Subgroups — C₂²×#6386 #11741168

Order 896.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#6386	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂³×C₄×C₇	C₄	1	1	yes	characteristic
5	C₂²×C₇×C₈	C₄	1	1	yes	characteristic
6	C₂²×C₇×C₈	C₄	1	6	yes	—
7	C₂²×C₇×C₈	C₂²	1	6	yes	—
8	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
9	#6386	C₂²	1	16	yes	direct product factor
10	C₂×C₇⋊C₁₆	C₂²	1	12	yes	complemented
11	C₂²×M₃₂	—	7	1	no	maximal, Sylow
12	C₂⁴×C₇	C₈	1	1	yes	characteristic
13	C₂²×C₄×C₇	C₈	1	6	yes	—
14	C₂²×C₄×C₇	C₈	1	1	yes	characteristic
15	C₂²×C₄×C₇	C₂×C₄	1	6	yes	—
16	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₇×C₈	C₂×C₄	1	12	yes	—
18	C₂×C₇×C₈	C₂×C₄	1	6	yes	—
19	C₂×C₇×C₈	C₂×C₄	1	3	yes	—
20	C₂×C₇×C₈	C₂³	1	3	yes	—
21	C₂×C₇×C₈	C₂³	1	4	yes	—
22	C₇⋊C₁₆	C₂³	1	8	yes	complemented
23	C₂³×C₈	D₇	1	1	yes	characteristic
24	C₂×M₃₂	—	7	12	no	—
25	C₂²×C₁₆	—	7	2	no	—
26	C₂³×C₇	—	2	4	no	—
27	C₂³×C₇	C₂×C₈	1	6	yes	—
28	C₂³×C₇	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₇	C₂×C₈	1	12	yes	—
30	C₂×C₄×C₇	C₂×C₈	1	6	yes	—
31	C₂×C₄×C₇	C₂×C₈	1	3	yes	—
32	C₂×C₄×C₇	C₂²×C₄	1	4	yes	—
33	C₂×C₄×C₇	C₂²×C₄	1	3	yes	—
34	C₇×C₈	C₂²×C₄	1	3	yes	—
35	C₇×C₈	C₂²×C₄	1	4	yes	—
36	C₇×C₈	C₂⁴	1	1	yes	characteristic
37	C₂³×C₄	Dic₇	1	1	yes	characteristic
38	C₂²×C₈	Dic₇	1	1	yes	characteristic
39	C₂²×C₈	Dic₇	1	6	yes	—
40	C₂²×C₈	C₂×D₇	1	6	yes	—
41	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
42	M₃₂	—	7	16	no	—
43	C₂×C₁₆	—	7	12	no	—
44	C₂²×C₇	—	2	12	no	—
45	C₂²×C₇	M₃₂	1	4	yes	complemented
46	C₂²×C₇	C₂²×C₈	1	4	yes	—
47	C₂²×C₇	C₂²×C₈	1	3	yes	—
48	C₄×C₇	C₂²×C₈	1	3	yes	—
49	C₄×C₇	C₂²×C₈	1	4	yes	—
50	C₄×C₇	C₂³×C₄	1	1	yes	characteristic
51	C₂⁴	C₇⋊C₈	1	1	yes	characteristic
52	C₂²×C₄	C₇⋊C₈	1	6	yes	—
53	C₂²×C₄	C₇⋊C₈	1	1	yes	characteristic
54	C₂²×C₄	C₂×Dic₇	1	6	yes	—
55	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
56	C₂×C₈	C₂×Dic₇	1	6	yes	—
57	C₂×C₈	C₂×Dic₇	1	12	yes	—
58	C₂×C₈	C₂×Dic₇	1	3	yes	—
59	C₂×C₈	C₂²×D₇	1	3	yes	—
60	C₂×C₈	C₂²×D₇	1	4	yes	—
61	C₁₆	—	7	8	no	—
62	C₂×C₇	—	2	4	no	—
63	C₂×C₇	C₂×M₃₂	1	6	yes	complemented
64	C₂×C₇	C₂³×C₈	1	1	yes	characteristic
65	C₂³	—	2	4	no	—
66	C₂³	C₂×C₇⋊C₈	1	6	yes	—
67	C₂³	C₂×C₇⋊C₈	1	1	yes	characteristic
68	C₂×C₄	C₂×C₇⋊C₈	1	12	yes	—
69	C₂×C₄	C₂×C₇⋊C₈	1	6	yes	—
70	C₂×C₄	C₂×C₇⋊C₈	1	3	yes	—
71	C₂×C₄	C₂²×Dic₇	1	4	yes	—
72	C₂×C₄	C₂²×Dic₇	1	3	yes	—
73	C₈	C₂²×Dic₇	1	4	yes	—
74	C₈	C₂²×Dic₇	1	3	yes	—
75	C₈	C₂³×D₇	1	1	yes	characteristic
76	C₇	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
77	C₂²	—	2	12	no	—
78	C₂²	#6386	1	4	yes	direct product factor
79	C₂²	C₂²×C₇⋊C₈	1	4	yes	—
80	C₂²	C₂²×C₇⋊C₈	1	3	yes	—
81	C₄	C₂²×C₇⋊C₈	1	3	yes	—
82	C₄	C₂²×C₇⋊C₈	1	4	yes	—
83	C₄	C₂³×Dic₇	1	1	yes	characteristic
84	C₂	—	2	4	no	—
85	C₂	C₂×#6386	1	6	yes	direct product factor
86	C₂	C₂³×C₇⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₇
commutator	C₂×C₇	C₂³×C₈
Frattini	C₈	C₂³×D₇
Fitting	C₂³×C₇×C₈	C₂
socle	C₂³×C₇	C₂×C₈
prime radical	C₂⁴×C₇	C₈
cosocle	C₇×C₈	C₂⁴
hypercenter	C₂³×C₈	D₇
hypocenter	C₇	C₂²×M₃₂
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#6386	C₂	direct (N × H)
3. C₂²×C₇⋊C₁₆	C₂	semidirect
9. #6386	C₂²	direct (N × H)
10. C₂×C₇⋊C₁₆	C₂²	semidirect
22. C₇⋊C₁₆	C₂³	semidirect
45. C₂²×C₇	M₃₂	semidirect
63. C₂×C₇	C₂×M₃₂	semidirect
76. C₇	C₂²×M₃₂	semidirect
78. C₂²	#6386	direct (N × H)
85. C₂	C₂×#6386	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×C₇×C₈	2
C₂×#6386	2
C₂²×C₇⋊C₁₆	2
C₂²×M₃₂	7

Subgroups — C22×#6386 #11741168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#6386 #11741168