Subgroups — #90422

Order 448.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#6509	C₂	1	2	yes	maximal
3	Q₈×Dic₇	C₂	1	2	yes	maximal, complemented
4	#6461	C₂	1	4	yes	maximal
5	#6452	C₂	1	4	yes	maximal, complemented
6	#6435	C₂	1	1	yes	maximal, characteristic
7	#6433	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×Q₈	C₂²	1	2	yes	—
9	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
10	C₇×C₄⋊C₄	C₂²	1	4	yes	—
11	C₄²×C₇	C₂²	1	1	yes	characteristic
12	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
13	(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
14	C₄⋊Dic₇	C₂²	1	2	yes	—
15	C₄⋊Dic₇	C₂²	1	4	yes	—
16	Dic₇⋊C₄	C₂²	1	4	yes	—
17	Dic₇⋊C₄	C₂²	1	2	yes	—
18	Dic₇⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₇	C₂²	1	1	yes	characteristic
20	C₄×Dic₇	C₂²	1	4	yes	—
21	#541	—	7	1	no	maximal, Sylow
22	C₂²×D₇	C₂³	1	1	yes	characteristic
23	C₇×Q₈	—	2	4	no	—
24	C₂×C₄×C₇	C₂³	1	4	yes	—
25	C₂×C₄×C₇	C₂³	1	2	yes	—
26	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₇	C₂³	1	2	yes	—
28	C₂×Dic₇	C₂³	1	4	yes	—
29	C₂×Dic₇	C₂³	1	1	yes	characteristic
30	C₄×D₇	—	2	2	no	—
31	C₄⋊Q₈	D₇	1	1	yes	characteristic, complemented
32	#126	—	7	4	no	—
33	#125	—	7	4	no	—
34	#125	—	7	1	no	—
35	C₂²⋊Q₈	—	7	2	no	—
36	C₄×Q₈	—	7	2	no	—
37	#117	—	7	1	no	—
38	C₂²×C₇	C₂⁴	1	1	yes	characteristic
39	C₂×D₇	—	2	1	no	—
40	C₂×D₇	—	2	2	no	—
41	C₄×C₇	—	2	4	no	—
42	C₄×C₇	—	2	2	no	—
43	C₄×C₇	#41	1	2	yes	—
44	Dic₇	—	2	1	no	—
45	Dic₇	—	2	2	no	—
46	Dic₇	—	2	4	no	—
47	C₂×Q₈	C₂×D₇	1	2	yes	—
48	C₂²×C₄	—	7	1	no	—
49	C₄⋊C₄	—	7	2	no	—
50	C₄⋊C₄	—	7	4	no	—
51	C₄⋊C₄	—	7	4	no	—
52	C₄⋊C₄	—	7	2	no	—
53	C₄⋊C₄	—	7	4	no	—
54	C₄⋊C₄	C₂×D₇	1	4	yes	—
55	C₂²⋊C₄	—	7	2	no	—
56	C₂²⋊C₄	—	7	4	no	—
57	C₄²	—	7	1	no	—
58	C₄²	—	7	4	no	—
59	C₄²	C₂×D₇	1	1	yes	characteristic
60	C₂×C₇	C₂×#41	1	1	yes	characteristic
61	C₂×C₇	2₋¹⁺⁴	1	2	yes	—
62	D₇	—	4	1	no	—
63	C₂³	—	7	1	no	—
64	Q₈	—	2	4	no	—
65	C₂×C₄	—	7	2	no	—
66	C₂×C₄	—	7	4	no	—
67	C₂×C₄	—	7	1	no	—
68	C₂×C₄	—	14	2	no	—
69	C₂×C₄	C₂²×D₇	1	4	yes	—
70	C₂×C₄	C₂²×D₇	1	2	yes	—
71	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
72	C₇	#541	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	14	1	no	—
74	C₂²	—	14	2	no	—
75	C₂²	C₂³×D₇	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	2	no	—
78	C₄	—	14	1	no	—
79	C₄	—	14	2	no	—
80	C₄	—	14	4	no	—
81	C₄	#1159	1	2	yes	—
82	C₂	—	28	1	no	—
83	C₂	C₂×#1159	1	1	yes	characteristic
84	C₂	#6551	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×C₄⋊Q₈	C₂
prime radical	C₂²×D₇	C₂³
hypercenter	C₄⋊Q₈	D₇
hypocenter	C₇	#541
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₇×C₄⋊Q₈	C₂	semidirect
3. Q₈×Dic₇	C₂	semidirect
5. #6452	C₂	semidirect
7. #6433	C₂	semidirect
31. C₄⋊Q₈	D₇	semidirect
72. C₇	#541	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₄⋊Q₈	2
#6509	2
Q₈×Dic₇	2
#6461	2
#6452	2
#6435	2
#6433	2
#541	7