Group #22620463 — subgroups

Subgroups — #22620463

Order 9344.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#17989215	C₂	1	2	yes	maximal
2	#17988995	C₂	1	4	yes	maximal, complemented
3	#17988987	C₂	1	2	yes	maximal, complemented
4	#17988984	C₂	1	2	yes	maximal, complemented
5	#17988630	C₂	1	2	yes	maximal
6	C₂×#14556523	C₂	1	1	yes	maximal, characteristic
7	#17988542	C₂	1	1	yes	maximal, characteristic
8	#427×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#11924012	C₂²	1	1	yes	characteristic
10	#14556591	—	2	2	no	—
11	#14556590	C₂²	1	4	yes	—
12	#14556583	—	2	2	no	—
13	#14556582	C₂²	1	4	yes	—
14	#14556568	—	2	4	no	—
15	#14556550	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
17	#14556524	C₂²	1	4	yes	—
18	#14556523	C₂²	1	2	yes	—
19	#14556523	C₂²	1	2	yes	—
20	#14556522	C₂²	1	4	yes	—
21	#14556501	C₂²	1	2	yes	—
22	C₈×Dic₇₃	C₂²	1	2	yes	—
23	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	#117×C₇₃	C₂²	1	2	yes	—
25	#107×C₇₃	C₂²	1	2	yes	—
26	#106×C₇₃	C₂²	1	1	yes	characteristic
27	#106×C₇₃	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₄₆	—	2	4	no	—
29	C₂²×Dic₇₃	—	2	2	no	—
30	#11924012	Q₈	1	4	yes	complemented
31	C₂×#10596831	C₂³	1	2	yes	—
32	C₄⋊Dic₇₃	—	2	2	no	—
33	C₄⋊Dic₇₃	—	2	4	no	—
34	C₄⋊Dic₇₃	D₄	1	2	yes	—
35	C₄⋊Dic₇₃	C₂³	1	4	yes	—
36	#11924007	—	2	4	no	—
37	#11924007	—	2	4	no	—
38	C₂²⋊Dic₇₃	D₄	1	2	yes	—
39	C₄×Dic₇₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₇₃	Q₈	1	4	yes	complemented
42	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
45	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
46	C₂²⋊C₄×C₇₃	—	2	2	no	—
47	C₄²×C₇₃	—	2	2	no	—
48	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₃	—	2	2	no	—
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₇₃	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₇₃	—	2	4	no	—
56	C₂×Dic₇₃	—	8	2	no	—
57	C₂×Dic₇₃	C₂×D₄	1	4	yes	—
58	Dic₁₄₆	—	8	4	no	—
59	C₈×C₇₃	C₂×Q₈	1	2	yes	—
60	C₈×C₇₃	C₂×Q₈	1	2	yes	—
61	#10596831	C₂×Q₈	1	4	yes	—
62	C₂²×C₇₃	—	2	2	no	—
63	C₂²×C₇₃	C₄⋊Q₈	1	2	yes	—
64	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₇₃	—	4	2	no	—
66	C₄×C₇₃	—	4	2	no	—
67	C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
68	C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
70	Dic₇₃	—	2	4	no	—
71	Dic₇₃	—	4	4	no	—
72	C₂×C₇₃	—	2	2	no	—
73	C₂×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₇₃	#577	1	2	yes	—
75	#3163	—	73	1	no	maximal, Sylow
76	C₇₃	#3163	1	1	yes	characteristic, complemented, Sylow
77	#544	—	73	2	no	—
78	#501	—	73	4	no	—
79	#499	—	73	2	no	—
80	#498	—	73	2	no	—
81	#428	—	73	2	no	—
82	#427	D₇₃	1	1	yes	characteristic
83	C₂×#106	—	73	1	no	—
84	#405	—	73	1	no	—
85	C₂×M₁₆	—	73	1	no	—
86	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
87	C₄⋊Q₈	—	73	4	no	—
88	C₄⋊Q₈	—	146	2	no	—
89	#125	—	73	4	no	—
90	#125	—	146	2	no	—
91	C₂²⋊Q₈	—	146	4	no	—
92	#117	—	73	1	no	—
93	#117	C₂×D₇₃	1	2	yes	—
94	C₂×C₄⋊C₄	—	73	2	no	—
95	#107	—	73	4	no	—
96	#107	C₂×D₇₃	1	2	yes	—
97	#106	—	73	2	no	—
98	#106	—	73	2	no	—
99	#106	—	73	4	no	—
100	#106	C₂×D₇₃	1	1	yes	characteristic
101	#106	C₂×D₇₃	1	1	yes	characteristic
102	#97	—	73	2	no	—
103	C₄×C₈	—	73	2	no	—
104	C₂×Q₈	—	146	4	no	—
105	C₂²×C₄	—	146	2	no	—
106	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
107	M₁₆	—	73	4	no	—
108	C₂×C₈	—	73	2	no	—
109	C₂×C₈	Dic₁₄₆	1	4	yes	complemented
110	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
111	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
112	C₄⋊C₄	—	73	2	no	—
113	C₄⋊C₄	—	73	4	no	—
114	C₄⋊C₄	—	146	4	no	—
115	C₄⋊C₄	—	146	4	no	—
116	C₄⋊C₄	—	146	4	no	—
117	C₄⋊C₄	—	146	2	no	—
118	C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
119	C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₂²⋊C₄	—	73	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	73	2	no	—
124	C₂³	D₄×D₇₃	1	1	yes	characteristic
125	Q₈	—	584	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	584	2	no	—
129	C₂×C₄	—	73	4	no	—
130	C₂×C₄	—	146	4	no	—
131	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
132	C₂×C₄	Q₈×D₇₃	1	2	yes	—
133	C₂×C₄	C₂×Dic₁₄₆	1	2	yes	—
134	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
135	C₈	—	73	4	no	—
136	C₈	C₂×Dic₁₄₆	1	2	yes	—
137	C₈	C₂×Dic₁₄₆	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	#14556590	1	2	yes	—
140	C₂²	C₂×D₄×D₇₃	1	1	yes	characteristic
141	C₄	—	4	2	no	—
142	C₄	—	4	2	no	—
143	C₄	—	292	4	no	—
144	C₄	—	146	4	no	—
145	C₄	#14556590	1	2	yes	—
146	C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
147	C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	C₂×#14556590	1	1	yes	characteristic
150	C₂	#17989394	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17989215

C₂

yes

maximal

#17988995

C₂

yes

maximal, complemented

#17988987

C₂

yes

maximal, complemented

#17988984

C₂

yes

maximal, complemented

#17988630

C₂

yes

maximal

C₂×#14556523

C₂

yes

maximal, characteristic

#17988542

C₂

yes

maximal, characteristic

#427×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924012

C₂²

yes

characteristic

#14556591

—

#14556590

C₂²

yes

—

#14556583

—

#14556582

C₂²

yes

—

#14556568

—

#14556550

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

—

#14556524

C₂²

yes

—

#14556523

C₂²

yes

—

#14556523

C₂²

yes

—

#14556522

C₂²

yes

—

#14556501

C₂²

yes

—

C₈×Dic₇₃

C₂²

yes

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

#117×C₇₃

C₂²

yes

—

#107×C₇₃

C₂²

yes

—

#106×C₇₃

C₂²

yes

characteristic

#106×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

#11924012

Q₈

yes

complemented

C₂×#10596831

C₂³

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

—

C₄⋊Dic₇₃

C₂³

yes

—

#11924007

—

#11924007

—

C₂²⋊Dic₇₃

D₄

yes

—

C₄×Dic₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

Q₈

yes

complemented

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

C₂³

yes

—

C₄⋊C₄×C₇₃

C₂³

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×D₄

yes

—

Dic₁₄₆

—

C₈×C₇₃

C₂×Q₈

yes

—

C₈×C₇₃

C₂×Q₈

yes

—

#10596831

C₂×Q₈

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

C₄⋊Q₈

yes

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄⋊Q₈

yes

—

C₄×C₇₃

C₂²×Q₈

yes

characteristic

C₄×C₇₃

C₂²×Q₈

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₇₃

#577

yes

—

#3163

—

maximal, Sylow

C₇₃

#3163

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₃

yes

characteristic

C₂×#106

—

#405

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₇₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

#125

—

#125

—

146

—

C₂²⋊Q₈

—

146

—

#117

—

#117

C₂×D₇₃

yes

—

C₂×C₄⋊C₄

—

#107

—

#107

C₂×D₇₃

yes

—

#106

—

#106

—

#106

—

100

#106

C₂×D₇₃

yes

characteristic

101

#106

C₂×D₇₃

yes

characteristic

102

#97

—

103

C₄×C₈

—

104

C₂×Q₈

—

146

—

105

C₂²×C₄

—

146

—

106

C₂²×C₄

C₂²×D₇₃

yes

characteristic

107

M₁₆

—

108

C₂×C₈

—

109

C₂×C₈

Dic₁₄₆

yes

complemented

110

C₂×C₈

C₂²×D₇₃

yes

characteristic

111

C₂×C₈

C₂²×D₇₃

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

146

—

115

C₄⋊C₄

—

146

—

116

C₄⋊C₄

—

146

—

117

C₄⋊C₄

—

146

—

118

C₄⋊C₄

C₂²×D₇₃

yes

—

119

C₄⋊C₄

C₂²×D₇₃

yes

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

—

123

C₄²

—

124

C₂³

D₄×D₇₃

yes

characteristic

125

Q₈

—

584

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

584

—

129

C₂×C₄

—

130

C₂×C₄

—

146

—

131

C₂×C₄

D₄×D₇₃

yes

characteristic

132

C₂×C₄

Q₈×D₇₃

yes

—

133

C₂×C₄

C₂×Dic₁₄₆

yes

—

134

C₂×C₄

C₂³×D₇₃

yes

characteristic

135

C₈

—

136

C₈

C₂×Dic₁₄₆

yes

—

137

C₈

C₂×Dic₁₄₆

yes

—

138

C₂²

—

139

C₂²

#14556590

yes

—

140

C₂²

C₂×D₄×D₇₃

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

292

—

144

C₄

—

146

—

145

C₄

#14556590

yes

—

146

C₄

C₂×Q₈×D₇₃

yes

characteristic

147

C₄

C₂²×Dic₁₄₆

yes

characteristic

148

C₂

—

149

C₂

C₂×#14556590

yes

characteristic

150

C₂

#17989394

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#427×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#427	D₇₃
hypocenter	C₇₃	#3163
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#427×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #17988995	C₂	semidirect
3. #17988987	C₂	semidirect
4. #17988984	C₂	semidirect
30. #11924012	Q₈	semidirect
41. C₂×C₈×C₇₃	Q₈	semidirect
76. C₇₃	#3163	semidirect
109. C₂×C₈	Dic₁₄₆	semidirect

H ≅ G/N

type

2. #17988995

C₂

semidirect

3. #17988987

C₂

semidirect

4. #17988984

C₂

semidirect

30. #11924012

Q₈

semidirect

41. C₂×C₈×C₇₃

Q₈

semidirect

76. C₇₃

#3163

semidirect

109. C₂×C₈

Dic₁₄₆

semidirect

Subgroups — #22620463

Conjugacy classes of subgroups

Complements