Group #26028249 — subgroups

Subgroups — #26028249

Order 11392.

712 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15667981	C₂	1	2	yes	maximal, complemented
2	#19151523	C₂	1	3	yes	maximal, complemented
3	#19151514	C₂	1	6	yes	maximal, complemented
4	#19151507	C₂	1	3	yes	maximal, complemented
5	#379×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic, complemented
7	C₂²×C₂²⋊D₈₉	C₂²	1	6	yes	—
8	#15667981	—	2	8	no	—
9	C₂×C₄⋊Dic₈₉	C₂²	1	3	yes	complemented
10	C₂×C₂²⋊Dic₈₉	C₂²	1	6	yes	—
11	C₂×#13083551	C₂²	1	6	yes	complemented
12	#15667919	C₂²	1	3	yes	complemented
13	#15667919	C₂²	1	3	yes	complemented
14	C₂⁵×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₈₉	C₂²	1	3	yes	—
16	C₂×C₂²⋊C₄×C₈₉	C₂²	1	3	yes	—
17	C₂³×D₈₉	C₂³	1	2	yes	—
18	C₂×D₃₅₆	—	4	3	no	—
19	C₂×C₂²⋊D₈₉	—	2	12	no	—
20	C₂×C₂²⋊D₈₉	—	4	6	no	—
21	C₂×C₂²⋊D₈₉	—	4	6	no	—
22	C₂²×Dic₈₉	C₂³	1	6	yes	—
23	C₄⋊Dic₈₉	—	4	3	no	—
24	C₂²⋊Dic₈₉	—	2	12	no	—
25	#13083551	—	4	6	no	—
26	C₂⁴×C₈₉	—	4	3	no	—
27	C₂⁴×C₈₉	—	4	3	no	—
28	C₂⁴×C₈₉	D₄	1	4	yes	complemented
29	C₂⁴×C₈₉	C₂³	1	3	yes	—
30	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
32	C₂²⋊C₄×C₈₉	—	4	3	no	—
33	C₂²⋊C₄×C₈₉	—	4	3	no	—
34	C₂³×C₈₉	—	2	3	no	—
35	C₂³×C₈₉	—	2	3	no	—
36	C₂³×C₈₉	—	2	6	no	—
37	C₂³×C₈₉	—	2	6	no	—
38	C₂³×C₈₉	—	4	12	no	—
39	C₂³×C₈₉	—	4	12	no	—
40	C₂³×C₈₉	—	4	4	no	—
41	C₂³×C₈₉	C₂×D₄	1	6	yes	—
42	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
43	C₂²×D₈₉	—	2	2	no	—
44	C₂²×D₈₉	—	2	6	no	—
45	C₂²×D₈₉	—	2	6	no	—
46	C₂×C₄×C₈₉	—	2	6	no	—
47	C₂×C₄×C₈₉	—	2	3	no	—
48	C₂×C₄×C₈₉	—	2	3	no	—
49	C₂²⋊D₈₉	—	4	24	no	—
50	C₂×Dic₈₉	—	2	6	no	—
51	C₂×Dic₈₉	—	2	6	no	—
52	C₂×Dic₈₉	—	2	6	no	—
53	D₃₅₆	—	4	4	no	—
54	C₂²×C₈₉	—	2	12	no	—
55	C₂²×C₈₉	—	2	6	no	—
56	C₂²×C₈₉	—	4	16	no	—
57	C₂²×C₈₉	—	4	6	no	—
58	C₂²×C₈₉	—	4	3	no	—
59	C₂²×C₈₉	—	4	3	no	—
60	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
61	C₂²×C₈₉	H(5,𝔽₂)	1	3	yes	—
62	C₂²×C₈₉	H(5,𝔽₂)	1	3	yes	—
63	C₂×D₈₉	—	4	6	no	—
64	C₂×D₈₉	—	4	6	no	—
65	C₂×D₈₉	—	4	2	no	—
66	C₄×C₈₉	—	4	3	no	—
67	C₄×C₈₉	—	4	1	no	—
68	Dic₈₉	—	4	6	no	—
69	C₂×C₈₉	—	4	6	no	—
70	C₂×C₈₉	#534	1	3	yes	—
71	C₂×C₈₉	#560	1	3	yes	—
72	C₂×C₈₉	H(3,𝔽₄)	1	1	yes	characteristic
73	D₈₉	—	8	2	no	—
74	#2812	—	89	1	no	maximal, Sylow
75	C₈₉	#2812	1	1	yes	characteristic, complemented, Sylow
76	C₂×#120	—	89	2	no	—
77	#397	—	89	3	no	—
78	#394	—	89	6	no	—
79	#392	—	89	3	no	—
80	#379	D₈₉	1	1	yes	characteristic, complemented
81	C₂⁵	C₂×D₈₉	1	1	yes	characteristic
82	C₂²×D₄	—	89	6	no	—
83	C₂²×D₄	—	89	1	no	—
84	#120	—	178	8	no	—
85	C₂×C₄⋊C₄	—	89	3	no	—
86	C₂×C₂²⋊C₄	—	89	6	no	—
87	C₂×C₂²⋊C₄	—	89	6	no	—
88	C₂×C₂²⋊C₄	C₂×D₈₉	1	3	yes	—
89	C₂×C₂²⋊C₄	C₂×D₈₉	1	3	yes	—
90	#95	—	89	3	no	—
91	#95	—	89	3	no	—
92	C₂⁴	—	4	3	no	—
93	C₂⁴	—	4	3	no	—
94	C₂⁴	—	89	2	no	—
95	C₂⁴	D₃₅₆	1	4	yes	complemented
96	C₂⁴	C₂²×D₈₉	1	3	yes	—
97	C₂×D₄	—	356	3	no	—
98	C₂×D₄	—	356	6	no	—
99	C₂×D₄	—	356	6	no	—
100	C₂×D₄	—	178	12	no	—
101	C₂²×C₄	—	89	6	no	—
102	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
104	C₄⋊C₄	—	356	3	no	—
105	C₂²⋊C₄	—	4	3	no	—
106	C₂²⋊C₄	—	4	3	no	—
107	C₂²⋊C₄	—	356	6	no	—
108	C₂²⋊C₄	—	178	12	no	—
109	C₂³	—	2	3	no	—
110	C₂³	—	2	3	no	—
111	C₂³	—	2	6	no	—
112	C₂³	—	2	6	no	—
113	C₂³	—	4	12	no	—
114	C₂³	—	4	12	no	—
115	C₂³	—	4	4	no	—
116	C₂³	—	178	2	no	—
117	C₂³	—	178	6	no	—
118	C₂³	—	178	6	no	—
119	C₂³	C₂×D₃₅₆	1	6	yes	—
120	C₂³	C₂³×D₈₉	1	1	yes	characteristic
121	D₄	—	356	4	no	—
122	D₄	—	356	24	no	—
123	C₂×C₄	—	2	6	no	—
124	C₂×C₄	—	2	3	no	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	178	6	no	—
127	C₂×C₄	—	178	6	no	—
128	C₂×C₄	—	178	6	no	—
129	C₂²	—	2	12	no	—
130	C₂²	—	2	6	no	—
131	C₂²	—	4	16	no	—
132	C₂²	—	4	6	no	—
133	C₂²	—	4	3	no	—
134	C₂²	—	4	3	no	—
135	C₂²	—	356	6	no	—
136	C₂²	—	356	6	no	—
137	C₂²	—	356	2	no	—
138	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
139	C₂²	#15668057	1	3	yes	—
140	C₂²	#15668057	1	3	yes	—
141	C₄	—	4	3	no	—
142	C₄	—	4	1	no	—
143	C₄	—	356	6	no	—
144	C₂	—	4	6	no	—
145	C₂	—	712	2	no	—
146	C₂	#19152160	1	3	yes	—
147	C₂	#19152337	1	3	yes	—
148	C₂	#19152341	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667981

C₂

yes

maximal, complemented

#19151523

C₂

yes

maximal, complemented

#19151514

C₂

yes

maximal, complemented

#19151507

C₂

yes

maximal, complemented

#379×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₅₆

C₂²

yes

characteristic, complemented

C₂²×C₂²⋊D₈₉

C₂²

yes

—

#15667981

—

C₂×C₄⋊Dic₈₉

C₂²

yes

complemented

C₂×C₂²⋊Dic₈₉

C₂²

yes

—

C₂×#13083551

C₂²

yes

complemented

#15667919

C₂²

yes

complemented

#15667919

C₂²

yes

complemented

C₂⁵×C₈₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₈₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₈₉

C₂²

yes

—

C₂³×D₈₉

C₂³

yes

—

C₂×D₃₅₆

—

C₂×C₂²⋊D₈₉

—

C₂×C₂²⋊D₈₉

—

C₂×C₂²⋊D₈₉

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

#13083551

—

C₂⁴×C₈₉

—

C₂⁴×C₈₉

—

C₂⁴×C₈₉

D₄

yes

complemented

C₂⁴×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#534

yes

—

C₂×C₈₉

#560

yes

—

C₂×C₈₉

H(3,𝔽₄)

yes

characteristic

D₈₉

—

#2812

—

maximal, Sylow

C₈₉

#2812

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂⁵

C₂×D₈₉

yes

characteristic

C₂²×D₄

—

C₂²×D₄

—

#120

—

178

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₈₉

yes

—

C₂×C₂²⋊C₄

C₂×D₈₉

yes

—

#95

—

#95

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

D₃₅₆

yes

complemented

C₂⁴

C₂²×D₈₉

yes

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

100

C₂×D₄

—

178

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₈₉

yes

characteristic

103

C₂²×C₄

C₂²×D₈₉

yes

—

104

C₄⋊C₄

—

356

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

356

—

108

C₂²⋊C₄

—

178

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

—

115

C₂³

—

116

C₂³

—

178

—

117

C₂³

—

178

—

118

C₂³

—

178

—

119

C₂³

C₂×D₃₅₆

yes

—

120

C₂³

C₂³×D₈₉

yes

characteristic

121

D₄

—

356

—

122

D₄

—

356

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

178

—

127

C₂×C₄

—

178

—

128

C₂×C₄

—

178

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

356

—

136

C₂²

—

356

—

137

C₂²

—

356

—

138

C₂²

C₂²×D₃₅₆

yes

characteristic

139

C₂²

#15668057

yes

—

140

C₂²

#15668057

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

356

—

144

C₂

—

145

C₂

—

712

—

146

C₂

#19152160

yes

—

147

C₂

#19152337

yes

—

148

C₂

#19152341

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#379×C₈₉	C₂
hypercenter	#379	D₈₉
hypocenter	C₈₉	#2812
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #379

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#379×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #379

N	H ≅ G/N	type
1. C₂×#15667981	C₂	semidirect
2. #19151523	C₂	semidirect
3. #19151514	C₂	semidirect
4. #19151507	C₂	semidirect
5. #379×C₈₉	C₂	semidirect
6. C₂²×D₃₅₆	C₂²	semidirect
9. C₂×C₄⋊Dic₈₉	C₂²	semidirect
11. C₂×#13083551	C₂²	semidirect
12. #15667919	C₂²	semidirect
13. #15667919	C₂²	semidirect
28. C₂⁴×C₈₉	D₄	semidirect
75. C₈₉	#2812	semidirect
80. #379	D₈₉	semidirect
95. C₂⁴	D₃₅₆	semidirect

H ≅ G/N

type

1. C₂×#15667981

C₂

semidirect

2. #19151523

C₂

semidirect

3. #19151514

C₂

semidirect

4. #19151507

C₂

semidirect

5. #379×C₈₉

C₂

semidirect

6. C₂²×D₃₅₆

C₂²

semidirect

9. C₂×C₄⋊Dic₈₉

C₂²

semidirect

11. C₂×#13083551

C₂²

semidirect

12. #15667919

C₂²

semidirect

13. #15667919

C₂²

semidirect

28. C₂⁴×C₈₉

D₄

semidirect

75. C₈₉

#2812

semidirect

80. #379

D₈₉

semidirect

95. C₂⁴

D₃₅₆

semidirect

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

subgroup	index
C₂×#15667981	2
#19151523	2
#19151514	2
#19151507	2
#379×C₈₉	2
#2812	89

subgroup

index

#379×C₈₉

Subgroups — #26028249

Conjugacy classes of subgroups

Complements

Maximal subgroups