Subgroups — #89846

Order 448.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#1142	C₂	1	2	yes	maximal, complemented
3	C₂×#1141	C₂	1	2	yes	maximal, complemented
4	#6438	C₂	1	1	yes	maximal, characteristic, complemented
5	#6377	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×D₄	—	2	2	no	—
7	C₂×C₇×D₄	C₂²	1	2	yes	—
8	C₂×D₂₈	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
10	C₄²×C₇	C₂²	1	1	yes	characteristic
11	#1142	—	2	4	no	—
12	#1141	—	2	4	no	—
13	(C₂×D₇)⋊C₄	—	2	2	no	—
14	C₂×C₇⋊C₈	C₂²	1	2	yes	—
15	#496	—	7	1	no	maximal, Sylow
16	C₂³×C₇	—	2	2	no	—
17	C₂²×D₇	—	2	1	no	—
18	C₇×D₄	—	2	2	no	—
19	C₇×D₄	—	2	2	no	—
20	C₇×D₄	—	4	4	no	—
21	C₂×C₄×C₇	D₄	1	2	yes	—
22	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂×Dic₇	—	2	1	no	—
24	D₂₈	—	2	2	no	—
25	Dic₁₄	—	2	2	no	—
26	C₇⋊C₈	D₄	1	4	yes	complemented
27	C₂×SD₁₆	—	7	2	no	—
28	C₂×D₈	—	7	2	no	—
29	#127	D₇	1	1	yes	characteristic, complemented
30	#124	—	7	1	no	—
31	#97	—	7	1	no	—
32	C₂²×C₇	—	4	2	no	—
33	C₂²×C₇	—	4	4	no	—
34	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
35	C₂×D₇	—	4	1	no	—
36	C₂×D₇	—	4	2	no	—
37	C₄×C₇	—	2	2	no	—
38	C₄×C₇	C₂×D₄	1	2	yes	—
39	Dic₇	—	4	1	no	—
40	C₂×Q₈	—	7	1	no	—
41	C₂×D₄	—	2	2	no	—
42	C₂×D₄	—	7	1	no	—
43	C₂×D₄	C₂×D₇	1	2	yes	—
44	SD₁₆	—	14	4	no	—
45	D₈	—	14	4	no	—
46	C₂×C₈	—	7	2	no	—
47	C₂²⋊C₄	—	14	2	no	—
48	C₄²	C₂×D₇	1	1	yes	characteristic
49	C₂×C₇	—	8	2	no	—
50	C₂×C₇	#127	1	1	yes	characteristic
51	C₂×C₇	AGL(1,ℤ/8)	1	2	yes	—
52	D₇	—	8	1	no	—
53	C₂³	—	2	2	no	—
54	C₂³	—	14	1	no	—
55	Q₈	—	14	2	no	—
56	D₄	—	2	2	no	—
57	D₄	—	2	2	no	—
58	D₄	—	4	4	no	—
59	D₄	—	14	2	no	—
60	C₂×C₄	—	14	1	no	—
61	C₂×C₄	C₂²⋊D₇	1	2	yes	—
62	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
63	C₈	—	7	4	no	—
64	C₇	#496	1	1	yes	characteristic, complemented, Sylow
65	C₂²	—	4	4	no	—
66	C₂²	—	4	2	no	—
67	C₂²	—	28	2	no	—
68	C₂²	—	28	1	no	—
69	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	28	1	no	—
72	C₄	D₄×D₇	1	2	yes	—
73	C₂	—	8	2	no	—
74	C₂	—	56	1	no	—
75	C₂	#6495	1	2	yes	—
76	C₂	#6503	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#127	C₂
socle	C₂²×C₇	C₂×D₄
hypercenter	#127	D₇
hypocenter	C₇	#496
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₇×#127	C₂	semidirect
2. C₂×#1142	C₂	semidirect
3. C₂×#1141	C₂	semidirect
4. #6438	C₂	semidirect
5. #6377	C₂	semidirect
26. C₇⋊C₈	D₄	semidirect
29. #127	D₇	semidirect
64. C₇	#496	semidirect

Maximal subgroups

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

subgroup	index
C₇×#127	2
C₂×#1142	2
C₂×#1141	2
#6438	2
#6377	2
#496	7