Subgroups — #89348

Order 448.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#6386	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×C₇	C₄	1	1	yes	characteristic
4	C₇×M₁₆	—	2	2	no	—
5	C₂×C₇×C₈	C₄	1	1	yes	characteristic
6	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₇⋊C₁₆	—	2	2	no	—
8	#349	—	7	1	no	maximal, Sylow
9	C₂³×C₇	C₈	1	1	yes	characteristic
10	C₂×C₄×C₇	—	2	1	no	—
11	C₂×C₄×C₇	—	2	1	no	—
12	C₂×C₄×C₇	C₈	1	1	yes	characteristic
13	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
14	C₇×C₈	—	2	1	no	—
15	C₇×C₈	D₄	1	2	yes	—
16	C₂×M₁₆	D₇	1	1	yes	characteristic
17	M₃₂	—	7	2	no	—
18	C₂²×C₇	—	2	1	no	—
19	C₂²×C₇	—	4	1	no	—
20	C₂²×C₇	C₂×C₈	1	1	yes	characteristic
21	C₄×C₇	—	2	1	no	—
22	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₇	M₁₆	1	1	yes	characteristic
24	C₂²×C₄	Dic₇	1	1	yes	characteristic
25	M₁₆	—	2	2	no	—
26	C₂×C₈	Dic₇	1	1	yes	characteristic
27	C₂×C₈	C₂×D₇	1	1	yes	characteristic
28	C₁₆	—	14	2	no	—
29	C₂×C₇	—	2	1	no	—
30	C₂×C₇	—	4	1	no	—
31	C₂×C₇	C₂²⋊C₈	1	1	yes	characteristic
32	C₂³	C₇⋊C₈	1	1	yes	characteristic
33	C₂×C₄	—	2	1	no	—
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	C₇⋊C₈	1	1	yes	characteristic
36	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
37	C₈	—	2	1	no	—
38	C₈	C₂²⋊D₇	1	2	yes	—
39	C₇	#349	1	1	yes	characteristic, complemented, Sylow
40	C₂²	—	2	1	no	—
41	C₂²	—	4	1	no	—
42	C₂²	C₂×C₇⋊C₈	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	#1136	1	1	yes	characteristic
45	C₄	C₂²⋊Dic₇	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	4	1	no	—
48	C₂	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊Dic₇
commutator	C₂²×C₇	C₂×C₈
Frattini	C₂×C₈	C₂×D₇
Fitting	C₂×C₇×M₁₆	C₂
socle	C₂×C₇	C₂²⋊C₈
prime radical	C₂³×C₇	C₈
cosocle	C₂×C₇×C₈	C₂²
hypercenter	C₂×M₁₆	D₇
hypocenter	C₇	#349
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. #6386	C₂	semidirect
39. C₇	#349	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×M₁₆	2
#6386	2
#349	7