Subgroups — #89897

Order 448.

102 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	C₂×#1136	C₂	1	2	yes	maximal
3	#6393	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
5	C₇×M₁₆	C₄	1	4	yes	—
6	C₂×C₇×C₈	C₂²	1	2	yes	—
7	#1136	—	2	2	no	—
8	#1136	C₂²	1	4	yes	—
9	C₂×C₇⋊C₈	—	2	2	no	—
10	#430	—	7	1	no	maximal, Sylow
11	C₂³×C₇	Q₈	1	1	yes	characteristic
12	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
13	C₂×C₄×C₇	D₄	1	2	yes	—
14	C₂×C₄×C₇	Q₈	1	1	yes	characteristic
15	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
16	C₇×C₈	C₂×C₄	1	4	yes	—
17	C₇⋊C₈	—	2	4	no	—
18	C₂×M₁₆	—	7	2	no	—
19	C₂×M₁₆	D₇	1	1	yes	characteristic
20	#108	—	7	4	no	—
21	C₂²×C₇	—	4	1	no	—
22	C₂²×C₇	C₄⋊C₄	1	2	yes	—
23	C₂²×C₇	C₂×Q₈	1	1	yes	characteristic
24	C₄×C₇	C₄⋊C₄	1	2	yes	—
25	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
26	C₄×C₇	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
28	M₁₆	—	7	4	no	—
29	M₁₆	—	14	2	no	—
30	M₁₆	Dic₇	1	4	yes	—
31	C₂×C₈	—	14	2	no	—
32	C₂×C₈	C₂×D₇	1	2	yes	—
33	C₂×C₇	—	2	2	no	—
34	C₂×C₇	—	2	1	no	—
35	C₂×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂³	Dic₁₄	1	1	yes	characteristic
37	C₂×C₄	Dic₁₄	1	1	yes	characteristic
38	C₂×C₄	D₂₈	1	2	yes	—
39	C₂×C₄	C₂×Dic₇	1	2	yes	—
40	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
41	C₈	—	14	4	no	—
42	C₈	C₂×Dic₇	1	4	yes	—
43	C₇	#430	1	1	yes	characteristic, complemented, Sylow
44	C₂²	—	4	1	no	—
45	C₂²	C₄⋊Dic₇	1	2	yes	—
46	C₂²	C₂×Dic₁₄	1	1	yes	characteristic
47	C₄	C₄⋊Dic₇	1	2	yes	—
48	C₄	C₂×D₂₈	1	1	yes	characteristic
49	C₄	C₂²×Dic₇	1	1	yes	characteristic
50	C₂	—	2	2	no	—
51	C₂	—	2	1	no	—
52	C₂	C₂×C₄⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₂₈
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂×C₇×M₁₆	C₂
socle	C₂×C₇	C₂×C₄⋊C₄
prime radical	C₂³×C₇	Q₈
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂×M₁₆	D₇
hypocenter	C₇	#430
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #6393	C₂	semidirect
43. C₇	#430	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
C₂×#1136	2
#6393	2
#430	7