Subgroups — #89779

Order 448.

108 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	#6432	C₂	1	1	yes	maximal, characteristic, complemented
3	#6406	C₂	1	2	yes	maximal
4	#6381	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₇	—	2	1	no	—
8	C₂×Dic₁₄	—	2	1	no	—
9	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₇×C₂²⋊C₄	—	2	1	no	—
11	C₄²×C₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₇	—	2	1	no	—
13	C₄⋊Dic₇	C₂²	1	2	yes	—
14	C₂×C₇⋊C₈	C₂²	1	2	yes	—
15	#474	—	7	1	no	maximal, Sylow
16	C₂³×C₇	—	2	1	no	—
17	C₇×D₄	—	2	1	no	—
18	C₇×D₄	Q₈	1	2	yes	complemented
19	C₂×C₄×C₇	—	2	1	no	—
20	C₂×C₄×C₇	—	4	2	no	—
21	C₂×C₄×C₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂×Dic₇	—	2	2	no	—
25	Dic₁₄	—	4	2	no	—
26	C₇⋊C₈	—	2	2	no	—
27	C₄⋊Q₈	—	7	1	no	—
28	C₄×D₄	D₇	1	1	yes	characteristic
29	#107	—	7	2	no	—
30	C₄⋊C₈	—	7	1	no	—
31	#102	—	7	2	no	—
32	C₂²×C₇	—	2	2	no	—
33	C₂²×C₇	—	4	1	no	—
34	C₂²×C₇	—	4	1	no	—
35	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
36	C₄×C₇	—	2	1	no	—
37	C₄×C₇	—	4	1	no	—
38	C₄×C₇	D₈	1	2	yes	—
39	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
40	C₄×C₇	#41	1	1	yes	characteristic
41	Dic₇	—	4	2	no	—
42	C₂×Q₈	—	14	1	no	—
43	C₂×D₄	C₂×D₇	1	1	yes	characteristic
44	C₂²×C₄	—	2	1	no	—
45	C₂×C₈	—	7	2	no	—
46	C₄⋊C₄	—	7	2	no	—
47	C₄⋊C₄	—	14	1	no	—
48	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
49	C₂²⋊C₄	—	2	1	no	—
50	C₄²	C₂×D₇	1	1	yes	characteristic
51	C₂×C₇	—	4	2	no	—
52	C₂×C₇	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂×C₇	C₂×D₈	1	1	yes	characteristic
54	C₂×C₇	#137	1	1	yes	characteristic
55	C₂³	—	2	1	no	—
56	Q₈	—	28	2	no	—
57	D₄	—	2	1	no	—
58	D₄	Dic₁₄	1	2	yes	complemented
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	14	2	no	—
62	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
65	C₈	—	14	2	no	—
66	C₇	#474	1	1	yes	characteristic, complemented, Sylow
67	C₂²	—	2	2	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	28	2	no	—
74	C₄	#1141	1	2	yes	—
75	C₄	C₂×Dic₁₄	1	1	yes	characteristic
76	C₄	#1157	1	1	yes	characteristic
77	C₂	—	4	2	no	—
78	C₂	#6487	1	1	yes	characteristic
79	C₂	C₂×#1141	1	1	yes	characteristic
80	C₂	#6514	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₄×C₇×D₄	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×C₇×D₄	C₂²
hypercenter	C₄×D₄	D₇
hypocenter	C₇	#474
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #6432	C₂	semidirect
4. #6381	C₂	semidirect
5. C₄⋊(C₇⋊C₈)	C₂	semidirect
18. C₇×D₄	Q₈	semidirect
58. D₄	Dic₁₄	semidirect
66. C₇	#474	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
#6432	2
#6406	2
#6381	2
C₄⋊(C₇⋊C₈)	2
#474	7