Subgroups — C₂³×#92470 #16860431

Order 3968.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₁₂₄	C₂	1	1	yes	maximal, characteristic
3	C₂²×#92470	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₃₁	—	2	1	no	—
6	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₂³×Dic₃₁	—	2	1	no	—
8	C₂²×D₁₂₄	—	2	7	no	—
9	C₂²×D₁₂₄	C₂²	1	14	yes	—
10	C₂²×Dic₆₂	—	2	7	no	—
11	C₂²×Dic₆₂	C₂²	1	14	yes	complemented
12	C₂²×C₈×C₃₁	C₂²	1	14	yes	complemented
13	C₂×#92470	C₂²	1	112	yes	direct product factor
14	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
15	C₂³×D₃₁	—	2	14	no	—
16	C₂³×D₃₁	—	4	8	no	—
17	C₂²×C₄×C₃₁	D₄	1	7	yes	—
18	C₂²×C₄×C₃₁	C₂³	1	7	yes	—
19	C₂²×Dic₃₁	—	2	14	no	—
20	C₂×D₁₂₄	—	2	42	no	—
21	C₂×D₁₂₄	C₂³	1	28	yes	—
22	C₂×Dic₆₂	—	2	42	no	—
23	C₂×Dic₆₂	C₂³	1	28	yes	complemented
24	C₂×C₈×C₃₁	C₂³	1	28	yes	complemented
25	#92470	C₂³	1	64	yes	direct product factor
26	C₂³×C₃₁	SD₁₆	1	8	yes	complemented
27	C₂³×C₃₁	C₂×D₄	1	7	yes	—
28	C₂²×D₃₁	—	2	28	no	—
29	C₂²×D₃₁	—	4	56	no	—
30	C₂×C₄×C₃₁	C₂×D₄	1	21	yes	—
31	C₂×C₄×C₃₁	C₂⁴	1	7	yes	—
32	C₂×Dic₃₁	—	2	28	no	—
33	D₁₂₄	—	2	28	no	—
34	D₁₂₄	C₂⁴	1	8	yes	—
35	Dic₆₂	—	2	28	no	—
36	Dic₆₂	C₂⁴	1	8	yes	complemented
37	C₈×C₃₁	C₂⁴	1	8	yes	complemented
38	C₂³×SD₁₆	—	31	1	no	maximal, Sylow
39	C₂²×C₃₁	C₂×SD₁₆	1	28	yes	complemented
40	C₂²×C₃₁	C₂²×D₄	1	7	yes	—
41	C₂×D₃₁	—	2	8	no	—
42	C₂×D₃₁	—	4	56	no	—
43	C₄×C₃₁	C₂²×D₄	1	7	yes	—
44	C₄×C₃₁	C₂⁵	1	1	yes	characteristic
45	Dic₃₁	—	2	8	no	—
46	C₂³×Q₈	—	31	1	no	—
47	C₂³×D₄	—	31	1	no	—
48	C₂²×SD₁₆	—	31	28	no	—
49	C₂³×C₈	D₃₁	1	1	yes	characteristic, complemented
50	C₂×C₃₁	C₂²×SD₁₆	1	14	yes	complemented
51	C₂×C₃₁	C₂³×D₄	1	1	yes	characteristic
52	D₃₁	—	4	8	no	—
53	C₂⁵	—	62	1	no	—
54	C₂²×Q₈	—	31	14	no	—
55	C₂²×Q₈	—	62	7	no	—
56	C₂²×D₄	—	31	14	no	—
57	C₂²×D₄	—	62	7	no	—
58	C₂³×C₄	—	62	1	no	—
59	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
60	C₂×SD₁₆	—	31	112	no	—
61	C₂²×C₈	C₂×D₃₁	1	14	yes	complemented
62	C₃₁	C₂³×SD₁₆	1	1	yes	characteristic, complemented, Sylow
63	C₂⁴	—	62	14	no	—
64	C₂⁴	—	124	8	no	—
65	C₂⁴	D₁₂₄	1	1	yes	characteristic
66	C₂×Q₈	—	31	28	no	—
67	C₂×Q₈	—	62	42	no	—
68	C₂×D₄	—	31	28	no	—
69	C₂×D₄	—	62	42	no	—
70	C₂²×C₄	—	62	14	no	—
71	C₂²×C₄	D₁₂₄	1	7	yes	—
72	C₂²×C₄	C₂²×D₃₁	1	7	yes	—
73	SD₁₆	—	31	64	no	—
74	C₂×C₈	C₂²×D₃₁	1	28	yes	complemented
75	C₂³	—	62	28	no	—
76	C₂³	—	124	56	no	—
77	C₂³	#92470	1	8	yes	direct product factor
78	C₂³	C₂×D₁₂₄	1	7	yes	—
79	Q₈	—	31	8	no	—
80	Q₈	—	62	28	no	—
81	D₄	—	31	8	no	—
82	D₄	—	62	28	no	—
83	C₂×C₄	—	62	28	no	—
84	C₂×C₄	C₂×D₁₂₄	1	21	yes	—
85	C₂×C₄	C₂³×D₃₁	1	7	yes	—
86	C₈	C₂³×D₃₁	1	8	yes	complemented
87	C₂²	—	62	8	no	—
88	C₂²	—	124	56	no	—
89	C₂²	C₂×#92470	1	28	yes	direct product factor
90	C₂²	C₂²×D₁₂₄	1	7	yes	—
91	C₄	—	62	8	no	—
92	C₄	C₂²×D₁₂₄	1	7	yes	—
93	C₄	C₂⁴×D₃₁	1	1	yes	characteristic
94	C₂	—	124	8	no	—
95	C₂	C₂²×#92470	1	14	yes	direct product factor
96	C₂	C₂³×D₁₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₁₂₄
commutator, cosocle	C₄×C₃₁	C₂⁵
Frattini	C₄	C₂⁴×D₃₁
Fitting	C₂³×C₈×C₃₁	C₂
socle	C₂⁴×C₃₁	D₄
prime radical	C₂³×D₁₂₄	C₂
hypercenter	C₂³×C₈	D₃₁
hypocenter	C₃₁	C₂³×SD₁₆
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₆₂	C₂	semidirect
3. C₂²×#92470	C₂	direct (N × H)
4. C₂³×C₈×C₃₁	C₂	semidirect
11. C₂²×Dic₆₂	C₂²	semidirect
12. C₂²×C₈×C₃₁	C₂²	semidirect
13. C₂×#92470	C₂²	direct (N × H)
23. C₂×Dic₆₂	C₂³	semidirect
24. C₂×C₈×C₃₁	C₂³	semidirect
25. #92470	C₂³	direct (N × H)
26. C₂³×C₃₁	SD₁₆	semidirect
36. Dic₆₂	C₂⁴	semidirect
37. C₈×C₃₁	C₂⁴	semidirect
39. C₂²×C₃₁	C₂×SD₁₆	semidirect
49. C₂³×C₈	D₃₁	semidirect
50. C₂×C₃₁	C₂²×SD₁₆	semidirect
61. C₂²×C₈	C₂×D₃₁	semidirect
62. C₃₁	C₂³×SD₁₆	semidirect
74. C₂×C₈	C₂²×D₃₁	semidirect
77. C₂³	#92470	direct (N × H)
86. C₈	C₂³×D₃₁	semidirect
89. C₂²	C₂×#92470	direct (N × H)
95. C₂	C₂²×#92470	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
C₂³×D₁₂₄	2
C₂²×#92470	2
C₂³×C₈×C₃₁	2
C₂³×SD₁₆	31

Subgroups — C23×#92470 #16860431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#92470 #16860431