Subgroups — #89421

Order 448.

194 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇⋊C₄	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₇	C₂	1	2	yes	maximal
3	C₂×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₇×#95	C₂	1	1	yes	maximal, characteristic
5	C₇⋊#95	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₇	C₂²	1	2	yes	—
8	C₂²×Dic₇	C₂²	1	2	yes	—
9	C₂²×Dic₇	C₂²	1	1	yes	characteristic
10	C₂²×Dic₇	C₂²	1	1	yes	characteristic
11	Dic₇⋊C₄	C₄	1	8	yes	complemented
12	C₄×Dic₇	—	2	4	no	—
13	C₄×Dic₇	C₄	1	4	yes	complemented
14	C₄×C₄⋊C₄	—	7	1	no	maximal, Sylow
15	C₂³×C₇	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₇	—	2	2	no	—
17	C₂×C₄×C₇	—	2	2	no	—
18	C₂×C₄×C₇	—	2	2	no	—
19	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₇	C₂×C₄	1	4	yes	—
21	C₂×Dic₇	—	2	2	no	—
22	C₂×Dic₇	—	2	2	no	—
23	C₂×Dic₇	C₂×C₄	1	4	yes	—
24	C₂×Dic₇	C₂×C₄	1	2	yes	—
25	C₂×Dic₇	C₂×C₄	1	4	yes	—
26	C₂×Dic₇	C₂×C₄	1	2	yes	—
27	C₂×Dic₇	D₄	1	2	yes	—
28	C₂×Dic₇	Q₈	1	2	yes	—
29	C₂×C₄⋊C₄	—	7	2	no	—
30	C₂×C₄²	—	7	1	no	—
31	C₂×C₄²	—	7	2	no	—
32	#95	—	7	1	no	—
33	#95	D₇	1	1	yes	characteristic
34	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₇	C₂²×C₄	1	2	yes	—
36	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₇	C₂×Q₈	1	1	yes	characteristic
38	C₂²×C₇	#41	1	2	yes	—
39	C₄×C₇	—	2	2	no	—
40	C₄×C₇	—	2	4	no	—
41	Dic₇	—	2	4	no	—
42	Dic₇	C₄²	1	4	yes	complemented
43	Dic₇	C₄⋊C₄	1	4	yes	complemented
44	C₂²×C₄	—	7	1	no	—
45	C₂²×C₄	—	7	1	no	—
46	C₂²×C₄	—	7	2	no	—
47	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
48	C₂²×C₄	C₂×D₇	1	2	yes	—
49	C₄⋊C₄	—	7	8	no	—
50	C₄²	—	7	4	no	—
51	C₄²	—	14	4	no	—
52	C₂×C₇	C₂×C₄²	1	1	yes	characteristic
53	C₂×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₇	#117	1	1	yes	characteristic
55	C₂×C₇	C₄×D₄	1	2	yes	—
56	C₂×C₇	C₄×Q₈	1	2	yes	—
57	C₂³	C₂²×D₇	1	1	yes	characteristic
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	7	2	no	—
62	C₂×C₄	—	7	2	no	—
63	C₂×C₄	—	7	4	no	—
64	C₂×C₄	—	7	2	no	—
65	C₂×C₄	—	7	2	no	—
66	C₂×C₄	—	7	4	no	—
67	C₂×C₄	—	14	2	no	—
68	C₂×C₄	—	14	2	no	—
69	C₂×C₄	C₄×D₇	1	2	yes	—
70	C₂×C₄	C₄×D₇	1	4	yes	—
71	C₇	C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
72	C₂²	C₂×C₄×D₇	1	1	yes	characteristic
73	C₂²	C₂×C₄×D₇	1	2	yes	—
74	C₂²	D₄×D₇	1	1	yes	characteristic
75	C₂²	#1159	1	2	yes	—
76	C₂²	Q₈×D₇	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	2	4	no	—
79	C₄	—	7	4	no	—
80	C₄	—	7	4	no	—
81	C₄	—	14	4	no	—
82	C₂	C₄²×D₇	1	1	yes	characteristic
83	C₂	#6440	1	1	yes	characteristic
84	C₂	(C₂²⋊D₇)⋊C₄	1	2	yes	—
85	C₂	#6450	1	2	yes	—
86	C₂	D₇×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₇
commutator	C₂×C₇	C₂×C₄²
Fitting	C₇×#95	C₂
socle, prime radical, cosocle	C₂³×C₇	C₂³
hypercenter	#95	D₇
hypocenter	C₇	C₄×C₄⋊C₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
11. Dic₇⋊C₄	C₄	semidirect
13. C₄×Dic₇	C₄	semidirect
42. Dic₇	C₄²	semidirect
43. Dic₇	C₄⋊C₄	semidirect
71. C₇	C₄×C₄⋊C₄	semidirect

Maximal subgroups

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

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