Group #26033117 — subgroups

Subgroups — #26033117

Order 11392.

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

normal subgroups only

This group has 352 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	#19152305	C₂	1	2	yes	maximal, complemented
2	#19152003	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151999	C₂	1	2	yes	maximal, complemented
4	#19151896	C₂	1	4	yes	maximal, complemented
5	#19151888	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×Dic₈₉	C₂	1	2	yes	maximal
8	C₄×D₈×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#15668012	C₂²	1	2	yes	—
10	#15668003	C₂²	1	4	yes	—
11	#15667991	—	2	2	no	—
12	#15667989	—	2	4	no	—
13	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
14	D₄×Dic₈₉	C₂²	1	4	yes	—
15	C₂×C₄⋊Dic₈₉	—	2	2	no	—
16	#15667946	C₂²	1	2	yes	—
17	#15667946	C₂²	1	2	yes	—
18	#15667943	C₂²	1	4	yes	—
19	C₄⋊#10623494	C₂²	1	1	yes	characteristic
20	#15667941	C₂²	1	2	yes	—
21	#15667933	C₂²	1	4	yes	—
22	C₈×Dic₈₉	C₂²	1	2	yes	—
23	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₈₉	C₂²	1	2	yes	—
25	#107×C₈₉	C₂²	1	1	yes	characteristic
26	#102×C₈₉	C₂²	1	2	yes	—
27	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₇₈	—	2	2	no	—
29	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
30	C₂²×Dic₈₉	—	2	4	no	—
31	C₂×#10623494	C₂³	1	2	yes	—
32	C₄⋊Dic₈₉	—	2	2	no	—
33	C₄⋊Dic₈₉	—	2	4	no	—
34	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
35	C₄⋊Dic₈₉	C₂³	1	4	yes	—
36	#13083553	—	2	4	no	—
37	#13083553	D₄	1	2	yes	—
38	C₂²⋊Dic₈₉	—	2	4	no	—
39	C₄×Dic₈₉	C₂³	1	2	yes	—
40	C₂×D₄×C₈₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₈₉	—	2	2	no	—
42	D₈×C₈₉	Q₈	1	4	yes	complemented
43	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
46	C₂²⋊C₄×C₈₉	—	2	2	no	—
47	C₄²×C₈₉	C₂³	1	1	yes	characteristic
48	C₂³×C₈₉	—	2	2	no	—
49	D₄×C₈₉	—	2	2	no	—
50	D₄×C₈₉	C₂×Q₈	1	4	yes	—
51	C₂×C₄×C₈₉	—	2	2	no	—
52	C₂×C₄×C₈₉	—	4	4	no	—
53	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₈₉	—	2	4	no	—
57	C₂×Dic₈₉	—	8	4	no	—
58	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
59	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
60	Dic₁₇₈	—	2	2	no	—
61	Dic₁₇₈	—	8	2	no	—
62	C₈×C₈₉	—	2	1	no	—
63	C₈×C₈₉	C₂×Q₈	1	2	yes	—
64	#10623494	—	2	2	no	—
65	C₂²×C₈₉	—	2	4	no	—
66	C₂²×C₈₉	—	4	2	no	—
67	C₂²×C₈₉	—	4	2	no	—
68	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
69	C₄×C₈₉	—	2	1	no	—
70	C₄×C₈₉	—	4	2	no	—
71	C₄×C₈₉	#135	1	2	yes	—
72	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
73	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
74	Dic₈₉	—	2	2	no	—
75	Dic₈₉	—	2	2	no	—
76	Dic₈₉	—	4	4	no	—
77	C₂×C₈₉	—	4	4	no	—
78	C₂×C₈₉	D₄×Q₈	1	1	yes	characteristic
79	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
80	C₂×C₈₉	#576	1	1	yes	characteristic
81	#3380	—	89	1	no	maximal, Sylow
82	C₈₉	#3380	1	1	yes	characteristic, complemented, Sylow
83	#554	—	89	2	no	—
84	#500	—	89	1	no	—
85	#499	—	89	2	no	—
86	#478	—	89	4	no	—
87	#476	—	89	2	no	—
88	C₈×Q₈	—	89	1	no	—
89	C₄×D₈	—	89	2	no	—
90	C₄×D₈	D₈₉	1	1	yes	characteristic
91	C₂×D₈	C₂×D₈₉	1	1	yes	characteristic
92	C₄⋊Q₈	—	89	2	no	—
93	#125	—	89	4	no	—
94	C₂²⋊Q₈	—	178	4	no	—
95	C₂²⋊Q₈	—	178	2	no	—
96	C₄×Q₈	—	89	1	no	—
97	C₄×D₄	—	89	4	no	—
98	C₄×D₄	C₂×D₈₉	1	2	yes	—
99	C₂×C₄⋊C₄	—	178	2	no	—
100	#107	—	89	2	no	—
101	#107	—	89	2	no	—
102	#107	C₂×D₈₉	1	1	yes	characteristic
103	#106	—	89	4	no	—
104	C₄⋊C₈	—	89	2	no	—
105	C₄⋊C₈	—	89	1	no	—
106	#102	—	89	4	no	—
107	#102	C₂×D₈₉	1	2	yes	—
108	C₄×C₈	—	89	2	no	—
109	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
110	C₂×Q₈	—	89	1	no	—
111	C₂×Q₈	—	178	2	no	—
112	C₂×D₄	C₂²×D₈₉	1	2	yes	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	—	178	4	no	—
115	D₈	Dic₁₇₈	1	4	yes	complemented
116	C₂×C₈	—	89	2	no	—
117	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
118	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
119	C₄⋊C₄	—	89	1	no	—
120	C₄⋊C₄	—	89	2	no	—
121	C₄⋊C₄	—	89	4	no	—
122	C₄⋊C₄	—	178	4	no	—
123	C₄⋊C₄	—	178	4	no	—
124	C₄⋊C₄	—	178	2	no	—
125	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
126	C₂²⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	—	178	4	no	—
128	C₄²	—	89	2	no	—
129	C₄²	C₂²×D₈₉	1	1	yes	characteristic
130	C₂³	—	2	2	no	—
131	Q₈	—	178	2	no	—
132	Q₈	—	712	2	no	—
133	D₄	—	2	2	no	—
134	D₄	C₂×Dic₁₇₈	1	4	yes	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	89	2	no	—
138	C₂×C₄	—	89	2	no	—
139	C₂×C₄	—	178	4	no	—
140	C₂×C₄	—	712	4	no	—
141	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
142	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	178	2	no	—
146	C₈	C₂×Dic₁₇₈	1	2	yes	—
147	C₂²	—	2	4	no	—
148	C₂²	—	4	2	no	—
149	C₂²	—	4	2	no	—
150	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	4	2	no	—
153	C₄	—	356	4	no	—
154	C₄	—	178	2	no	—
155	C₄	—	178	2	no	—
156	C₄	#15668031	1	2	yes	—
157	C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
158	C₄	#15668059	1	1	yes	characteristic
159	C₂	—	4	4	no	—
160	C₂	D₄×Dic₁₇₈	1	1	yes	characteristic
161	C₂	C₂×#15668031	1	1	yes	characteristic
162	C₂	#19152400	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152305

C₂

yes

maximal, complemented

#19152003

C₂

yes

maximal, characteristic, complemented

#19151999

C₂

yes

maximal, complemented

#19151896

C₂

yes

maximal, complemented

#19151888

C₂

yes

maximal, complemented

C₈×Dic₁₇₈

C₂

yes

maximal, characteristic, complemented

D₈×Dic₈₉

C₂

yes

maximal

C₄×D₈×C₈₉

C₂

yes

maximal, characteristic

#15668012

C₂²

yes

—

#15668003

C₂²

yes

—

#15667991

—

#15667989

—

C₄×Dic₁₇₈

C₂²

yes

characteristic

D₄×Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

—

#15667946

C₂²

yes

—

#15667946

C₂²

yes

—

#15667943

C₂²

yes

—

C₄⋊#10623494

C₂²

yes

characteristic

#15667941

C₂²

yes

—

#15667933

C₂²

yes

—

C₈×Dic₈₉

C₂²

yes

—

C₂×D₈×C₈₉

C₂²

yes

characteristic

C₄×D₄×C₈₉

C₂²

yes

—

#107×C₈₉

C₂²

yes

characteristic

#102×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

characteristic

C₂²×Dic₈₉

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

characteristic

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

#13083553

D₄

yes

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

C₂³

yes

—

C₂×D₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

—

D₈×C₈₉

Q₈

yes

complemented

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

C₂³

yes

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

C₂×D₄

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×Q₈

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#135

yes

—

C₄×C₈₉

C₂²×Q₈

yes

characteristic

C₄×C₈₉

2₋¹⁺⁴

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

D₄×Q₈

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₈×Q₈

—

C₄×D₈

—

C₄×D₈

D₈₉

yes

characteristic

C₂×D₈

C₂×D₈₉

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

178

—

C₂²⋊Q₈

—

178

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

C₂×D₈₉

yes

—

C₂×C₄⋊C₄

—

178

—

100

#107

—

101

#107

—

102

#107

C₂×D₈₉

yes

characteristic

103

#106

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

#102

—

107

#102

C₂×D₈₉

yes

—

108

C₄×C₈

—

109

C₄×C₈

C₂×D₈₉

yes

characteristic

110

C₂×Q₈

—

111

C₂×Q₈

—

178

—

112

C₂×D₄

C₂²×D₈₉

yes

—

113

C₂²×C₄

—

114

C₂²×C₄

—

178

—

115

D₈

Dic₁₇₈

yes

complemented

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₈₉

yes

characteristic

118

C₂×C₈

C₂²×D₈₉

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

178

—

123

C₄⋊C₄

—

178

—

124

C₄⋊C₄

—

178

—

125

C₄⋊C₄

C₂²×D₈₉

yes

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

178

—

128

C₄²

—

129

C₄²

C₂²×D₈₉

yes

characteristic

130

C₂³

—

131

Q₈

—

178

—

132

Q₈

—

712

—

133

D₄

—

134

D₄

C₂×Dic₁₇₈

yes

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

178

—

140

C₂×C₄

—

712

—

141

C₂×C₄

D₄×D₈₉

yes

characteristic

142

C₂×C₄

D₄×D₈₉

yes

characteristic

143

C₂×C₄

C₂³×D₈₉

yes

characteristic

144

C₈

—

145

C₈

—

178

—

146

C₈

C₂×Dic₁₇₈

yes

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂×D₄×D₈₉

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

356

—

154

C₄

—

178

—

155

C₄

—

178

—

156

C₄

#15668031

yes

—

157

C₄

C₂²×Dic₁₇₈

yes

characteristic

158

C₄

#15668059

yes

characteristic

159

C₂

—

160

C₂

D₄×Dic₁₇₈

yes

characteristic

161

C₂

C₂×#15668031

yes

characteristic

162

C₂

#19152400

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₄×D₈×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×D₈×C₈₉	C₂²
hypercenter	C₄×D₈	D₈₉
hypocenter	C₈₉	#3380
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂²

C₂×D₄×D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₄×D₈×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

prime radical

C₂×D₈×C₈₉

C₂²

hypercenter

C₄×D₈

D₈₉

hypocenter

C₈₉

#3380

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. #19152305	C₂	semidirect
2. #19152003	C₂	semidirect
3. #19151999	C₂	semidirect
4. #19151896	C₂	semidirect
5. #19151888	C₂	semidirect
6. C₈×Dic₁₇₈	C₂	semidirect
42. D₈×C₈₉	Q₈	semidirect
82. C₈₉	#3380	semidirect
115. D₈	Dic₁₇₈	semidirect

H ≅ G/N

type

1. #19152305

C₂

semidirect

2. #19152003

C₂

semidirect

3. #19151999

C₂

semidirect

4. #19151896

C₂

semidirect

5. #19151888

C₂

semidirect

6. C₈×Dic₁₇₈

C₂

semidirect

42. D₈×C₈₉

Q₈

semidirect

82. C₈₉

#3380

semidirect

115. D₈

Dic₁₇₈

semidirect

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

subgroup	index
#19152305	2
#19152003	2
#19151999	2
#19151896	2
#19151888	2
C₈×Dic₁₇₈	2
D₈×Dic₈₉	2
C₄×D₈×C₈₉	2
#3380	89

subgroup

index

C₈×Dic₁₇₈

D₈×Dic₈₉

C₄×D₈×C₈₉

#3380

Subgroups — #26033117

Conjugacy classes of subgroups

Complements

Maximal subgroups