Group #36481690 — subgroups

Subgroups — #36481690

Order 22784.

1550 conjugacy classes of subgroups; only the 651 normal subgroups are shown.

normal subgroups only

This group has 1550 conjugacy classes of subgroups — too many to draw. Only its 651 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×#19151468	C₂	1	3	yes	maximal
2	C₂×#19151464	C₂	1	3	yes	maximal
3	#26018517	C₂	1	8	yes	maximal
4	C₂×#376×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
6	C₂²×C₄⋊Dic₈₉	C₂²	1	3	yes	—
7	C₂²×C₄⋊Dic₈₉	C₂²	1	12	yes	—
8	#19151468	—	2	12	no	—
9	#19151464	—	2	12	no	—
10	C₂×#15667919	C₂²	1	12	yes	—
11	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
12	#376×C₈₉	C₄	1	8	yes	complemented
13	C₂×#95×C₈₉	C₂²	1	4	yes	—
14	C₂³×Dic₈₉	C₂³	1	2	yes	—
15	C₂³×Dic₈₉	C₂³	1	6	yes	—
16	C₂×C₄⋊Dic₈₉	—	2	6	no	—
17	C₂×C₄⋊Dic₈₉	—	2	6	no	—
18	C₂×C₄⋊Dic₈₉	—	2	24	no	—
19	C₂×C₄⋊Dic₈₉	—	2	12	no	—
20	C₂×C₄⋊Dic₈₉	—	4	24	no	—
21	#15667919	—	2	24	no	—
22	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
23	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₈₉	C₂×C₄	1	12	yes	—
25	C₂×C₄²×C₈₉	D₄	1	12	yes	—
26	C₂×C₄²×C₈₉	Q₈	1	12	yes	—
27	#95×C₈₉	C₂×C₄	1	16	yes	—
28	C₂²×Dic₈₉	—	2	6	no	—
29	C₂²×Dic₈₉	—	2	6	no	—
30	C₂²×Dic₈₉	—	2	12	no	—
31	C₂²×Dic₈₉	—	2	24	no	—
32	C₂²×Dic₈₉	—	2	8	no	—
33	C₄⋊Dic₈₉	—	4	48	no	—
34	C₄⋊Dic₈₉	—	4	4	no	—
35	C₄⋊Dic₈₉	—	4	12	no	—
36	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
37	C₂²×C₄×C₈₉	—	2	12	no	—
38	C₂²×C₄×C₈₉	—	2	12	no	—
39	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
40	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₈₉	C₂×D₄	1	12	yes	—
42	C₂²×C₄×C₈₉	C₂×D₄	1	6	yes	—
43	C₂²×C₄×C₈₉	C₂×Q₈	1	12	yes	—
44	C₂²×C₄×C₈₉	C₂×Q₈	1	6	yes	—
45	C₄²×C₈₉	C₄⋊C₄	1	48	yes	complemented
46	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
47	C₂³×C₈₉	C₂²×D₄	1	3	yes	—
48	C₂³×C₈₉	C₂²×Q₈	1	3	yes	—
49	C₂³×C₈₉	H(5,𝔽₂)	1	4	yes	—
50	C₂³×C₈₉	2₋¹⁺⁴	1	4	yes	—
51	C₂×C₄×C₈₉	—	2	24	no	—
52	C₂×C₄×C₈₉	—	2	12	no	—
53	C₂×C₄×C₈₉	—	2	12	no	—
54	C₂×C₄×C₈₉	—	2	24	no	—
55	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	24	yes	—
56	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	12	yes	—
57	C₂×C₄×C₈₉	#127	1	4	yes	—
58	C₂×C₄×C₈₉	C₄⋊Q₈	1	12	yes	—
59	C₂×Dic₈₉	—	4	6	no	—
60	C₂×Dic₈₉	—	4	2	no	—
61	C₂×Dic₈₉	—	4	24	no	—
62	C₂×Dic₈₉	—	4	12	no	—
63	C₂×Dic₈₉	—	4	12	no	—
64	C₂²×C₈₉	C₂²×C₄⋊C₄	1	3	yes	—
65	C₂²×C₈₉	#520	1	4	yes	—
66	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
67	C₂²×C₈₉	C₂×C₄⋊Q₈	1	3	yes	—
68	C₂²×C₈₉	#535	1	6	yes	—
69	C₂²×C₈₉	#536	1	6	yes	—
70	C₂²×C₈₉	#544	1	12	yes	—
71	C₄×C₈₉	—	2	24	no	—
72	C₄×C₈₉	#384	1	8	yes	—
73	Dic₈₉	—	8	6	no	—
74	Dic₈₉	—	8	2	no	—
75	#14354	—	89	1	no	maximal, Sylow
76	C₂×C₈₉	C₂×#384	1	1	yes	characteristic
77	C₂×C₈₉	#2317	1	6	yes	—
78	C₂×C₈₉	#2657	1	1	yes	characteristic
79	C₂×C₈₉	#2658	1	1	yes	characteristic
80	C₂×C₈₉	#2660	1	6	yes	—
81	C₂×#384	—	89	3	no	—
82	C₂×#382	—	89	3	no	—
83	C₂×#376	D₈₉	1	1	yes	characteristic
84	#1444	—	89	8	no	—
85	C₈₉	#14354	1	1	yes	characteristic, complemented, Sylow
86	C₂²×C₄⋊C₄	—	89	12	no	—
87	C₂²×C₄⋊C₄	—	89	1	no	—
88	C₂²×C₄⋊C₄	—	89	3	no	—
89	C₂²×C₄²	C₂×D₈₉	1	3	yes	—
90	#384	—	178	12	no	—
91	#382	—	178	12	no	—
92	#376	Dic₈₉	1	8	yes	complemented
93	C₂×#95	—	89	12	no	—
94	C₂×#95	C₂×D₈₉	1	4	yes	—
95	C₂³×C₄	—	89	6	no	—
96	C₂³×C₄	—	89	2	no	—
97	C₂³×C₄	C₂²×D₈₉	1	6	yes	—
98	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
99	C₂×C₄⋊C₄	—	356	24	no	—
100	C₂×C₄⋊C₄	—	178	6	no	—
101	C₂×C₄⋊C₄	—	178	6	no	—
102	C₂×C₄⋊C₄	—	178	24	no	—
103	C₂×C₄⋊C₄	—	178	12	no	—
104	C₂×C₄²	Dic₁₇₈	1	12	yes	—
105	C₂×C₄²	D₃₅₆	1	12	yes	—
106	C₂×C₄²	C₂×Dic₈₉	1	12	yes	—
107	#95	—	178	24	no	—
108	#95	C₂×Dic₈₉	1	16	yes	—
109	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
110	C₂²×C₄	—	2	12	no	—
111	C₂²×C₄	—	2	12	no	—
112	C₂²×C₄	—	178	6	no	—
113	C₂²×C₄	—	178	6	no	—
114	C₂²×C₄	—	178	12	no	—
115	C₂²×C₄	—	178	24	no	—
116	C₂²×C₄	—	178	8	no	—
117	C₂²×C₄	C₂²×Dic₈₉	1	12	yes	—
118	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
119	C₂²×C₄	C₂×D₃₅₆	1	12	yes	—
120	C₂²×C₄	C₂×D₃₅₆	1	6	yes	—
121	C₂²×C₄	C₂×Dic₁₇₈	1	12	yes	—
122	C₂²×C₄	C₂×Dic₁₇₈	1	6	yes	—
123	C₄⋊C₄	—	356	48	no	—
124	C₄⋊C₄	—	356	4	no	—
125	C₄⋊C₄	—	356	12	no	—
126	C₄²	C₄⋊Dic₈₉	1	48	yes	complemented
127	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
128	C₂³	C₂²×D₃₅₆	1	3	yes	—
129	C₂³	C₂²×Dic₁₇₈	1	3	yes	—
130	C₂³	#15668058	1	4	yes	—
131	C₂³	#15668059	1	4	yes	—
132	C₂×C₄	—	2	12	no	—
133	C₂×C₄	—	2	12	no	—
134	C₂×C₄	—	2	24	no	—
135	C₂×C₄	—	2	24	no	—
136	C₂×C₄	—	356	6	no	—
137	C₂×C₄	—	356	2	no	—
138	C₂×C₄	—	356	24	no	—
139	C₂×C₄	—	356	12	no	—
140	C₂×C₄	—	356	12	no	—
141	C₂×C₄	C₂×C₄⋊Dic₈₉	1	24	yes	—
142	C₂×C₄	C₂×C₄⋊Dic₈₉	1	12	yes	—
143	C₂×C₄	#15668009	1	4	yes	—
144	C₂×C₄	#15668012	1	12	yes	—
145	C₂²	C₂²×C₄⋊Dic₈₉	1	3	yes	—
146	C₂²	#19152081	1	4	yes	—
147	C₂²	C₂×#15668009	1	1	yes	characteristic
148	C₂²	C₂×#15668012	1	3	yes	—
149	C₂²	#19152167	1	6	yes	—
150	C₂²	#19152174	1	6	yes	—
151	C₂²	#19152226	1	12	yes	—
152	C₄	—	2	24	no	—
153	C₄	—	712	6	no	—
154	C₄	—	712	2	no	—
155	C₄	#19151468	1	8	yes	—
156	C₂	C₂×#19151468	1	1	yes	characteristic
157	C₂	#26022875	1	6	yes	—
158	C₂	#26026358	1	1	yes	characteristic
159	C₂	#26026362	1	1	yes	characteristic
160	C₂	#26026371	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151468

C₂

yes

maximal

C₂×#19151464

C₂

yes

maximal

#26018517

C₂

yes

maximal

C₂×#376×C₈₉

C₂

yes

maximal, characteristic

C₂²×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂²×C₄⋊Dic₈₉

C₂²

yes

—

C₂²×C₄⋊Dic₈₉

C₂²

yes

—

#19151468

—

#19151464

—

C₂×#15667919

C₂²

yes

—

C₂²×C₄²×C₈₉

C₂²

yes

—

#376×C₈₉

C₄

yes

complemented

C₂×#95×C₈₉

C₂²

yes

—

C₂³×Dic₈₉

C₂³

yes

—

C₂³×Dic₈₉

C₂³

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

#15667919

—

C₂³×C₄×C₈₉

C₂³

yes

—

C₂³×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

Q₈

yes

—

#95×C₈₉

C₂×C₄

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂⁴×C₈₉

C₂⁴

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂²×C₄×C₈₉

C₂×Q₈

yes

—

C₂²×C₄×C₈₉

C₂×Q₈

yes

—

C₄²×C₈₉

C₄⋊C₄

yes

complemented

C₂³×C₈₉

C₂³×C₄

yes

characteristic

C₂³×C₈₉

C₂²×D₄

yes

—

C₂³×C₈₉

C₂²×Q₈

yes

—

C₂³×C₈₉

H(5,𝔽₂)

yes

—

C₂³×C₈₉

2₋¹⁺⁴

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₉

#127

yes

—

C₂×C₄×C₈₉

C₄⋊Q₈

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂²×C₈₉

C₂²×C₄⋊C₄

yes

—

C₂²×C₈₉

#520

yes

—

C₂²×C₈₉

C₂×#127

yes

characteristic

C₂²×C₈₉

C₂×C₄⋊Q₈

yes

—

C₂²×C₈₉

#535

yes

—

C₂²×C₈₉

#536

yes

—

C₂²×C₈₉

#544

yes

—

C₄×C₈₉

—

C₄×C₈₉

#384

yes

—

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×#384

—

C₂×#382

—

C₂×#376

D₈₉

yes

characteristic

#1444

—

C₈₉

#14354

yes

characteristic, complemented, Sylow

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

—

C₂²×C₄²

C₂×D₈₉

yes

—

#384

—

178

—

#382

—

178

—

#376

Dic₈₉

yes

complemented

C₂×#95

—

C₂×#95

C₂×D₈₉

yes

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₈₉

yes

—

C₂³×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₄⋊C₄

—

356

—

100

C₂×C₄⋊C₄

—

178

—

101

C₂×C₄⋊C₄

—

178

—

102

C₂×C₄⋊C₄

—

178

—

103

C₂×C₄⋊C₄

—

178

—

104

C₂×C₄²

Dic₁₇₈

yes

—

105

C₂×C₄²

D₃₅₆

yes

—

106

C₂×C₄²

C₂×Dic₈₉

yes

—

107

#95

—

178

—

108

#95

C₂×Dic₈₉

yes

—

109

C₂⁴

C₂³×D₈₉

yes

characteristic

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

178

—

113

C₂²×C₄

—

178

—

114

C₂²×C₄

—

178

—

115

C₂²×C₄

—

178

—

116

C₂²×C₄

—

178

—

117

C₂²×C₄

C₂²×Dic₈₉

yes

—

118

C₂²×C₄

C₂²×Dic₈₉

yes

—

119

C₂²×C₄

C₂×D₃₅₆

yes

—

120

C₂²×C₄

C₂×D₃₅₆

yes

—

121

C₂²×C₄

C₂×Dic₁₇₈

yes

—

122

C₂²×C₄

C₂×Dic₁₇₈

yes

—

123

C₄⋊C₄

—

356

—

124

C₄⋊C₄

—

356

—

125

C₄⋊C₄

—

356

—

126

C₄²

C₄⋊Dic₈₉

yes

complemented

127

C₂³

C₂³×Dic₈₉

yes

characteristic

128

C₂³

C₂²×D₃₅₆

yes

—

129

C₂³

C₂²×Dic₁₇₈

yes

—

130

C₂³

#15668058

yes

—

131

C₂³

#15668059

yes

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

356

—

137

C₂×C₄

—

356

—

138

C₂×C₄

—

356

—

139

C₂×C₄

—

356

—

140

C₂×C₄

—

356

—

141

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

—

142

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

—

143

C₂×C₄

#15668009

yes

—

144

C₂×C₄

#15668012

yes

—

145

C₂²

C₂²×C₄⋊Dic₈₉

yes

—

146

C₂²

#19152081

yes

—

147

C₂²

C₂×#15668009

yes

characteristic

148

C₂²

C₂×#15668012

yes

—

149

C₂²

#19152167

yes

—

150

C₂²

#19152174

yes

—

151

C₂²

#19152226

yes

—

152

C₄

—

153

C₄

—

712

—

154

C₄

—

712

—

155

C₄

#19151468

yes

—

156

C₂

C₂×#19151468

yes

characteristic

157

C₂

#26022875

yes

—

158

C₂

#26026358

yes

characteristic

159

C₂

#26026362

yes

characteristic

160

C₂

#26026371

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₈₉
commutator	C₂³×C₈₉	C₂³×C₄
Fitting	C₂×#376×C₈₉	C₂
socle, prime radical, cosocle	C₂⁴×C₈₉	C₂⁴
hypercenter	C₂×#376	D₈₉
hypocenter	C₈₉	#14354
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₈₉

commutator

C₂³×C₈₉

C₂³×C₄

Fitting

C₂×#376×C₈₉

C₂

socle, prime radical, cosocle

C₂⁴×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂⁴ ◁ C₂×#376

N	H ≅ G/N	type
12. #376×C₈₉	C₄	semidirect
45. C₄²×C₈₉	C₄⋊C₄	semidirect
85. C₈₉	#14354	semidirect
92. #376	Dic₈₉	semidirect
126. C₄²	C₄⋊Dic₈₉	semidirect

H ≅ G/N

type

12. #376×C₈₉

C₄

semidirect

45. C₄²×C₈₉

C₄⋊C₄

semidirect

85. C₈₉

#14354

semidirect

92. #376

Dic₈₉

semidirect

126. C₄²

C₄⋊Dic₈₉

semidirect

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

subgroup	index
C₂×#19151468	2
C₂×#19151464	2
#26018517	2
C₂×#376×C₈₉	2
#14354	89

subgroup

index

C₂×#376×C₈₉

Subgroups — #36481690

Conjugacy classes of subgroups

Complements

Maximal subgroups