Subgroups — C₂²×#11758400 #16858807

Order 3968.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758400	C₂	1	12	yes	maximal, direct product factor
2	C₂²×C₃₁⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂³×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×C₃₁	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₃₁	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₃₁	C₄	1	6	yes	—
7	C₂²×C₈×C₃₁	C₂²	1	6	yes	—
8	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	#11758400	C₂²	1	16	yes	direct product factor
10	C₂×C₃₁⋊C₁₆	C₂²	1	12	yes	complemented
11	C₂⁴×C₃₁	C₈	1	1	yes	characteristic
12	C₂²×C₄×C₃₁	C₈	1	6	yes	—
13	C₂²×C₄×C₃₁	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₃₁	C₂×C₄	1	6	yes	—
15	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₃₁	C₂×C₄	1	12	yes	—
17	C₂×C₈×C₃₁	C₂×C₄	1	6	yes	—
18	C₂×C₈×C₃₁	C₂×C₄	1	3	yes	—
19	C₂×C₈×C₃₁	C₂³	1	3	yes	—
20	C₂×C₈×C₃₁	C₂³	1	4	yes	—
21	C₃₁⋊C₁₆	C₂³	1	8	yes	complemented
22	C₂³×C₃₁	—	2	4	no	—
23	C₂³×C₃₁	C₂×C₈	1	6	yes	—
24	C₂³×C₃₁	C₂×C₈	1	1	yes	characteristic
25	C₂×C₄×C₃₁	C₂×C₈	1	12	yes	—
26	C₂×C₄×C₃₁	C₂×C₈	1	6	yes	—
27	C₂×C₄×C₃₁	C₂×C₈	1	3	yes	—
28	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
29	C₂×C₄×C₃₁	C₂²×C₄	1	3	yes	—
30	C₈×C₃₁	C₂²×C₄	1	3	yes	—
31	C₈×C₃₁	C₂²×C₄	1	4	yes	—
32	C₈×C₃₁	C₂⁴	1	1	yes	characteristic
33	C₂²×M₃₂	—	31	1	no	maximal, Sylow
34	C₂²×C₃₁	—	2	12	no	—
35	C₂²×C₃₁	M₃₂	1	4	yes	complemented
36	C₂²×C₃₁	C₂²×C₈	1	4	yes	—
37	C₂²×C₃₁	C₂²×C₈	1	3	yes	—
38	C₄×C₃₁	C₂²×C₈	1	3	yes	—
39	C₄×C₃₁	C₂²×C₈	1	4	yes	—
40	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
41	C₂³×C₈	D₃₁	1	1	yes	characteristic
42	C₂×M₃₂	—	31	12	no	—
43	C₂²×C₁₆	—	31	2	no	—
44	C₂×C₃₁	—	2	4	no	—
45	C₂×C₃₁	C₂×M₃₂	1	6	yes	complemented
46	C₂×C₃₁	C₂³×C₈	1	1	yes	characteristic
47	C₂³×C₄	Dic₃₁	1	1	yes	characteristic
48	C₂²×C₈	Dic₃₁	1	1	yes	characteristic
49	C₂²×C₈	Dic₃₁	1	6	yes	—
50	C₂²×C₈	C₂×D₃₁	1	6	yes	—
51	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
52	M₃₂	—	31	16	no	—
53	C₂×C₁₆	—	31	12	no	—
54	C₃₁	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
55	C₂⁴	C₃₁⋊C₈	1	1	yes	characteristic
56	C₂²×C₄	C₃₁⋊C₈	1	6	yes	—
57	C₂²×C₄	C₃₁⋊C₈	1	1	yes	characteristic
58	C₂²×C₄	C₂×Dic₃₁	1	6	yes	—
59	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
60	C₂×C₈	C₂×Dic₃₁	1	6	yes	—
61	C₂×C₈	C₂×Dic₃₁	1	12	yes	—
62	C₂×C₈	C₂×Dic₃₁	1	3	yes	—
63	C₂×C₈	C₂²×D₃₁	1	3	yes	—
64	C₂×C₈	C₂²×D₃₁	1	4	yes	—
65	C₁₆	—	31	8	no	—
66	C₂³	—	2	4	no	—
67	C₂³	C₂×C₃₁⋊C₈	1	6	yes	—
68	C₂³	C₂×C₃₁⋊C₈	1	1	yes	characteristic
69	C₂×C₄	C₂×C₃₁⋊C₈	1	12	yes	—
70	C₂×C₄	C₂×C₃₁⋊C₈	1	6	yes	—
71	C₂×C₄	C₂×C₃₁⋊C₈	1	3	yes	—
72	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
73	C₂×C₄	C₂²×Dic₃₁	1	3	yes	—
74	C₈	C₂²×Dic₃₁	1	4	yes	—
75	C₈	C₂²×Dic₃₁	1	3	yes	—
76	C₈	C₂³×D₃₁	1	1	yes	characteristic
77	C₂²	—	2	12	no	—
78	C₂²	#11758400	1	4	yes	direct product factor
79	C₂²	C₂²×C₃₁⋊C₈	1	4	yes	—
80	C₂²	C₂²×C₃₁⋊C₈	1	3	yes	—
81	C₄	C₂²×C₃₁⋊C₈	1	3	yes	—
82	C₄	C₂²×C₃₁⋊C₈	1	4	yes	—
83	C₄	C₂³×Dic₃₁	1	1	yes	characteristic
84	C₂	—	2	4	no	—
85	C₂	C₂×#11758400	1	6	yes	direct product factor
86	C₂	C₂³×C₃₁⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₃₁
commutator	C₂×C₃₁	C₂³×C₈
Frattini	C₈	C₂³×D₃₁
Fitting	C₂³×C₈×C₃₁	C₂
socle	C₂³×C₃₁	C₂×C₈
prime radical	C₂⁴×C₃₁	C₈
cosocle	C₈×C₃₁	C₂⁴
hypercenter	C₂³×C₈	D₃₁
hypocenter	C₃₁	C₂²×M₃₂
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#11758400	C₂	direct (N × H)
2. C₂²×C₃₁⋊C₁₆	C₂	semidirect
9. #11758400	C₂²	direct (N × H)
10. C₂×C₃₁⋊C₁₆	C₂²	semidirect
21. C₃₁⋊C₁₆	C₂³	semidirect
35. C₂²×C₃₁	M₃₂	semidirect
45. C₂×C₃₁	C₂×M₃₂	semidirect
54. C₃₁	C₂²×M₃₂	semidirect
78. C₂²	#11758400	direct (N × H)
85. C₂	C₂×#11758400	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×#11758400	2
C₂²×C₃₁⋊C₁₆	2
C₂³×C₈×C₃₁	2
C₂²×M₃₂	31

Subgroups — C22×#11758400 #16858807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11758400 #16858807