Subgroups — #89969

Order 448.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#135	C₂	1	1	yes	maximal, characteristic
2	#6511	C₂	1	2	yes	maximal, complemented
3	#6411	C₂	1	2	yes	maximal, complemented
4	#6393	C₂	1	1	yes	maximal, characteristic, complemented
5	#6389	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#41	C₂²	1	2	yes	—
7	C₇×Q₁₆	C₄	1	1	yes	characteristic, complemented
8	C₇×SD₁₆	C₄	1	2	yes	complemented
9	C₇×D₈	C₄	1	1	yes	characteristic, complemented
10	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₄×Dic₇	C₂²	1	1	yes	characteristic
12	#1136	—	2	2	no	—
13	#1136	C₂²	1	2	yes	—
14	C₂×C₇⋊C₈	—	2	2	no	—
15	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
16	#444	—	7	1	no	maximal, Sylow
17	C₇×Q₈	C₂×C₄	1	2	yes	—
18	C₇×D₄	—	2	2	no	—
19	C₇×D₄	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₇	—	2	2	no	—
21	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂×Dic₇	—	2	1	no	—
23	C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₇⋊C₈	—	2	2	no	—
26	C₇⋊C₈	D₄	1	2	yes	—
27	#135	D₇	1	1	yes	characteristic
28	#131	—	7	2	no	—
29	#108	—	7	1	no	—
30	#104	—	7	2	no	—
31	#97	—	7	1	no	—
32	C₂²×C₇	—	2	2	no	—
33	C₂²×C₇	#41	1	1	yes	characteristic
34	C₄×C₇	—	2	2	no	—
35	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₇	C₂×D₄	1	1	yes	characteristic
37	Dic₇	—	4	1	no	—
38	#41	C₂×D₇	1	2	yes	—
39	Q₁₆	Dic₇	1	1	yes	characteristic, complemented
40	SD₁₆	Dic₇	1	2	yes	complemented
41	D₈	Dic₇	1	1	yes	characteristic, complemented
42	M₁₆	—	7	2	no	—
43	M₁₆	—	14	2	no	—
44	C₂×C₈	—	7	1	no	—
45	C₂×C₈	—	14	2	no	—
46	C₂×C₈	C₂×D₇	1	1	yes	characteristic
47	C₄²	—	7	1	no	—
48	C₂×C₇	—	2	1	no	—
49	C₂×C₇	—	4	2	no	—
50	C₂×C₇	C₄×D₄	1	1	yes	characteristic
51	Q₈	C₂×Dic₇	1	2	yes	—
52	D₄	—	2	2	no	—
53	D₄	C₂×Dic₇	1	2	yes	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	14	1	no	—
56	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
57	C₈	—	7	2	no	—
58	C₈	—	14	2	no	—
59	C₈	C₂×Dic₇	1	1	yes	characteristic
60	C₈	C₂×Dic₇	1	1	yes	characteristic
61	C₇	#444	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	2	2	no	—
63	C₂²	#1159	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	28	1	no	—
66	C₄	D₄×D₇	1	1	yes	characteristic
67	C₄	C₂²×Dic₇	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	2	no	—
70	C₂	D₄×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting, prime radical	C₇×#135	C₂
socle	C₂×C₇	C₄×D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#135	D₇
hypocenter	C₇	#444
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
2. #6511	C₂	semidirect
3. #6411	C₂	semidirect
4. #6393	C₂	semidirect
5. #6389	C₂	semidirect
7. C₇×Q₁₆	C₄	semidirect
8. C₇×SD₁₆	C₄	semidirect
9. C₇×D₈	C₄	semidirect
39. Q₁₆	Dic₇	semidirect
40. SD₁₆	Dic₇	semidirect
41. D₈	Dic₇	semidirect
61. C₇	#444	semidirect

Maximal subgroups

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

subgroup	index
C₇×#135	2
#6511	2
#6411	2
#6393	2
#6389	2
#444	7