Subgroups — #89673

Order 448.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6464	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#6386	C₂	1	1	yes	maximal, characteristic, complemented
4	#6372	C₂	1	2	yes	maximal, complemented
5	D₇×C₁₆	C₂	1	2	yes	maximal, complemented
6	#1157	C₄	1	1	yes	characteristic
7	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
8	#1136	C₄	1	1	yes	characteristic
9	#1131	C₄	1	2	yes	—
10	C₈×D₇	C₂²	1	2	yes	—
11	C₇×C₁₆	C₂²	1	1	yes	characteristic
12	C₇×C₁₆	C₂²	1	1	yes	characteristic
13	C₇⋊C₁₆	C₂²	1	2	yes	—
14	#504	—	7	1	no	maximal, Sylow
15	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
16	C₂²⋊D₇	C₈	1	2	yes	—
17	D₂₈	C₈	1	1	yes	characteristic
18	C₄×D₇	C₂×C₄	1	2	yes	—
19	Dic₁₄	C₈	1	1	yes	characteristic
20	C₇×C₈	C₂×C₄	1	1	yes	characteristic
21	C₇×C₈	C₂³	1	1	yes	characteristic
22	C₇⋊C₈	C₂×C₄	1	2	yes	—
23	#131	—	7	1	no	—
24	M₃₂	—	7	1	no	—
25	M₃₂	—	7	2	no	—
26	C₂×C₁₆	—	7	2	no	—
27	C₂×C₁₆	D₇	1	1	yes	characteristic, complemented
28	C₂²×C₇	C₂×C₈	1	1	yes	characteristic
29	C₂×D₇	C₂×C₈	1	2	yes	—
30	C₄×C₇	C₂×C₈	1	1	yes	characteristic
31	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
32	Dic₇	C₂×C₈	1	2	yes	—
33	#41	—	7	1	no	—
34	M₁₆	—	7	1	no	—
35	M₁₆	—	7	2	no	—
36	C₂×C₈	—	7	2	no	—
37	C₂×C₈	C₂×D₇	1	1	yes	characteristic
38	C₁₆	—	7	2	no	—
39	C₁₆	C₂×D₇	1	1	yes	characteristic
40	C₁₆	C₂×D₇	1	1	yes	characteristic
41	C₂×C₇	—	2	1	no	—
42	C₂×C₇	C₂²×C₈	1	1	yes	characteristic
43	D₇	—	2	2	no	—
44	Q₈	—	7	1	no	—
45	D₄	—	7	2	no	—
46	D₄	—	7	1	no	—
47	C₂×C₄	—	7	2	no	—
48	C₂×C₄	C₄×D₇	1	1	yes	characteristic
49	C₈	—	7	2	no	—
50	C₈	C₄×D₇	1	1	yes	characteristic
51	C₈	C₂²×D₇	1	1	yes	characteristic
52	C₇	#504	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	7	2	no	—
54	C₂²	C₈×D₇	1	1	yes	characteristic
55	C₄	—	7	2	no	—
56	C₄	C₈×D₇	1	1	yes	characteristic
57	C₄	C₂×C₄×D₇	1	1	yes	characteristic
58	C₂	—	2	1	no	—
59	C₂	—	14	2	no	—
60	C₂	C₂×C₈×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂×D₇
commutator, socle	C₂×C₇	C₂²×C₈
Frattini	C₈	C₂²×D₇
Fitting	C₂×C₇×C₁₆	C₂
prime radical	#1157	C₄
cosocle	C₇×C₈	C₂³
hypercenter	C₂×C₁₆	D₇
hypocenter	C₇	#504
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×C₁₆	C₂	semidirect
3. #6386	C₂	semidirect
4. #6372	C₂	semidirect
5. D₇×C₁₆	C₂	semidirect
27. C₂×C₁₆	D₇	semidirect
52. C₇	#504	semidirect

Maximal subgroups

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

subgroup	index
#6464	2
C₂×C₇×C₁₆	2
#6386	2
#6372	2
D₇×C₁₆	2
#504	7