Group #26028901 — subgroups

Subgroups — #26028901

Order 11392.

384 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 173 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	C₄×#13083574	C₂	1	1	yes	maximal, characteristic
2	#19151660	C₂	1	4	yes	maximal, complemented
3	#19151653	C₂	1	4	yes	maximal, complemented
4	#19151624	C₂	1	2	yes	maximal, complemented
5	#19151564	C₂	1	2	yes	maximal
6	C₄×#13083558	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#97×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13083574	C₂²	1	1	yes	characteristic
9	C₂×#13083558	C₂²	1	2	yes	—
10	C₄×Dic₁₇₈	C₄	1	2	yes	—
11	C₄×D₃₅₆	C₄	1	2	yes	—
12	C₄×C₂²⋊D₈₉	C₄	1	4	yes	—
13	#15667971	C₂²	1	1	yes	characteristic
14	#15667969	C₂²	1	2	yes	—
15	C₄²×D₈₉	C₂²	1	2	yes	—
16	#15667941	C₂²	1	8	yes	—
17	#15667924	C₂²	1	8	yes	—
18	#15667923	C₂²	1	2	yes	—
19	C₄×#10623494	C₂²	1	2	yes	—
20	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
22	#97×C₈₉	C₂²	1	2	yes	—
23	#97×C₈₉	C₂²	1	2	yes	—
24	#13083574	—	2	4	no	—
25	C₂×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
26	C₂×D₃₅₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₈₉	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₈₉	C₂³	1	2	yes	—
30	#13083558	—	2	4	no	—
31	C₂×#10623494	C₂³	1	4	yes	—
32	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
33	#13083553	C₂×C₄	1	4	yes	—
34	C₂²⋊Dic₈₉	C₂×C₄	1	2	yes	—
35	#13083551	C₂×C₄	1	4	yes	—
36	C₄×Dic₈₉	C₂×C₄	1	4	yes	—
37	C₄×Dic₈₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₈₉	—	2	4	no	—
41	C₂×C₈×C₈₉	C₂³	1	4	yes	—
42	C₄²×C₈₉	C₂×C₄	1	2	yes	—
43	C₄²×C₈₉	C₂³	1	1	yes	characteristic
44	C₄²×C₈₉	C₂³	1	1	yes	characteristic
45	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₈₉	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₈₉	—	4	2	no	—
48	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₈₉	#41	1	4	yes	—
52	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
53	C₂²⋊D₈₉	—	2	4	no	—
54	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
55	C₂×Dic₈₉	C₂²×C₄	1	4	yes	—
56	D₃₅₆	—	2	2	no	—
57	C₄×D₈₉	—	2	4	no	—
58	C₄×D₈₉	—	4	4	no	—
59	Dic₁₇₈	—	2	2	no	—
60	C₈×C₈₉	—	2	4	no	—
61	#10623494	—	2	4	no	—
62	C₂²×C₈₉	—	2	2	no	—
63	C₂²×C₈₉	#117	1	2	yes	—
64	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₈₉	—	2	2	no	—
66	C₂×D₈₉	—	2	4	no	—
67	C₄×C₈₉	—	2	2	no	—
68	C₄×C₈₉	—	2	2	no	—
69	C₄×C₈₉	#117	1	2	yes	—
70	C₄×C₈₉	C₂×#41	1	2	yes	—
71	Dic₈₉	—	2	2	no	—
72	Dic₈₉	—	2	4	no	—
73	C₂×C₈₉	—	2	2	no	—
74	C₂×C₈₉	C₂×#117	1	1	yes	characteristic
75	C₂×C₈₉	#568	1	2	yes	—
76	D₈₉	—	4	2	no	—
77	#2923	—	89	1	no	maximal, Sylow
78	C₈₉	#2923	1	1	yes	characteristic, complemented, Sylow
79	C₄×#41	—	89	1	no	—
80	#433	—	89	4	no	—
81	#432	—	89	4	no	—
82	#424	—	89	2	no	—
83	#408	—	89	2	no	—
84	C₄×M₁₆	—	89	1	no	—
85	C₂×#97	D₈₉	1	1	yes	characteristic, complemented
86	C₂×#41	—	89	1	no	—
87	C₂×M₁₆	—	89	2	no	—
88	C₂²×C₈	C₂×D₈₉	1	2	yes	—
89	C₄×Q₈	—	89	2	no	—
90	C₄×D₄	—	89	4	no	—
91	C₄×D₄	—	89	2	no	—
92	#117	—	89	1	no	—
93	#117	—	89	2	no	—
94	C₂×C₄²	—	89	2	no	—
95	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
96	C₄⋊C₈	—	89	8	no	—
97	C₂²⋊C₈	—	89	8	no	—
98	#97	—	89	2	no	—
99	#97	C₂×D₈₉	1	2	yes	—
100	#97	C₂×D₈₉	1	2	yes	—
101	C₄×C₈	—	89	2	no	—
102	#41	—	178	4	no	—
103	C₂×Q₈	—	89	1	no	—
104	C₂×D₄	—	89	2	no	—
105	C₂×D₄	—	89	1	no	—
106	C₂²×C₄	—	89	4	no	—
107	C₂²×C₄	—	89	2	no	—
108	C₂²×C₄	C₄×D₈₉	1	2	yes	—
109	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
110	M₁₆	—	178	4	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	89	4	no	—
113	C₂×C₈	C₂²×D₈₉	1	4	yes	—
114	C₄⋊C₄	—	89	4	no	—
115	C₄⋊C₄	—	89	2	no	—
116	C₂²⋊C₄	—	89	2	no	—
117	C₂²⋊C₄	—	89	4	no	—
118	C₄²	—	89	4	no	—
119	C₄²	—	89	2	no	—
120	C₄²	C₄×D₈₉	1	2	yes	—
121	C₄²	C₂²×D₈₉	1	1	yes	characteristic
122	C₄²	C₂²×D₈₉	1	1	yes	characteristic
123	C₂³	—	89	2	no	—
124	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
125	Q₈	—	178	2	no	—
126	D₄	—	178	4	no	—
127	D₄	—	178	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	89	4	no	—
130	C₂×C₄	—	89	2	no	—
131	C₂×C₄	—	356	4	no	—
132	C₂×C₄	—	178	4	no	—
133	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
135	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
136	C₂×C₄	#13083574	1	4	yes	—
137	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	178	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	178	2	no	—
142	C₂²	—	178	4	no	—
143	C₂²	#15667969	1	2	yes	—
144	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	178	2	no	—
148	C₄	—	178	4	no	—
149	C₄	#15667969	1	2	yes	—
150	C₄	C₂×#13083574	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	356	2	no	—
153	C₂	C₂×#15667969	1	1	yes	characteristic
154	C₂	#19152364	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13083574

C₂

yes

maximal, characteristic

#19151660

C₂

yes

maximal, complemented

#19151653

C₂

yes

maximal, complemented

#19151624

C₂

yes

maximal, complemented

#19151564

C₂

yes

maximal

C₄×#13083558

C₂

yes

maximal, characteristic, complemented

C₂×#97×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

C₂²

yes

characteristic

C₂×#13083558

C₂²

yes

—

C₄×Dic₁₇₈

C₄

yes

—

C₄×D₃₅₆

C₄

yes

—

C₄×C₂²⋊D₈₉

C₄

yes

—

#15667971

C₂²

yes

characteristic

#15667969

C₂²

yes

—

C₄²×D₈₉

C₂²

yes

—

#15667941

C₂²

yes

—

#15667924

C₂²

yes

—

#15667923

C₂²

yes

—

C₄×#10623494

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#97×C₈₉

C₂²

yes

—

#97×C₈₉

C₂²

yes

—

#13083574

—

C₂×Dic₁₇₈

C₂×C₄

yes

characteristic

C₂×D₃₅₆

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₈₉

C₂×C₄

yes

—

C₂×C₄×D₈₉

C₂×C₄

yes

—

C₂×C₄×D₈₉

C₂³

yes

—

#13083558

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

#13083553

C₂×C₄

yes

—

C₂²⋊Dic₈₉

C₂×C₄

yes

—

#13083551

C₂×C₄

yes

—

C₄×Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄²×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

C₂²×C₄

yes

characteristic

C₂²×D₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

C₂²×C₄

yes

—

C₂×Dic₈₉

C₂²×C₄

yes

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#117

yes

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#117

yes

—

C₄×C₈₉

C₂×#41

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×#117

yes

characteristic

C₂×C₈₉

#568

yes

—

D₈₉

—

#2923

—

maximal, Sylow

C₈₉

#2923

yes

characteristic, complemented, Sylow

C₄×#41

—

—

—

—

—

C₄×M₁₆

—

C₂×#97

D₈₉

yes

characteristic, complemented

C₂×#41

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₈₉

yes

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

C₂×D₈₉

yes

—

100

#97

C₂×D₈₉

yes

—

101

C₄×C₈

—

102

#41

—

178

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₈₉

yes

—

109

C₂²×C₄

C₂²×D₈₉

yes

characteristic

110

M₁₆

—

178

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂²×D₈₉

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₄×D₈₉

yes

—

121

C₄²

C₂²×D₈₉

yes

characteristic

122

C₄²

C₂²×D₈₉

yes

characteristic

123

C₂³

—

124

C₂³

C₂×C₄×D₈₉

yes

characteristic

125

Q₈

—

178

—

126

D₄

—

178

—

127

D₄

—

178

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

356

—

132

C₂×C₄

—

178

—

133

C₂×C₄

C₂×C₄×D₈₉

yes

—

134

C₂×C₄

C₂×C₄×D₈₉

yes

—

135

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

136

C₂×C₄

#13083574

yes

—

137

C₂×C₄

C₂³×D₈₉

yes

characteristic

138

C₈

—

139

C₈

—

178

—

140

C₂²

—

141

C₂²

—

178

—

142

C₂²

—

178

—

143

C₂²

#15667969

yes

—

144

C₂²

C₂²×C₄×D₈₉

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

178

—

148

C₄

—

178

—

149

C₄

#15667969

yes

—

150

C₄

C₂×#13083574

yes

—

151

C₂

—

152

C₂

—

356

—

153

C₂

C₂×#15667969

yes

characteristic

154

C₂

#19152364

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	C₂×#97×C₈₉	C₂
prime radical	C₂×#13083574	C₂²
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₂×#97	D₈₉
hypocenter	C₈₉	#2923
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂³×C₄

Fitting

C₂×#97×C₈₉

C₂

prime radical

C₂×#13083574

C₂²

cosocle

C₂×C₄×C₈₉

C₂⁴

hypercenter

C₂×#97

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₄ ◁ C₂×#97

N	H ≅ G/N	type
2. #19151660	C₂	semidirect
3. #19151653	C₂	semidirect
4. #19151624	C₂	semidirect
6. C₄×#13083558	C₂	semidirect
7. C₂×#97×C₈₉	C₂	semidirect
78. C₈₉	#2923	semidirect
85. C₂×#97	D₈₉	semidirect

H ≅ G/N

type

2. #19151660

C₂

semidirect

3. #19151653

C₂

semidirect

4. #19151624

C₂

semidirect

6. C₄×#13083558

C₂

semidirect

7. C₂×#97×C₈₉

C₂

semidirect

78. C₈₉

#2923

semidirect

85. C₂×#97

D₈₉

semidirect

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

subgroup	index
C₄×#13083574	2
#19151660	2
#19151653	2
#19151624	2
#19151564	2
C₄×#13083558	2
C₂×#97×C₈₉	2
#2923	89

subgroup

index

C₂×#97×C₈₉

Subgroups — #26028901

Conjugacy classes of subgroups

Complements

Maximal subgroups