Group #11734304 — subgroups

Subgroups — #11734304

Order 896.

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

normal subgroups only

This group has 380 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×#393	C₂	1	1	yes	maximal, characteristic
2	#90001	C₂	1	2	yes	maximal, complemented
3	#89725	C₂	1	3	yes	maximal
4	#89718	C₂	1	3	yes	maximal
5	#89432	C₂	1	6	yes	maximal, complemented
6	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₂³×Dic₇	C₂²	1	1	yes	characteristic
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	3	yes	—
13	C₇⋊#95	C₂²	1	6	yes	—
14	C₇⋊#95	C₂²	1	3	yes	—
15	C₇⋊#95	C₂²	1	6	yes	—
16	#2835	—	7	1	no	maximal, Sylow
17	C₂⁴×C₇	C₂³	1	1	yes	characteristic
18	C₂×C₇×Q₈	—	4	3	no	—
19	C₂²×C₄×C₇	C₂³	1	3	yes	—
20	C₂²×C₄×C₇	C₂³	1	3	yes	—
21	C₂²×Dic₇	—	4	3	no	—
22	C₂²×Dic₇	C₂³	1	6	yes	—
23	C₂²×Dic₇	C₂³	1	2	yes	—
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	#400	—	7	6	no	—
29	#397	—	7	3	no	—
30	#395	—	7	2	no	—
31	#393	D₇	1	1	yes	characteristic
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	C₇×Q₈	—	4	4	no	—
38	C₂×C₄×C₇	—	2	6	no	—
39	C₂×C₄×C₇	—	2	3	no	—
40	C₂×C₄×C₇	—	2	3	no	—
41	C₂×C₄×C₇	—	2	6	no	—
42	C₂×Dic₇	—	2	6	no	—
43	C₂×Dic₇	—	2	6	no	—
44	C₂×Dic₇	—	2	6	no	—
45	C₂×Dic₇	—	2	6	no	—
46	C₂×Dic₇	—	4	4	no	—
47	C₂²×Q₈	C₂×D₇	1	1	yes	characteristic
48	C₂³×C₄	—	7	1	no	—
49	C₂×C₄⋊C₄	—	7	6	no	—
50	C₂×C₄⋊C₄	—	7	3	no	—
51	C₂×C₂²⋊C₄	—	7	3	no	—
52	C₂×C₂²⋊C₄	C₂×D₇	1	3	yes	—
53	#95	—	7	6	no	—
54	#95	—	7	6	no	—
55	#95	—	7	3	no	—
56	#95	C₂×D₇	1	3	yes	—
57	C₂²×C₇	—	2	6	no	—
58	C₂²×C₇	—	4	1	no	—
59	C₂²×C₇	—	4	3	no	—
60	C₂²×C₇	C₂×#41	1	1	yes	characteristic
61	C₂²×C₇	H(5,𝔽₂)	1	3	yes	—
62	C₂²×C₇	2₋¹⁺⁴	1	3	yes	—
63	C₄×C₇	—	4	3	no	—
64	C₄×C₇	—	4	3	no	—
65	Dic₇	—	4	6	no	—
66	Dic₇	—	4	2	no	—
67	C₂⁴	C₂²×D₇	1	1	yes	characteristic
68	C₂×Q₈	—	4	3	no	—
69	C₂²×C₄	—	7	6	no	—
70	C₂²×C₄	—	7	2	no	—
71	C₂²×C₄	—	28	3	no	—
72	C₂²×C₄	C₂²×D₇	1	3	yes	—
73	C₂²×C₄	C₂²×D₇	1	3	yes	—
74	C₄⋊C₄	—	28	6	no	—
75	C₄⋊C₄	—	28	3	no	—
76	C₂²⋊C₄	—	4	3	no	—
77	C₂²⋊C₄	—	14	6	no	—
78	C₂×C₇	—	4	2	no	—
79	C₂×C₇	#539	1	3	yes	—
80	C₂×C₇	H(3,𝔽₄)	1	1	yes	characteristic
81	C₂×C₇	#563	1	3	yes	—
82	C₂³	—	2	3	no	—
83	C₂³	—	2	3	no	—
84	C₂³	#1161	1	2	yes	—
85	C₂³	C₂³×D₇	1	1	yes	characteristic
86	Q₈	—	4	4	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	2	3	no	—
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	2	6	no	—
91	C₂×C₄	—	14	6	no	—
92	C₂×C₄	—	14	6	no	—
93	C₂×C₄	—	14	6	no	—
94	C₂×C₄	—	14	6	no	—
95	C₂×C₄	—	28	4	no	—
96	C₇	#2835	1	1	yes	characteristic, complemented, Sylow
97	C₂²	—	2	6	no	—
98	C₂²	—	4	1	no	—
99	C₂²	—	4	3	no	—
100	C₂²	#6548	1	3	yes	—
101	C₂²	C₂×#1161	1	1	yes	characteristic
102	C₂²	#6554	1	3	yes	—
103	C₄	—	4	3	no	—
104	C₄	—	4	3	no	—
105	C₄	—	28	2	no	—
106	C₄	—	28	6	no	—
107	C₂	—	4	2	no	—
108	C₂	#90285	1	1	yes	characteristic
109	C₂	#90336	1	3	yes	—
110	C₂	#90363	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#393

C₂

yes

maximal, characteristic

#90001

C₂

yes

maximal, complemented

#89725

C₂

yes

maximal

#89718

C₂

yes

maximal

#89432

C₂

yes

maximal, complemented

C₂²×C₇×Q₈

C₂²

yes

characteristic

C₂³×Dic₇

C₂²

yes

characteristic

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

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

#2835

—

maximal, Sylow

C₂⁴×C₇

C₂³

yes

characteristic

C₂×C₇×Q₈

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×Dic₇

—

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₇×C₂²⋊C₄

—

C₂²⋊Dic₇

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

—

—

—

D₇

yes

characteristic

#385

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

#41

yes

—

C₂³×C₇

C₂⁴

yes

characteristic

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂²×Q₈

C₂×D₇

yes

characteristic

C₂³×C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₇

yes

—

—

—

—

C₂×D₇

yes

—

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

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂×C₇

—

C₂×C₇

#539

yes

—

C₂×C₇

H(3,𝔽₄)

yes

characteristic

C₂×C₇

#563

yes

—

C₂³

—

C₂³

—

C₂³

#1161

yes

—

C₂³

C₂³×D₇

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₇

#2835

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

—

100

C₂²

#6548

yes

—

101

C₂²

C₂×#1161

yes

characteristic

102

C₂²

#6554

yes

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₂

—

108

C₂

#90285

yes

characteristic

109

C₂

#90336

yes

—

110

C₂

#90363

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, cosocle	C₂³×C₇	C₂⁴
Fitting	C₇×#393	C₂
prime radical	C₂⁴×C₇	C₂³
hypercenter	#393	D₇
hypocenter	C₇	#2835
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #393

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇

commutator, socle, cosocle

C₂³×C₇

C₂⁴

Fitting

C₇×#393

C₂

prime radical

C₂⁴×C₇

C₂³

D₇

C₇

G ▷ C₂³×C₇ ▷ C₇

C₁ ◁ C₂³ ◁ #393

N	H ≅ G/N	type
2. #90001	C₂	semidirect
5. #89432	C₂	semidirect
96. C₇	#2835	semidirect

H ≅ G/N

type

2. #90001

C₂

semidirect

5. #89432

C₂

semidirect

96. C₇

#2835

semidirect

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

subgroup	index
C₇×#393	2
#90001	2
#89725	2
#89718	2
#89432	2
#2835	7

subgroup

index

Subgroups — #11734304

Conjugacy classes of subgroups

Complements

Maximal subgroups