Subgroups — #89274

Order 448.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×C₈	—	2	1	no	—
5	C₄²×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₇⋊C₈	—	2	1	no	—
7	C₂×C₇⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₇⋊C₈	C₄	1	1	yes	characteristic
9	#335	—	7	1	no	maximal, Sylow
10	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₄×C₇	D₄	1	1	yes	characteristic
12	C₂×C₄×C₇	Q₈	1	1	yes	characteristic
13	C₇×C₈	—	4	1	no	—
14	C₇⋊C₈	—	2	1	no	—
15	C₇⋊C₈	—	4	1	no	—
16	C₇⋊C₈	C₈	1	2	yes	complemented
17	C₄⋊C₈	—	7	1	no	—
18	C₄⋊C₈	D₇	1	1	yes	characteristic
19	C₄×C₈	—	7	1	no	—
20	C₂²×C₇	C₄⋊C₄	1	1	yes	characteristic
21	C₄×C₇	—	2	1	no	—
22	C₄×C₇	C₂×C₈	1	1	yes	characteristic
23	C₄×C₇	M₁₆	1	1	yes	characteristic
24	C₄×C₇	D₈	1	1	yes	characteristic
25	C₄×C₇	Q₁₆	1	1	yes	characteristic
26	C₂×C₈	—	2	1	no	—
27	C₂×C₈	—	7	1	no	—
28	C₂×C₈	—	7	1	no	—
29	C₂×C₈	—	14	1	no	—
30	C₄²	C₂×D₇	1	1	yes	characteristic
31	C₂×C₇	C₄⋊C₈	1	1	yes	characteristic
32	C₂×C₇	#107	1	1	yes	characteristic
33	C₂×C₇	#108	1	1	yes	characteristic
34	C₂×C₄	Dic₁₄	1	1	yes	characteristic
35	C₂×C₄	C₄×D₇	1	1	yes	characteristic
36	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
37	C₈	—	4	1	no	—
38	C₈	—	7	2	no	—
39	C₈	—	14	1	no	—
40	C₈	—	28	1	no	—
41	C₇	#335	1	1	yes	characteristic, complemented, Sylow
42	C₂²	Dic₇⋊C₄	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	C₈×D₇	1	1	yes	characteristic
45	C₄	#1131	1	1	yes	characteristic
46	C₄	#1141	1	1	yes	characteristic
47	C₄	#1144	1	1	yes	characteristic
48	C₂	#6381	1	1	yes	characteristic
49	C₂	Dic₇⋊C₈	1	1	yes	characteristic
50	C₂	#6396	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₇
commutator	C₄×C₇	C₂×C₈
Frattini	C₄²	C₂×D₇
Fitting	C₇×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₇	C₄⋊C₄
cosocle	C₄²×C₇	C₂²
hypercenter	C₄⋊C₈	D₇
hypocenter	C₇	#335
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
16. C₇⋊C₈	C₈	semidirect
41. C₇	#335	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₄⋊C₈	2
C₄⋊(C₇⋊C₈)	2
C₄×C₇⋊C₈	2
#335	7