Group #13073825 — subgroups

Subgroups — #13073825

Order 1408.

500 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 189 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×C₁₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₂	C₂	1	1	yes	maximal, characteristic
3	#10622842	C₂	1	4	yes	maximal
4	#10622589	C₂	1	2	yes	maximal
5	#10622587	C₂	1	2	yes	maximal
6	#10622586	C₂	1	4	yes	maximal
7	C₄×C₄⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₁₁	C₂²	1	2	yes	—
9	C₂²×Dic₂₂	C₂²	1	1	yes	characteristic
10	C₁₁×C₄⋊Q₈	—	2	4	no	—
11	C₂×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
16	C₂×Dic₁₁⋊C₄	C₂²	1	2	yes	—
17	C₂×Dic₁₁⋊C₄	C₂²	1	8	yes	—
18	C₂×C₄×Dic₁₁	C₂²	1	2	yes	—
19	C₂×C₄×Dic₁₁	C₂²	1	2	yes	—
20	C₄×Dic₂₂	—	2	4	no	—
21	C₁₁⋊#95	C₂²	1	2	yes	—
22	C₁₁⋊#95	C₂²	1	8	yes	—
23	C₂×Q₈×C₁₁	—	2	4	no	—
24	C₂×Q₈×C₁₁	—	4	4	no	—
25	C₂²×C₄×C₁₁	C₂³	1	4	yes	—
26	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
29	C₂²×Dic₁₁	C₂³	1	2	yes	—
30	C₂²×Dic₁₁	C₂³	1	2	yes	—
31	C₂²×Dic₁₁	C₂³	1	4	yes	—
32	C₂×Dic₂₂	—	2	2	no	—
33	C₂×Dic₂₂	—	2	2	no	—
34	C₂×Dic₂₂	Q₈	1	4	yes	complemented
35	C₁₁×C₄⋊C₄	—	2	4	no	—
36	C₁₁×C₄⋊C₄	—	2	4	no	—
37	C₄²×C₁₁	D₄	1	4	yes	—
38	C₄⋊Dic₁₁	—	2	2	no	—
39	C₄⋊Dic₁₁	Q₈	1	8	yes	complemented
40	Dic₁₁⋊C₄	—	2	4	no	—
41	Dic₁₁⋊C₄	—	4	8	no	—
42	C₄×Dic₁₁	—	2	4	no	—
43	C₄×Dic₁₁	—	2	4	no	—
44	#2567	—	11	1	no	maximal, Sylow
45	C₂³×C₁₁	C₂⁴	1	1	yes	characteristic
46	Q₈×C₁₁	—	4	8	no	—
47	C₂×C₄×C₁₁	—	2	2	no	—
48	C₂×C₄×C₁₁	—	2	4	no	—
49	C₂×C₄×C₁₁	—	2	8	no	—
50	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₁₁	C₂×Q₈	1	4	yes	—
55	C₂×C₄×C₁₁	#41	1	2	yes	—
56	C₂×Dic₁₁	—	2	2	no	—
57	C₂×Dic₁₁	—	2	4	no	—
58	C₂×Dic₁₁	—	2	2	no	—
59	C₂×Dic₁₁	—	2	4	no	—
60	C₂×Dic₁₁	—	2	2	no	—
61	C₂×Dic₁₁	—	2	4	no	—
62	C₂×Dic₁₁	C₂×Q₈	1	8	yes	—
63	C₂×Dic₁₁	C₂×Q₈	1	4	yes	—
64	Dic₂₂	—	2	8	no	—
65	C₂×C₄⋊Q₈	D₁₁	1	1	yes	characteristic
66	C₂×C₄×Q₈	—	11	1	no	—
67	#395	—	11	4	no	—
68	#391	—	11	2	no	—
69	#389	—	11	4	no	—
70	#384	—	11	2	no	—
71	C₄×C₄⋊C₄	—	11	1	no	—
72	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₁₁	C₂²×Q₈	1	1	yes	characteristic
74	C₂²×C₁₁	C₂²×Q₈	1	2	yes	—
75	C₂²×C₁₁	C₂×#41	1	1	yes	characteristic
76	C₂²×C₁₁	H(5,𝔽₂)	1	2	yes	—
77	C₄×C₁₁	—	2	2	no	—
78	C₄×C₁₁	—	4	4	no	—
79	C₄×C₁₁	C₂²⋊Q₈	1	4	yes	—
80	C₄×C₁₁	C₄⋊Q₈	1	4	yes	—
81	Dic₁₁	—	2	4	no	—
82	Dic₁₁	—	2	8	no	—
83	Dic₁₁	—	4	2	no	—
84	C₂²×Q₈	—	11	1	no	—
85	C₂²×Q₈	C₂×D₁₁	1	2	yes	—
86	C₄⋊Q₈	—	2	4	no	—
87	C₄×Q₈	—	22	4	no	—
88	C₂×C₄⋊C₄	—	11	2	no	—
89	C₂×C₄⋊C₄	—	11	1	no	—
90	C₂×C₄⋊C₄	—	11	2	no	—
91	C₂×C₄⋊C₄	—	11	8	no	—
92	C₂×C₄⋊C₄	C₂×D₁₁	1	2	yes	—
93	C₂×C₄⋊C₄	C₂×D₁₁	1	2	yes	—
94	C₂×C₄²	—	11	2	no	—
95	C₂×C₄²	—	11	2	no	—
96	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
97	#95	—	11	2	no	—
98	#95	—	11	8	no	—
99	C₂×C₁₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₁₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₁₁	#535	1	1	yes	characteristic
102	C₂×C₁₁	#554	1	2	yes	—
103	C₂×C₁₁	Q₈²	1	2	yes	—
104	C₂×Q₈	—	2	4	no	—
105	C₂×Q₈	—	4	4	no	—
106	C₂×Q₈	—	11	4	no	—
107	C₂×Q₈	—	22	2	no	—
108	C₂×Q₈	—	22	2	no	—
109	C₂²×C₄	—	11	2	no	—
110	C₂²×C₄	—	11	4	no	—
111	C₂²×C₄	—	11	2	no	—
112	C₂²×C₄	C₂²×D₁₁	1	4	yes	—
113	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	11	8	no	—
119	C₄⋊C₄	—	22	4	no	—
120	C₄⋊C₄	—	22	2	no	—
121	C₄⋊C₄	—	44	8	no	—
122	C₄²	—	22	4	no	—
123	C₄²	—	22	4	no	—
124	C₄²	C₂²⋊D₁₁	1	4	yes	—
125	C₁₁	#2567	1	1	yes	characteristic, complemented, Sylow
126	C₂³	C₂³×D₁₁	1	1	yes	characteristic
127	Q₈	—	4	8	no	—
128	Q₈	—	22	8	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	11	8	no	—
133	C₂×C₄	—	11	4	no	—
134	C₂×C₄	—	22	4	no	—
135	C₂×C₄	—	22	4	no	—
136	C₂×C₄	—	22	2	no	—
137	C₂×C₄	—	22	2	no	—
138	C₂×C₄	—	22	4	no	—
139	C₂×C₄	—	22	2	no	—
140	C₂×C₄	Q₈×D₁₁	1	4	yes	—
141	C₂×C₄	Q₈×D₁₁	1	2	yes	—
142	C₂×C₄	#4343	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
144	C₂×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
145	C₂×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
146	C₂²	#67037	1	2	yes	—
147	C₂²	C₂×Q₈×D₁₁	1	1	yes	characteristic
148	C₂²	C₂×Q₈×D₁₁	1	2	yes	—
149	C₂²	C₂×#4343	1	1	yes	characteristic
150	C₂²	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	4	4	no	—
153	C₄	—	22	8	no	—
154	C₄	—	22	4	no	—
155	C₄	—	44	2	no	—
156	C₄	#66997	1	4	yes	—
157	C₄	#66999	1	4	yes	—
158	C₂	#10623253	1	2	yes	—
159	C₂	#10623261	1	2	yes	—
160	C₂	#10623336	1	1	yes	characteristic
161	C₂	C₂×#66997	1	1	yes	characteristic
162	C₂	C₂×#66999	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₁×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×C₄×Dic₂₂

C₂

yes

maximal, characteristic

#10622842

C₂

yes

maximal

#10622589

C₂

yes

maximal

#10622587

C₂

yes

maximal

#10622586

C₂

yes

maximal

C₄×C₄⋊Dic₁₁

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₁

C₂²

yes

—

C₂²×Dic₂₂

C₂²

yes

characteristic

C₁₁×C₄⋊Q₈

—

C₂×C₁₁×C₄⋊C₄

C₂²

yes

—

C₂×C₁₁×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₁

C₂²

yes

—

C₂×Dic₁₁⋊C₄

C₂²

yes

—

C₂×Dic₁₁⋊C₄

C₂²

yes

—

C₂×C₄×Dic₁₁

C₂²

yes

—

C₂×C₄×Dic₁₁

C₂²

yes

—

C₄×Dic₂₂

—

C₁₁⋊#95

C₂²

yes

—

C₁₁⋊#95

C₂²

yes

—

C₂×Q₈×C₁₁

—

C₂×Q₈×C₁₁

—

C₂²×C₄×C₁₁

C₂³

yes

—

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×Dic₁₁

C₂³

yes

—

C₂²×Dic₁₁

C₂³

yes

—

C₂²×Dic₁₁

C₂³

yes

—

C₂×Dic₂₂

—

C₂×Dic₂₂

—

C₂×Dic₂₂

Q₈

yes

complemented

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₄²×C₁₁

D₄

yes

—

C₄⋊Dic₁₁

—

C₄⋊Dic₁₁

Q₈

yes

complemented

Dic₁₁⋊C₄

—

Dic₁₁⋊C₄

—

C₄×Dic₁₁

—

C₄×Dic₁₁

—

#2567

—

maximal, Sylow

C₂³×C₁₁

C₂⁴

yes

characteristic

Q₈×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂×Q₈

yes

—

C₂×C₄×C₁₁

C₂×Q₈

yes

—

C₂×C₄×C₁₁

#41

yes

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

C₂×Q₈

yes

—

C₂×Dic₁₁

C₂×Q₈

yes

—

Dic₂₂

—

C₂×C₄⋊Q₈

D₁₁

yes

characteristic

C₂×C₄×Q₈

—

—

—

—

—

C₄×C₄⋊C₄

—

C₂²×C₁₁

C₂²×D₄

yes

characteristic

C₂²×C₁₁

C₂²×Q₈

yes

characteristic

C₂²×C₁₁

C₂²×Q₈

yes

—

C₂²×C₁₁

C₂×#41

yes

characteristic

C₂²×C₁₁

H(5,𝔽₂)

yes

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

C₂²⋊Q₈

yes

—

C₄×C₁₁

C₄⋊Q₈

yes

—

Dic₁₁

—

Dic₁₁

—

Dic₁₁

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₁₁

yes

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₁

yes

—

C₂×C₄⋊C₄

C₂×D₁₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₁

yes

characteristic

#95

—

#95

—

C₂×C₁₁

C₂×C₂²⋊Q₈

yes

characteristic

100

C₂×C₁₁

C₂×C₄⋊Q₈

yes

characteristic

101

C₂×C₁₁

#535

yes

characteristic

102

C₂×C₁₁

#554

yes

—

103

C₂×C₁₁

Q₈²

yes

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂²×D₁₁

yes

—

113

C₂²×C₄

C₂²×D₁₁

yes

characteristic

114

C₂²×C₄

C₂²×D₁₁

yes

characteristic

115

C₂²×C₄

C₂²×D₁₁

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄²

—

123

C₄²

—

124

C₄²

C₂²⋊D₁₁

yes

—

125

C₁₁

#2567

yes

characteristic, complemented, Sylow

126

C₂³

C₂³×D₁₁

yes

characteristic

127

Q₈

—

128

Q₈

—

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

—

139

C₂×C₄

—

140

C₂×C₄

Q₈×D₁₁

yes

—

141

C₂×C₄

Q₈×D₁₁

yes

—

142

C₂×C₄

#4343

yes

—

143

C₂×C₄

C₂×C₂²⋊D₁₁

yes

—

144

C₂×C₄

C₂×C₂²⋊D₁₁

yes

—

145

C₂×C₄

C₂×C₂²⋊D₁₁

yes

—

146

C₂²

#67037

yes

—

147

C₂²

C₂×Q₈×D₁₁

yes

characteristic

148

C₂²

C₂×Q₈×D₁₁

yes

—

149

C₂²

C₂×#4343

yes

characteristic

150

C₂²

C₂²×C₂²⋊D₁₁

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#66997

yes

—

157

C₄

#66999

yes

—

158

C₂

#10623253

yes

—

159

C₂

#10623261

yes

—

160

C₂

#10623336

yes

characteristic

161

C₂

C₂×#66997

yes

characteristic

162

C₂

C₂×#66999

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₁
commutator, socle, prime radical, cosocle	C₂³×C₁₁	C₂⁴
Fitting	C₂×C₁₁×C₄⋊Q₈	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₁
hypocenter	C₁₁	#2567
LCS	G ▷ C₂³×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₁

commutator, socle, prime radical, cosocle

C₂³×C₁₁

C₂⁴

Fitting

C₂×C₁₁×C₄⋊Q₈

C₂

hypercenter

C₂×C₄⋊Q₈

D₁₁

C₁₁

G ▷ C₂³×C₁₁ ▷ C₁₁

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
34. C₂×Dic₂₂	Q₈	semidirect
39. C₄⋊Dic₁₁	Q₈	semidirect
125. C₁₁	#2567	semidirect

H ≅ G/N

type

34. C₂×Dic₂₂

Q₈

semidirect

39. C₄⋊Dic₁₁

Q₈

semidirect

125. C₁₁

#2567

semidirect

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

subgroup	index
C₂×C₁₁×C₄⋊Q₈	2
C₂×C₄×Dic₂₂	2
#10622842	2
#10622589	2
#10622587	2
#10622586	2
C₄×C₄⋊Dic₁₁	2
#2567	11

subgroup

index

C₂×C₁₁×C₄⋊Q₈

C₂×C₄×Dic₂₂

C₄×C₄⋊Dic₁₁

Subgroups — #13073825

Conjugacy classes of subgroups

Complements

Maximal subgroups