Subgroups — #89784

Order 448.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#1136	C₂	1	2	yes	maximal, complemented
3	C₂²⋊(C₇⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×D₄	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₇	C₂²	1	2	yes	—
8	C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
9	C₇×C₂²⋊C₄	C₄	1	2	yes	—
10	C₄²×C₇	C₂²	1	1	yes	characteristic
11	#1136	—	2	4	no	—
12	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₇⋊C₈	C₂²	1	2	yes	—
14	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
15	#436	—	7	1	no	maximal, Sylow
16	C₂³×C₇	C₂×C₄	1	2	yes	—
17	C₇×D₄	—	2	2	no	—
18	C₂×C₄×C₇	—	2	2	no	—
19	C₂×C₄×C₇	—	2	2	no	—
20	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₇⋊C₈	—	2	2	no	—
25	C₇⋊C₈	—	2	1	no	—
26	C₇⋊C₈	D₄	1	2	yes	complemented
27	C₂×M₁₆	—	7	2	no	—
28	C₄×D₄	D₇	1	1	yes	characteristic
29	C₄⋊C₈	—	7	1	no	—
30	C₂²⋊C₈	—	7	2	no	—
31	C₄×C₈	—	7	1	no	—
32	C₂²×C₇	—	2	2	no	—
33	C₂²×C₇	—	2	2	no	—
34	C₂²×C₇	—	2	2	no	—
35	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₇	—	2	2	no	—
37	C₄×C₇	—	2	1	no	—
38	C₄×C₇	M₁₆	1	2	yes	complemented
39	C₄×C₇	C₂×D₄	1	1	yes	characteristic
40	C₄×C₇	#41	1	1	yes	characteristic
41	C₂×D₄	Dic₇	1	1	yes	characteristic
42	C₂²×C₄	C₂×D₇	1	2	yes	—
43	M₁₆	—	14	4	no	—
44	C₂×C₈	—	7	1	no	—
45	C₂×C₈	—	7	1	no	—
46	C₂×C₈	—	7	2	no	—
47	C₄⋊C₄	Dic₇	1	1	yes	characteristic
48	C₂²⋊C₄	Dic₇	1	2	yes	—
49	C₄²	C₂×D₇	1	1	yes	characteristic
50	C₂×C₇	—	4	2	no	—
51	C₂×C₇	C₄×D₄	1	1	yes	characteristic
52	C₂×C₇	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₇	#131	1	1	yes	characteristic
54	C₂³	C₂×Dic₇	1	2	yes	—
55	D₄	—	2	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
59	C₂×C₄	C₂×Dic₇	1	2	yes	—
60	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
62	C₈	—	7	2	no	—
63	C₈	—	14	2	no	—
64	C₈	—	14	1	no	—
65	C₇	#436	1	1	yes	characteristic, complemented, Sylow
66	C₂²	—	2	2	no	—
67	C₂²	—	2	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	C₂²×Dic₇	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	#1136	1	2	yes	complemented
73	C₄	D₄×D₇	1	1	yes	characteristic
74	C₄	#1159	1	1	yes	characteristic
75	C₂	—	4	2	no	—
76	C₂	C₂×#1136	1	1	yes	characteristic
77	C₂	D₄×Dic₇	1	1	yes	characteristic
78	C₂	#6511	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator, socle	C₂²×C₇	C₂²×C₄
Fitting	C₄×C₇×D₄	C₂
prime radical	C₂×C₇×D₄	C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₄×D₄	D₇
hypocenter	C₇	#436
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×#1136	C₂	semidirect
3. C₂²⋊(C₇⋊C₈)	C₂	semidirect
4. C₄⋊(C₇⋊C₈)	C₂	semidirect
5. C₄×C₇⋊C₈	C₂	semidirect
26. C₇⋊C₈	D₄	semidirect
38. C₄×C₇	M₁₆	semidirect
65. C₇	#436	semidirect
72. C₄	#1136	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×D₄	2
C₂×#1136	2
C₂²⋊(C₇⋊C₈)	2
C₄⋊(C₇⋊C₈)	2
C₄×C₇⋊C₈	2
#436	7