Group #22620590 — subgroups

Subgroups — #22620590

Order 9344.

338 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 125 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	#17989199	C₂	1	2	yes	maximal
2	#17988989	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988967	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988956	C₂	1	2	yes	maximal, complemented
5	#17988845	C₂	1	2	yes	maximal
6	#17988828	C₂	1	2	yes	maximal
7	#17988695	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₂₉₂	C₂	1	2	yes	maximal, complemented
9	#405×C₇₃	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₂₉₂	C₂²	1	2	yes	—
11	C₂×Dic₂₉₂	C₂²	1	2	yes	—
12	#14556591	C₂²	1	2	yes	—
13	#14556584	—	2	4	no	—
14	#14556583	C₂²	1	2	yes	—
15	#14556581	—	2	4	no	—
16	#14556570	C₂²	1	4	yes	—
17	C₄×Dic₁₄₆	C₂²	1	4	yes	—
18	#14556525	C₂²	1	2	yes	—
19	#14556523	C₂²	1	2	yes	—
20	#14556516	C₂²	1	4	yes	—
21	#14556516	C₂²	1	4	yes	—
22	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
23	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	#117×C₇₃	C₂²	1	1	yes	characteristic
25	#97×C₇₃	C₂²	1	1	yes	characteristic
26	#97×C₇₃	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
29	C₂×Dic₁₄₆	C₂³	1	4	yes	—
30	Dic₂₉₂	—	2	4	no	—
31	Dic₂₉₂	—	2	4	no	—
32	C₄⋊Dic₇₃	—	2	4	no	—
33	C₄⋊Dic₇₃	C₂³	1	4	yes	—
34	#11924007	—	2	4	no	—
35	#11924007	—	2	4	no	—
36	#11924007	—	2	4	no	—
37	C₂²⋊Dic₇₃	—	2	4	no	—
38	C₄×Dic₇₃	—	2	4	no	—
39	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
40	M₁₆×C₇₃	—	2	2	no	—
41	C₂×C₈×C₇₃	—	2	2	no	—
42	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₇₃	D₄	1	2	yes	—
47	C₂²⋊C₄×C₇₃	D₄	1	2	yes	—
48	C₄²×C₇₃	C₂³	1	1	yes	characteristic
49	C₄²×C₇₃	C₂³	1	1	yes	characteristic
50	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
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	1	yes	characteristic
56	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₇₃	—	2	4	no	—
58	C₂×Dic₇₃	—	2	4	no	—
59	Dic₁₄₆	—	2	4	no	—
60	Dic₁₄₆	—	2	4	no	—
61	C₈×C₇₃	—	2	1	no	—
62	C₈×C₇₃	—	2	1	no	—
63	C₈×C₇₃	#41	1	2	yes	—
64	C₈×C₇₃	#41	1	2	yes	—
65	C₂²×C₇₃	—	2	1	no	—
66	C₂²×C₇₃	—	2	2	no	—
67	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₇₃	—	2	1	no	—
69	C₄×C₇₃	—	2	2	no	—
70	C₄×C₇₃	—	2	2	no	—
71	C₄×C₇₃	C₂×#41	1	1	yes	characteristic
72	C₄×C₇₃	C₂×#41	1	1	yes	characteristic
73	Dic₇₃	—	4	4	no	—
74	Dic₇₃	—	4	4	no	—
75	C₂×C₇₃	—	4	1	no	—
76	C₂×C₇₃	#532	1	1	yes	characteristic
77	C₂×C₇₃	#578	1	2	yes	—
78	#3177	—	73	1	no	maximal, Sylow
79	C₇₃	#3177	1	1	yes	characteristic, complemented, Sylow
80	#541	—	73	2	no	—
81	#499	—	73	1	no	—
82	#494	—	73	1	no	—
83	#491	—	73	2	no	—
84	#470	—	73	2	no	—
85	#467	—	73	2	no	—
86	#441	—	73	2	no	—
87	C₄×Q₁₆	—	73	2	no	—
88	#405	D₇₃	1	1	yes	characteristic
89	C₂×Q₁₆	—	73	2	no	—
90	C₂×Q₁₆	—	73	2	no	—
91	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
92	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
93	C₄⋊Q₈	—	73	2	no	—
94	#126	—	146	4	no	—
95	#125	—	73	2	no	—
96	#125	—	146	4	no	—
97	C₂²⋊Q₈	—	73	4	no	—
98	C₄×Q₈	—	73	4	no	—
99	#117	C₂×D₇₃	1	1	yes	characteristic
100	#107	—	73	2	no	—
101	#106	—	73	2	no	—
102	#103	—	73	4	no	—
103	#103	—	73	4	no	—
104	#97	C₂×D₇₃	1	1	yes	characteristic
105	#97	C₂×D₇₃	1	1	yes	characteristic
106	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
107	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
108	C₂×Q₈	—	73	4	no	—
109	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
110	Q₁₆	—	146	4	no	—
111	Q₁₆	—	146	4	no	—
112	M₁₆	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
115	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
116	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
117	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
118	C₄⋊C₄	—	73	4	no	—
119	C₄⋊C₄	—	146	4	no	—
120	C₄⋊C₄	—	146	4	no	—
121	C₄⋊C₄	—	146	4	no	—
122	C₄⋊C₄	—	146	4	no	—
123	C₄⋊C₄	D₂₉₂	1	2	yes	—
124	C₂²⋊C₄	—	146	4	no	—
125	C₂²⋊C₄	D₂₉₂	1	2	yes	—
126	C₄²	—	146	4	no	—
127	C₄²	C₂²×D₇₃	1	1	yes	characteristic
128	C₄²	C₂²×D₇₃	1	1	yes	characteristic
129	C₂³	C₂×D₂₉₂	1	1	yes	characteristic
130	Q₈	—	146	4	no	—
131	Q₈	—	146	4	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	146	4	no	—
135	C₂×C₄	—	146	4	no	—
136	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
137	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
138	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
139	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	#11924028	1	2	yes	—
143	C₈	#11924028	1	2	yes	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	2	no	—
146	C₂²	C₂²×D₂₉₂	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	—	292	4	no	—
151	C₄	—	292	4	no	—
152	C₄	C₂×#11924028	1	1	yes	characteristic
153	C₄	C₂×#11924028	1	1	yes	characteristic
154	C₂	—	4	1	no	—
155	C₂	#17989137	1	1	yes	characteristic
156	C₂	#17989404	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17989199

C₂

yes

maximal

#17988989

C₂

yes

maximal, characteristic, complemented

#17988967

C₂

yes

maximal, characteristic, complemented

#17988956

C₂

yes

maximal, complemented

#17988845

C₂

yes

maximal

#17988828

C₂

yes

maximal

#17988695

C₂

yes

maximal, complemented

C₄×Dic₂₉₂

C₂

yes

maximal, complemented

#405×C₇₃

C₂

yes

maximal, characteristic

C₂×Dic₂₉₂

C₂²

yes

—

C₂×Dic₂₉₂

C₂²

yes

—

#14556591

C₂²

yes

—

#14556584

—

#14556583

C₂²

yes

—

#14556581

—

#14556570

C₂²

yes

—

C₄×Dic₁₄₆

C₂²

yes

—

#14556525

C₂²

yes

—

#14556523

C₂²

yes

—

#14556516

C₂²

yes

—

#14556516

C₂²

yes

—

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

#117×C₇₃

C₂²

yes

characteristic

#97×C₇₃

C₂²

yes

characteristic

#97×C₇₃

C₂²

yes

characteristic

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

C₂³

yes

—

Dic₂₉₂

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂³

yes

—

#11924007

—

#11924007

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

D₄

yes

—

C₂²⋊C₄×C₇₃

D₄

yes

—

C₄²×C₇₃

C₂³

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#41

yes

—

C₈×C₇₃

#41

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#41

yes

characteristic

C₄×C₇₃

C₂×#41

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#532

yes

characteristic

C₂×C₇₃

#578

yes

—

#3177

—

maximal, Sylow

C₇₃

#3177

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

—

C₄×Q₁₆

—

#405

D₇₃

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₇₃

yes

characteristic

C₂²×C₈

C₂×D₇₃

yes

characteristic

C₄⋊Q₈

—

#126

—

146

—

#125

—

#125

—

146

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₇₃

yes

characteristic

100

#107

—

101

#106

—

102

#103

—

103

#103

—

104

#97

C₂×D₇₃

yes

characteristic

105

#97

C₂×D₇₃

yes

characteristic

106

C₄×C₈

C₂×D₇₃

yes

characteristic

107

C₄×C₈

C₂×D₇₃

yes

characteristic

108

C₂×Q₈

—

109

C₂²×C₄

C₂²×D₇₃

yes

characteristic

110

Q₁₆

—

146

—

111

Q₁₆

—

146

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

C₂²×D₇₃

yes

characteristic

115

C₂×C₈

C₂²×D₇₃

yes

characteristic

116

C₂×C₈

C₂²×D₇₃

yes

characteristic

117

C₂×C₈

C₂²×D₇₃

yes

characteristic

118

C₄⋊C₄

—

119

C₄⋊C₄

—

146

—

120

C₄⋊C₄

—

146

—

121

C₄⋊C₄

—

146

—

122

C₄⋊C₄

—

146

—

123

C₄⋊C₄

D₂₉₂

yes

—

124

C₂²⋊C₄

—

146

—

125

C₂²⋊C₄

D₂₉₂

yes

—

126

C₄²

—

146

—

127

C₄²

C₂²×D₇₃

yes

characteristic

128

C₄²

C₂²×D₇₃

yes

characteristic

129

C₂³

C₂×D₂₉₂

yes

characteristic

130

Q₈

—

146

—

131

Q₈

—

146

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

146

—

135

C₂×C₄

—

146

—

136

C₂×C₄

C₂×D₂₉₂

yes

—

137

C₂×C₄

C₂×D₂₉₂

yes

—

138

C₂×C₄

C₂×D₂₉₂

yes

characteristic

139

C₂×C₄

C₂³×D₇₃

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

#11924028

yes

—

143

C₈

#11924028

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂²×D₂₉₂

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

292

—

151

C₄

—

292

—

152

C₄

C₂×#11924028

yes

characteristic

153

C₄

C₂×#11924028

yes

characteristic

154

C₂

—

155

C₂

#17989137

yes

characteristic

156

C₂

#17989404

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#405×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#405	D₇₃
hypocenter	C₇₃	#3177
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

subgroup

G/H

center

C₂²

C₂²×D₂₉₂

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#405×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405

N	H ≅ G/N	type
2. #17988989	C₂	semidirect
3. #17988967	C₂	semidirect
4. #17988956	C₂	semidirect
7. #17988695	C₂	semidirect
8. C₄×Dic₂₉₂	C₂	semidirect
79. C₇₃	#3177	semidirect

H ≅ G/N

type

2. #17988989

C₂

semidirect

3. #17988967

C₂

semidirect

4. #17988956

C₂

semidirect

7. #17988695

C₂

semidirect

8. C₄×Dic₂₉₂

C₂

semidirect

79. C₇₃

#3177

semidirect

Subgroups — #22620590

Conjugacy classes of subgroups

Complements