Group #36717057 — subgroups

Subgroups — C₂×#26023428 #36717057

Order 22784.

3798 conjugacy classes of subgroups; only the 923 normal subgroups are shown.

normal subgroups only

This group has 3798 conjugacy classes of subgroups — too many to draw. Only its 923 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	C₂²×#15668009	C₂	1	1	yes	maximal, characteristic
2	#26023428	C₂	1	16	yes	maximal, direct product factor
3	C₂×#19151476	C₂	1	12	yes	maximal, complemented
4	C₂×#19151466	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#383×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₃₅₆	C₂²	1	6	yes	—
7	C₂³×C₄×D₈₉	C₂²	1	4	yes	—
8	C₂×#15668009	C₄	1	16	yes	complemented
9	C₂×#15668009	C₂²	1	8	yes	—
10	C₂²×#13083551	C₂²	1	12	yes	—
11	#19151476	C₂²	1	96	yes	complemented
12	#19151466	C₂²	1	8	yes	complemented
13	C₂×#15667919	C₂²	1	6	yes	—
14	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
15	#383×C₈₉	C₂²	1	8	yes	complemented
16	C₂×#95×C₈₉	C₂²	1	6	yes	—
17	C₂⁴×D₈₉	C₂³	1	4	yes	—
18	C₂²×D₃₅₆	—	4	12	no	—
19	C₂²×D₃₅₆	—	4	12	no	—
20	C₂²×D₃₅₆	C₂×C₄	1	48	yes	complemented
21	C₂²×D₃₅₆	C₂³	1	24	yes	—
22	C₂³×Dic₈₉	C₂³	1	4	yes	—
23	C₂²×C₄×D₈₉	—	4	24	no	—
24	C₂²×C₄×D₈₉	C₂³	1	16	yes	—
25	#15668009	C₂×C₄	1	64	yes	complemented
26	C₂×#13083551	—	4	24	no	—
27	C₂×#13083551	C₂³	1	48	yes	—
28	#15667919	C₂³	1	24	yes	—
29	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
30	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
31	C₂×C₄²×C₈₉	C₂×C₄	1	8	yes	complemented
32	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
33	#95×C₈₉	C₂³	1	24	yes	—
34	C₂³×D₈₉	—	2	24	no	—
35	C₂³×D₈₉	—	2	24	no	—
36	C₂³×D₈₉	C₂²×C₄	1	16	yes	—
37	C₂³×D₈₉	C₂⁴	1	8	yes	—
38	C₂×D₃₅₆	—	4	96	no	—
39	C₂×D₃₅₆	—	4	24	no	—
40	C₂×D₃₅₆	—	4	24	no	—
41	C₂×D₃₅₆	C₂²×C₄	1	96	yes	complemented
42	C₂²×Dic₈₉	—	2	24	no	—
43	C₂²×Dic₈₉	C₂⁴	1	8	yes	—
44	C₂×C₄×D₈₉	—	4	48	no	—
45	C₂×C₄×D₈₉	—	4	64	no	—
46	#13083551	—	4	48	no	—
47	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	—	2	12	no	—
49	C₂²×C₄×C₈₉	—	2	24	no	—
50	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
51	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
52	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
53	C₄²×C₈₉	C₂²×C₄	1	16	yes	complemented
54	C₂³×C₈₉	C₂³×C₄	1	2	yes	—
55	C₂³×C₈₉	H(5,𝔽₂)	1	12	yes	—
56	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
57	C₂²×D₈₉	—	2	96	no	—
58	C₂²×D₈₉	—	2	24	no	—
59	C₂²×D₈₉	—	2	24	no	—
60	C₂²×D₈₉	—	4	16	no	—
61	C₂²×D₈₉	—	4	48	no	—
62	C₂²×D₈₉	C₂³×C₄	1	16	yes	—
63	C₂×C₄×C₈₉	—	2	24	no	—
64	C₂×C₄×C₈₉	—	2	12	no	—
65	C₂×C₄×C₈₉	—	2	24	no	—
66	C₂×C₄×C₈₉	—	4	16	no	—
67	C₂×C₄×C₈₉	C₂³×C₄	1	12	yes	—
68	C₂×Dic₈₉	—	2	24	no	—
69	C₂×Dic₈₉	—	4	16	no	—
70	D₃₅₆	—	4	96	no	—
71	C₄×D₈₉	—	4	64	no	—
72	C₂²×C₈₉	#518	1	12	yes	—
73	C₂²×C₈₉	#560	1	16	yes	—
74	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
75	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	6	yes	—
76	C₂×D₈₉	—	2	48	no	—
77	C₂×D₈₉	—	4	64	no	—
78	C₂×D₈₉	—	4	8	no	—
79	C₂×D₈₉	—	4	24	no	—
80	C₄×C₈₉	—	2	12	no	—
81	C₄×C₈₉	—	4	8	no	—
82	Dic₈₉	—	4	8	no	—
83	C₂×#2380	—	89	1	no	maximal, Sylow
84	C₂×C₈₉	#2380	1	8	yes	complemented
85	C₂×C₈₉	C₂×#518	1	3	yes	—
86	C₂×C₈₉	C₂×#560	1	4	yes	—
87	D₈₉	—	4	16	no	—
88	C₂²×#127	—	89	1	no	—
89	#2380	—	89	16	no	—
90	C₂×#386	—	89	12	no	—
91	C₂×#383	—	89	1	no	—
92	C₂×#383	D₈₉	1	1	yes	characteristic, complemented
93	C₈₉	C₂×#2380	1	1	yes	characteristic, complemented, Sylow
94	C₂³×D₄	—	89	6	no	—
95	C₂⁴×C₄	—	89	4	no	—
96	C₂×#127	—	89	16	no	—
97	C₂×#127	—	89	8	no	—
98	C₂²×C₂²⋊C₄	—	89	12	no	—
99	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
100	#386	—	89	96	no	—
101	#383	—	89	8	no	—
102	#383	C₂×D₈₉	1	8	yes	complemented
103	C₂×#95	—	89	6	no	—
104	C₂×#95	C₂×D₈₉	1	6	yes	—
105	C₂⁵	—	89	4	no	—
106	C₂²×D₄	—	89	48	no	—
107	C₂²×D₄	—	89	24	no	—
108	C₂²×D₄	—	356	12	no	—
109	C₂²×D₄	—	356	12	no	—
110	C₂³×C₄	—	89	4	no	—
111	C₂³×C₄	—	89	16	no	—
112	C₂³×C₄	—	356	24	no	—
113	C₂³×C₄	C₂²×D₈₉	1	4	yes	—
114	C₂³×C₄	C₂²×D₈₉	1	3	yes	—
115	#127	—	89	64	no	—
116	C₂×C₂²⋊C₄	—	89	48	no	—
117	C₂×C₂²⋊C₄	—	356	24	no	—
118	C₂×C₄²	C₄×D₈₉	1	8	yes	complemented
119	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
120	#95	—	89	24	no	—
121	#95	C₂²×D₈₉	1	24	yes	—
122	C₂⁴	—	89	16	no	—
123	C₂⁴	—	89	8	no	—
124	C₂⁴	—	178	24	no	—
125	C₂⁴	—	178	24	no	—
126	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
127	C₂×D₄	—	89	96	no	—
128	C₂×D₄	—	356	24	no	—
129	C₂×D₄	—	356	24	no	—
130	C₂×D₄	—	356	96	no	—
131	C₂²×C₄	—	2	12	no	—
132	C₂²×C₄	—	2	24	no	—
133	C₂²×C₄	—	89	8	no	—
134	C₂²×C₄	—	356	48	no	—
135	C₂²×C₄	—	356	64	no	—
136	C₂²×C₄	—	178	24	no	—
137	C₂²×C₄	C₂×C₄×D₈₉	1	12	yes	—
138	C₂²×C₄	C₂³×D₈₉	1	8	yes	—
139	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
140	C₂²⋊C₄	—	356	48	no	—
141	C₄²	C₂×C₄×D₈₉	1	16	yes	complemented
142	C₂³	—	89	16	no	—
143	C₂³	—	356	16	no	—
144	C₂³	—	356	48	no	—
145	C₂³	—	178	24	no	—
146	C₂³	—	178	24	no	—
147	C₂³	—	178	96	no	—
148	C₂³	C₂²×C₄×D₈₉	1	2	yes	—
149	C₂³	#15668058	1	12	yes	—
150	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
151	D₄	—	356	96	no	—
152	C₂×C₄	—	2	12	no	—
153	C₂×C₄	—	2	24	no	—
154	C₂×C₄	—	2	24	no	—
155	C₂×C₄	—	4	16	no	—
156	C₂×C₄	—	356	64	no	—
157	C₂×C₄	—	356	16	no	—
158	C₂×C₄	—	178	24	no	—
159	C₂×C₄	C₂²×C₄×D₈₉	1	12	yes	—
160	C₂²	—	356	64	no	—
161	C₂²	—	356	8	no	—
162	C₂²	—	356	24	no	—
163	C₂²	—	178	48	no	—
164	C₂²	#19152072	1	12	yes	—
165	C₂²	#19152339	1	16	yes	—
166	C₂²	C₂³×C₄×D₈₉	1	1	yes	characteristic
167	C₂²	C₂×#15668058	1	6	yes	—
168	C₄	—	2	12	no	—
169	C₄	—	4	8	no	—
170	C₄	—	356	8	no	—
171	C₂	—	356	16	no	—
172	C₂	#26023428	1	8	yes	direct product factor
173	C₂	C₂×#19152072	1	3	yes	—
174	C₂	C₂×#19152339	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#15668009

C₂

yes

maximal, characteristic

#26023428

C₂

yes

maximal, direct product factor

C₂×#19151476

C₂

yes

maximal, complemented

C₂×#19151466

C₂

yes

maximal, characteristic, complemented

C₂×#383×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₃₅₆

C₂²

yes

—

C₂³×C₄×D₈₉

C₂²

yes

—

C₂×#15668009

C₄

yes

complemented

C₂×#15668009

C₂²

yes

—

C₂²×#13083551

C₂²

yes

—

#19151476

C₂²

yes

complemented

#19151466

C₂²

yes

complemented

C₂×#15667919

C₂²

yes

—

C₂²×C₄²×C₈₉

C₂²

yes

characteristic

#383×C₈₉

C₂²

yes

complemented

C₂×#95×C₈₉

C₂²

yes

—

C₂⁴×D₈₉

C₂³

yes

—

C₂²×D₃₅₆

—

C₂²×D₃₅₆

—

C₂²×D₃₅₆

C₂×C₄

yes

complemented

C₂²×D₃₅₆

C₂³

yes

—

C₂³×Dic₈₉

C₂³

yes

—

C₂²×C₄×D₈₉

—

C₂²×C₄×D₈₉

C₂³

yes

—

#15668009

C₂×C₄

yes

complemented

C₂×#13083551

—

C₂×#13083551

C₂³

yes

—

#15667919

C₂³

yes

—

C₂³×C₄×C₈₉

C₂³

yes

—

C₂³×C₄×C₈₉

C₂³

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

complemented

C₂×C₄²×C₈₉

C₂³

yes

—

#95×C₈₉

C₂³

yes

—

C₂³×D₈₉

—

C₂³×D₈₉

—

C₂³×D₈₉

C₂²×C₄

yes

—

C₂³×D₈₉

C₂⁴

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

C₂²×C₄

yes

complemented

C₂²×Dic₈₉

—

C₂²×Dic₈₉

C₂⁴

yes

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

#13083551

—

C₂⁴×C₈₉

C₂⁴

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

—

C₂²×C₄×C₈₉

C₂⁴

yes

—

C₂²×C₄×C₈₉

C₂⁴

yes

—

C₄²×C₈₉

C₂²×C₄

yes

complemented

C₂³×C₈₉

C₂³×C₄

yes

—

C₂³×C₈₉

H(5,𝔽₂)

yes

—

C₂³×C₈₉

C₂⁵

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

C₂³×C₄

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂³×C₄

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

C₂²×C₈₉

#518

yes

—

C₂²×C₈₉

#560

yes

—

C₂²×C₈₉

C₂⁴×C₄

yes

characteristic

C₂²×C₈₉

C₂×H(5,𝔽₂)

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

C₂×#560

yes

—

D₈₉

—

C₂²×#127

—

—

—

—

D₈₉

yes

characteristic, complemented

C₈₉

C₂×#2380

yes

characteristic, complemented, Sylow

C₂³×D₄

—

C₂⁴×C₄

—

C₂×#127

—

C₂×#127

—

C₂²×C₂²⋊C₄

—

C₂²×C₄²

C₂×D₈₉

yes

characteristic

100

#386

—

101

#383

—

102

#383

C₂×D₈₉

yes

complemented

103

C₂×#95

—

104

C₂×#95

C₂×D₈₉

yes

—

105

C₂⁵

—

106

C₂²×D₄

—

107

C₂²×D₄

—

108

C₂²×D₄

—

356

—

109

C₂²×D₄

—

356

—

110

C₂³×C₄

—

111

C₂³×C₄

—

112

C₂³×C₄

—

356

—

113

C₂³×C₄

C₂²×D₈₉

yes

—

114

C₂³×C₄

C₂²×D₈₉

yes

—

115

#127

—

116

C₂×C₂²⋊C₄

—

117

C₂×C₂²⋊C₄

—

356

—

118

C₂×C₄²

C₄×D₈₉

yes

complemented

119

C₂×C₄²

C₂²×D₈₉

yes

—

120

#95

—

121

#95

C₂²×D₈₉

yes

—

122

C₂⁴

—

123

C₂⁴

—

124

C₂⁴

—

178

—

125

C₂⁴

—

178

—

126

C₂⁴

C₂³×D₈₉

yes

characteristic

127

C₂×D₄

—

128

C₂×D₄

—

356

—

129

C₂×D₄

—

356

—

130

C₂×D₄

—

356

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

356

—

135

C₂²×C₄

—

356

—

136

C₂²×C₄

—

178

—

137

C₂²×C₄

C₂×C₄×D₈₉

yes

—

138

C₂²×C₄

C₂³×D₈₉

yes

—

139

C₂²×C₄

C₂³×D₈₉

yes

—

140

C₂²⋊C₄

—

356

—

141

C₄²

C₂×C₄×D₈₉

yes

complemented

142

C₂³

—

143

C₂³

—

356

—

144

C₂³

—

356

—

145

C₂³

—

178

—

146

C₂³

—

178

—

147

C₂³

—

178

—

148

C₂³

C₂²×C₄×D₈₉

yes

—

149

C₂³

#15668058

yes

—

150

C₂³

C₂⁴×D₈₉

yes

characteristic

151

D₄

—

356

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

356

—

157

C₂×C₄

—

356

—

158

C₂×C₄

—

178

—

159

C₂×C₄

C₂²×C₄×D₈₉

yes

—

160

C₂²

—

356

—

161

C₂²

—

356

—

162

C₂²

—

356

—

163

C₂²

—

178

—

164

C₂²

#19152072

yes

—

165

C₂²

#19152339

yes

—

166

C₂²

C₂³×C₄×D₈₉

yes

characteristic

167

C₂²

C₂×#15668058

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

356

—

171

C₂

—

356

—

172

C₂

#26023428

yes

direct product factor

173

C₂

C₂×#19152072

yes

—

174

C₂

C₂×#19152339

yes

—

	subgroup	G/H
center	C₂⁴	C₂³×D₈₉
commutator	C₂²×C₈₉	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂×#383×C₈₉	C₂
socle	C₂⁴×C₈₉	C₂⁴
prime radical	C₂²×#15668009	C₂
cosocle	C₂³×C₈₉	C₂⁵
hypercenter	C₂×#383	D₈₉
hypocenter	C₈₉	C₂×#2380
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×#383

subgroup

G/H

center

C₂⁴

C₂³×D₈₉

commutator

C₂²×C₈₉

C₂⁴×C₄

Frattini

C₂³

C₂⁴×D₈₉

Fitting

C₂×#383×C₈₉

C₂

socle

C₂⁴×C₈₉

C₂⁴

prime radical

C₂²×#15668009

C₂

cosocle

C₂³×C₈₉

C₂⁵

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂⁴ ◁ C₂×#383

N	H ≅ G/N	type
2. #26023428	C₂	direct (N × H)
3. C₂×#19151476	C₂	semidirect
4. C₂×#19151466	C₂	semidirect
5. C₂×#383×C₈₉	C₂	semidirect
8. C₂×#15668009	C₄	semidirect
11. #19151476	C₂²	semidirect
12. #19151466	C₂²	semidirect
15. #383×C₈₉	C₂²	semidirect
20. C₂²×D₃₅₆	C₂×C₄	semidirect
25. #15668009	C₂×C₄	semidirect
31. C₂×C₄²×C₈₉	C₂×C₄	semidirect
41. C₂×D₃₅₆	C₂²×C₄	semidirect
53. C₄²×C₈₉	C₂²×C₄	semidirect
84. C₂×C₈₉	#2380	semidirect
92. C₂×#383	D₈₉	semidirect
93. C₈₉	C₂×#2380	semidirect
102. #383	C₂×D₈₉	semidirect
118. C₂×C₄²	C₄×D₈₉	semidirect
141. C₄²	C₂×C₄×D₈₉	semidirect
172. C₂	#26023428	direct (N × H)

H ≅ G/N

type

2. #26023428

C₂

direct (N × H)

3. C₂×#19151476

C₂

semidirect

4. C₂×#19151466

C₂

semidirect

5. C₂×#383×C₈₉

C₂

semidirect

8. C₂×#15668009

C₄

semidirect

11. #19151476

C₂²

semidirect

12. #19151466

C₂²

semidirect

15. #383×C₈₉

C₂²

semidirect

20. C₂²×D₃₅₆

C₂×C₄

semidirect

25. #15668009

C₂×C₄

semidirect

31. C₂×C₄²×C₈₉

C₂×C₄

semidirect

41. C₂×D₃₅₆

C₂²×C₄

semidirect

53. C₄²×C₈₉

C₂²×C₄

semidirect

84. C₂×C₈₉

#2380

semidirect

92. C₂×#383

D₈₉

semidirect

93. C₈₉

C₂×#2380

semidirect

102. #383

C₂×D₈₉

semidirect

118. C₂×C₄²

C₄×D₈₉

semidirect

141. C₄²

C₂×C₄×D₈₉

semidirect

172. C₂

#26023428

direct (N × H)

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

subgroup	index
C₂²×#15668009	2
#26023428	2
C₂×#19151476	2
C₂×#19151466	2
C₂×#383×C₈₉	2
C₂×#2380	89

subgroup

index

C₂×#383×C₈₉

Subgroups — C₂×#26023428 #36717057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#26023428 #36717057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26023428 #36717057