Subgroups — C₃₁×#19766 #21476525

Order 7936.

650 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3384	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3335	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3298	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3283	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1716	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1690	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1679	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1671	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1671	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1654	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1651	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1643	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1635	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1582	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1580	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#135	—	2	1	no	—
17	C₃₁×#554	C₂²	1	1	yes	characteristic
18	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#547	C₂²	1	1	yes	characteristic
20	C₃₁×#547	C₂²	1	1	yes	characteristic
21	C₃₁×#501	—	2	1	no	—
22	C₃₁×#498	C₂²	1	1	yes	characteristic
23	C₃₁×#498	C₂²	1	1	yes	characteristic
24	C₃₁×#498	C₂²	1	1	yes	characteristic
25	C₃₁×#494	C₂²	1	1	yes	characteristic
26	C₃₁×#487	C₂²	1	1	yes	characteristic
27	C₃₁×#487	C₂²	1	1	yes	characteristic
28	C₃₁×#483	—	2	1	no	—
29	C₃₁×#483	—	2	1	no	—
30	C₃₁×#481	—	2	1	no	—
31	C₃₁×#478	—	2	1	no	—
32	C₃₁×#476	—	2	1	no	—
33	C₃₁×#476	C₂²	1	1	yes	characteristic
34	C₃₁×#476	C₂²	1	1	yes	characteristic
35	C₃₁×#475	C₂²	1	1	yes	characteristic
36	C₃₁×#475	C₂²	1	1	yes	characteristic
37	C₃₁×#474	C₂²	1	1	yes	characteristic
38	C₃₁×#467	—	2	1	no	—
39	C₃₁×#465	—	2	1	no	—
40	C₃₁×#465	—	2	1	no	—
41	C₃₁×#461	C₂²	1	1	yes	characteristic
42	C₃₁×#461	C₂²	1	1	yes	characteristic
43	C₃₁×#461	C₂²	1	1	yes	characteristic
44	C₃₁×#452	—	2	1	no	—
45	C₃₁×#452	—	2	1	no	—
46	C₃₁×#448	—	2	1	no	—
47	C₃₁×#446	C₂²	1	1	yes	characteristic
48	C₃₁×#442	—	2	1	no	—
49	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
50	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
51	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
52	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
53	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
54	C₂×C₃₁×#106	—	2	1	no	—
55	C₂×C₃₁×#103	—	2	1	no	—
56	C₃₁×#334	C₂²	1	1	yes	characteristic
57	C₃₁×#334	C₂²	1	1	yes	characteristic
58	C₃₁×#333	C₂²	1	1	yes	characteristic
59	C₃₁×#332	C₂²	1	1	yes	characteristic
60	C₃₁×#332	C₂²	1	1	yes	characteristic
61	C₃₁×#332	C₂²	1	1	yes	characteristic
62	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
64	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
65	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₃₁×#322	C₂²	1	1	yes	characteristic
67	C₂×#41×C₃₁	—	2	1	no	—
68	C₂×#41×C₃₁	—	2	1	no	—
69	C₂²×Q₈×C₃₁	—	2	1	no	—
70	C₃₁×#135	—	4	4	no	—
71	C₂×Q₁₆×C₃₁	—	2	1	no	—
72	C₂×Q₁₆×C₃₁	—	2	1	no	—
73	C₂×SD₁₆×C₃₁	—	2	1	no	—
74	C₂×SD₁₆×C₃₁	—	2	1	no	—
75	C₂×SD₁₆×C₃₁	—	2	1	no	—
76	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
77	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₃₁	—	2	1	no	—
79	C₂²×C₈×C₃₁	—	2	1	no	—
80	C₂²×C₈×C₃₁	—	2	1	no	—
81	C₃₁×C₄⋊Q₈	—	2	1	no	—
82	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₃₁×#125	—	2	1	no	—
87	C₃₁×#123	—	4	1	no	—
88	C₃₁×#123	—	4	1	no	—
89	C₃₁×C₂²⋊Q₈	—	2	1	no	—
90	C₃₁×C₂²⋊Q₈	—	2	1	no	—
91	C₃₁×C₂²⋊Q₈	—	2	1	no	—
92	C₃₁×C₂²⋊Q₈	—	2	1	no	—
93	C₃₁×C₂²⋊Q₈	—	4	1	no	—
94	C₃₁×C₂²⋊Q₈	—	4	1	no	—
95	C₃₁×#121	—	2	1	no	—
96	C₃₁×#121	—	2	1	no	—
97	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
98	C₄×D₄×C₃₁	—	2	1	no	—
99	C₄×D₄×C₃₁	—	2	1	no	—
100	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
101	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
102	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
106	C₃₁×#107	—	2	1	no	—
107	C₃₁×#107	D₄	1	1	yes	characteristic
108	C₃₁×#106	—	2	1	no	—
109	C₃₁×#106	—	2	1	no	—
110	C₃₁×#106	—	2	1	no	—
111	C₃₁×#106	—	2	1	no	—
112	C₃₁×#106	—	2	1	no	—
113	C₃₁×#106	—	2	1	no	—
114	C₃₁×#106	—	2	1	no	—
115	C₃₁×#106	—	2	1	no	—
116	C₃₁×#106	—	4	1	no	—
117	C₃₁×#106	D₄	1	1	yes	characteristic
118	C₃₁×C₄⋊C₈	—	2	1	no	—
119	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
120	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
121	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
122	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
123	C₃₁×#103	—	2	1	no	—
124	C₃₁×#103	—	2	1	no	—
125	C₃₁×#103	—	2	1	no	—
126	C₃₁×#103	—	2	1	no	—
127	C₃₁×#103	—	2	1	no	—
128	C₃₁×#103	—	2	1	no	—
129	C₃₁×#103	—	2	1	no	—
130	C₃₁×#103	—	4	1	no	—
131	C₃₁×#103	D₄	1	1	yes	characteristic
132	C₃₁×#102	—	2	1	no	—
133	C₃₁×#102	—	2	1	no	—
134	C₃₁×#102	—	2	1	no	—
135	C₃₁×#102	D₄	1	1	yes	characteristic
136	C₃₁×#102	D₄	1	1	yes	characteristic
137	C₃₁×#102	D₄	1	1	yes	characteristic
138	C₃₁×C₂²⋊C₈	—	2	1	no	—
139	C₃₁×C₂²⋊C₈	—	2	1	no	—
140	C₃₁×C₂²⋊C₈	—	2	1	no	—
141	C₃₁×#97	—	2	1	no	—
142	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
145	#41×C₃₁	—	4	2	no	—
146	#41×C₃₁	—	4	2	no	—
147	#41×C₃₁	—	8	1	no	—
148	#41×C₃₁	—	8	1	no	—
149	C₂×Q₈×C₃₁	—	2	1	no	—
150	C₂×Q₈×C₃₁	—	2	1	no	—
151	C₂×Q₈×C₃₁	—	2	1	no	—
152	C₂×Q₈×C₃₁	—	2	1	no	—
153	C₂×Q₈×C₃₁	—	4	1	no	—
154	C₂×Q₈×C₃₁	—	4	2	no	—
155	C₂×Q₈×C₃₁	—	8	1	no	—
156	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂×D₄×C₃₁	—	4	1	no	—
158	C₂×D₄×C₃₁	—	4	1	no	—
159	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	—	2	1	no	—
163	C₂²×C₄×C₃₁	—	2	1	no	—
164	C₂²×C₄×C₃₁	—	2	1	no	—
165	C₂²×C₄×C₃₁	—	4	1	no	—
166	C₂²×C₄×C₃₁	—	4	1	no	—
167	C₂²×C₄×C₃₁	—	4	1	no	—
168	C₂²×C₄×C₃₁	—	4	1	no	—
169	Q₁₆×C₃₁	—	4	2	no	—
170	Q₁₆×C₃₁	—	8	1	no	—
171	SD₁₆×C₃₁	—	4	1	no	—
172	SD₁₆×C₃₁	—	4	2	no	—
173	SD₁₆×C₃₁	—	4	2	no	—
174	SD₁₆×C₃₁	—	8	1	no	—
175	D₈×C₃₁	SD₁₆	1	4	yes	complemented
176	C₂×C₈×C₃₁	—	2	1	no	—
177	C₂×C₈×C₃₁	—	2	1	no	—
178	C₂×C₈×C₃₁	—	2	1	no	—
179	C₂×C₈×C₃₁	—	2	1	no	—
180	C₂×C₈×C₃₁	—	2	1	no	—
181	C₂×C₈×C₃₁	—	4	2	no	—
182	C₂×C₈×C₃₁	—	4	2	no	—
183	C₂×C₈×C₃₁	—	8	1	no	—
184	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
185	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
186	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
187	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₄⋊C₄×C₃₁	—	2	1	no	—
189	C₄⋊C₄×C₃₁	—	2	1	no	—
190	C₄⋊C₄×C₃₁	—	2	1	no	—
191	C₄⋊C₄×C₃₁	—	2	1	no	—
192	C₄⋊C₄×C₃₁	—	2	1	no	—
193	C₄⋊C₄×C₃₁	—	2	1	no	—
194	C₄⋊C₄×C₃₁	—	2	1	no	—
195	C₄⋊C₄×C₃₁	—	2	1	no	—
196	C₄⋊C₄×C₃₁	—	2	1	no	—
197	C₄⋊C₄×C₃₁	—	4	1	no	—
198	C₄⋊C₄×C₃₁	—	4	1	no	—
199	C₄⋊C₄×C₃₁	—	4	1	no	—
200	C₄⋊C₄×C₃₁	—	4	1	no	—
201	C₄⋊C₄×C₃₁	—	4	1	no	—
202	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
203	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
204	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
205	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
206	C₂²⋊C₄×C₃₁	—	2	1	no	—
207	C₂²⋊C₄×C₃₁	—	2	1	no	—
208	C₂²⋊C₄×C₃₁	—	2	1	no	—
209	C₂²⋊C₄×C₃₁	—	4	1	no	—
210	C₂²⋊C₄×C₃₁	—	4	1	no	—
211	C₂²⋊C₄×C₃₁	—	4	1	no	—
212	C₂²⋊C₄×C₃₁	—	4	1	no	—
213	C₄²×C₃₁	—	2	1	no	—
214	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
215	#19766	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂³×C₃₁	—	2	1	no	—
217	C₂³×C₃₁	—	2	1	no	—
218	C₂³×C₃₁	—	2	1	no	—
219	Q₈×C₃₁	—	2	1	no	—
220	Q₈×C₃₁	—	2	1	no	—
221	Q₈×C₃₁	—	4	1	no	—
222	Q₈×C₃₁	—	4	1	no	—
223	Q₈×C₃₁	—	4	1	no	—
224	Q₈×C₃₁	—	4	1	no	—
225	Q₈×C₃₁	—	4	1	no	—
226	Q₈×C₃₁	—	4	1	no	—
227	Q₈×C₃₁	—	8	1	no	—
228	Q₈×C₃₁	—	8	2	no	—
229	D₄×C₃₁	—	2	1	no	—
230	D₄×C₃₁	—	2	1	no	—
231	D₄×C₃₁	—	2	1	no	—
232	D₄×C₃₁	—	2	1	no	—
233	D₄×C₃₁	—	8	2	no	—
234	D₄×C₃₁	—	8	2	no	—
235	D₄×C₃₁	C₂×SD₁₆	1	2	yes	—
236	D₄×C₃₁	C₂×SD₁₆	1	2	yes	—
237	C₂×C₄×C₃₁	—	2	1	no	—
238	C₂×C₄×C₃₁	—	2	1	no	—
239	C₂×C₄×C₃₁	—	2	1	no	—
240	C₂×C₄×C₃₁	—	2	1	no	—
241	C₂×C₄×C₃₁	—	2	1	no	—
242	C₂×C₄×C₃₁	—	4	1	no	—
243	C₂×C₄×C₃₁	—	4	1	no	—
244	C₂×C₄×C₃₁	—	4	1	no	—
245	C₂×C₄×C₃₁	—	4	1	no	—
246	C₂×C₄×C₃₁	—	4	1	no	—
247	C₂×C₄×C₃₁	—	4	1	no	—
248	C₂×C₄×C₃₁	—	4	1	no	—
249	C₂×C₄×C₃₁	—	4	1	no	—
250	C₂×C₄×C₃₁	—	4	1	no	—
251	C₂×C₄×C₃₁	—	4	1	no	—
252	C₂×C₄×C₃₁	—	8	2	no	—
253	C₂×C₄×C₃₁	—	8	2	no	—
254	C₂×C₄×C₃₁	—	8	2	no	—
255	C₂×C₄×C₃₁	—	16	1	no	—
256	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
257	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
259	C₈×C₃₁	—	2	1	no	—
260	C₈×C₃₁	—	2	1	no	—
261	C₈×C₃₁	—	2	1	no	—
262	C₈×C₃₁	—	4	1	no	—
263	C₈×C₃₁	—	4	1	no	—
264	C₈×C₃₁	—	4	1	no	—
265	C₈×C₃₁	—	4	1	no	—
266	C₈×C₃₁	—	4	1	no	—
267	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
268	#3384	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#3335	C₂×C₃₁	1	1	yes	characteristic, complemented
270	#3298	C₂×C₃₁	1	1	yes	characteristic, complemented
271	#3283	C₂×C₃₁	1	1	yes	characteristic
272	#1716	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#1690	C₂×C₃₁	1	1	yes	characteristic, complemented
274	#1679	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#1671	C₂×C₃₁	1	1	yes	characteristic, complemented
276	#1671	C₂×C₃₁	1	1	yes	characteristic, complemented
277	#1654	C₂×C₃₁	1	1	yes	characteristic, complemented
278	#1651	C₂×C₃₁	1	1	yes	characteristic, complemented
279	#1643	C₂×C₃₁	1	1	yes	characteristic, complemented
280	#1635	C₂×C₃₁	1	1	yes	characteristic, complemented
281	#1582	C₂×C₃₁	1	1	yes	characteristic, complemented
282	#1580	C₂×C₃₁	1	1	yes	characteristic, complemented
283	C₂²×C₃₁	—	2	2	no	—
284	C₂²×C₃₁	—	2	2	no	—
285	C₂²×C₃₁	—	4	1	no	—
286	C₂²×C₃₁	—	4	1	no	—
287	C₂²×C₃₁	—	4	1	no	—
288	C₂²×C₃₁	—	4	1	no	—
289	C₂²×C₃₁	—	4	1	no	—
290	C₂²×C₃₁	—	4	1	no	—
291	C₂²×C₃₁	—	4	1	no	—
292	C₂²×C₃₁	D₄²	1	1	yes	characteristic
293	C₄×C₃₁	—	2	1	no	—
294	C₄×C₃₁	—	4	1	no	—
295	C₄×C₃₁	—	4	1	no	—
296	C₄×C₃₁	—	4	1	no	—
297	C₄×C₃₁	—	4	1	no	—
298	C₄×C₃₁	—	4	1	no	—
299	C₄×C₃₁	—	8	1	no	—
300	C₄×C₃₁	—	8	1	no	—
301	C₄×C₃₁	—	8	1	no	—
302	C₄×C₃₁	—	8	1	no	—
303	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
304	C₄×C₃₁	#575	1	1	yes	characteristic
305	C₄×C₃₁	#577	1	1	yes	characteristic
306	C₄×C₃₁	#577	1	1	yes	characteristic
307	C₂×#135	—	2	1	no	—
308	#554	C₂²×C₃₁	1	1	yes	characteristic
309	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
310	#547	C₂²×C₃₁	1	1	yes	characteristic
311	#547	C₂²×C₃₁	1	1	yes	characteristic
312	#501	—	2	1	no	—
313	#498	C₂²×C₃₁	1	1	yes	characteristic
314	#498	C₂²×C₃₁	1	1	yes	characteristic
315	#498	C₂²×C₃₁	1	1	yes	characteristic
316	#494	C₂²×C₃₁	1	1	yes	characteristic
317	#487	C₂²×C₃₁	1	1	yes	characteristic
318	#487	C₂²×C₃₁	1	1	yes	characteristic
319	#483	—	2	1	no	—
320	#483	—	2	1	no	—
321	#481	—	2	1	no	—
322	#478	—	2	1	no	—
323	#476	—	2	1	no	—
324	#476	C₂²×C₃₁	1	1	yes	characteristic
325	#476	C₂²×C₃₁	1	1	yes	characteristic
326	#475	C₂²×C₃₁	1	1	yes	characteristic
327	#475	C₂²×C₃₁	1	1	yes	characteristic
328	#474	C₂²×C₃₁	1	1	yes	characteristic
329	#467	—	2	1	no	—
330	#465	—	2	1	no	—
331	#465	—	2	1	no	—
332	#461	C₂²×C₃₁	1	1	yes	characteristic
333	#461	C₂²×C₃₁	1	1	yes	characteristic
334	#461	C₂²×C₃₁	1	1	yes	characteristic
335	#452	—	2	1	no	—
336	#452	—	2	1	no	—
337	#448	—	2	1	no	—
338	#446	C₂²×C₃₁	1	1	yes	characteristic
339	#442	—	2	1	no	—
340	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
341	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
342	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
343	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
344	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
345	C₂×#106	—	2	1	no	—
346	C₂×#103	—	2	1	no	—
347	#334	C₂²×C₃₁	1	1	yes	characteristic
348	#334	C₂²×C₃₁	1	1	yes	characteristic
349	#333	C₂²×C₃₁	1	1	yes	characteristic
350	#332	C₂²×C₃₁	1	1	yes	characteristic
351	#332	C₂²×C₃₁	1	1	yes	characteristic
352	#332	C₂²×C₃₁	1	1	yes	characteristic
353	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
354	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
355	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
356	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
357	#322	C₂²×C₃₁	1	1	yes	characteristic
358	C₂×C₃₁	—	4	2	no	—
359	C₂×C₃₁	—	4	2	no	—
360	C₂×C₃₁	—	8	1	no	—
361	C₂×C₃₁	#3267	1	1	yes	characteristic
362	C₂×C₃₁	D₄×SD₁₆	1	1	yes	characteristic
363	C₂×C₃₁	#3289	1	1	yes	characteristic
364	C₂×#41	—	2	1	no	—
365	C₂×#41	—	2	1	no	—
366	C₂²×Q₈	—	2	1	no	—
367	#135	—	4	4	no	—
368	C₂×Q₁₆	—	2	1	no	—
369	C₂×Q₁₆	—	2	1	no	—
370	C₂×SD₁₆	—	2	1	no	—
371	C₂×SD₁₆	—	2	1	no	—
372	C₂×SD₁₆	—	2	1	no	—
373	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
374	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
375	C₂²×C₈	—	2	1	no	—
376	C₂²×C₈	—	2	1	no	—
377	C₂²×C₈	—	2	1	no	—
378	C₄⋊Q₈	—	2	1	no	—
379	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
380	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
381	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
382	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
383	#125	—	2	1	no	—
384	#123	—	4	1	no	—
385	#123	—	4	1	no	—
386	C₂²⋊Q₈	—	2	1	no	—
387	C₂²⋊Q₈	—	2	1	no	—
388	C₂²⋊Q₈	—	2	1	no	—
389	C₂²⋊Q₈	—	2	1	no	—
390	C₂²⋊Q₈	—	4	1	no	—
391	C₂²⋊Q₈	—	4	1	no	—
392	#121	—	2	1	no	—
393	#121	—	2	1	no	—
394	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
395	C₄×D₄	—	2	1	no	—
396	C₄×D₄	—	2	1	no	—
397	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
398	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
399	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
400	C₂×C₄⋊C₄	—	2	1	no	—
401	C₂×C₄⋊C₄	—	2	1	no	—
402	C₂×C₄⋊C₄	—	2	1	no	—
403	#107	—	2	1	no	—
404	#107	D₄×C₃₁	1	1	yes	characteristic
405	#106	—	2	1	no	—
406	#106	—	2	1	no	—
407	#106	—	2	1	no	—
408	#106	—	2	1	no	—
409	#106	—	2	1	no	—
410	#106	—	2	1	no	—
411	#106	—	2	1	no	—
412	#106	—	2	1	no	—
413	#106	—	4	1	no	—
414	#106	D₄×C₃₁	1	1	yes	characteristic
415	C₄⋊C₈	—	2	1	no	—
416	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
417	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
418	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
419	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
420	#103	—	2	1	no	—
421	#103	—	2	1	no	—
422	#103	—	2	1	no	—
423	#103	—	2	1	no	—
424	#103	—	2	1	no	—
425	#103	—	2	1	no	—
426	#103	—	2	1	no	—
427	#103	—	4	1	no	—
428	#103	D₄×C₃₁	1	1	yes	characteristic
429	#102	—	2	1	no	—
430	#102	—	2	1	no	—
431	#102	—	2	1	no	—
432	#102	D₄×C₃₁	1	1	yes	characteristic
433	#102	D₄×C₃₁	1	1	yes	characteristic
434	#102	D₄×C₃₁	1	1	yes	characteristic
435	C₂²⋊C₈	—	2	1	no	—
436	C₂²⋊C₈	—	2	1	no	—
437	C₂²⋊C₈	—	2	1	no	—
438	#97	—	2	1	no	—
439	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
440	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
441	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
442	C₃₁	#19766	1	1	yes	characteristic, direct product factor, Sylow
443	#41	—	4	2	no	—
444	#41	—	4	2	no	—
445	#41	—	8	1	no	—
446	#41	—	8	1	no	—
447	C₂×Q₈	—	2	1	no	—
448	C₂×Q₈	—	2	1	no	—
449	C₂×Q₈	—	2	1	no	—
450	C₂×Q₈	—	2	1	no	—
451	C₂×Q₈	—	4	1	no	—
452	C₂×Q₈	—	4	2	no	—
453	C₂×Q₈	—	8	1	no	—
454	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
455	C₂×D₄	—	4	1	no	—
456	C₂×D₄	—	4	1	no	—
457	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
458	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
459	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	4	1	no	—
464	C₂²×C₄	—	4	1	no	—
465	C₂²×C₄	—	4	1	no	—
466	C₂²×C₄	—	4	1	no	—
467	Q₁₆	—	4	2	no	—
468	Q₁₆	—	8	1	no	—
469	SD₁₆	—	4	1	no	—
470	SD₁₆	—	4	2	no	—
471	SD₁₆	—	4	2	no	—
472	SD₁₆	—	8	1	no	—
473	D₈	SD₁₆×C₃₁	1	4	yes	complemented
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	4	2	no	—
480	C₂×C₈	—	4	2	no	—
481	C₂×C₈	—	8	1	no	—
482	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
483	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
484	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
485	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	4	1	no	—
496	C₄⋊C₄	—	4	1	no	—
497	C₄⋊C₄	—	4	1	no	—
498	C₄⋊C₄	—	4	1	no	—
499	C₄⋊C₄	—	4	1	no	—
500	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
501	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
502	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
503	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
504	C₂²⋊C₄	—	2	1	no	—
505	C₂²⋊C₄	—	2	1	no	—
506	C₂²⋊C₄	—	2	1	no	—
507	C₂²⋊C₄	—	4	1	no	—
508	C₂²⋊C₄	—	4	1	no	—
509	C₂²⋊C₄	—	4	1	no	—
510	C₂²⋊C₄	—	4	1	no	—
511	C₄²	—	2	1	no	—
512	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
513	C₂³	—	2	1	no	—
514	C₂³	—	2	1	no	—
515	C₂³	—	2	1	no	—
516	Q₈	—	2	1	no	—
517	Q₈	—	2	1	no	—
518	Q₈	—	4	1	no	—
519	Q₈	—	4	1	no	—
520	Q₈	—	4	1	no	—
521	Q₈	—	4	1	no	—
522	Q₈	—	4	1	no	—
523	Q₈	—	4	1	no	—
524	Q₈	—	8	1	no	—
525	Q₈	—	8	2	no	—
526	D₄	—	2	1	no	—
527	D₄	—	2	1	no	—
528	D₄	—	2	1	no	—
529	D₄	—	2	1	no	—
530	D₄	—	8	2	no	—
531	D₄	—	8	2	no	—
532	D₄	C₂×SD₁₆×C₃₁	1	2	yes	—
533	D₄	C₂×SD₁₆×C₃₁	1	2	yes	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	8	2	no	—
550	C₂×C₄	—	8	2	no	—
551	C₂×C₄	—	8	2	no	—
552	C₂×C₄	—	16	1	no	—
553	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
554	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
555	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
556	C₈	—	2	1	no	—
557	C₈	—	2	1	no	—
558	C₈	—	2	1	no	—
559	C₈	—	4	1	no	—
560	C₈	—	4	1	no	—
561	C₈	—	4	1	no	—
562	C₈	—	4	1	no	—
563	C₈	—	4	1	no	—
564	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
565	C₂²	—	2	2	no	—
566	C₂²	—	2	2	no	—
567	C₂²	—	4	1	no	—
568	C₂²	—	4	1	no	—
569	C₂²	—	4	1	no	—
570	C₂²	—	4	1	no	—
571	C₂²	—	4	1	no	—
572	C₂²	—	4	1	no	—
573	C₂²	—	4	1	no	—
574	C₂²	D₄²×C₃₁	1	1	yes	characteristic
575	C₄	—	2	1	no	—
576	C₄	—	4	1	no	—
577	C₄	—	4	1	no	—
578	C₄	—	4	1	no	—
579	C₄	—	4	1	no	—
580	C₄	—	4	1	no	—
581	C₄	—	8	1	no	—
582	C₄	—	8	1	no	—
583	C₄	—	8	1	no	—
584	C₄	—	8	1	no	—
585	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
586	C₄	C₃₁×#575	1	1	yes	characteristic
587	C₄	C₃₁×#577	1	1	yes	characteristic
588	C₄	C₃₁×#577	1	1	yes	characteristic
589	C₂	—	4	2	no	—
590	C₂	—	4	2	no	—
591	C₂	—	8	1	no	—
592	C₂	C₃₁×#3267	1	1	yes	characteristic
593	C₂	D₄×SD₁₆×C₃₁	1	1	yes	characteristic
594	C₂	C₃₁×#3289	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3283	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3384	C₂	semidirect
2. C₃₁×#3335	C₂	semidirect
3. C₃₁×#3298	C₂	semidirect
5. C₃₁×#1716	C₂	semidirect
6. C₃₁×#1690	C₂	semidirect
7. C₃₁×#1679	C₂	semidirect
8. C₃₁×#1671	C₂	semidirect
9. C₃₁×#1671	C₂	semidirect
10. C₃₁×#1654	C₂	semidirect
11. C₃₁×#1651	C₂	semidirect
12. C₃₁×#1643	C₂	semidirect
13. C₃₁×#1635	C₂	semidirect
14. C₃₁×#1582	C₂	semidirect
15. C₃₁×#1580	C₂	semidirect
175. D₈×C₃₁	SD₁₆	semidirect
215. #19766	C₃₁	direct (N × H)
268. #3384	C₂×C₃₁	semidirect
269. #3335	C₂×C₃₁	semidirect
270. #3298	C₂×C₃₁	semidirect
272. #1716	C₂×C₃₁	semidirect
273. #1690	C₂×C₃₁	semidirect
274. #1679	C₂×C₃₁	semidirect
275. #1671	C₂×C₃₁	semidirect
276. #1671	C₂×C₃₁	semidirect
277. #1654	C₂×C₃₁	semidirect
278. #1651	C₂×C₃₁	semidirect
279. #1643	C₂×C₃₁	semidirect
280. #1635	C₂×C₃₁	semidirect
281. #1582	C₂×C₃₁	semidirect
282. #1580	C₂×C₃₁	semidirect
442. C₃₁	#19766	direct (N × H)
473. D₈	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3384	2
C₃₁×#3335	2
C₃₁×#3298	2
C₃₁×#3283	2
C₃₁×#1716	2
C₃₁×#1690	2
C₃₁×#1679	2
C₃₁×#1671	2
C₃₁×#1654	2
C₃₁×#1651	2
C₃₁×#1643	2
C₃₁×#1635	2
C₃₁×#1582	2
C₃₁×#1580	2
#19766	31

Subgroups — C31×#19766 #21476525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19766 #21476525