Group #11734678 — subgroups

Subgroups — #11734678

Order 896.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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, complemented
2	#90003	C₂	1	2	yes	maximal, complemented
3	#89763	C₂	1	3	yes	maximal
4	#89749	C₂	1	3	yes	maximal, complemented
5	#89442	C₂	1	6	yes	maximal, complemented
6	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₂²×D₂₈	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₄⋊C₄	C₂²	1	3	yes	—
9	C₂×(C₂×D₇)⋊C₄	C₂²	1	6	yes	—
10	C₂×(C₂×D₇)⋊C₄	C₂²	1	6	yes	—
11	C₂×C₄⋊Dic₇	C₂²	1	3	yes	—
12	C₂×C₄×Dic₇	C₂²	1	3	yes	—
13	C₇×#95	C₂²	1	3	yes	—
14	C₇⋊#95	C₂²	1	6	yes	—
15	C₇⋊#95	C₂²	1	3	yes	—
16	#2785	—	7	1	no	maximal, Sylow
17	C₂³×D₇	C₂³	1	2	yes	—
18	C₂×C₇×Q₈	—	4	3	no	—
19	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₇	C₂³	1	3	yes	—
21	C₂²×C₄×C₇	C₂³	1	3	yes	—
22	C₂²×Dic₇	C₂³	1	6	yes	—
23	C₂×D₂₈	—	4	3	no	—
24	C₇×C₄⋊C₄	—	4	3	no	—
25	(C₂×D₇)⋊C₄	—	2	12	no	—
26	(C₂×D₇)⋊C₄	—	4	6	no	—
27	C₄⋊Dic₇	—	4	3	no	—
28	C₄×Dic₇	—	2	6	no	—
29	#395	D₇	1	1	yes	characteristic, complemented
30	#394	—	7	3	no	—
31	#393	—	7	2	no	—
32	#388	—	7	6	no	—
33	#382	—	7	3	no	—
34	C₂³×C₇	C₂⁴	1	1	yes	characteristic
35	C₂²×D₇	—	2	6	no	—
36	C₂²×D₇	—	2	6	no	—
37	C₂²×D₇	—	2	2	no	—
38	C₇×Q₈	—	4	4	no	—
39	C₂×C₄×C₇	—	2	3	no	—
40	C₂×C₄×C₇	—	2	3	no	—
41	C₂×C₄×C₇	—	2	6	no	—
42	C₂×C₄×C₇	—	2	3	no	—
43	C₂×C₄×C₇	—	2	3	no	—
44	C₂×C₄×C₇	#41	1	6	yes	—
45	C₂×Dic₇	—	2	6	no	—
46	C₂×Dic₇	—	2	6	no	—
47	C₂×Dic₇	—	2	6	no	—
48	D₂₈	—	4	4	no	—
49	C₂²×Q₈	C₂×D₇	1	1	yes	characteristic
50	C₂²×D₄	—	7	1	no	—
51	C₂×C₄⋊C₄	—	7	3	no	—
52	C₂×C₄⋊C₄	C₂×D₇	1	3	yes	—
53	C₂×C₂²⋊C₄	—	7	6	no	—
54	C₂×C₂²⋊C₄	—	7	6	no	—
55	C₂×C₄²	—	7	3	no	—
56	#95	—	7	3	no	—
57	#95	—	7	6	no	—
58	#95	C₂×D₇	1	3	yes	—
59	C₂²×C₇	C₂×#41	1	3	yes	—
60	C₂²×C₇	H(5,𝔽₂)	1	3	yes	—
61	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
62	C₂×D₇	—	4	6	no	—
63	C₂×D₇	—	4	6	no	—
64	C₂×D₇	—	4	2	no	—
65	C₄×C₇	—	2	6	no	—
66	C₄×C₇	—	4	1	no	—
67	C₄×C₇	—	4	3	no	—
68	Dic₇	—	4	6	no	—
69	C₂⁴	—	7	2	no	—
70	C₂×Q₈	—	4	3	no	—
71	C₂×D₄	—	28	3	no	—
72	C₂²×C₄	—	7	6	no	—
73	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
74	C₂²×C₄	C₂²×D₇	1	3	yes	—
75	C₂²×C₄	C₂²×D₇	1	3	yes	—
76	C₄⋊C₄	—	4	3	no	—
77	C₄⋊C₄	—	28	3	no	—
78	C₂²⋊C₄	—	14	12	no	—
79	C₂²⋊C₄	—	28	6	no	—
80	C₄²	—	14	6	no	—
81	C₂×C₇	#539	1	3	yes	—
82	C₂×C₇	#555	1	3	yes	—
83	C₂×C₇	H(3,𝔽₄)	1	1	yes	characteristic
84	D₇	—	8	2	no	—
85	C₂³	—	14	6	no	—
86	C₂³	—	14	6	no	—
87	C₂³	—	14	2	no	—
88	C₂³	C₂³×D₇	1	1	yes	characteristic
89	Q₈	—	4	4	no	—
90	D₄	—	28	4	no	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	2	6	no	—
94	C₂×C₄	—	2	3	no	—
95	C₂×C₄	—	2	3	no	—
96	C₂×C₄	—	14	6	no	—
97	C₂×C₄	—	14	6	no	—
98	C₂×C₄	—	14	6	no	—
99	C₂×C₄	#1161	1	6	yes	—
100	C₇	#2785	1	1	yes	characteristic, complemented, Sylow
101	C₂²	—	28	6	no	—
102	C₂²	—	28	2	no	—
103	C₂²	—	28	6	no	—
104	C₂²	#6548	1	3	yes	—
105	C₂²	C₂×#1161	1	3	yes	—
106	C₂²	#6554	1	1	yes	characteristic
107	C₄	—	2	6	no	—
108	C₄	—	4	1	no	—
109	C₄	—	4	3	no	—
110	C₄	—	28	6	no	—
111	C₂	—	56	2	no	—
112	C₂	#90285	1	1	yes	characteristic
113	C₂	#90336	1	3	yes	—
114	C₂	#90425	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#395

C₂

yes

maximal, characteristic, complemented

#90003

C₂

yes

maximal, complemented

#89763

C₂

yes

maximal

#89749

C₂

yes

maximal, complemented

#89442

C₂

yes

maximal, complemented

C₂²×C₇×Q₈

C₂²

yes

characteristic

C₂²×D₂₈

C₂²

yes

characteristic

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×(C₂×D₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₇)⋊C₄

C₂²

yes

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

—

C₇×#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

#2785

—

maximal, Sylow

C₂³×D₇

C₂³

yes

—

C₂×C₇×Q₈

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₂×D₂₈

—

C₇×C₄⋊C₄

—

(C₂×D₇)⋊C₄

—

(C₂×D₇)⋊C₄

—

C₄⋊Dic₇

—

C₄×Dic₇

—

#395

D₇

yes

characteristic, complemented

—

—

—

—

C₂³×C₇

C₂⁴

yes

characteristic

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#41

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

D₂₈

—

C₂²×Q₈

C₂×D₇

yes

characteristic

C₂²×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

#95

—

#95

C₂×D₇

yes

—

C₂²×C₇

C₂×#41

yes

—

C₂²×C₇

H(5,𝔽₂)

yes

—

C₂²×C₇

2₋¹⁺⁴

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂×C₇

#539

yes

—

C₂×C₇

#555

yes

—

C₂×C₇

H(3,𝔽₄)

yes

characteristic

D₇

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₇

yes

characteristic

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#1161

yes

—

100

C₇

#2785

yes

characteristic, complemented, Sylow

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

#6548

yes

—

105

C₂²

C₂×#1161

yes

—

106

C₂²

#6554

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₂

—

112

C₂

#90285

yes

characteristic

113

C₂

#90336

yes

—

114

C₂

#90425

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, cosocle	C₂³×C₇	C₂⁴
Fitting	C₇×#395	C₂
prime radical	C₂²×D₂₈	C₂²
hypercenter	#395	D₇
hypocenter	C₇	#2785
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #395

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇

commutator, socle, cosocle

C₂³×C₇

C₂⁴

Fitting

C₇×#395

C₂

prime radical

C₂²×D₂₈

C₂²

D₇

C₇

G ▷ C₂³×C₇ ▷ C₇

C₁ ◁ C₂³ ◁ #395

N	H ≅ G/N	type
1. C₇×#395	C₂	semidirect
2. #90003	C₂	semidirect
4. #89749	C₂	semidirect
5. #89442	C₂	semidirect
29. #395	D₇	semidirect
100. C₇	#2785	semidirect

H ≅ G/N

type

1. C₇×#395

C₂

semidirect

2. #90003

C₂

semidirect

4. #89749

C₂

semidirect

5. #89442

C₂

semidirect

29. #395

D₇

semidirect

100. C₇

#2785

semidirect

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

subgroup	index
C₇×#395	2
#90003	2
#89763	2
#89749	2
#89442	2
#2785	7

subgroup

index

Subgroups — #11734678

Conjugacy classes of subgroups

Complements

Maximal subgroups