Subgroups — #89448

Order 448.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×(C₂×D₇)⋊C₄	C₂	1	3	yes	maximal
2	C₂×Dic₇⋊C₄	C₂	1	3	yes	maximal, complemented
3	C₇×#95	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₇	C₂²	1	1	yes	characteristic
5	C₂²×C₄×C₇	C₂²	1	3	yes	—
6	C₂²×Dic₇	C₂²	1	3	yes	—
7	(C₂×D₇)⋊C₄	—	2	6	no	—
8	Dic₇⋊C₄	—	2	6	no	—
9	#396	—	7	1	no	maximal, Sylow
10	C₂³×C₇	C₂³	1	1	yes	characteristic
11	C₂²×D₇	—	2	3	no	—
12	C₂²×D₇	—	2	3	no	—
13	C₂²×D₇	Q₈	1	2	yes	—
14	C₂×C₄×C₇	—	2	3	no	—
15	C₂×C₄×C₇	—	2	3	no	—
16	C₂×C₄×C₇	—	2	3	no	—
17	C₂×Dic₇	—	2	3	no	—
18	C₂×Dic₇	—	2	3	no	—
19	C₂×Dic₇	D₄	1	6	yes	—
20	C₂×C₄⋊C₄	—	7	3	no	—
21	C₂×C₂²⋊C₄	—	7	3	no	—
22	#95	D₇	1	1	yes	characteristic, complemented
23	C₂²×C₇	C₂×D₄	1	3	yes	—
24	C₂²×C₇	C₂×Q₈	1	1	yes	characteristic
25	C₂²×C₇	#41	1	3	yes	—
26	C₂×D₇	—	2	6	no	—
27	C₂×D₇	—	4	3	no	—
28	C₂×D₇	—	4	1	no	—
29	C₄×C₇	—	4	3	no	—
30	Dic₇	—	2	6	no	—
31	C₂⁴	—	7	1	no	—
32	C₂²×C₄	—	7	3	no	—
33	C₂²×C₄	C₂×D₇	1	3	yes	—
34	C₄⋊C₄	—	14	6	no	—
35	C₂²⋊C₄	—	14	6	no	—
36	C₂×C₇	#121	1	3	yes	—
37	C₂×C₇	C₂²⋊Q₈	1	3	yes	—
38	C₂×C₇	#126	1	1	yes	characteristic
39	D₇	—	4	2	no	—
40	C₂³	—	7	2	no	—
41	C₂³	—	14	3	no	—
42	C₂³	—	14	3	no	—
43	C₂³	C₂²×D₇	1	1	yes	characteristic
44	C₂×C₄	—	2	3	no	—
45	C₂×C₄	—	2	3	no	—
46	C₂×C₄	—	2	3	no	—
47	C₂×C₄	—	7	6	no	—
48	C₂×C₄	—	14	3	no	—
49	C₂×C₄	—	14	3	no	—
50	C₇	#396	1	1	yes	characteristic, complemented, Sylow
51	C₂²	—	14	6	no	—
52	C₂²	—	28	1	no	—
53	C₂²	—	28	3	no	—
54	C₂²	#1157	1	3	yes	—
55	C₂²	D₄×D₇	1	3	yes	—
56	C₂²	Q₈×D₇	1	1	yes	characteristic
57	C₄	—	4	3	no	—
58	C₄	—	14	6	no	—
59	C₂	—	28	2	no	—
60	C₂	#6439	1	1	yes	characteristic
61	C₂	#6447	1	3	yes	—
62	C₂	#6459	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₇
commutator, socle, cosocle	C₂³×C₇	C₂³
Fitting	C₇×#95	C₂
prime radical	C₂³×D₇	C₂²
hypercenter	#95	D₇
hypocenter	C₇	#396
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₇⋊C₄	C₂	semidirect
3. C₇×#95	C₂	semidirect
22. #95	D₇	semidirect
50. C₇	#396	semidirect

Maximal subgroups

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

subgroup	index
C₂×(C₂×D₇)⋊C₄	2
C₂×Dic₇⋊C₄	2
C₇×#95	2
#396	7