Group #26025460 — subgroups

Subgroups — #26025460

Order 11392.

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

normal subgroups only

This group has 500 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×#15668012	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
3	#19151519	C₂	1	4	yes	maximal
4	#19151504	C₂	1	2	yes	maximal
5	#19151497	C₂	1	4	yes	maximal
6	#19151468	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
10	#15668012	—	2	4	no	—
11	C₄×Dic₁₇₈	—	2	4	no	—
12	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₈₉	C₂²	1	8	yes	—
16	C₂×#13083553	C₂²	1	2	yes	—
17	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
18	#15667919	C₂²	1	8	yes	—
19	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
20	C₂×C₄²×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
22	#95×C₈₉	C₂²	1	2	yes	—
23	C₂×Dic₁₇₈	—	2	2	no	—
24	C₂×Dic₁₇₈	—	2	2	no	—
25	C₂×Dic₁₇₈	—	2	4	no	—
26	C₂×Dic₁₇₈	—	4	4	no	—
27	C₂×Dic₁₇₈	Q₈	1	4	yes	complemented
28	C₂²×Dic₈₉	C₂³	1	4	yes	—
29	C₂²×Dic₈₉	C₂³	1	2	yes	—
30	C₂²×Dic₈₉	C₂³	1	2	yes	—
31	C₄⋊Dic₈₉	—	2	2	no	—
32	C₄⋊Dic₈₉	—	2	4	no	—
33	C₄⋊Dic₈₉	—	2	4	no	—
34	C₄⋊Dic₈₉	—	4	8	no	—
35	#13083553	—	2	4	no	—
36	C₄×Dic₈₉	—	2	4	no	—
37	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
39	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₈₉	Q₈	1	8	yes	complemented
42	C₄²×C₈₉	—	2	4	no	—
43	C₄²×C₈₉	D₄	1	4	yes	—
44	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
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	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₈₉	C₂×Q₈	1	8	yes	—
53	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
54	C₂×C₄×C₈₉	#41	1	2	yes	—
55	C₂×Dic₈₉	—	2	2	no	—
56	C₂×Dic₈₉	—	2	2	no	—
57	C₂×Dic₈₉	—	2	2	no	—
58	C₂×Dic₈₉	—	2	4	no	—
59	C₂×Dic₈₉	—	2	4	no	—
60	C₂×Dic₈₉	—	2	8	no	—
61	C₂×Dic₈₉	C₂×Q₈	1	4	yes	—
62	Dic₁₇₈	—	2	8	no	—
63	Dic₁₇₈	—	4	8	no	—
64	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
65	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
66	C₂²×C₈₉	C₂²×Q₈	1	2	yes	—
67	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
68	C₂²×C₈₉	H(5,𝔽₂)	1	2	yes	—
69	C₄×C₈₉	—	2	2	no	—
70	C₄×C₈₉	—	2	8	no	—
71	C₄×C₈₉	C₂²⋊Q₈	1	4	yes	—
72	C₄×C₈₉	C₄⋊Q₈	1	4	yes	—
73	Dic₈₉	—	2	4	no	—
74	Dic₈₉	—	4	4	no	—
75	Dic₈₉	—	4	2	no	—
76	C₂×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₈₉	#535	1	1	yes	characteristic
79	C₂×C₈₉	#554	1	2	yes	—
80	C₂×C₈₉	Q₈²	1	2	yes	—
81	#2567	—	89	1	no	maximal, Sylow
82	C₈₉	#2567	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄⋊Q₈	—	89	1	no	—
84	C₂×C₄×Q₈	—	89	1	no	—
85	#395	—	89	4	no	—
86	#391	—	89	2	no	—
87	#389	—	89	4	no	—
88	#384	—	89	2	no	—
89	C₄×C₄⋊C₄	D₈₉	1	1	yes	characteristic
90	C₂²×Q₈	—	89	1	no	—
91	C₂²×Q₈	—	89	2	no	—
92	C₄⋊Q₈	—	178	4	no	—
93	C₄×Q₈	—	178	4	no	—
94	C₂×C₄⋊C₄	—	89	8	no	—
95	C₂×C₄⋊C₄	—	89	1	no	—
96	C₂×C₄⋊C₄	—	89	2	no	—
97	C₂×C₄⋊C₄	—	89	2	no	—
98	C₂×C₄⋊C₄	—	89	2	no	—
99	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
100	C₂×C₄²	—	89	2	no	—
101	C₂×C₄²	C₂×D₈₉	1	2	yes	—
102	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
103	#95	—	89	8	no	—
104	#95	C₂×D₈₉	1	2	yes	—
105	C₂×Q₈	—	89	4	no	—
106	C₂×Q₈	—	356	4	no	—
107	C₂×Q₈	—	178	4	no	—
108	C₂×Q₈	—	178	2	no	—
109	C₂×Q₈	—	178	2	no	—
110	C₂²×C₄	—	89	2	no	—
111	C₂²×C₄	—	89	4	no	—
112	C₂²×C₄	—	89	2	no	—
113	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
115	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
117	C₄⋊C₄	—	356	8	no	—
118	C₄⋊C₄	—	178	4	no	—
119	C₄⋊C₄	—	178	2	no	—
120	C₄⋊C₄	—	178	4	no	—
121	C₄⋊C₄	—	178	4	no	—
122	C₄⋊C₄	Dic₁₇₈	1	8	yes	complemented
123	C₄²	—	2	4	no	—
124	C₄²	—	178	4	no	—
125	C₄²	D₃₅₆	1	4	yes	—
126	C₂³	C₂³×D₈₉	1	1	yes	characteristic
127	Q₈	—	356	8	no	—
128	Q₈	—	178	8	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	89	4	no	—
133	C₂×C₄	—	178	2	no	—
134	C₂×C₄	—	178	4	no	—
135	C₂×C₄	—	178	4	no	—
136	C₂×C₄	—	178	8	no	—
137	C₂×C₄	—	178	2	no	—
138	C₂×C₄	—	178	2	no	—
139	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
140	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
141	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
142	C₂×C₄	Q₈×D₈₉	1	2	yes	—
143	C₂×C₄	C₂×Dic₁₇₈	1	8	yes	—
144	C₂×C₄	C₂×Dic₁₇₈	1	4	yes	—
145	C₂×C₄	#13083573	1	2	yes	—
146	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
147	C₂²	C₂×Q₈×D₈₉	1	1	yes	characteristic
148	C₂²	C₂²×Dic₁₇₈	1	2	yes	—
149	C₂²	C₂×#13083573	1	1	yes	characteristic
150	C₂²	#15668058	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	2	8	no	—
153	C₄	—	356	4	no	—
154	C₄	—	356	2	no	—
155	C₄	—	178	4	no	—
156	C₄	#15667992	1	4	yes	—
157	C₄	#15668012	1	4	yes	—
158	C₂	C₂×#15667992	1	1	yes	characteristic
159	C₂	C₂×#15668012	1	1	yes	characteristic
160	C₂	#19152167	1	1	yes	characteristic
161	C₂	#19152305	1	2	yes	—
162	C₂	Q₈×Dic₁₇₈	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668012

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₇₈

C₂

yes

maximal, characteristic

#19151519

C₂

yes

maximal

#19151504

C₂

yes

maximal

#19151497

C₂

yes

maximal

#19151468

C₂

yes

maximal

C₄×C₄⋊C₄×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

#15668012

—

C₄×Dic₁₇₈

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#95×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

Q₈

yes

complemented

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

Q₈

yes

complemented

C₄²×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂×C₄×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

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×Q₈

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂²×Q₈

yes

characteristic

C₂²×C₈₉

C₂²×Q₈

yes

—

C₂²×C₈₉

C₂×#41

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

C₄⋊Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₈₉

C₂×C₄⋊Q₈

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

Q₈²

yes

—

#2567

—

maximal, Sylow

C₈₉

#2567

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

—

—

—

—

C₄×C₄⋊C₄

D₈₉

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

—

C₄⋊Q₈

—

178

—

C₄×Q₈

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

100

C₂×C₄²

—

101

C₂×C₄²

C₂×D₈₉

yes

—

102

C₂×C₄²

C₂×D₈₉

yes

characteristic

103

#95

—

104

#95

C₂×D₈₉

yes

—

105

C₂×Q₈

—

106

C₂×Q₈

—

356

—

107

C₂×Q₈

—

178

—

108

C₂×Q₈

—

178

—

109

C₂×Q₈

—

178

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂²×D₈₉

yes

characteristic

114

C₂²×C₄

C₂²×D₈₉

yes

—

115

C₂²×C₄

C₂²×D₈₉

yes

characteristic

116

C₂²×C₄

C₂²×D₈₉

yes

characteristic

117

C₄⋊C₄

—

356

—

118

C₄⋊C₄

—

178

—

119

C₄⋊C₄

—

178

—

120

C₄⋊C₄

—

178

—

121

C₄⋊C₄

—

178

—

122

C₄⋊C₄

Dic₁₇₈

yes

complemented

123

C₄²

—

124

C₄²

—

178

—

125

C₄²

D₃₅₆

yes

—

126

C₂³

C₂³×D₈₉

yes

characteristic

127

Q₈

—

356

—

128

Q₈

—

178

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

178

—

134

C₂×C₄

—

178

—

135

C₂×C₄

—

178

—

136

C₂×C₄

—

178

—

137

C₂×C₄

—

178

—

138

C₂×C₄

—

178

—

139

C₂×C₄

C₂×D₃₅₆

yes

—

140

C₂×C₄

C₂×D₃₅₆

yes

—

141

C₂×C₄

C₂×D₃₅₆

yes

—

142

C₂×C₄

Q₈×D₈₉

yes

—

143

C₂×C₄

C₂×Dic₁₇₈

yes

—

144

C₂×C₄

C₂×Dic₁₇₈

yes

—

145

C₂×C₄

#13083573

yes

—

146

C₂²

C₂²×D₃₅₆

yes

characteristic

147

C₂²

C₂×Q₈×D₈₉

yes

characteristic

148

C₂²

C₂²×Dic₁₇₈

yes

—

149

C₂²

C₂×#13083573

yes

characteristic

150

C₂²

#15668058

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

356

—

154

C₄

—

356

—

155

C₄

—

178

—

156

C₄

#15667992

yes

—

157

C₄

#15668012

yes

—

158

C₂

C₂×#15667992

yes

characteristic

159

C₂

C₂×#15668012

yes

characteristic

160

C₂

#19152167

yes

characteristic

161

C₂

#19152305

yes

—

162

C₂

Q₈×Dic₁₇₈

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₄×C₄⋊C₄×C₈₉	C₂
hypercenter	C₄×C₄⋊C₄	D₈₉
hypocenter	C₈₉	#2567
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, prime radical, cosocle

C₂³×C₈₉

C₂⁴

Fitting

C₄×C₄⋊C₄×C₈₉

C₂

hypercenter

C₄×C₄⋊C₄

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
27. C₂×Dic₁₇₈	Q₈	semidirect
41. C₄⋊C₄×C₈₉	Q₈	semidirect
82. C₈₉	#2567	semidirect
122. C₄⋊C₄	Dic₁₇₈	semidirect

H ≅ G/N

type

27. C₂×Dic₁₇₈

Q₈

semidirect

41. C₄⋊C₄×C₈₉

Q₈

semidirect

82. C₈₉

#2567

semidirect

122. C₄⋊C₄

Dic₁₇₈

semidirect

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

subgroup	index
C₂×#15668012	2
C₂×C₄×Dic₁₇₈	2
#19151519	2
#19151504	2
#19151497	2
#19151468	2
C₄×C₄⋊C₄×C₈₉	2
#2567	89

subgroup

index

C₂×C₄×Dic₁₇₈

C₄×C₄⋊C₄×C₈₉

Subgroups — #26025460

Conjugacy classes of subgroups

Complements

Maximal subgroups