Subgroups — #90019

Order 448.

506 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₇	C₂	1	6	yes	maximal, complemented
3	C₂⁴×C₇	—	2	12	no	—
4	C₂⁴×C₇	C₄	1	4	yes	complemented
5	C₂⁴×C₇	C₂²	1	3	yes	—
6	C₂²×Dic₇	C₂²	1	4	yes	complemented
7	C₂²⋊Dic₇	—	2	12	no	—
8	#379	—	7	1	no	maximal, Sylow
9	C₂³×C₇	—	2	32	no	—
10	C₂³×C₇	—	2	24	no	—
11	C₂³×C₇	—	2	12	no	—
12	C₂³×C₇	C₂×C₄	1	6	yes	—
13	C₂³×C₇	D₄	1	12	yes	—
14	C₂³×C₇	C₂³	1	1	yes	characteristic
15	C₂×Dic₇	—	2	12	no	—
16	C₂⁵	D₇	1	1	yes	characteristic
17	C₂×C₂²⋊C₄	—	7	6	no	—
18	C₂²×C₇	—	2	32	no	—
19	C₂²×C₇	—	2	12	no	—
20	C₂²×C₇	—	2	24	no	—
21	C₂²×C₇	C₂²⋊C₄	1	12	yes	complemented
22	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
23	C₂²×C₇	C₂×D₄	1	6	yes	—
24	Dic₇	—	4	4	no	—
25	C₂⁴	—	2	12	no	—
26	C₂⁴	Dic₇	1	4	yes	complemented
27	C₂⁴	C₂×D₇	1	3	yes	—
28	C₂²×C₄	—	7	4	no	—
29	C₂²⋊C₄	—	14	12	no	—
30	C₂×C₇	—	2	12	no	—
31	C₂×C₇	C₂×C₂²⋊C₄	1	3	yes	—
32	C₂×C₇	#120	1	4	yes	—
33	C₂³	—	2	32	no	—
34	C₂³	—	2	24	no	—
35	C₂³	—	2	12	no	—
36	C₂³	C₂×Dic₇	1	6	yes	—
37	C₂³	C₂²⋊D₇	1	12	yes	—
38	C₂³	C₂²×D₇	1	1	yes	characteristic
39	C₂×C₄	—	14	12	no	—
40	C₇	#379	1	1	yes	characteristic, complemented, Sylow
41	C₂²	—	2	32	no	—
42	C₂²	—	2	12	no	—
43	C₂²	—	2	24	no	—
44	C₂²	C₂²⋊Dic₇	1	12	yes	complemented
45	C₂²	C₂²×Dic₇	1	1	yes	characteristic
46	C₂²	C₂×C₂²⋊D₇	1	6	yes	—
47	C₄	—	28	4	no	—
48	C₂	—	2	12	no	—
49	C₂	C₂×C₂²⋊Dic₇	1	3	yes	—
50	C₂	#6516	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₇
commutator	C₂²×C₇	C₂²×C₄
Fitting, prime radical	C₂⁵×C₇	C₂
socle, cosocle	C₂³×C₇	C₂³
hypercenter	C₂⁵	D₇
hypocenter	C₇	#379
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
2. C₂×C₂²⋊Dic₇	C₂	semidirect
4. C₂⁴×C₇	C₄	semidirect
6. C₂²×Dic₇	C₂²	semidirect
21. C₂²×C₇	C₂²⋊C₄	semidirect
26. C₂⁴	Dic₇	semidirect
40. C₇	#379	semidirect
44. C₂²	C₂²⋊Dic₇	semidirect

Maximal subgroups

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

subgroup	index
C₂⁵×C₇	2
C₂×C₂²⋊Dic₇	2
#379	7