Subgroups — C₃₁×#25592 #21482351

Order 7936.

1134 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3510	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3140	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3040	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2858	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2005	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2004	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2003	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2002	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1994	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1967	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1888	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1887	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1886	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1885	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₃₁×#104	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#137	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
18	C₃₁×#559	—	2	2	no	—
19	C₃₁×#556	—	2	2	no	—
20	C₃₁×#551	—	2	4	no	—
21	C₃₁×#542	—	2	2	no	—
22	C₃₁×#536	C₂²	1	1	yes	characteristic
23	C₃₁×#535	C₂²	1	1	yes	characteristic
24	C₃₁×#533	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
27	C₃₁×#529	—	2	2	no	—
28	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
32	C₃₁×#519	C₂²	1	1	yes	characteristic
33	C₃₁×#518	C₂²	1	1	yes	characteristic
34	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#489	—	2	2	no	—
36	C₃₁×#488	—	2	2	no	—
37	C₃₁×#464	—	2	2	no	—
38	C₃₁×#463	—	2	2	no	—
39	C₃₁×#423	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
41	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
42	C₃₁×#419	C₂²	1	1	yes	characteristic
43	C₃₁×#419	C₂²	1	1	yes	characteristic
44	C₃₁×#419	C₂²	1	1	yes	characteristic
45	C₃₁×#418	C₂²	1	1	yes	characteristic
46	C₃₁×#418	C₂²	1	1	yes	characteristic
47	C₃₁×#418	C₂²	1	1	yes	characteristic
48	C₂×C₃₁×#101	C₂²	1	1	yes	characteristic
49	C₂×C₃₁×#101	C₂²	1	1	yes	characteristic
50	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic
51	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic
52	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
53	C₃₁×#390	—	2	4	no	—
54	C₄³×C₃₁	C₂²	1	1	yes	characteristic
55	C₃₁×#339	C₂²	1	1	yes	characteristic
56	C₃₁×#339	C₂²	1	1	yes	characteristic
57	C₃₁×#339	C₂²	1	1	yes	characteristic
58	C₃₁×#339	C₂²	1	1	yes	characteristic
59	C₃₁×#339	C₂²	1	1	yes	characteristic
60	C₃₁×#339	C₂²	1	1	yes	characteristic
61	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₃₁	—	2	2	no	—
65	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#137	—	4	1	no	—
68	C₃₁×#137	—	4	1	no	—
69	C₃₁×AGL(1,ℤ/8)	—	4	1	no	—
70	C₃₁×AGL(1,ℤ/8)	—	4	1	no	—
71	C₂×Q₁₆×C₃₁	—	2	1	no	—
72	C₂×SD₁₆×C₃₁	—	2	1	no	—
73	C₂×SD₁₆×C₃₁	—	2	1	no	—
74	C₂×D₈×C₃₁	—	2	1	no	—
75	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊Q₈	—	2	1	no	—
80	C₃₁×C₄⋊Q₈	—	2	2	no	—
81	C₃₁×C₄⋊Q₈	—	4	1	no	—
82	C₃₁×C₄⋊Q₈	D₄	1	2	yes	complemented
83	C₃₁×#127	—	2	1	no	—
84	C₃₁×#127	—	2	2	no	—
85	C₃₁×#127	—	4	1	no	—
86	C₃₁×#126	—	2	4	no	—
87	C₃₁×#126	—	2	4	no	—
88	C₃₁×#125	D₄	1	2	yes	complemented
89	C₃₁×#124	—	2	4	no	—
90	C₃₁×#124	—	2	1	no	—
91	C₃₁×#124	—	2	1	no	—
92	C₃₁×#124	—	2	1	no	—
93	C₃₁×#124	—	2	1	no	—
94	C₃₁×#124	—	2	2	no	—
95	C₃₁×#124	—	2	2	no	—
96	C₃₁×#124	—	2	1	no	—
97	C₃₁×#124	—	2	1	no	—
98	C₃₁×#124	—	2	2	no	—
99	C₃₁×#124	—	2	2	no	—
100	C₃₁×#124	—	4	1	no	—
101	C₃₁×#124	—	4	1	no	—
102	C₃₁×#123	—	2	4	no	—
103	C₃₁×#123	—	2	4	no	—
104	C₃₁×#123	—	2	2	no	—
105	C₃₁×C₂²⋊Q₈	—	2	4	no	—
106	C₃₁×C₂²⋊Q₈	—	2	2	no	—
107	C₃₁×C₂²⋊Q₈	—	2	2	no	—
108	C₃₁×#121	—	2	2	no	—
109	C₃₁×#120	—	2	2	no	—
110	C₄×Q₈×C₃₁	—	2	2	no	—
111	C₄×Q₈×C₃₁	—	2	2	no	—
112	C₄×Q₈×C₃₁	—	2	2	no	—
113	C₄×Q₈×C₃₁	D₄	1	2	yes	complemented
114	C₄×D₄×C₃₁	—	2	4	no	—
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	D₄	1	2	yes	complemented
119	C₃₁×#117	—	2	4	no	—
120	C₃₁×#117	—	2	2	no	—
121	C₃₁×#117	—	2	1	no	—
122	C₃₁×#117	C₂³	1	1	yes	characteristic
123	C₃₁×#117	C₂³	1	1	yes	characteristic
124	C₃₁×#117	C₂³	1	1	yes	characteristic
125	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
127	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
128	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
129	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
130	C₂×C₂²⋊C₄×C₃₁	D₄	1	4	yes	—
131	C₂×C₄²×C₃₁	—	2	1	no	—
132	C₂×C₄²×C₃₁	—	2	2	no	—
133	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
136	C₃₁×C₄⋊C₈	—	2	2	no	—
137	C₃₁×#104	—	2	2	no	—
138	C₃₁×#104	—	2	2	no	—
139	C₃₁×#103	—	2	2	no	—
140	C₃₁×#103	—	2	2	no	—
141	C₃₁×#103	—	2	2	no	—
142	C₃₁×#102	—	2	2	no	—
143	C₃₁×#102	—	2	2	no	—
144	C₃₁×#102	—	2	2	no	—
145	C₃₁×#101	—	4	1	no	—
146	C₃₁×#101	—	4	1	no	—
147	C₃₁×#100	—	4	1	no	—
148	C₃₁×#100	—	4	1	no	—
149	C₃₁×#97	—	2	1	no	—
150	C₄×C₈×C₃₁	—	2	1	no	—
151	C₂⁴×C₃₁	—	2	1	no	—
152	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
153	#41×C₃₁	—	2	1	no	—
154	#41×C₃₁	—	2	1	no	—
155	#41×C₃₁	—	2	1	no	—
156	#41×C₃₁	—	2	1	no	—
157	C₂×Q₈×C₃₁	—	2	2	no	—
158	C₂×Q₈×C₃₁	—	2	1	no	—
159	C₂×Q₈×C₃₁	—	2	1	no	—
160	C₂×Q₈×C₃₁	—	2	1	no	—
161	C₂×Q₈×C₃₁	—	2	1	no	—
162	C₂×Q₈×C₃₁	—	2	1	no	—
163	C₂×Q₈×C₃₁	—	2	1	no	—
164	C₂×Q₈×C₃₁	—	2	1	no	—
165	C₂×Q₈×C₃₁	—	2	1	no	—
166	C₂×Q₈×C₃₁	—	2	1	no	—
167	C₂×Q₈×C₃₁	—	2	1	no	—
168	C₂×Q₈×C₃₁	—	2	1	no	—
169	C₂×Q₈×C₃₁	—	2	1	no	—
170	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
171	C₂×D₄×C₃₁	—	2	1	no	—
172	C₂×D₄×C₃₁	—	2	4	no	—
173	C₂×D₄×C₃₁	—	2	1	no	—
174	C₂×D₄×C₃₁	—	2	1	no	—
175	C₂×D₄×C₃₁	—	2	1	no	—
176	C₂×D₄×C₃₁	—	2	1	no	—
177	C₂×D₄×C₃₁	—	2	1	no	—
178	C₂×D₄×C₃₁	—	2	1	no	—
179	C₂×D₄×C₃₁	—	2	1	no	—
180	C₂×D₄×C₃₁	—	2	1	no	—
181	C₂×D₄×C₃₁	—	2	1	no	—
182	C₂×D₄×C₃₁	—	2	1	no	—
183	C₂×D₄×C₃₁	—	2	1	no	—
184	C₂×D₄×C₃₁	—	2	1	no	—
185	C₂×D₄×C₃₁	—	4	4	no	—
186	C₂×D₄×C₃₁	—	4	8	no	—
187	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₃₁	—	2	1	no	—
189	C₂²×C₄×C₃₁	—	2	1	no	—
190	C₂²×C₄×C₃₁	—	2	4	no	—
191	C₂²×C₄×C₃₁	—	2	1	no	—
192	C₂²×C₄×C₃₁	—	2	2	no	—
193	C₂²×C₄×C₃₁	—	2	1	no	—
194	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
195	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
196	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
197	Q₁₆×C₃₁	—	4	2	no	—
198	SD₁₆×C₃₁	—	4	2	no	—
199	SD₁₆×C₃₁	—	4	2	no	—
200	D₈×C₃₁	—	4	2	no	—
201	M₁₆×C₃₁	—	2	1	no	—
202	M₁₆×C₃₁	—	2	1	no	—
203	M₁₆×C₃₁	—	2	1	no	—
204	M₁₆×C₃₁	—	2	1	no	—
205	M₁₆×C₃₁	—	2	1	no	—
206	M₁₆×C₃₁	—	2	1	no	—
207	M₁₆×C₃₁	—	2	1	no	—
208	M₁₆×C₃₁	—	2	1	no	—
209	C₂×C₈×C₃₁	—	2	1	no	—
210	C₂×C₈×C₃₁	—	2	1	no	—
211	C₂×C₈×C₃₁	—	2	1	no	—
212	C₂×C₈×C₃₁	—	2	1	no	—
213	C₄⋊C₄×C₃₁	—	2	4	no	—
214	C₄⋊C₄×C₃₁	—	2	4	no	—
215	C₄⋊C₄×C₃₁	—	2	2	no	—
216	C₄⋊C₄×C₃₁	—	2	2	no	—
217	C₄⋊C₄×C₃₁	—	2	2	no	—
218	C₄⋊C₄×C₃₁	—	2	2	no	—
219	C₄⋊C₄×C₃₁	—	4	4	no	—
220	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
221	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
222	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
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	1	no	—
227	C₂²⋊C₄×C₃₁	—	2	1	no	—
228	C₂²⋊C₄×C₃₁	—	2	1	no	—
229	C₂²⋊C₄×C₃₁	—	2	4	no	—
230	C₂²⋊C₄×C₃₁	—	2	4	no	—
231	C₂²⋊C₄×C₃₁	—	2	2	no	—
232	C₂²⋊C₄×C₃₁	—	4	4	no	—
233	C₂²⋊C₄×C₃₁	—	4	4	no	—
234	C₂²⋊C₄×C₃₁	—	4	2	no	—
235	C₂²⋊C₄×C₃₁	—	4	2	no	—
236	C₄²×C₃₁	—	2	4	no	—
237	C₄²×C₃₁	—	2	2	no	—
238	C₄²×C₃₁	—	2	1	no	—
239	C₄²×C₃₁	—	2	1	no	—
240	C₄²×C₃₁	—	2	1	no	—
241	C₄²×C₃₁	—	2	1	no	—
242	C₄²×C₃₁	—	2	1	no	—
243	C₄²×C₃₁	—	4	4	no	—
244	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
245	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
246	C₄²×C₃₁	C₂×D₄	1	2	yes	—
247	C₄²×C₃₁	#41	1	2	yes	—
248	C₄²×C₃₁	#41	1	2	yes	—
249	#25592	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
250	C₂³×C₃₁	—	2	2	no	—
251	C₂³×C₃₁	—	2	1	no	—
252	C₂³×C₃₁	—	2	2	no	—
253	C₂³×C₃₁	—	2	2	no	—
254	C₂³×C₃₁	—	4	4	no	—
255	C₂³×C₃₁	—	4	4	no	—
256	C₂³×C₃₁	—	4	1	no	—
257	C₂³×C₃₁	—	4	1	no	—
258	C₂³×C₃₁	—	4	1	no	—
259	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
260	Q₈×C₃₁	—	2	1	no	—
261	Q₈×C₃₁	—	2	1	no	—
262	Q₈×C₃₁	—	4	1	no	—
263	Q₈×C₃₁	—	4	1	no	—
264	Q₈×C₃₁	—	4	1	no	—
265	Q₈×C₃₁	—	4	1	no	—
266	Q₈×C₃₁	—	4	1	no	—
267	Q₈×C₃₁	—	4	1	no	—
268	Q₈×C₃₁	—	4	1	no	—
269	Q₈×C₃₁	—	4	1	no	—
270	Q₈×C₃₁	—	8	2	no	—
271	D₄×C₃₁	—	2	1	no	—
272	D₄×C₃₁	—	2	1	no	—
273	D₄×C₃₁	—	4	1	no	—
274	D₄×C₃₁	—	4	1	no	—
275	D₄×C₃₁	—	4	1	no	—
276	D₄×C₃₁	—	4	1	no	—
277	D₄×C₃₁	—	4	1	no	—
278	D₄×C₃₁	—	4	1	no	—
279	D₄×C₃₁	—	4	1	no	—
280	D₄×C₃₁	—	4	1	no	—
281	D₄×C₃₁	—	4	1	no	—
282	D₄×C₃₁	—	4	1	no	—
283	D₄×C₃₁	—	8	8	no	—
284	D₄×C₃₁	—	8	4	no	—
285	C₂×C₄×C₃₁	—	2	4	no	—
286	C₂×C₄×C₃₁	—	2	2	no	—
287	C₂×C₄×C₃₁	—	2	2	no	—
288	C₂×C₄×C₃₁	—	2	2	no	—
289	C₂×C₄×C₃₁	—	2	2	no	—
290	C₂×C₄×C₃₁	—	2	1	no	—
291	C₂×C₄×C₃₁	—	2	2	no	—
292	C₂×C₄×C₃₁	—	2	2	no	—
293	C₂×C₄×C₃₁	—	2	2	no	—
294	C₂×C₄×C₃₁	—	2	2	no	—
295	C₂×C₄×C₃₁	—	4	2	no	—
296	C₂×C₄×C₃₁	—	4	2	no	—
297	C₂×C₄×C₃₁	—	4	1	no	—
298	C₂×C₄×C₃₁	—	4	1	no	—
299	C₂×C₄×C₃₁	—	4	1	no	—
300	C₂×C₄×C₃₁	—	4	1	no	—
301	C₂×C₄×C₃₁	—	4	1	no	—
302	C₂×C₄×C₃₁	—	8	2	no	—
303	C₂×C₄×C₃₁	—	8	4	no	—
304	C₂×C₄×C₃₁	#120	1	2	yes	—
305	C₂×C₄×C₃₁	#120	1	2	yes	—
306	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
307	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
308	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
309	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
310	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
311	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
312	C₈×C₃₁	—	4	1	no	—
313	C₈×C₃₁	—	4	1	no	—
314	C₈×C₃₁	—	4	1	no	—
315	C₈×C₃₁	—	4	1	no	—
316	#3510	C₂×C₃₁	1	1	yes	characteristic, complemented
317	#3140	C₂×C₃₁	1	1	yes	characteristic, complemented
318	#3040	C₂×C₃₁	1	1	yes	characteristic, complemented
319	#2858	C₂×C₃₁	1	1	yes	characteristic, complemented
320	#2005	C₂×C₃₁	1	1	yes	characteristic, complemented
321	#2004	C₂×C₃₁	1	1	yes	characteristic, complemented
322	#2003	C₂×C₃₁	1	1	yes	characteristic, complemented
323	#2002	C₂×C₃₁	1	1	yes	characteristic
324	#1994	C₂×C₃₁	1	1	yes	characteristic, complemented
325	#1967	C₂×C₃₁	1	1	yes	characteristic, complemented
326	#1888	C₂×C₃₁	1	1	yes	characteristic, complemented
327	#1887	C₂×C₃₁	1	1	yes	characteristic, complemented
328	#1886	C₂×C₃₁	1	1	yes	characteristic, complemented
329	#1885	C₂×C₃₁	1	1	yes	characteristic, complemented
330	C₄×#104	C₂×C₃₁	1	1	yes	characteristic, complemented
331	C₂²×C₃₁	—	2	2	no	—
332	C₂²×C₃₁	—	4	1	no	—
333	C₂²×C₃₁	—	4	2	no	—
334	C₂²×C₃₁	—	4	2	no	—
335	C₂²×C₃₁	—	4	4	no	—
336	C₂²×C₃₁	—	8	2	no	—
337	C₂²×C₃₁	—	8	2	no	—
338	C₂²×C₃₁	—	8	2	no	—
339	C₂²×C₃₁	—	8	2	no	—
340	C₂²×C₃₁	—	8	1	no	—
341	C₂²×C₃₁	—	8	2	no	—
342	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
343	C₂²×C₃₁	#546	1	1	yes	characteristic
344	C₂²×C₃₁	#546	1	1	yes	characteristic
345	C₄×C₃₁	—	2	4	no	—
346	C₄×C₃₁	—	2	2	no	—
347	C₄×C₃₁	—	2	2	no	—
348	C₄×C₃₁	—	4	2	no	—
349	C₄×C₃₁	—	4	2	no	—
350	C₄×C₃₁	—	4	1	no	—
351	C₄×C₃₁	—	4	2	no	—
352	C₄×C₃₁	—	4	2	no	—
353	C₄×C₃₁	—	4	2	no	—
354	C₄×C₃₁	—	8	1	no	—
355	C₄×C₃₁	#532	1	1	yes	characteristic
356	C₄×C₃₁	#535	1	1	yes	characteristic
357	C₄×C₃₁	#550	1	1	yes	characteristic
358	C₄×C₃₁	#550	1	1	yes	characteristic
359	C₂×#137	C₂²×C₃₁	1	1	yes	characteristic
360	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
361	#559	—	2	2	no	—
362	#556	—	2	2	no	—
363	#551	—	2	4	no	—
364	#542	—	2	2	no	—
365	#536	C₂²×C₃₁	1	1	yes	characteristic
366	#535	C₂²×C₃₁	1	1	yes	characteristic
367	#533	C₂²×C₃₁	1	1	yes	characteristic
368	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
369	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
370	#529	—	2	2	no	—
371	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
372	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
373	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
374	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
375	#519	C₂²×C₃₁	1	1	yes	characteristic
376	#518	C₂²×C₃₁	1	1	yes	characteristic
377	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
378	#489	—	2	2	no	—
379	#488	—	2	2	no	—
380	#464	—	2	2	no	—
381	#463	—	2	2	no	—
382	#423	C₂²×C₃₁	1	1	yes	characteristic
383	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
384	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
385	#419	C₂²×C₃₁	1	1	yes	characteristic
386	#419	C₂²×C₃₁	1	1	yes	characteristic
387	#419	C₂²×C₃₁	1	1	yes	characteristic
388	#418	C₂²×C₃₁	1	1	yes	characteristic
389	#418	C₂²×C₃₁	1	1	yes	characteristic
390	#418	C₂²×C₃₁	1	1	yes	characteristic
391	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic
392	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic
393	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
394	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
395	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
396	#390	—	2	4	no	—
397	C₄³	C₂²×C₃₁	1	1	yes	characteristic
398	#339	C₂²×C₃₁	1	1	yes	characteristic
399	#339	C₂²×C₃₁	1	1	yes	characteristic
400	#339	C₂²×C₃₁	1	1	yes	characteristic
401	#339	C₂²×C₃₁	1	1	yes	characteristic
402	#339	C₂²×C₃₁	1	1	yes	characteristic
403	#339	C₂²×C₃₁	1	1	yes	characteristic
404	C₂×C₃₁	—	2	1	no	—
405	C₂×C₃₁	—	2	1	no	—
406	C₂×C₃₁	—	8	1	no	—
407	C₂×C₃₁	—	8	2	no	—
408	C₂×C₃₁	—	16	1	no	—
409	C₂×C₃₁	#2431	1	1	yes	characteristic
410	C₂×C₃₁	#3018	1	2	yes	—
411	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
412	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
413	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
414	C₂²×D₄	—	2	2	no	—
415	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
416	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
417	#137	—	4	1	no	—
418	#137	—	4	1	no	—
419	AGL(1,ℤ/8)	—	4	1	no	—
420	AGL(1,ℤ/8)	—	4	1	no	—
421	C₂×Q₁₆	—	2	1	no	—
422	C₂×SD₁₆	—	2	1	no	—
423	C₂×SD₁₆	—	2	1	no	—
424	C₂×D₈	—	2	1	no	—
425	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
426	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
427	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
428	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
429	C₄⋊Q₈	—	2	1	no	—
430	C₄⋊Q₈	—	2	2	no	—
431	C₄⋊Q₈	—	4	1	no	—
432	C₄⋊Q₈	D₄×C₃₁	1	2	yes	complemented
433	#127	—	2	1	no	—
434	#127	—	2	2	no	—
435	#127	—	4	1	no	—
436	#126	—	2	4	no	—
437	#126	—	2	4	no	—
438	#125	D₄×C₃₁	1	2	yes	complemented
439	#124	—	2	4	no	—
440	#124	—	2	1	no	—
441	#124	—	2	1	no	—
442	#124	—	2	1	no	—
443	#124	—	2	1	no	—
444	#124	—	2	2	no	—
445	#124	—	2	2	no	—
446	#124	—	2	1	no	—
447	#124	—	2	1	no	—
448	#124	—	2	2	no	—
449	#124	—	2	2	no	—
450	#124	—	4	1	no	—
451	#124	—	4	1	no	—
452	#123	—	2	4	no	—
453	#123	—	2	4	no	—
454	#123	—	2	2	no	—
455	C₂²⋊Q₈	—	2	4	no	—
456	C₂²⋊Q₈	—	2	2	no	—
457	C₂²⋊Q₈	—	2	2	no	—
458	#121	—	2	2	no	—
459	#120	—	2	2	no	—
460	C₄×Q₈	—	2	2	no	—
461	C₄×Q₈	—	2	2	no	—
462	C₄×Q₈	—	2	2	no	—
463	C₄×Q₈	D₄×C₃₁	1	2	yes	complemented
464	C₄×D₄	—	2	4	no	—
465	C₄×D₄	—	2	2	no	—
466	C₄×D₄	—	2	2	no	—
467	C₄×D₄	—	2	2	no	—
468	C₄×D₄	D₄×C₃₁	1	2	yes	complemented
469	#117	—	2	4	no	—
470	#117	—	2	2	no	—
471	#117	—	2	1	no	—
472	#117	C₂³×C₃₁	1	1	yes	characteristic
473	#117	C₂³×C₃₁	1	1	yes	characteristic
474	#117	C₂³×C₃₁	1	1	yes	characteristic
475	C₂×C₄⋊C₄	—	2	2	no	—
476	C₂×C₂²⋊C₄	—	2	2	no	—
477	C₂×C₂²⋊C₄	—	2	2	no	—
478	C₂×C₂²⋊C₄	—	2	2	no	—
479	C₂×C₂²⋊C₄	—	2	2	no	—
480	C₂×C₂²⋊C₄	D₄×C₃₁	1	4	yes	—
481	C₂×C₄²	—	2	1	no	—
482	C₂×C₄²	—	2	2	no	—
483	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
484	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
485	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
486	C₄⋊C₈	—	2	2	no	—
487	#104	—	2	2	no	—
488	#104	—	2	2	no	—
489	#103	—	2	2	no	—
490	#103	—	2	2	no	—
491	#103	—	2	2	no	—
492	#102	—	2	2	no	—
493	#102	—	2	2	no	—
494	#102	—	2	2	no	—
495	#101	—	4	1	no	—
496	#101	—	4	1	no	—
497	#100	—	4	1	no	—
498	#100	—	4	1	no	—
499	#97	—	2	1	no	—
500	C₄×C₈	—	2	1	no	—
501	C₃₁	#25592	1	1	yes	characteristic, direct product factor, Sylow
502	C₂⁴	—	2	1	no	—
503	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
504	#41	—	2	1	no	—
505	#41	—	2	1	no	—
506	#41	—	2	1	no	—
507	#41	—	2	1	no	—
508	C₂×Q₈	—	2	2	no	—
509	C₂×Q₈	—	2	1	no	—
510	C₂×Q₈	—	2	1	no	—
511	C₂×Q₈	—	2	1	no	—
512	C₂×Q₈	—	2	1	no	—
513	C₂×Q₈	—	2	1	no	—
514	C₂×Q₈	—	2	1	no	—
515	C₂×Q₈	—	2	1	no	—
516	C₂×Q₈	—	2	1	no	—
517	C₂×Q₈	—	2	1	no	—
518	C₂×Q₈	—	2	1	no	—
519	C₂×Q₈	—	2	1	no	—
520	C₂×Q₈	—	2	1	no	—
521	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
522	C₂×D₄	—	2	1	no	—
523	C₂×D₄	—	2	4	no	—
524	C₂×D₄	—	2	1	no	—
525	C₂×D₄	—	2	1	no	—
526	C₂×D₄	—	2	1	no	—
527	C₂×D₄	—	2	1	no	—
528	C₂×D₄	—	2	1	no	—
529	C₂×D₄	—	2	1	no	—
530	C₂×D₄	—	2	1	no	—
531	C₂×D₄	—	2	1	no	—
532	C₂×D₄	—	2	1	no	—
533	C₂×D₄	—	2	1	no	—
534	C₂×D₄	—	2	1	no	—
535	C₂×D₄	—	2	1	no	—
536	C₂×D₄	—	4	4	no	—
537	C₂×D₄	—	4	8	no	—
538	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
539	C₂²×C₄	—	2	1	no	—
540	C₂²×C₄	—	2	1	no	—
541	C₂²×C₄	—	2	4	no	—
542	C₂²×C₄	—	2	1	no	—
543	C₂²×C₄	—	2	2	no	—
544	C₂²×C₄	—	2	1	no	—
545	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
546	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
547	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
548	Q₁₆	—	4	2	no	—
549	SD₁₆	—	4	2	no	—
550	SD₁₆	—	4	2	no	—
551	D₈	—	4	2	no	—
552	M₁₆	—	2	1	no	—
553	M₁₆	—	2	1	no	—
554	M₁₆	—	2	1	no	—
555	M₁₆	—	2	1	no	—
556	M₁₆	—	2	1	no	—
557	M₁₆	—	2	1	no	—
558	M₁₆	—	2	1	no	—
559	M₁₆	—	2	1	no	—
560	C₂×C₈	—	2	1	no	—
561	C₂×C₈	—	2	1	no	—
562	C₂×C₈	—	2	1	no	—
563	C₂×C₈	—	2	1	no	—
564	C₄⋊C₄	—	2	4	no	—
565	C₄⋊C₄	—	2	4	no	—
566	C₄⋊C₄	—	2	2	no	—
567	C₄⋊C₄	—	2	2	no	—
568	C₄⋊C₄	—	2	2	no	—
569	C₄⋊C₄	—	2	2	no	—
570	C₄⋊C₄	—	4	4	no	—
571	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
572	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
573	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
574	C₂²⋊C₄	—	2	4	no	—
575	C₂²⋊C₄	—	2	4	no	—
576	C₂²⋊C₄	—	2	4	no	—
577	C₂²⋊C₄	—	2	1	no	—
578	C₂²⋊C₄	—	2	1	no	—
579	C₂²⋊C₄	—	2	1	no	—
580	C₂²⋊C₄	—	2	4	no	—
581	C₂²⋊C₄	—	2	4	no	—
582	C₂²⋊C₄	—	2	2	no	—
583	C₂²⋊C₄	—	4	4	no	—
584	C₂²⋊C₄	—	4	4	no	—
585	C₂²⋊C₄	—	4	2	no	—
586	C₂²⋊C₄	—	4	2	no	—
587	C₄²	—	2	4	no	—
588	C₄²	—	2	2	no	—
589	C₄²	—	2	1	no	—
590	C₄²	—	2	1	no	—
591	C₄²	—	2	1	no	—
592	C₄²	—	2	1	no	—
593	C₄²	—	2	1	no	—
594	C₄²	—	4	4	no	—
595	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
596	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
597	C₄²	C₂×D₄×C₃₁	1	2	yes	—
598	C₄²	#41×C₃₁	1	2	yes	—
599	C₄²	#41×C₃₁	1	2	yes	—
600	C₂³	—	2	2	no	—
601	C₂³	—	2	1	no	—
602	C₂³	—	2	2	no	—
603	C₂³	—	2	2	no	—
604	C₂³	—	4	4	no	—
605	C₂³	—	4	4	no	—
606	C₂³	—	4	1	no	—
607	C₂³	—	4	1	no	—
608	C₂³	—	4	1	no	—
609	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
610	Q₈	—	2	1	no	—
611	Q₈	—	2	1	no	—
612	Q₈	—	4	1	no	—
613	Q₈	—	4	1	no	—
614	Q₈	—	4	1	no	—
615	Q₈	—	4	1	no	—
616	Q₈	—	4	1	no	—
617	Q₈	—	4	1	no	—
618	Q₈	—	4	1	no	—
619	Q₈	—	4	1	no	—
620	Q₈	—	8	2	no	—
621	D₄	—	2	1	no	—
622	D₄	—	2	1	no	—
623	D₄	—	4	1	no	—
624	D₄	—	4	1	no	—
625	D₄	—	4	1	no	—
626	D₄	—	4	1	no	—
627	D₄	—	4	1	no	—
628	D₄	—	4	1	no	—
629	D₄	—	4	1	no	—
630	D₄	—	4	1	no	—
631	D₄	—	4	1	no	—
632	D₄	—	4	1	no	—
633	D₄	—	8	8	no	—
634	D₄	—	8	4	no	—
635	C₂×C₄	—	2	4	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	2	1	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	4	2	no	—
646	C₂×C₄	—	4	2	no	—
647	C₂×C₄	—	4	1	no	—
648	C₂×C₄	—	4	1	no	—
649	C₂×C₄	—	4	1	no	—
650	C₂×C₄	—	4	1	no	—
651	C₂×C₄	—	4	1	no	—
652	C₂×C₄	—	8	2	no	—
653	C₂×C₄	—	8	4	no	—
654	C₂×C₄	C₃₁×#120	1	2	yes	—
655	C₂×C₄	C₃₁×#120	1	2	yes	—
656	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
657	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
658	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
659	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
660	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
661	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
662	C₈	—	4	1	no	—
663	C₈	—	4	1	no	—
664	C₈	—	4	1	no	—
665	C₈	—	4	1	no	—
666	C₂²	—	2	2	no	—
667	C₂²	—	4	1	no	—
668	C₂²	—	4	2	no	—
669	C₂²	—	4	2	no	—
670	C₂²	—	4	4	no	—
671	C₂²	—	8	2	no	—
672	C₂²	—	8	2	no	—
673	C₂²	—	8	2	no	—
674	C₂²	—	8	2	no	—
675	C₂²	—	8	1	no	—
676	C₂²	—	8	2	no	—
677	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
678	C₂²	C₃₁×#546	1	1	yes	characteristic
679	C₂²	C₃₁×#546	1	1	yes	characteristic
680	C₄	—	2	4	no	—
681	C₄	—	2	2	no	—
682	C₄	—	2	2	no	—
683	C₄	—	4	2	no	—
684	C₄	—	4	2	no	—
685	C₄	—	4	1	no	—
686	C₄	—	4	2	no	—
687	C₄	—	4	2	no	—
688	C₄	—	4	2	no	—
689	C₄	—	8	1	no	—
690	C₄	C₃₁×#532	1	1	yes	characteristic
691	C₄	C₃₁×#535	1	1	yes	characteristic
692	C₄	C₃₁×#550	1	1	yes	characteristic
693	C₄	C₃₁×#550	1	1	yes	characteristic
694	C₂	—	2	1	no	—
695	C₂	—	2	1	no	—
696	C₂	—	8	2	no	—
697	C₂	—	8	1	no	—
698	C₂	—	16	1	no	—
699	C₂	C₃₁×#2431	1	1	yes	characteristic
700	C₂	C₃₁×#3018	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#2002	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3510	C₂	semidirect
2. C₃₁×#3140	C₂	semidirect
3. C₃₁×#3040	C₂	semidirect
4. C₃₁×#2858	C₂	semidirect
5. C₃₁×#2005	C₂	semidirect
6. C₃₁×#2004	C₂	semidirect
7. C₃₁×#2003	C₂	semidirect
9. C₃₁×#1994	C₂	semidirect
10. C₃₁×#1967	C₂	semidirect
11. C₃₁×#1888	C₂	semidirect
12. C₃₁×#1887	C₂	semidirect
13. C₃₁×#1886	C₂	semidirect
14. C₃₁×#1885	C₂	semidirect
15. C₄×C₃₁×#104	C₂	semidirect
82. C₃₁×C₄⋊Q₈	D₄	semidirect
88. C₃₁×#125	D₄	semidirect
113. C₄×Q₈×C₃₁	D₄	semidirect
118. C₄×D₄×C₃₁	D₄	semidirect
249. #25592	C₃₁	direct (N × H)
316. #3510	C₂×C₃₁	semidirect
317. #3140	C₂×C₃₁	semidirect
318. #3040	C₂×C₃₁	semidirect
319. #2858	C₂×C₃₁	semidirect
320. #2005	C₂×C₃₁	semidirect
321. #2004	C₂×C₃₁	semidirect
322. #2003	C₂×C₃₁	semidirect
324. #1994	C₂×C₃₁	semidirect
325. #1967	C₂×C₃₁	semidirect
326. #1888	C₂×C₃₁	semidirect
327. #1887	C₂×C₃₁	semidirect
328. #1886	C₂×C₃₁	semidirect
329. #1885	C₂×C₃₁	semidirect
330. C₄×#104	C₂×C₃₁	semidirect
432. C₄⋊Q₈	D₄×C₃₁	semidirect
438. #125	D₄×C₃₁	semidirect
463. C₄×Q₈	D₄×C₃₁	semidirect
468. C₄×D₄	D₄×C₃₁	semidirect
501. C₃₁	#25592	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3510	2
C₃₁×#3140	2
C₃₁×#3040	2
C₃₁×#2858	2
C₃₁×#2005	2
C₃₁×#2004	2
C₃₁×#2003	2
C₃₁×#2002	2
C₃₁×#1994	2
C₃₁×#1967	2
C₃₁×#1888	2
C₃₁×#1887	2
C₃₁×#1886	2
C₃₁×#1885	2
C₄×C₃₁×#104	2
#25592	31

Subgroups — C31×#25592 #21482351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25592 #21482351