Group #11734685 — subgroups

Subgroups — #11734685

Order 896.

432 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#395	C₂	1	1	yes	maximal, characteristic
2	#90001	C₂	1	2	yes	maximal
3	#89751	C₂	1	3	yes	maximal
4	#89746	C₂	1	3	yes	maximal
5	#89418	C₂	1	6	yes	maximal
6	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₄	C₂²	1	1	yes	characteristic
8	C₂²×Dic₁₄	C₂²	1	3	yes	—
9	C₂×C₇×C₄⋊C₄	C₂²	1	3	yes	—
10	C₂×Dic₇⋊C₄	C₂²	1	6	yes	—
11	C₂×C₄×Dic₇	C₂²	1	3	yes	—
12	C₇×#95	C₂²	1	3	yes	—
13	C₇⋊#95	C₂²	1	6	yes	—
14	C₇⋊#95	C₂²	1	6	yes	—
15	C₇⋊#95	C₂²	1	3	yes	—
16	#2811	—	7	1	no	maximal, Sylow
17	C₂×C₇×Q₈	—	4	3	no	—
18	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₇	C₂³	1	3	yes	—
20	C₂²×C₄×C₇	C₂³	1	3	yes	—
21	C₂²×Dic₇	C₂³	1	2	yes	—
22	C₂²×Dic₇	C₂³	1	6	yes	—
23	C₂×Dic₁₄	—	4	3	no	—
24	C₂×Dic₁₄	—	4	3	no	—
25	C₂×Dic₁₄	—	4	3	no	—
26	C₂×Dic₁₄	Q₈	1	12	yes	complemented
27	C₇×C₄⋊C₄	—	4	3	no	—
28	Dic₇⋊C₄	—	4	6	no	—
29	C₄×Dic₇	—	4	3	no	—
30	#395	—	7	2	no	—
31	#395	—	7	3	no	—
32	#395	D₇	1	1	yes	characteristic
33	#391	—	7	3	no	—
34	#391	—	7	6	no	—
35	C₂³×C₇	C₂⁴	1	1	yes	characteristic
36	C₇×Q₈	—	4	4	no	—
37	C₂×C₄×C₇	—	2	3	no	—
38	C₂×C₄×C₇	—	2	3	no	—
39	C₂×C₄×C₇	—	2	6	no	—
40	C₂×C₄×C₇	—	2	3	no	—
41	C₂×C₄×C₇	—	2	3	no	—
42	C₂×C₄×C₇	C₂×Q₈	1	6	yes	—
43	C₂×Dic₇	—	2	6	no	—
44	C₂×Dic₇	—	2	6	no	—
45	C₂×Dic₇	—	2	6	no	—
46	C₂×Dic₇	C₂×Q₈	1	12	yes	—
47	Dic₁₄	—	4	12	no	—
48	Dic₁₄	—	4	4	no	—
49	C₂²×Q₈	—	7	3	no	—
50	C₂²×Q₈	—	7	1	no	—
51	C₂²×Q₈	C₂×D₇	1	1	yes	characteristic
52	C₂×C₄⋊C₄	—	7	6	no	—
53	C₂×C₄⋊C₄	C₂×D₇	1	3	yes	—
54	C₂×C₄²	—	7	3	no	—
55	#95	—	7	3	no	—
56	#95	—	7	6	no	—
57	#95	—	7	6	no	—
58	#95	C₂×D₇	1	3	yes	—
59	C₂²×C₇	C₂²×Q₈	1	3	yes	—
60	C₂²×C₇	H(5,𝔽₂)	1	3	yes	—
61	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
62	C₄×C₇	—	2	6	no	—
63	C₄×C₇	—	4	1	no	—
64	C₄×C₇	—	4	3	no	—
65	Dic₇	—	2	12	no	—
66	Dic₇	—	4	2	no	—
67	C₂×Q₈	—	4	3	no	—
68	C₂×Q₈	—	7	12	no	—
69	C₂×Q₈	—	28	3	no	—
70	C₂×Q₈	—	28	3	no	—
71	C₂×Q₈	—	28	3	no	—
72	C₂²×C₄	—	7	6	no	—
73	C₂²×C₄	—	7	2	no	—
74	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
75	C₂²×C₄	C₂²×D₇	1	3	yes	—
76	C₂²×C₄	C₂²×D₇	1	3	yes	—
77	C₄⋊C₄	—	4	3	no	—
78	C₄⋊C₄	—	28	6	no	—
79	C₄²	—	28	3	no	—
80	C₂×C₇	#543	1	3	yes	—
81	C₂×C₇	Q₈²	1	3	yes	—
82	C₂×C₇	H(3,𝔽₄)	1	1	yes	characteristic
83	C₂³	C₂³×D₇	1	1	yes	characteristic
84	Q₈	—	4	4	no	—
85	Q₈	—	28	12	no	—
86	Q₈	—	28	4	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	2	3	no	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	2	3	no	—
91	C₂×C₄	—	2	6	no	—
92	C₂×C₄	—	7	12	no	—
93	C₂×C₄	—	14	6	no	—
94	C₂×C₄	—	14	6	no	—
95	C₂×C₄	—	14	6	no	—
96	C₂×C₄	Q₈×D₇	1	6	yes	—
97	C₇	#2811	1	1	yes	characteristic, complemented, Sylow
98	C₂²	#6548	1	3	yes	—
99	C₂²	C₂×Q₈×D₇	1	3	yes	—
100	C₂²	#6553	1	1	yes	characteristic
101	C₄	—	2	6	no	—
102	C₄	—	4	3	no	—
103	C₄	—	4	1	no	—
104	C₄	—	14	12	no	—
105	C₄	—	28	2	no	—
106	C₂	#90285	1	1	yes	characteristic
107	C₂	#90325	1	3	yes	—
108	C₂	#90413	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#395

C₂

yes

maximal, characteristic

#90001

C₂

yes

maximal

#89751

C₂

yes

maximal

#89746

C₂

yes

maximal

#89418

C₂

yes

maximal

C₂²×C₇×Q₈

C₂²

yes

characteristic

C₂²×Dic₁₄

C₂²

yes

characteristic

C₂²×Dic₁₄

C₂²

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×Dic₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

—

C₇×#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

#2811

—

maximal, Sylow

C₂×C₇×Q₈

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

Q₈

yes

complemented

C₇×C₄⋊C₄

—

Dic₇⋊C₄

—

C₄×Dic₇

—

#395

—

#395

—

#395

D₇

yes

characteristic

#391

—

#391

—

C₂³×C₇

C₂⁴

yes

characteristic

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

C₂×Q₈

yes

—

Dic₁₄

—

Dic₁₄

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₄²

—

—

—

—

C₂×D₇

yes

—

C₂²×C₇

C₂²×Q₈

yes

—

C₂²×C₇

H(5,𝔽₂)

yes

—

C₂²×C₇

H(5,𝔽₂)

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₂×C₇

#543

yes

—

C₂×C₇

Q₈²

yes

—

C₂×C₇

H(3,𝔽₄)

yes

characteristic

C₂³

C₂³×D₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Q₈×D₇

yes

—

C₇

#2811

yes

characteristic, complemented, Sylow

C₂²

#6548

yes

—

C₂²

C₂×Q₈×D₇

yes

—

100

C₂²

#6553

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₂

#90285

yes

characteristic

107

C₂

#90325

yes

—

108

C₂

#90413

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, prime radical, cosocle	C₂³×C₇	C₂⁴
Fitting	C₇×#395	C₂
hypercenter	#395	D₇
hypocenter	C₇	#2811
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #395

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇

commutator, socle, prime radical, cosocle

C₂³×C₇

C₂⁴

C₂

D₇

C₇

G ▷ C₂³×C₇ ▷ C₇

C₁ ◁ C₂³ ◁ #395

N	H ≅ G/N	type
26. C₂×Dic₁₄	Q₈	semidirect
97. C₇	#2811	semidirect

H ≅ G/N

type

26. C₂×Dic₁₄

Q₈

semidirect

97. C₇

#2811

semidirect

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

subgroup	index
C₇×#395	2
#90001	2
#89751	2
#89746	2
#89418	2
#2811	7

subgroup

index

Subgroups — #11734685

Conjugacy classes of subgroups

Complements

Maximal subgroups