Subgroups — C₃₁×#11705 #21468464

Order 7936.

508 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2112	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1735	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1347	C₂	1	4	yes	maximal, complemented
5	C₂²×D₈×C₃₁	C₄	1	1	yes	characteristic, complemented
6	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×#121	—	2	1	no	—
8	C₂×C₃₁×#117	—	2	1	no	—
9	C₂×C₃₁×M₃₂	—	2	1	no	—
10	C₃₁×#466	—	2	1	no	—
11	C₃₁×#466	—	2	1	no	—
12	C₃₁×#466	C₄	1	2	yes	complemented
13	C₃₁×#466	C₂²	1	2	yes	—
14	C₂×C₃₁×#107	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃₁×#102	—	2	1	no	—
16	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
17	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
18	C₃₁×#340	—	2	1	no	—
19	C₃₁×#340	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₃₁	—	2	1	no	—
21	C₂²×D₄×C₃₁	—	4	1	no	—
22	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
23	C₂×D₈×C₃₁	—	2	1	no	—
24	C₂×D₈×C₃₁	—	4	2	no	—
25	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₃₁	—	4	2	no	—
27	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
33	C₃₁×#121	—	2	2	no	—
34	C₃₁×#121	—	2	2	no	—
35	C₃₁×#121	—	4	1	no	—
36	C₃₁×#121	—	4	1	no	—
37	C₃₁×#117	—	2	4	no	—
38	C₃₁×#117	—	4	2	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
42	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
43	C₂×C₄²×C₃₁	—	4	1	no	—
44	C₃₁×M₃₂	—	4	2	no	—
45	C₂×C₁₆×C₃₁	—	2	2	no	—
46	C₃₁×#107	—	2	1	no	—
47	C₃₁×#107	C₂×C₄	1	2	yes	—
48	C₃₁×#102	—	2	2	no	—
49	C₃₁×#102	—	2	2	no	—
50	C₂⁴×C₃₁	—	4	1	no	—
51	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
52	C₂×D₄×C₃₁	—	2	2	no	—
53	C₂×D₄×C₃₁	—	4	2	no	—
54	C₂×D₄×C₃₁	—	4	2	no	—
55	C₂×D₄×C₃₁	—	4	2	no	—
56	C₂×D₄×C₃₁	—	4	1	no	—
57	C₂×D₄×C₃₁	—	4	2	no	—
58	C₂×D₄×C₃₁	—	8	4	no	—
59	C₂²×C₄×C₃₁	—	2	2	no	—
60	C₂²×C₄×C₃₁	—	2	2	no	—
61	C₂²×C₄×C₃₁	—	4	1	no	—
62	C₂²×C₄×C₃₁	—	4	2	no	—
63	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	D₈×C₃₁	—	4	2	no	—
70	D₈×C₃₁	—	4	2	no	—
71	C₂×C₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	4	1	no	—
74	C₂×C₈×C₃₁	—	4	1	no	—
75	C₂×C₈×C₃₁	—	4	1	no	—
76	C₂×C₈×C₃₁	—	4	1	no	—
77	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₄⋊C₄×C₃₁	—	4	2	no	—
84	C₄⋊C₄×C₃₁	—	4	1	no	—
85	C₂²⋊C₄×C₃₁	—	4	4	no	—
86	C₂²⋊C₄×C₃₁	—	4	2	no	—
87	C₂²⋊C₄×C₃₁	—	4	2	no	—
88	C₄²×C₃₁	—	4	4	no	—
89	C₁₆×C₃₁	—	4	2	no	—
90	#11705	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₃₁	—	2	2	no	—
92	C₂³×C₃₁	—	2	2	no	—
93	C₂³×C₃₁	—	4	1	no	—
94	C₂³×C₃₁	—	4	2	no	—
95	C₂³×C₃₁	—	8	2	no	—
96	C₂³×C₃₁	—	8	2	no	—
97	C₂³×C₃₁	—	8	2	no	—
98	C₂³×C₃₁	#102	1	1	yes	characteristic
99	C₂³×C₃₁	#102	1	1	yes	characteristic
100	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
101	D₄×C₃₁	—	4	2	no	—
102	D₄×C₃₁	—	4	2	no	—
103	D₄×C₃₁	—	8	4	no	—
104	D₄×C₃₁	—	8	4	no	—
105	D₄×C₃₁	—	8	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	1	no	—
110	C₂×C₄×C₃₁	—	2	1	no	—
111	C₂×C₄×C₃₁	—	2	1	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	—	8	2	no	—
116	C₂×C₄×C₃₁	—	8	2	no	—
117	C₂×C₄×C₃₁	—	8	2	no	—
118	C₂×C₄×C₃₁	#99	1	2	yes	—
119	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
120	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
121	C₂×C₄×C₃₁	#104	1	2	yes	—
122	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
124	C₈×C₃₁	—	2	1	no	—
125	C₈×C₃₁	—	2	1	no	—
126	C₈×C₃₁	—	2	1	no	—
127	C₈×C₃₁	—	2	1	no	—
128	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
129	#2112	C₂×C₃₁	1	1	yes	characteristic, complemented
130	#1735	C₂×C₃₁	1	1	yes	characteristic, complemented
131	#1347	C₂×C₃₁	1	4	yes	complemented
132	C₂²×C₃₁	—	2	1	no	—
133	C₂²×C₃₁	—	2	1	no	—
134	C₂²×C₃₁	—	2	1	no	—
135	C₂²×C₃₁	—	2	1	no	—
136	C₂²×C₃₁	—	2	1	no	—
137	C₂²×C₃₁	—	2	1	no	—
138	C₂²×C₃₁	—	2	1	no	—
139	C₂²×C₃₁	—	4	4	no	—
140	C₂²×C₃₁	—	8	2	no	—
141	C₂²×C₃₁	—	8	2	no	—
142	C₂²×C₃₁	—	8	2	no	—
143	C₂²×C₃₁	—	16	4	no	—
144	C₂²×C₃₁	#327	1	2	yes	—
145	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
146	C₂²×C₃₁	#509	1	2	yes	—
147	C₄×C₃₁	—	2	1	no	—
148	C₄×C₃₁	—	2	1	no	—
149	C₄×C₃₁	—	8	4	no	—
150	C₄×C₃₁	—	8	2	no	—
151	C₄×C₃₁	#327	1	2	yes	—
152	C₄×C₃₁	C₂×#99	1	1	yes	characteristic
153	C₄×C₃₁	C₂×#104	1	1	yes	characteristic
154	C₂²×D₈	C₄×C₃₁	1	1	yes	characteristic, complemented
155	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
156	C₂×#121	—	2	1	no	—
157	C₂×#117	—	2	1	no	—
158	C₂×M₃₂	—	2	1	no	—
159	#466	—	2	1	no	—
160	#466	—	2	1	no	—
161	#466	C₄×C₃₁	1	2	yes	complemented
162	#466	C₂²×C₃₁	1	2	yes	—
163	C₂×#107	C₄×C₃₁	1	1	yes	characteristic, complemented
164	C₂×#102	—	2	1	no	—
165	C₂²⋊C₁₆	—	2	1	no	—
166	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
167	#340	—	2	1	no	—
168	#340	C₂²×C₃₁	1	2	yes	complemented
169	C₂×C₃₁	—	2	2	no	—
170	C₂×C₃₁	—	2	2	no	—
171	C₂×C₃₁	—	4	1	no	—
172	C₂×C₃₁	—	16	2	no	—
173	C₂×C₃₁	C₂×#327	1	1	yes	characteristic
174	C₂×C₃₁	#2140	1	1	yes	characteristic
175	C₂×C₃₁	#2145	1	1	yes	characteristic
176	C₂²×D₄	—	2	1	no	—
177	C₂²×D₄	—	4	1	no	—
178	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
179	C₂×D₈	—	2	1	no	—
180	C₂×D₈	—	4	2	no	—
181	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
182	C₂²×C₈	—	4	2	no	—
183	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
184	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
185	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
186	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
187	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
189	#121	—	2	2	no	—
190	#121	—	2	2	no	—
191	#121	—	4	1	no	—
192	#121	—	4	1	no	—
193	#117	—	2	4	no	—
194	#117	—	4	2	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₂²⋊C₄	—	4	1	no	—
198	C₂×C₂²⋊C₄	—	4	1	no	—
199	C₂×C₄²	—	4	1	no	—
200	M₃₂	—	4	2	no	—
201	C₂×C₁₆	—	2	2	no	—
202	#107	—	2	1	no	—
203	#107	C₂×C₄×C₃₁	1	2	yes	—
204	#102	—	2	2	no	—
205	#102	—	2	2	no	—
206	C₃₁	#11705	1	1	yes	characteristic, direct product factor, Sylow
207	C₂⁴	—	4	1	no	—
208	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	8	4	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
221	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
222	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
224	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
226	D₈	—	4	2	no	—
227	D₈	—	4	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
236	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₂²⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₄²	—	4	4	no	—
246	C₁₆	—	4	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	4	1	no	—
250	C₂³	—	4	2	no	—
251	C₂³	—	8	2	no	—
252	C₂³	—	8	2	no	—
253	C₂³	—	8	2	no	—
254	C₂³	C₃₁×#102	1	1	yes	characteristic
255	C₂³	C₃₁×#102	1	1	yes	characteristic
256	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
257	D₄	—	4	2	no	—
258	D₄	—	4	2	no	—
259	D₄	—	8	4	no	—
260	D₄	—	8	4	no	—
261	D₄	—	8	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	8	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	C₃₁×#99	1	2	yes	—
275	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
276	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
277	C₂×C₄	C₃₁×#104	1	2	yes	—
278	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	4	4	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	16	4	no	—
296	C₂²	C₃₁×#327	1	2	yes	—
297	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
298	C₂²	C₃₁×#509	1	2	yes	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	8	4	no	—
302	C₄	—	8	2	no	—
303	C₄	C₃₁×#327	1	2	yes	—
304	C₄	C₂×C₃₁×#99	1	1	yes	characteristic
305	C₄	C₂×C₃₁×#104	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	2	2	no	—
308	C₂	—	4	1	no	—
309	C₂	—	16	2	no	—
310	C₂	C₂×C₃₁×#327	1	1	yes	characteristic
311	C₂	C₃₁×#2140	1	1	yes	characteristic
312	C₂	C₃₁×#2145	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂×C₃₁×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2112	C₂	semidirect
3. C₃₁×#1735	C₂	semidirect
4. C₃₁×#1347	C₂	semidirect
5. C₂²×D₈×C₃₁	C₄	semidirect
12. C₃₁×#466	C₄	semidirect
14. C₂×C₃₁×#107	C₄	semidirect
17. C₃₁×C₂²⋊C₁₆	C₂²	semidirect
19. C₃₁×#340	C₂²	semidirect
90. #11705	C₃₁	direct (N × H)
129. #2112	C₂×C₃₁	semidirect
130. #1735	C₂×C₃₁	semidirect
131. #1347	C₂×C₃₁	semidirect
154. C₂²×D₈	C₄×C₃₁	semidirect
161. #466	C₄×C₃₁	semidirect
163. C₂×#107	C₄×C₃₁	semidirect
166. C₂²⋊C₁₆	C₂²×C₃₁	semidirect
168. #340	C₂²×C₃₁	semidirect
206. C₃₁	#11705	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×#466	2
C₃₁×#2112	2
C₃₁×#1735	2
C₃₁×#1347	2
#11705	31

Subgroups — C31×#11705 #21468464

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11705 #21468464