Group #26030600 — subgroups

Subgroups — #26030600

Order 11392.

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

normal subgroups only

This group has 392 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	#19152148	C₂	1	1	yes	maximal, characteristic
2	#19151968	C₂	1	2	yes	maximal, complemented
3	#19151964	C₂	1	2	yes	maximal, complemented
4	#19151960	C₂	1	4	yes	maximal, complemented
5	#19151592	C₂	1	4	yes	maximal, complemented
6	C₂×#15667923	C₂	1	1	yes	maximal, characteristic, complemented
7	#533×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13083574	C₂²	1	2	yes	—
9	C₂²×#10623494	C₂²	1	2	yes	—
10	#15668012	C₂²	1	1	yes	characteristic
11	#15668009	C₂²	1	1	yes	characteristic
12	#15668000	C₂²	1	2	yes	—
13	#15667986	—	2	2	no	—
14	C₄×D₃₅₆	—	2	2	no	—
15	#15667935	C₂²	1	8	yes	—
16	#15667932	C₂²	1	8	yes	—
17	#15667923	C₂²	1	2	yes	—
18	#15667923	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
20	#125×C₈₉	C₂²	1	2	yes	—
21	C₂²⋊Q₈×C₈₉	—	2	2	no	—
22	C₄×Q₈×C₈₉	—	2	2	no	—
23	#117×C₈₉	C₂²	1	2	yes	—
24	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
25	#13083574	—	2	8	no	—
26	C₂×Dic₁₇₈	C₂³	1	2	yes	—
27	C₂×D₃₅₆	—	2	2	no	—
28	C₂×D₃₅₆	C₂³	1	2	yes	—
29	C₂×C₂²⋊D₈₉	—	2	2	no	—
30	C₂×C₄×D₈₉	—	2	2	no	—
31	C₂×#10623494	—	2	4	no	—
32	C₂×#10623494	C₂³	1	4	yes	—
33	C₄⋊Dic₈₉	—	2	2	no	—
34	#13083551	—	2	4	no	—
35	C₂×Q₈×C₈₉	—	2	2	no	—
36	C₂²×C₄×C₈₉	D₄	1	2	yes	—
37	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₈₉	—	2	4	no	—
39	C₄⋊C₄×C₈₉	—	2	2	no	—
40	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
41	C₂²⋊C₄×C₈₉	—	2	2	no	—
42	C₄²×C₈₉	—	2	2	no	—
43	C₄²×C₈₉	D₄	1	2	yes	—
44	C₄²×C₈₉	C₂³	1	1	yes	characteristic
45	C₄²×C₈₉	C₂³	1	1	yes	characteristic
46	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×D₈₉	—	2	2	no	—
48	Q₈×C₈₉	—	4	4	no	—
49	C₂×C₄×C₈₉	—	2	4	no	—
50	C₂×C₄×C₈₉	—	4	2	no	—
51	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₈₉	#41	1	4	yes	—
55	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
56	C₂²⋊D₈₉	—	4	4	no	—
57	C₂×Dic₈₉	—	2	2	no	—
58	D₃₅₆	—	2	4	no	—
59	D₃₅₆	—	4	4	no	—
60	C₄×D₈₉	—	4	4	no	—
61	Dic₁₇₈	—	2	4	no	—
62	#10623494	—	2	4	no	—
63	C₂²×C₈₉	—	2	2	no	—
64	C₂²×C₈₉	#124	1	2	yes	—
65	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₈₉	—	4	2	no	—
67	C₂×D₈₉	—	4	4	no	—
68	C₄×C₈₉	—	2	2	no	—
69	C₄×C₈₉	—	2	2	no	—
70	C₄×C₈₉	—	4	4	no	—
71	C₄×C₈₉	#124	1	2	yes	—
72	C₄×C₈₉	C₂×#41	1	2	yes	—
73	Dic₈₉	—	4	2	no	—
74	C₂×C₈₉	—	2	2	no	—
75	C₂×C₈₉	C₂×#124	1	1	yes	characteristic
76	C₂×C₈₉	#575	1	2	yes	—
77	D₈₉	—	8	2	no	—
78	#3135	—	89	1	no	maximal, Sylow
79	C₈₉	#3135	1	1	yes	characteristic, complemented, Sylow
80	#533	D₈₉	1	1	yes	characteristic, complemented
81	#532	—	89	1	no	—
82	#491	—	89	2	no	—
83	#490	—	89	2	no	—
84	#489	—	89	4	no	—
85	#416	—	89	4	no	—
86	C₂×#97	—	89	1	no	—
87	C₂×#41	—	89	2	no	—
88	C₂²×C₈	—	89	2	no	—
89	C₄⋊Q₈	—	89	1	no	—
90	C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
91	#127	—	89	1	no	—
92	#125	C₂×D₈₉	1	2	yes	—
93	#124	—	89	2	no	—
94	C₂²⋊Q₈	—	2	2	no	—
95	#121	—	178	2	no	—
96	C₄×Q₈	—	2	2	no	—
97	C₄×D₄	—	178	2	no	—
98	#117	C₂×D₈₉	1	2	yes	—
99	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
100	#103	—	89	8	no	—
101	#102	—	89	8	no	—
102	#97	—	89	2	no	—
103	#97	—	89	2	no	—
104	#41	—	178	8	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	89	2	no	—
107	C₂×D₄	—	89	2	no	—
108	C₂×D₄	—	178	2	no	—
109	C₂×D₄	—	178	2	no	—
110	C₂²×C₄	—	178	2	no	—
111	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
112	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
113	C₂×C₈	—	89	4	no	—
114	C₂×C₈	—	178	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	178	2	no	—
118	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	178	4	no	—
121	C₄²	—	2	2	no	—
122	C₄²	C₂²⋊D₈₉	1	2	yes	—
123	C₄²	C₂²×D₈₉	1	1	yes	characteristic
124	C₄²	C₂²×D₈₉	1	1	yes	characteristic
125	C₂³	—	178	2	no	—
126	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
127	Q₈	—	4	4	no	—
128	Q₈	—	178	4	no	—
129	D₄	—	356	4	no	—
130	D₄	—	356	4	no	—
131	D₄	—	178	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	356	4	no	—
135	C₂×C₄	—	178	2	no	—
136	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
139	C₂×C₄	#13083575	1	4	yes	—
140	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
141	C₈	—	178	4	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	356	4	no	—
144	C₂²	—	356	2	no	—
145	C₂²	#15668001	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	4	no	—
150	C₄	—	356	2	no	—
151	C₄	#15668001	1	2	yes	—
152	C₄	C₂×#13083575	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	712	2	no	—
155	C₂	C₂×#15668001	1	1	yes	characteristic
156	C₂	#19152396	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152148

C₂

yes

maximal, characteristic

#19151968

C₂

yes

maximal, complemented

#19151964

C₂

yes

maximal, complemented

#19151960

C₂

yes

maximal, complemented

#19151592

C₂

yes

maximal, complemented

C₂×#15667923

C₂

yes

maximal, characteristic, complemented

#533×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

C₂²

yes

—

C₂²×#10623494

C₂²

yes

—

#15668012

C₂²

yes

characteristic

#15668009

C₂²

yes

characteristic

#15668000

C₂²

yes

—

#15667986

—

C₄×D₃₅₆

—

#15667935

C₂²

yes

—

#15667932

C₂²

yes

—

#15667923

C₂²

yes

—

#15667923

C₂²

yes

—

C₄⋊Q₈×C₈₉

C₂²

yes

—

#125×C₈₉

C₂²

yes

—

C₂²⋊Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

#117×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#13083574

—

C₂×Dic₁₇₈

C₂³

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

C₂³

yes

—

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

C₂×#10623494

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

#13083551

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#124

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#124

yes

—

C₄×C₈₉

C₂×#41

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×#124

yes

characteristic

C₂×C₈₉

#575

yes

—

D₈₉

—

#3135

—

maximal, Sylow

C₈₉

#3135

yes

characteristic, complemented, Sylow

#533

D₈₉

yes

characteristic, complemented

—

—

—

—

—

C₂×#97

—

C₂×#41

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₉

yes

—

#127

—

#125

C₂×D₈₉

yes

—

#124

—

C₂²⋊Q₈

—

#121

—

178

—

C₄×Q₈

—

C₄×D₄

—

178

—

#117

C₂×D₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

100

#103

—

101

#102

—

102

#97

—

103

#97

—

104

#41

—

178

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×D₄

—

108

C₂×D₄

—

178

—

109

C₂×D₄

—

178

—

110

C₂²×C₄

—

178

—

111

C₂²×C₄

C₂²⋊D₈₉

yes

—

112

C₂²×C₄

C₂²×D₈₉

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

178

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

178

—

118

C₄⋊C₄

C₂²×D₈₉

yes

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

178

—

121

C₄²

—

122

C₄²

C₂²⋊D₈₉

yes

—

123

C₄²

C₂²×D₈₉

yes

characteristic

124

C₄²

C₂²×D₈₉

yes

characteristic

125

C₂³

—

178

—

126

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

127

Q₈

—

128

Q₈

—

178

—

129

D₄

—

356

—

130

D₄

—

356

—

131

D₄

—

178

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

356

—

135

C₂×C₄

—

178

—

136

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

137

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

138

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

139

C₂×C₄

#13083575

yes

—

140

C₂×C₄

C₂³×D₈₉

yes

characteristic

141

C₈

—

178

—

142

C₂²

—

143

C₂²

—

356

—

144

C₂²

—

356

—

145

C₂²

#15668001

yes

—

146

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

356

—

151

C₄

#15668001

yes

—

152

C₄

C₂×#13083575

yes

—

153

C₂

—

154

C₂

—

712

—

155

C₂

C₂×#15668001

yes

characteristic

156

C₂

#19152396

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#533×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152148	C₂
hypercenter	#533	D₈₉
hypocenter	C₈₉	#3135
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#533×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

C₂

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #19151968	C₂	semidirect
3. #19151964	C₂	semidirect
4. #19151960	C₂	semidirect
5. #19151592	C₂	semidirect
6. C₂×#15667923	C₂	semidirect
7. #533×C₈₉	C₂	semidirect
79. C₈₉	#3135	semidirect
80. #533	D₈₉	semidirect

H ≅ G/N

type

2. #19151968

C₂

semidirect

3. #19151964

C₂

semidirect

4. #19151960

C₂

semidirect

5. #19151592

C₂

semidirect

6. C₂×#15667923

C₂

semidirect

7. #533×C₈₉

C₂

semidirect

79. C₈₉

#3135

semidirect

80. #533

D₈₉

semidirect

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

subgroup	index
#19152148	2
#19151968	2
#19151964	2
#19151960	2
#19151592	2
C₂×#15667923	2
#533×C₈₉	2
#3135	89

subgroup

index

#533×C₈₉

Subgroups — #26030600

Conjugacy classes of subgroups

Complements

Maximal subgroups