Subgroups — #89259

Order 448.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6438	C₂	1	1	yes	maximal, characteristic
2	C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
3	#6377	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₂₈	C₄	1	1	yes	characteristic
5	C₂×Dic₁₄	C₄	1	1	yes	characteristic
6	C₂×C₇×C₈	—	2	1	no	—
7	C₄²×C₇	C₂²	1	1	yes	characteristic
8	(C₂×D₇)⋊C₄	—	2	1	no	—
9	(C₂×D₇)⋊C₄	—	2	1	no	—
10	C₂×C₇⋊C₈	—	2	1	no	—
11	#329	—	7	1	no	maximal, Sylow
12	C₂²×D₇	—	2	1	no	—
13	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₇	D₄	1	1	yes	characteristic
15	C₂×C₄×C₇	D₄	1	1	yes	characteristic
16	C₂×Dic₇	—	2	1	no	—
17	D₂₈	—	4	1	no	—
18	Dic₁₄	—	4	1	no	—
19	C₇×C₈	—	2	2	no	—
20	C₇⋊C₈	—	2	2	no	—
21	#124	—	7	1	no	—
22	#97	—	7	1	no	—
23	#97	D₇	1	1	yes	characteristic, complemented
24	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×D₇	—	4	1	no	—
26	C₂×D₇	—	4	1	no	—
27	C₂×D₇	—	4	1	no	—
28	C₄×C₇	—	2	1	no	—
29	C₄×C₇	—	2	1	no	—
30	C₄×C₇	—	2	1	no	—
31	Dic₇	—	4	1	no	—
32	C₂×Q₈	—	7	1	no	—
33	C₂×D₄	—	7	1	no	—
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	—	14	1	no	—
36	C₂²⋊C₄	—	14	1	no	—
37	C₂²⋊C₄	—	14	1	no	—
38	C₄²	C₂×D₇	1	1	yes	characteristic
39	C₂×C₇	#100	1	1	yes	characteristic
40	C₂×C₇	#104	1	1	yes	characteristic
41	C₂×C₇	#104	1	1	yes	characteristic
42	D₇	—	8	1	no	—
43	C₂³	—	14	1	no	—
44	Q₈	—	28	1	no	—
45	D₄	—	28	1	no	—
46	C₂×C₄	—	14	1	no	—
47	C₂×C₄	C₄×D₇	1	1	yes	characteristic
48	C₂×C₄	D₂₈	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₈	—	14	2	no	—
52	C₇	#329	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	28	1	no	—
54	C₂²	—	28	1	no	—
55	C₂²	—	28	1	no	—
56	C₂²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	28	1	no	—
61	C₂	—	56	1	no	—
62	C₂	#6379	1	1	yes	characteristic
63	C₂	#6397	1	1	yes	characteristic
64	C₂	#6399	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₇)⋊C₄
commutator	C₂×C₄×C₇	C₂×C₄
Frattini	C₄²	C₂×D₇
Fitting	C₇×#97	C₂
socle	C₂²×C₇	C₂²⋊C₄
prime radical	C₂×D₂₈	C₄
cosocle	C₄²×C₇	C₂²
hypercenter	#97	D₇
hypocenter	C₇	#329
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
2. C₇×#97	C₂	semidirect
3. #6377	C₂	semidirect
23. #97	D₇	semidirect
52. C₇	#329	semidirect

Maximal subgroups

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

subgroup	index
#6438	2
C₇×#97	2
#6377	2
#329	7