Subgroups — #89832

Order 448.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×#1144	C₂	1	2	yes	maximal, complemented
3	C₂×#1142	C₂	1	2	yes	maximal
4	#6432	C₂	1	1	yes	maximal, characteristic, complemented
5	#6377	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
8	C₂×Dic₁₄	C₂²	1	2	yes	—
9	C₇×C₂²⋊C₄	—	2	2	no	—
10	C₄²×C₇	C₂²	1	1	yes	characteristic
11	#1144	—	2	4	no	—
12	#1142	—	2	4	no	—
13	C₄⋊Dic₇	—	2	2	no	—
14	C₂×C₇⋊C₈	C₂²	1	2	yes	—
15	#497	—	7	1	no	maximal, Sylow
16	C₂³×C₇	—	2	1	no	—
17	C₇×Q₈	—	2	2	no	—
18	C₇×D₄	—	2	2	no	—
19	C₂×C₄×C₇	—	2	1	no	—
20	C₂×C₄×C₇	D₄	1	2	yes	—
21	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂×Dic₇	—	2	2	no	—
23	Dic₁₄	—	2	2	no	—
24	Dic₁₄	—	2	2	no	—
25	C₇⋊C₈	D₄	1	4	yes	—
26	C₂×Q₁₆	—	7	2	no	—
27	C₂×SD₁₆	—	7	2	no	—
28	C₄⋊Q₈	—	7	1	no	—
29	#124	D₇	1	1	yes	characteristic
30	#97	—	7	1	no	—
31	C₂²×C₇	—	4	1	no	—
32	C₂²×C₇	—	4	2	no	—
33	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
34	C₄×C₇	—	2	2	no	—
35	C₄×C₇	—	4	1	no	—
36	C₄×C₇	C₂×D₄	1	2	yes	—
37	Dic₇	—	4	2	no	—
38	C₂×Q₈	—	7	2	no	—
39	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
40	C₂×D₄	C₂×D₇	1	1	yes	characteristic
41	Q₁₆	—	14	4	no	—
42	SD₁₆	—	14	4	no	—
43	C₂×C₈	—	7	2	no	—
44	C₄⋊C₄	—	14	2	no	—
45	C₂²⋊C₄	—	2	2	no	—
46	C₄²	C₂×D₇	1	1	yes	characteristic
47	C₂×C₇	—	8	1	no	—
48	C₂×C₇	#127	1	1	yes	characteristic
49	C₂×C₇	#137	1	2	yes	—
50	C₂³	—	2	1	no	—
51	Q₈	—	2	2	no	—
52	Q₈	—	14	2	no	—
53	Q₈	—	14	2	no	—
54	D₄	—	2	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	14	2	no	—
57	C₂×C₄	C₂²⋊D₇	1	2	yes	—
58	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
59	C₈	—	7	4	no	—
60	C₇	#497	1	1	yes	characteristic, complemented, Sylow
61	C₂²	—	4	1	no	—
62	C₂²	—	4	2	no	—
63	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	4	1	no	—
66	C₄	—	28	2	no	—
67	C₄	D₄×D₇	1	2	yes	—
68	C₂	—	8	1	no	—
69	C₂	#6503	1	1	yes	characteristic
70	C₂	#6514	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#124	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×C₇×D₄	C₂²
hypercenter	#124	D₇
hypocenter	C₇	#497
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
2. C₂×#1144	C₂	semidirect
4. #6432	C₂	semidirect
5. #6377	C₂	semidirect
60. C₇	#497	semidirect

Maximal subgroups

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

subgroup	index
C₇×#124	2
C₂×#1144	2
C₂×#1142	2
#6432	2
#6377	2
#497	7