Subgroups — C₃₁×#3196 #16843129

Order 3968.

414 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#525	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#485	C₂	1	2	yes	maximal, complemented
4	C₃₁×#484	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#481	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#470	C₂	1	2	yes	maximal, complemented
7	C₃₁×#468	C₂	1	2	yes	maximal, complemented
8	C₃₁×#452	C₂	1	2	yes	maximal, complemented
9	C₃₁×#451	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#448	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#407	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
16	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
17	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#123	—	2	1	no	—
19	C₃₁×C₂²⋊Q₈	—	2	1	no	—
20	C₃₁×C₂²⋊Q₈	—	2	1	no	—
21	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
24	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₃₁×#121	C₂²	1	1	yes	characteristic
26	C₃₁×#121	C₂²	1	1	yes	characteristic
27	C₃₁×#120	—	2	1	no	—
28	C₄×D₄×C₃₁	—	2	1	no	—
29	C₄×D₄×C₃₁	—	2	1	no	—
30	C₃₁×#117	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
34	C₃₁×#107	C₂²	1	2	yes	complemented
35	C₃₁×#106	C₂²	1	2	yes	complemented
36	C₃₁×#103	C₂²	1	2	yes	complemented
37	C₃₁×#103	C₂²	1	2	yes	complemented
38	C₃₁×#103	C₂²	1	2	yes	complemented
39	C₃₁×#102	C₂²	1	2	yes	complemented
40	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
43	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
44	#41×C₃₁	—	4	2	no	—
45	C₂×Q₈×C₃₁	—	2	1	no	—
46	C₂×Q₈×C₃₁	—	4	2	no	—
47	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
48	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₃₁	—	2	1	no	—
50	C₂×D₄×C₃₁	—	2	1	no	—
51	C₂×D₄×C₃₁	—	2	1	no	—
52	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	—	2	1	no	—
54	C₂²×C₄×C₃₁	—	2	1	no	—
55	C₂²×C₄×C₃₁	—	2	1	no	—
56	C₂²×C₄×C₃₁	—	4	2	no	—
57	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
63	Q₁₆×C₃₁	—	4	2	no	—
64	SD₁₆×C₃₁	—	4	2	no	—
65	M₁₆×C₃₁	—	4	2	no	—
66	C₂×C₈×C₃₁	C₂³	1	2	yes	—
67	C₂×C₈×C₃₁	C₂³	1	2	yes	—
68	C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₄⋊C₄×C₃₁	—	2	1	no	—
71	C₄⋊C₄×C₃₁	—	2	1	no	—
72	C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
74	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₃₁	—	2	2	no	—
77	C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂²⋊C₄×C₃₁	—	2	1	no	—
79	C₂²⋊C₄×C₃₁	—	2	1	no	—
80	C₂²⋊C₄×C₃₁	—	2	1	no	—
81	C₂²⋊C₄×C₃₁	—	2	1	no	—
82	C₂²⋊C₄×C₃₁	—	2	1	no	—
83	C₄²×C₃₁	—	2	1	no	—
84	C₂³×C₃₁	—	2	2	no	—
85	C₂³×C₃₁	—	2	1	no	—
86	C₂³×C₃₁	—	4	1	no	—
87	C₂³×C₃₁	—	4	1	no	—
88	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
91	Q₈×C₃₁	—	2	2	no	—
92	Q₈×C₃₁	—	2	2	no	—
93	Q₈×C₃₁	—	2	2	no	—
94	Q₈×C₃₁	—	4	2	no	—
95	D₄×C₃₁	—	2	2	no	—
96	D₄×C₃₁	—	4	2	no	—
97	D₄×C₃₁	—	4	2	no	—
98	D₄×C₃₁	—	4	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	1	no	—
103	C₂×C₄×C₃₁	—	2	1	no	—
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	1	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	4	1	no	—
108	C₂×C₄×C₃₁	—	4	1	no	—
109	C₂×C₄×C₃₁	—	4	1	no	—
110	C₂×C₄×C₃₁	—	4	2	no	—
111	C₂×C₄×C₃₁	—	4	2	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
117	C₈×C₃₁	—	2	2	no	—
118	C₈×C₃₁	—	2	2	no	—
119	#3196	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₃₁	—	2	1	no	—
121	C₂²×C₃₁	—	2	1	no	—
122	C₂²×C₃₁	—	2	1	no	—
123	C₂²×C₃₁	—	2	1	no	—
124	C₂²×C₃₁	—	2	1	no	—
125	C₂²×C₃₁	—	2	1	no	—
126	C₂²×C₃₁	—	2	1	no	—
127	C₂²×C₃₁	—	2	1	no	—
128	C₂²×C₃₁	—	4	4	no	—
129	C₂²×C₃₁	—	4	1	no	—
130	C₂²×C₃₁	—	4	1	no	—
131	C₂²×C₃₁	—	4	1	no	—
132	C₂²×C₃₁	#137	1	2	yes	—
133	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
134	C₄×C₃₁	—	2	1	no	—
135	C₄×C₃₁	—	2	1	no	—
136	C₄×C₃₁	—	2	1	no	—
137	C₄×C₃₁	—	4	1	no	—
138	C₄×C₃₁	—	4	2	no	—
139	C₄×C₃₁	—	4	1	no	—
140	C₄×C₃₁	—	4	2	no	—
141	C₄×C₃₁	—	4	1	no	—
142	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
143	#525	C₂×C₃₁	1	1	yes	characteristic
144	C₂×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#485	C₂×C₃₁	1	2	yes	complemented
146	#484	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#481	C₂×C₃₁	1	1	yes	characteristic, complemented
148	#470	C₂×C₃₁	1	2	yes	complemented
149	#468	C₂×C₃₁	1	2	yes	complemented
150	#452	C₂×C₃₁	1	2	yes	complemented
151	#451	C₂×C₃₁	1	1	yes	characteristic, complemented
152	#448	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#407	C₂×C₃₁	1	1	yes	characteristic, complemented
154	C₂×C₃₁	—	2	2	no	—
155	C₂×C₃₁	—	4	1	no	—
156	C₂×C₃₁	—	4	1	no	—
157	C₂×C₃₁	—	8	1	no	—
158	C₂×C₃₁	#534	1	1	yes	characteristic
159	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
160	C₂×C₃₁	#575	1	1	yes	characteristic
161	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
162	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
163	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
164	C₂×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
165	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
166	C₂×M₁₆	C₂²×C₃₁	1	2	yes	—
167	#123	—	2	1	no	—
168	C₂²⋊Q₈	—	2	1	no	—
169	C₂²⋊Q₈	—	2	1	no	—
170	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
171	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
172	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
173	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
174	#121	C₂²×C₃₁	1	1	yes	characteristic
175	#121	C₂²×C₃₁	1	1	yes	characteristic
176	#120	—	2	1	no	—
177	C₄×D₄	—	2	1	no	—
178	C₄×D₄	—	2	1	no	—
179	#117	C₂²×C₃₁	1	1	yes	characteristic
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
182	C₂×C₂²⋊C₄	—	2	1	no	—
183	#107	C₂²×C₃₁	1	2	yes	complemented
184	#106	C₂²×C₃₁	1	2	yes	complemented
185	#103	C₂²×C₃₁	1	2	yes	complemented
186	#103	C₂²×C₃₁	1	2	yes	complemented
187	#103	C₂²×C₃₁	1	2	yes	complemented
188	#102	C₂²×C₃₁	1	2	yes	complemented
189	C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
190	C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
191	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
192	C₃₁	#3196	1	1	yes	characteristic, direct product factor, Sylow
193	C₂⁴	D₄×C₃₁	1	1	yes	characteristic
194	#41	—	4	2	no	—
195	C₂×Q₈	—	2	1	no	—
196	C₂×Q₈	—	4	2	no	—
197	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
198	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
208	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
209	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
210	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
211	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
213	Q₁₆	—	4	2	no	—
214	SD₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	C₂×C₈	C₂³×C₃₁	1	2	yes	—
217	C₂×C₈	C₂³×C₃₁	1	2	yes	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
224	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
225	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
226	C₂²⋊C₄	—	2	2	no	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	2	1	no	—
231	C₂²⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	4	1	no	—
237	C₂³	—	4	1	no	—
238	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
239	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
240	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
241	Q₈	—	2	2	no	—
242	Q₈	—	2	2	no	—
243	Q₈	—	2	2	no	—
244	Q₈	—	4	2	no	—
245	D₄	—	2	2	no	—
246	D₄	—	4	2	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
264	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
265	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
266	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	2	2	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	C₃₁×#137	1	2	yes	—
282	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	4	2	no	—
290	C₄	—	4	1	no	—
291	C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
292	C₂	—	2	2	no	—
293	C₂	—	4	1	no	—
294	C₂	—	4	1	no	—
295	C₂	—	8	1	no	—
296	C₂	C₃₁×#534	1	1	yes	characteristic
297	C₂	C₂×C₃₁×#137	1	1	yes	characteristic
298	C₂	C₃₁×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×C₂²⋊Q₈	C₂	semidirect
3. C₃₁×#485	C₂	semidirect
4. C₃₁×#484	C₂	semidirect
5. C₃₁×#481	C₂	semidirect
6. C₃₁×#470	C₂	semidirect
7. C₃₁×#468	C₂	semidirect
8. C₃₁×#452	C₂	semidirect
9. C₃₁×#451	C₂	semidirect
10. C₃₁×#448	C₂	semidirect
11. C₃₁×#407	C₂	semidirect
15. C₂×Q₁₆×C₃₁	C₂²	semidirect
16. C₂×SD₁₆×C₃₁	C₂²	semidirect
34. C₃₁×#107	C₂²	semidirect
35. C₃₁×#106	C₂²	semidirect
36. C₃₁×#103	C₂²	semidirect
37. C₃₁×#103	C₂²	semidirect
38. C₃₁×#103	C₂²	semidirect
39. C₃₁×#102	C₂²	semidirect
119. #3196	C₃₁	direct (N × H)
144. C₂×C₂²⋊Q₈	C₂×C₃₁	semidirect
145. #485	C₂×C₃₁	semidirect
146. #484	C₂×C₃₁	semidirect
147. #481	C₂×C₃₁	semidirect
148. #470	C₂×C₃₁	semidirect
149. #468	C₂×C₃₁	semidirect
150. #452	C₂×C₃₁	semidirect
151. #451	C₂×C₃₁	semidirect
152. #448	C₂×C₃₁	semidirect
153. #407	C₂×C₃₁	semidirect
164. C₂×Q₁₆	C₂²×C₃₁	semidirect
165. C₂×SD₁₆	C₂²×C₃₁	semidirect
183. #107	C₂²×C₃₁	semidirect
184. #106	C₂²×C₃₁	semidirect
185. #103	C₂²×C₃₁	semidirect
186. #103	C₂²×C₃₁	semidirect
187. #103	C₂²×C₃₁	semidirect
188. #102	C₂²×C₃₁	semidirect
192. C₃₁	#3196	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×#525	2
C₂×C₃₁×C₂²⋊Q₈	2
C₃₁×#485	2
C₃₁×#484	2
C₃₁×#481	2
C₃₁×#470	2
C₃₁×#468	2
C₃₁×#452	2
C₃₁×#451	2
C₃₁×#448	2
C₃₁×#407	2
#3196	31

Subgroups — C31×#3196 #16843129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3196 #16843129