semidirect
53. C₃₁×#448	C₂²	semidirect
54. C₃₁×#447	C₂²	semidirect
55. C₃₁×#433	C₂²	semidirect
56. C₃₁×#423	C₂²	semidirect
57. C₃₁×#417	C₂²	semidirect
58. C₃₁×#417	C₂²	semidirect
60. C₄×M₁₆×C₃₁	C₂²	semidirect
235. #60810	C₃₁	direct (N × H)
294. C₂×#542	C₂×C₃₁	semidirect
295. #3449	C₂×C₃₁	semidirect
296. #3225	C₂×C₃₁	semidirect
297. #3224	C₂×C₃₁	semidirect
298. #3153	C₂×C₃₁	semidirect
299. #3141	C₂×C₃₁	semidirect
300. #3037	C₂×C₃₁	semidirect
301. #3036	C₂×C₃₁	semidirect
302. #3007	C₂×C₃₁	semidirect
303. #3006	C₂×C₃₁	semidirect
304. #2921	C₂×C₃₁	semidirect
333. C₂×#137	C₂²×C₃₁	semidirect
334. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
336. #542	C₂²×C₃₁	semidirect
339. #536	C₂²×C₃₁	semidirect
340. #535	C₂²×C₃₁	semidirect
345. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
348. C₂×#124	C₂²×C₃₁	semidirect
354. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
355. C₂×C₄×D₄	C₂²×C₃₁	semidirect
357. #497	C₂²×C₃₁	semidirect
358. #496	C₂²×C₃₁	semidirect
359. #495	C₂²×C₃₁	semidirect
360. #489	C₂²×C₃₁	semidirect
361. #487	C₂²×C₃₁	semidirect
362. #486	C₂²×C₃₁	semidirect
363. #464	C₂²×C₃₁	semidirect
364. #463	C₂²×C₃₁	semidirect
365. #462	C₂²×C₃₁	semidirect
366. #461	C₂²×C₃₁	semidirect
367. #460	C₂²×C₃₁	semidirect
368. #459	C₂²×C₃₁	semidirect
369. #452	C₂²×C₃₁	semidirect
370. #451	C₂²×C₃₁	semidirect
371. #450	C₂²×C₃₁	semidirect
373. #448	C₂²×C₃₁	semidirect
374. #447	C₂²×C₃₁	semidirect
375. #433	C₂²×C₃₁	semidirect
376. #423	C₂²×C₃₁	semidirect
377. #417	C₂²×C₃₁	semidirect
378. #417	C₂²×C₃₁	semidirect
380. C₄×M₁₆	C₂²×C₃₁	semidirect
477. C₃₁	#60810	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#542	2
C₃₁×#3449	2
C₃₁×#3225	2
C₃₁×#3224	2
C₃₁×#3153	2
C₃₁×#3141	2
C₃₁×#3037	2
C₃₁×#3036	2
C₃₁×#3007	2
C₃₁×#3006	2
C₃₁×#2921	2
#60810	31

Subgroups — C31×#60810 #21517569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60810 #21517569