Subgroups — #89327

Order 448.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#1136	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
5	C₂²×Dic₇	—	2	1	no	—
6	C₇×C₄⋊C₄	C₄	1	2	yes	complemented
7	C₇×C₂²⋊C₄	—	2	1	no	—
8	C₄²×C₇	—	2	1	no	—
9	C₄×Dic₇	—	2	1	no	—
10	C₄×Dic₇	C₄	1	2	yes	complemented
11	#1136	—	2	1	no	—
12	#1136	C₄	1	2	yes	complemented
13	C₂×C₇⋊C₈	—	2	1	no	—
14	#339	—	7	1	no	maximal, Sylow
15	C₂³×C₇	D₄	1	1	yes	characteristic
16	C₂×C₄×C₇	—	2	2	no	—
17	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₇	Q₈	1	1	yes	characteristic
23	C₂×Dic₇	—	2	4	no	—
24	C₂×Dic₇	—	2	2	no	—
25	C₇⋊C₈	—	2	2	no	—
26	C₂×M₁₆	—	7	1	no	—
27	#117	D₇	1	1	yes	characteristic
28	C₂×C₄²	—	7	1	no	—
29	C₂²×C₇	—	2	2	no	—
30	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₇	C₄⋊C₄	1	1	yes	characteristic
33	C₄×C₇	—	4	2	no	—
34	C₄×C₇	C₄²	1	1	yes	characteristic
35	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
37	C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
38	Dic₇	—	2	4	no	—
39	C₂²×C₄	—	14	1	no	—
40	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
41	M₁₆	—	7	2	no	—
42	M₁₆	—	14	1	no	—
43	C₂×C₈	—	14	1	no	—
44	C₄⋊C₄	Dic₇	1	2	yes	complemented
45	C₂²⋊C₄	—	2	1	no	—
46	C₄²	—	2	1	no	—
47	C₄²	—	7	2	no	—
48	C₄²	—	14	1	no	—
49	C₂×C₇	—	2	1	no	—
50	C₂×C₇	—	2	1	no	—
51	C₂×C₇	#95	1	1	yes	characteristic
52	C₂×C₇	#104	1	2	yes	—
53	C₂³	C₂²⋊D₇	1	1	yes	characteristic
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	14	4	no	—
56	C₂×C₄	—	14	2	no	—
57	C₂×C₄	Dic₁₄	1	1	yes	characteristic
58	C₂×C₄	C₄×D₇	1	1	yes	characteristic
59	C₂×C₄	C₄×D₇	1	1	yes	characteristic
60	C₂×C₄	D₂₈	1	1	yes	characteristic
61	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
62	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
63	C₈	—	14	2	no	—
64	C₇	#339	1	1	yes	characteristic, complemented, Sylow
65	C₂²	—	2	2	no	—
66	C₂²	Dic₇⋊C₄	1	1	yes	characteristic
67	C₂²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
68	C₂²	C₂²⋊Dic₇	1	1	yes	characteristic
69	C₄	—	4	2	no	—
70	C₄	—	14	4	no	—
71	C₄	C₄×Dic₇	1	1	yes	characteristic
72	C₄	Dic₇⋊C₄	1	1	yes	characteristic
73	C₄	C₄⋊Dic₇	1	1	yes	characteristic
74	C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	2	1	no	—
77	C₂	C₇⋊#95	1	1	yes	characteristic
78	C₂	#6411	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₇
commutator	C₄×C₇	C₄²
Frattini	C₂²×C₄	C₂×D₇
Fitting	C₇×#117	C₂
socle	C₂²×C₇	C₂²⋊C₄
prime radical	C₂³×C₇	D₄
cosocle	C₂²×C₄×C₇	C₂²
hypercenter	#117	D₇
hypocenter	C₇	#339
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
6. C₇×C₄⋊C₄	C₄	semidirect
10. C₄×Dic₇	C₄	semidirect
12. #1136	C₄	semidirect
44. C₄⋊C₄	Dic₇	semidirect
64. C₇	#339	semidirect

Maximal subgroups

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

subgroup	index
C₇×#117	2
C₂×C₄×Dic₇	2
C₂×#1136	2
#339	7