Group #11735883 — subgroups

Subgroups — #11735883

Order 896.

372 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 115 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×#401	C₂	1	1	yes	maximal, characteristic, complemented
2	#89449	C₂	1	1	yes	maximal, characteristic
3	#89447	C₂	1	3	yes	maximal
4	#89440	C₂	1	3	yes	maximal
5	C₇⋊#401	C₂	1	1	yes	maximal, characteristic, complemented
6	#89433	C₂	1	3	yes	maximal, complemented
7	#89429	C₂	1	3	yes	maximal, complemented
8	C₂²×C₄×D₇	C₂²	1	1	yes	characteristic
9	C₂×(C₂×D₇)⋊C₄	C₂²	1	3	yes	—
10	C₂×(C₂×D₇)⋊C₄	C₂²	1	3	yes	—
11	C₂×C₄⋊Dic₇	C₂²	1	3	yes	—
12	C₂×C₄⋊Dic₇	C₂²	1	3	yes	—
13	C₂×Dic₇⋊C₄	C₂²	1	3	yes	—
14	C₂×Dic₇⋊C₄	C₂²	1	3	yes	—
15	C₇×#95	C₂²	1	3	yes	—
16	C₇×#95	C₂²	1	3	yes	—
17	C₇×#95	C₂²	1	1	yes	characteristic
18	C₇⋊#95	C₂²	1	3	yes	—
19	C₇⋊#95	C₂²	1	3	yes	—
20	C₇⋊#95	C₂²	1	3	yes	—
21	#2834	—	7	1	no	maximal, Sylow
22	C₂³×D₇	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₇	C₂³	1	3	yes	—
24	C₂²×C₄×C₇	C₂³	1	3	yes	—
25	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₇	C₂³	1	3	yes	—
27	C₂²×Dic₇	C₂³	1	1	yes	characteristic
28	C₂²×Dic₇	C₂³	1	3	yes	—
29	C₂×C₄×D₇	—	4	3	no	—
30	(C₂×D₇)⋊C₄	—	2	6	no	—
31	(C₂×D₇)⋊C₄	—	4	3	no	—
32	C₄⋊Dic₇	—	4	3	no	—
33	C₄⋊Dic₇	—	4	3	no	—
34	Dic₇⋊C₄	—	4	3	no	—
35	Dic₇⋊C₄	—	4	3	no	—
36	#401	—	7	1	no	—
37	#401	D₇	1	1	yes	characteristic, complemented
38	#399	—	7	1	no	—
39	#398	—	7	3	no	—
40	#398	—	7	3	no	—
41	#397	—	7	3	no	—
42	#385	—	7	3	no	—
43	C₂³×C₇	C₂⁴	1	1	yes	characteristic
44	C₂²×D₇	—	2	3	no	—
45	C₂²×D₇	—	2	3	no	—
46	C₂²×D₇	#41	1	2	yes	—
47	C₂×C₄×C₇	—	2	3	no	—
48	C₂×C₄×C₇	—	2	3	no	—
49	C₂×C₄×C₇	—	2	3	no	—
50	C₂×C₄×C₇	—	2	3	no	—
51	C₂×C₄×C₇	—	2	3	no	—
52	C₂×C₄×C₇	—	2	6	no	—
53	C₂×Dic₇	—	2	3	no	—
54	C₂×Dic₇	—	2	3	no	—
55	C₂×Dic₇	—	2	3	no	—
56	C₂×Dic₇	—	2	3	no	—
57	C₂×Dic₇	—	2	3	no	—
58	C₂×Dic₇	—	2	3	no	—
59	C₂×Dic₇	—	2	3	no	—
60	C₄×D₇	—	4	4	no	—
61	C₂³×C₄	—	7	1	no	—
62	C₂×C₄⋊C₄	—	7	3	no	—
63	C₂×C₄⋊C₄	—	7	3	no	—
64	C₂×C₄⋊C₄	—	7	3	no	—
65	C₂×C₄⋊C₄	—	7	3	no	—
66	C₂×C₂²⋊C₄	—	7	3	no	—
67	C₂×C₂²⋊C₄	—	7	3	no	—
68	#95	—	7	3	no	—
69	#95	—	7	3	no	—
70	#95	—	7	3	no	—
71	#95	C₂×D₇	1	3	yes	—
72	#95	C₂×D₇	1	3	yes	—
73	#95	C₂×D₇	1	1	yes	characteristic
74	C₂²×C₇	C₂×#41	1	1	yes	characteristic
75	C₂²×C₇	H(5,𝔽₂)	1	3	yes	—
76	C₂²×C₇	2₋¹⁺⁴	1	3	yes	—
77	C₂×D₇	—	2	6	no	—
78	C₂×D₇	—	4	1	no	—
79	C₂×D₇	—	4	3	no	—
80	C₄×C₇	—	4	3	no	—
81	C₄×C₇	—	4	1	no	—
82	C₄×C₇	—	4	3	no	—
83	Dic₇	—	4	3	no	—
84	Dic₇	—	4	1	no	—
85	Dic₇	—	4	3	no	—
86	C₂⁴	—	7	1	no	—
87	C₂²×C₄	—	7	3	no	—
88	C₂²×C₄	—	7	1	no	—
89	C₂²×C₄	—	7	3	no	—
90	C₂²×C₄	—	28	3	no	—
91	C₂²×C₄	C₂²×D₇	1	3	yes	—
92	C₂²×C₄	C₂²×D₇	1	3	yes	—
93	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
94	C₄⋊C₄	—	28	3	no	—
95	C₄⋊C₄	—	28	3	no	—
96	C₄⋊C₄	—	28	3	no	—
97	C₄⋊C₄	—	28	3	no	—
98	C₂²⋊C₄	—	14	6	no	—
99	C₂²⋊C₄	—	28	3	no	—
100	C₂×C₇	#539	1	3	yes	—
101	C₂×C₇	#560	1	1	yes	characteristic
102	C₂×C₇	#563	1	3	yes	—
103	D₇	—	4	2	no	—
104	C₂³	—	7	2	no	—
105	C₂³	—	14	3	no	—
106	C₂³	—	14	3	no	—
107	C₂³	C₂³×D₇	1	1	yes	characteristic
108	C₂×C₄	—	2	3	no	—
109	C₂×C₄	—	2	3	no	—
110	C₂×C₄	—	2	3	no	—
111	C₂×C₄	—	2	3	no	—
112	C₂×C₄	—	2	3	no	—
113	C₂×C₄	—	2	6	no	—
114	C₂×C₄	—	14	3	no	—
115	C₂×C₄	—	14	3	no	—
116	C₂×C₄	—	14	3	no	—
117	C₂×C₄	—	14	3	no	—
118	C₂×C₄	—	14	3	no	—
119	C₂×C₄	—	14	3	no	—
120	C₂×C₄	—	14	3	no	—
121	C₂×C₄	—	28	4	no	—
122	C₇	#2834	1	1	yes	characteristic, complemented, Sylow
123	C₂²	—	14	6	no	—
124	C₂²	—	28	1	no	—
125	C₂²	—	28	3	no	—
126	C₂²	D₇×#41	1	1	yes	characteristic
127	C₂²	#6553	1	3	yes	—
128	C₂²	#6554	1	3	yes	—
129	C₄	—	4	3	no	—
130	C₄	—	4	1	no	—
131	C₄	—	4	3	no	—
132	C₄	—	28	3	no	—
133	C₄	—	28	1	no	—
134	C₄	—	28	3	no	—
135	C₂	—	28	2	no	—
136	C₂	#90398	1	3	yes	—
137	C₂	#90402	1	1	yes	characteristic
138	C₂	#90403	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#401

C₂

yes

maximal, characteristic, complemented

#89449

C₂

yes

maximal, characteristic

#89447

C₂

yes

maximal

#89440

C₂

yes

maximal

C₇⋊#401

C₂

yes

maximal, characteristic, complemented

#89433

C₂

yes

maximal, complemented

#89429

C₂

yes

maximal, complemented

C₂²×C₄×D₇

C₂²

yes

characteristic

C₂×(C₂×D₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₇)⋊C₄

C₂²

yes

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×Dic₇⋊C₄

C₂²

yes

—

C₂×Dic₇⋊C₄

C₂²

yes

—

C₇×#95

C₂²

yes

—

C₇×#95

C₂²

yes

—

C₇×#95

C₂²

yes

characteristic

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

#2834

—

maximal, Sylow

C₂³×D₇

C₂³

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

characteristic

C₂²×Dic₇

C₂³

yes

—

C₂×C₄×D₇

—

(C₂×D₇)⋊C₄

—

(C₂×D₇)⋊C₄

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

Dic₇⋊C₄

—

#401

—

#401

D₇

yes

characteristic, complemented

—

—

—

—

—

C₂³×C₇

C₂⁴

yes

characteristic

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

#41

yes

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₄×D₇

—

C₂³×C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

—

—

—

C₂×D₇

yes

—

#95

C₂×D₇

yes

—

#95

C₂×D₇

yes

characteristic

C₂²×C₇

C₂×#41

yes

characteristic

C₂²×C₇

H(5,𝔽₂)

yes

—

C₂²×C₇

2₋¹⁺⁴

yes

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂×C₇

#539

yes

—

101

C₂×C₇

#560

yes

characteristic

102

C₂×C₇

#563

yes

—

103

D₇

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

C₂³×D₇

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₇

#2834

yes

characteristic, complemented, Sylow

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

D₇×#41

yes

characteristic

127

C₂²

#6553

yes

—

128

C₂²

#6554

yes

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₂

—

136

C₂

#90398

yes

—

137

C₂

#90402

yes

characteristic

138

C₂

#90403

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, cosocle	C₂³×C₇	C₂⁴
Fitting	C₇×#401	C₂
prime radical	C₂³×D₇	C₂³
hypercenter	#401	D₇
hypocenter	C₇	#2834
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #401

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇

commutator, socle, cosocle

C₂³×C₇

C₂⁴

Fitting

C₇×#401

C₂

prime radical

C₂³×D₇

C₂³

D₇

C₇

G ▷ C₂³×C₇ ▷ C₇

C₁ ◁ C₂³ ◁ #401

N	H ≅ G/N	type
1. C₇×#401	C₂	semidirect
5. C₇⋊#401	C₂	semidirect
6. #89433	C₂	semidirect
7. #89429	C₂	semidirect
37. #401	D₇	semidirect
122. C₇	#2834	semidirect

H ≅ G/N

type

1. C₇×#401

C₂

semidirect

5. C₇⋊#401

C₂

semidirect

6. #89433

C₂

semidirect

7. #89429

C₂

semidirect

37. #401

D₇

semidirect

122. C₇

#2834

semidirect

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

subgroup	index
C₇×#401	2
#89449	2
#89447	2
#89440	2
C₇⋊#401	2
#89433	2
#89429	2
#2834	7

subgroup

index

Subgroups — #11735883

Conjugacy classes of subgroups

Complements

Maximal subgroups