Group #22615146 — subgroups

Subgroups — #22615146

Order 9344.

468 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 173 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×#14556591	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic
3	#17988527	C₂	1	4	yes	maximal
4	#17988493	C₂	1	2	yes	maximal
5	#17988492	C₂	1	4	yes	maximal
6	#17988453	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₄×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₄₆	C₂²	1	2	yes	—
9	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
10	#14556591	—	2	4	no	—
11	C₄×Dic₁₄₆	—	2	4	no	—
12	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
15	C₂×#11924007	C₂²	1	2	yes	—
16	C₂×C₄×Dic₇₃	C₂²	1	2	yes	—
17	#14556498	C₂²	1	8	yes	—
18	#14556498	C₂²	1	8	yes	—
19	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
20	C₂×C₄²×C₇₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
22	#95×C₇₃	C₂²	1	2	yes	—
23	C₂×Dic₁₄₆	—	2	2	no	—
24	C₂×Dic₁₄₆	—	2	2	no	—
25	C₂×Dic₁₄₆	—	2	4	no	—
26	C₂×Dic₁₄₆	—	2	8	no	—
27	C₂×Dic₁₄₆	Q₈	1	4	yes	complemented
28	C₂²×Dic₇₃	C₂³	1	2	yes	—
29	C₂²×Dic₇₃	C₂³	1	4	yes	—
30	C₂²×Dic₇₃	C₂³	1	2	yes	—
31	C₄⋊Dic₇₃	—	2	2	no	—
32	C₄⋊Dic₇₃	—	2	4	no	—
33	C₄⋊Dic₇₃	—	2	4	no	—
34	#11924007	—	2	4	no	—
35	C₄×Dic₇₃	—	2	4	no	—
36	C₂²×C₄×C₇₃	C₂³	1	4	yes	—
37	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₇₃	—	2	4	no	—
41	C₄²×C₇₃	—	2	4	no	—
42	C₄²×C₇₃	D₄	1	4	yes	—
43	C₂³×C₇₃	C₂⁴	1	1	yes	characteristic
44	C₂×C₄×C₇₃	—	2	2	no	—
45	C₂×C₄×C₇₃	—	2	4	no	—
46	C₂×C₄×C₇₃	—	2	4	no	—
47	C₂×C₄×C₇₃	—	2	4	no	—
48	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₇₃	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₇₃	#41	1	2	yes	—
53	C₂×C₄×C₇₃	#41	1	4	yes	—
54	C₂×Dic₇₃	—	2	2	no	—
55	C₂×Dic₇₃	—	2	2	no	—
56	C₂×Dic₇₃	—	2	2	no	—
57	C₂×Dic₇₃	—	2	4	no	—
58	C₂×Dic₇₃	—	2	8	no	—
59	C₂×Dic₇₃	—	2	4	no	—
60	C₂×Dic₇₃	C₂×Q₈	1	4	yes	—
61	Dic₁₄₆	—	2	8	no	—
62	Dic₁₄₆	—	4	8	no	—
63	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
64	C₂²×C₇₃	C₂²×Q₈	1	1	yes	characteristic
65	C₂²×C₇₃	C₂×#41	1	1	yes	characteristic
66	C₂²×C₇₃	C₂×#41	1	2	yes	—
67	C₂²×C₇₃	2₋¹⁺⁴	1	2	yes	—
68	C₄×C₇₃	—	2	2	no	—
69	C₄×C₇₃	—	4	4	no	—
70	C₄×C₇₃	C₂²⋊Q₈	1	4	yes	—
71	C₄×C₇₃	#124	1	4	yes	—
72	Dic₇₃	—	2	4	no	—
73	Dic₇₃	—	4	4	no	—
74	Dic₇₃	—	4	2	no	—
75	C₂×C₇₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₇₃	C₂×#124	1	1	yes	characteristic
77	C₂×C₇₃	#536	1	1	yes	characteristic
78	C₂×C₇₃	#556	1	2	yes	—
79	C₂×C₇₃	#557	1	2	yes	—
80	#2568	—	73	1	no	maximal, Sylow
81	C₇₃	#2568	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₄⋊Q₈	—	73	1	no	—
83	C₂×C₄×Q₈	—	73	1	no	—
84	#400	—	73	4	no	—
85	#391	—	73	4	no	—
86	#391	—	73	2	no	—
87	#382	—	73	2	no	—
88	C₄×C₄⋊C₄	D₇₃	1	1	yes	characteristic
89	C₂²×Q₈	—	73	1	no	—
90	C₂²×Q₈	—	73	2	no	—
91	C₄⋊Q₈	—	146	4	no	—
92	C₄×Q₈	—	146	4	no	—
93	C₂×C₄⋊C₄	—	73	1	no	—
94	C₂×C₄⋊C₄	—	73	2	no	—
95	C₂×C₄⋊C₄	—	73	2	no	—
96	C₂×C₄⋊C₄	—	73	2	no	—
97	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
98	C₂×C₄²	—	73	2	no	—
99	C₂×C₄²	C₂×D₇₃	1	2	yes	—
100	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
101	#95	—	73	8	no	—
102	#95	—	73	8	no	—
103	#95	C₂×D₇₃	1	2	yes	—
104	C₂×Q₈	—	73	4	no	—
105	C₂×Q₈	—	146	4	no	—
106	C₂×Q₈	—	146	2	no	—
107	C₂×Q₈	—	146	2	no	—
108	C₂×Q₈	—	146	8	no	—
109	C₂²×C₄	—	73	2	no	—
110	C₂²×C₄	—	73	2	no	—
111	C₂²×C₄	—	73	4	no	—
112	C₂²×C₄	C₂²×D₇₃	1	4	yes	—
113	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	146	4	no	—
118	C₄⋊C₄	—	146	2	no	—
119	C₄⋊C₄	—	146	4	no	—
120	C₄⋊C₄	—	146	4	no	—
121	C₄²	—	2	4	no	—
122	C₄²	—	146	4	no	—
123	C₄²	D₂₉₂	1	4	yes	—
124	C₂³	C₂³×D₇₃	1	1	yes	characteristic
125	Q₈	—	292	8	no	—
126	Q₈	—	146	8	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	73	4	no	—
132	C₂×C₄	—	146	2	no	—
133	C₂×C₄	—	146	4	no	—
134	C₂×C₄	—	146	8	no	—
135	C₂×C₄	—	146	4	no	—
136	C₂×C₄	—	146	2	no	—
137	C₂×C₄	—	146	2	no	—
138	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
139	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
140	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
141	C₂×C₄	Q₈×D₇₃	1	2	yes	—
142	C₂×C₄	#11924027	1	2	yes	—
143	C₂×C₄	#11924028	1	4	yes	—
144	C₂²	C₂²×D₂₉₂	1	1	yes	characteristic
145	C₂²	C₂×Q₈×D₇₃	1	1	yes	characteristic
146	C₂²	C₂×#11924027	1	1	yes	characteristic
147	C₂²	C₂×#11924028	1	2	yes	—
148	C₂²	#14556638	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	4	4	no	—
151	C₄	—	292	2	no	—
152	C₄	—	292	4	no	—
153	C₄	—	146	4	no	—
154	C₄	#14556571	1	4	yes	—
155	C₄	#14556579	1	4	yes	—
156	C₂	C₂×#14556571	1	1	yes	characteristic
157	C₂	C₂×#14556579	1	1	yes	characteristic
158	C₂	#17989163	1	1	yes	characteristic
159	C₂	#17989307	1	2	yes	—
160	C₂	#17989315	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556591

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₄₆

C₂

yes

maximal, characteristic

#17988527

C₂

yes

maximal

#17988493

C₂

yes

maximal

#17988492

C₂

yes

maximal

#17988453

C₂

yes

maximal

C₄×C₄⋊C₄×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

—

C₂²×Dic₁₄₆

C₂²

yes

characteristic

#14556591

—

C₄×Dic₁₄₆

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×#11924007

C₂²

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

—

#14556498

C₂²

yes

—

#14556498

C₂²

yes

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

#95×C₇₃

C₂²

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

Q₈

yes

complemented

C₂²×Dic₇₃

C₂³

yes

—

C₂²×Dic₇₃

C₂³

yes

—

C₂²×Dic₇₃

C₂³

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

—

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

—

C₄²×C₇₃

—

C₄²×C₇₃

D₄

yes

—

C₂³×C₇₃

C₂⁴

yes

characteristic

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×Q₈

yes

—

C₂×C₄×C₇₃

#41

yes

—

C₂×C₄×C₇₃

#41

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×Q₈

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂²×C₇₃

C₂²×Q₈

yes

characteristic

C₂²×C₇₃

C₂×#41

yes

characteristic

C₂²×C₇₃

C₂×#41

yes

—

C₂²×C₇₃

2₋¹⁺⁴

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊Q₈

yes

—

C₄×C₇₃

#124

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

C₂×C₂²⋊Q₈

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#557

yes

—

#2568

—

maximal, Sylow

C₇₃

#2568

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

—

—

—

—

C₄×C₄⋊C₄

D₇₃

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

—

C₄⋊Q₈

—

146

—

C₄×Q₈

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₃

yes

—

100

C₂×C₄²

C₂×D₇₃

yes

characteristic

101

#95

—

102

#95

—

103

#95

C₂×D₇₃

yes

—

104

C₂×Q₈

—

105

C₂×Q₈

—

146

—

106

C₂×Q₈

—

146

—

107

C₂×Q₈

—

146

—

108

C₂×Q₈

—

146

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂²×D₇₃

yes

—

113

C₂²×C₄

C₂²×D₇₃

yes

characteristic

114

C₂²×C₄

C₂²×D₇₃

yes

characteristic

115

C₂²×C₄

C₂²×D₇₃

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

146

—

118

C₄⋊C₄

—

146

—

119

C₄⋊C₄

—

146

—

120

C₄⋊C₄

—

146

—

121

C₄²

—

122

C₄²

—

146

—

123

C₄²

D₂₉₂

yes

—

124

C₂³

C₂³×D₇₃

yes

characteristic

125

Q₈

—

292

—

126

Q₈

—

146

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

146

—

133

C₂×C₄

—

146

—

134

C₂×C₄

—

146

—

135

C₂×C₄

—

146

—

136

C₂×C₄

—

146

—

137

C₂×C₄

—

146

—

138

C₂×C₄

C₂×D₂₉₂

yes

—

139

C₂×C₄

C₂×D₂₉₂

yes

—

140

C₂×C₄

C₂×D₂₉₂

yes

—

141

C₂×C₄

Q₈×D₇₃

yes

—

142

C₂×C₄

#11924027

yes

—

143

C₂×C₄

#11924028

yes

—

144

C₂²

C₂²×D₂₉₂

yes

characteristic

145

C₂²

C₂×Q₈×D₇₃

yes

characteristic

146

C₂²

C₂×#11924027

yes

characteristic

147

C₂²

C₂×#11924028

yes

—

148

C₂²

#14556638

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

292

—

152

C₄

—

292

—

153

C₄

—

146

—

154

C₄

#14556571

yes

—

155

C₄

#14556579

yes

—

156

C₂

C₂×#14556571

yes

characteristic

157

C₂

C₂×#14556579

yes

characteristic

158

C₂

#17989163

yes

characteristic

159

C₂

#17989307

yes

—

160

C₂

#17989315

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₃
commutator, socle	C₂³×C₇₃	C₂⁴
Fitting	C₄×C₄⋊C₄×C₇₃	C₂
hypercenter	C₄×C₄⋊C₄	D₇₃
hypocenter	C₇₃	#2568
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₃

commutator, socle

C₂³×C₇₃

C₂⁴

Fitting

C₄×C₄⋊C₄×C₇₃

C₂

hypercenter

C₄×C₄⋊C₄

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
27. C₂×Dic₁₄₆	Q₈	semidirect
81. C₇₃	#2568	semidirect

H ≅ G/N

type

27. C₂×Dic₁₄₆

Q₈

semidirect

81. C₇₃

#2568

semidirect

Subgroups — #22615146

Conjugacy classes of subgroups

Complements