Subgroups — #89449

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×C₄⋊Dic₇	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	C₄⋊Dic₇	—	2	6	no	—
9	#399	—	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	6	no	—
15	C₂×C₄×C₇	D₄	1	6	yes	—
16	C₂×Dic₇	—	2	6	no	—
17	C₂×Dic₇	—	2	3	no	—
18	C₂×C₄⋊C₄	—	7	3	no	—
19	C₂×C₂²⋊C₄	—	7	3	no	—
20	#95	D₇	1	1	yes	characteristic, complemented
21	C₂²×C₇	C₂×D₄	1	3	yes	—
22	C₂²×C₇	C₂×Q₈	1	1	yes	characteristic
23	C₂²×C₇	#41	1	3	yes	—
24	C₂×D₇	—	2	6	no	—
25	C₂×D₇	—	4	3	no	—
26	C₂×D₇	—	4	1	no	—
27	C₄×C₇	—	2	6	no	—
28	Dic₇	—	4	3	no	—
29	C₂⁴	—	7	1	no	—
30	C₂²×C₄	—	7	3	no	—
31	C₂²×C₄	C₂×D₇	1	3	yes	—
32	C₄⋊C₄	—	14	6	no	—
33	C₂²⋊C₄	—	14	6	no	—
34	C₂×C₇	C₂²⋊Q₈	1	3	yes	—
35	C₂×C₇	#123	1	3	yes	—
36	C₂×C₇	#127	1	1	yes	characteristic
37	D₇	—	4	2	no	—
38	C₂³	—	7	2	no	—
39	C₂³	—	14	3	no	—
40	C₂³	—	14	3	no	—
41	C₂³	C₂²×D₇	1	1	yes	characteristic
42	C₂×C₄	—	2	6	no	—
43	C₂×C₄	—	14	3	no	—
44	C₂×C₄	—	14	6	no	—
45	C₂×C₄	D₂₈	1	6	yes	—
46	C₇	#399	1	1	yes	characteristic, complemented, Sylow
47	C₂²	—	14	6	no	—
48	C₂²	—	28	1	no	—
49	C₂²	—	28	3	no	—
50	C₂²	C₂×D₂₈	1	3	yes	—
51	C₂²	#1159	1	3	yes	—
52	C₂²	Q₈×D₇	1	1	yes	characteristic
53	C₄	—	2	6	no	—
54	C₄	—	28	3	no	—
55	C₂	—	28	2	no	—
56	C₂	#6437	1	1	yes	characteristic
57	C₂	#6449	1	3	yes	—
58	C₂	#6460	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	#399
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×C₄⋊Dic₇	C₂	semidirect
3. C₇×#95	C₂	semidirect
20. #95	D₇	semidirect
46. C₇	#399	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×C₄⋊Dic₇	2
C₇×#95	2
#399	7