Subgroups — C₃₁×#29359 #21486118

Order 7936.

642 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2642	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2331	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2074	C₂	1	2	yes	maximal
4	C₃₁×#2024	C₂	1	2	yes	maximal
5	C₃₁×#1989	C₂	1	2	yes	maximal
6	C₃₁×#1981	C₂	1	2	yes	maximal
7	C₃₁×#1935	C₂	1	2	yes	maximal
8	C₃₁×#1933	C₂	1	2	yes	maximal
9	C₃₁×#1768	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃₁×#103	C₂²	1	2	yes	—
15	C₂×C₃₁×#103	C₂²	1	2	yes	—
16	C₂×C₃₁×#103	C₂²	1	2	yes	—
17	C₂×C₃₁×#103	C₂²	1	2	yes	—
18	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#400	—	2	2	no	—
21	C₃₁×#400	—	2	2	no	—
22	C₃₁×#391	—	2	2	no	—
23	C₃₁×#391	C₂²	1	2	yes	—
24	C₃₁×#391	C₂²	1	2	yes	—
25	C₃₁×#389	C₂²	1	2	yes	—
26	C₃₁×#387	—	2	2	no	—
27	C₃₁×#382	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#376	C₂²	1	1	yes	characteristic
30	C₃₁×#340	C₂²	1	2	yes	—
31	C₃₁×#340	C₂²	1	2	yes	—
32	C₃₁×#340	C₂²	1	2	yes	—
33	C₃₁×#340	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₃₁	—	2	1	no	—
36	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₃₁×C₄⋊Q₈	—	2	4	no	—
40	C₃₁×C₄⋊Q₈	—	2	4	no	—
41	C₃₁×#125	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
49	C₂×C₄²×C₃₁	—	2	2	no	—
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×C₄⋊C₈	—	2	4	no	—
54	C₃₁×#103	—	2	4	no	—
55	C₃₁×#103	—	2	4	no	—
56	C₃₁×#103	—	4	2	no	—
57	C₃₁×#103	—	4	2	no	—
58	C₃₁×#97	—	4	1	no	—
59	C₄×C₈×C₃₁	—	4	1	no	—
60	C₃₁×#95	—	2	2	no	—
61	C₃₁×#95	—	2	4	no	—
62	C₃₁×#95	—	2	2	no	—
63	C₃₁×#95	—	2	2	no	—
64	C₃₁×#95	—	2	1	no	—
65	C₃₁×#95	—	2	1	no	—
66	C₃₁×#95	—	2	2	no	—
67	C₃₁×#95	D₄	1	2	yes	—
68	C₃₁×#95	D₄	1	2	yes	—
69	C₂×Q₈×C₃₁	—	2	4	no	—
70	C₂×Q₈×C₃₁	—	2	2	no	—
71	C₂×Q₈×C₃₁	—	2	2	no	—
72	C₂×Q₈×C₃₁	—	2	2	no	—
73	C₂×Q₈×C₃₁	—	2	2	no	—
74	C₂×Q₈×C₃₁	—	2	2	no	—
75	C₂×Q₈×C₃₁	—	2	2	no	—
76	C₂×Q₈×C₃₁	—	8	2	no	—
77	C₂²×C₄×C₃₁	—	2	2	no	—
78	C₂²×C₄×C₃₁	—	2	2	no	—
79	C₂²×C₄×C₃₁	—	2	2	no	—
80	C₂²×C₄×C₃₁	—	2	2	no	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
86	C₂×C₈×C₃₁	—	2	2	no	—
87	C₂×C₈×C₃₁	—	2	2	no	—
88	C₂×C₈×C₃₁	—	2	2	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	2	2	no	—
91	C₂×C₈×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	4	no	—
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	2	4	no	—
95	C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	4	4	no	—
98	C₄⋊C₄×C₃₁	—	4	4	no	—
99	C₄⋊C₄×C₃₁	—	8	2	no	—
100	C₄⋊C₄×C₃₁	#41	1	4	yes	—
101	C₄²×C₃₁	—	2	2	no	—
102	C₄²×C₃₁	—	2	1	no	—
103	C₄²×C₃₁	—	2	1	no	—
104	C₄²×C₃₁	—	8	2	no	—
105	C₄²×C₃₁	#41	1	4	yes	—
106	#29359	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
108	Q₈×C₃₁	—	4	2	no	—
109	Q₈×C₃₁	—	4	2	no	—
110	Q₈×C₃₁	—	4	2	no	—
111	Q₈×C₃₁	—	4	2	no	—
112	Q₈×C₃₁	—	8	4	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	4	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	2	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	4	2	no	—
126	C₂×C₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	4	2	no	—
134	C₂×C₄×C₃₁	#124	1	4	yes	—
135	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
136	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
137	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
139	C₈×C₃₁	—	4	2	no	—
140	C₈×C₃₁	—	4	2	no	—
141	#2642	C₂×C₃₁	1	1	yes	characteristic
142	#2331	C₂×C₃₁	1	1	yes	characteristic
143	#2074	C₂×C₃₁	1	2	yes	—
144	#2024	C₂×C₃₁	1	2	yes	—
145	#1989	C₂×C₃₁	1	2	yes	—
146	#1981	C₂×C₃₁	1	2	yes	—
147	#1935	C₂×C₃₁	1	2	yes	—
148	#1933	C₂×C₃₁	1	2	yes	—
149	#1768	C₂×C₃₁	1	1	yes	characteristic
150	C₂²×C₃₁	#525	1	1	yes	characteristic
151	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
152	C₂²×C₃₁	#537	1	1	yes	characteristic
153	C₂²×C₃₁	#577	1	2	yes	—
154	C₂²×C₃₁	#578	1	2	yes	—
155	C₄×C₃₁	—	2	4	no	—
156	C₄×C₃₁	—	4	1	no	—
157	C₄×C₃₁	—	4	4	no	—
158	C₄×C₃₁	—	4	1	no	—
159	C₄×C₃₁	—	4	2	no	—
160	C₄×C₃₁	—	8	2	no	—
161	C₄×C₃₁	—	8	2	no	—
162	C₄×C₃₁	—	8	2	no	—
163	C₄×C₃₁	#556	1	2	yes	—
164	C₄×C₃₁	#559	1	2	yes	—
165	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
167	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
168	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
169	C₂×#103	C₂²×C₃₁	1	2	yes	—
170	C₂×#103	C₂²×C₃₁	1	2	yes	—
171	C₂×#103	C₂²×C₃₁	1	2	yes	—
172	C₂×#103	C₂²×C₃₁	1	2	yes	—
173	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
174	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
175	#400	—	2	2	no	—
176	#400	—	2	2	no	—
177	#391	—	2	2	no	—
178	#391	C₂²×C₃₁	1	2	yes	—
179	#391	C₂²×C₃₁	1	2	yes	—
180	#389	C₂²×C₃₁	1	2	yes	—
181	#387	—	2	2	no	—
182	#382	C₂²×C₃₁	1	1	yes	characteristic
183	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
184	#376	C₂²×C₃₁	1	1	yes	characteristic
185	#340	C₂²×C₃₁	1	2	yes	—
186	#340	C₂²×C₃₁	1	2	yes	—
187	#340	C₂²×C₃₁	1	2	yes	—
188	#340	C₂²×C₃₁	1	2	yes	—
189	#336	C₂²×C₃₁	1	2	yes	—
190	C₂×C₃₁	#2592	1	1	yes	characteristic
191	C₂×C₃₁	#3123	1	1	yes	characteristic
192	C₂×C₃₁	#3127	1	1	yes	characteristic
193	C₂×C₃₁	#3137	1	2	yes	—
194	C₂×C₃₁	#3258	1	2	yes	—
195	C₂²×Q₈	—	2	1	no	—
196	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
197	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
198	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
199	C₄⋊Q₈	—	2	4	no	—
200	C₄⋊Q₈	—	2	4	no	—
201	#125	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
207	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
211	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
212	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
213	C₄⋊C₈	—	2	4	no	—
214	#103	—	2	4	no	—
215	#103	—	2	4	no	—
216	#103	—	4	2	no	—
217	#103	—	4	2	no	—
218	#97	—	4	1	no	—
219	C₄×C₈	—	4	1	no	—
220	#95	—	2	2	no	—
221	#95	—	2	1	no	—
222	#95	—	2	1	no	—
223	#95	—	2	2	no	—
224	#95	—	2	2	no	—
225	#95	—	2	4	no	—
226	#95	—	2	2	no	—
227	#95	D₄×C₃₁	1	2	yes	—
228	#95	D₄×C₃₁	1	2	yes	—
229	C₃₁	#29359	1	1	yes	characteristic, direct product factor, Sylow
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	8	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
246	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	4	4	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₄⋊C₄	—	8	2	no	—
261	C₄⋊C₄	#41×C₃₁	1	4	yes	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	8	2	no	—
266	C₄²	#41×C₃₁	1	4	yes	—
267	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	Q₈	—	4	2	no	—
272	Q₈	—	8	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	C₃₁×#124	1	4	yes	—
295	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
296	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
297	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
298	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
299	C₈	—	4	2	no	—
300	C₈	—	4	2	no	—
301	C₂²	C₃₁×#525	1	1	yes	characteristic
302	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
303	C₂²	C₃₁×#537	1	1	yes	characteristic
304	C₂²	C₃₁×#577	1	2	yes	—
305	C₂²	C₃₁×#578	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	4	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	2	no	—
314	C₄	C₃₁×#556	1	2	yes	—
315	C₄	C₃₁×#559	1	2	yes	—
316	C₂	C₃₁×#2592	1	1	yes	characteristic
317	C₂	C₃₁×#3123	1	1	yes	characteristic
318	C₂	C₃₁×#3127	1	1	yes	characteristic
319	C₂	C₃₁×#3137	1	2	yes	—
320	C₂	C₃₁×#3258	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
106. #29359	C₃₁	direct (N × H)
229. C₃₁	#29359	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×#2642	2
C₃₁×#2331	2
C₃₁×#2074	2
C₃₁×#2024	2
C₃₁×#1989	2
C₃₁×#1981	2
C₃₁×#1935	2
C₃₁×#1933	2
C₃₁×#1768	2
#29359	31

Subgroups — C31×#29359 #21486118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29359 #21486118