Group #11726618 — subgroups

Subgroups — #11726618

Order 896.

288 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×Dic₂₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#89746	C₂	1	2	yes	maximal
4	C₂×#6384	C₂	1	2	yes	maximal
5	C₇×#340	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₄	C₂²	1	2	yes	—
7	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
10	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₈	—	2	4	no	—
12	C₂×Dic₂₈	—	2	2	no	—
13	C₇⋊#95	—	2	2	no	—
14	C₇⋊#95	—	2	2	no	—
15	Dic₇⋊C₈	—	2	2	no	—
16	#6384	—	2	4	no	—
17	#2056	—	7	1	no	maximal, Sylow
18	C₂²×C₄×C₇	—	2	2	no	—
19	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂²×Dic₇	—	2	2	no	—
21	C₂²×Dic₇	D₄	1	1	yes	characteristic
22	C₂²×Dic₇	D₄	1	1	yes	characteristic
23	C₂×Dic₁₄	—	2	2	no	—
24	C₂×Dic₁₄	—	2	2	no	—
25	C₂×Dic₁₄	—	2	2	no	—
26	C₂×Dic₁₄	—	2	2	no	—
27	C₂×Dic₁₄	—	2	2	no	—
28	C₂×Dic₁₄	D₄	1	4	yes	—
29	C₂×C₇×C₈	—	2	2	no	—
30	C₂×C₇×C₈	D₄	1	2	yes	—
31	C₇×C₄⋊C₄	—	2	2	no	—
32	C₇×C₄⋊C₄	—	2	2	no	—
33	C₄×Dic₇	—	2	1	no	—
34	C₄×Dic₇	—	2	1	no	—
35	C₂×C₇⋊C₈	—	2	2	no	—
36	C₂×C₇⋊C₈	—	2	1	no	—
37	Dic₂₈	—	2	8	no	—
38	C₂²×Q₁₆	—	7	1	no	—
39	C₂×C₄⋊C₈	—	7	1	no	—
40	C₂×#103	—	7	2	no	—
41	#391	—	7	2	no	—
42	#340	D₇	1	1	yes	characteristic
43	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇	—	4	2	no	—
45	C₂×C₄×C₇	—	4	2	no	—
46	C₂×C₄×C₇	—	4	2	no	—
47	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₇	#41	1	2	yes	—
50	C₂×C₄×C₇	#41	1	1	yes	characteristic
51	C₂×Dic₇	—	2	1	no	—
52	C₂×Dic₇	—	2	1	no	—
53	C₂×Dic₇	—	2	1	no	—
54	C₂×Dic₇	—	2	4	no	—
55	C₂×Dic₇	—	2	2	no	—
56	C₂×Dic₇	—	4	2	no	—
57	C₂×Dic₇	—	4	2	no	—
58	C₂×Dic₇	Q₁₆	1	2	yes	complemented
59	Dic₁₄	—	2	4	no	—
60	Dic₁₄	—	2	4	no	—
61	Dic₁₄	—	4	2	no	—
62	Dic₁₄	—	4	2	no	—
63	C₇×C₈	—	2	2	no	—
64	C₇⋊C₈	—	4	1	no	—
65	C₂²×Q₈	—	7	2	no	—
66	C₂×Q₁₆	—	14	4	no	—
67	C₂×Q₁₆	—	14	2	no	—
68	C₂²×C₈	—	7	1	no	—
69	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
70	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
71	C₂×C₄²	—	7	1	no	—
72	C₄⋊C₈	—	14	2	no	—
73	#103	—	14	4	no	—
74	#95	—	14	2	no	—
75	#95	—	14	2	no	—
76	C₂²×C₇	#121	1	2	yes	—
77	C₂²×C₇	#121	1	1	yes	characteristic
78	C₂²×C₇	C₂×Q₁₆	1	1	yes	characteristic
79	C₂²×C₇	#135	1	1	yes	characteristic
80	C₂²×C₇	#137	1	2	yes	—
81	C₄×C₇	—	8	2	no	—
82	C₄×C₇	#120	1	1	yes	characteristic
83	C₄×C₇	#123	1	2	yes	—
84	C₄×C₇	#124	1	1	yes	characteristic
85	Dic₇	—	2	2	no	—
86	Dic₇	—	4	4	no	—
87	Dic₇	—	4	1	no	—
88	C₂×Q₈	—	7	4	no	—
89	C₂×Q₈	—	14	2	no	—
90	C₂×Q₈	—	14	2	no	—
91	C₂×Q₈	—	14	2	no	—
92	C₂×Q₈	—	14	2	no	—
93	C₂×Q₈	—	14	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	7	1	no	—
96	C₂²×C₄	—	7	1	no	—
97	C₂²×C₄	—	14	2	no	—
98	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
99	Q₁₆	—	14	8	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	14	1	no	—
102	C₂×C₈	—	14	2	no	—
103	C₂×C₈	D₂₈	1	2	yes	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄²	—	14	1	no	—
107	C₄²	—	14	1	no	—
108	C₂×C₇	#394	1	1	yes	characteristic
109	C₂×C₇	#462	1	2	yes	—
110	C₂×C₇	#464	1	2	yes	—
111	C₂×C₇	#467	1	1	yes	characteristic
112	C₂×C₇	#470	1	1	yes	characteristic
113	C₂³	D₄×D₇	1	1	yes	characteristic
114	Q₈	—	14	4	no	—
115	Q₈	—	14	4	no	—
116	Q₈	—	28	2	no	—
117	Q₈	—	28	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	7	2	no	—
122	C₂×C₄	—	14	1	no	—
123	C₂×C₄	—	14	1	no	—
124	C₂×C₄	—	14	4	no	—
125	C₂×C₄	—	14	1	no	—
126	C₂×C₄	—	14	2	no	—
127	C₂×C₄	—	28	2	no	—
128	C₂×C₄	—	28	2	no	—
129	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
130	C₂×C₄	#1157	1	2	yes	—
131	C₂×C₄	D₄×D₇	1	2	yes	—
132	C₂×C₄	#1161	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	28	1	no	—
135	C₇	#2056	1	1	yes	characteristic, complemented, Sylow
136	C₂²	#6447	1	2	yes	—
137	C₂²	#6458	1	1	yes	characteristic
138	C₂²	#6475	1	1	yes	characteristic
139	C₂²	#6478	1	2	yes	—
140	C₂²	D₇×Q₁₆	1	1	yes	characteristic
141	C₄	—	8	2	no	—
142	C₄	—	14	2	no	—
143	C₄	—	28	1	no	—
144	C₄	—	28	4	no	—
145	C₄	#6438	1	1	yes	characteristic
146	C₄	#6445	1	1	yes	characteristic
147	C₄	#6457	1	2	yes	—
148	C₂	#89444	1	1	yes	characteristic
149	C₂	#89539	1	2	yes	—
150	C₂	#89565	1	2	yes	—
151	C₂	#89640	1	1	yes	characteristic
152	C₂	#89659	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₈

C₂

yes

maximal, characteristic

C₂×Dic₇⋊C₈

C₂

yes

maximal, characteristic

#89746

C₂

yes

maximal

C₂×#6384

C₂

yes

maximal

C₇×#340

C₂

yes

maximal, characteristic

C₂²×Dic₁₄

C₂²

yes

—

C₂²×C₇×C₈

C₂²

yes

characteristic

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

characteristic

C₂²×C₇⋊C₈

C₂²

yes

characteristic

C₂×Dic₂₈

—

C₂×Dic₂₈

—

C₇⋊#95

—

C₇⋊#95

—

Dic₇⋊C₈

—

#6384

—

#2056

—

maximal, Sylow

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

—

C₂²×Dic₇

D₄

yes

characteristic

C₂²×Dic₇

D₄

yes

characteristic

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

D₄

yes

—

C₂×C₇×C₈

—

C₂×C₇×C₈

D₄

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄×Dic₇

—

C₄×Dic₇

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

Dic₂₈

—

C₂²×Q₁₆

—

C₂×C₄⋊C₈

—

C₂×#103

—

#391

—

#340

D₇

yes

characteristic

C₂³×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

#41

yes

—

C₂×C₄×C₇

#41

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

Q₁₆

yes

complemented

Dic₁₄

—

Dic₁₄

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇⋊C₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇

yes

characteristic

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₄²

—

C₄⋊C₈

—

#103

—

#95

—

#95

—

C₂²×C₇

#121

yes

—

C₂²×C₇

#121

yes

characteristic

C₂²×C₇

C₂×Q₁₆

yes

C₂²×C₇

yes

C₂²×C₇

yes

—

C₄×C₇

—

C₄×C₇

#120

yes

characteristic

C₄×C₇

#123

yes

—

C₄×C₇

#124

yes

characteristic

Dic₇

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

characteristic

Q₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

D₂₈

yes

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₂×C₇

#394

yes

characteristic

109

C₂×C₇

#462

yes

—

110

C₂×C₇

#464

yes

—

111

C₂×C₇

#467

yes

characteristic

112

C₂×C₇

#470

yes

characteristic

113

C₂³

D₄×D₇

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×D₂₈

yes

characteristic

130

C₂×C₄

#1157

yes

—

131

C₂×C₄

D₄×D₇

yes

—

132

C₂×C₄

#1161

yes

characteristic

133

C₈

—

134

C₈

—

135

C₇

#2056

yes

characteristic, complemented, Sylow

136

C₂²

#6447

yes

—

137

C₂²

#6458

yes

characteristic

138

C₂²

#6475

yes

characteristic

139

C₂²

#6478

yes

—

140

C₂²

D₇×Q₁₆

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#6438

yes

characteristic

146

C₄

#6445

yes

characteristic

147

C₄

#6457

yes

—

148

C₂

#89444

yes

characteristic

149

C₂

#89539

yes

—

150

C₂

#89565

yes

—

151

C₂

#89640

yes

characteristic

152

C₂

#89659

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇
commutator, cosocle	C₂²×C₄×C₇	C₂³
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₇×#340	C₂
socle, prime radical	C₂³×C₇	C₂×D₄
hypercenter	#340	D₇
hypocenter	C₇	#2056
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₇

commutator, cosocle

C₂²×C₄×C₇

C₂³

Frattini

C₂²×C₄

C₂²×D₇

Fitting

C₇×#340

C₂

socle, prime radical

C₂³×C₇

C₂×D₄

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
58. C₂×Dic₇	Q₁₆	semidirect
135. C₇	#2056	semidirect

H ≅ G/N

type

58. C₂×Dic₇

Q₁₆

semidirect

135. C₇

#2056

semidirect

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

subgroup	index
C₂²×Dic₂₈	2
C₂×Dic₇⋊C₈	2
#89746	2
C₂×#6384	2
C₇×#340	2
#2056	7

subgroup

index

C₂²×Dic₂₈

C₂×Dic₇⋊C₈

Subgroups — #11726618

Conjugacy classes of subgroups

Complements

Maximal subgroups