Group #22614920 — subgroups

Subgroups — #22614920

Order 9344.

540 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 189 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×#14556589	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic
3	#17988518	C₂	1	4	yes	maximal
4	#17988494	C₂	1	2	yes	maximal
5	#17988483	C₂	1	4	yes	maximal
6	C₄×C₄⋊Dic₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
9	#14556589	—	2	8	no	—
10	C₄×Dic₁₄₆	—	2	4	no	—
11	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
13	C₂×#11924007	C₂²	1	8	yes	—
14	C₂×#11924007	C₂²	1	2	yes	—
15	C₂×C₄×Dic₇₃	C₂²	1	2	yes	—
16	C₂×C₄×Dic₇₃	C₂²	1	2	yes	—
17	#14556498	C₂²	1	2	yes	—
18	#14556498	C₂²	1	8	yes	—
19	C₂²×Q₈×C₇₃	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₇₃	—	2	4	no	—
21	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
23	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₄₆	—	2	2	no	—
25	C₂×Dic₁₄₆	—	2	2	no	—
26	C₂×Dic₁₄₆	D₄	1	4	yes	—
27	C₂²×Dic₇₃	C₂³	1	2	yes	—
28	C₂²×Dic₇₃	C₂³	1	2	yes	—
29	C₂²×Dic₇₃	C₂³	1	4	yes	—
30	C₄⋊Dic₇₃	—	2	2	no	—
31	C₄⋊Dic₇₃	Q₈	1	8	yes	complemented
32	#11924007	—	2	4	no	—
33	#11924007	—	2	16	no	—
34	C₄×Dic₇₃	—	2	4	no	—
35	C₄×Dic₇₃	—	2	4	no	—
36	C₂×Q₈×C₇₃	—	2	4	no	—
37	C₂×Q₈×C₇₃	—	2	4	no	—
38	C₂×Q₈×C₇₃	—	2	4	no	—
39	C₂²×C₄×C₇₃	C₂³	1	4	yes	—
40	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₇₃	—	2	4	no	—
44	C₄⋊C₄×C₇₃	—	2	4	no	—
45	C₄²×C₇₃	D₄	1	4	yes	—
46	C₂³×C₇₃	C₂⁴	1	1	yes	characteristic
47	Q₈×C₇₃	—	4	8	no	—
48	C₂×C₄×C₇₃	—	2	2	no	—
49	C₂×C₄×C₇₃	—	2	4	no	—
50	C₂×C₄×C₇₃	—	2	4	no	—
51	C₂×C₄×C₇₃	—	2	4	no	—
52	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₇₃	C₂×Q₈	1	4	yes	—
57	C₂×C₄×C₇₃	#41	1	2	yes	—
58	C₂×Dic₇₃	—	2	4	no	—
59	C₂×Dic₇₃	—	2	2	no	—
60	C₂×Dic₇₃	—	2	2	no	—
61	C₂×Dic₇₃	—	2	4	no	—
62	C₂×Dic₇₃	—	2	2	no	—
63	C₂×Dic₇₃	—	2	4	no	—
64	C₂×Dic₇₃	C₂×D₄	1	4	yes	—
65	C₂×Dic₇₃	C₂×Q₈	1	8	yes	—
66	Dic₁₄₆	—	2	8	no	—
67	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
68	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
69	C₂²×C₇₃	C₂²×Q₈	1	2	yes	—
70	C₂²×C₇₃	C₂×#41	1	1	yes	characteristic
71	C₂²×C₇₃	2₋¹⁺⁴	1	2	yes	—
72	C₄×C₇₃	—	2	2	no	—
73	C₄×C₇₃	—	4	4	no	—
74	C₄×C₇₃	#121	1	4	yes	—
75	C₄×C₇₃	C₄⋊Q₈	1	4	yes	complemented
76	Dic₇₃	—	2	4	no	—
77	Dic₇₃	—	2	8	no	—
78	Dic₇₃	—	4	2	no	—
79	C₂×C₇₃	C₂×#121	1	1	yes	characteristic
80	C₂×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₇₃	#536	1	1	yes	characteristic
82	C₂×C₇₃	D₄×Q₈	1	2	yes	—
83	C₂×C₇₃	#557	1	2	yes	—
84	#2546	—	73	1	no	maximal, Sylow
85	C₇₃	#2546	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₄⋊Q₈	—	73	2	no	—
87	C₂×C₄⋊Q₈	D₇₃	1	1	yes	characteristic
88	C₂×C₄×Q₈	—	73	1	no	—
89	#398	—	73	4	no	—
90	#391	—	73	2	no	—
91	#389	—	73	4	no	—
92	C₄×C₄⋊C₄	—	73	1	no	—
93	C₂²×Q₈	—	73	1	no	—
94	C₂²×Q₈	C₂×D₇₃	1	2	yes	—
95	C₄⋊Q₈	—	2	4	no	—
96	C₄⋊Q₈	—	146	8	no	—
97	C₄×Q₈	—	146	4	no	—
98	C₂×C₄⋊C₄	—	73	2	no	—
99	C₂×C₄⋊C₄	—	73	2	no	—
100	C₂×C₄⋊C₄	—	73	1	no	—
101	C₂×C₄⋊C₄	—	73	8	no	—
102	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
103	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
104	C₂×C₄²	—	73	2	no	—
105	C₂×C₄²	—	73	2	no	—
106	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
107	#95	—	73	2	no	—
108	#95	—	73	8	no	—
109	C₂×Q₈	—	2	4	no	—
110	C₂×Q₈	—	2	4	no	—
111	C₂×Q₈	—	2	4	no	—
112	C₂×Q₈	—	73	4	no	—
113	C₂×Q₈	—	146	2	no	—
114	C₂×Q₈	—	146	2	no	—
115	C₂²×C₄	—	73	4	no	—
116	C₂²×C₄	—	73	2	no	—
117	C₂²×C₄	—	73	2	no	—
118	C₂²×C₄	C₂²×D₇₃	1	4	yes	—
119	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
120	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
121	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	2	4	no	—
124	C₄⋊C₄	—	73	8	no	—
125	C₄⋊C₄	—	146	16	no	—
126	C₄⋊C₄	—	146	4	no	—
127	C₄⋊C₄	—	146	2	no	—
128	C₄²	—	146	4	no	—
129	C₄²	—	146	4	no	—
130	C₄²	C₂²⋊D₇₃	1	4	yes	—
131	C₂³	C₂³×D₇₃	1	1	yes	characteristic
132	Q₈	—	4	8	no	—
133	Q₈	—	146	8	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	73	4	no	—
139	C₂×C₄	—	73	8	no	—
140	C₂×C₄	—	146	2	no	—
141	C₂×C₄	—	146	4	no	—
142	C₂×C₄	—	146	4	no	—
143	C₂×C₄	—	146	2	no	—
144	C₂×C₄	—	146	2	no	—
145	C₂×C₄	—	146	4	no	—
146	C₂×C₄	D₄×D₇₃	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
149	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
150	C₂×C₄	Q₈×D₇₃	1	4	yes	—
151	C₂×C₄	#11924027	1	2	yes	—
152	C₂²	C₂×D₄×D₇₃	1	1	yes	characteristic
153	C₂²	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
154	C₂²	C₂×Q₈×D₇₃	1	2	yes	—
155	C₂²	C₂×#11924027	1	1	yes	characteristic
156	C₂²	#14556639	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	4	4	no	—
159	C₄	—	292	2	no	—
160	C₄	—	146	8	no	—
161	C₄	—	146	4	no	—
162	C₄	#14556562	1	4	yes	—
163	C₄	#14556589	1	4	yes	complemented
164	C₂	C₂×#14556562	1	1	yes	characteristic
165	C₂	C₂×#14556589	1	1	yes	characteristic
166	C₂	#17989161	1	1	yes	characteristic
167	C₂	#17989252	1	2	yes	—
168	C₂	#17989312	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556589

C₂

yes

maximal

C₂×C₄×Dic₁₄₆

C₂

yes

maximal, characteristic

#17988518

C₂

yes

maximal

#17988494

C₂

yes

maximal

#17988483

C₂

yes

maximal

C₄×C₄⋊Dic₇₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

characteristic

#14556589

—

C₄×Dic₁₄₆

—

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×#11924007

C₂²

yes

—

C₂×#11924007

C₂²

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

—

#14556498

C₂²

yes

—

#14556498

C₂²

yes

—

C₂²×Q₈×C₇₃

C₂²

yes

—

C₄⋊Q₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

D₄

yes

—

C₂²×Dic₇₃

C₂³

yes

—

C₂²×Dic₇₃

C₂³

yes

—

C₂²×Dic₇₃

C₂³

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

Q₈

yes

complemented

#11924007

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

D₄

yes

—

C₂³×C₇₃

C₂⁴

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

#41

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×D₄

yes

—

C₂×Dic₇₃

C₂×Q₈

yes

—

Dic₁₄₆

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂²×C₇₃

C₂²×Q₈

yes

—

C₂²×C₇₃

C₂×#41

yes

characteristic

C₂²×C₇₃

2₋¹⁺⁴

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#121

yes

—

C₄×C₇₃

C₄⋊Q₈

yes

complemented

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

C₂×#121

yes

characteristic

C₂×C₇₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₇₃

#536

yes

characteristic

C₂×C₇₃

D₄×Q₈

yes

—

C₂×C₇₃

#557

yes

—

#2546

—

maximal, Sylow

C₇₃

#2546

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₇₃

yes

characteristic

C₂×C₄×Q₈

—

#398

—

#391

—

#389

—

C₄×C₄⋊C₄

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₇₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₄×Q₈

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

103

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

104

C₂×C₄²

—

105

C₂×C₄²

—

106

C₂×C₄²

C₂×D₇₃

yes

characteristic

107

#95

—

108

#95

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

146

—

114

C₂×Q₈

—

146

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

C₂²×D₇₃

yes

—

119

C₂²×C₄

C₂²×D₇₃

yes

characteristic

120

C₂²×C₄

C₂²×D₇₃

yes

characteristic

121

C₂²×C₄

C₂²×D₇₃

yes

characteristic

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

146

—

126

C₄⋊C₄

—

146

—

127

C₄⋊C₄

—

146

—

128

C₄²

—

146

—

129

C₄²

—

146

—

130

C₄²

C₂²⋊D₇₃

yes

—

131

C₂³

C₂³×D₇₃

yes

characteristic

132

Q₈

—

133

Q₈

—

146

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

146

—

141

C₂×C₄

—

146

—

142

C₂×C₄

—

146

—

143

C₂×C₄

—

146

—

144

C₂×C₄

—

146

—

145

C₂×C₄

—

146

—

146

C₂×C₄

D₄×D₇₃

yes

—

147

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

148

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

149

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

150

C₂×C₄

Q₈×D₇₃

yes

—

151

C₂×C₄

#11924027

yes

—

152

C₂²

C₂×D₄×D₇₃

yes

characteristic

153

C₂²

C₂²×C₂²⋊D₇₃

yes

characteristic

154

C₂²

C₂×Q₈×D₇₃

yes

—

155

C₂²

C₂×#11924027

yes

characteristic

156

C₂²

#14556639

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

292

—

160

C₄

—

146

—

161

C₄

—

146

—

162

C₄

#14556562

yes

—

163

C₄

#14556589

yes

complemented

164

C₂

C₂×#14556562

yes

characteristic

165

C₂

C₂×#14556589

yes

characteristic

166

C₂

#17989161

yes

characteristic

167

C₂

#17989252

yes

—

168

C₂

#17989312

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₃
commutator, socle	C₂³×C₇₃	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₇₃	C₂
hypercenter	C₂×C₄⋊Q₈	D₇₃
hypocenter	C₇₃	#2546
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₃

commutator, socle

C₂³×C₇₃

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₇₃

C₂

hypercenter

C₂×C₄⋊Q₈

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
31. C₄⋊Dic₇₃	Q₈	semidirect
75. C₄×C₇₃	C₄⋊Q₈	semidirect
85. C₇₃	#2546	semidirect
163. C₄	#14556589	semidirect

H ≅ G/N

type

31. C₄⋊Dic₇₃

Q₈

semidirect

75. C₄×C₇₃

C₄⋊Q₈

semidirect

85. C₇₃

#2546

semidirect

163. C₄

#14556589

semidirect

Subgroups — #22614920

Conjugacy classes of subgroups

Complements