Subgroups — C₃₁×#9671 #21466430

Order 7936.

434 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1479	C₂	1	2	yes	maximal
4	C₃₁×#1461	C₂	1	2	yes	maximal
5	C₃₁×#1452	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₃₁	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
8	C₄×Q₁₆×C₃₁	—	2	1	no	—
9	C₄×Q₁₆×C₃₁	—	2	1	no	—
10	C₄×Q₁₆×C₃₁	—	2	1	no	—
11	C₄×Q₁₆×C₃₁	—	2	1	no	—
12	C₃₁×#431	C₂²	1	2	yes	—
13	C₃₁×#431	C₂²	1	1	yes	characteristic
14	C₃₁×#431	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#107	C₄	1	1	yes	characteristic
16	C₂×C₃₁×#103	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#335	—	2	1	no	—
19	C₃₁×#334	—	2	1	no	—
20	C₃₁×Q₈⋊C₈	—	2	4	no	—
21	C₃₁×#324	—	2	4	no	—
22	C₃₁×#322	—	2	1	no	—
23	C₃₁×#322	—	2	1	no	—
24	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
25	C₂×Q₁₆×C₃₁	—	2	4	no	—
26	C₂×Q₁₆×C₃₁	—	2	1	no	—
27	C₂×Q₁₆×C₃₁	—	2	1	no	—
28	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₄×Q₈×C₃₁	—	2	2	no	—
31	C₄×Q₈×C₃₁	—	2	2	no	—
32	C₄×Q₈×C₃₁	—	2	2	no	—
33	C₄×Q₈×C₃₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₃₁	—	2	2	no	—
37	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
38	C₃₁×#107	—	2	1	no	—
39	C₃₁×#107	—	2	1	no	—
40	C₃₁×C₄⋊C₈	—	2	2	no	—
41	C₃₁×C₄⋊C₈	—	2	1	no	—
42	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
43	C₃₁×#103	—	2	2	no	—
44	C₃₁×#103	—	2	2	no	—
45	C₃₁×C₂²⋊C₈	—	2	1	no	—
46	C₃₁×C₂²⋊C₈	—	2	2	no	—
47	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
48	C₄×C₈×C₃₁	—	2	1	no	—
49	C₄×C₈×C₃₁	—	2	1	no	—
50	C₄×C₈×C₃₁	—	2	2	no	—
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	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
55	C₂×Q₈×C₃₁	—	2	2	no	—
56	C₂×Q₈×C₃₁	—	2	2	no	—
57	C₂×Q₈×C₃₁	—	2	2	no	—
58	C₂×Q₈×C₃₁	—	2	2	no	—
59	C₂×Q₈×C₃₁	—	2	2	no	—
60	C₂×Q₈×C₃₁	M₁₆	1	4	yes	—
61	C₂²×C₄×C₃₁	—	2	2	no	—
62	C₂²×C₄×C₃₁	—	2	2	no	—
63	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
66	Q₁₆×C₃₁	—	2	4	no	—
67	Q₁₆×C₃₁	—	2	4	no	—
68	C₂×C₈×C₃₁	—	2	2	no	—
69	C₂×C₈×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	—	2	2	no	—
71	C₂×C₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	4	2	no	—
73	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₄⋊C₄×C₃₁	—	2	2	no	—
78	C₄⋊C₄×C₃₁	—	2	2	no	—
79	C₄⋊C₄×C₃₁	—	4	2	no	—
80	C₄⋊C₄×C₃₁	—	4	2	no	—
81	C₄²×C₃₁	—	4	2	no	—
82	C₄²×C₃₁	—	4	2	no	—
83	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₃₁	#41	1	1	yes	characteristic
87	#9671	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
89	Q₈×C₃₁	—	2	4	no	—
90	Q₈×C₃₁	—	2	4	no	—
91	Q₈×C₃₁	—	4	2	no	—
92	Q₈×C₃₁	—	4	2	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	4	no	—
95	C₂×C₄×C₃₁	—	4	1	no	—
96	C₂×C₄×C₃₁	—	4	2	no	—
97	C₂×C₄×C₃₁	—	4	2	no	—
98	C₂×C₄×C₃₁	—	4	2	no	—
99	C₂×C₄×C₃₁	—	8	2	no	—
100	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
103	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
104	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
105	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₃₁	#131	1	2	yes	—
108	C₈×C₃₁	—	2	1	no	—
109	C₈×C₃₁	—	2	1	no	—
110	C₈×C₃₁	—	2	1	no	—
111	C₈×C₃₁	—	2	1	no	—
112	C₈×C₃₁	—	4	2	no	—
113	C₈×C₃₁	—	4	2	no	—
114	C₈×C₃₁	—	8	2	no	—
115	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic
116	#1564	C₂×C₃₁	1	1	yes	characteristic
117	#1479	C₂×C₃₁	1	2	yes	—
118	#1461	C₂×C₃₁	1	2	yes	—
119	#1452	C₂×C₃₁	1	1	yes	characteristic
120	C₂²×C₃₁	—	2	1	no	—
121	C₂²×C₃₁	—	2	1	no	—
122	C₂²×C₃₁	#390	1	1	yes	characteristic
123	C₂²×C₃₁	#435	1	2	yes	—
124	C₄×C₃₁	—	2	1	no	—
125	C₄×C₃₁	—	2	1	no	—
126	C₄×C₃₁	—	2	1	no	—
127	C₄×C₃₁	—	4	4	no	—
128	C₄×C₃₁	—	8	2	no	—
129	C₄×C₃₁	#407	1	1	yes	characteristic
130	C₄×C₃₁	#408	1	1	yes	characteristic
131	C₄×C₃₁	#435	1	2	yes	—
132	C₄×C₃₁	#473	1	2	yes	—
133	C₂²×Q₁₆	C₄×C₃₁	1	1	yes	characteristic
134	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
135	C₄×Q₁₆	—	2	1	no	—
136	C₄×Q₁₆	—	2	1	no	—
137	C₄×Q₁₆	—	2	1	no	—
138	C₄×Q₁₆	—	2	1	no	—
139	#431	C₂²×C₃₁	1	2	yes	—
140	#431	C₂²×C₃₁	1	1	yes	characteristic
141	#431	C₂²×C₃₁	1	1	yes	characteristic
142	C₂×#107	C₄×C₃₁	1	1	yes	characteristic
143	C₂×#103	C₄×C₃₁	1	2	yes	—
144	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
145	#335	—	2	1	no	—
146	#334	—	2	1	no	—
147	Q₈⋊C₈	—	2	4	no	—
148	#324	—	2	4	no	—
149	#322	—	2	1	no	—
150	#322	—	2	1	no	—
151	C₂×C₃₁	—	2	2	no	—
152	C₂×C₃₁	#1955	1	1	yes	characteristic
153	C₂×C₃₁	#1976	1	1	yes	characteristic
154	C₂×C₃₁	#1979	1	1	yes	characteristic
155	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
156	C₂×Q₁₆	—	2	4	no	—
157	C₂×Q₁₆	—	2	1	no	—
158	C₂×Q₁₆	—	2	1	no	—
159	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
160	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
161	C₄×Q₈	—	2	2	no	—
162	C₄×Q₈	—	2	2	no	—
163	C₄×Q₈	—	2	2	no	—
164	C₄×Q₈	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
169	#107	—	2	1	no	—
170	#107	—	2	1	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	2	1	no	—
173	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
174	#103	—	2	2	no	—
175	#103	—	2	2	no	—
176	C₂²⋊C₈	—	2	1	no	—
177	C₂²⋊C₈	—	2	2	no	—
178	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	—	2	2	no	—
182	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
183	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
184	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
185	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
186	C₃₁	#9671	1	1	yes	characteristic, direct product factor, Sylow
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	M₁₆×C₃₁	1	4	yes	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
197	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
198	Q₁₆	—	2	4	no	—
199	Q₁₆	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄²	—	4	2	no	—
214	C₄²	—	4	2	no	—
215	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
216	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
217	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
218	C₄²	#41×C₃₁	1	1	yes	characteristic
219	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
220	Q₈	—	2	4	no	—
221	Q₈	—	2	4	no	—
222	Q₈	—	4	2	no	—
223	Q₈	—	4	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
232	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
233	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
234	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
235	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
236	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
237	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
238	C₂×C₄	C₃₁×#131	1	2	yes	—
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	4	2	no	—
244	C₈	—	4	2	no	—
245	C₈	—	8	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	C₃₁×#390	1	1	yes	characteristic
249	C₂²	C₃₁×#435	1	2	yes	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	2	no	—
255	C₄	C₃₁×#407	1	1	yes	characteristic
256	C₄	C₃₁×#408	1	1	yes	characteristic
257	C₄	C₃₁×#435	1	2	yes	—
258	C₄	C₃₁×#473	1	2	yes	—
259	C₂	—	2	2	no	—
260	C₂	C₃₁×#1955	1	1	yes	characteristic
261	C₂	C₃₁×#1976	1	1	yes	characteristic
262	C₂	C₃₁×#1979	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#390
prime radical	C₂³×C₃₁	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
87. #9671	C₃₁	direct (N × H)
186. C₃₁	#9671	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×C₄×Q₁₆×C₃₁	2
C₃₁×#1564	2
C₃₁×#1479	2
C₃₁×#1461	2
C₃₁×#1452	2
#9671	31

Subgroups — C31×#9671 #21466430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9671 #21466430