Subgroups — #89983

Order 448.

218 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₇	C₂	1	2	yes	maximal
3	C₇⋊#95	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₇	C₂²	1	1	yes	characteristic
5	C₂²×C₄×C₇	—	2	4	no	—
6	C₂²×C₄×C₇	C₄	1	4	yes	complemented
7	C₂²×C₄×C₇	C₂²	1	2	yes	—
8	C₂²×Dic₇	C₂²	1	4	yes	—
9	C₂²⋊Dic₇	—	2	4	no	—
10	#381	—	7	1	no	maximal, Sylow
11	C₂³×C₇	—	2	4	no	—
12	C₂³×C₇	C₂×C₄	1	2	yes	—
13	C₂³×C₇	D₄	1	4	yes	—
14	C₂³×C₇	C₂³	1	1	yes	characteristic
15	C₂×C₄×C₇	—	2	8	no	—
16	C₂×C₄×C₇	—	2	4	no	—
17	C₂×C₄×C₇	C₂×C₄	1	4	yes	—
18	C₂×Dic₇	—	2	4	no	—
19	C₂×Dic₇	—	2	8	no	—
20	C₂³×C₄	D₇	1	1	yes	characteristic
21	C₂×C₂²⋊C₄	—	7	2	no	—
22	#95	—	7	4	no	—
23	C₂²×C₇	—	2	4	no	—
24	C₂²×C₇	—	2	8	no	—
25	C₂²×C₇	C₂²⋊C₄	1	4	yes	—
26	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₇	C₂×D₄	1	2	yes	—
28	C₂²×C₇	#41	1	4	yes	—
29	C₄×C₇	—	2	4	no	—
30	Dic₇	—	4	4	no	—
31	C₂⁴	C₂×D₇	1	1	yes	characteristic
32	C₂²×C₄	—	2	4	no	—
33	C₂²×C₄	—	7	4	no	—
34	C₂²×C₄	Dic₇	1	4	yes	complemented
35	C₂²×C₄	C₂×D₇	1	2	yes	—
36	C₂²⋊C₄	—	14	4	no	—
37	C₂×C₇	—	2	4	no	—
38	C₂×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₇	#117	1	2	yes	—
40	C₂×C₇	#123	1	4	yes	—
41	C₂³	—	2	4	no	—
42	C₂³	C₂×Dic₇	1	2	yes	—
43	C₂³	C₂²⋊D₇	1	4	yes	—
44	C₂³	C₂²×D₇	1	1	yes	characteristic
45	C₂×C₄	—	2	8	no	—
46	C₂×C₄	—	2	4	no	—
47	C₂×C₄	—	14	8	no	—
48	C₂×C₄	—	14	4	no	—
49	C₂×C₄	C₂×Dic₇	1	4	yes	—
50	C₇	#381	1	1	yes	characteristic, complemented, Sylow
51	C₂²	—	2	4	no	—
52	C₂²	—	2	8	no	—
53	C₂²	C₂²⋊Dic₇	1	4	yes	—
54	C₂²	#1157	1	4	yes	—
55	C₂²	C₂²×Dic₇	1	1	yes	characteristic
56	C₂²	C₂×C₂²⋊D₇	1	2	yes	—
57	C₄	—	2	4	no	—
58	C₄	—	28	4	no	—
59	C₂	—	2	4	no	—
60	C₂	#6489	1	2	yes	—
61	C₂	#6492	1	4	yes	—
62	C₂	C₂×C₂²⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₇
commutator	C₂²×C₇	C₂²×C₄
Fitting	C₂³×C₄×C₇	C₂
socle, cosocle	C₂³×C₇	C₂³
prime radical	C₂⁴×C₇	C₂²
hypercenter	C₂³×C₄	D₇
hypocenter	C₇	#381
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
3. C₇⋊#95	C₂	semidirect
6. C₂²×C₄×C₇	C₄	semidirect
34. C₂²×C₄	Dic₇	semidirect
50. C₇	#381	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₇	2
C₂×C₂²⋊Dic₇	2
C₇⋊#95	2
#381	7