Subgroups — #89618

Order 448.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₂₈	C₂	1	1	yes	maximal, characteristic
2	#6432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#6395	C₂	1	2	yes	maximal
5	#6392	C₂	1	2	yes	maximal, complemented
6	C₂×D₂₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₇	—	2	1	no	—
8	C₂×Dic₁₄	—	2	1	no	—
9	C₂×C₇×C₈	C₂²	1	2	yes	—
10	C₄²×C₇	C₂²	1	1	yes	characteristic
11	(C₂×D₇)⋊C₄	—	2	1	no	—
12	C₄⋊Dic₇	—	2	1	no	—
13	C₄⋊Dic₇	C₂²	1	2	yes	—
14	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
15	#474	—	7	1	no	maximal, Sylow
16	C₂²×D₇	—	2	1	no	—
17	C₂×C₄×C₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₇	D₄	1	1	yes	characteristic
19	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂×Dic₇	—	2	2	no	—
21	C₂×Dic₇	—	2	1	no	—
22	D₂₈	—	2	1	no	—
23	D₂₈	Q₈	1	2	yes	complemented
24	C₄×D₇	—	4	2	no	—
25	Dic₁₄	—	4	2	no	—
26	C₇×C₈	—	2	2	no	—
27	C₄⋊Q₈	—	7	1	no	—
28	C₄×D₄	—	7	1	no	—
29	#107	—	7	2	no	—
30	C₄⋊C₈	D₇	1	1	yes	characteristic, complemented
31	#102	—	7	2	no	—
32	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
33	C₂×D₇	—	2	2	no	—
34	C₂×D₇	—	4	1	no	—
35	C₂×D₇	—	4	1	no	—
36	C₄×C₇	—	2	1	no	—
37	C₄×C₇	D₈	1	2	yes	—
38	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
39	C₄×C₇	#41	1	1	yes	characteristic
40	Dic₇	—	4	2	no	—
41	Dic₇	—	4	1	no	—
42	C₂×Q₈	—	14	1	no	—
43	C₂×D₄	—	7	1	no	—
44	C₂²×C₄	—	14	1	no	—
45	C₂×C₈	C₂×D₇	1	2	yes	—
46	C₄⋊C₄	—	7	2	no	—
47	C₄⋊C₄	—	7	1	no	—
48	C₄⋊C₄	—	14	1	no	—
49	C₂²⋊C₄	—	14	1	no	—
50	C₄²	C₂×D₇	1	1	yes	characteristic
51	C₂×C₇	C₂²⋊Q₈	1	1	yes	characteristic
52	C₂×C₇	C₂×D₈	1	1	yes	characteristic
53	C₂×C₇	#137	1	1	yes	characteristic
54	D₇	—	4	2	no	—
55	C₂³	—	14	1	no	—
56	Q₈	—	28	2	no	—
57	D₄	—	7	2	no	—
58	D₄	—	14	1	no	—
59	C₂×C₄	—	14	1	no	—
60	C₂×C₄	—	14	2	no	—
61	C₂×C₄	—	28	2	no	—
62	C₂×C₄	D₂₈	1	1	yes	characteristic
63	C₂×C₄	D₂₈	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
65	C₈	—	2	2	no	—
66	C₇	#474	1	1	yes	characteristic, complemented, Sylow
67	C₂²	—	14	2	no	—
68	C₂²	—	28	1	no	—
69	C₂²	—	28	1	no	—
70	C₂²	C₂×D₂₈	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	28	1	no	—
73	C₄	—	28	2	no	—
74	C₄	D₅₆	1	2	yes	—
75	C₄	#1159	1	1	yes	characteristic
76	C₄	Q₈×D₇	1	1	yes	characteristic
77	C₂	—	28	2	no	—
78	C₂	#6460	1	1	yes	characteristic
79	C₂	C₂×D₅₆	1	1	yes	characteristic
80	C₂	#6472	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₈
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×C₄⋊C₈	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×D₂₈	C₂²
hypercenter	C₄⋊C₈	D₇
hypocenter	C₇	#474
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #6432	C₂	semidirect
3. C₇×C₄⋊C₈	C₂	semidirect
5. #6392	C₂	semidirect
23. D₂₈	Q₈	semidirect
30. C₄⋊C₈	D₇	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×D₂₈	2
#6432	2
C₇×C₄⋊C₈	2
#6395	2
#6392	2
#474	7