Subgroups — C₃₁×#61287 #21518046

Order 7936.

1846 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3511	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3357	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3353	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3342	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3341	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3338	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3334	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3098	C₂	1	4	yes	maximal, complemented
10	C₃₁×#3079	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3061	C₂	1	2	yes	maximal, complemented
12	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#2913	C₂	1	2	yes	maximal, complemented
15	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#131	C₂²	1	2	yes	—
17	C₃₁×#556	C₂²	1	4	yes	—
18	C₃₁×#555	C₂²	1	4	yes	—
19	C₃₁×#554	C₂²	1	4	yes	—
20	C₃₁×#553	—	2	4	no	—
21	C₃₁×#548	—	2	4	no	—
22	C₃₁×#547	C₂²	1	4	yes	—
23	C₃₁×#542	—	2	4	no	—
24	C₃₁×#537	C₂²	1	4	yes	—
25	C₃₁×#533	—	2	2	no	—
26	C₃₁×#532	—	2	2	no	—
27	C₃₁×#532	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
29	C₃₁×#520	C₂²	1	4	yes	—
30	C₄×#41×C₃₁	C₂²	1	2	yes	—
31	C₃₁×#500	C₂²	1	2	yes	—
32	C₃₁×#500	C₂²	1	2	yes	—
33	C₃₁×#500	C₂²	1	2	yes	—
34	C₃₁×#500	C₂²	1	2	yes	—
35	C₃₁×#485	C₂²	1	8	yes	—
36	C₃₁×#484	C₂²	1	8	yes	—
37	C₃₁×#482	C₂²	1	8	yes	—
38	C₃₁×#480	C₂²	1	8	yes	—
39	C₃₁×#477	C₂²	1	4	yes	—
40	C₃₁×#476	C₂²	1	4	yes	—
41	C₃₁×#475	C₂²	1	4	yes	—
42	C₃₁×#474	C₂²	1	4	yes	—
43	C₃₁×#468	C₂²	1	8	yes	—
44	C₃₁×#467	C₂²	1	4	yes	—
45	C₃₁×#466	C₂²	1	4	yes	—
46	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
47	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
48	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
49	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
50	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
51	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
52	C₃₁×#436	C₂²	1	2	yes	—
53	C₃₁×#436	C₂²	1	2	yes	—
54	C₈×D₄×C₃₁	C₂²	1	2	yes	—
55	C₈×D₄×C₃₁	C₂²	1	2	yes	—
56	C₃₁×#428	C₂²	1	4	yes	—
57	C₃₁×#428	C₂²	1	8	yes	—
58	C₃₁×#427	C₂²	1	4	yes	—
59	C₃₁×#427	C₂²	1	1	yes	characteristic
60	C₂×C₃₁×#107	C₂²	1	4	yes	—
61	C₂×C₃₁×#107	C₂²	1	2	yes	—
62	C₂×C₃₁×#107	C₂²	1	2	yes	—
63	C₃₁×#423	C₂²	1	1	yes	characteristic
64	C₃₁×#417	C₂²	1	4	yes	—
65	C₃₁×#416	C₂²	1	2	yes	—
66	C₃₁×#408	C₂²	1	2	yes	—
67	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
68	C₂×#41×C₃₁	—	2	4	no	—
69	C₂×#41×C₃₁	C₂³	1	2	yes	—
70	C₂×#41×C₃₁	C₂³	1	2	yes	—
71	C₂²×Q₈×C₃₁	—	2	2	no	—
72	C₃₁×#135	—	2	4	no	—
73	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
75	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#131	—	2	8	no	—
77	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
78	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
79	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
81	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₃₁×C₄⋊Q₈	—	2	2	no	—
83	C₃₁×C₄⋊Q₈	—	2	2	no	—
84	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
85	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
86	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
87	C₃₁×#127	D₄	1	1	yes	characteristic
88	C₃₁×#126	—	2	8	no	—
89	C₃₁×#125	—	2	4	no	—
90	C₃₁×#124	—	2	4	no	—
91	C₃₁×#124	—	2	8	no	—
92	C₃₁×#124	D₄	1	2	yes	—
93	C₃₁×#123	—	2	8	no	—
94	C₃₁×C₂²⋊Q₈	—	2	4	no	—
95	C₃₁×C₂²⋊Q₈	—	2	4	no	—
96	C₃₁×C₂²⋊Q₈	—	2	8	no	—
97	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
98	C₃₁×#121	—	2	4	no	—
99	C₃₁×#121	—	2	4	no	—
100	C₃₁×#121	—	2	8	no	—
101	C₃₁×#121	D₄	1	2	yes	—
102	C₃₁×#121	D₄	1	2	yes	—
103	C₃₁×#121	C₂³	1	8	yes	—
104	C₄×Q₈×C₃₁	—	2	4	no	—
105	C₄×Q₈×C₃₁	—	2	4	no	—
106	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
107	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
108	C₄×D₄×C₃₁	—	2	4	no	—
109	C₄×D₄×C₃₁	—	2	8	no	—
110	C₄×D₄×C₃₁	—	2	4	no	—
111	C₄×D₄×C₃₁	—	2	2	no	—
112	C₄×D₄×C₃₁	—	2	2	no	—
113	C₄×D₄×C₃₁	C₂³	1	2	yes	—
114	C₄×D₄×C₃₁	C₂³	1	2	yes	—
115	C₄×D₄×C₃₁	C₂³	1	2	yes	—
116	C₄×D₄×C₃₁	C₂³	1	2	yes	—
117	C₃₁×#117	—	2	4	no	—
118	C₃₁×#117	—	2	2	no	—
119	C₃₁×#117	C₂³	1	2	yes	—
120	C₃₁×#117	C₂³	1	8	yes	—
121	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
123	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
124	C₂×C₄²×C₃₁	—	2	2	no	—
125	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
126	C₃₁×#107	C₂³	1	1	yes	characteristic
127	C₃₁×#107	C₂³	1	4	yes	—
128	C₃₁×#107	C₂³	1	4	yes	—
129	C₃₁×#107	C₂³	1	8	yes	—
130	C₃₁×#107	C₂³	1	4	yes	—
131	C₃₁×#107	C₂³	1	4	yes	—
132	C₃₁×#107	C₂³	1	1	yes	characteristic
133	C₃₁×#106	C₂³	1	2	yes	—
134	C₃₁×#106	C₂³	1	8	yes	—
135	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
136	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
137	C₃₁×#103	C₂³	1	2	yes	—
138	C₃₁×#103	C₂³	1	2	yes	—
139	C₃₁×#103	C₂³	1	8	yes	—
140	C₃₁×#102	C₂³	1	2	yes	—
141	C₃₁×#102	C₂³	1	2	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	#41×C₃₁	—	2	4	no	—
150	#41×C₃₁	—	2	4	no	—
151	#41×C₃₁	—	2	4	no	—
152	#41×C₃₁	—	2	4	no	—
153	#41×C₃₁	—	8	8	no	—
154	C₂×Q₈×C₃₁	—	2	4	no	—
155	C₂×Q₈×C₃₁	—	2	4	no	—
156	C₂×Q₈×C₃₁	—	8	4	no	—
157	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
158	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
159	C₂×D₄×C₃₁	—	2	2	no	—
160	C₂×D₄×C₃₁	—	2	8	no	—
161	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
162	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
163	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
164	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
165	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₃₁	—	2	4	no	—
167	C₂²×C₄×C₃₁	—	2	8	no	—
168	C₂²×C₄×C₃₁	—	2	2	no	—
169	C₂²×C₄×C₃₁	—	2	2	no	—
170	C₂²×C₄×C₃₁	—	2	2	no	—
171	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
174	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
175	Q₁₆×C₃₁	—	2	2	no	—
176	SD₁₆×C₃₁	—	2	4	no	—
177	D₈×C₃₁	—	2	2	no	—
178	M₁₆×C₃₁	—	2	4	no	—
179	M₁₆×C₃₁	—	2	8	no	—
180	C₂×C₈×C₃₁	—	2	2	no	—
181	C₂×C₈×C₃₁	—	2	8	no	—
182	C₂×C₈×C₃₁	#41	1	4	yes	complemented
183	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
184	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
185	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
186	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
187	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
188	C₄⋊C₄×C₃₁	—	2	4	no	—
189	C₄⋊C₄×C₃₁	—	2	4	no	—
190	C₄⋊C₄×C₃₁	—	2	4	no	—
191	C₄⋊C₄×C₃₁	—	2	8	no	—
192	C₄⋊C₄×C₃₁	—	2	2	no	—
193	C₄⋊C₄×C₃₁	—	2	2	no	—
194	C₄⋊C₄×C₃₁	—	2	8	no	—
195	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
196	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
197	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
198	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
199	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
200	C₂²⋊C₄×C₃₁	—	2	2	no	—
201	C₂²⋊C₄×C₃₁	—	2	8	no	—
202	C₂²⋊C₄×C₃₁	—	2	8	no	—
203	C₂²⋊C₄×C₃₁	—	2	2	no	—
204	C₂²⋊C₄×C₃₁	—	2	2	no	—
205	C₂²⋊C₄×C₃₁	—	2	8	no	—
206	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
207	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
208	C₄²×C₃₁	—	2	2	no	—
209	C₄²×C₃₁	—	2	2	no	—
210	C₄²×C₃₁	—	2	2	no	—
211	C₄²×C₃₁	—	2	8	no	—
212	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
213	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
214	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
215	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
216	#61287	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂³×C₃₁	—	2	2	no	—
218	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
219	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	Q₈×C₃₁	—	2	4	no	—
221	Q₈×C₃₁	—	2	2	no	—
222	Q₈×C₃₁	—	2	2	no	—
223	Q₈×C₃₁	—	4	8	no	—
224	Q₈×C₃₁	—	8	4	no	—
225	D₄×C₃₁	—	2	8	no	—
226	D₄×C₃₁	—	2	4	no	—
227	D₄×C₃₁	—	2	4	no	—
228	D₄×C₃₁	—	2	4	no	—
229	D₄×C₃₁	—	2	2	no	—
230	D₄×C₃₁	—	2	2	no	—
231	D₄×C₃₁	—	4	4	no	—
232	D₄×C₃₁	—	8	8	no	—
233	C₂×C₄×C₃₁	—	2	2	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	4	no	—
236	C₂×C₄×C₃₁	—	2	4	no	—
237	C₂×C₄×C₃₁	—	2	2	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	8	no	—
241	C₂×C₄×C₃₁	—	4	4	no	—
242	C₂×C₄×C₃₁	—	4	4	no	—
243	C₂×C₄×C₃₁	—	4	4	no	—
244	C₂×C₄×C₃₁	—	8	4	no	—
245	C₂×C₄×C₃₁	—	8	8	no	—
246	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
247	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
253	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
254	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
255	C₈×C₃₁	—	2	4	no	—
256	C₈×C₃₁	—	2	1	no	—
257	C₈×C₃₁	—	2	1	no	—
258	C₈×C₃₁	C₂×#41	1	2	yes	—
259	C₈×C₃₁	C₂×#41	1	2	yes	—
260	#3511	C₂×C₃₁	1	2	yes	complemented
261	#3357	C₂×C₃₁	1	4	yes	complemented
262	#3353	C₂×C₃₁	1	4	yes	complemented
263	#3342	C₂×C₃₁	1	2	yes	complemented
264	#3341	C₂×C₃₁	1	2	yes	complemented
265	#3338	C₂×C₃₁	1	2	yes	complemented
266	#3334	C₂×C₃₁	1	2	yes	complemented
267	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#3098	C₂×C₃₁	1	4	yes	complemented
269	#3079	C₂×C₃₁	1	2	yes	complemented
270	#3061	C₂×C₃₁	1	2	yes	complemented
271	C₄×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
272	#2931	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#2913	C₂×C₃₁	1	2	yes	complemented
274	C₂²×C₃₁	—	2	1	no	—
275	C₂²×C₃₁	—	2	1	no	—
276	C₂²×C₃₁	—	2	4	no	—
277	C₂²×C₃₁	—	2	4	no	—
278	C₂²×C₃₁	—	2	4	no	—
279	C₂²×C₃₁	—	4	2	no	—
280	C₂²×C₃₁	—	4	2	no	—
281	C₂²×C₃₁	—	4	2	no	—
282	C₂²×C₃₁	#547	1	2	yes	—
283	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
284	C₄×C₃₁	—	2	4	no	—
285	C₄×C₃₁	—	2	1	no	—
286	C₄×C₃₁	—	2	1	no	—
287	C₄×C₃₁	—	4	2	no	—
288	C₄×C₃₁	—	4	2	no	—
289	C₄×C₃₁	—	4	8	no	—
290	C₄×C₃₁	—	4	2	no	—
291	C₄×C₃₁	#547	1	2	yes	—
292	C₄×C₃₁	#576	1	2	yes	—
293	C₄×C₃₁	#578	1	2	yes	—
294	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
295	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
296	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
297	C₂×#131	C₂²×C₃₁	1	2	yes	—
298	#556	C₂²×C₃₁	1	4	yes	—
299	#555	C₂²×C₃₁	1	4	yes	—
300	#554	C₂²×C₃₁	1	4	yes	—
301	#553	—	2	4	no	—
302	#548	—	2	4	no	—
303	#547	C₂²×C₃₁	1	4	yes	—
304	#542	—	2	4	no	—
305	#537	C₂²×C₃₁	1	4	yes	—
306	#533	—	2	2	no	—
307	#532	—	2	2	no	—
308	#532	C₂²×C₃₁	1	1	yes	characteristic
309	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
310	#520	C₂²×C₃₁	1	4	yes	—
311	C₄×#41	C₂²×C₃₁	1	2	yes	—
312	#500	C₂²×C₃₁	1	2	yes	—
313	#500	C₂²×C₃₁	1	2	yes	—
314	#500	C₂²×C₃₁	1	2	yes	—
315	#500	C₂²×C₃₁	1	2	yes	—
316	#485	C₂²×C₃₁	1	8	yes	—
317	#484	C₂²×C₃₁	1	8	yes	—
318	#482	C₂²×C₃₁	1	8	yes	—
319	#480	C₂²×C₃₁	1	8	yes	—
320	#477	C₂²×C₃₁	1	4	yes	—
321	#476	C₂²×C₃₁	1	4	yes	—
322	#475	C₂²×C₃₁	1	4	yes	—
323	#474	C₂²×C₃₁	1	4	yes	—
324	#468	C₂²×C₃₁	1	8	yes	—
325	#467	C₂²×C₃₁	1	4	yes	—
326	#466	C₂²×C₃₁	1	4	yes	—
327	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
328	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
329	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
330	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
331	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
332	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
333	#436	C₂²×C₃₁	1	2	yes	—
334	#436	C₂²×C₃₁	1	2	yes	—
335	C₈×D₄	C₂²×C₃₁	1	2	yes	—
336	C₈×D₄	C₂²×C₃₁	1	2	yes	—
337	#428	C₂²×C₃₁	1	4	yes	—
338	#428	C₂²×C₃₁	1	8	yes	—
339	#427	C₂²×C₃₁	1	4	yes	—
340	#427	C₂²×C₃₁	1	1	yes	characteristic
341	C₂×#107	C₂²×C₃₁	1	4	yes	—
342	C₂×#107	C₂²×C₃₁	1	2	yes	—
343	C₂×#107	C₂²×C₃₁	1	2	yes	—
344	#423	C₂²×C₃₁	1	1	yes	characteristic
345	#417	C₂²×C₃₁	1	4	yes	—
346	#416	C₂²×C₃₁	1	2	yes	—
347	#408	C₂²×C₃₁	1	2	yes	—
348	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
349	C₂×C₃₁	—	2	2	no	—
350	C₂×C₃₁	—	4	4	no	—
351	C₂×C₃₁	—	8	2	no	—
352	C₂×C₃₁	C₂×#547	1	1	yes	characteristic
353	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
354	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
355	C₂×#41	—	2	4	no	—
356	C₂×#41	C₂³×C₃₁	1	2	yes	—
357	C₂×#41	C₂³×C₃₁	1	2	yes	—
358	C₂²×Q₈	—	2	2	no	—
359	#135	—	2	4	no	—
360	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
361	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
362	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
363	#131	—	2	8	no	—
364	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
365	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
366	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
367	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
368	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
369	C₄⋊Q₈	—	2	2	no	—
370	C₄⋊Q₈	—	2	2	no	—
371	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
372	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
373	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
374	#127	D₄×C₃₁	1	1	yes	characteristic
375	#126	—	2	8	no	—
376	#125	—	2	4	no	—
377	#124	—	2	4	no	—
378	#124	—	2	8	no	—
379	#124	D₄×C₃₁	1	2	yes	—
380	#123	—	2	8	no	—
381	C₂²⋊Q₈	—	2	4	no	—
382	C₂²⋊Q₈	—	2	4	no	—
383	C₂²⋊Q₈	—	2	8	no	—
384	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
385	#121	—	2	4	no	—
386	#121	—	2	4	no	—
387	#121	—	2	8	no	—
388	#121	D₄×C₃₁	1	2	yes	—
389	#121	D₄×C₃₁	1	2	yes	—
390	#121	C₂³×C₃₁	1	8	yes	—
391	C₄×Q₈	—	2	4	no	—
392	C₄×Q₈	—	2	4	no	—
393	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
394	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
395	C₄×D₄	—	2	4	no	—
396	C₄×D₄	—	2	8	no	—
397	C₄×D₄	—	2	4	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	—	2	2	no	—
400	C₄×D₄	C₂³×C₃₁	1	2	yes	—
401	C₄×D₄	C₂³×C₃₁	1	2	yes	—
402	C₄×D₄	C₂³×C₃₁	1	2	yes	—
403	C₄×D₄	C₂³×C₃₁	1	2	yes	—
404	#117	—	2	4	no	—
405	#117	—	2	2	no	—
406	#117	C₂³×C₃₁	1	2	yes	—
407	#117	C₂³×C₃₁	1	8	yes	—
408	C₂×C₄⋊C₄	—	2	2	no	—
409	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
410	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
411	C₂×C₄²	—	2	2	no	—
412	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
413	#107	C₂³×C₃₁	1	1	yes	characteristic
414	#107	C₂³×C₃₁	1	4	yes	—
415	#107	C₂³×C₃₁	1	4	yes	—
416	#107	C₂³×C₃₁	1	8	yes	—
417	#107	C₂³×C₃₁	1	4	yes	—
418	#107	C₂³×C₃₁	1	4	yes	—
419	#107	C₂³×C₃₁	1	1	yes	characteristic
420	#106	C₂³×C₃₁	1	2	yes	—
421	#106	C₂³×C₃₁	1	8	yes	—
422	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
423	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
424	#103	C₂³×C₃₁	1	2	yes	—
425	#103	C₂³×C₃₁	1	2	yes	—
426	#103	C₂³×C₃₁	1	8	yes	—
427	#102	C₂³×C₃₁	1	2	yes	—
428	#102	C₂³×C₃₁	1	2	yes	—
429	#102	C₂³×C₃₁	1	8	yes	—
430	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
431	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
432	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
433	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
434	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
435	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
436	C₃₁	#61287	1	1	yes	characteristic, direct product factor, Sylow
437	#41	—	2	4	no	—
438	#41	—	2	4	no	—
439	#41	—	2	4	no	—
440	#41	—	2	4	no	—
441	#41	—	8	8	no	—
442	C₂×Q₈	—	2	4	no	—
443	C₂×Q₈	—	2	4	no	—
444	C₂×Q₈	—	8	4	no	—
445	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
446	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	2	8	no	—
449	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
450	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
451	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
452	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
453	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
454	C₂²×C₄	—	2	4	no	—
455	C₂²×C₄	—	2	8	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
460	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
461	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
462	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
463	Q₁₆	—	2	2	no	—
464	SD₁₆	—	2	4	no	—
465	D₈	—	2	2	no	—
466	M₁₆	—	2	4	no	—
467	M₁₆	—	2	8	no	—
468	C₂×C₈	—	2	2	no	—
469	C₂×C₈	—	2	8	no	—
470	C₂×C₈	#41×C₃₁	1	4	yes	complemented
471	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
472	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
473	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
474	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
475	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
476	C₄⋊C₄	—	2	4	no	—
477	C₄⋊C₄	—	2	4	no	—
478	C₄⋊C₄	—	2	4	no	—
479	C₄⋊C₄	—	2	8	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	8	no	—
483	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
484	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
485	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
486	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
487	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
488	C₂²⋊C₄	—	2	2	no	—
489	C₂²⋊C₄	—	2	8	no	—
490	C₂²⋊C₄	—	2	8	no	—
491	C₂²⋊C₄	—	2	2	no	—
492	C₂²⋊C₄	—	2	2	no	—
493	C₂²⋊C₄	—	2	8	no	—
494	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
495	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
496	C₄²	—	2	2	no	—
497	C₄²	—	2	2	no	—
498	C₄²	—	2	2	no	—
499	C₄²	—	2	8	no	—
500	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
501	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
502	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
503	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
504	C₂³	—	2	2	no	—
505	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
506	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
507	Q₈	—	2	4	no	—
508	Q₈	—	2	2	no	—
509	Q₈	—	2	2	no	—
510	Q₈	—	4	8	no	—
511	Q₈	—	8	4	no	—
512	D₄	—	2	8	no	—
513	D₄	—	2	4	no	—
514	D₄	—	2	4	no	—
515	D₄	—	2	4	no	—
516	D₄	—	2	2	no	—
517	D₄	—	2	2	no	—
518	D₄	—	4	4	no	—
519	D₄	—	8	8	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	8	no	—
528	C₂×C₄	—	4	4	no	—
529	C₂×C₄	—	4	4	no	—
530	C₂×C₄	—	4	4	no	—
531	C₂×C₄	—	8	4	no	—
532	C₂×C₄	—	8	8	no	—
533	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
534	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
535	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
536	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
537	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
538	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
539	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
540	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
541	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
542	C₈	—	2	4	no	—
543	C₈	—	2	1	no	—
544	C₈	—	2	1	no	—
545	C₈	C₂×#41×C₃₁	1	2	yes	—
546	C₈	C₂×#41×C₃₁	1	2	yes	—
547	C₂²	—	2	1	no	—
548	C₂²	—	2	1	no	—
549	C₂²	—	2	4	no	—
550	C₂²	—	2	4	no	—
551	C₂²	—	2	4	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	4	2	no	—
554	C₂²	—	4	2	no	—
555	C₂²	C₃₁×#547	1	2	yes	—
556	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
557	C₄	—	2	4	no	—
558	C₄	—	2	1	no	—
559	C₄	—	2	1	no	—
560	C₄	—	4	2	no	—
561	C₄	—	4	2	no	—
562	C₄	—	4	8	no	—
563	C₄	—	4	2	no	—
564	C₄	C₃₁×#547	1	2	yes	—
565	C₄	C₃₁×#576	1	2	yes	—
566	C₄	C₃₁×#578	1	2	yes	—
567	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
568	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
569	C₂	—	2	2	no	—
570	C₂	—	4	4	no	—
571	C₂	—	8	2	no	—
572	C₂	C₂×C₃₁×#547	1	1	yes	characteristic
573	C₂	C₂×C₃₁×#576	1	1	yes	characteristic
574	C₂	C₂×C₃₁×#578	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. C₃₁×#3511	C₂	semidirect
2. C₃₁×#3357	C₂	semidirect
3. C₃₁×#3353	C₂	semidirect
4. C₃₁×#3342	C₂	semidirect
5. C₃₁×#3341	C₂	semidirect
6. C₃₁×#3338	C₂	semidirect
7. C₃₁×#3334	C₂	semidirect
8. C₂×C₃₁×#500	C₂	semidirect
9. C₃₁×#3098	C₂	semidirect
10. C₃₁×#3079	C₂	semidirect
11. C₃₁×#3061	C₂	semidirect
12. C₄×C₃₁×#135	C₂	semidirect
13. C₃₁×#2931	C₂	semidirect
14. C₃₁×#2913	C₂	semidirect
182. C₂×C₈×C₃₁	#41	semidirect
216. #61287	C₃₁	direct (N × H)
260. #3511	C₂×C₃₁	semidirect
261. #3357	C₂×C₃₁	semidirect
262. #3353	C₂×C₃₁	semidirect
263. #3342	C₂×C₃₁	semidirect
264. #3341	C₂×C₃₁	semidirect
265. #3338	C₂×C₃₁	semidirect
266. #3334	C₂×C₃₁	semidirect
267. C₂×#500	C₂×C₃₁	semidirect
268. #3098	C₂×C₃₁	semidirect
269. #3079	C₂×C₃₁	semidirect
270. #3061	C₂×C₃₁	semidirect
271. C₄×#135	C₂×C₃₁	semidirect
272. #2931	C₂×C₃₁	semidirect
273. #2913	C₂×C₃₁	semidirect
436. C₃₁	#61287	direct (N × H)
470. C₂×C₈	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3511	2
C₃₁×#3357	2
C₃₁×#3353	2
C₃₁×#3342	2
C₃₁×#3341	2
C₃₁×#3338	2
C₃₁×#3334	2
C₂×C₃₁×#500	2
C₃₁×#3098	2
C₃₁×#3079	2
C₃₁×#3061	2
C₄×C₃₁×#135	2
C₃₁×#2931	2
C₃₁×#2913	2
#61287	31

Subgroups — C31×#61287 #21518046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61287 #21518046