Group #26029331 — subgroups

Subgroups — #26029331

Order 11392.

396 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 217 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×Q₈×D₈₉	C₂	1	1	yes	maximal, characteristic
2	Q₈×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₁₇₈	C₂	1	3	yes	maximal, complemented
4	#19151722	C₂	1	3	yes	maximal, complemented
5	#19151650	C₂	1	3	yes	maximal
6	#19151649	C₂	1	3	yes	maximal
7	C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₈₉	C₄	1	1	yes	characteristic
9	Q₈×Dic₈₉	C₄	1	1	yes	characteristic
10	C₄×Dic₁₇₈	C₂²	1	3	yes	—
11	#15667977	C₄	1	3	yes	—
12	C₄⋊C₄×D₈₉	C₄	1	3	yes	—
13	C₄²×D₈₉	C₂²	1	3	yes	—
14	C₄⋊#10623494	C₂²	1	3	yes	—
15	#15667941	C₂²	1	3	yes	—
16	#15667941	C₂²	1	3	yes	—
17	#15667924	C₂²	1	1	yes	characteristic
18	#15667924	C₂²	1	3	yes	—
19	C₄×#10623494	C₂²	1	3	yes	—
20	C₈×Dic₈₉	C₂²	1	6	yes	—
21	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
23	C₄×C₈×C₈₉	C₂²	1	3	yes	—
24	C₂×Dic₁₇₈	C₂×C₄	1	3	yes	—
25	Q₈×D₈₉	C₈	1	8	yes	complemented
26	C₂×C₄×D₈₉	C₂×C₄	1	3	yes	—
27	C₂×C₄×D₈₉	C₂×C₄	1	3	yes	—
28	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
29	C₂×#10623494	C₂³	1	3	yes	—
30	C₂×#10623494	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₈₉	C₂×C₄	1	3	yes	—
32	#13083553	C₂×C₄	1	6	yes	—
33	C₄×Dic₈₉	C₂×C₄	1	3	yes	—
34	C₄×Dic₈₉	C₂×C₄	1	3	yes	—
35	C₄×Dic₈₉	C₂³	1	3	yes	—
36	C₂×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₈₉	C₂³	1	3	yes	—
39	C₄⋊C₄×C₈₉	C₂×C₄	1	3	yes	—
40	C₄²×C₈₉	C₂³	1	3	yes	—
41	C₂²×D₈₉	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₈₉	C₂×C₈	1	4	yes	—
43	C₂×C₄×C₈₉	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₈₉	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₈₉	C₂²×C₄	1	3	yes	—
48	C₂×Dic₈₉	C₂²×C₄	1	3	yes	—
49	C₄×D₈₉	—	2	1	no	—
50	C₄×D₈₉	—	2	1	no	—
51	C₄×D₈₉	—	4	3	no	—
52	C₄×D₈₉	C₂×C₈	1	12	yes	—
53	Dic₁₇₈	C₂×C₈	1	12	yes	—
54	C₈×C₈₉	—	2	3	no	—
55	C₈×C₈₉	—	2	1	no	—
56	#10623494	—	2	1	no	—
57	#10623494	—	2	3	no	—
58	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
59	C₂×D₈₉	—	2	1	no	—
60	C₂×D₈₉	—	2	1	no	—
61	C₂×D₈₉	C₂²×C₈	1	2	yes	—
62	C₄×C₈₉	—	2	3	no	—
63	C₄×C₈₉	C₂²×C₈	1	6	yes	—
64	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
65	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
66	Dic₈₉	—	2	1	no	—
67	Dic₈₉	—	2	3	no	—
68	Dic₈₉	C₂²×C₈	1	6	yes	—
69	C₂×C₈₉	#519	1	1	yes	characteristic
70	C₂×C₈₉	C₂³×C₈	1	1	yes	characteristic
71	C₂×C₈₉	#568	1	1	yes	characteristic
72	D₈₉	—	2	2	no	—
73	#2982	—	89	1	no	maximal, Sylow
74	C₈₉	#2982	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₄×Q₈	—	89	1	no	—
76	C₈×Q₈	—	89	1	no	—
77	C₈×Q₈	—	89	3	no	—
78	C₈×Q₈	—	89	3	no	—
79	C₈×Q₈	D₈₉	1	1	yes	characteristic, complemented
80	#431	—	89	3	no	—
81	#431	—	89	3	no	—
82	C₂²×Q₈	—	89	1	no	—
83	C₄×Q₈	—	89	1	no	—
84	C₄×Q₈	—	89	3	no	—
85	C₄×Q₈	—	89	3	no	—
86	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
87	C₂×C₄⋊C₄	—	89	3	no	—
88	C₂×C₄²	—	89	3	no	—
89	C₄⋊C₈	—	89	3	no	—
90	C₄⋊C₈	—	89	3	no	—
91	C₄⋊C₈	—	89	3	no	—
92	C₄⋊C₈	C₂×D₈₉	1	3	yes	—
93	C₂²⋊C₈	—	89	3	no	—
94	C₂²⋊C₈	—	89	1	no	—
95	C₄×C₈	—	89	6	no	—
96	C₄×C₈	—	89	3	no	—
97	C₄×C₈	C₂×D₈₉	1	3	yes	—
98	C₂×Q₈	—	89	8	no	—
99	C₂×Q₈	—	89	3	no	—
100	C₂×Q₈	C₄×D₈₉	1	1	yes	characteristic
101	C₂²×C₄	—	89	3	no	—
102	C₂²×C₄	—	89	3	no	—
103	C₂²×C₄	—	89	1	no	—
104	C₂×C₈	—	89	1	no	—
105	C₂×C₈	—	89	3	no	—
106	C₂×C₈	C₂²×D₈₉	1	3	yes	—
107	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
108	C₄⋊C₄	—	89	6	no	—
109	C₄⋊C₄	—	89	3	no	—
110	C₄⋊C₄	C₄×D₈₉	1	3	yes	—
111	C₄²	—	89	3	no	—
112	C₄²	—	89	3	no	—
113	C₄²	—	89	3	no	—
114	C₄²	C₂²×D₈₉	1	3	yes	—
115	C₂³	—	89	1	no	—
116	Q₈	—	89	12	no	—
117	Q₈	C₈×D₈₉	1	4	yes	—
118	C₂×C₄	—	89	3	no	—
119	C₂×C₄	—	89	12	no	—
120	C₂×C₄	—	89	1	no	—
121	C₂×C₄	—	89	3	no	—
122	C₂×C₄	—	356	3	no	—
123	C₂×C₄	—	178	1	no	—
124	C₂×C₄	—	178	1	no	—
125	C₂×C₄	C₂×C₄×D₈₉	1	3	yes	—
126	C₂×C₄	C₂×C₄×D₈₉	1	3	yes	—
127	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
128	C₈	—	2	3	no	—
129	C₈	—	2	1	no	—
130	C₈	—	178	1	no	—
131	C₈	—	178	3	no	—
132	C₂²	—	89	2	no	—
133	C₂²	—	178	1	no	—
134	C₂²	—	178	1	no	—
135	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
136	C₄	—	2	3	no	—
137	C₄	—	89	6	no	—
138	C₄	—	178	1	no	—
139	C₄	—	178	3	no	—
140	C₄	C₂×C₈×D₈₉	1	6	yes	—
141	C₄	#15668059	1	1	yes	characteristic
142	C₄	#15668060	1	1	yes	characteristic
143	C₂	—	178	2	no	—
144	C₂	#19152074	1	1	yes	characteristic
145	C₂	C₂²×C₈×D₈₉	1	1	yes	characteristic
146	C₂	#19152363	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Q₈×D₈₉

C₂

yes

maximal, characteristic

Q₈×#10623494

C₂

yes

maximal, characteristic, complemented

C₈×Dic₁₇₈

C₂

yes

maximal, complemented

#19151722

C₂

yes

maximal, complemented

#19151650

C₂

yes

maximal

#19151649

C₂

yes

maximal

C₈×Q₈×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₈₉

C₄

yes

characteristic

Q₈×Dic₈₉

C₄

yes

characteristic

C₄×Dic₁₇₈

C₂²

yes

—

#15667977

C₄

yes

—

C₄⋊C₄×D₈₉

C₄

yes

—

C₄²×D₈₉

C₂²

yes

—

C₄⋊#10623494

C₂²

yes

—

#15667941

C₂²

yes

—

#15667941

C₂²

yes

—

#15667924

C₂²

yes

characteristic

#15667924

C₂²

yes

—

C₄×#10623494

C₂²

yes

—

C₈×Dic₈₉

C₂²

yes

—

C₄×Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

C₂×C₄

yes

—

Q₈×D₈₉

C₈

yes

complemented

C₂×C₄×D₈₉

C₂×C₄

yes

—

C₂×C₄×D₈₉

C₂×C₄

yes

—

C₂×C₄×D₈₉

C₂³

yes

characteristic

C₂×#10623494

C₂³

yes

—

C₂×#10623494

C₂³

yes

characteristic

C₄⋊Dic₈₉

C₂×C₄

yes

—

#13083553

C₂×C₄

yes

—

C₄×Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

C₂³

yes

—

C₂×Q₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂³

yes

—

C₂²×D₈₉

C₂²×C₄

yes

characteristic

Q₈×C₈₉

C₂×C₈

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

C₂²×C₄

yes

characteristic

C₂×Dic₈₉

C₂²×C₄

yes

—

C₂×Dic₈₉

C₂²×C₄

yes

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

C₂×C₈

yes

—

Dic₁₇₈

C₂×C₈

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

C₂²×C₈

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂²×C₈

yes

—

C₄×C₈₉

2₋¹⁺⁴

yes

characteristic

C₄×C₈₉

2₋¹⁺⁴

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

C₂²×C₈

yes

—

C₂×C₈₉

#519

yes

characteristic

C₂×C₈₉

C₂³×C₈

yes

characteristic

C₂×C₈₉

#568

yes

characteristic

D₈₉

—

#2982

—

maximal, Sylow

C₈₉

#2982

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

D₈₉

yes

characteristic, complemented

#431

—

#431

—

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₈₉

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₈₉

yes

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

C₄×D₈₉

yes

characteristic

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂²×D₈₉

yes

—

107

C₂×C₈

C₂²×D₈₉

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

C₄×D₈₉

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂²×D₈₉

yes

—

115

C₂³

—

116

Q₈

—

117

Q₈

C₈×D₈₉

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

356

—

123

C₂×C₄

—

178

—

124

C₂×C₄

—

178

—

125

C₂×C₄

C₂×C₄×D₈₉

yes

—

126

C₂×C₄

C₂×C₄×D₈₉

yes

—

127

C₂×C₄

C₂³×D₈₉

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

178

—

131

C₈

—

178

—

132

C₂²

—

133

C₂²

—

178

—

134

C₂²

—

178

—

135

C₂²

C₂²×C₄×D₈₉

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

178

—

139

C₄

—

178

—

140

C₄

C₂×C₈×D₈₉

yes

—

141

C₄

#15668059

yes

characteristic

142

C₄

#15668060

yes

characteristic

143

C₂

—

178

—

144

C₂

#19152074

yes

characteristic

145

C₂

C₂²×C₈×D₈₉

yes

characteristic

146

C₂

#19152363

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator	C₂×C₈₉	C₂³×C₈
Fitting	C₈×Q₈×C₈₉	C₂
socle	C₂²×C₈₉	C₂³×C₄
prime radical	C₂²×D₈₉	C₂²×C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₈×Q₈	D₈₉
hypocenter	C₈₉	#2982
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₈₉

commutator

C₂×C₈₉

C₂³×C₈

Fitting

C₈×Q₈×C₈₉

C₂

socle

C₂²×C₈₉

C₂³×C₄

prime radical

C₂²×D₈₉

C₂²×C₄

cosocle

C₂×C₄×C₈₉

C₂⁴

hypercenter

C₈×Q₈

D₈₉

C₈₉

G ▷ C₂×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₄ ◁ C₈×Q₈

N	H ≅ G/N	type
2. Q₈×#10623494	C₂	semidirect
3. C₈×Dic₁₇₈	C₂	semidirect
4. #19151722	C₂	semidirect
7. C₈×Q₈×C₈₉	C₂	semidirect
25. Q₈×D₈₉	C₈	semidirect
74. C₈₉	#2982	semidirect
79. C₈×Q₈	D₈₉	semidirect

H ≅ G/N

type

2. Q₈×#10623494

C₂

semidirect

3. C₈×Dic₁₇₈

C₂

semidirect

4. #19151722

C₂

semidirect

7. C₈×Q₈×C₈₉

C₂

semidirect

25. Q₈×D₈₉

C₈

semidirect

74. C₈₉

#2982

semidirect

79. C₈×Q₈

D₈₉

semidirect

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

subgroup	index
C₄×Q₈×D₈₉	2
Q₈×#10623494	2
C₈×Dic₁₇₈	2
#19151722	2
#19151650	2
#19151649	2
C₈×Q₈×C₈₉	2
#2982	89

subgroup

index

C₄×Q₈×D₈₉

C₈×Dic₁₇₈

C₈×Q₈×C₈₉

Subgroups — #26029331

Conjugacy classes of subgroups

Complements

Maximal subgroups