Subgroups — C₃₁×#9641 #21466400

Order 7936.

530 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×Q₈⋊C₈	C₂	1	4	yes	maximal
5	C₂×C₃₁×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
7	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
9	C₃₁×#391	C₄	1	2	yes	—
10	C₃₁×#389	—	2	2	no	—
11	C₃₁×#384	C₄	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
13	C₄³×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#336	—	2	2	no	—
15	C₃₁×#336	—	2	2	no	—
16	C₃₁×Q₈⋊C₈	—	2	8	no	—
17	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₃₁	—	2	2	no	—
19	C₂²×C₈×C₃₁	—	2	2	no	—
20	C₃₁×C₄⋊Q₈	—	2	2	no	—
21	C₃₁×C₄⋊Q₈	C₈	1	4	yes	complemented
22	C₄×Q₈×C₃₁	—	2	4	no	—
23	C₄×Q₈×C₃₁	—	2	4	no	—
24	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₃₁	—	2	2	no	—
29	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₃₁	D₄	1	2	yes	—
32	C₂×C₄²×C₃₁	D₄	1	2	yes	—
33	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₃₁×C₄⋊C₈	—	2	4	no	—
35	C₄×C₈×C₃₁	—	2	2	no	—
36	C₄×C₈×C₃₁	D₄	1	4	yes	—
37	C₃₁×#95	—	2	4	no	—
38	C₂×Q₈×C₃₁	—	2	2	no	—
39	C₂×Q₈×C₃₁	—	2	4	no	—
40	C₂×Q₈×C₃₁	—	2	4	no	—
41	C₂×Q₈×C₃₁	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₃₁	—	2	2	no	—
43	C₂²×C₄×C₃₁	—	2	2	no	—
44	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₃₁	—	2	4	no	—
50	C₂×C₈×C₃₁	—	4	4	no	—
51	C₂×C₈×C₃₁	—	4	2	no	—
52	C₂×C₈×C₃₁	—	4	4	no	—
53	C₄⋊C₄×C₃₁	—	2	8	no	—
54	C₄⋊C₄×C₃₁	—	2	4	no	—
55	C₄⋊C₄×C₃₁	—	4	4	no	—
56	C₄⋊C₄×C₃₁	—	4	4	no	—
57	C₄²×C₃₁	—	2	2	no	—
58	C₄²×C₃₁	—	2	4	no	—
59	C₄²×C₃₁	—	4	2	no	—
60	C₄²×C₃₁	—	4	2	no	—
61	C₄²×C₃₁	—	4	2	no	—
62	C₄²×C₃₁	C₂×C₈	1	2	yes	—
63	C₄²×C₃₁	M₁₆	1	2	yes	—
64	C₄²×C₃₁	C₂×D₄	1	2	yes	—
65	C₄²×C₃₁	#41	1	2	yes	—
66	#9641	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	Q₈×C₃₁	—	2	8	no	—
69	Q₈×C₃₁	—	4	4	no	—
70	C₂×C₄×C₃₁	—	2	2	no	—
71	C₂×C₄×C₃₁	—	2	2	no	—
72	C₂×C₄×C₃₁	—	2	2	no	—
73	C₂×C₄×C₃₁	—	2	4	no	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	—	2	2	no	—
76	C₂×C₄×C₃₁	—	2	2	no	—
77	C₂×C₄×C₃₁	—	4	2	no	—
78	C₂×C₄×C₃₁	—	4	2	no	—
79	C₂×C₄×C₃₁	—	4	2	no	—
80	C₂×C₄×C₃₁	—	4	4	no	—
81	C₂×C₄×C₃₁	C₂²⋊C₈	1	4	yes	—
82	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₃₁	#121	1	2	yes	—
84	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
85	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₃₁	#131	1	2	yes	—
89	C₂×C₄×C₃₁	#137	1	4	yes	—
90	C₈×C₃₁	—	4	4	no	—
91	C₈×C₃₁	—	8	2	no	—
92	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
93	#1831	C₂×C₃₁	1	1	yes	characteristic
94	#1744	C₂×C₃₁	1	1	yes	characteristic
95	C₂×Q₈⋊C₈	C₂×C₃₁	1	4	yes	—
96	C₂²×C₃₁	#390	1	1	yes	characteristic
97	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
98	C₂²×C₃₁	#408	1	1	yes	characteristic
99	C₂²×C₃₁	#419	1	2	yes	—
100	C₂²×C₃₁	#421	1	2	yes	—
101	C₄×C₃₁	—	2	2	no	—
102	C₄×C₃₁	—	2	4	no	—
103	C₄×C₃₁	—	4	4	no	—
104	C₄×C₃₁	—	4	2	no	—
105	C₄×C₃₁	—	8	2	no	—
106	C₄×C₃₁	C₈×D₄	1	2	yes	—
107	C₄×C₃₁	#436	1	2	yes	—
108	C₄×C₃₁	#491	1	2	yes	—
109	C₄×C₃₁	#497	1	2	yes	—
110	C₂×C₄⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
111	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
112	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
113	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
114	#391	C₄×C₃₁	1	2	yes	—
115	#389	—	2	2	no	—
116	#384	C₄×C₃₁	1	1	yes	characteristic
117	C₄×C₄⋊C₄	—	2	2	no	—
118	C₄³	C₂²×C₃₁	1	1	yes	characteristic
119	#336	—	2	2	no	—
120	#336	—	2	2	no	—
121	Q₈⋊C₈	—	2	8	no	—
122	C₂×C₃₁	#1479	1	2	yes	—
123	C₂×C₃₁	#1497	1	2	yes	—
124	C₂×C₃₁	#1954	1	1	yes	characteristic
125	C₂×C₃₁	#1960	1	1	yes	characteristic
126	C₂×C₃₁	#1963	1	1	yes	characteristic
127	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	C₈×C₃₁	1	4	yes	complemented
132	C₄×Q₈	—	2	4	no	—
133	C₄×Q₈	—	2	4	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
138	C₂×C₄²	—	2	2	no	—
139	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
140	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
141	C₂×C₄²	D₄×C₃₁	1	2	yes	—
142	C₂×C₄²	D₄×C₃₁	1	2	yes	—
143	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
144	C₄⋊C₈	—	2	4	no	—
145	C₄×C₈	—	2	2	no	—
146	C₄×C₈	D₄×C₃₁	1	4	yes	—
147	#95	—	2	4	no	—
148	C₃₁	#9641	1	1	yes	characteristic, direct product factor, Sylow
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×Q₈	C₂×C₈×C₃₁	1	4	yes	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
156	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
157	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
159	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	4	4	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	4	no	—
164	C₄⋊C₄	—	2	8	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	4	4	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	4	no	—
170	C₄²	—	4	2	no	—
171	C₄²	—	4	2	no	—
172	C₄²	—	4	2	no	—
173	C₄²	C₂×C₈×C₃₁	1	2	yes	—
174	C₄²	M₁₆×C₃₁	1	2	yes	—
175	C₄²	C₂×D₄×C₃₁	1	2	yes	—
176	C₄²	#41×C₃₁	1	2	yes	—
177	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
178	Q₈	—	2	8	no	—
179	Q₈	—	4	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	C₃₁×C₂²⋊C₈	1	4	yes	—
192	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
193	C₂×C₄	C₃₁×#121	1	2	yes	—
194	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
195	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
196	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
197	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
198	C₂×C₄	C₃₁×#131	1	2	yes	—
199	C₂×C₄	C₃₁×#137	1	4	yes	—
200	C₈	—	4	4	no	—
201	C₈	—	8	2	no	—
202	C₂²	C₃₁×#390	1	1	yes	characteristic
203	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
204	C₂²	C₃₁×#408	1	1	yes	characteristic
205	C₂²	C₃₁×#419	1	2	yes	—
206	C₂²	C₃₁×#421	1	2	yes	—
207	C₄	—	2	2	no	—
208	C₄	—	2	4	no	—
209	C₄	—	4	4	no	—
210	C₄	—	4	2	no	—
211	C₄	—	8	2	no	—
212	C₄	C₈×D₄×C₃₁	1	2	yes	—
213	C₄	C₃₁×#436	1	2	yes	—
214	C₄	C₃₁×#491	1	2	yes	—
215	C₄	C₃₁×#497	1	2	yes	—
216	C₂	C₃₁×#1479	1	2	yes	—
217	C₂	C₃₁×#1497	1	2	yes	—
218	C₂	C₃₁×#1954	1	1	yes	characteristic
219	C₂	C₃₁×#1960	1	1	yes	characteristic
220	C₂	C₃₁×#1963	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
21. C₃₁×C₄⋊Q₈	C₈	semidirect
66. #9641	C₃₁	direct (N × H)
131. C₄⋊Q₈	C₈×C₃₁	semidirect
148. C₃₁	#9641	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×C₄⋊Q₈	2
C₃₁×#1831	2
C₃₁×#1744	2
C₂×C₃₁×Q₈⋊C₈	2
#9641	31

Subgroups — C31×#9641 #21466400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9641 #21466400