Group #22609976 — subgroups

Subgroups — #22609976

Order 9344.

260 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 81 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×#14556599	C₂	1	1	yes	maximal, characteristic
2	#17988623	C₂	1	2	yes	maximal, complemented
3	#17988560	C₂	1	2	yes	maximal
4	#17988344	C₂	1	1	yes	maximal, characteristic, complemented
5	H(3,ℤ/4)×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11924028	C₂²	1	1	yes	characteristic
7	#14556599	—	2	2	no	—
8	#14556599	—	2	2	no	—
9	C₂×#11924011	C₄	1	2	yes	complemented
10	C₂²×#10596831	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₇₃	C₄	1	2	yes	complemented
12	#14556550	C₂²	1	2	yes	—
13	#14556524	—	2	2	no	—
14	#14556522	—	2	2	no	—
15	#14556505	—	2	4	no	—
16	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	#117×C₇₃	C₂²	1	2	yes	—
18	#11924028	—	2	2	no	—
19	#11924028	—	2	2	no	—
20	C₂×Dic₁₄₆	Q₈	1	1	yes	characteristic
21	C₂×D₂₉₂	Q₈	1	1	yes	characteristic
22	C₂×C₂²⋊D₇₃	D₄	1	2	yes	—
23	C₂×C₄×D₇₃	C₂×C₄	1	2	yes	—
24	#11924011	—	2	4	no	—
25	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₂×#10596831	D₄	1	2	yes	—
28	C₂×#10596831	D₄	1	2	yes	—
29	C₈×D₇₃	—	2	2	no	—
30	C₈×D₇₃	—	2	2	no	—
31	C₄⋊Dic₇₃	—	2	2	no	—
32	C₂²⋊Dic₇₃	—	2	2	no	—
33	C₄×Dic₇₃	—	2	2	no	—
34	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
35	M₁₆×C₇₃	—	2	2	no	—
36	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₇₃	—	2	2	no	—
38	C₂²⋊C₄×C₇₃	—	2	2	no	—
39	C₄²×C₇₃	—	2	2	no	—
40	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₇₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₇₃	—	4	2	no	—
43	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₇₃	#41	1	2	yes	—
47	C₂²⋊D₇₃	—	2	4	no	—
48	C₂×Dic₇₃	—	4	2	no	—
49	C₂×Dic₇₃	C₄⋊C₄	1	2	yes	—
50	D₂₉₂	—	2	2	no	—
51	C₄×D₇₃	—	2	2	no	—
52	C₄×D₇₃	—	2	2	no	—
53	Dic₁₄₆	—	2	2	no	—
54	C₈×C₇₃	—	2	2	no	—
55	#10596831	C₂²⋊C₄	1	1	yes	characteristic
56	#10596831	C₂²⋊C₄	1	1	yes	characteristic
57	#10596831	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₇₃	—	4	1	no	—
59	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₇₃	#121	1	2	yes	—
61	C₂×D₇₃	—	2	2	no	—
62	C₂×D₇₃	—	4	2	no	—
63	C₄×C₇₃	—	8	2	no	—
64	C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₇₃	#117	1	1	yes	characteristic
66	C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
67	Dic₇₃	—	2	2	no	—
68	Dic₇₃	—	8	2	no	—
69	C₂×C₇₃	—	2	2	no	—
70	C₂×C₇₃	—	2	1	no	—
71	C₂×C₇₃	#380	1	1	yes	characteristic
72	D₇₃	—	4	2	no	—
73	#1812	—	73	1	no	maximal, Sylow
74	C₇₃	#1812	1	1	yes	characteristic, complemented, Sylow
75	C₂×#131	—	73	1	no	—
76	#427	—	73	2	no	—
77	#410	—	73	2	no	—
78	H(3,ℤ/4)	—	73	1	no	—
79	H(3,ℤ/4)	D₇₃	1	1	yes	characteristic, complemented
80	C₂×#41	—	73	1	no	—
81	#131	—	146	2	no	—
82	#131	—	146	2	no	—
83	C₂×M₁₆	—	73	2	no	—
84	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
85	C₂²×C₈	—	73	1	no	—
86	C₂²×C₈	—	73	2	no	—
87	#117	—	73	2	no	—
88	#117	C₂×D₇₃	1	2	yes	—
89	#107	—	146	2	no	—
90	#106	—	146	2	no	—
91	#99	—	146	4	no	—
92	#41	—	146	2	no	—
93	#41	—	146	2	no	—
94	C₂×Q₈	—	73	1	no	—
95	C₂×D₄	—	73	2	no	—
96	C₂×D₄	—	73	1	no	—
97	C₂²×C₄	—	73	2	no	—
98	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	—	146	4	no	—
101	C₂×C₈	—	73	1	no	—
102	C₂×C₈	—	73	1	no	—
103	C₂×C₈	—	73	2	no	—
104	C₂×C₈	—	73	2	no	—
105	C₂×C₈	—	146	2	no	—
106	C₂×C₈	—	146	2	no	—
107	C₂×C₈	C₄×D₇₃	1	2	yes	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	146	2	no	—
110	C₂²⋊C₄	—	2	2	no	—
111	C₂²⋊C₄	—	146	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	—	146	2	no	—
114	C₂³	—	73	2	no	—
115	C₂³	D₄×D₇₃	1	1	yes	characteristic
116	Q₈	—	146	2	no	—
117	D₄	—	146	4	no	—
118	D₄	—	146	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	292	2	no	—
121	C₂×C₄	—	73	2	no	—
122	C₂×C₄	—	146	2	no	—
123	C₂×C₄	—	146	2	no	—
124	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
125	C₂×C₄	D₄×D₇₃	1	2	yes	—
126	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
127	C₂×C₄	#11924028	1	2	yes	—
128	C₈	—	2	2	no	—
129	C₈	—	73	1	no	—
130	C₈	—	73	1	no	—
131	C₈	—	73	2	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	292	2	no	—
134	C₂²	—	146	2	no	—
135	C₂²	C₄⋊C₄×D₇₃	1	1	yes	characteristic
136	C₂²	#14556564	1	2	yes	—
137	C₄	—	8	2	no	—
138	C₄	—	584	2	no	—
139	C₄	—	146	2	no	—
140	C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
141	C₄	#14556548	1	1	yes	characteristic
142	C₄	#14556569	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	—	2	1	no	—
145	C₂	—	292	2	no	—
146	C₂	#17988446	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556599

C₂

yes

maximal, characteristic

#17988623

C₂

yes

maximal, complemented

#17988560

C₂

yes

maximal

#17988344

C₂

yes

maximal, characteristic, complemented

H(3,ℤ/4)×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924028

C₂²

yes

—

—

C₄

yes

complemented

C₂²×#10596831

C₂²

yes

characteristic

C₂×C₈×D₇₃

C₄

yes

complemented

#14556550

C₂²

yes

—

#14556524

—

#14556522

—

#14556505

—

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

#117×C₇₃

C₂²

yes

—

#11924028

—

#11924028

—

C₂×Dic₁₄₆

Q₈

yes

characteristic

C₂×D₂₉₂

Q₈

yes

characteristic

C₂×C₂²⋊D₇₃

D₄

yes

—

C₂×C₄×D₇₃

C₂×C₄

yes

—

#11924011

—

C₂×#10596831

C₂×C₄

yes

characteristic

C₂×#10596831

C₂×C₄

yes

characteristic

C₂×#10596831

D₄

yes

—

C₂×#10596831

D₄

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₃

#41

yes

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₄⋊C₄

yes

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

C₂²⋊C₄

yes

characteristic

#10596831

C₂²⋊C₄

yes

characteristic

#10596831

C₂²⋊C₄

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₃

#121

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

#117

yes

characteristic

C₄×C₇₃

C₂²⋊Q₈

yes

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#380

yes

characteristic

D₇₃

—

#1812

—

maximal, Sylow

C₇₃

#1812

yes

characteristic, complemented, Sylow

—

—

—

—

D₇₃

yes

characteristic, complemented

C₂×#41

—

#131

—

146

—

#131

—

146

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₇₃

yes

—

#107

—

146

—

#106

—

146

—

#99

—

146

—

#41

—

146

—

#41

—

146

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇₃

yes

characteristic

M₁₆

—

100

M₁₆

—

146

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

146

—

106

C₂×C₈

—

146

—

107

C₂×C₈

C₄×D₇₃

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

146

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

146

—

112

C₄²

—

113

C₄²

—

146

—

114

C₂³

—

115

C₂³

D₄×D₇₃

yes

characteristic

116

Q₈

—

146

—

117

D₄

—

146

—

118

D₄

—

146

—

119

C₂×C₄

—

120

C₂×C₄

—

292

—

121

C₂×C₄

—

122

C₂×C₄

—

146

—

123

C₂×C₄

—

146

—

124

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

125

C₂×C₄

D₄×D₇₃

yes

—

126

C₂×C₄

Q₈×D₇₃

yes

characteristic

127

C₂×C₄

#11924028

yes

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂²

—

133

C₂²

—

292

—

134

C₂²

—

146

—

135

C₂²

C₄⋊C₄×D₇₃

yes

characteristic

136

C₂²

#14556564

yes

—

137

C₄

—

138

C₄

—

584

—

139

C₄

—

146

—

140

C₄

C₂²⋊C₄×D₇₃

yes

characteristic

141

C₄

#14556548

yes

characteristic

142

C₄

#14556569

yes

—

143

C₂

—

144

C₂

—

145

C₂

—

292

—

146

C₂

#17988446

yes

characteristic

	subgroup	G/H
center	C₄	#14556548
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	H(3,ℤ/4)×C₇₃	C₂
socle	C₂×C₇₃	#380
hypercenter	H(3,ℤ/4)	D₇₃
hypocenter	C₇₃	#1812
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#14556548

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

H(3,ℤ/4)×C₇₃

C₂

C₂×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #17988623	C₂	semidirect
4. #17988344	C₂	semidirect
5. H(3,ℤ/4)×C₇₃	C₂	semidirect
9. C₂×#11924011	C₄	semidirect
11. C₂×C₈×D₇₃	C₄	semidirect
74. C₇₃	#1812	semidirect
79. H(3,ℤ/4)	D₇₃	semidirect

H ≅ G/N

type

2. #17988623

C₂

semidirect

4. #17988344

C₂

semidirect

5. H(3,ℤ/4)×C₇₃

C₂

semidirect

9. C₂×#11924011

C₄

semidirect

11. C₂×C₈×D₇₃

C₄

semidirect

74. C₇₃

#1812

semidirect

79. H(3,ℤ/4)

D₇₃

semidirect

Subgroups — #22609976

Conjugacy classes of subgroups

Complements