Group #11739923 — subgroups

Subgroups — #11739923

Order 896.

500 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 197 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	C₂×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#6432	C₂	1	1	yes	maximal, characteristic
3	#90002	C₂	1	2	yes	maximal
4	#89751	C₂	1	4	yes	maximal
5	#89748	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
7	#89700	C₂	1	4	yes	maximal
8	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₄	C₂²	1	2	yes	—
10	C₄×C₇×Q₈	—	2	4	no	—
11	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄²×C₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
18	C₂×Dic₇⋊C₄	C₂²	1	8	yes	—
19	C₂×C₄×Dic₇	C₂²	1	2	yes	—
20	#6432	—	2	4	no	—
21	C₇⋊#95	C₂²	1	8	yes	—
22	C₇⋊#95	C₂²	1	2	yes	—
23	#2567	—	7	1	no	maximal, Sylow
24	C₂×C₇×Q₈	—	2	2	no	—
25	C₂×C₇×Q₈	—	2	2	no	—
26	C₂×C₇×Q₈	Q₈	1	4	yes	complemented
27	C₂²×C₄×C₇	C₂³	1	2	yes	—
28	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₇	C₂³	1	2	yes	—
30	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
32	C₂²×Dic₇	C₂³	1	4	yes	—
33	C₂²×Dic₇	C₂³	1	4	yes	—
34	C₂×Dic₁₄	—	2	4	no	—
35	C₂×Dic₁₄	—	4	4	no	—
36	C₇×C₄⋊C₄	—	2	4	no	—
37	C₇×C₄⋊C₄	—	2	2	no	—
38	C₄²×C₇	—	2	4	no	—
39	C₄²×C₇	D₄	1	4	yes	—
40	C₄⋊Dic₇	—	2	4	no	—
41	C₄⋊Dic₇	—	2	4	no	—
42	C₄⋊Dic₇	Q₈	1	8	yes	complemented
43	Dic₇⋊C₄	—	4	8	no	—
44	C₄×Dic₇	—	2	4	no	—
45	C₂×C₄⋊Q₈	—	7	1	no	—
46	C₂×C₄×Q₈	D₇	1	1	yes	characteristic
47	#395	—	7	4	no	—
48	#391	—	7	2	no	—
49	#389	—	7	4	no	—
50	#384	—	7	2	no	—
51	C₄×C₄⋊C₄	—	7	1	no	—
52	C₂³×C₇	C₂⁴	1	1	yes	characteristic
53	C₇×Q₈	—	2	8	no	—
54	C₂×C₄×C₇	—	2	2	no	—
55	C₂×C₄×C₇	—	2	2	no	—
56	C₂×C₄×C₇	—	2	2	no	—
57	C₂×C₄×C₇	—	2	2	no	—
58	C₂×C₄×C₇	—	2	4	no	—
59	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
63	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
65	C₂×C₄×C₇	#41	1	2	yes	—
66	C₂×Dic₇	—	2	4	no	—
67	C₂×Dic₇	—	2	4	no	—
68	C₂×Dic₇	—	2	4	no	—
69	C₂×Dic₇	—	2	8	no	—
70	C₂×Dic₇	C₂×Q₈	1	8	yes	—
71	Dic₁₄	—	4	8	no	—
72	C₂²×Q₈	—	7	2	no	—
73	C₂²×Q₈	C₂×D₇	1	1	yes	characteristic
74	C₄⋊Q₈	—	14	4	no	—
75	C₄×Q₈	—	2	4	no	—
76	C₂×C₄⋊C₄	—	7	2	no	—
77	C₂×C₄⋊C₄	—	7	2	no	—
78	C₂×C₄⋊C₄	—	7	2	no	—
79	C₂×C₄⋊C₄	—	7	8	no	—
80	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
81	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
82	C₂×C₄²	—	7	2	no	—
83	C₂×C₄²	C₂×D₇	1	2	yes	—
84	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
85	#95	—	7	2	no	—
86	#95	—	7	8	no	—
87	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
88	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
89	C₂²×C₇	C₂²×Q₈	1	2	yes	—
90	C₂²×C₇	C₂×#41	1	1	yes	characteristic
91	C₂²×C₇	H(5,𝔽₂)	1	2	yes	—
92	C₄×C₇	—	2	4	no	—
93	C₄×C₇	—	2	2	no	—
94	C₄×C₇	—	4	2	no	—
95	C₄×C₇	C₂²⋊Q₈	1	4	yes	—
96	C₄×C₇	C₄⋊Q₈	1	4	yes	—
97	Dic₇	—	2	8	no	—
98	Dic₇	—	4	4	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	14	4	no	—
102	C₂×Q₈	—	28	4	no	—
103	C₂×Q₈	Dic₁₄	1	4	yes	complemented
104	C₂²×C₄	—	7	4	no	—
105	C₂²×C₄	—	7	4	no	—
106	C₂²×C₄	C₂²×D₇	1	2	yes	—
107	C₂²×C₄	C₂²×D₇	1	2	yes	—
108	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
109	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
111	C₄⋊C₄	—	2	4	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	7	8	no	—
114	C₄⋊C₄	—	14	4	no	—
115	C₄⋊C₄	—	14	4	no	—
116	C₄⋊C₄	—	28	8	no	—
117	C₄²	—	2	4	no	—
118	C₄²	—	14	4	no	—
119	C₄²	C₂²⋊D₇	1	4	yes	—
120	C₂×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₂×C₇	#535	1	1	yes	characteristic
123	C₂×C₇	#554	1	2	yes	—
124	C₂×C₇	Q₈²	1	2	yes	—
125	C₂³	C₂³×D₇	1	1	yes	characteristic
126	Q₈	—	2	8	no	—
127	Q₈	—	28	8	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	7	8	no	—
134	C₂×C₄	—	14	4	no	—
135	C₂×C₄	—	14	4	no	—
136	C₂×C₄	—	14	8	no	—
137	C₂×C₄	—	14	4	no	—
138	C₂×C₄	C₂×Dic₁₄	1	4	yes	—
139	C₂×C₄	C₂×Dic₁₄	1	2	yes	—
140	C₂×C₄	#1157	1	2	yes	—
141	C₂×C₄	Q₈×D₇	1	4	yes	—
142	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
144	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
145	C₇	#2567	1	1	yes	characteristic, complemented, Sylow
146	C₂²	C₂²×Dic₁₄	1	1	yes	characteristic
147	C₂²	C₂×#1157	1	1	yes	characteristic
148	C₂²	C₂×Q₈×D₇	1	2	yes	—
149	C₂²	#6553	1	2	yes	—
150	C₂²	C₂²×C₂²⋊D₇	1	1	yes	characteristic
151	C₄	—	2	4	no	—
152	C₄	—	2	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	14	8	no	—
155	C₄	—	28	4	no	—
156	C₄	#6487	1	4	yes	—
157	C₄	#6507	1	4	yes	—
158	C₂	Q₈×Dic₁₄	1	2	yes	—
159	C₂	#90270	1	2	yes	—
160	C₂	C₂×#6487	1	1	yes	characteristic
161	C₂	C₂×#6507	1	1	yes	characteristic
162	C₂	#90521	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₇×Q₈

C₂

yes

maximal, characteristic

C₂×#6432

C₂

yes

maximal, characteristic

#90002

C₂

yes

maximal

#89751

C₂

yes

maximal

#89748

C₂

yes

maximal

C₄×C₄⋊Dic₇

C₂

yes

maximal, characteristic

#89700

C₂

yes

maximal

C₂²×C₇×Q₈

C₂²

yes

characteristic

C₂²×Dic₁₄

C₂²

yes

—

C₄×C₇×Q₈

—

C₂×C₇×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₇

C₂²

yes

—

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

—

C₂×C₄×Dic₇

C₂²

yes

—

#6432

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

#2567

—

maximal, Sylow

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

Q₈

yes

complemented

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

—

C₄²×C₇

D₄

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

Q₈

yes

complemented

Dic₇⋊C₄

—

C₄×Dic₇

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

D₇

yes

—

—

—

—

C₄×C₄⋊C₄

—

C₂³×C₇

C₂⁴

yes

characteristic

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×C₄×C₇

#41

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

C₂×Q₈

yes

—

Dic₁₄

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₇

yes

characteristic

C₄⋊Q₈

—

C₄×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

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇

yes

—

C₂×C₄²

C₂×D₇

yes

characteristic

#95

—

#95

—

C₂²×C₇

C₂²×D₄

yes

characteristic

C₂²×C₇

C₂²×Q₈

yes

characteristic

C₂²×C₇

C₂²×Q₈

yes

—

C₂²×C₇

C₂×#41

yes

characteristic

C₂²×C₇

H(5,𝔽₂)

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂²⋊Q₈

yes

—

C₄×C₇

C₄⋊Q₈

yes

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

Dic₁₄

yes

complemented

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂²×D₇

yes

—

107

C₂²×C₄

C₂²×D₇

yes

—

108

C₂²×C₄

C₂²×D₇

yes

characteristic

109

C₂²×C₄

C₂²×D₇

yes

characteristic

110

C₂²×C₄

C₂²×D₇

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

C₂²⋊D₇

yes

—

120

C₂×C₇

C₂×C₂²⋊Q₈

yes

characteristic

121

C₂×C₇

C₂×C₄⋊Q₈

yes

characteristic

122

C₂×C₇

#535

yes

characteristic

123

C₂×C₇

#554

yes

—

124

C₂×C₇

Q₈²

yes

—

125

C₂³

C₂³×D₇

yes

characteristic

126

Q₈

—

127

Q₈

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₂×Dic₁₄

yes

—

139

C₂×C₄

C₂×Dic₁₄

yes

—

140

C₂×C₄

#1157

yes

—

141

C₂×C₄

Q₈×D₇

yes

—

142

C₂×C₄

C₂×C₂²⋊D₇

yes

—

143

C₂×C₄

C₂×C₂²⋊D₇

yes

—

144

C₂×C₄

C₂×C₂²⋊D₇

yes

—

145

C₇

#2567

yes

characteristic, complemented, Sylow

146

C₂²

C₂²×Dic₁₄

yes

characteristic

147

C₂²

C₂×#1157

yes

characteristic

148

C₂²

C₂×Q₈×D₇

yes

—

149

C₂²

#6553

yes

—

150

C₂²

C₂²×C₂²⋊D₇

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#6487

yes

—

157

C₄

#6507

yes

—

158

C₂

Q₈×Dic₁₄

yes

—

159

C₂

#90270

yes

—

160

C₂

C₂×#6487

yes

characteristic

161

C₂

C₂×#6507

yes

characteristic

162

C₂

#90521

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, prime radical, cosocle	C₂³×C₇	C₂⁴
Fitting	C₂×C₄×C₇×Q₈	C₂
hypercenter	C₂×C₄×Q₈	D₇
hypocenter	C₇	#2567
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇

commutator, socle, prime radical, cosocle

C₂³×C₇

C₂⁴

Fitting

C₂×C₄×C₇×Q₈

C₂

hypercenter

C₂×C₄×Q₈

D₇

C₇

G ▷ C₂³×C₇ ▷ C₇

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
26. C₂×C₇×Q₈	Q₈	semidirect
42. C₄⋊Dic₇	Q₈	semidirect
103. C₂×Q₈	Dic₁₄	semidirect
145. C₇	#2567	semidirect

H ≅ G/N

type

26. C₂×C₇×Q₈

Q₈

semidirect

42. C₄⋊Dic₇

Q₈

semidirect

103. C₂×Q₈

Dic₁₄

semidirect

145. C₇

#2567

semidirect

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

subgroup	index
C₂×C₄×C₇×Q₈	2
C₂×#6432	2
#90002	2
#89751	2
#89748	2
C₄×C₄⋊Dic₇	2
#89700	2
#2567	7

subgroup

index

C₂×C₄×C₇×Q₈

C₄×C₄⋊Dic₇

Subgroups — #11739923

Conjugacy classes of subgroups

Complements

Maximal subgroups