Group #11735539 — subgroups

Subgroups — #11735539

Order 896.

372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 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×#399	C₂	1	1	yes	maximal, characteristic
2	#89756	C₂	1	6	yes	maximal, complemented
3	#89725	C₂	1	3	yes	maximal
4	#89718	C₂	1	3	yes	maximal
5	C₇⋊#401	C₂	1	2	yes	maximal, complemented
6	C₂³×Dic₇	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₄⋊C₄	C₂²	1	3	yes	—
8	C₂×C₇×C₂²⋊C₄	C₂²	1	3	yes	—
9	C₂×C₂²⋊Dic₇	C₂²	1	3	yes	—
10	C₂×C₄⋊Dic₇	C₂²	1	3	yes	—
11	C₂×Dic₇⋊C₄	C₂²	1	6	yes	—
12	C₇×#95	C₂²	1	1	yes	characteristic
13	C₇⋊#95	C₂²	1	6	yes	—
14	C₇⋊#95	C₂²	1	6	yes	—
15	C₇⋊#95	C₂²	1	3	yes	—
16	#2834	—	7	1	no	maximal, Sylow
17	C₂⁴×C₇	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₇	C₂³	1	3	yes	—
19	C₂²×C₄×C₇	C₂³	1	3	yes	—
20	C₂²×Dic₇	—	4	3	no	—
21	C₂²×Dic₇	C₂³	1	6	yes	—
22	C₂²×Dic₇	C₂³	1	2	yes	—
23	C₇×C₄⋊C₄	—	4	3	no	—
24	C₇×C₂²⋊C₄	—	4	3	no	—
25	C₂²⋊Dic₇	—	2	6	no	—
26	C₄⋊Dic₇	—	4	3	no	—
27	Dic₇⋊C₄	—	4	6	no	—
28	#401	—	7	2	no	—
29	#399	D₇	1	1	yes	characteristic
30	#398	—	7	6	no	—
31	#397	—	7	3	no	—
32	#385	—	7	3	no	—
33	C₂³×C₇	—	2	3	no	—
34	C₂³×C₇	—	2	3	no	—
35	C₂³×C₇	#41	1	2	yes	—
36	C₂³×C₇	C₂⁴	1	1	yes	characteristic
37	C₂×C₄×C₇	—	2	6	no	—
38	C₂×C₄×C₇	—	2	3	no	—
39	C₂×C₄×C₇	—	2	3	no	—
40	C₂×C₄×C₇	—	2	6	no	—
41	C₂×Dic₇	—	2	6	no	—
42	C₂×Dic₇	—	2	6	no	—
43	C₂×Dic₇	—	2	6	no	—
44	C₂×Dic₇	—	2	6	no	—
45	C₂×Dic₇	—	4	4	no	—
46	C₂³×C₄	—	7	1	no	—
47	C₂×C₄⋊C₄	—	7	6	no	—
48	C₂×C₄⋊C₄	—	7	3	no	—
49	C₂×C₄⋊C₄	C₂×D₇	1	3	yes	—
50	C₂×C₂²⋊C₄	—	7	3	no	—
51	C₂×C₂²⋊C₄	C₂×D₇	1	3	yes	—
52	#95	—	7	6	no	—
53	#95	—	7	3	no	—
54	#95	—	7	6	no	—
55	#95	C₂×D₇	1	1	yes	characteristic
56	C₂²×C₇	—	2	6	no	—
57	C₂²×C₇	—	4	1	no	—
58	C₂²×C₇	—	4	3	no	—
59	C₂²×C₇	C₂×#41	1	1	yes	characteristic
60	C₂²×C₇	H(5,𝔽₂)	1	3	yes	—
61	C₂²×C₇	2₋¹⁺⁴	1	3	yes	—
62	C₄×C₇	—	4	3	no	—
63	C₄×C₇	—	4	3	no	—
64	Dic₇	—	4	6	no	—
65	Dic₇	—	4	2	no	—
66	C₂⁴	C₂²×D₇	1	1	yes	characteristic
67	C₂²×C₄	—	7	6	no	—
68	C₂²×C₄	—	7	2	no	—
69	C₂²×C₄	—	28	3	no	—
70	C₂²×C₄	C₂²×D₇	1	3	yes	—
71	C₂²×C₄	C₂²×D₇	1	3	yes	—
72	C₄⋊C₄	—	4	3	no	—
73	C₄⋊C₄	—	28	6	no	—
74	C₄⋊C₄	—	28	3	no	—
75	C₂²⋊C₄	—	4	3	no	—
76	C₂²⋊C₄	—	14	6	no	—
77	C₂×C₇	—	4	2	no	—
78	C₂×C₇	#539	1	3	yes	—
79	C₂×C₇	#560	1	1	yes	characteristic
80	C₂×C₇	#563	1	3	yes	—
81	C₂³	—	2	3	no	—
82	C₂³	—	2	3	no	—
83	C₂³	#1161	1	2	yes	—
84	C₂³	C₂³×D₇	1	1	yes	characteristic
85	C₂×C₄	—	2	3	no	—
86	C₂×C₄	—	2	3	no	—
87	C₂×C₄	—	2	6	no	—
88	C₂×C₄	—	2	6	no	—
89	C₂×C₄	—	14	6	no	—
90	C₂×C₄	—	14	6	no	—
91	C₂×C₄	—	14	6	no	—
92	C₂×C₄	—	14	6	no	—
93	C₂×C₄	—	28	4	no	—
94	C₇	#2834	1	1	yes	characteristic, complemented, Sylow
95	C₂²	—	2	6	no	—
96	C₂²	—	4	1	no	—
97	C₂²	—	4	3	no	—
98	C₂²	#6548	1	3	yes	—
99	C₂²	C₂×#1161	1	1	yes	characteristic
100	C₂²	#6554	1	3	yes	—
101	C₄	—	4	3	no	—
102	C₄	—	4	3	no	—
103	C₄	—	28	2	no	—
104	C₄	—	28	6	no	—
105	C₂	—	4	2	no	—
106	C₂	#90336	1	3	yes	—
107	C₂	#90342	1	3	yes	—
108	C₂	#90411	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#399

C₂

yes

maximal, characteristic

#89756

C₂

yes

maximal, complemented

#89725

C₂

yes

maximal

#89718

C₂

yes

maximal

C₇⋊#401

C₂

yes

maximal, complemented

C₂³×Dic₇

C₂²

yes

characteristic

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₇×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₇

C₂²

yes

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×Dic₇⋊C₄

C₂²

yes

—

C₇×#95

C₂²

yes

characteristic

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

#2834

—

maximal, Sylow

C₂⁴×C₇

C₂³

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×Dic₇

—

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₇×C₄⋊C₄

—

C₇×C₂²⋊C₄

—

C₂²⋊Dic₇

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

#401

—

#399

D₇

yes

—

—

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

#41

yes

—

C₂³×C₇

C₂⁴

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂³×C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₇

yes

—

—

—

—

C₂×D₇

yes

characteristic

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂×#41

yes

characteristic

C₂²×C₇

H(5,𝔽₂)

yes

—

C₂²×C₇

2₋¹⁺⁴

yes

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

Dic₇

—

C₂⁴

C₂²×D₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂×C₇

—

C₂×C₇

#539

yes

—

C₂×C₇

#560

yes

characteristic

C₂×C₇

#563

yes

—

C₂³

—

C₂³

—

C₂³

#1161

yes

—

C₂³

C₂³×D₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₇

#2834

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

—

C₂²

#6548

yes

—

C₂²

C₂×#1161

yes

characteristic

100

C₂²

#6554

yes

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₂

—

106

C₂

#90336

yes

—

107

C₂

#90342

yes

—

108

C₂

#90411

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, cosocle	C₂³×C₇	C₂⁴
Fitting	C₇×#399	C₂
prime radical	C₂⁴×C₇	C₂³
hypercenter	#399	D₇
hypocenter	C₇	#2834
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #399

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇

commutator, socle, cosocle

C₂³×C₇

C₂⁴

Fitting

C₇×#399

C₂

prime radical

C₂⁴×C₇

C₂³

D₇

C₇

G ▷ C₂³×C₇ ▷ C₇

C₁ ◁ C₂³ ◁ #399

N	H ≅ G/N	type
2. #89756	C₂	semidirect
5. C₇⋊#401	C₂	semidirect
94. C₇	#2834	semidirect

H ≅ G/N

type

2. #89756

C₂

semidirect

5. C₇⋊#401

C₂

semidirect

94. C₇

#2834

semidirect

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

subgroup	index
C₇×#399	2
#89756	2
#89725	2
#89718	2
C₇⋊#401	2
#2834	7

subgroup

index

Subgroups — #11735539

Conjugacy classes of subgroups

Complements

Maximal subgroups