Group #26031691 — subgroups

Subgroups — #26031691

Order 11392.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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	#19152203	C₂	1	2	yes	maximal, complemented
2	#19151899	C₂	1	4	yes	maximal, complemented
3	#19151851	C₂	1	2	yes	maximal, complemented
4	#19151834	C₂	1	2	yes	maximal, complemented
5	#19151794	C₂	1	4	yes	maximal, complemented
6	#424×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₇₁₂	C₂²	1	2	yes	—
8	C₂×D₇₁₂	C₂²	1	2	yes	—
9	#15668009	C₂²	1	2	yes	—
10	#15668004	C₂²	1	2	yes	—
11	#15667997	—	2	4	no	—
12	#15667986	C₂²	1	4	yes	—
13	#15667985	—	2	4	no	—
14	C₄×D₃₅₆	C₂²	1	4	yes	—
15	#15667946	C₂²	1	2	yes	—
16	#15667944	C₂²	1	2	yes	—
17	#15667934	C₂²	1	4	yes	—
18	#15667934	C₂²	1	4	yes	—
19	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	#117×C₈₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₈₉	C₂²	1	4	yes	—
23	C₂×D₃₅₆	—	2	4	no	—
24	C₂×D₃₅₆	C₂³	1	4	yes	—
25	C₂×C₂²⋊D₈₉	—	2	4	no	—
26	C₂×C₄×D₈₉	—	2	4	no	—
27	D₇₁₂	—	4	2	no	—
28	D₇₁₂	—	4	2	no	—
29	C₄⋊Dic₈₉	C₂³	1	4	yes	—
30	#13083553	—	2	4	no	—
31	#13083551	—	2	4	no	—
32	#13083551	—	2	4	no	—
33	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
34	M₁₆×C₈₉	—	4	1	no	—
35	C₂×C₈×C₈₉	—	4	1	no	—
36	C₂×C₈×C₈₉	C₂³	1	2	yes	—
37	C₂×C₈×C₈₉	C₂³	1	2	yes	—
38	C₄⋊C₄×C₈₉	D₄	1	2	yes	—
39	C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
40	C₄²×C₈₉	C₂³	1	2	yes	—
41	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₈₉	—	2	4	no	—
43	C₂×C₄×C₈₉	—	2	1	no	—
44	C₂×C₄×C₈₉	—	2	1	no	—
45	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
46	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
48	C₂²⋊D₈₉	—	8	4	no	—
49	C₂×Dic₈₉	—	2	4	no	—
50	D₃₅₆	—	2	4	no	—
51	D₃₅₆	—	2	4	no	—
52	D₃₅₆	—	8	4	no	—
53	C₄×D₈₉	—	8	4	no	—
54	C₈×C₈₉	—	2	2	no	—
55	C₈×C₈₉	—	2	2	no	—
56	C₂²×C₈₉	—	2	1	no	—
57	C₂²×C₈₉	—	2	2	no	—
58	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
59	C₂×D₈₉	—	4	4	no	—
60	C₂×D₈₉	—	4	4	no	—
61	C₂×D₈₉	—	4	4	no	—
62	C₄×C₈₉	—	2	1	no	—
63	C₄×C₈₉	—	2	4	no	—
64	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
65	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
66	Dic₈₉	—	4	4	no	—
67	C₂×C₈₉	—	4	1	no	—
68	C₂×C₈₉	#537	1	1	yes	characteristic
69	C₂×C₈₉	#576	1	2	yes	—
70	D₈₉	—	8	4	no	—
71	#3259	—	89	1	no	maximal, Sylow
72	C₈₉	#3259	1	1	yes	characteristic, complemented, Sylow
73	#540	—	89	2	no	—
74	#478	—	89	4	no	—
75	#469	—	89	2	no	—
76	#466	—	89	2	no	—
77	#459	—	89	4	no	—
78	#424	D₈₉	1	1	yes	characteristic, complemented
79	C₂×D₈	—	89	2	no	—
80	C₂×D₈	—	89	2	no	—
81	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
83	#127	—	89	2	no	—
84	#125	—	89	2	no	—
85	#123	—	178	4	no	—
86	#121	—	89	4	no	—
87	#121	—	178	4	no	—
88	C₄×D₄	—	89	4	no	—
89	#117	C₂×D₈₉	1	1	yes	characteristic
90	#107	—	89	2	no	—
91	#106	—	89	2	no	—
92	C₄⋊C₈	C₂×D₈₉	1	4	yes	—
93	#102	—	89	4	no	—
94	#102	—	89	4	no	—
95	C₂×D₄	—	89	4	no	—
96	C₂×D₄	—	178	4	no	—
97	C₂×D₄	—	178	4	no	—
98	C₂²×C₄	—	178	4	no	—
99	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
100	D₈	—	356	2	no	—
101	D₈	—	356	2	no	—
102	M₁₆	—	4	1	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	C₂²×D₈₉	1	2	yes	—
105	C₂×C₈	C₂²×D₈₉	1	2	yes	—
106	C₄⋊C₄	—	89	4	no	—
107	C₄⋊C₄	—	178	4	no	—
108	C₄⋊C₄	D₃₅₆	1	2	yes	—
109	C₂²⋊C₄	—	178	4	no	—
110	C₂²⋊C₄	—	178	4	no	—
111	C₂²⋊C₄	D₃₅₆	1	2	yes	—
112	C₄²	C₂²×D₈₉	1	2	yes	—
113	C₂³	—	178	4	no	—
114	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
115	D₄	—	178	4	no	—
116	D₄	—	178	4	no	—
117	D₄	—	712	4	no	—
118	D₄	—	712	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	178	4	no	—
122	C₂×C₄	—	712	4	no	—
123	C₂×C₄	C₂×D₃₅₆	1	4	yes	—
124	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
125	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	356	4	no	—
131	C₂²	—	356	4	no	—
132	C₂²	—	356	4	no	—
133	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
134	C₄	—	2	4	no	—
135	C₄	—	2	1	no	—
136	C₄	—	356	4	no	—
137	C₄	#15668057	1	1	yes	characteristic
138	C₄	#15668060	1	1	yes	characteristic
139	C₂	—	4	1	no	—
140	C₂	—	712	4	no	—
141	C₂	#19152179	1	1	yes	characteristic
142	C₂	#19152403	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152203

C₂

yes

maximal, complemented

#19151899

C₂

yes

maximal, complemented

#19151851

C₂

yes

maximal, complemented

#19151834

C₂

yes

maximal, complemented

#19151794

C₂

yes

maximal, complemented

#424×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×D₇₁₂

C₂²

yes

—

C₂×D₇₁₂

C₂²

yes

—

#15668009

C₂²

yes

—

#15668004

C₂²

yes

—

#15667997

—

#15667986

C₂²

yes

—

#15667985

—

C₄×D₃₅₆

C₂²

yes

—

#15667946

C₂²

yes

—

#15667944

C₂²

yes

—

#15667934

C₂²

yes

—

#15667934

C₂²

yes

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

C₂³

yes

—

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

D₇₁₂

—

D₇₁₂

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

#13083551

—

#13083551

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

D₄

yes

—

C₂²⋊C₄×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

H(5,𝔽₂)

yes

characteristic

C₄×C₈₉

2₋¹⁺⁴

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#537

yes

characteristic

C₂×C₈₉

#576

yes

—

D₈₉

—

#3259

—

maximal, Sylow

C₈₉

#3259

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂²×C₈

C₂×D₈₉

yes

—

—

—

178

—

#121

—

#121

—

178

—

C₄×D₄

—

#117

C₂×D₈₉

yes

characteristic

#107

—

#106

—

C₄⋊C₈

C₂×D₈₉

yes

—

#102

—

#102

—

C₂×D₄

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

100

D₈

—

356

—

101

D₈

—

356

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

C₂²×D₈₉

yes

—

105

C₂×C₈

C₂²×D₈₉

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

178

—

108

C₄⋊C₄

D₃₅₆

yes

—

109

C₂²⋊C₄

—

178

—

110

C₂²⋊C₄

—

178

—

111

C₂²⋊C₄

D₃₅₆

yes

—

112

C₄²

C₂²×D₈₉

yes

—

113

C₂³

—

178

—

114

C₂³

C₂×D₃₅₆

yes

characteristic

115

D₄

—

178

—

116

D₄

—

178

—

117

D₄

—

712

—

118

D₄

—

712

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

178

—

122

C₂×C₄

—

712

—

123

C₂×C₄

C₂×D₃₅₆

yes

—

124

C₂×C₄

C₂×D₃₅₆

yes

characteristic

125

C₂×C₄

C₂³×D₈₉

yes

characteristic

126

C₈

—

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

356

—

131

C₂²

—

356

—

132

C₂²

—

356

—

133

C₂²

C₂²×D₃₅₆

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

356

—

137

C₄

#15668057

yes

characteristic

138

C₄

#15668060

yes

characteristic

139

C₂

—

140

C₂

—

712

—

141

C₂

#19152179

yes

characteristic

142

C₂

#19152403

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₃₅₆
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#424×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#424	D₈₉
hypocenter	C₈₉	#3259
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₃₅₆

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#424×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
1. #19152203	C₂	semidirect
2. #19151899	C₂	semidirect
3. #19151851	C₂	semidirect
4. #19151834	C₂	semidirect
5. #19151794	C₂	semidirect
6. #424×C₈₉	C₂	semidirect
72. C₈₉	#3259	semidirect
78. #424	D₈₉	semidirect

H ≅ G/N

type

1. #19152203

C₂

semidirect

2. #19151899

C₂

semidirect

3. #19151851

C₂

semidirect

4. #19151834

C₂

semidirect

5. #19151794

C₂

semidirect

6. #424×C₈₉

C₂

semidirect

72. C₈₉

#3259

semidirect

78. #424

D₈₉

semidirect

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

subgroup	index
#19152203	2
#19151899	2
#19151851	2
#19151834	2
#19151794	2
#424×C₈₉	2
#3259	89

subgroup

index

#424×C₈₉

Subgroups — #26031691

Conjugacy classes of subgroups

Complements

Maximal subgroups