Group #26022977 — subgroups

Subgroups — #26022977

Order 11392.

740 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 205 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×#15668009	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₃₅₆	C₂	1	2	yes	maximal, complemented
3	#19151500	C₂	1	2	yes	maximal, complemented
4	#19151476	C₂	1	8	yes	maximal, complemented
5	#19151464	C₂	1	1	yes	maximal, characteristic, complemented
6	#376×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₃₅₆	C₂²	1	2	yes	—
8	C₂²×D₃₅₆	C₂²	1	4	yes	—
9	C₂²×C₄×D₈₉	C₂²	1	4	yes	—
10	#15668009	C₄	1	8	yes	complemented
11	C₄×D₃₅₆	—	2	8	no	—
12	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
13	C₂×#13083551	C₂²	1	4	yes	—
14	C₂×#13083551	C₂²	1	8	yes	—
15	#15667919	C₂²	1	4	yes	—
16	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
19	#95×C₈₉	C₂²	1	4	yes	—
20	C₂³×D₈₉	C₂³	1	4	yes	—
21	C₂×D₃₅₆	—	2	4	no	—
22	C₂×D₃₅₆	—	2	4	no	—
23	C₂×D₃₅₆	—	4	8	no	—
24	C₂×D₃₅₆	C₂×C₄	1	8	yes	complemented
25	C₂×D₃₅₆	C₂×C₄	1	16	yes	complemented
26	C₂²×Dic₈₉	C₂³	1	4	yes	—
27	C₂×C₄×D₈₉	—	2	8	no	—
28	C₂×C₄×D₈₉	—	4	8	no	—
29	C₄⋊Dic₈₉	—	2	4	no	—
30	#13083551	—	2	8	no	—
31	#13083551	—	4	8	no	—
32	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
33	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
34	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
35	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
36	C₄²×C₈₉	—	2	2	no	—
37	C₄²×C₈₉	C₂×C₄	1	4	yes	complemented
38	C₄²×C₈₉	D₄	1	4	yes	complemented
39	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
40	C₂²×D₈₉	—	2	4	no	—
41	C₂²×D₈₉	—	2	4	no	—
42	C₂²×D₈₉	—	2	8	no	—
43	C₂²×D₈₉	—	2	8	no	—
44	C₂²×D₈₉	C₂²×C₄	1	8	yes	—
45	C₂×C₄×C₈₉	—	2	4	no	—
46	C₂×C₄×C₈₉	—	2	4	no	—
47	C₂×C₄×C₈₉	—	2	2	no	—
48	C₂×C₄×C₈₉	—	2	4	no	—
49	C₂×C₄×C₈₉	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
52	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₈₉	#41	1	2	yes	—
54	C₂×Dic₈₉	—	2	4	no	—
55	C₂×Dic₈₉	—	2	8	no	—
56	D₃₅₆	—	2	16	no	—
57	D₃₅₆	—	4	16	no	—
58	C₄×D₈₉	—	4	16	no	—
59	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
60	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
61	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
62	C₂²×C₈₉	H(5,𝔽₂)	1	4	yes	—
63	C₂×D₈₉	—	2	8	no	—
64	C₂×D₈₉	—	2	16	no	—
65	C₂×D₈₉	—	4	4	no	—
66	C₂×D₈₉	—	4	4	no	—
67	C₂×D₈₉	—	4	8	no	—
68	C₄×C₈₉	—	2	4	no	—
69	C₄×C₈₉	—	2	4	no	—
70	C₄×C₈₉	—	4	2	no	—
71	C₄×C₈₉	C₄×D₄	1	4	yes	—
72	Dic₈₉	—	4	4	no	—
73	C₂×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
74	C₂×C₈₉	#518	1	2	yes	—
75	C₂×C₈₉	#535	1	2	yes	—
76	C₂×C₈₉	#540	1	2	yes	—
77	D₈₉	—	4	8	no	—
78	#2328	—	89	1	no	maximal, Sylow
79	C₈₉	#2328	1	1	yes	characteristic, complemented, Sylow
80	C₂×#127	—	89	1	no	—
81	C₂×C₄×D₄	—	89	2	no	—
82	#390	—	89	2	no	—
83	#386	—	89	8	no	—
84	#382	—	89	1	no	—
85	#376	D₈₉	1	1	yes	characteristic, complemented
86	C₂²×D₄	—	89	4	no	—
87	C₂²×D₄	—	89	2	no	—
88	C₂³×C₄	—	89	4	no	—
89	#127	—	89	8	no	—
90	C₄×D₄	—	178	8	no	—
91	C₂×C₄⋊C₄	—	89	2	no	—
92	C₂×C₂²⋊C₄	—	89	8	no	—
93	C₂×C₂²⋊C₄	—	89	4	no	—
94	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
96	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
97	#95	—	89	4	no	—
98	#95	C₂×D₈₉	1	4	yes	—
99	C₂⁴	—	89	4	no	—
100	C₂×D₄	—	89	16	no	—
101	C₂×D₄	—	89	8	no	—
102	C₂×D₄	—	356	8	no	—
103	C₂×D₄	—	178	4	no	—
104	C₂×D₄	—	178	4	no	—
105	C₂²×C₄	—	89	4	no	—
106	C₂²×C₄	—	356	8	no	—
107	C₂²×C₄	—	178	8	no	—
108	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
109	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
110	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
111	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
112	C₄⋊C₄	—	178	4	no	—
113	C₂²⋊C₄	—	356	8	no	—
114	C₂²⋊C₄	—	178	8	no	—
115	C₄²	—	2	2	no	—
116	C₄²	C₄×D₈₉	1	4	yes	complemented
117	C₄²	D₃₅₆	1	4	yes	complemented
118	C₂³	—	89	8	no	—
119	C₂³	—	178	4	no	—
120	C₂³	—	178	4	no	—
121	C₂³	—	178	8	no	—
122	C₂³	—	178	8	no	—
123	C₂³	C₂³×D₈₉	1	1	yes	characteristic
124	D₄	—	356	16	no	—
125	D₄	—	178	16	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	356	16	no	—
131	C₂×C₄	—	178	4	no	—
132	C₂×C₄	—	178	8	no	—
133	C₂×C₄	C₂×C₄×D₈₉	1	4	yes	—
134	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
135	C₂×C₄	C₂×D₃₅₆	1	4	yes	—
136	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
137	C₂×C₄	#13083574	1	2	yes	—
138	C₂²	—	356	4	no	—
139	C₂²	—	356	4	no	—
140	C₂²	—	356	8	no	—
141	C₂²	—	178	8	no	—
142	C₂²	—	178	16	no	—
143	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
144	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
145	C₂²	C₂×#13083574	1	1	yes	characteristic
146	C₂²	#15668058	1	4	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	2	4	no	—
149	C₄	—	4	2	no	—
150	C₄	—	356	4	no	—
151	C₄	C₄×D₃₅₆	1	4	yes	—
152	C₂	—	356	8	no	—
153	C₂	C₂×C₄×D₃₅₆	1	1	yes	characteristic
154	C₂	#19152072	1	2	yes	—
155	C₂	#19152167	1	2	yes	—
156	C₂	#19152203	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668009

C₂

yes

maximal, characteristic

C₂×C₄×D₃₅₆

C₂

yes

maximal, complemented

#19151500

C₂

yes

maximal, complemented

#19151476

C₂

yes

maximal, complemented

#19151464

C₂

yes

maximal, characteristic, complemented

#376×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₅₆

C₂²

yes

—

C₂²×D₃₅₆

C₂²

yes

—

C₂²×C₄×D₈₉

C₂²

yes

—

#15668009

C₄

yes

complemented

C₄×D₃₅₆

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

#15667919

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#95×C₈₉

C₂²

yes

—

C₂³×D₈₉

C₂³

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

C₂×C₄

yes

complemented

C₂×D₃₅₆

C₂×C₄

yes

complemented

C₂²×Dic₈₉

C₂³

yes

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

C₄⋊Dic₈₉

—

#13083551

—

#13083551

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

C₂×C₄

yes

complemented

C₄²×C₈₉

D₄

yes

complemented

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄×D₄

yes

—

Dic₈₉

—

C₂×C₈₉

C₂×C₄×D₄

yes

characteristic

C₂×C₈₉

#518

yes

—

C₂×C₈₉

#535

yes

—

C₂×C₈₉

#540

yes

—

D₈₉

—

#2328

—

maximal, Sylow

C₈₉

#2328

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×C₄×D₄

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂²×D₄

—

C₂²×D₄

—

C₂³×C₄

—

#127

—

C₄×D₄

—

178

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₂×C₄²

C₂×D₈₉

yes

characteristic

#95

—

#95

C₂×D₈₉

yes

—

C₂⁴

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

356

—

103

C₂×D₄

—

178

—

104

C₂×D₄

—

178

—

105

C₂²×C₄

—

106

C₂²×C₄

—

356

—

107

C₂²×C₄

—

178

—

108

C₂²×C₄

C₂²×D₈₉

yes

—

109

C₂²×C₄

C₂²×D₈₉

yes

—

110

C₂²×C₄

C₂²×D₈₉

yes

—

111

C₂²×C₄

C₂²×D₈₉

yes

characteristic

112

C₄⋊C₄

—

178

—

113

C₂²⋊C₄

—

356

—

114

C₂²⋊C₄

—

178

—

115

C₄²

—

116

C₄²

C₄×D₈₉

yes

complemented

117

C₄²

D₃₅₆

yes

complemented

118

C₂³

—

119

C₂³

—

178

—

120

C₂³

—

178

—

121

C₂³

—

178

—

122

C₂³

—

178

—

123

C₂³

C₂³×D₈₉

yes

characteristic

124

D₄

—

356

—

125

D₄

—

178

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

356

—

131

C₂×C₄

—

178

—

132

C₂×C₄

—

178

—

133

C₂×C₄

C₂×C₄×D₈₉

yes

—

134

C₂×C₄

C₂×C₄×D₈₉

yes

—

135

C₂×C₄

C₂×D₃₅₆

yes

—

136

C₂×C₄

C₂×D₃₅₆

yes

—

137

C₂×C₄

#13083574

yes

—

138

C₂²

—

356

—

139

C₂²

—

356

—

140

C₂²

—

356

—

141

C₂²

—

178

—

142

C₂²

—

178

—

143

C₂²

C₂²×C₄×D₈₉

yes

characteristic

144

C₂²

C₂²×D₃₅₆

yes

characteristic

145

C₂²

C₂×#13083574

yes

characteristic

146

C₂²

#15668058

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

356

—

151

C₄

C₄×D₃₅₆

yes

—

152

C₂

—

356

—

153

C₂

C₂×C₄×D₃₅₆

yes

characteristic

154

C₂

#19152072

yes

—

155

C₂

#19152167

yes

—

156

C₂

#19152203

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator	C₂²×C₈₉	C₂³×C₄
Fitting	#376×C₈₉	C₂
socle, cosocle	C₂³×C₈₉	C₂⁴
prime radical	C₂×#15668009	C₂
hypercenter	#376	D₈₉
hypocenter	C₈₉	#2328
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #376

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator

C₂²×C₈₉

C₂³×C₄

Fitting

#376×C₈₉

C₂

socle, cosocle

C₂³×C₈₉

C₂⁴

C₂

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #376

N	H ≅ G/N	type
2. C₂×C₄×D₃₅₆	C₂	semidirect
3. #19151500	C₂	semidirect
4. #19151476	C₂	semidirect
5. #19151464	C₂	semidirect
6. #376×C₈₉	C₂	semidirect
10. #15668009	C₄	semidirect
24. C₂×D₃₅₆	C₂×C₄	semidirect
25. C₂×D₃₅₆	C₂×C₄	semidirect
37. C₄²×C₈₉	C₂×C₄	semidirect
38. C₄²×C₈₉	D₄	semidirect
79. C₈₉	#2328	semidirect
85. #376	D₈₉	semidirect
116. C₄²	C₄×D₈₉	semidirect
117. C₄²	D₃₅₆	semidirect

H ≅ G/N

type

2. C₂×C₄×D₃₅₆

C₂

semidirect

3. #19151500

C₂

semidirect

4. #19151476

C₂

semidirect

5. #19151464

C₂

semidirect

6. #376×C₈₉

C₂

semidirect

10. #15668009

C₄

semidirect

24. C₂×D₃₅₆

C₂×C₄

semidirect

25. C₂×D₃₅₆

C₂×C₄

semidirect

37. C₄²×C₈₉

C₂×C₄

semidirect

38. C₄²×C₈₉

D₄

semidirect

79. C₈₉

#2328

semidirect

85. #376

D₈₉

semidirect

116. C₄²

C₄×D₈₉

semidirect

117. C₄²

D₃₅₆

semidirect

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

subgroup	index
C₂×#15668009	2
C₂×C₄×D₃₅₆	2
#19151500	2
#19151476	2
#19151464	2
#376×C₈₉	2
#2328	89

subgroup

index

C₂×C₄×D₃₅₆

#376×C₈₉

Subgroups — #26022977

Conjugacy classes of subgroups

Complements

Maximal subgroups