Group #26030911 — subgroups

Subgroups — #26030911

Order 11392.

338 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 125 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	#19152210	C₂	1	2	yes	maximal
2	#19152000	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151978	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151967	C₂	1	2	yes	maximal, complemented
5	#19151856	C₂	1	2	yes	maximal
6	#19151839	C₂	1	2	yes	maximal
7	#19151706	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₃₅₆	C₂	1	2	yes	maximal, complemented
9	#405×C₈₉	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₃₅₆	C₂²	1	2	yes	—
11	C₂×Dic₃₅₆	C₂²	1	2	yes	—
12	#15668012	C₂²	1	2	yes	—
13	#15668005	—	2	4	no	—
14	#15668004	C₂²	1	2	yes	—
15	#15668002	—	2	4	no	—
16	#15667991	C₂²	1	4	yes	—
17	C₄×Dic₁₇₈	C₂²	1	4	yes	—
18	#15667946	C₂²	1	2	yes	—
19	#15667944	C₂²	1	2	yes	—
20	#15667937	C₂²	1	4	yes	—
21	#15667937	C₂²	1	4	yes	—
22	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
23	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	#117×C₈₉	C₂²	1	1	yes	characteristic
25	#97×C₈₉	C₂²	1	1	yes	characteristic
26	#97×C₈₉	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×Dic₁₇₈	C₂³	1	4	yes	—
30	Dic₃₅₆	—	2	4	no	—
31	Dic₃₅₆	—	2	4	no	—
32	C₄⋊Dic₈₉	—	2	4	no	—
33	C₄⋊Dic₈₉	C₂³	1	4	yes	—
34	#13083553	—	2	4	no	—
35	#13083553	—	2	4	no	—
36	#13083553	—	2	4	no	—
37	C₂²⋊Dic₈₉	—	2	4	no	—
38	C₄×Dic₈₉	—	2	4	no	—
39	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	M₁₆×C₈₉	—	2	2	no	—
41	C₂×C₈×C₈₉	—	2	2	no	—
42	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₈₉	D₄	1	2	yes	—
47	C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
48	C₄²×C₈₉	C₂³	1	1	yes	characteristic
49	C₄²×C₈₉	C₂³	1	1	yes	characteristic
50	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₈₉	—	2	1	no	—
52	C₂×C₄×C₈₉	—	2	1	no	—
53	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₈₉	—	2	4	no	—
58	C₂×Dic₈₉	—	2	4	no	—
59	Dic₁₇₈	—	2	4	no	—
60	Dic₁₇₈	—	2	4	no	—
61	C₈×C₈₉	—	2	1	no	—
62	C₈×C₈₉	—	2	1	no	—
63	C₈×C₈₉	#41	1	2	yes	—
64	C₈×C₈₉	#41	1	2	yes	—
65	C₂²×C₈₉	—	2	1	no	—
66	C₂²×C₈₉	—	2	2	no	—
67	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₈₉	—	2	1	no	—
69	C₄×C₈₉	—	2	2	no	—
70	C₄×C₈₉	—	2	2	no	—
71	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
72	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
73	Dic₈₉	—	4	4	no	—
74	Dic₈₉	—	4	4	no	—
75	C₂×C₈₉	—	4	1	no	—
76	C₂×C₈₉	#532	1	1	yes	characteristic
77	C₂×C₈₉	#578	1	2	yes	—
78	#3177	—	89	1	no	maximal, Sylow
79	C₈₉	#3177	1	1	yes	characteristic, complemented, Sylow
80	#541	—	89	2	no	—
81	#499	—	89	1	no	—
82	#494	—	89	1	no	—
83	#491	—	89	2	no	—
84	#470	—	89	2	no	—
85	#467	—	89	2	no	—
86	#441	—	89	2	no	—
87	C₄×Q₁₆	—	89	2	no	—
88	#405	D₈₉	1	1	yes	characteristic
89	C₂×Q₁₆	—	89	2	no	—
90	C₂×Q₁₆	—	89	2	no	—
91	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
92	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
93	C₄⋊Q₈	—	89	2	no	—
94	#126	—	178	4	no	—
95	#125	—	89	2	no	—
96	#125	—	178	4	no	—
97	C₂²⋊Q₈	—	89	4	no	—
98	C₄×Q₈	—	89	4	no	—
99	#117	C₂×D₈₉	1	1	yes	characteristic
100	#107	—	89	2	no	—
101	#106	—	89	2	no	—
102	#103	—	89	4	no	—
103	#103	—	89	4	no	—
104	#97	C₂×D₈₉	1	1	yes	characteristic
105	#97	C₂×D₈₉	1	1	yes	characteristic
106	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
107	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
108	C₂×Q₈	—	89	4	no	—
109	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
110	Q₁₆	—	178	4	no	—
111	Q₁₆	—	178	4	no	—
112	M₁₆	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
115	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
116	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
117	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
118	C₄⋊C₄	—	89	4	no	—
119	C₄⋊C₄	—	178	4	no	—
120	C₄⋊C₄	—	178	4	no	—
121	C₄⋊C₄	—	178	4	no	—
122	C₄⋊C₄	—	178	4	no	—
123	C₄⋊C₄	D₃₅₆	1	2	yes	—
124	C₂²⋊C₄	—	178	4	no	—
125	C₂²⋊C₄	D₃₅₆	1	2	yes	—
126	C₄²	—	178	4	no	—
127	C₄²	C₂²×D₈₉	1	1	yes	characteristic
128	C₄²	C₂²×D₈₉	1	1	yes	characteristic
129	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
130	Q₈	—	178	4	no	—
131	Q₈	—	178	4	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	178	4	no	—
135	C₂×C₄	—	178	4	no	—
136	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
137	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
138	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
139	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	#13083574	1	2	yes	—
143	C₈	#13083574	1	2	yes	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	2	no	—
146	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	—	356	4	no	—
151	C₄	—	356	4	no	—
152	C₄	C₂×#13083574	1	1	yes	characteristic
153	C₄	C₂×#13083574	1	1	yes	characteristic
154	C₂	—	4	1	no	—
155	C₂	#19152148	1	1	yes	characteristic
156	C₂	#19152415	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152210

C₂

yes

maximal

#19152000

C₂

yes

maximal, characteristic, complemented

#19151978

C₂

yes

maximal, characteristic, complemented

#19151967

C₂

yes

maximal, complemented

#19151856

C₂

yes

maximal

#19151839

C₂

yes

maximal

#19151706

C₂

yes

maximal, complemented

C₄×Dic₃₅₆

C₂

yes

maximal, complemented

#405×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₃₅₆

C₂²

yes

—

C₂×Dic₃₅₆

C₂²

yes

—

#15668012

C₂²

yes

—

#15668005

—

#15668004

C₂²

yes

—

#15668002

—

#15667991

C₂²

yes

—

C₄×Dic₁₇₈

C₂²

yes

—

#15667946

C₂²

yes

—

#15667944

C₂²

yes

—

#15667937

C₂²

yes

—

#15667937

C₂²

yes

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

characteristic

#97×C₈₉

C₂²

yes

characteristic

#97×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

C₂³

yes

—

Dic₃₅₆

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

D₄

yes

—

C₂²⋊C₄×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#41

yes

—

C₈×C₈₉

#41

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#41

yes

characteristic

C₄×C₈₉

C₂×#41

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#532

yes

characteristic

C₂×C₈₉

#578

yes

—

#3177

—

maximal, Sylow

C₈₉

#3177

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

—

C₄×Q₁₆

—

#405

D₈₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₄⋊Q₈

—

#126

—

178

—

#125

—

#125

—

178

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₈₉

yes

characteristic

100

#107

—

101

#106

—

102

#103

—

103

#103

—

104

#97

C₂×D₈₉

yes

characteristic

105

#97

C₂×D₈₉

yes

characteristic

106

C₄×C₈

C₂×D₈₉

yes

characteristic

107

C₄×C₈

C₂×D₈₉

yes

characteristic

108

C₂×Q₈

—

109

C₂²×C₄

C₂²×D₈₉

yes

characteristic

110

Q₁₆

—

178

—

111

Q₁₆

—

178

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

C₂²×D₈₉

yes

characteristic

115

C₂×C₈

C₂²×D₈₉

yes

characteristic

116

C₂×C₈

C₂²×D₈₉

yes

characteristic

117

C₂×C₈

C₂²×D₈₉

yes

characteristic

118

C₄⋊C₄

—

119

C₄⋊C₄

—

178

—

120

C₄⋊C₄

—

178

—

121

C₄⋊C₄

—

178

—

122

C₄⋊C₄

—

178

—

123

C₄⋊C₄

D₃₅₆

yes

—

124

C₂²⋊C₄

—

178

—

125

C₂²⋊C₄

D₃₅₆

yes

—

126

C₄²

—

178

—

127

C₄²

C₂²×D₈₉

yes

characteristic

128

C₄²

C₂²×D₈₉

yes

characteristic

129

C₂³

C₂×D₃₅₆

yes

characteristic

130

Q₈

—

178

—

131

Q₈

—

178

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

178

—

135

C₂×C₄

—

178

—

136

C₂×C₄

C₂×D₃₅₆

yes

—

137

C₂×C₄

C₂×D₃₅₆

yes

—

138

C₂×C₄

C₂×D₃₅₆

yes

characteristic

139

C₂×C₄

C₂³×D₈₉

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

#13083574

yes

—

143

C₈

#13083574

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂²×D₃₅₆

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

356

—

151

C₄

—

356

—

152

C₄

C₂×#13083574

yes

characteristic

153

C₄

C₂×#13083574

yes

characteristic

154

C₂

—

155

C₂

#19152148

yes

characteristic

156

C₂

#19152415

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₃₅₆
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#405×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	#405	D₈₉
hypocenter	C₈₉	#3177
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

subgroup

G/H

center

C₂²

C₂²×D₃₅₆

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#405×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405

N	H ≅ G/N	type
2. #19152000	C₂	semidirect
3. #19151978	C₂	semidirect
4. #19151967	C₂	semidirect
7. #19151706	C₂	semidirect
8. C₄×Dic₃₅₆	C₂	semidirect
79. C₈₉	#3177	semidirect

H ≅ G/N

type

2. #19152000

C₂

semidirect

3. #19151978

C₂

semidirect

4. #19151967

C₂

semidirect

7. #19151706

C₂

semidirect

8. C₄×Dic₃₅₆

C₂

semidirect

79. C₈₉

#3177

semidirect

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

subgroup	index
#19152210	2
#19152000	2
#19151978	2
#19151967	2
#19151856	2
#19151839	2
#19151706	2
C₄×Dic₃₅₆	2
#405×C₈₉	2
#3177	89

subgroup

index

C₄×Dic₃₅₆

#405×C₈₉

#3177

Subgroups — #26030911

Conjugacy classes of subgroups

Complements

Maximal subgroups