Group #22620256 — subgroups

Subgroups — #22620256

Order 9344.

400 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 141 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	#17989137	C₂	1	1	yes	maximal, characteristic
2	#17988944	C₂	1	4	yes	maximal, complemented
3	#17988939	C₂	1	2	yes	maximal, complemented
4	#17988935	C₂	1	2	yes	maximal, complemented
5	#17988581	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
7	#533×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924028	C₂²	1	2	yes	—
9	C₂²×#10596831	C₂²	1	2	yes	—
10	#14556591	C₂²	1	1	yes	characteristic
11	#14556588	C₂²	1	1	yes	characteristic
12	#14556579	C₂²	1	2	yes	—
13	#14556565	—	2	2	no	—
14	C₄×D₂₉₂	—	2	2	no	—
15	#14556514	C₂²	1	8	yes	—
16	#14556511	C₂²	1	8	yes	—
17	C₄×#10596831	C₂²	1	2	yes	—
18	C₄×#10596831	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
20	#125×C₇₃	C₂²	1	2	yes	—
21	C₂²⋊Q₈×C₇₃	—	2	2	no	—
22	C₄×Q₈×C₇₃	—	2	2	no	—
23	#117×C₇₃	C₂²	1	2	yes	—
24	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
25	#11924028	—	2	8	no	—
26	C₂×Dic₁₄₆	C₂³	1	2	yes	—
27	C₂×D₂₉₂	—	2	2	no	—
28	C₂×D₂₉₂	C₂³	1	2	yes	—
29	C₂×C₂²⋊D₇₃	—	2	2	no	—
30	C₂×C₄×D₇₃	—	2	2	no	—
31	C₂×#10596831	—	2	4	no	—
32	C₂×#10596831	C₂³	1	4	yes	—
33	C₄⋊Dic₇₃	—	2	2	no	—
34	#11924005	—	2	4	no	—
35	C₂×Q₈×C₇₃	—	2	2	no	—
36	C₂²×C₄×C₇₃	D₄	1	2	yes	—
37	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₇₃	—	2	4	no	—
39	C₄⋊C₄×C₇₃	—	2	2	no	—
40	C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
41	C₂²⋊C₄×C₇₃	—	2	2	no	—
42	C₄²×C₇₃	—	2	2	no	—
43	C₄²×C₇₃	D₄	1	2	yes	—
44	C₄²×C₇₃	C₂³	1	1	yes	characteristic
45	C₄²×C₇₃	C₂³	1	1	yes	characteristic
46	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×D₇₃	—	2	2	no	—
48	Q₈×C₇₃	—	4	4	no	—
49	C₂×C₄×C₇₃	—	2	4	no	—
50	C₂×C₄×C₇₃	—	2	4	no	—
51	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#41	1	4	yes	—
55	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
56	C₂²⋊D₇₃	—	4	4	no	—
57	C₂×Dic₇₃	—	2	2	no	—
58	D₂₉₂	—	2	4	no	—
59	D₂₉₂	—	4	4	no	—
60	C₄×D₇₃	—	4	4	no	—
61	Dic₁₄₆	—	2	4	no	—
62	#10596831	—	2	4	no	—
63	C₂²×C₇₃	—	2	2	no	—
64	C₂²×C₇₃	#124	1	2	yes	—
65	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₇₃	—	4	2	no	—
67	C₂×D₇₃	—	4	4	no	—
68	C₄×C₇₃	—	2	2	no	—
69	C₄×C₇₃	—	4	4	no	—
70	C₄×C₇₃	#124	1	2	yes	—
71	C₄×C₇₃	#135	1	4	yes	—
72	C₄×C₇₃	C₂×#41	1	2	yes	—
73	Dic₇₃	—	4	2	no	—
74	C₂×C₇₃	—	2	2	no	—
75	C₂×C₇₃	C₂×#124	1	1	yes	characteristic
76	C₂×C₇₃	C₂×#135	1	2	yes	—
77	D₇₃	—	8	2	no	—
78	#3131	—	73	1	no	maximal, Sylow
79	C₇₃	#3131	1	1	yes	characteristic, complemented, Sylow
80	#533	D₇₃	1	1	yes	characteristic, complemented
81	#532	—	73	1	no	—
82	#488	—	73	4	no	—
83	#487	—	73	2	no	—
84	#486	—	73	2	no	—
85	#416	—	73	4	no	—
86	C₂×C₄×C₈	—	73	1	no	—
87	C₂×#41	—	73	2	no	—
88	C₂²×C₈	—	73	2	no	—
89	C₄⋊Q₈	—	73	1	no	—
90	C₄⋊Q₈	C₂×D₇₃	1	2	yes	—
91	#127	—	73	1	no	—
92	#125	C₂×D₇₃	1	2	yes	—
93	#124	—	73	2	no	—
94	C₂²⋊Q₈	—	2	2	no	—
95	#121	—	146	2	no	—
96	C₄×Q₈	—	2	2	no	—
97	C₄×D₄	—	146	2	no	—
98	#117	C₂×D₇₃	1	2	yes	—
99	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
100	#103	—	73	8	no	—
101	#102	—	73	8	no	—
102	C₄×C₈	—	73	2	no	—
103	C₄×C₈	—	73	2	no	—
104	#41	—	146	8	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	73	2	no	—
107	C₂×D₄	—	73	2	no	—
108	C₂×D₄	—	146	2	no	—
109	C₂×D₄	—	146	2	no	—
110	C₂²×C₄	—	146	2	no	—
111	C₂²×C₄	C₂²⋊D₇₃	1	2	yes	—
112	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
113	C₂×C₈	—	73	4	no	—
114	C₂×C₈	—	146	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	146	2	no	—
118	C₄⋊C₄	C₂²×D₇₃	1	4	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	146	4	no	—
121	C₄²	—	2	2	no	—
122	C₄²	C₂²⋊D₇₃	1	2	yes	—
123	C₄²	C₂²×D₇₃	1	1	yes	characteristic
124	C₄²	C₂²×D₇₃	1	1	yes	characteristic
125	C₂³	—	146	2	no	—
126	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
127	Q₈	—	4	4	no	—
128	Q₈	—	146	4	no	—
129	D₄	—	292	4	no	—
130	D₄	—	292	4	no	—
131	D₄	—	146	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	292	4	no	—
135	C₂×C₄	—	146	2	no	—
136	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
139	C₂×C₄	#11924029	1	4	yes	—
140	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
141	C₈	—	146	4	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	292	2	no	—
144	C₂²	—	292	4	no	—
145	C₂²	#14556580	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	4	4	no	—
149	C₄	—	292	2	no	—
150	C₄	#14556580	1	2	yes	—
151	C₄	#14556611	1	4	yes	—
152	C₄	C₂×#11924029	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	584	2	no	—
155	C₂	C₂×#14556580	1	1	yes	characteristic
156	C₂	C₂×#14556611	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17989137

C₂

yes

maximal, characteristic

#17988944

C₂

yes

maximal, complemented

#17988939

C₂

yes

maximal, complemented

#17988935

C₂

yes

maximal, complemented

#17988581

C₂

yes

maximal, complemented

C₂×C₄×#10596831

C₂

yes

maximal, characteristic, complemented

#533×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924028

C₂²

yes

—

C₂²×#10596831

C₂²

yes

—

#14556591

C₂²

yes

characteristic

#14556588

C₂²

yes

characteristic

#14556579

C₂²

yes

—

#14556565

—

C₄×D₂₉₂

—

#14556514

C₂²

yes

—

#14556511

C₂²

yes

—

C₄×#10596831

C₂²

yes

—

C₄×#10596831

C₂²

yes

—

C₄⋊Q₈×C₇₃

C₂²

yes

—

#125×C₇₃

C₂²

yes

—

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

#117×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

#11924028

—

C₂×Dic₁₄₆

C₂³

yes

—

C₂×D₂₉₂

—

C₂×D₂₉₂

C₂³

yes

—

C₂×C₂²⋊D₇₃

—

C₂×C₄×D₇₃

—

C₂×#10596831

—

C₂×#10596831

C₂³

yes

—

C₄⋊Dic₇₃

—

#11924005

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂³

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

D₄

yes

—

C₄²×C₇₃

C₂³

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

Q₈×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

characteristic

C₂×C₄×C₇₃

#41

yes

—

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

Dic₁₄₆

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#124

yes

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#124

yes

—

C₄×C₇₃

#135

yes

—

C₄×C₇₃

C₂×#41

yes

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₂×#124

yes

characteristic

C₂×C₇₃

C₂×#135

yes

—

D₇₃

—

#3131

—

maximal, Sylow

C₇₃

#3131

yes

characteristic, complemented, Sylow

#533

D₇₃

yes

characteristic, complemented

—

—

—

—

—

C₂×C₄×C₈

—

C₂×#41

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₃

yes

—

#127

—

#125

C₂×D₇₃

yes

—

#124

—

C₂²⋊Q₈

—

#121

—

146

—

C₄×Q₈

—

C₄×D₄

—

146

—

#117

C₂×D₇₃

yes

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

100

#103

—

101

#102

—

102

C₄×C₈

—

103

C₄×C₈

—

104

#41

—

146

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×D₄

—

108

C₂×D₄

—

146

—

109

C₂×D₄

—

146

—

110

C₂²×C₄

—

146

—

111

C₂²×C₄

C₂²⋊D₇₃

yes

—

112

C₂²×C₄

C₂²×D₇₃

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

146

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

146

—

118

C₄⋊C₄

C₂²×D₇₃

yes

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

146

—

121

C₄²

—

122

C₄²

C₂²⋊D₇₃

yes

—

123

C₄²

C₂²×D₇₃

yes

characteristic

124

C₄²

C₂²×D₇₃

yes

characteristic

125

C₂³

—

146

—

126

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

127

Q₈

—

128

Q₈

—

146

—

129

D₄

—

292

—

130

D₄

—

292

—

131

D₄

—

146

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

292

—

135

C₂×C₄

—

146

—

136

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

137

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

138

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

139

C₂×C₄

#11924029

yes

—

140

C₂×C₄

C₂³×D₇₃

yes

characteristic

141

C₈

—

146

—

142

C₂²

—

143

C₂²

—

292

—

144

C₂²

—

292

—

145

C₂²

#14556580

yes

—

146

C₂²

C₂²×C₂²⋊D₇₃

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

292

—

150

C₄

#14556580

yes

—

151

C₄

#14556611

yes

—

152

C₄

C₂×#11924029

yes

—

153

C₂

—

154

C₂

—

584

—

155

C₂

C₂×#14556580

yes

characteristic

156

C₂

C₂×#14556611

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#533×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#533	D₇₃
hypocenter	C₇₃	#3131
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#533×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #17988944	C₂	semidirect
3. #17988939	C₂	semidirect
4. #17988935	C₂	semidirect
5. #17988581	C₂	semidirect
6. C₂×C₄×#10596831	C₂	semidirect
7. #533×C₇₃	C₂	semidirect
79. C₇₃	#3131	semidirect
80. #533	D₇₃	semidirect

H ≅ G/N

type

2. #17988944

C₂

semidirect

3. #17988939

C₂

semidirect

4. #17988935

C₂

semidirect

5. #17988581

C₂

semidirect

6. C₂×C₄×#10596831

C₂

semidirect

7. #533×C₇₃

C₂

semidirect

79. C₇₃

#3131

semidirect

80. #533

D₇₃

semidirect

Subgroups — #22620256

Conjugacy classes of subgroups

Complements