Group #22626237 — subgroups

Subgroups — C₂×#17989500 #22626237

Order 9344.

232 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 74 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×#14556664	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556660	C₂	1	1	yes	maximal, characteristic
3	#17989500	C₂	1	4	yes	maximal, direct product factor
4	C₂×Q₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#64408	—	2	1	no	—
6	#14556664	—	2	1	no	—
7	#14556664	C₂²	1	2	yes	—
8	C₂×#11924031	—	2	1	no	—
9	#14556660	—	2	1	no	—
10	#14556660	C₂²	1	2	yes	—
11	C₂×Q₈×D₇₃	—	2	1	no	—
12	C₂×Dic₂₉₂	C₄	1	1	yes	characteristic, complemented
13	C₂×#11924018	—	2	1	no	—
14	Q₁₆×D₇₃	—	2	1	no	—
15	Q₁₆×D₇₃	—	2	1	no	—
16	Q₁₆×D₇₃	C₄	1	2	yes	complemented
17	Q₁₆×D₇₃	C₂²	1	2	yes	—
18	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆×C₇₃	C₄	1	1	yes	characteristic, complemented
20	C₂²×#64408	—	4	1	no	—
21	C₄⋊#64408	—	2	2	no	—
22	C₄⋊#64408	—	4	1	no	—
23	#11924031	—	2	2	no	—
24	C₂×Dic₁₄₆	—	2	1	no	—
25	C₂×Dic₁₄₆	—	4	1	no	—
26	Q₈×D₇₃	—	2	2	no	—
27	Q₈×D₇₃	—	2	2	no	—
28	Q₈×D₇₃	—	4	1	no	—
29	Q₈×D₇₃	—	4	1	no	—
30	C₂×C₄×D₇₃	—	4	1	no	—
31	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
32	Dic₂₉₂	—	2	1	no	—
33	Dic₂₉₂	C₂×C₄	1	2	yes	complemented
34	#11924018	—	2	2	no	—
35	#11924018	—	2	2	no	—
36	C₂×#10596831	D₄	1	1	yes	characteristic
37	C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
38	C₈×D₇₃	D₄	1	1	yes	characteristic
39	C₈×D₇₃	D₄	1	1	yes	characteristic
40	C₈×D₇₃	C₂³	1	1	yes	characteristic
41	C₂×Q₈×C₇₃	—	2	1	no	—
42	Q₁₆×C₇₃	—	2	1	no	—
43	Q₁₆×C₇₃	C₂×C₄	1	2	yes	complemented
44	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₇₃	D₈	1	1	yes	characteristic
46	C₂×#64408	—	4	2	no	—
47	C₂×#64408	—	8	2	no	—
48	Q₈×C₇₃	—	2	2	no	—
49	Q₈×C₇₃	—	4	1	no	—
50	C₂×C₄×C₇₃	—	4	1	no	—
51	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Dic₇₃	—	4	1	no	—
53	C₂×Dic₇₃	SD₁₆	1	1	yes	characteristic
54	C₄×D₇₃	—	4	2	no	—
55	C₄×D₇₃	—	4	2	no	—
56	C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₄×D₇₃	D₈	1	1	yes	characteristic
58	C₄×D₇₃	SD₁₆	1	1	yes	characteristic
59	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
60	Dic₁₄₆	—	2	2	no	—
61	Dic₁₄₆	—	4	1	no	—
62	Dic₁₄₆	—	4	2	no	—
63	Dic₁₄₆	—	4	2	no	—
64	C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
66	#10596831	C₂²⋊C₄	1	1	yes	characteristic
67	#10596831	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₇₃	#102	1	1	yes	characteristic
69	C₂×D₇₃	#102	1	1	yes	characteristic
70	C₂×D₇₃	SD₃₂	1	2	yes	—
71	C₂×D₇₃	Q₃₂	1	2	yes	—
72	C₂×D₇₃	C₂×D₈	1	1	yes	characteristic
73	#64408	—	8	2	no	—
74	C₄×C₇₃	—	4	2	no	—
75	C₄×C₇₃	#102	1	1	yes	characteristic
76	C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	Dic₇₃	—	4	2	no	—
78	Dic₇₃	#102	1	1	yes	characteristic
79	Dic₇₃	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₇₃	#358	1	2	yes	complemented
81	C₂×C₇₃	C₂×#102	1	1	yes	characteristic
82	D₇₃	#358	1	2	yes	complemented
83	D₇₃	C₂×SD₃₂	1	1	yes	characteristic
84	D₇₃	C₂×Q₃₂	1	1	yes	characteristic
85	C₂×#358	—	73	1	no	maximal, Sylow
86	C₇₃	C₂×#358	1	1	yes	characteristic, complemented, Sylow
87	C₂²×Q₁₆	—	73	1	no	—
88	C₂²×C₁₆	—	73	1	no	—
89	C₂×#107	—	73	1	no	—
90	#358	—	73	4	no	—
91	C₂²×Q₈	—	146	1	no	—
92	C₂×Q₁₆	—	73	2	no	—
93	C₂×Q₁₆	—	73	2	no	—
94	C₂×Q₁₆	—	73	1	no	—
95	C₂×Q₁₆	—	146	1	no	—
96	C₂×Q₁₆	—	146	1	no	—
97	C₂×Q₁₆	—	146	1	no	—
98	C₂×Q₁₆	#64408	1	1	yes	characteristic, complemented
99	C₂²×C₈	—	73	1	no	—
100	C₂×C₄⋊C₄	—	146	1	no	—
101	C₂×C₁₆	—	73	2	no	—
102	C₂×C₁₆	—	146	1	no	—
103	C₂×C₁₆	—	146	1	no	—
104	#107	—	73	2	no	—
105	#107	—	146	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	—	292	1	no	—
108	C₂×Q₈	—	292	1	no	—
109	C₂×Q₈	—	292	1	no	—
110	C₂×Q₈	—	146	2	no	—
111	C₂×Q₈	—	146	2	no	—
112	C₂×Q₈	—	146	1	no	—
113	C₂²×C₄	—	292	1	no	—
114	C₂²×C₄	—	292	1	no	—
115	C₂²×C₄	—	73	1	no	—
116	Q₁₆	—	2	1	no	—
117	Q₁₆	—	73	2	no	—
118	Q₁₆	—	146	2	no	—
119	Q₁₆	—	146	2	no	—
120	Q₁₆	—	146	1	no	—
121	Q₁₆	C₂×#64408	1	2	yes	complemented
122	C₂×C₈	—	73	1	no	—
123	C₂×C₈	—	73	1	no	—
124	C₂×C₈	—	73	1	no	—
125	C₂×C₈	—	73	1	no	—
126	C₂×C₈	—	73	1	no	—
127	C₂×C₈	C₂×#64408	1	1	yes	characteristic
128	C₄⋊C₄	—	292	1	no	—
129	C₄⋊C₄	—	146	2	no	—
130	C₁₆	—	146	2	no	—
131	C₂³	—	73	1	no	—
132	Q₈	—	2	2	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	292	2	no	—
135	Q₈	—	292	2	no	—
136	Q₈	—	292	1	no	—
137	Q₈	—	146	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	292	2	no	—
140	C₂×C₄	—	292	2	no	—
141	C₂×C₄	—	292	1	no	—
142	C₂×C₄	—	292	2	no	—
143	C₂×C₄	—	584	2	no	—
144	C₂×C₄	—	73	1	no	—
145	C₂×C₄	—	73	1	no	—
146	C₂×C₄	—	73	1	no	—
147	C₂×C₄	—	73	1	no	—
148	C₂×C₄	—	73	1	no	—
149	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
150	C₈	—	73	1	no	—
151	C₈	—	73	1	no	—
152	C₈	C₂²⋊#64408	1	1	yes	characteristic
153	C₈	C₂²×#64408	1	1	yes	characteristic
154	C₂²	—	73	2	no	—
155	C₂²	—	73	2	no	—
156	C₂²	—	73	1	no	—
157	C₂²	—	73	1	no	—
158	C₂²	#14556653	1	1	yes	characteristic
159	C₄	—	4	2	no	—
160	C₄	—	292	2	no	—
161	C₄	—	584	2	no	—
162	C₄	—	73	1	no	—
163	C₄	—	73	1	no	—
164	C₄	#14556653	1	1	yes	characteristic
165	C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
166	C₂	—	73	1	no	—
167	C₂	—	73	2	no	—
168	C₂	—	73	1	no	—
169	C₂	#17989500	1	2	yes	direct product factor
170	C₂	C₂×#14556653	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556664

C₂

yes

maximal, characteristic

C₂×#14556660

C₂

yes

maximal, characteristic

#17989500

C₂

yes

maximal, direct product factor

C₂×Q₁₆×D₇₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#64408

—

#14556664

—

#14556664

C₂²

yes

—

C₂×#11924031

—

#14556660

—

#14556660

C₂²

yes

—

C₂×Q₈×D₇₃

—

C₂×Dic₂₉₂

C₄

yes

characteristic, complemented

C₂×#11924018

—

Q₁₆×D₇₃

—

Q₁₆×D₇₃

—

Q₁₆×D₇₃

C₄

yes

complemented

Q₁₆×D₇₃

C₂²

yes

—

C₂×C₈×D₇₃

C₂²

yes

characteristic

C₂×Q₁₆×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

—

C₄⋊#64408

—

C₄⋊#64408

—

#11924031

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

Q₈×D₇₃

—

Q₈×D₇₃

—

Q₈×D₇₃

—

Q₈×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

D₄

yes

characteristic

Dic₂₉₂

—

Dic₂₉₂

C₂×C₄

yes

—

—

D₄

yes

characteristic

C₈×D₇₃

C₂×C₄

yes

characteristic

C₈×D₇₃

D₄

yes

characteristic

C₈×D₇₃

D₄

yes

characteristic

C₈×D₇₃

C₂³

yes

characteristic

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

C₂×C₄

yes

complemented

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

D₈

yes

characteristic

C₂×#64408

—

C₂×#64408

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

SD₁₆

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂²⋊C₄

yes

characteristic

C₄×D₇₃

D₈

yes

characteristic

C₄×D₇₃

SD₁₆

yes

characteristic

C₄×D₇₃

C₂×D₄

yes

characteristic

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₈×C₇₃

C₂²×C₄

yes

characteristic

#10596831

C₂²⋊C₄

yes

characteristic

#10596831

C₂×D₄

yes

C₂²×C₇₃

yes

C₂×D₇₃

yes

C₂×D₇₃

SD₃₂

yes

—

C₂×D₇₃

Q₃₂

yes

—

C₂×D₇₃

C₂×D₈

yes

characteristic

#64408

—

C₄×C₇₃

—

C₄×C₇₃

#102

yes

characteristic

C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

Dic₇₃

—

Dic₇₃

#102

yes

characteristic

Dic₇₃

C₂×SD₁₆

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

yes

D₇₃

C₂×SD₃₂

yes

characteristic

D₇₃

C₂×Q₃₂

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂²×C₁₆

—

C₂×#107

—

#358

—

C₂²×Q₈

—

146

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

146

—

C₂×Q₁₆

—

146

—

C₂×Q₁₆

—

146

—

C₂×Q₁₆

#64408

yes

characteristic, complemented

C₂²×C₈

—

100

C₂×C₄⋊C₄

—

146

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

146

—

103

C₂×C₁₆

—

146

—

104

#107

—

105

#107

—

146

—

106

C₂×Q₈

—

107

C₂×Q₈

—

292

—

108

C₂×Q₈

—

292

—

109

C₂×Q₈

—

292

—

110

C₂×Q₈

—

146

—

111

C₂×Q₈

—

146

—

112

C₂×Q₈

—

146

—

113

C₂²×C₄

—

292

—

114

C₂²×C₄

—

292

—

115

C₂²×C₄

—

116

Q₁₆

—

117

Q₁₆

—

118

Q₁₆

—

146

—

119

Q₁₆

—

146

—

120

Q₁₆

—

146

—

121

Q₁₆

C₂×#64408

yes

complemented

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₂×#64408

yes

characteristic

128

C₄⋊C₄

—

292

—

129

C₄⋊C₄

—

146

—

130

C₁₆

—

146

—

131

C₂³

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

292

—

135

Q₈

—

292

—

136

Q₈

—

292

—

137

Q₈

—

146

—

138

C₂×C₄

—

139

C₂×C₄

—

292

—

140

C₂×C₄

—

292

—

141

C₂×C₄

—

292

—

142

C₂×C₄

—

292

—

143

C₂×C₄

—

584

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

C₂²⋊#64408

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

C₂²⋊#64408

yes

characteristic

153

C₈

C₂²×#64408

yes

characteristic

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

#14556653

yes

characteristic

159

C₄

—

160

C₄

—

292

—

161

C₄

—

584

—

162

C₄

—

163

C₄

—

164

C₄

#14556653

yes

characteristic

165

C₄

C₂×C₂²⋊#64408

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

#17989500

yes

direct product factor

170

C₂

C₂×#14556653

yes

characteristic

	subgroup	G/H
center	C₂²	#14556653
commutator	C₈×C₇₃	C₂²×C₄
Frattini	C₈	C₂²×#64408
Fitting	C₂×Q₁₆×C₇₃	C₄
socle	C₂²×C₇₃	#102
hypercenter	C₂×Q₁₆	#64408
hypocenter	C₇₃	C₂×#358
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

#14556653

commutator

C₈×C₇₃

C₂²×C₄

Frattini

C₈

C₂²×#64408

Fitting

C₂×Q₁₆×C₇₃

C₄

C₂²×C₇₃

C₂×Q₁₆

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

N	H ≅ G/N	type
3. #17989500	C₂	direct (N × H)
12. C₂×Dic₂₉₂	C₄	semidirect
16. Q₁₆×D₇₃	C₄	semidirect
19. C₂×Q₁₆×C₇₃	C₄	semidirect
33. Dic₂₉₂	C₂×C₄	semidirect
43. Q₁₆×C₇₃	C₂×C₄	semidirect
80. C₂×C₇₃	#358	semidirect
82. D₇₃	#358	semidirect
86. C₇₃	C₂×#358	semidirect
98. C₂×Q₁₆	#64408	semidirect
121. Q₁₆	C₂×#64408	semidirect
169. C₂	#17989500	direct (N × H)

H ≅ G/N

type

3. #17989500

C₂

direct (N × H)

12. C₂×Dic₂₉₂

C₄

semidirect

16. Q₁₆×D₇₃

C₄

semidirect

19. C₂×Q₁₆×C₇₃

C₄

semidirect

33. Dic₂₉₂

C₂×C₄

semidirect

43. Q₁₆×C₇₃

C₂×C₄

semidirect

80. C₂×C₇₃

#358

semidirect

82. D₇₃

#358

semidirect

86. C₇₃

C₂×#358

semidirect

98. C₂×Q₁₆

#64408

semidirect

121. Q₁₆

C₂×#64408

semidirect

169. C₂

#17989500

direct (N × H)

Subgroups — C₂×#17989500 #22626237

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17989500 #22626237

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17989500 #22626237