Subgroups — C₃₁×#19297 #21476056

Order 7936.

616 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3079	C₂	1	2	yes	maximal
3	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1696	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1695	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1548	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#500	—	2	1	no	—
11	C₃₁×#500	—	2	1	no	—
12	C₃₁×#500	C₂²	1	4	yes	—
13	C₃₁×#477	C₂²	1	4	yes	—
14	C₃₁×#476	C₂²	1	4	yes	—
15	C₃₁×#475	—	2	2	no	—
16	C₃₁×#474	—	2	2	no	—
17	C₃₁×#446	C₂²	1	2	yes	—
18	C₃₁×#436	C₂²	1	2	yes	—
19	C₃₁×#435	—	2	2	no	—
20	C₃₁×#431	—	2	1	no	—
21	C₃₁×#431	C₂²	1	1	yes	characteristic
22	C₃₁×#428	—	2	2	no	—
23	C₂×C₃₁×#107	—	2	2	no	—
24	C₃₁×#423	C₂²	1	2	yes	—
25	C₃₁×#417	—	2	2	no	—
26	C₄×M₁₆×C₃₁	—	2	1	no	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#334	C₂²	1	4	yes	—
29	C₃₁×#333	C₂²	1	8	yes	—
30	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×Q₈×C₃₁	—	2	2	no	—
32	C₂×M₁₆×C₃₁	—	2	2	no	—
33	C₂×M₁₆×C₃₁	—	2	2	no	—
34	C₂²×C₈×C₃₁	D₄	1	2	yes	—
35	C₃₁×C₄⋊Q₈	—	2	2	no	—
36	C₃₁×C₄⋊Q₈	—	2	2	no	—
37	C₃₁×C₄⋊Q₈	—	2	2	no	—
38	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
39	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
40	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₄×D₄×C₃₁	—	2	1	no	—
42	C₄×D₄×C₃₁	—	2	1	no	—
43	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
44	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₃₁×#117	—	2	1	no	—
46	C₃₁×#117	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₄²×C₃₁	—	2	1	no	—
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×#107	—	2	4	no	—
52	C₃₁×#107	—	2	4	no	—
53	C₃₁×#107	—	2	4	no	—
54	C₃₁×#106	—	2	4	no	—
55	C₃₁×C₄⋊C₈	—	2	2	no	—
56	C₃₁×C₄⋊C₈	—	2	2	no	—
57	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
58	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
59	C₃₁×#103	—	2	4	no	—
60	C₃₁×#102	—	2	4	no	—
61	C₃₁×C₂²⋊C₈	—	2	1	no	—
62	C₃₁×C₂²⋊C₈	—	2	2	no	—
63	C₃₁×#97	—	2	2	no	—
64	C₄×C₈×C₃₁	—	2	1	no	—
65	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
66	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
67	C₄×C₈×C₃₁	C₂³	1	2	yes	—
68	#41×C₃₁	—	2	4	no	—
69	C₂×Q₈×C₃₁	—	2	4	no	—
70	C₂×Q₈×C₃₁	—	4	2	no	—
71	C₂×Q₈×C₃₁	—	8	4	no	—
72	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂×D₄×C₃₁	—	2	1	no	—
74	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	—	2	1	no	—
76	C₂²×C₄×C₃₁	—	2	1	no	—
77	C₂²×C₄×C₃₁	—	2	1	no	—
78	C₂²×C₄×C₃₁	—	4	2	no	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
82	M₁₆×C₃₁	—	4	4	no	—
83	M₁₆×C₃₁	—	4	4	no	—
84	C₂×C₈×C₃₁	—	2	4	no	—
85	C₂×C₈×C₃₁	—	2	2	no	—
86	C₂×C₈×C₃₁	—	2	4	no	—
87	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
89	C₄⋊C₄×C₃₁	—	2	1	no	—
90	C₄⋊C₄×C₃₁	—	2	1	no	—
91	C₄⋊C₄×C₃₁	—	2	4	no	—
92	C₄⋊C₄×C₃₁	—	2	4	no	—
93	C₄⋊C₄×C₃₁	—	4	2	no	—
94	C₄⋊C₄×C₃₁	—	4	2	no	—
95	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂²⋊C₄×C₃₁	—	2	1	no	—
98	C₂²⋊C₄×C₃₁	—	2	1	no	—
99	C₂²⋊C₄×C₃₁	—	2	1	no	—
100	C₄²×C₃₁	—	2	1	no	—
101	C₄²×C₃₁	—	2	1	no	—
102	C₄²×C₃₁	—	2	1	no	—
103	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₃₁	#41	1	1	yes	characteristic
106	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
107	#19297	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₃₁	—	2	1	no	—
109	C₂³×C₃₁	#121	1	1	yes	characteristic
110	Q₈×C₃₁	—	2	2	no	—
111	Q₈×C₃₁	—	8	4	no	—
112	Q₈×C₃₁	—	8	4	no	—
113	D₄×C₃₁	—	2	2	no	—
114	D₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	2	1	no	—
119	C₂×C₄×C₃₁	—	2	1	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	4	no	—
124	C₂×C₄×C₃₁	—	8	4	no	—
125	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
126	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
127	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
128	C₂×C₄×C₃₁	#137	1	4	yes	—
129	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
132	C₈×C₃₁	—	2	2	no	—
133	C₈×C₃₁	—	4	2	no	—
134	C₈×C₃₁	—	4	4	no	—
135	C₈×C₃₁	#137	1	4	yes	—
136	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#3079	C₂×C₃₁	1	2	yes	—
138	#2965	C₂×C₃₁	1	1	yes	characteristic
139	#1696	C₂×C₃₁	1	4	yes	complemented
140	#1695	C₂×C₃₁	1	4	yes	complemented
141	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
142	#1548	C₂×C₃₁	1	2	yes	complemented
143	C₂²×C₃₁	—	2	1	no	—
144	C₂²×C₃₁	—	2	1	no	—
145	C₂²×C₃₁	—	4	1	no	—
146	C₂²×C₃₁	—	4	1	no	—
147	C₂²×C₃₁	—	4	1	no	—
148	C₂²×C₃₁	#470	1	2	yes	—
149	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
150	C₄×C₃₁	—	2	1	no	—
151	C₄×C₃₁	—	2	1	no	—
152	C₄×C₃₁	—	4	1	no	—
153	C₄×C₃₁	—	4	1	no	—
154	C₄×C₃₁	—	4	1	no	—
155	C₄×C₃₁	—	8	4	no	—
156	C₄×C₃₁	#470	1	2	yes	—
157	C₄×C₃₁	#472	1	2	yes	—
158	C₄×C₃₁	C₂×#137	1	2	yes	—
159	C₄×C₃₁	C₂×#137	1	2	yes	—
160	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
161	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
162	#500	—	2	1	no	—
163	#500	—	2	1	no	—
164	#500	C₂²×C₃₁	1	4	yes	—
165	#477	C₂²×C₃₁	1	4	yes	—
166	#476	C₂²×C₃₁	1	4	yes	—
167	#475	—	2	2	no	—
168	#474	—	2	2	no	—
169	#446	C₂²×C₃₁	1	2	yes	—
170	#436	C₂²×C₃₁	1	2	yes	—
171	#435	—	2	2	no	—
172	#431	—	2	1	no	—
173	#431	C₂²×C₃₁	1	1	yes	characteristic
174	#428	—	2	2	no	—
175	C₂×#107	—	2	2	no	—
176	#423	C₂²×C₃₁	1	2	yes	—
177	#417	—	2	2	no	—
178	C₄×M₁₆	—	2	1	no	—
179	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
180	#334	C₂²×C₃₁	1	4	yes	—
181	#333	C₂²×C₃₁	1	8	yes	—
182	C₂×C₃₁	—	2	2	no	—
183	C₂×C₃₁	—	8	1	no	—
184	C₂×C₃₁	#3039	1	1	yes	characteristic
185	C₂×C₃₁	C₂×#470	1	1	yes	characteristic
186	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
187	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
188	C₂²×Q₈	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂²×C₈	D₄×C₃₁	1	2	yes	—
192	C₄⋊Q₈	—	2	2	no	—
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
196	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
197	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
201	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
202	#117	—	2	1	no	—
203	#117	D₄×C₃₁	1	1	yes	characteristic
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
208	#107	—	2	4	no	—
209	#107	—	2	4	no	—
210	#107	—	2	4	no	—
211	#106	—	2	4	no	—
212	C₄⋊C₈	—	2	2	no	—
213	C₄⋊C₈	—	2	2	no	—
214	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
215	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
216	#103	—	2	4	no	—
217	#102	—	2	4	no	—
218	C₂²⋊C₈	—	2	1	no	—
219	C₂²⋊C₈	—	2	2	no	—
220	#97	—	2	2	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
223	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
224	C₄×C₈	C₂³×C₃₁	1	2	yes	—
225	C₃₁	#19297	1	1	yes	characteristic, direct product factor, Sylow
226	#41	—	2	4	no	—
227	C₂×Q₈	—	2	4	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×Q₈	—	8	4	no	—
230	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
239	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
240	M₁₆	—	4	4	no	—
241	M₁₆	—	4	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
246	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
254	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
255	C₂²⋊C₄	—	2	1	no	—
256	C₂²⋊C₄	—	2	1	no	—
257	C₂²⋊C₄	—	2	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
262	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
263	C₄²	#41×C₃₁	1	1	yes	characteristic
264	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
265	C₂³	—	2	1	no	—
266	C₂³	C₃₁×#121	1	1	yes	characteristic
267	Q₈	—	2	2	no	—
268	Q₈	—	8	4	no	—
269	Q₈	—	8	4	no	—
270	D₄	—	2	2	no	—
271	D₄	—	4	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	8	4	no	—
282	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
283	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
284	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
285	C₂×C₄	C₃₁×#137	1	4	yes	—
286	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
288	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	4	2	no	—
291	C₈	—	4	4	no	—
292	C₈	C₃₁×#137	1	4	yes	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	C₃₁×#470	1	2	yes	—
299	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	8	4	no	—
306	C₄	C₃₁×#470	1	2	yes	—
307	C₄	C₃₁×#472	1	2	yes	—
308	C₄	C₂×C₃₁×#137	1	2	yes	—
309	C₄	C₂×C₃₁×#137	1	2	yes	—
310	C₂	—	2	2	no	—
311	C₂	—	8	1	no	—
312	C₂	C₃₁×#3039	1	1	yes	characteristic
313	C₂	C₂×C₃₁×#470	1	1	yes	characteristic
314	C₂	C₂×C₃₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ 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. C₂×C₃₁×#500	C₂	semidirect
4. C₃₁×#1696	C₂	semidirect
5. C₃₁×#1695	C₂	semidirect
6. C₃₁×#1566	C₂	semidirect
7. C₃₁×#1548	C₂	semidirect
107. #19297	C₃₁	direct (N × H)
136. C₂×#500	C₂×C₃₁	semidirect
139. #1696	C₂×C₃₁	semidirect
140. #1695	C₂×C₃₁	semidirect
141. #1566	C₂×C₃₁	semidirect
142. #1548	C₂×C₃₁	semidirect
225. C₃₁	#19297	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#500	2
C₃₁×#3079	2
C₃₁×#2965	2
C₃₁×#1696	2
C₃₁×#1695	2
C₃₁×#1566	2
C₃₁×#1548	2
#19297	31

Subgroups — C31×#19297 #21476056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19297 #21476056