Group #26031027 — subgroups

Subgroups — #26031027

Order 11392.

416 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 137 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×#15667991	C₂	1	2	yes	maximal
2	#19151934	C₂	1	4	yes	maximal, complemented
3	#19151839	C₂	1	4	yes	maximal, complemented
4	#19151756	C₂	1	4	yes	maximal, complemented
5	C₂×C₂²⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
7	C₂×Dic₃₅₆	C₂²	1	4	yes	complemented
8	#15667991	—	2	2	no	—
9	#15667991	—	2	2	no	—
10	#15667991	C₂²	1	4	yes	—
11	#15667991	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
13	C₂×#13083553	—	2	2	no	—
14	C₂×C₂²⋊Dic₈₉	—	2	2	no	—
15	#15667946	C₂²	1	4	yes	complemented
16	#15667937	C₂²	1	8	yes	complemented
17	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
19	C₂²⋊C₈×C₈₉	C₂²	1	4	yes	—
20	C₂×Dic₁₇₈	—	2	2	no	—
21	C₂×Dic₁₇₈	—	4	4	no	—
22	C₂×Dic₁₇₈	C₂³	1	4	yes	—
23	C₂²×Dic₈₉	—	2	2	no	—
24	C₂²×Dic₈₉	—	2	2	no	—
25	Dic₃₅₆	—	4	4	no	—
26	C₄⋊Dic₈₉	—	2	2	no	—
27	C₄⋊Dic₈₉	C₂³	1	4	yes	—
28	#13083553	—	2	4	no	—
29	#13083553	—	2	4	no	—
30	C₂²⋊Dic₈₉	—	2	4	no	—
31	C₂²⋊Dic₈₉	—	2	4	no	—
32	C₂⁴×C₈₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₈₉	—	4	2	no	—
34	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₈₉	D₄	1	2	yes	—
36	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
38	C₂×C₈×C₈₉	—	4	2	no	—
39	C₂×C₈×C₈₉	C₂³	1	4	yes	—
40	C₂³×C₈₉	—	4	1	no	—
41	C₂³×C₈₉	—	4	1	no	—
42	C₂³×C₈₉	Q₁₆	1	4	yes	complemented
43	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂³×C₈₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₈₉	—	2	2	no	—
46	C₂×C₄×C₈₉	—	2	4	no	—
47	C₂×C₄×C₈₉	—	4	1	no	—
48	C₂×C₄×C₈₉	—	4	2	no	—
49	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₈₉	—	2	4	no	—
53	C₂×Dic₈₉	—	2	4	no	—
54	C₂×Dic₈₉	—	4	4	no	—
55	C₂×Dic₈₉	—	4	4	no	—
56	Dic₁₇₈	—	2	8	no	—
57	Dic₁₇₈	—	4	4	no	—
58	C₈×C₈₉	—	2	4	no	—
59	C₂²×C₈₉	—	2	2	no	—
60	C₂²×C₈₉	—	2	1	no	—
61	C₂²×C₈₉	—	2	2	no	—
62	C₂²×C₈₉	—	2	1	no	—
63	C₂²×C₈₉	—	4	4	no	—
64	C₂²×C₈₉	C₂×Q₁₆	1	2	yes	—
65	C₂²×C₈₉	C₂×Q₁₆	1	4	yes	—
66	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
67	C₄×C₈₉	—	2	2	no	—
68	C₄×C₈₉	—	2	1	no	—
69	C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
70	Dic₈₉	—	4	4	no	—
71	Dic₈₉	—	4	4	no	—
72	C₂×C₈₉	—	2	4	no	—
73	C₂×C₈₉	—	2	2	no	—
74	C₂×C₈₉	#534	1	1	yes	characteristic
75	C₂×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
76	C₂×C₈₉	#575	1	1	yes	characteristic
77	#3189	—	89	1	no	maximal, Sylow
78	C₈₉	#3189	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₂²⋊Q₈	—	89	2	no	—
80	#484	—	89	4	no	—
81	#467	—	89	4	no	—
82	#451	—	89	4	no	—
83	C₂×C₂²⋊C₈	D₈₉	1	1	yes	characteristic
84	C₂²×Q₈	—	89	2	no	—
85	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
86	C₂×Q₁₆	—	89	4	no	—
87	C₂²×C₈	C₂×D₈₉	1	2	yes	—
88	C₂²⋊Q₈	—	89	4	no	—
89	C₂²⋊Q₈	—	89	4	no	—
90	C₂²⋊Q₈	—	178	2	no	—
91	C₂²⋊Q₈	—	178	2	no	—
92	C₂×C₄⋊C₄	—	89	2	no	—
93	C₂×C₄⋊C₄	—	178	2	no	—
94	C₂×C₂²⋊C₄	—	178	2	no	—
95	#107	—	89	4	no	—
96	#103	—	89	8	no	—
97	C₂²⋊C₈	C₂×D₈₉	1	4	yes	—
98	C₂⁴	D₃₅₆	1	1	yes	characteristic
99	C₂×Q₈	—	89	4	no	—
100	C₂×Q₈	—	356	4	no	—
101	C₂×Q₈	—	178	2	no	—
102	C₂²×C₄	—	4	2	no	—
103	C₂²×C₄	—	178	2	no	—
104	C₂²×C₄	—	178	2	no	—
105	C₂²×C₄	D₃₅₆	1	2	yes	—
106	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
107	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
108	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
109	Q₁₆	—	356	4	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	C₂²×D₈₉	1	4	yes	—
112	C₄⋊C₄	—	89	4	no	—
113	C₄⋊C₄	—	178	2	no	—
114	C₄⋊C₄	—	178	4	no	—
115	C₄⋊C₄	—	178	4	no	—
116	C₂²⋊C₄	—	178	4	no	—
117	C₂²⋊C₄	—	178	4	no	—
118	C₂³	—	4	1	no	—
119	C₂³	—	4	1	no	—
120	C₂³	Dic₃₅₆	1	4	yes	complemented
121	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
122	C₂³	C₂×D₃₅₆	1	2	yes	—
123	Q₈	—	356	4	no	—
124	Q₈	—	178	8	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	356	4	no	—
130	C₂×C₄	—	356	4	no	—
131	C₂×C₄	—	178	4	no	—
132	C₂×C₄	—	178	4	no	—
133	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
134	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
135	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
136	C₈	—	2	4	no	—
137	C₂²	—	2	2	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	4	4	no	—
142	C₂²	C₂×Dic₃₅₆	1	2	yes	—
143	C₂²	C₂×Dic₃₅₆	1	4	yes	—
144	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	1	no	—
147	C₄	—	356	4	no	—
148	C₄	—	356	4	no	—
149	C₄	#15668057	1	2	yes	—
150	C₂	—	2	4	no	—
151	C₂	—	2	2	no	—
152	C₂	#19152160	1	1	yes	characteristic
153	C₂	C₂²×Dic₃₅₆	1	1	yes	characteristic
154	C₂	#19152398	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667991

C₂

yes

maximal

#19151934

C₂

yes

maximal, complemented

#19151839

C₂

yes

maximal, complemented

#19151756

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

C₂×Dic₃₅₆

C₂²

yes

—

—

C₂²

yes

—

#15667991

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

—

C₂×C₂²⋊Dic₈₉

—

#15667946

C₂²

yes

complemented

#15667937

C₂²

yes

complemented

C₂³×C₄×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

—

C₂²⋊C₈×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂³

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₂⁴×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

Q₁₆

yes

complemented

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂³×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×Q₁₆

yes

—

C₂²×C₈₉

C₂×Q₁₆

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

H(5,𝔽₂)

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#534

yes

characteristic

C₂×C₈₉

C₂²×Q₁₆

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×C₂²⋊Q₈

—

#484

—

#467

—

#451

—

C₂×C₂²⋊C₈

D₈₉

yes

characteristic

C₂²×Q₈

—

C₂³×C₄

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

C₂²×C₈

C₂×D₈₉

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

178

—

C₂²⋊Q₈

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₂²⋊C₄

—

178

—

#107

—

#103

—

C₂²⋊C₈

C₂×D₈₉

yes

—

C₂⁴

D₃₅₆

yes

characteristic

C₂×Q₈

—

100

C₂×Q₈

—

356

—

101

C₂×Q₈

—

178

—

102

C₂²×C₄

—

103

C₂²×C₄

—

178

—

104

C₂²×C₄

—

178

—

105

C₂²×C₄

D₃₅₆

yes

—

106

C₂²×C₄

D₃₅₆

yes

characteristic

107

C₂²×C₄

C₂²×D₈₉

yes

—

108

C₂²×C₄

C₂²×D₈₉

yes

characteristic

109

Q₁₆

—

356

—

110

C₂×C₈

—

111

C₂×C₈

C₂²×D₈₉

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

178

—

114

C₄⋊C₄

—

178

—

115

C₄⋊C₄

—

178

—

116

C₂²⋊C₄

—

178

—

117

C₂²⋊C₄

—

178

—

118

C₂³

—

119

C₂³

—

120

C₂³

Dic₃₅₆

yes

complemented

121

C₂³

C₂×D₃₅₆

yes

characteristic

122

C₂³

C₂×D₃₅₆

yes

—

123

Q₈

—

356

—

124

Q₈

—

178

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

356

—

130

C₂×C₄

—

356

—

131

C₂×C₄

—

178

—

132

C₂×C₄

—

178

—

133

C₂×C₄

C₂×D₃₅₆

yes

characteristic

134

C₂×C₄

C₂×D₃₅₆

yes

—

135

C₂×C₄

C₂³×D₈₉

yes

characteristic

136

C₈

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂×Dic₃₅₆

yes

—

143

C₂²

C₂×Dic₃₅₆

yes

—

144

C₂²

C₂²×D₃₅₆

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

356

—

148

C₄

—

356

—

149

C₄

#15668057

yes

—

150

C₂

—

151

C₂

—

152

C₂

#19152160

yes

characteristic

153

C₂

C₂²×Dic₃₅₆

yes

characteristic

154

C₂

#19152398

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₅₆
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×C₂²⋊C₈×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂⁴×C₈₉	D₄
hypercenter	C₂×C₂²⋊C₈	D₈₉
hypocenter	C₈₉	#3189
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂²

C₂²×D₃₅₆

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₂×C₂²⋊C₈×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

prime radical

C₂⁴×C₈₉

D₄

hypercenter

C₂×C₂²⋊C₈

D₈₉

hypocenter

C₈₉

#3189

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
2. #19151934	C₂	semidirect
3. #19151839	C₂	semidirect
4. #19151756	C₂	semidirect
7. C₂×Dic₃₅₆	C₂²	semidirect
15. #15667946	C₂²	semidirect
16. #15667937	C₂²	semidirect
42. C₂³×C₈₉	Q₁₆	semidirect
78. C₈₉	#3189	semidirect
120. C₂³	Dic₃₅₆	semidirect

H ≅ G/N

type

2. #19151934

C₂

semidirect

3. #19151839

C₂

semidirect

4. #19151756

C₂

semidirect

7. C₂×Dic₃₅₆

C₂²

semidirect

15. #15667946

C₂²

semidirect

16. #15667937

C₂²

semidirect

42. C₂³×C₈₉

Q₁₆

semidirect

78. C₈₉

#3189

semidirect

120. C₂³

Dic₃₅₆

semidirect

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

subgroup	index
C₂×#15667991	2
#19151934	2
#19151839	2
#19151756	2
C₂×C₂²⋊C₈×C₈₉	2
#3189	89

subgroup

index

C₂×C₂²⋊C₈×C₈₉

Subgroups — #26031027

Conjugacy classes of subgroups

Complements

Maximal subgroups