Group #26029335 — subgroups

Subgroups — #26029335

Order 11392.

356 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 177 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×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	#19151728	C₂	1	4	yes	maximal, complemented
3	#19151725	C₂	1	4	yes	maximal, complemented
4	#19151652	C₂	1	4	yes	maximal
5	#19151651	C₂	1	1	yes	maximal, characteristic
6	#431×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₇₈	C₄	1	1	yes	characteristic
8	C₄×Dic₁₇₈	C₄	1	2	yes	—
9	C₄×Dic₁₇₈	C₄	1	2	yes	—
10	C₄×Dic₁₇₈	C₂²	1	2	yes	—
11	C₄×Dic₁₇₈	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
13	C₂×#13083553	C₄	1	2	yes	—
14	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
15	C₄⋊#10623494	C₂²	1	2	yes	—
16	#15667941	C₂²	1	8	yes	—
17	C₂²⋊#10623494	C₂²	1	2	yes	—
18	#15667922	C₂²	1	8	yes	—
19	C₄×#10623494	C₂²	1	2	yes	—
20	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
22	C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
23	C₄×C₈×C₈₉	C₂²	1	2	yes	—
24	C₂×Dic₁₇₈	—	2	4	no	—
25	C₂×Dic₁₇₈	C₂×C₄	1	2	yes	—
26	C₂×Dic₁₇₈	C₂×C₄	1	2	yes	—
27	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
28	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
29	C₂×#10623494	C₂³	1	4	yes	—
30	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
31	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
32	#13083553	C₂×C₄	1	4	yes	—
33	#13083553	C₂×C₄	1	4	yes	—
34	C₄×Dic₈₉	C₂×C₄	1	4	yes	—
35	C₄×Dic₈₉	C₂³	1	4	yes	—
36	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₂³	1	4	yes	—
40	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₈₉	C₂³	1	1	yes	characteristic
43	C₄²×C₈₉	C₂³	1	1	yes	characteristic
44	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈₉	—	2	2	no	—
46	C₂×C₄×C₈₉	—	4	1	no	—
47	C₂×C₄×C₈₉	M₁₆	1	4	yes	—
48	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₈₉	—	2	4	no	—
55	C₂×Dic₈₉	—	4	2	no	—
56	C₂×Dic₈₉	C₂²×C₄	1	4	yes	—
57	C₂×Dic₈₉	C₂²×C₄	1	4	yes	—
58	Dic₁₇₈	—	2	4	no	—
59	Dic₁₇₈	—	2	4	no	—
60	C₈×C₈₉	—	2	4	no	—
61	#10623494	—	2	4	no	—
62	C₂²×C₈₉	—	2	1	no	—
63	C₂²×C₈₉	—	2	1	no	—
64	C₂²×C₈₉	C₂×M₁₆	1	2	yes	—
65	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₈₉	—	2	1	no	—
67	C₄×C₈₉	—	2	1	no	—
68	C₄×C₈₉	—	2	1	no	—
69	C₄×C₈₉	C₂×M₁₆	1	2	yes	—
70	C₄×C₈₉	C₂×M₁₆	1	2	yes	—
71	C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
72	Dic₈₉	—	2	4	no	—
73	Dic₈₉	—	2	4	no	—
74	C₂×C₈₉	—	2	2	no	—
75	C₂×C₈₉	#519	1	1	yes	characteristic
76	C₂×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
77	C₂×C₈₉	#568	1	1	yes	characteristic
78	#2983	—	89	1	no	maximal, Sylow
79	C₈₉	#2983	1	1	yes	characteristic, complemented, Sylow
80	C₂×C₄×Q₈	—	89	1	no	—
81	#446	—	89	4	no	—
82	#446	—	89	4	no	—
83	#432	—	89	4	no	—
84	#431	—	89	1	no	—
85	#431	D₈₉	1	1	yes	characteristic
86	C₂²×Q₈	—	89	1	no	—
87	C₄×Q₈	—	89	2	no	—
88	C₄×Q₈	—	89	2	no	—
89	C₄×Q₈	—	89	2	no	—
90	C₄×Q₈	—	89	2	no	—
91	C₂×C₄⋊C₄	—	89	2	no	—
92	C₂×C₄⋊C₄	—	89	1	no	—
93	C₂×C₄²	—	89	2	no	—
94	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
95	C₄⋊C₈	—	89	8	no	—
96	C₄⋊C₈	—	89	2	no	—
97	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
98	C₂²⋊C₈	—	89	2	no	—
99	C₂²⋊C₈	C₂×D₈₉	1	2	yes	—
100	#97	—	89	8	no	—
101	C₄×C₈	—	89	2	no	—
102	C₄×C₈	C₂×D₈₉	1	2	yes	—
103	C₂×Q₈	—	89	2	no	—
104	C₂×Q₈	—	89	2	no	—
105	C₂×Q₈	—	178	4	no	—
106	C₂²×C₄	—	89	2	no	—
107	C₂²×C₄	—	89	2	no	—
108	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
109	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
111	C₂×C₈	—	89	4	no	—
112	C₂×C₈	C₂²×D₈₉	1	4	yes	—
113	C₄⋊C₄	—	89	4	no	—
114	C₄⋊C₄	—	89	4	no	—
115	C₄⋊C₄	—	89	2	no	—
116	C₄⋊C₄	—	89	2	no	—
117	C₄²	—	89	4	no	—
118	C₄²	—	89	4	no	—
119	C₄²	C₄×D₈₉	1	1	yes	characteristic
120	C₄²	C₄×D₈₉	1	1	yes	characteristic
121	C₄²	C₂²×D₈₉	1	1	yes	characteristic
122	C₄²	C₂²×D₈₉	1	1	yes	characteristic
123	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
124	Q₈	—	178	4	no	—
125	Q₈	—	178	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	89	4	no	—
129	C₂×C₄	—	89	4	no	—
130	C₂×C₄	—	356	2	no	—
131	C₂×C₄	—	178	4	no	—
132	C₂×C₄	#13083557	1	4	yes	—
133	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	178	4	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×#13083557	1	2	yes	—
144	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	178	4	no	—
149	C₄	—	178	4	no	—
150	C₄	C₂×#13083557	1	2	yes	—
151	C₄	C₂×#13083557	1	2	yes	—
152	C₄	#15668059	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	#19152075	1	1	yes	characteristic
155	C₂	C₂²×#13083557	1	1	yes	characteristic
156	C₂	#19152364	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₇₈

C₂

yes

maximal, characteristic

#19151728

C₂

yes

maximal, complemented

#19151725

C₂

yes

maximal, complemented

#19151652

C₂

yes

maximal

#19151651

C₂

yes

maximal, characteristic

#431×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₄

yes

characteristic

C₄×Dic₁₇₈

C₄

yes

—

C₄×Dic₁₇₈

C₄

yes

—

C₄×Dic₁₇₈

C₂²

yes

—

C₄×Dic₁₇₈

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₄

yes

characteristic

C₂×#13083553

C₄

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

C₄⋊#10623494

C₂²

yes

—

#15667941

C₂²

yes

—

C₂²⋊#10623494

C₂²

yes

—

#15667922

C₂²

yes

—

C₄×#10623494

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₂²⋊C₈×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂×C₄

yes

—

C₂×Dic₁₇₈

C₂×C₄

yes

—

C₂²×Dic₈₉

C₂×C₄

yes

—

C₂²×Dic₈₉

C₂×C₄

yes

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

#13083553

C₂×C₄

yes

—

#13083553

C₂×C₄

yes

—

C₄×Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

—

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

M₁₆

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂²×C₄

yes

—

C₂×Dic₈₉

C₂²×C₄

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×M₁₆

yes

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×M₁₆

yes

—

C₄×C₈₉

C₂×M₁₆

yes

—

C₄×C₈₉

2₋¹⁺⁴

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#519

yes

characteristic

C₂×C₈₉

C₂²×M₁₆

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

—

—

—

—

D₈₉

yes

characteristic

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₈₉

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₈₉

yes

—

100

#97

—

101

C₄×C₈

—

102

C₄×C₈

C₂×D₈₉

yes

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

178

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₈₉

yes

characteristic

109

C₂²×C₄

C₄×D₈₉

yes

characteristic

110

C₂²×C₄

C₂²×D₈₉

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₈₉

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

C₄×D₈₉

yes

characteristic

120

C₄²

C₄×D₈₉

yes

characteristic

121

C₄²

C₂²×D₈₉

yes

characteristic

122

C₄²

C₂²×D₈₉

yes

characteristic

123

C₂³

C₂×C₄×D₈₉

yes

characteristic

124

Q₈

—

178

—

125

Q₈

—

178

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

356

—

131

C₂×C₄

—

178

—

132

C₂×C₄

#13083557

yes

—

133

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

134

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

135

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

136

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

137

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

138

C₂×C₄

C₂³×D₈₉

yes

characteristic

139

C₈

—

140

C₈

—

178

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×#13083557

yes

—

144

C₂²

C₂²×C₄×D₈₉

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

178

—

149

C₄

—

178

—

150

C₄

C₂×#13083557

yes

—

151

C₄

C₂×#13083557

yes

—

152

C₄

#15668059

yes

—

153

C₂

—

154

C₂

#19152075

yes

characteristic

155

C₂

C₂²×#13083557

yes

characteristic

156

C₂

#19152364

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	#431×C₈₉	C₂
prime radical	C₂³×C₈₉	C₂²×C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#431	D₈₉
hypocenter	C₈₉	#2983
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂³×C₄

Fitting

#431×C₈₉

C₂

prime radical

C₂³×C₈₉

C₂²×C₄

cosocle

C₂×C₄×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. #19151728	C₂	semidirect
3. #19151725	C₂	semidirect
79. C₈₉	#2983	semidirect

H ≅ G/N

type

2. #19151728

C₂

semidirect

3. #19151725

C₂

semidirect

79. C₈₉

#2983

semidirect

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

subgroup	index
C₂×C₄×Dic₁₇₈	2
#19151728	2
#19151725	2
#19151652	2
#19151651	2
#431×C₈₉	2
#2983	89

subgroup

index

C₂×C₄×Dic₁₇₈

#431×C₈₉

Subgroups — #26029335

Conjugacy classes of subgroups

Complements

Maximal subgroups