Group #11731663 — subgroups

Subgroups — #11731663

Order 896.

356 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 133 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×#382	C₂	1	1	yes	maximal, characteristic
2	#89758	C₂	1	2	yes	maximal
3	#89751	C₂	1	2	yes	maximal
4	#89708	C₂	1	2	yes	maximal
5	#89432	C₂	1	4	yes	maximal
6	#89428	C₂	1	4	yes	maximal
7	C₂²×Dic₁₄	C₂²	1	2	yes	—
8	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
14	C₂×Dic₇⋊C₄	C₂²	1	4	yes	—
15	C₇×#95	C₂²	1	4	yes	—
16	C₇⋊#95	C₂²	1	4	yes	—
17	C₇⋊#95	C₂²	1	4	yes	—
18	C₇⋊#95	C₂²	1	2	yes	—
19	C₇⋊#95	C₂²	1	2	yes	—
20	C₇⋊#95	C₂²	1	4	yes	—
21	#2844	—	7	1	no	maximal, Sylow
22	C₂²×C₄×C₇	C₂³	1	2	yes	—
23	C₂²×C₄×C₇	C₂³	1	2	yes	—
24	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₇	C₂³	1	2	yes	—
26	C₂²×Dic₇	C₂³	1	4	yes	—
27	C₂²×Dic₇	C₂³	1	4	yes	—
28	C₂×Dic₁₄	—	4	4	no	—
29	C₂×Dic₁₄	—	4	2	no	—
30	C₇×C₄⋊C₄	—	4	1	no	—
31	C₇×C₄⋊C₄	—	4	1	no	—
32	C₄²×C₇	Q₈	1	4	yes	complemented
33	C₄⋊Dic₇	—	4	2	no	—
34	C₄⋊Dic₇	—	4	2	no	—
35	C₄⋊Dic₇	—	4	2	no	—
36	Dic₇⋊C₄	—	4	4	no	—
37	#400	—	7	2	no	—
38	#400	—	7	4	no	—
39	#395	—	7	2	no	—
40	#395	—	7	4	no	—
41	#382	—	7	2	no	—
42	#382	D₇	1	1	yes	characteristic
43	C₂³×C₇	C₂⁴	1	1	yes	characteristic
44	C₂×C₄×C₇	—	2	2	no	—
45	C₂×C₄×C₇	—	2	1	no	—
46	C₂×C₄×C₇	—	2	1	no	—
47	C₂×C₄×C₇	—	2	2	no	—
48	C₂×C₄×C₇	—	2	4	no	—
49	C₂×C₄×C₇	—	2	2	no	—
50	C₂×C₄×C₇	—	2	2	no	—
51	C₂×C₄×C₇	—	2	4	no	—
52	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
54	C₂×Dic₇	—	2	4	no	—
55	C₂×Dic₇	—	2	4	no	—
56	C₂×Dic₇	—	2	4	no	—
57	C₂×Dic₇	—	2	4	no	—
58	C₂×Dic₇	—	2	4	no	—
59	C₂×Dic₇	—	2	4	no	—
60	Dic₁₄	—	4	8	no	—
61	C₂²×Q₈	—	7	2	no	—
62	C₂×C₄⋊C₄	—	7	4	no	—
63	C₂×C₄⋊C₄	—	7	2	no	—
64	C₂×C₄⋊C₄	—	7	2	no	—
65	C₂×C₄⋊C₄	—	7	2	no	—
66	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
67	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
68	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
69	#95	—	7	4	no	—
70	#95	—	7	4	no	—
71	#95	—	7	2	no	—
72	#95	—	7	4	no	—
73	#95	—	7	2	no	—
74	#95	C₂×D₇	1	4	yes	—
75	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
76	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
77	C₂²×C₇	H(5,𝔽₂)	1	2	yes	—
78	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
79	C₂²×C₇	2₋¹⁺⁴	1	2	yes	—
80	C₄×C₇	—	2	2	no	—
81	C₄×C₇	—	2	4	no	—
82	C₄×C₇	—	4	2	no	—
83	C₄×C₇	—	4	2	no	—
84	Dic₇	—	4	4	no	—
85	Dic₇	—	4	4	no	—
86	C₂×Q₈	—	28	2	no	—
87	C₂×Q₈	—	28	4	no	—
88	C₂²×C₄	—	7	4	no	—
89	C₂²×C₄	—	7	4	no	—
90	C₂²×C₄	C₂²×D₇	1	2	yes	—
91	C₂²×C₄	C₂²×D₇	1	2	yes	—
92	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₇	1	2	yes	—
94	C₄⋊C₄	—	4	1	no	—
95	C₄⋊C₄	—	4	1	no	—
96	C₄⋊C₄	—	28	4	no	—
97	C₄⋊C₄	—	28	2	no	—
98	C₄⋊C₄	—	28	2	no	—
99	C₄⋊C₄	—	28	2	no	—
100	C₄²	Dic₁₄	1	4	yes	complemented
101	C₂×C₇	#544	1	1	yes	characteristic
102	C₂×C₇	#544	1	2	yes	—
103	C₂×C₇	H(3,𝔽₄)	1	1	yes	characteristic
104	C₂×C₇	#563	1	2	yes	—
105	C₂×C₇	#563	1	1	yes	characteristic
106	C₂³	C₂³×D₇	1	1	yes	characteristic
107	Q₈	—	28	8	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	14	4	no	—
117	C₂×C₄	—	14	4	no	—
118	C₂×C₄	—	14	4	no	—
119	C₂×C₄	—	14	4	no	—
120	C₂×C₄	—	14	4	no	—
121	C₂×C₄	—	14	4	no	—
122	C₂×C₄	C₂×Dic₁₄	1	2	yes	—
123	C₂×C₄	C₂×Dic₁₄	1	4	yes	—
124	C₇	#2844	1	1	yes	characteristic, complemented, Sylow
125	C₂²	C₂²×Dic₁₄	1	1	yes	characteristic
126	C₂²	#6548	1	1	yes	characteristic
127	C₂²	#6551	1	1	yes	characteristic
128	C₂²	#6553	1	2	yes	—
129	C₂²	#6554	1	2	yes	—
130	C₄	—	2	2	no	—
131	C₄	—	2	4	no	—
132	C₄	—	4	2	no	—
133	C₄	—	4	2	no	—
134	C₄	—	28	4	no	—
135	C₄	—	28	4	no	—
136	C₂	#90191	1	1	yes	characteristic
137	C₂	#90210	1	2	yes	—
138	C₂	#90363	1	1	yes	characteristic
139	C₂	#90374	1	1	yes	characteristic
140	C₂	#90390	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#382

C₂

yes

maximal, characteristic

#89758

C₂

yes

maximal

#89751

C₂

yes

maximal

#89708

C₂

yes

maximal

#89432

C₂

yes

maximal

#89428

C₂

yes

maximal

C₂²×Dic₁₄

C₂²

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₇×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄²×C₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×C₄⋊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

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

#2844

—

maximal, Sylow

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

Q₈

yes

complemented

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

—

—

—

—

—

D₇

yes

characteristic

C₂³×C₇

C₂⁴

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

characteristic

C₂×C₄⋊C₄

C₂×D₇

yes

characteristic

C₂×C₄²

C₂×D₇

yes

—

—

—

—

—

C₂×D₇

yes

—

C₂²×C₇

C₂²×Q₈

yes

C₂²×C₇

yes

C₂²×C₇

yes

—

C₂²×C₇

2₋¹⁺⁴

yes

characteristic

C₂²×C₇

2₋¹⁺⁴

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂²×C₄

C₂²×D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

Dic₁₄

yes

complemented

101

C₂×C₇

#544

yes

characteristic

102

C₂×C₇

#544

yes

—

103

C₂×C₇

H(3,𝔽₄)

yes

characteristic

104

C₂×C₇

#563

yes

—

105

C₂×C₇

#563

yes

characteristic

106

C₂³

C₂³×D₇

yes

characteristic

107

Q₈

—

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

C₂×C₄

C₂×Dic₁₄

yes

—

123

C₂×C₄

C₂×Dic₁₄

yes

—

124

C₇

#2844

yes

characteristic, complemented, Sylow

125

C₂²

C₂²×Dic₁₄

yes

characteristic

126

C₂²

#6548

yes

characteristic

127

C₂²

#6551

yes

characteristic

128

C₂²

#6553

yes

—

129

C₂²

#6554

yes

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₂

#90191

yes

characteristic

137

C₂

#90210

yes

—

138

C₂

#90363

yes

characteristic

139

C₂

#90374

yes

characteristic

140

C₂

#90390

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, prime radical, cosocle	C₂³×C₇	C₂⁴
Fitting	C₇×#382	C₂
hypercenter	#382	D₇
hypocenter	C₇	#2844
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #382

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 ◁ #382

N	H ≅ G/N	type
32. C₄²×C₇	Q₈	semidirect
100. C₄²	Dic₁₄	semidirect
124. C₇	#2844	semidirect

H ≅ G/N

type

32. C₄²×C₇

Q₈

semidirect

100. C₄²

Dic₁₄

semidirect

124. C₇

#2844

semidirect

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

subgroup	index
C₇×#382	2
#89758	2
#89751	2
#89708	2
#89432	2
#89428	2
#2844	7

subgroup

index

Subgroups — #11731663

Conjugacy classes of subgroups

Complements

Maximal subgroups