Subgroups — #89749

Order 448.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₇⋊#95	C₂	1	4	yes	maximal
5	C₂²×C₄×C₇	C₂²	1	2	yes	—
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂²×Dic₇	C₂²	1	2	yes	—
8	C₂²×Dic₇	C₂²	1	2	yes	—
9	C₇×C₄⋊C₄	—	2	2	no	—
10	C₄⋊Dic₇	—	2	2	no	—
11	C₄×Dic₇	C₄	1	4	yes	complemented
12	#382	—	7	1	no	maximal, Sylow
13	C₂³×C₇	C₂³	1	1	yes	characteristic
14	C₂×C₄×C₇	—	2	2	no	—
15	C₂×C₄×C₇	—	2	4	no	—
16	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₇	D₄	1	2	yes	—
18	C₂×C₄×C₇	Q₈	1	2	yes	—
19	C₂×Dic₇	—	2	2	no	—
20	C₂×Dic₇	—	2	2	no	—
21	C₂×Dic₇	—	2	2	no	—
22	C₂×Dic₇	—	2	4	no	—
23	C₂×Dic₇	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄	—	7	1	no	—
25	C₂×C₄⋊C₄	D₇	1	1	yes	characteristic
26	C₂×C₄²	—	7	1	no	—
27	#95	—	7	4	no	—
28	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₇	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₇	#41	1	2	yes	—
32	C₂²×C₇	#41	1	2	yes	—
33	C₄×C₇	—	4	2	no	—
34	C₄×C₇	C₄⋊C₄	1	4	yes	—
35	Dic₇	—	2	4	no	—
36	Dic₇	—	4	2	no	—
37	C₂²×C₄	—	7	2	no	—
38	C₂²×C₄	—	7	2	no	—
39	C₂²×C₄	C₂×D₇	1	2	yes	—
40	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
41	C₄⋊C₄	—	2	2	no	—
42	C₄⋊C₄	—	14	2	no	—
43	C₄²	—	7	4	no	—
44	C₂×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₇	#117	1	2	yes	—
46	C₂×C₇	#124	1	1	yes	characteristic
47	C₂×C₇	#124	1	1	yes	characteristic
48	C₂×C₇	#125	1	2	yes	—
49	C₂³	C₂²×D₇	1	1	yes	characteristic
50	C₂×C₄	—	2	4	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	7	4	no	—
53	C₂×C₄	—	14	2	no	—
54	C₂×C₄	—	14	2	no	—
55	C₂×C₄	—	14	4	no	—
56	C₂×C₄	—	14	2	no	—
57	C₂×C₄	Dic₁₄	1	2	yes	—
58	C₂×C₄	C₄×D₇	1	2	yes	—
59	C₂×C₄	C₂²⋊D₇	1	2	yes	—
60	C₇	#382	1	1	yes	characteristic, complemented, Sylow
61	C₂²	C₂×Dic₁₄	1	1	yes	characteristic
62	C₂²	C₂×C₄×D₇	1	1	yes	characteristic
63	C₂²	#1159	1	2	yes	—
64	C₂²	#1161	1	2	yes	—
65	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
66	C₄	—	4	2	no	—
67	C₄	—	14	4	no	—
68	C₄	—	28	2	no	—
69	C₄	Dic₇⋊C₄	1	4	yes	—
70	C₂	#6453	1	2	yes	—
71	C₂	#6455	1	2	yes	—
72	C₂	C₂×Dic₇⋊C₄	1	1	yes	characteristic
73	C₂	#6499	1	1	yes	characteristic
74	C₂	#6510	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₇
commutator	C₂²×C₇	C₂²×C₄
Fitting	C₂×C₇×C₄⋊C₄	C₂
socle, prime radical, cosocle	C₂³×C₇	C₂³
hypercenter	C₂×C₄⋊C₄	D₇
hypocenter	C₇	#382
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
11. C₄×Dic₇	C₄	semidirect
60. C₇	#382	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₄⋊C₄	2
C₂×C₄⋊Dic₇	2
C₂×C₄×Dic₇	2
C₇⋊#95	2
#382	7