Subgroups — C₃₁×#11464 #21468223

Order 7936.

310 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2922	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×#1323	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1315	C₂	1	2	yes	maximal
6	C₄×#41×C₃₁	C₄	1	1	yes	characteristic
7	C₃₁×#433	—	2	1	no	—
8	C₃₁×#433	—	2	1	no	—
9	C₃₁×#431	—	2	1	no	—
10	C₃₁×#431	—	2	1	no	—
11	C₃₁×#424	C₄	1	1	yes	characteristic
12	C₃₁×#424	C₂²	1	1	yes	characteristic
13	C₃₁×#408	C₄	1	1	yes	characteristic
14	C₃₁×#408	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₃₁	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	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂×M₁₆×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	C₄×Q₈×C₃₁	C₈	1	1	yes	characteristic
29	C₄×Q₈×C₃₁	C₈	1	1	yes	characteristic
30	C₄×D₄×C₃₁	—	2	1	no	—
31	C₄×D₄×C₃₁	—	2	1	no	—
32	C₄×D₄×C₃₁	C₈	1	1	yes	characteristic
33	C₄×D₄×C₃₁	C₈	1	1	yes	characteristic
34	C₃₁×#117	—	2	1	no	—
35	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	—	2	1	no	—
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	C₄×C₈×C₃₁	—	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	#41×C₃₁	—	4	1	no	—
55	#41×C₃₁	—	4	1	no	—
56	C₂×Q₈×C₃₁	C₂×C₈	1	1	yes	characteristic
57	C₂×D₄×C₃₁	—	2	1	no	—
58	C₂×D₄×C₃₁	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	—	2	1	no	—
60	C₂²×C₄×C₃₁	—	2	1	no	—
61	C₂²×C₄×C₃₁	—	2	1	no	—
62	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	M₁₆×C₃₁	—	4	1	no	—
66	M₁₆×C₃₁	—	4	1	no	—
67	C₂×C₈×C₃₁	—	2	1	no	—
68	C₂×C₈×C₃₁	—	2	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	C₂×C₈	1	1	yes	characteristic
78	C₄⋊C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
79	C₂²⋊C₄×C₃₁	—	2	1	no	—
80	C₂²⋊C₄×C₃₁	—	2	1	no	—
81	C₂²⋊C₄×C₃₁	—	2	1	no	—
82	C₄²×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	C₂×C₈	1	1	yes	characteristic
88	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
89	C₁₆×C₃₁	—	4	2	no	—
90	C₁₆×C₃₁	—	4	2	no	—
91	#11464	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₃₁	—	2	1	no	—
93	C₂³×C₃₁	C₂×M₁₆	1	1	yes	characteristic
94	Q₈×C₃₁	—	4	1	no	—
95	D₄×C₃₁	—	4	1	no	—
96	D₄×C₃₁	—	4	2	no	—
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	—	4	1	no	—
104	C₂×C₄×C₃₁	—	4	2	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	C₂²⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
114	C₈×C₃₁	—	2	1	no	—
115	C₈×C₃₁	—	2	1	no	—
116	C₈×C₃₁	—	2	1	no	—
117	C₈×C₃₁	—	2	1	no	—
118	C₈×C₃₁	—	4	1	no	—
119	C₈×C₃₁	—	4	1	no	—
120	#2922	C₂×C₃₁	1	1	yes	characteristic
121	#2163	C₂×C₃₁	1	1	yes	characteristic, complemented
122	#2162	C₂×C₃₁	1	1	yes	characteristic, complemented
123	#1323	C₂×C₃₁	1	2	yes	complemented
124	#1315	C₂×C₃₁	1	2	yes	—
125	C₂²×C₃₁	—	2	1	no	—
126	C₂²×C₃₁	—	2	1	no	—
127	C₂²×C₃₁	—	2	1	no	—
128	C₂²×C₃₁	—	4	1	no	—
129	C₂²×C₃₁	—	4	1	no	—
130	C₂²×C₃₁	—	4	1	no	—
131	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₄×C₃₁	—	2	1	no	—
133	C₄×C₃₁	—	2	1	no	—
134	C₄×C₃₁	—	2	1	no	—
135	C₄×C₃₁	—	2	1	no	—
136	C₄×C₃₁	—	4	1	no	—
137	C₄×C₃₁	—	4	1	no	—
138	C₄×C₃₁	—	4	1	no	—
139	C₄×C₃₁	#350	1	2	yes	—
140	C₄×C₃₁	#410	1	1	yes	characteristic
141	C₄×C₃₁	#413	1	1	yes	characteristic
142	C₄×#41	C₄×C₃₁	1	1	yes	characteristic
143	#433	—	2	1	no	—
144	#433	—	2	1	no	—
145	#431	—	2	1	no	—
146	#431	—	2	1	no	—
147	#424	C₄×C₃₁	1	1	yes	characteristic
148	#424	C₂²×C₃₁	1	1	yes	characteristic
149	#408	C₄×C₃₁	1	1	yes	characteristic
150	#408	C₂²×C₃₁	1	1	yes	characteristic
151	C₄×M₁₆	C₄×C₃₁	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
153	C₄⋊C₁₆	—	2	1	no	—
154	C₄⋊C₁₆	—	2	1	no	—
155	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
156	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
157	#346	—	2	1	no	—
158	C₄×C₁₆	—	2	1	no	—
159	C₂×C₃₁	—	4	1	no	—
160	C₂×C₃₁	—	8	1	no	—
161	C₂×C₃₁	#1458	1	1	yes	characteristic
162	C₂×C₃₁	C₂×#350	1	1	yes	characteristic
163	C₂×C₃₁	#2117	1	1	yes	characteristic
164	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
165	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
166	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
167	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
169	C₄×Q₈	C₈×C₃₁	1	1	yes	characteristic
170	C₄×Q₈	C₈×C₃₁	1	1	yes	characteristic
171	C₄×D₄	—	2	1	no	—
172	C₄×D₄	—	2	1	no	—
173	C₄×D₄	C₈×C₃₁	1	1	yes	characteristic
174	C₄×D₄	C₈×C₃₁	1	1	yes	characteristic
175	#117	—	2	1	no	—
176	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	2	2	no	—
181	C₄⋊C₈	—	2	1	no	—
182	C₄⋊C₈	—	2	1	no	—
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	—	2	1	no	—
185	C₂²⋊C₈	—	2	1	no	—
186	C₂²⋊C₈	—	2	1	no	—
187	C₂²⋊C₈	—	2	1	no	—
188	C₂²⋊C₈	—	2	1	no	—
189	#97	—	2	1	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
192	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
193	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
194	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
195	C₃₁	#11464	1	1	yes	characteristic, direct product factor, Sylow
196	#41	—	4	1	no	—
197	#41	—	4	1	no	—
198	C₂×Q₈	C₂×C₈×C₃₁	1	1	yes	characteristic
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	C₂×C₈×C₃₁	1	1	yes	characteristic
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
205	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
207	M₁₆	—	4	1	no	—
208	M₁₆	—	4	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
214	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
215	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
216	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
217	C₄⋊C₄	—	2	1	no	—
218	C₄⋊C₄	—	2	1	no	—
219	C₄⋊C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
220	C₄⋊C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
221	C₂²⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	1	no	—
227	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
228	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
229	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
230	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
231	C₁₆	—	4	2	no	—
232	C₁₆	—	4	2	no	—
233	C₂³	—	2	1	no	—
234	C₂³	C₂×M₁₆×C₃₁	1	1	yes	characteristic
235	Q₈	—	4	1	no	—
236	D₄	—	4	1	no	—
237	D₄	—	4	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
250	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
251	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
252	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
253	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	4	1	no	—
260	C₈	—	4	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	C₃₁×#350	1	2	yes	—
276	C₄	C₃₁×#410	1	1	yes	characteristic
277	C₄	C₃₁×#413	1	1	yes	characteristic
278	C₂	—	4	1	no	—
279	C₂	—	8	1	no	—
280	C₂	C₃₁×#1458	1	1	yes	characteristic
281	C₂	C₂×C₃₁×#350	1	1	yes	characteristic
282	C₂	C₃₁×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₃₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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×#1323	C₂	semidirect
91. #11464	C₃₁	direct (N × H)
121. #2163	C₂×C₃₁	semidirect
122. #2162	C₂×C₃₁	semidirect
123. #1323	C₂×C₃₁	semidirect
195. C₃₁	#11464	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×#2922	2
C₃₁×#2163	2
C₃₁×#2162	2
C₃₁×#1323	2
C₃₁×#1315	2
#11464	31

Subgroups — C31×#11464 #21468223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11464 #21468223