Subgroups — C₃₁×#29322 #21486081

Order 7936.

614 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2647	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2331	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2085	C₂	1	2	yes	maximal
4	C₃₁×#2057	C₂	1	2	yes	maximal
5	C₃₁×#1949	C₂	1	2	yes	maximal
6	C₃₁×#1947	C₂	1	2	yes	maximal
7	C₃₁×#1943	C₂	1	2	yes	maximal
8	C₃₁×#1942	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×#107	C₂²	1	2	yes	—
12	C₂×C₃₁×#107	C₂²	1	2	yes	—
13	C₂×C₃₁×#107	C₂²	1	2	yes	—
14	C₂×C₃₁×#106	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#400	—	2	2	no	—
19	C₃₁×#398	—	2	2	no	—
20	C₃₁×#391	—	2	2	no	—
21	C₃₁×#389	C₂²	1	2	yes	—
22	C₃₁×#389	C₂²	1	2	yes	—
23	C₃₁×#389	C₂²	1	2	yes	—
24	C₃₁×#387	—	2	2	no	—
25	C₃₁×#384	C₂²	1	1	yes	characteristic
26	C₃₁×#382	C₂²	1	1	yes	characteristic
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×C₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
38	C₃₁×C₄⋊Q₈	—	2	4	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
48	C₂×C₄²×C₃₁	—	2	2	no	—
49	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#107	—	2	4	no	—
53	C₃₁×#107	—	2	4	no	—
54	C₃₁×#107	—	4	2	no	—
55	C₃₁×#106	—	4	2	no	—
56	C₃₁×C₄⋊C₈	Q₈	1	8	yes	complemented
57	C₃₁×#97	—	4	1	no	—
58	C₄×C₈×C₃₁	—	4	1	no	—
59	C₃₁×#95	—	2	2	no	—
60	C₃₁×#95	—	2	2	no	—
61	C₃₁×#95	—	2	2	no	—
62	C₃₁×#95	—	2	2	no	—
63	C₃₁×#95	—	2	1	no	—
64	C₃₁×#95	—	2	1	no	—
65	C₃₁×#95	—	2	2	no	—
66	C₃₁×#95	D₄	1	2	yes	—
67	C₃₁×#95	D₄	1	2	yes	—
68	C₂×Q₈×C₃₁	—	2	4	no	—
69	C₂×Q₈×C₃₁	—	8	2	no	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	—	2	2	no	—
72	C₂²×C₄×C₃₁	—	2	2	no	—
73	C₂²×C₄×C₃₁	—	2	2	no	—
74	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	—	2	2	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
84	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
85	C₄⋊C₄×C₃₁	—	2	4	no	—
86	C₄⋊C₄×C₃₁	—	2	2	no	—
87	C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₄⋊C₄×C₃₁	—	2	2	no	—
89	C₄⋊C₄×C₃₁	—	2	2	no	—
90	C₄⋊C₄×C₃₁	—	2	2	no	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	8	2	no	—
94	C₄⋊C₄×C₃₁	—	8	2	no	—
95	C₄⋊C₄×C₃₁	—	8	2	no	—
96	C₄⋊C₄×C₃₁	—	8	2	no	—
97	C₄⋊C₄×C₃₁	#41	1	4	yes	—
98	C₄²×C₃₁	—	2	2	no	—
99	C₄²×C₃₁	—	2	1	no	—
100	C₄²×C₃₁	—	2	1	no	—
101	C₄²×C₃₁	—	8	2	no	—
102	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
103	#29322	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
105	Q₈×C₃₁	—	8	4	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	1	no	—
109	C₂×C₄×C₃₁	—	2	1	no	—
110	C₂×C₄×C₃₁	—	2	4	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	4	2	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	4	2	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	4	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	C₄⋊Q₈	1	4	yes	—
128	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
129	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
130	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
132	C₈×C₃₁	—	2	4	no	—
133	C₈×C₃₁	—	2	4	no	—
134	#2647	C₂×C₃₁	1	1	yes	characteristic
135	#2331	C₂×C₃₁	1	1	yes	characteristic
136	#2085	C₂×C₃₁	1	2	yes	—
137	#2057	C₂×C₃₁	1	2	yes	—
138	#1949	C₂×C₃₁	1	2	yes	—
139	#1947	C₂×C₃₁	1	2	yes	—
140	#1943	C₂×C₃₁	1	2	yes	—
141	#1942	C₂×C₃₁	1	2	yes	—
142	#1768	C₂×C₃₁	1	1	yes	characteristic
143	C₂²×C₃₁	#525	1	1	yes	characteristic
144	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
145	C₂²×C₃₁	#537	1	1	yes	characteristic
146	C₂²×C₃₁	#576	1	2	yes	—
147	C₂²×C₃₁	#578	1	2	yes	—
148	C₄×C₃₁	—	2	4	no	—
149	C₄×C₃₁	—	4	1	no	—
150	C₄×C₃₁	—	4	4	no	—
151	C₄×C₃₁	—	4	1	no	—
152	C₄×C₃₁	—	4	2	no	—
153	C₄×C₃₁	—	8	2	no	—
154	C₄×C₃₁	—	8	2	no	—
155	C₄×C₃₁	—	8	2	no	—
156	C₄×C₃₁	#554	1	2	yes	—
157	C₄×C₃₁	#557	1	2	yes	—
158	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
159	C₂×#107	C₂²×C₃₁	1	2	yes	—
160	C₂×#107	C₂²×C₃₁	1	2	yes	—
161	C₂×#107	C₂²×C₃₁	1	2	yes	—
162	C₂×#106	C₂²×C₃₁	1	2	yes	—
163	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
164	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
166	#400	—	2	2	no	—
167	#398	—	2	2	no	—
168	#391	—	2	2	no	—
169	#389	C₂²×C₃₁	1	2	yes	—
170	#389	C₂²×C₃₁	1	2	yes	—
171	#389	C₂²×C₃₁	1	2	yes	—
172	#387	—	2	2	no	—
173	#384	C₂²×C₃₁	1	1	yes	characteristic
174	#382	C₂²×C₃₁	1	1	yes	characteristic
175	#382	C₂²×C₃₁	1	1	yes	characteristic
176	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
177	#376	C₂²×C₃₁	1	1	yes	characteristic
178	#340	C₂²×C₃₁	1	2	yes	—
179	#340	C₂²×C₃₁	1	2	yes	—
180	#340	C₂²×C₃₁	1	2	yes	—
181	#340	C₂²×C₃₁	1	2	yes	—
182	#336	C₂²×C₃₁	1	2	yes	—
183	C₂×C₃₁	#2584	1	1	yes	characteristic
184	C₂×C₃₁	#3122	1	1	yes	characteristic
185	C₂×C₃₁	#3127	1	1	yes	characteristic
186	C₂×C₃₁	#3164	1	2	yes	—
187	C₂×C₃₁	#3262	1	2	yes	—
188	C₂²×Q₈	—	2	1	no	—
189	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
190	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
191	C₄⋊Q₈	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
198	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
199	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
200	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
203	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
204	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
205	#107	—	2	4	no	—
206	#107	—	2	4	no	—
207	#107	—	4	2	no	—
208	#106	—	4	2	no	—
209	C₄⋊C₈	Q₈×C₃₁	1	8	yes	complemented
210	#97	—	4	1	no	—
211	C₄×C₈	—	4	1	no	—
212	#95	—	2	2	no	—
213	#95	—	2	1	no	—
214	#95	—	2	1	no	—
215	#95	—	2	2	no	—
216	#95	—	2	2	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	#95	D₄×C₃₁	1	2	yes	—
220	#95	D₄×C₃₁	1	2	yes	—
221	C₃₁	#29322	1	1	yes	characteristic, direct product factor, Sylow
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	8	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
232	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
238	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	8	2	no	—
248	C₄⋊C₄	—	8	2	no	—
249	C₄⋊C₄	—	8	2	no	—
250	C₄⋊C₄	—	8	2	no	—
251	C₄⋊C₄	#41×C₃₁	1	4	yes	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	8	2	no	—
256	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
257	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
258	Q₈	—	8	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
281	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
282	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
283	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
284	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
285	C₈	—	2	4	no	—
286	C₈	—	2	4	no	—
287	C₂²	C₃₁×#525	1	1	yes	characteristic
288	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
289	C₂²	C₃₁×#537	1	1	yes	characteristic
290	C₂²	C₃₁×#576	1	2	yes	—
291	C₂²	C₃₁×#578	1	2	yes	—
292	C₄	—	2	4	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	4	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	2	no	—
297	C₄	—	8	2	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	2	no	—
300	C₄	C₃₁×#554	1	2	yes	—
301	C₄	C₃₁×#557	1	2	yes	—
302	C₂	C₃₁×#2584	1	1	yes	characteristic
303	C₂	C₃₁×#3122	1	1	yes	characteristic
304	C₂	C₃₁×#3127	1	1	yes	characteristic
305	C₂	C₃₁×#3164	1	2	yes	—
306	C₂	C₃₁×#3262	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
56. C₃₁×C₄⋊C₈	Q₈	semidirect
103. #29322	C₃₁	direct (N × H)
209. C₄⋊C₈	Q₈×C₃₁	semidirect
221. C₃₁	#29322	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×#2647	2
C₃₁×#2331	2
C₃₁×#2085	2
C₃₁×#2057	2
C₃₁×#1949	2
C₃₁×#1947	2
C₃₁×#1943	2
C₃₁×#1942	2
C₃₁×#1768	2
#29322	31

Subgroups — C31×#29322 #21486081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29322 #21486081