Subgroups — #89665

Order 448.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₇×M₁₆	C₂	1	2	yes	maximal
2	C₂×#1131	C₂	1	1	yes	maximal, characteristic
3	C₇×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	#6396	C₂	1	2	yes	maximal, complemented
5	#6393	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
7	C₇×M₁₆	C₂²	1	2	yes	—
8	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
9	#1136	C₂²	1	2	yes	—
10	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
11	#1131	—	2	2	no	—
12	#1131	C₄	1	4	yes	—
13	C₈×D₇	—	2	2	no	—
14	#430	—	7	1	no	maximal, Sylow
15	C₂²×D₇	Q₈	1	1	yes	characteristic
16	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
17	C₂×Dic₇	Q₈	1	1	yes	characteristic
18	C₄×D₇	C₂×C₄	1	2	yes	—
19	C₄×D₇	D₄	1	2	yes	—
20	C₇×C₈	—	2	2	no	—
21	C₇×C₈	C₂×C₄	1	2	yes	—
22	C₇⋊C₈	—	2	2	no	—
23	C₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₂×M₁₆	—	7	1	no	—
25	C₂×M₁₆	—	7	2	no	—
26	#108	—	7	2	no	—
27	#108	—	7	1	no	—
28	#108	D₇	1	1	yes	characteristic, complemented
29	C₂²×C₇	C₂×Q₈	1	1	yes	characteristic
30	C₂×D₇	—	4	1	no	—
31	C₂×D₇	C₄⋊C₄	1	2	yes	—
32	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
33	C₄×C₇	C₂×D₄	1	1	yes	characteristic
34	Dic₇	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄	—	7	1	no	—
36	M₁₆	—	7	4	no	—
37	M₁₆	—	7	2	no	—
38	M₁₆	—	14	2	no	—
39	M₁₆	C₂×D₇	1	2	yes	—
40	C₂×C₈	—	7	1	no	—
41	C₂×C₈	—	14	2	no	—
42	C₂×C₈	C₂×D₇	1	1	yes	characteristic
43	C₂×C₇	—	2	1	no	—
44	C₂×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	D₇	—	2	2	no	—
46	C₂³	—	7	1	no	—
47	C₂×C₄	—	7	1	no	—
48	C₂×C₄	—	7	2	no	—
49	C₂×C₄	—	7	2	no	—
50	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	7	2	no	—
53	C₈	—	14	2	no	—
54	C₈	C₄×D₇	1	2	yes	—
55	C₇	#430	1	1	yes	characteristic, complemented, Sylow
56	C₂²	—	7	2	no	—
57	C₂²	—	28	1	no	—
58	C₂²	Q₈×D₇	1	1	yes	characteristic
59	C₄	—	7	2	no	—
60	C₄	C₂×C₄×D₇	1	1	yes	characteristic
61	C₄	D₄×D₇	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	14	2	no	—
64	C₂	D₇×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#108	C₂
socle	C₂×C₇	C₂×C₄⋊C₄
prime radical	C₂²×D₇	Q₈
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#108	D₇
hypocenter	C₇	#430
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
3. C₇×#108	C₂	semidirect
4. #6396	C₂	semidirect
5. #6393	C₂	semidirect
28. #108	D₇	semidirect
55. C₇	#430	semidirect

Maximal subgroups

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

subgroup	index
D₇×M₁₆	2
C₂×#1131	2
C₇×#108	2
#6396	2
#6393	2
#430	7