Subgroups — C₃₁×#33687 #21490446

Order 7936.

520 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3177	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2950	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2237	C₂	1	2	yes	maximal
4	C₃₁×#2229	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2209	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2188	C₂	1	2	yes	maximal
7	C₃₁×#2176	C₂	1	2	yes	maximal, complemented
8	C₄×C₃₁×Q₃₂	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2164	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#541	—	2	1	no	—
12	C₃₁×#519	—	2	1	no	—
13	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
14	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
15	C₃₁×#499	C₂²	1	1	yes	characteristic
16	C₃₁×#494	C₂²	1	1	yes	characteristic
17	C₃₁×#491	—	2	1	no	—
18	C₃₁×#470	—	2	1	no	—
19	C₃₁×#467	C₂²	1	2	yes	—
20	C₃₁×#441	—	2	1	no	—
21	C₃₁×#441	—	2	1	no	—
22	C₃₁×#441	—	2	1	no	—
23	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#427	C₂²	1	1	yes	characteristic
27	C₃₁×#419	—	2	1	no	—
28	C₃₁×#405	C₂²	1	1	yes	characteristic
29	C₃₁×#367	C₂²	1	2	yes	—
30	C₃₁×#366	C₂²	1	2	yes	—
31	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
32	C₃₁×#358	C₂²	1	2	yes	—
33	C₃₁×#358	C₂²	1	2	yes	—
34	C₃₁×#358	C₂²	1	2	yes	—
35	C₃₁×#358	C₂²	1	2	yes	—
36	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
37	C₃₁×#346	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₃₁	—	2	1	no	—
40	C₂×Q₁₆×C₃₁	—	2	4	no	—
41	C₂×Q₁₆×C₃₁	—	2	1	no	—
42	C₂×Q₁₆×C₃₁	—	2	1	no	—
43	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
44	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
45	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×C₄⋊Q₈	—	2	1	no	—
48	C₃₁×#126	—	4	2	no	—
49	C₃₁×#125	—	2	1	no	—
50	C₃₁×#125	—	4	2	no	—
51	C₃₁×C₂²⋊Q₈	—	2	2	no	—
52	C₄×Q₈×C₃₁	—	2	2	no	—
53	C₄×Q₈×C₃₁	—	2	2	no	—
54	C₄×Q₈×C₃₁	—	2	2	no	—
55	C₄×Q₈×C₃₁	—	4	2	no	—
56	C₃₁×#117	—	2	1	no	—
57	C₃₁×#117	—	4	2	no	—
58	C₃₁×#117	D₄	1	1	yes	characteristic
59	C₃₁×Q₃₂	—	2	4	no	—
60	C₃₁×Q₃₂	—	2	4	no	—
61	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
63	C₃₁×#107	C₂³	1	2	yes	—
64	C₃₁×#107	C₂³	1	2	yes	—
65	C₃₁×#106	—	2	1	no	—
66	C₃₁×#106	—	2	1	no	—
67	C₃₁×#103	—	2	2	no	—
68	C₃₁×#103	—	2	2	no	—
69	C₃₁×#103	—	2	2	no	—
70	C₃₁×#103	—	2	2	no	—
71	C₃₁×#97	D₄	1	1	yes	characteristic
72	C₃₁×#97	D₄	1	1	yes	characteristic
73	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₃₁	—	2	4	no	—
76	C₂×Q₈×C₃₁	—	2	2	no	—
77	C₂×Q₈×C₃₁	—	2	2	no	—
78	C₂×Q₈×C₃₁	—	4	2	no	—
79	C₂×Q₈×C₃₁	—	4	1	no	—
80	C₂²×C₄×C₃₁	—	4	1	no	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	Q₁₆×C₃₁	—	2	4	no	—
83	Q₁₆×C₃₁	—	2	2	no	—
84	Q₁₆×C₃₁	—	2	2	no	—
85	Q₁₆×C₃₁	—	2	2	no	—
86	Q₁₆×C₃₁	—	4	2	no	—
87	Q₁₆×C₃₁	#41	1	4	yes	—
88	M₁₆×C₃₁	—	4	1	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₄⋊C₄×C₃₁	—	4	2	no	—
97	C₄⋊C₄×C₃₁	—	4	2	no	—
98	C₄⋊C₄×C₃₁	—	4	2	no	—
99	C₄⋊C₄×C₃₁	—	4	2	no	—
100	C₄⋊C₄×C₃₁	—	4	2	no	—
101	C₄⋊C₄×C₃₁	—	4	2	no	—
102	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
103	C₂²⋊C₄×C₃₁	—	4	1	no	—
104	C₂²⋊C₄×C₃₁	—	4	2	no	—
105	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
106	C₄²×C₃₁	—	4	1	no	—
107	C₄²×C₃₁	—	4	2	no	—
108	C₄²×C₃₁	—	4	2	no	—
109	C₄²×C₃₁	—	4	2	no	—
110	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₃₁	—	2	2	no	—
113	C₁₆×C₃₁	—	2	2	no	—
114	#33687	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
116	Q₈×C₃₁	—	2	4	no	—
117	Q₈×C₃₁	—	4	4	no	—
118	Q₈×C₃₁	—	4	2	no	—
119	Q₈×C₃₁	—	4	2	no	—
120	Q₈×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	—	2	1	no	—
122	C₂×C₄×C₃₁	—	2	1	no	—
123	C₂×C₄×C₃₁	—	4	4	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	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
132	C₈×C₃₁	—	2	1	no	—
133	C₈×C₃₁	—	2	1	no	—
134	C₈×C₃₁	—	2	1	no	—
135	C₈×C₃₁	C₂×#41	1	2	yes	—
136	#3177	C₂×C₃₁	1	1	yes	characteristic
137	#2950	C₂×C₃₁	1	1	yes	characteristic
138	#2237	C₂×C₃₁	1	2	yes	—
139	#2229	C₂×C₃₁	1	2	yes	complemented
140	#2209	C₂×C₃₁	1	2	yes	complemented
141	#2188	C₂×C₃₁	1	2	yes	—
142	#2176	C₂×C₃₁	1	2	yes	complemented
143	C₄×Q₃₂	C₂×C₃₁	1	2	yes	complemented
144	#2164	C₂×C₃₁	1	1	yes	characteristic
145	C₂²×C₃₁	—	2	1	no	—
146	C₂²×C₃₁	—	2	2	no	—
147	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
148	C₄×C₃₁	—	2	1	no	—
149	C₄×C₃₁	—	2	2	no	—
150	C₄×C₃₁	—	2	2	no	—
151	C₄×C₃₁	—	4	4	no	—
152	C₄×C₃₁	—	8	2	no	—
153	C₄×C₃₁	—	8	2	no	—
154	C₄×C₃₁	—	8	2	no	—
155	C₄×C₃₁	#525	1	1	yes	characteristic
156	C₄×C₃₁	#575	1	1	yes	characteristic
157	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
158	#541	—	2	1	no	—
159	#519	—	2	1	no	—
160	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
161	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
162	#499	C₂²×C₃₁	1	1	yes	characteristic
163	#494	C₂²×C₃₁	1	1	yes	characteristic
164	#491	—	2	1	no	—
165	#470	—	2	1	no	—
166	#467	C₂²×C₃₁	1	2	yes	—
167	#441	—	2	1	no	—
168	#441	—	2	1	no	—
169	#441	—	2	1	no	—
170	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
171	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
172	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
173	#427	C₂²×C₃₁	1	1	yes	characteristic
174	#419	—	2	1	no	—
175	#405	C₂²×C₃₁	1	1	yes	characteristic
176	#367	C₂²×C₃₁	1	2	yes	—
177	#366	C₂²×C₃₁	1	2	yes	—
178	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
179	#358	C₂²×C₃₁	1	2	yes	—
180	#358	C₂²×C₃₁	1	2	yes	—
181	#358	C₂²×C₃₁	1	2	yes	—
182	#358	C₂²×C₃₁	1	2	yes	—
183	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
184	#346	C₂²×C₃₁	1	1	yes	characteristic
185	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
186	C₂×C₃₁	—	4	1	no	—
187	C₂×C₃₁	#3100	1	1	yes	characteristic
188	C₂×C₃₁	#3417	1	2	yes	—
189	C₂²×Q₈	—	2	1	no	—
190	C₂×Q₁₆	—	2	4	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×Q₁₆	—	2	1	no	—
193	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
194	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
195	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
197	C₄⋊Q₈	—	2	1	no	—
198	#126	—	4	2	no	—
199	#125	—	2	1	no	—
200	#125	—	4	2	no	—
201	C₂²⋊Q₈	—	2	2	no	—
202	C₄×Q₈	—	2	2	no	—
203	C₄×Q₈	—	2	2	no	—
204	C₄×Q₈	—	2	2	no	—
205	C₄×Q₈	—	4	2	no	—
206	#117	—	2	1	no	—
207	#117	—	4	2	no	—
208	#117	D₄×C₃₁	1	1	yes	characteristic
209	Q₃₂	—	2	4	no	—
210	Q₃₂	—	2	4	no	—
211	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
212	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
213	#107	C₂³×C₃₁	1	2	yes	—
214	#107	C₂³×C₃₁	1	2	yes	—
215	#106	—	2	1	no	—
216	#106	—	2	1	no	—
217	#103	—	2	2	no	—
218	#103	—	2	2	no	—
219	#103	—	2	2	no	—
220	#103	—	2	2	no	—
221	#97	D₄×C₃₁	1	1	yes	characteristic
222	#97	D₄×C₃₁	1	1	yes	characteristic
223	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
224	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
225	C₃₁	#33687	1	1	yes	characteristic, direct product factor, Sylow
226	C₂×Q₈	—	2	4	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
233	Q₁₆	—	2	4	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	2	2	no	—
237	Q₁₆	—	4	2	no	—
238	Q₁₆	#41×C₃₁	1	4	yes	—
239	M₁₆	—	4	1	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
244	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
254	C₂²⋊C₄	—	4	1	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
257	C₄²	—	4	1	no	—
258	C₄²	—	4	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	—	4	2	no	—
261	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
262	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
263	C₁₆	—	2	2	no	—
264	C₁₆	—	2	2	no	—
265	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
266	Q₈	—	2	4	no	—
267	Q₈	—	4	4	no	—
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	4	4	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	C₂×D₈×C₃₁	1	2	yes	—
279	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
280	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	C₂×#41×C₃₁	1	2	yes	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	2	no	—
288	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	4	no	—
293	C₄	—	8	2	no	—
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	C₃₁×#525	1	1	yes	characteristic
297	C₄	C₃₁×#575	1	1	yes	characteristic
298	C₂	—	4	1	no	—
299	C₂	C₃₁×#3100	1	1	yes	characteristic
300	C₂	C₃₁×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂³×C₃₁	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#117 ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#2229	C₂	semidirect
5. C₃₁×#2209	C₂	semidirect
7. C₃₁×#2176	C₂	semidirect
8. C₄×C₃₁×Q₃₂	C₂	semidirect
114. #33687	C₃₁	direct (N × H)
139. #2229	C₂×C₃₁	semidirect
140. #2209	C₂×C₃₁	semidirect
142. #2176	C₂×C₃₁	semidirect
143. C₄×Q₃₂	C₂×C₃₁	semidirect
225. C₃₁	#33687	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×#3177	2
C₃₁×#2950	2
C₃₁×#2237	2
C₃₁×#2229	2
C₃₁×#2209	2
C₃₁×#2188	2
C₃₁×#2176	2
C₄×C₃₁×Q₃₂	2
C₃₁×#2164	2
#33687	31

Subgroups — C31×#33687 #21490446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33687 #21490446