Group #22573499 — subgroups

Subgroups — C₂×#16863952 #22573499

Order 8576.

1168 conjugacy classes of subgroups; only the 541 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 541 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	#16863952	C₂	1	16	yes	maximal, direct product factor
2	C₂×#14524042	C₂	1	2	yes	maximal, complemented
3	C₂×#14524035	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14524021	C₂	1	4	yes	maximal
5	C₂×#14524008	C₂	1	4	yes	maximal, complemented
6	C₂×#14524001	C₂	1	2	yes	maximal
7	C₂×C₄²×D₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#125×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₁₃₄	C₂²	1	4	yes	—
10	C₂²×C₄×D₆₇	C₂²	1	2	yes	—
11	C₂²×C₄×D₆₇	C₂²	1	1	yes	characteristic
12	#14524042	C₂²	1	16	yes	complemented
13	#14524035	C₂²	1	8	yes	complemented
14	#14524021	C₂²	1	32	yes	—
15	#14524008	C₂²	1	32	yes	complemented
16	#14524001	C₂²	1	16	yes	—
17	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
18	C₂×Dic₆₇⋊C₄	C₂²	1	4	yes	—
19	C₂×Dic₆₇⋊C₄	C₂²	1	4	yes	—
20	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	4	yes	—
21	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
22	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₆₇	C₂²	1	4	yes	—
24	C₄²×D₆₇	C₂²	1	8	yes	—
25	#125×C₆₇	C₂²	1	8	yes	complemented
26	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
28	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
29	C₂³×D₆₇	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₃₄	—	2	16	no	—
31	C₂×Dic₁₃₄	C₂³	1	16	yes	—
32	C₂²×Dic₆₇	C₂³	1	4	yes	—
33	C₂²×Dic₆₇	C₂³	1	1	yes	characteristic
34	C₂²×Dic₆₇	C₂³	1	2	yes	—
35	C₂×C₄×D₆₇	—	2	8	no	—
36	C₂×C₄×D₆₇	Q₈	1	8	yes	complemented
37	C₂×C₄×D₆₇	C₂³	1	8	yes	—
38	C₂×C₄×D₆₇	C₂³	1	4	yes	—
39	C₄⋊Dic₆₇	C₂³	1	8	yes	—
40	Dic₆₇⋊C₄	C₂³	1	16	yes	—
41	Dic₆₇⋊C₄	C₂³	1	16	yes	—
42	(C₂×D₆₇)⋊C₄	C₂³	1	16	yes	—
43	C₄×Dic₆₇	C₂³	1	8	yes	—
44	C₄×Dic₆₇	C₂³	1	4	yes	—
45	C₄×Dic₆₇	C₂³	1	16	yes	—
46	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
48	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
50	C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
51	C₄²×C₆₇	C₂³	1	4	yes	—
52	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
53	C₂²×D₆₇	—	2	4	no	—
54	C₂²×D₆₇	C₂×Q₈	1	4	yes	—
55	C₂²×D₆₇	C₂⁴	1	2	yes	—
56	C₂×C₄×C₆₇	—	2	4	no	—
57	C₂×C₄×C₆₇	—	2	8	no	—
58	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
59	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
60	C₂×C₄×C₆₇	C₂⁴	1	8	yes	—
61	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
62	C₂×Dic₆₇	—	2	8	no	—
63	C₂×Dic₆₇	C₂×Q₈	1	4	yes	—
64	C₂×Dic₆₇	#41	1	8	yes	—
65	C₂×Dic₆₇	C₂⁴	1	8	yes	—
66	C₂×Dic₆₇	C₂⁴	1	2	yes	—
67	C₂×Dic₆₇	C₂⁴	1	4	yes	—
68	C₄×D₆₇	—	2	16	no	—
69	C₄×D₆₇	C₂×Q₈	1	16	yes	complemented
70	Dic₁₃₄	—	2	32	no	—
71	C₂²×C₆₇	C₂²×Q₈	1	2	yes	—
72	C₂²×C₆₇	C₂×#41	1	4	yes	—
73	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
74	C₂×D₆₇	—	2	8	no	—
75	C₂×D₆₇	—	2	4	no	—
76	C₂×D₆₇	C₂²×Q₈	1	4	yes	—
77	C₄×C₆₇	—	2	8	no	—
78	C₄×C₆₇	—	2	4	no	—
79	C₄×C₆₇	C₂²×Q₈	1	4	yes	—
80	Dic₆₇	—	2	8	no	—
81	Dic₆₇	C₂²×Q₈	1	4	yes	—
82	Dic₆₇	C₂×#41	1	8	yes	—
83	C₂×C₆₇	#533	1	4	yes	complemented
84	C₂×C₆₇	C₂³×Q₈	1	1	yes	characteristic
85	C₂×C₆₇	C₂²×#41	1	2	yes	—
86	D₆₇	—	2	4	no	—
87	C₂×#533	—	67	1	no	maximal, Sylow
88	C₆₇	C₂×#533	1	1	yes	characteristic, complemented, Sylow
89	#533	—	67	16	no	—
90	C₂×C₄⋊Q₈	—	67	2	no	—
91	C₂×#125	—	67	1	no	—
92	C₂×#125	D₆₇	1	1	yes	characteristic, complemented
93	C₂×C₂²⋊Q₈	—	67	4	no	—
94	C₂×C₄×Q₈	—	67	4	no	—
95	C₂×#117	—	67	2	no	—
96	C₂²×C₄²	—	67	1	no	—
97	C₂²×Q₈	—	67	4	no	—
98	C₂³×C₄	—	67	1	no	—
99	C₂³×C₄	—	67	2	no	—
100	C₄⋊Q₈	—	67	16	no	—
101	#125	—	67	8	no	—
102	#125	C₂×D₆₇	1	8	yes	complemented
103	C₂²⋊Q₈	—	67	32	no	—
104	C₄×Q₈	—	67	32	no	—
105	#117	—	67	16	no	—
106	C₂×C₄⋊C₄	—	67	2	no	—
107	C₂×C₄⋊C₄	—	67	4	no	—
108	C₂×C₄⋊C₄	—	67	4	no	—
109	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
110	C₂×C₄⋊C₄	C₂×D₆₇	1	4	yes	—
111	C₂×C₂²⋊C₄	—	67	4	no	—
112	C₂×C₄²	—	67	4	no	—
113	C₂×C₄²	—	67	2	no	—
114	C₂×C₄²	—	67	1	no	—
115	C₂×C₄²	—	67	8	no	—
116	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
117	C₂⁴	—	67	1	no	—
118	C₂×Q₈	—	67	16	no	—
119	C₂×Q₈	—	134	16	no	—
120	C₂²×C₄	—	67	1	no	—
121	C₂²×C₄	—	67	2	no	—
122	C₂²×C₄	—	67	8	no	—
123	C₂²×C₄	—	67	4	no	—
124	C₂²×C₄	—	67	4	no	—
125	C₂²×C₄	—	67	8	no	—
126	C₂²×C₄	—	134	8	no	—
127	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
128	C₂²×C₄	C₂²×D₆₇	1	4	yes	—
129	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
130	C₄⋊C₄	—	67	8	no	—
131	C₄⋊C₄	—	67	16	no	—
132	C₄⋊C₄	—	67	16	no	—
133	C₄⋊C₄	C₂²×D₆₇	1	8	yes	—
134	C₄⋊C₄	C₂²×D₆₇	1	16	yes	—
135	C₂²⋊C₄	—	67	16	no	—
136	C₄²	—	67	8	no	—
137	C₄²	—	67	4	no	—
138	C₄²	—	67	16	no	—
139	C₄²	C₂²×D₆₇	1	4	yes	—
140	C₂³	—	67	4	no	—
141	C₂³	—	67	2	no	—
142	C₂³	—	134	4	no	—
143	C₂³	C₂³×D₆₇	1	1	yes	characteristic
144	Q₈	—	134	32	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	8	no	—
147	C₂×C₄	—	67	4	no	—
148	C₂×C₄	—	67	8	no	—
149	C₂×C₄	—	67	2	no	—
150	C₂×C₄	—	67	4	no	—
151	C₂×C₄	—	67	16	no	—
152	C₂×C₄	—	67	8	no	—
153	C₂×C₄	—	134	16	no	—
154	C₂×C₄	—	134	8	no	—
155	C₂×C₄	Q₈×D₆₇	1	4	yes	—
156	C₂×C₄	C₂³×D₆₇	1	4	yes	—
157	C₂×C₄	C₂³×D₆₇	1	8	yes	—
158	C₂×C₄	C₂³×D₆₇	1	2	yes	—
159	C₂²	—	67	4	no	—
160	C₂²	—	134	8	no	—
161	C₂²	—	134	4	no	—
162	C₂²	C₂×Q₈×D₆₇	1	2	yes	—
163	C₂²	#41×D₆₇	1	4	yes	—
164	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
165	C₄	—	2	4	no	—
166	C₄	—	2	8	no	—
167	C₄	—	67	4	no	—
168	C₄	—	67	8	no	—
169	C₄	—	134	8	no	—
170	C₄	C₂×Q₈×D₆₇	1	4	yes	—
171	C₂	—	134	4	no	—
172	C₂	#16863952	1	4	yes	direct product factor
173	C₂	C₂²×Q₈×D₆₇	1	1	yes	characteristic
174	C₂	C₂×#41×D₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863952

C₂

yes

maximal, direct product factor

C₂×#14524042

C₂

yes

maximal, complemented

C₂×#14524035

C₂

yes

maximal, characteristic, complemented

C₂×#14524021

C₂

yes

maximal

C₂×#14524008

C₂

yes

maximal, complemented

C₂×#14524001

C₂

yes

maximal

C₂×C₄²×D₆₇

C₂

yes

maximal, characteristic

C₂×#125×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₃₄

C₂²

yes

—

C₂²×C₄×D₆₇

C₂²

yes

—

C₂²×C₄×D₆₇

C₂²

yes

characteristic

#14524042

C₂²

yes

complemented

#14524035

C₂²

yes

complemented

#14524021

C₂²

yes

—

#14524008

C₂²

yes

complemented

#14524001

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×(C₂×D₆₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

C₂×C₄×Dic₆₇

C₂²

yes

—

C₄²×D₆₇

C₂²

yes

—

#125×C₆₇

C₂²

yes

complemented

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂³×D₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

characteristic

C₂²×Dic₆₇

C₂³

yes

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

Q₈

yes

complemented

C₂×C₄×D₆₇

C₂³

yes

—

C₂×C₄×D₆₇

C₂³

yes

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

C₂³

yes

—

Dic₆₇⋊C₄

C₂³

yes

—

(C₂×D₆₇)⋊C₄

C₂³

yes

—

C₄×Dic₆₇

C₂³

yes

—

C₄×Dic₆₇

C₂³

yes

—

C₄×Dic₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

C₂×Q₈

yes

—

C₂²×D₆₇

C₂⁴

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×Q₈

yes

—

C₂×Dic₆₇

#41

yes

—

C₂×Dic₆₇

C₂⁴

yes

—

C₂×Dic₆₇

C₂⁴

yes

—

C₂×Dic₆₇

C₂⁴

yes

—

C₄×D₆₇

—

C₄×D₆₇

C₂×Q₈

yes

complemented

Dic₁₃₄

—

C₂²×C₆₇

C₂²×Q₈

yes

—

C₂²×C₆₇

C₂×#41

yes

—

C₂²×C₆₇

C₂⁵

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

C₂²×Q₈

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²×Q₈

yes

—

Dic₆₇

—

Dic₆₇

C₂²×Q₈

yes

—

Dic₆₇

C₂×#41

yes

—

C₂×C₆₇

#533

yes

complemented

C₂×C₆₇

C₂³×Q₈

yes

characteristic

C₂×C₆₇

C₂²×#41

yes

—

D₆₇

—

C₂×#533

—

maximal, Sylow

C₆₇

C₂×#533

yes

characteristic, complemented, Sylow

#533

—

C₂×C₄⋊Q₈

—

C₂×#125

—

C₂×#125

D₆₇

yes

characteristic, complemented

C₂×C₂²⋊Q₈

—

C₂×C₄×Q₈

—

C₂×#117

—

C₂²×C₄²

—

C₂²×Q₈

—

C₂³×C₄

—

C₂³×C₄

—

100

C₄⋊Q₈

—

101

#125

—

102

#125

C₂×D₆₇

yes

complemented

103

C₂²⋊Q₈

—

104

C₄×Q₈

—

105

#117

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

110

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

111

C₂×C₂²⋊C₄

—

112

C₂×C₄²

—

113

C₂×C₄²

—

114

C₂×C₄²

—

115

C₂×C₄²

—

116

C₂×C₄²

C₂×D₆₇

yes

characteristic

117

C₂⁴

—

118

C₂×Q₈

—

119

C₂×Q₈

—

134

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

134

—

127

C₂²×C₄

C₂²×D₆₇

yes

—

128

C₂²×C₄

C₂²×D₆₇

yes

—

129

C₂²×C₄

C₂²×D₆₇

yes

characteristic

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

C₂²×D₆₇

yes

—

134

C₄⋊C₄

C₂²×D₆₇

yes

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

C₂²×D₆₇

yes

—

140

C₂³

—

141

C₂³

—

142

C₂³

—

134

—

143

C₂³

C₂³×D₆₇

yes

characteristic

144

Q₈

—

134

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

134

—

154

C₂×C₄

—

134

—

155

C₂×C₄

Q₈×D₆₇

yes

—

156

C₂×C₄

C₂³×D₆₇

yes

—

157

C₂×C₄

C₂³×D₆₇

yes

—

158

C₂×C₄

C₂³×D₆₇

yes

—

159

C₂²

—

160

C₂²

—

134

—

161

C₂²

—

134

—

162

C₂²

C₂×Q₈×D₆₇

yes

—

163

C₂²

#41×D₆₇

yes

—

164

C₂²

C₂⁴×D₆₇

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

134

—

170

C₄

C₂×Q₈×D₆₇

yes

—

171

C₂

—

134

—

172

C₂

#16863952

yes

direct product factor

173

C₂

C₂²×Q₈×D₆₇

yes

characteristic

174

C₂

C₂×#41×D₆₇

yes

—

	subgroup	G/H
center	C₂³	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂⁵
Frattini	C₂²	C₂⁴×D₆₇
Fitting	C₂×#125×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁴
hypercenter	C₂×#125	D₆₇
hypocenter	C₆₇	C₂×#533
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂³×D₆₇

commutator

C₂²×C₆₇

C₂⁵

Frattini

C₂²

C₂⁴×D₆₇

Fitting

C₂×#125×C₆₇

C₂

socle

C₂³×C₆₇

C₂⁴

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
1. #16863952	C₂	direct (N × H)
2. C₂×#14524042	C₂	semidirect
3. C₂×#14524035	C₂	semidirect
5. C₂×#14524008	C₂	semidirect
8. C₂×#125×C₆₇	C₂	semidirect
12. #14524042	C₂²	semidirect
13. #14524035	C₂²	semidirect
15. #14524008	C₂²	semidirect
25. #125×C₆₇	C₂²	semidirect
36. C₂×C₄×D₆₇	Q₈	semidirect
69. C₄×D₆₇	C₂×Q₈	semidirect
83. C₂×C₆₇	#533	semidirect
88. C₆₇	C₂×#533	semidirect
92. C₂×#125	D₆₇	semidirect
102. #125	C₂×D₆₇	semidirect
172. C₂	#16863952	direct (N × H)

H ≅ G/N

type

1. #16863952

C₂

direct (N × H)

2. C₂×#14524042

C₂

semidirect

3. C₂×#14524035

C₂

semidirect

5. C₂×#14524008

C₂

semidirect

8. C₂×#125×C₆₇

C₂

semidirect

12. #14524042

C₂²

semidirect

13. #14524035

C₂²

semidirect

15. #14524008

C₂²

semidirect

25. #125×C₆₇

C₂²

semidirect

36. C₂×C₄×D₆₇

Q₈

semidirect

69. C₄×D₆₇

C₂×Q₈

semidirect

83. C₂×C₆₇

#533

semidirect

88. C₆₇

C₂×#533

semidirect

92. C₂×#125

D₆₇

semidirect

102. #125

C₂×D₆₇

semidirect

172. C₂

#16863952

direct (N × H)

Subgroups — C₂×#16863952 #22573499

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16863952 #22573499

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16863952 #22573499