Subgroups — #89347

Order 448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#1136	C₂	1	2	yes	maximal
3	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
4	C₂×C₇×C₈	—	2	2	no	—
5	C₂×C₇×C₈	C₄	1	2	yes	—
6	#1136	—	2	2	no	—
7	#1136	C₄	1	4	yes	—
8	C₂×C₇⋊C₈	—	2	2	no	—
9	#341	—	7	1	no	maximal, Sylow
10	C₂³×C₇	Q₈	1	1	yes	characteristic
11	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
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	2	yes	—
15	C₇×C₈	—	2	2	no	—
16	C₇⋊C₈	—	2	4	no	—
17	C₂×M₁₆	—	7	2	no	—
18	C₂²×C₈	D₇	1	1	yes	characteristic
19	C₂²×C₇	—	2	2	no	—
20	C₂²×C₇	C₄⋊C₄	1	2	yes	—
21	C₂²×C₇	C₄⋊C₄	1	1	yes	characteristic
22	C₄×C₇	C₄²	1	1	yes	characteristic
23	C₄×C₇	C₂²⋊C₄	1	2	yes	—
24	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
26	M₁₆	—	7	4	no	—
27	M₁₆	—	14	2	no	—
28	C₂×C₈	—	2	2	no	—
29	C₂×C₈	—	14	2	no	—
30	C₂×C₈	Dic₇	1	2	yes	—
31	C₂×C₇	—	2	2	no	—
32	C₂×C₇	#95	1	1	yes	characteristic
33	C₂×C₇	#108	1	2	yes	—
34	C₂³	Dic₁₄	1	1	yes	characteristic
35	C₂×C₄	C₄×D₇	1	2	yes	—
36	C₂×C₄	D₂₈	1	1	yes	characteristic
37	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
38	C₂×C₄	C₂²⋊D₇	1	2	yes	—
39	C₈	—	2	2	no	—
40	C₈	—	14	4	no	—
41	C₇	#341	1	1	yes	characteristic, complemented, Sylow
42	C₂²	—	2	2	no	—
43	C₂²	Dic₇⋊C₄	1	2	yes	—
44	C₂²	C₄⋊Dic₇	1	1	yes	characteristic
45	C₄	C₄×Dic₇	1	1	yes	characteristic
46	C₄	(C₂×D₇)⋊C₄	1	2	yes	—
47	C₄	C₂²⋊Dic₇	1	1	yes	characteristic
48	C₂	—	2	2	no	—
49	C₂	#6393	1	2	yes	—
50	C₂	C₇⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₂₈
commutator	C₄×C₇	C₄²
Frattini	C₂²×C₄	C₂×D₇
Fitting	C₂²×C₇×C₈	C₂
socle	C₂²×C₇	C₄⋊C₄
prime radical	C₂³×C₇	Q₈
cosocle	C₂²×C₄×C₇	C₂²
hypercenter	C₂²×C₈	D₇
hypocenter	C₇	#341
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
41. C₇	#341	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₈	2
C₂×#1136	2
#341	7