Subgroups — C₂²×#11758446 #16858987

Order 3968.

1624 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₆₂	C₂	1	2	yes	maximal
2	C₂×#11758446	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊Dic₃₁	C₂	1	4	yes	maximal
4	C₂²×C₄²×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
6	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
7	C₂³×Dic₃₁	C₂²	1	4	yes	—
8	C₂²×Dic₆₂	—	2	16	no	—
9	C₂²×Dic₆₂	C₂²	1	24	yes	—
10	C₂×C₄²×C₃₁	C₂²	1	12	yes	—
11	C₂×C₄⋊Dic₃₁	C₂²	1	48	yes	—
12	#11758446	C₂²	1	64	yes	direct product factor
13	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
14	C₂²×C₄×C₃₁	D₄	1	8	yes	—
15	C₂²×C₄×C₃₁	Q₈	1	16	yes	complemented
16	C₂²×C₄×C₃₁	C₂³	1	6	yes	—
17	C₂²×C₄×C₃₁	C₂³	1	12	yes	—
18	C₂²×Dic₃₁	—	2	16	no	—
19	C₂²×Dic₃₁	C₂³	1	24	yes	—
20	C₂×Dic₆₂	—	2	96	no	—
21	C₂×Dic₆₂	C₂³	1	32	yes	—
22	C₄²×C₃₁	C₂³	1	16	yes	—
23	C₄⋊Dic₃₁	C₂³	1	64	yes	—
24	C₂³×C₃₁	C₂×D₄	1	4	yes	—
25	C₂³×C₃₁	C₂×Q₈	1	8	yes	—
26	C₂³×C₃₁	C₂⁴	1	3	yes	—
27	C₂×C₄×C₃₁	C₂×D₄	1	24	yes	—
28	C₂×C₄×C₃₁	C₂×Q₈	1	48	yes	complemented
29	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
30	C₂×C₄×C₃₁	C₂⁴	1	8	yes	—
31	C₂×Dic₃₁	—	2	48	no	—
32	C₂×Dic₃₁	C₂⁴	1	16	yes	—
33	Dic₆₂	—	2	64	no	—
34	C₂²×C₄⋊Q₈	—	31	1	no	maximal, Sylow
35	C₂²×C₃₁	C₄⋊Q₈	1	16	yes	complemented
36	C₂²×C₃₁	C₂²×D₄	1	6	yes	—
37	C₂²×C₃₁	C₂²×Q₈	1	12	yes	—
38	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
39	C₄×C₃₁	C₂²×D₄	1	8	yes	—
40	C₄×C₃₁	C₂²×Q₈	1	16	yes	complemented
41	Dic₃₁	—	2	16	no	—
42	C₂³×Q₈	—	31	2	no	—
43	C₂×C₄⋊Q₈	—	31	24	no	—
44	C₂²×C₄⋊C₄	—	31	4	no	—
45	C₂²×C₄²	D₃₁	1	1	yes	characteristic
46	C₂×C₃₁	C₂×C₄⋊Q₈	1	12	yes	complemented
47	C₂×C₃₁	C₂³×D₄	1	1	yes	characteristic
48	C₂×C₃₁	C₂³×Q₈	1	2	yes	—
49	C₂²×Q₈	—	31	24	no	—
50	C₂²×Q₈	—	62	16	no	—
51	C₂³×C₄	—	31	4	no	—
52	C₂³×C₄	C₂×D₃₁	1	2	yes	—
53	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
54	C₄⋊Q₈	—	31	64	no	—
55	C₂×C₄⋊C₄	—	31	48	no	—
56	C₂×C₄²	C₂×D₃₁	1	12	yes	—
57	C₃₁	C₂²×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
58	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
59	C₂×Q₈	—	31	32	no	—
60	C₂×Q₈	—	62	96	no	—
61	C₂²×C₄	—	31	24	no	—
62	C₂²×C₄	—	62	16	no	—
63	C₂²×C₄	Dic₆₂	1	16	yes	complemented
64	C₂²×C₄	D₁₂₄	1	8	yes	—
65	C₂²×C₄	C₂²×D₃₁	1	12	yes	—
66	C₂²×C₄	C₂²×D₃₁	1	6	yes	—
67	C₄⋊C₄	—	31	64	no	—
68	C₄²	C₂²×D₃₁	1	16	yes	—
69	C₂³	C₂×Dic₆₂	1	8	yes	—
70	C₂³	C₂×D₁₂₄	1	4	yes	—
71	C₂³	C₂³×D₃₁	1	3	yes	—
72	Q₈	—	62	64	no	—
73	C₂×C₄	—	31	16	no	—
74	C₂×C₄	—	62	48	no	—
75	C₂×C₄	C₂×Dic₆₂	1	48	yes	complemented
76	C₂×C₄	C₂×D₁₂₄	1	24	yes	—
77	C₂×C₄	C₂³×D₃₁	1	8	yes	—
78	C₂×C₄	C₂³×D₃₁	1	4	yes	—
79	C₂²	#11758446	1	16	yes	direct product factor
80	C₂²	C₂²×Dic₆₂	1	12	yes	—
81	C₂²	C₂²×D₁₂₄	1	6	yes	—
82	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
83	C₄	—	62	16	no	—
84	C₄	C₂²×Dic₆₂	1	16	yes	complemented
85	C₄	C₂²×D₁₂₄	1	8	yes	—
86	C₂	C₂×#11758446	1	12	yes	direct product factor
87	C₂	C₂³×D₁₂₄	1	1	yes	characteristic
88	C₂	C₂³×Dic₆₂	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₃₁
commutator, cosocle	C₂²×C₃₁	C₂⁵
Frattini	C₂²	C₂⁴×D₃₁
Fitting	C₂²×C₄²×C₃₁	C₂
socle, prime radical	C₂⁴×C₃₁	C₂³
hypercenter	C₂²×C₄²	D₃₁
hypocenter	C₃₁	C₂²×C₄⋊Q₈
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#11758446	C₂	direct (N × H)
12. #11758446	C₂²	direct (N × H)
15. C₂²×C₄×C₃₁	Q₈	semidirect
28. C₂×C₄×C₃₁	C₂×Q₈	semidirect
35. C₂²×C₃₁	C₄⋊Q₈	semidirect
40. C₄×C₃₁	C₂²×Q₈	semidirect
46. C₂×C₃₁	C₂×C₄⋊Q₈	semidirect
57. C₃₁	C₂²×C₄⋊Q₈	semidirect
63. C₂²×C₄	Dic₆₂	semidirect
75. C₂×C₄	C₂×Dic₆₂	semidirect
79. C₂²	#11758446	direct (N × H)
84. C₄	C₂²×Dic₆₂	semidirect
86. C₂	C₂×#11758446	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₆₂	2
C₂×#11758446	2
C₂²×C₄⋊Dic₃₁	2
C₂²×C₄²×C₃₁	2
C₂²×C₄⋊Q₈	31

Subgroups — C22×#11758446 #16858987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11758446 #16858987