Subgroups — C₃₁×#11461 #21468220

Order 7936.

320 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#433	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2163	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2162	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1322	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1314	C₂	1	2	yes	maximal
6	C₂×C₃₁×#117	C₄	1	1	yes	characteristic
7	C₃₁×#433	—	2	1	no	—
8	C₃₁×#433	—	2	1	no	—
9	C₃₁×#433	—	2	1	no	—
10	C₃₁×#433	—	2	1	no	—
11	C₂×C₃₁×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₂²⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#97	C₄	1	1	yes	characteristic
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×C₄⋊C₁₆	—	2	1	no	—
18	C₃₁×C₄⋊C₁₆	—	2	1	no	—
19	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
20	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
21	C₃₁×#346	—	2	1	no	—
22	C₄×C₁₆×C₃₁	—	2	1	no	—
23	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
28	C₃₁×#117	—	2	1	no	—
29	C₃₁×#117	—	2	1	no	—
30	C₃₁×#117	—	2	1	no	—
31	C₃₁×#117	—	2	1	no	—
32	C₂×C₄⋊C₄×C₃₁	C₈	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₃₁	C₈	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₃₁	C₈	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₃₁	C₈	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆×C₃₁	—	2	2	no	—
39	C₂×C₁₆×C₃₁	—	2	2	no	—
40	C₃₁×C₄⋊C₈	—	2	1	no	—
41	C₃₁×C₄⋊C₈	—	2	1	no	—
42	C₃₁×C₄⋊C₈	—	2	1	no	—
43	C₃₁×C₄⋊C₈	—	2	1	no	—
44	C₃₁×C₂²⋊C₈	—	2	1	no	—
45	C₃₁×C₂²⋊C₈	—	2	1	no	—
46	C₃₁×C₂²⋊C₈	—	2	1	no	—
47	C₃₁×C₂²⋊C₈	—	2	1	no	—
48	C₃₁×#97	—	2	1	no	—
49	C₃₁×#97	—	2	1	no	—
50	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
52	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
53	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
54	C₂⁴×C₃₁	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	—	2	1	no	—
56	C₂²×C₄×C₃₁	—	2	1	no	—
57	C₂²×C₄×C₃₁	—	4	1	no	—
58	C₂²×C₄×C₃₁	—	4	1	no	—
59	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₃₁	—	2	1	no	—
66	C₂×C₈×C₃₁	—	2	1	no	—
67	C₂×C₈×C₃₁	—	4	1	no	—
68	C₂×C₈×C₃₁	—	4	1	no	—
69	C₂×C₈×C₃₁	—	4	1	no	—
70	C₂×C₈×C₃₁	—	4	1	no	—
71	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₃₁	—	2	1	no	—
76	C₄⋊C₄×C₃₁	—	2	1	no	—
77	C₄⋊C₄×C₃₁	—	2	1	no	—
78	C₄⋊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	C₄²×C₃₁	—	2	1	no	—
85	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₃₁	M₁₆	1	1	yes	characteristic
88	C₄²×C₃₁	M₁₆	1	1	yes	characteristic
89	C₁₆×C₃₁	—	4	2	no	—
90	C₁₆×C₃₁	—	4	2	no	—
91	#11461	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₃₁	—	2	1	no	—
93	C₂³×C₃₁	—	2	2	no	—
94	C₂³×C₃₁	—	4	1	no	—
95	C₂³×C₃₁	—	4	1	no	—
96	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	1	no	—
99	C₂×C₄×C₃₁	—	2	1	no	—
100	C₂×C₄×C₃₁	—	2	1	no	—
101	C₂×C₄×C₃₁	—	2	1	no	—
102	C₂×C₄×C₃₁	—	2	1	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	4	1	no	—
105	C₂×C₄×C₃₁	—	4	2	no	—
106	C₂×C₄×C₃₁	—	4	1	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	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
116	C₈×C₃₁	—	2	1	no	—
117	C₈×C₃₁	—	2	1	no	—
118	C₈×C₃₁	—	2	1	no	—
119	C₈×C₃₁	—	2	1	no	—
120	C₈×C₃₁	—	4	1	no	—
121	C₈×C₃₁	—	4	1	no	—
122	C₂×#433	C₂×C₃₁	1	1	yes	characteristic
123	#2163	C₂×C₃₁	1	1	yes	characteristic, complemented
124	#2162	C₂×C₃₁	1	1	yes	characteristic, complemented
125	#1322	C₂×C₃₁	1	2	yes	complemented
126	#1314	C₂×C₃₁	1	2	yes	—
127	C₂²×C₃₁	—	2	1	no	—
128	C₂²×C₃₁	—	2	1	no	—
129	C₂²×C₃₁	—	4	1	no	—
130	C₂²×C₃₁	—	4	1	no	—
131	C₂²×C₃₁	—	4	1	no	—
132	C₂²×C₃₁	—	8	2	no	—
133	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₂²×C₃₁	#504	1	2	yes	—
135	C₄×C₃₁	—	2	1	no	—
136	C₄×C₃₁	—	2	1	no	—
137	C₄×C₃₁	—	2	1	no	—
138	C₄×C₃₁	—	2	1	no	—
139	C₄×C₃₁	—	2	1	no	—
140	C₄×C₃₁	—	4	1	no	—
141	C₄×C₃₁	—	4	1	no	—
142	C₄×C₃₁	—	4	1	no	—
143	C₄×C₃₁	#410	1	1	yes	characteristic
144	C₄×C₃₁	#413	1	1	yes	characteristic
145	C₂×#117	C₄×C₃₁	1	1	yes	characteristic
146	#433	—	2	1	no	—
147	#433	—	2	1	no	—
148	#433	—	2	1	no	—
149	#433	—	2	1	no	—
150	C₂×C₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
151	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
152	C₂×C₂²⋊C₈	C₄×C₃₁	1	1	yes	characteristic
153	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
154	C₂×#97	C₄×C₃₁	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
156	C₄⋊C₁₆	—	2	1	no	—
157	C₄⋊C₁₆	—	2	1	no	—
158	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
159	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
160	#346	—	2	1	no	—
161	C₄×C₁₆	—	2	1	no	—
162	C₂×C₃₁	—	2	2	no	—
163	C₂×C₃₁	—	8	1	no	—
164	C₂×C₃₁	#1458	1	1	yes	characteristic
165	C₂×C₃₁	#2113	1	1	yes	characteristic
166	C₂×C₃₁	#2115	1	1	yes	characteristic
167	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
168	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
169	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
170	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
172	#117	—	2	1	no	—
173	#117	—	2	1	no	—
174	#117	—	2	1	no	—
175	#117	—	2	1	no	—
176	C₂×C₄⋊C₄	C₈×C₃₁	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₈×C₃₁	1	1	yes	characteristic
178	C₂×C₂²⋊C₄	C₈×C₃₁	1	1	yes	characteristic
179	C₂×C₂²⋊C₄	C₈×C₃₁	1	1	yes	characteristic
180	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
181	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
182	C₂×C₁₆	—	2	2	no	—
183	C₂×C₁₆	—	2	2	no	—
184	C₄⋊C₈	—	2	1	no	—
185	C₄⋊C₈	—	2	1	no	—
186	C₄⋊C₈	—	2	1	no	—
187	C₄⋊C₈	—	2	1	no	—
188	C₂²⋊C₈	—	2	1	no	—
189	C₂²⋊C₈	—	2	1	no	—
190	C₂²⋊C₈	—	2	1	no	—
191	C₂²⋊C₈	—	2	1	no	—
192	#97	—	2	1	no	—
193	#97	—	2	1	no	—
194	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
195	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
196	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
197	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
198	C₃₁	#11461	1	1	yes	characteristic, direct product factor, Sylow
199	C₂⁴	C₂×C₈×C₃₁	1	1	yes	characteristic
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	—	4	1	no	—
204	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
217	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
218	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	—	2	1	no	—
225	C₂²⋊C₄	—	2	1	no	—
226	C₂²⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	2	1	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	2	1	no	—
230	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
231	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
232	C₄²	M₁₆×C₃₁	1	1	yes	characteristic
233	C₄²	M₁₆×C₃₁	1	1	yes	characteristic
234	C₁₆	—	4	2	no	—
235	C₁₆	—	4	2	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	4	1	no	—
239	C₂³	—	4	1	no	—
240	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
255	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
256	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
257	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
259	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	4	1	no	—
265	C₈	—	4	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	8	2	no	—
272	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
273	C₂²	C₃₁×#504	1	2	yes	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	1	no	—
282	C₄	C₃₁×#410	1	1	yes	characteristic
283	C₄	C₃₁×#413	1	1	yes	characteristic
284	C₂	—	2	2	no	—
285	C₂	—	8	1	no	—
286	C₂	C₃₁×#1458	1	1	yes	characteristic
287	C₂	C₃₁×#2113	1	1	yes	characteristic
288	C₂	C₃₁×#2115	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2163	C₂	semidirect
3. C₃₁×#2162	C₂	semidirect
4. C₃₁×#1322	C₂	semidirect
91. #11461	C₃₁	direct (N × H)
123. #2163	C₂×C₃₁	semidirect
124. #2162	C₂×C₃₁	semidirect
125. #1322	C₂×C₃₁	semidirect
198. C₃₁	#11461	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×#433	2
C₃₁×#2163	2
C₃₁×#2162	2
C₃₁×#1322	2
C₃₁×#1314	2
#11461	31

Subgroups — C31×#11461 #21468220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11461 #21468220