Subgroups — #89280

Order 448.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6437	C₂	1	1	yes	maximal, characteristic
2	C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₂₈	—	2	1	no	—
5	C₂×D₂₈	—	2	1	no	—
6	C₂×D₂₈	C₄	1	2	yes	—
7	C₂×C₇×C₈	—	2	1	no	—
8	C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₇⋊C₈	—	2	1	no	—
10	#331	—	7	1	no	maximal, Sylow
11	C₂²×D₇	—	2	2	no	—
12	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₇	D₄	1	1	yes	characteristic
14	C₂×C₄×C₇	D₄	1	1	yes	characteristic
15	D₂₈	—	4	2	no	—
16	D₂₈	—	4	1	no	—
17	D₂₈	—	4	1	no	—
18	D₂₈	—	4	1	no	—
19	D₂₈	—	4	1	no	—
20	C₇×C₈	—	4	1	no	—
21	C₇⋊C₈	—	4	1	no	—
22	#127	—	7	1	no	—
23	C₄⋊C₈	—	7	1	no	—
24	C₄⋊C₈	D₇	1	1	yes	characteristic, complemented
25	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×D₇	—	4	2	no	—
27	C₂×D₇	—	4	2	no	—
28	C₂×D₇	—	4	2	no	—
29	C₄×C₇	—	2	1	no	—
30	C₄×C₇	D₈	1	1	yes	characteristic
31	C₄×C₇	D₈	1	1	yes	characteristic
32	C₄×C₇	SD₁₆	1	1	yes	characteristic
33	C₄×C₇	SD₁₆	1	1	yes	characteristic
34	C₂×D₄	—	7	2	no	—
35	C₂×D₄	—	14	1	no	—
36	C₂×D₄	—	14	1	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	14	1	no	—
39	C₄²	C₂×D₇	1	1	yes	characteristic
40	C₂×C₇	#100	1	1	yes	characteristic
41	C₂×C₇	#102	1	1	yes	characteristic
42	C₂×C₇	#102	1	1	yes	characteristic
43	D₇	—	8	2	no	—
44	C₂³	—	14	2	no	—
45	D₄	—	28	2	no	—
46	D₄	—	28	1	no	—
47	D₄	—	28	1	no	—
48	D₄	—	28	1	no	—
49	D₄	—	28	1	no	—
50	C₂×C₄	C₄×D₇	1	1	yes	characteristic
51	C₂×C₄	D₂₈	1	1	yes	characteristic
52	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
53	C₈	—	4	1	no	—
54	C₈	—	28	1	no	—
55	C₇	#331	1	1	yes	characteristic, complemented, Sylow
56	C₂²	—	28	2	no	—
57	C₂²	—	28	2	no	—
58	C₂²	—	28	2	no	—
59	C₂²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	#1132	1	1	yes	characteristic
62	C₄	D₅₆	1	1	yes	characteristic
63	C₄	#1141	1	1	yes	characteristic
64	C₄	#1143	1	1	yes	characteristic
65	C₂	—	56	2	no	—
66	C₂	#6383	1	1	yes	characteristic
67	C₂	#6395	1	1	yes	characteristic
68	C₂	#6397	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₇)⋊C₄
commutator	C₂×C₄×C₇	C₂×C₄
Frattini	C₄²	C₂×D₇
Fitting	C₇×C₄⋊C₈	C₂
socle	C₂²×C₇	C₂²⋊C₄
prime radical	#6437	C₂
cosocle	C₄²×C₇	C₂²
hypercenter	C₄⋊C₈	D₇
hypocenter	C₇	#331
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×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. C₇×C₄⋊C₈	C₂	semidirect
3. C₄⋊(C₇⋊C₈)	C₂	semidirect
24. C₄⋊C₈	D₇	semidirect
55. C₇	#331	semidirect

Maximal subgroups

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

subgroup	index
#6437	2
C₇×C₄⋊C₈	2
C₄⋊(C₇⋊C₈)	2
#331	7