Subgroups — C₂²×C₈×Q₈×C₃₁ #21516944

Order 7936.

2478 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₃₁	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₃₁×C₄⋊C₈	C₂	1	3	yes	maximal
4	C₂²×C₄×C₈×C₃₁	C₂	1	3	yes	maximal
5	C₂³×Q₈×C₃₁	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₃₁	C₂²	1	3	yes	—
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₃₁	C₄	1	12	yes	—
9	C₂×C₄×Q₈×C₃₁	C₂²	1	12	yes	—
10	C₂²×C₄⋊C₄×C₃₁	C₄	1	3	yes	—
11	C₂²×C₄²×C₃₁	C₂²	1	3	yes	—
12	C₈×Q₈×C₃₁	C₂²	1	64	yes	direct product factor
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	36	yes	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	36	yes	—
15	C₂²×Q₈×C₃₁	C₈	1	16	yes	direct product factor
16	C₂²×Q₈×C₃₁	C₂×C₄	1	12	yes	—
17	C₂³×C₄×C₃₁	C₂×C₄	1	3	yes	—
18	C₂³×C₄×C₃₁	C₂×C₄	1	3	yes	—
19	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₃₁	—	2	12	no	—
21	C₂²×C₈×C₃₁	Q₈	1	8	yes	direct product factor
22	C₂²×C₈×C₃₁	C₂³	1	18	yes	—
23	C₂²×C₈×C₃₁	C₂³	1	6	yes	—
24	C₄×Q₈×C₃₁	C₂×C₄	1	48	yes	—
25	C₄×Q₈×C₃₁	C₂³	1	16	yes	—
26	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	36	yes	—
27	C₂×C₄²×C₃₁	C₂×C₄	1	18	yes	—
28	C₂×C₄²×C₃₁	C₂³	1	18	yes	—
29	C₃₁×C₄⋊C₈	C₂³	1	48	yes	—
30	C₄×C₈×C₃₁	C₂³	1	48	yes	—
31	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×Q₈×C₃₁	C₂×C₈	1	96	yes	direct product factor
33	C₂×Q₈×C₃₁	C₂²×C₄	1	16	yes	—
34	C₂²×C₄×C₃₁	—	2	12	no	—
35	C₂²×C₄×C₃₁	C₂×C₈	1	24	yes	—
36	C₂²×C₄×C₃₁	C₂²×C₄	1	18	yes	—
37	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
38	C₂²×C₄×C₃₁	C₂²×C₄	1	18	yes	—
39	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
40	C₂²×C₄×C₃₁	#41	1	4	yes	—
41	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
42	C₂×C₈×C₃₁	—	2	36	no	—
43	C₂×C₈×C₃₁	C₂×Q₈	1	24	yes	direct product factor
44	C₂×C₈×C₃₁	C₂⁴	1	12	yes	—
45	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
46	C₄⋊C₄×C₃₁	C₂²×C₄	1	48	yes	—
47	C₄²×C₃₁	C₂²×C₄	1	36	yes	—
48	C₄²×C₃₁	C₂⁴	1	12	yes	—
49	C₂²×C₈×Q₈	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₃₁	C₄×Q₈	1	4	yes	—
51	C₂³×C₃₁	C₂²×C₈	1	4	yes	—
52	C₂³×C₃₁	#131	1	4	yes	—
53	C₂³×C₃₁	C₂³×C₄	1	3	yes	—
54	Q₈×C₃₁	C₂²×C₈	1	64	yes	direct product factor
55	C₂×C₄×C₃₁	—	2	36	no	—
56	C₂×C₄×C₃₁	C₄×Q₈	1	12	yes	—
57	C₂×C₄×C₃₁	C₂²×C₈	1	72	yes	—
58	C₂×C₄×C₃₁	C₂³×C₄	1	12	yes	—
59	C₂×C₄×C₃₁	C₂³×C₄	1	3	yes	—
60	C₂×C₄×C₃₁	C₂³×C₄	1	12	yes	—
61	C₂×C₄×C₃₁	C₂²×Q₈	1	6	yes	—
62	C₂×C₄×C₃₁	C₂×#41	1	6	yes	—
63	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
64	C₈×C₃₁	—	2	12	no	—
65	C₈×C₃₁	C₂²×Q₈	1	8	yes	direct product factor
66	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic
67	C₂×C₈×Q₈	C₂×C₃₁	1	24	yes	direct product factor
68	C₂²×C₄⋊C₈	C₂×C₃₁	1	3	yes	—
69	C₂²×C₄×C₈	C₂×C₃₁	1	3	yes	—
70	C₂²×C₃₁	C₈×Q₈	1	16	yes	direct product factor
71	C₂²×C₃₁	C₂×C₄×Q₈	1	6	yes	—
72	C₂²×C₃₁	C₂³×C₈	1	6	yes	—
73	C₂²×C₃₁	C₂×#131	1	6	yes	—
74	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
75	C₄×C₃₁	—	2	12	no	—
76	C₄×C₃₁	C₂×C₄×Q₈	1	6	yes	—
77	C₄×C₃₁	C₂³×C₈	1	24	yes	—
78	C₄×C₃₁	C₂³×Q₈	1	1	yes	characteristic
79	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
80	C₂³×Q₈	C₄×C₃₁	1	1	yes	characteristic
81	C₂³×C₈	C₂²×C₃₁	1	3	yes	—
82	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
83	C₂×C₄×Q₈	C₄×C₃₁	1	12	yes	—
84	C₂×C₄×Q₈	C₂²×C₃₁	1	12	yes	—
85	C₂²×C₄⋊C₄	C₄×C₃₁	1	3	yes	—
86	C₂²×C₄²	C₂²×C₃₁	1	3	yes	—
87	C₈×Q₈	C₂²×C₃₁	1	64	yes	direct product factor
88	C₂×C₄⋊C₈	C₂²×C₃₁	1	36	yes	—
89	C₂×C₄×C₈	C₂²×C₃₁	1	36	yes	—
90	C₂×C₃₁	C₂×C₈×Q₈	1	12	yes	direct product factor
91	C₂×C₃₁	C₂²×C₄×Q₈	1	1	yes	characteristic
92	C₂×C₃₁	C₂⁴×C₈	1	1	yes	characteristic
93	C₂×C₃₁	C₂²×#131	1	1	yes	characteristic
94	C₂²×Q₈	C₈×C₃₁	1	16	yes	direct product factor
95	C₂²×Q₈	C₂×C₄×C₃₁	1	12	yes	—
96	C₂³×C₄	C₂×C₄×C₃₁	1	3	yes	—
97	C₂³×C₄	C₂×C₄×C₃₁	1	3	yes	—
98	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
99	C₂²×C₈	—	2	12	no	—
100	C₂²×C₈	Q₈×C₃₁	1	8	yes	direct product factor
101	C₂²×C₈	C₂³×C₃₁	1	18	yes	—
102	C₂²×C₈	C₂³×C₃₁	1	6	yes	—
103	C₄×Q₈	C₂×C₄×C₃₁	1	48	yes	—
104	C₄×Q₈	C₂³×C₃₁	1	16	yes	—
105	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	36	yes	—
106	C₂×C₄²	C₂×C₄×C₃₁	1	18	yes	—
107	C₂×C₄²	C₂³×C₃₁	1	18	yes	—
108	C₄⋊C₈	C₂³×C₃₁	1	48	yes	—
109	C₄×C₈	C₂³×C₃₁	1	48	yes	—
110	C₃₁	C₂²×C₈×Q₈	1	1	yes	characteristic, direct product factor, Sylow
111	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
112	C₂×Q₈	C₂×C₈×C₃₁	1	96	yes	direct product factor
113	C₂×Q₈	C₂²×C₄×C₃₁	1	16	yes	—
114	C₂²×C₄	—	2	12	no	—
115	C₂²×C₄	C₂×C₈×C₃₁	1	24	yes	—
116	C₂²×C₄	C₂²×C₄×C₃₁	1	18	yes	—
117	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
118	C₂²×C₄	C₂²×C₄×C₃₁	1	18	yes	—
119	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
120	C₂²×C₄	#41×C₃₁	1	4	yes	—
121	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
122	C₂×C₈	—	2	36	no	—
123	C₂×C₈	C₂×Q₈×C₃₁	1	24	yes	direct product factor
124	C₂×C₈	C₂⁴×C₃₁	1	12	yes	—
125	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
126	C₄⋊C₄	C₂²×C₄×C₃₁	1	48	yes	—
127	C₄²	C₂²×C₄×C₃₁	1	36	yes	—
128	C₄²	C₂⁴×C₃₁	1	12	yes	—
129	C₂³	C₄×Q₈×C₃₁	1	4	yes	—
130	C₂³	C₂²×C₈×C₃₁	1	4	yes	—
131	C₂³	C₃₁×#131	1	4	yes	—
132	C₂³	C₂³×C₄×C₃₁	1	3	yes	—
133	Q₈	C₂²×C₈×C₃₁	1	64	yes	direct product factor
134	C₂×C₄	—	2	36	no	—
135	C₂×C₄	C₄×Q₈×C₃₁	1	12	yes	—
136	C₂×C₄	C₂²×C₈×C₃₁	1	72	yes	—
137	C₂×C₄	C₂³×C₄×C₃₁	1	12	yes	—
138	C₂×C₄	C₂³×C₄×C₃₁	1	3	yes	—
139	C₂×C₄	C₂³×C₄×C₃₁	1	12	yes	—
140	C₂×C₄	C₂²×Q₈×C₃₁	1	6	yes	—
141	C₂×C₄	C₂×#41×C₃₁	1	6	yes	—
142	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
143	C₈	—	2	12	no	—
144	C₈	C₂²×Q₈×C₃₁	1	8	yes	direct product factor
145	C₂²	C₈×Q₈×C₃₁	1	16	yes	direct product factor
146	C₂²	C₂×C₄×Q₈×C₃₁	1	6	yes	—
147	C₂²	C₂³×C₈×C₃₁	1	6	yes	—
148	C₂²	C₂×C₃₁×#131	1	6	yes	—
149	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
150	C₄	—	2	12	no	—
151	C₄	C₂×C₄×Q₈×C₃₁	1	6	yes	—
152	C₄	C₂³×C₈×C₃₁	1	24	yes	—
153	C₄	C₂³×Q₈×C₃₁	1	1	yes	characteristic
154	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
155	C₂	C₂×C₈×Q₈×C₃₁	1	12	yes	direct product factor
156	C₂	C₂²×C₄×Q₈×C₃₁	1	1	yes	characteristic
157	C₂	C₂⁴×C₈×C₃₁	1	1	yes	characteristic
158	C₂	C₂²×C₃₁×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈×C₃₁	C₂²
commutator	C₂	C₂⁴×C₈×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle, prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Q₈×C₃₁	C₂	direct (N × H)
12. C₈×Q₈×C₃₁	C₂²	direct (N × H)
15. C₂²×Q₈×C₃₁	C₈	direct (N × H)
21. C₂²×C₈×C₃₁	Q₈	direct (N × H)
32. C₂×Q₈×C₃₁	C₂×C₈	direct (N × H)
43. C₂×C₈×C₃₁	C₂×Q₈	direct (N × H)
49. C₂²×C₈×Q₈	C₃₁	direct (N × H)
54. Q₈×C₃₁	C₂²×C₈	direct (N × H)
65. C₈×C₃₁	C₂²×Q₈	direct (N × H)
67. C₂×C₈×Q₈	C₂×C₃₁	direct (N × H)
70. C₂²×C₃₁	C₈×Q₈	direct (N × H)
87. C₈×Q₈	C₂²×C₃₁	direct (N × H)
90. C₂×C₃₁	C₂×C₈×Q₈	direct (N × H)
94. C₂²×Q₈	C₈×C₃₁	direct (N × H)
100. C₂²×C₈	Q₈×C₃₁	direct (N × H)
110. C₃₁	C₂²×C₈×Q₈	direct (N × H)
112. C₂×Q₈	C₂×C₈×C₃₁	direct (N × H)
123. C₂×C₈	C₂×Q₈×C₃₁	direct (N × H)
133. Q₈	C₂²×C₈×C₃₁	direct (N × H)
144. C₈	C₂²×Q₈×C₃₁	direct (N × H)
145. C₂²	C₈×Q₈×C₃₁	direct (N × H)
155. C₂	C₂×C₈×Q₈×C₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₃₁	2
C₂×C₈×Q₈×C₃₁	2
C₂²×C₃₁×C₄⋊C₈	2
C₂²×C₄×C₈×C₃₁	2
C₂²×C₈×Q₈	31

Subgroups — C22×C8×Q8×C31 #21516944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×Q₈×C₃₁ #21516944