Group #22609975 — subgroups

Subgroups — #22609975

Order 9344.

260 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14556598	C₂	1	1	yes	maximal, characteristic
2	#17988623	C₂	1	2	yes	maximal, complemented
3	#17988561	C₂	1	1	yes	maximal, characteristic
4	#17988343	C₂	1	2	yes	maximal, complemented
5	#410×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#14556598	—	2	2	no	—
7	#14556598	—	2	2	no	—
8	C₂×#11924012	C₄	1	2	yes	complemented
9	C₂×#11924012	C₂²	1	1	yes	characteristic
10	C₂²×#10596831	C₄	1	2	yes	complemented
11	C₂²×#10596831	C₂²	1	1	yes	characteristic
12	#14556550	C₂²	1	2	yes	—
13	#14556524	—	2	2	no	—
14	#14556522	—	2	2	no	—
15	#14556506	—	2	2	no	—
16	C₂×#41×C₇₃	C₂²	1	1	yes	characteristic
17	#117×C₇₃	C₂²	1	2	yes	—
18	#99×C₇₃	—	2	2	no	—
19	#11924012	—	2	2	no	—
20	#11924012	—	2	4	no	—
21	C₂×#10596831	—	2	2	no	—
22	C₂×#10596831	—	2	2	no	—
23	C₂×#10596831	C₂×C₄	1	2	yes	—
24	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
25	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₂×#10596831	D₄	1	2	yes	—
27	C₂×#10596831	D₄	1	2	yes	—
28	C₄⋊Dic₇₃	—	2	2	no	—
29	C₂²⋊Dic₇₃	—	2	2	no	—
30	C₄×Dic₇₃	—	2	2	no	—
31	#41×C₇₃	—	2	2	no	—
32	#41×C₇₃	—	2	2	no	—
33	C₂×Q₈×C₇₃	Q₈	1	1	yes	characteristic
34	C₂×D₄×C₇₃	D₄	1	2	yes	—
35	C₂×D₄×C₇₃	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₇₃	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₇₃	—	2	2	no	—
39	C₂²⋊C₄×C₇₃	—	2	2	no	—
40	C₄²×C₇₃	—	2	2	no	—
41	C₂³×C₇₃	C₄⋊C₄	1	2	yes	—
42	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₇₃	—	2	2	no	—
44	D₄×C₇₃	—	2	4	no	—
45	D₄×C₇₃	—	2	2	no	—
46	C₂×C₄×C₇₃	—	2	2	no	—
47	C₂×C₄×C₇₃	—	2	2	no	—
48	C₂×C₄×C₇₃	—	4	2	no	—
49	C₂×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#41	1	2	yes	—
54	C₂×Dic₇₃	—	4	2	no	—
55	#10596831	—	2	2	no	—
56	#10596831	C₂²⋊C₄	1	1	yes	characteristic
57	#10596831	C₂²⋊C₄	1	1	yes	characteristic
58	#10596831	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₇₃	—	2	2	no	—
60	C₂²×C₇₃	—	4	1	no	—
61	C₂²×C₇₃	—	4	2	no	—
62	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₇₃	#121	1	2	yes	—
64	C₄×C₇₃	—	2	2	no	—
65	C₄×C₇₃	—	8	2	no	—
66	C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₇₃	#117	1	1	yes	characteristic
68	C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
69	Dic₇₃	—	8	2	no	—
70	C₂×C₇₃	—	2	2	no	—
71	C₂×C₇₃	—	2	1	no	—
72	C₂×C₇₃	—	4	2	no	—
73	C₂×C₇₃	#380	1	1	yes	characteristic
74	#1812	—	73	1	no	maximal, Sylow
75	C₇₃	#1812	1	1	yes	characteristic, complemented, Sylow
76	C₂×#131	—	73	1	no	—
77	#427	—	73	2	no	—
78	#410	—	73	1	no	—
79	#410	D₇₃	1	1	yes	characteristic
80	H(3,ℤ/4)	—	73	2	no	—
81	C₂×#41	C₂×D₇₃	1	1	yes	characteristic
82	#131	—	146	2	no	—
83	#131	—	146	2	no	—
84	C₂×M₁₆	—	73	2	no	—
85	C₂×M₁₆	—	73	1	no	—
86	C₂²×C₈	—	73	1	no	—
87	C₂²×C₈	—	73	2	no	—
88	#117	—	73	2	no	—
89	#117	C₂×D₇₃	1	2	yes	—
90	#107	—	146	2	no	—
91	#106	—	146	2	no	—
92	#99	—	2	2	no	—
93	#99	—	146	2	no	—
94	#41	—	2	2	no	—
95	#41	—	2	2	no	—
96	C₂×Q₈	Dic₁₄₆	1	1	yes	characteristic
97	C₂×D₄	Dic₁₄₆	1	1	yes	characteristic
98	C₂×D₄	C₂²⋊D₇₃	1	2	yes	—
99	C₂²×C₄	C₄×D₇₃	1	2	yes	—
100	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
101	M₁₆	—	146	4	no	—
102	M₁₆	—	146	2	no	—
103	C₂×C₈	—	73	1	no	—
104	C₂×C₈	—	73	1	no	—
105	C₂×C₈	—	73	2	no	—
106	C₂×C₈	—	73	2	no	—
107	C₂×C₈	—	73	2	no	—
108	C₂×C₈	—	146	2	no	—
109	C₂×C₈	—	146	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	146	2	no	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₂²⋊C₄	—	146	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	146	2	no	—
116	C₂³	#11924007	1	2	yes	—
117	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
118	Q₈	—	2	2	no	—
119	D₄	—	2	4	no	—
120	D₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	292	2	no	—
125	C₂×C₄	#11924007	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
127	C₂×C₄	D₄×D₇₃	1	2	yes	—
128	C₂×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
129	C₂×C₄	#11924027	1	2	yes	—
130	C₈	—	73	1	no	—
131	C₈	—	73	1	no	—
132	C₈	—	73	2	no	—
133	C₈	—	146	2	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	2	no	—
137	C₂²	C₂×#11924007	1	1	yes	characteristic
138	C₂²	#14556563	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	8	2	no	—
141	C₄	—	584	2	no	—
142	C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
143	C₄	#14556547	1	1	yes	characteristic
144	C₄	#14556568	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	#17988445	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556598

C₂

yes

maximal, characteristic

#17988623

C₂

yes

maximal, complemented

#17988561

C₂

yes

maximal, characteristic

#17988343

C₂

yes

maximal, complemented

#410×C₇₃

C₂

yes

maximal, characteristic

#14556598

—

#14556598

—

C₂×#11924012

C₄

yes

complemented

C₂×#11924012

C₂²

yes

characteristic

C₂²×#10596831

C₄

yes

complemented

C₂²×#10596831

C₂²

yes

characteristic

#14556550

C₂²

yes

—

#14556524

—

#14556522

—

#14556506

—

C₂×#41×C₇₃

C₂²

yes

characteristic

#117×C₇₃

C₂²

yes

—

#99×C₇₃

—

—

—

—

—

C₂×C₄

yes

—

C₂×#10596831

C₂×C₄

yes

characteristic

C₂×#10596831

C₂×C₄

yes

characteristic

C₂×#10596831

D₄

yes

—

C₂×#10596831

D₄

yes

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

Q₈

yes

characteristic

C₂×D₄×C₇₃

D₄

yes

—

C₂×D₄×C₇₃

Q₈

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₄⋊C₄

yes

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₄

yes

—

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

—

C₂×Dic₇₃

—

#10596831

—

#10596831

C₂²⋊C₄

yes

characteristic

#10596831

C₂²⋊C₄

yes

characteristic

#10596831

C₂²⋊C₄

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₃

#121

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

#117

yes

characteristic

C₄×C₇₃

C₂²⋊Q₈

yes

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

—

—

—

D₇₃

yes

characteristic

H(3,ℤ/4)

—

C₂×#41

C₂×D₇₃

yes

characteristic

#131

—

146

—

#131

—

146

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₇₃

yes

—

#107

—

146

—

#106

—

146

—

#99

—

#99

—

146

—

#41

—

#41

—

C₂×Q₈

Dic₁₄₆

yes

characteristic

C₂×D₄

Dic₁₄₆

yes

characteristic

C₂×D₄

C₂²⋊D₇₃

yes

—

C₂²×C₄

C₄×D₇₃

yes

—

100

C₂²×C₄

C₂²×D₇₃

yes

characteristic

101

M₁₆

—

146

—

102

M₁₆

—

146

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

146

—

109

C₂×C₈

—

146

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

146

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

146

—

114

C₄²

—

115

C₄²

—

146

—

116

C₂³

#11924007

yes

—

117

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

118

Q₈

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

292

—

125

C₂×C₄

#11924007

yes

—

126

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

127

C₂×C₄

D₄×D₇₃

yes

—

128

C₂×C₄

C₂×Dic₁₄₆

yes

characteristic

129

C₂×C₄

#11924027

yes

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

146

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×#11924007

yes

characteristic

138

C₂²

#14556563

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

584

—

142

C₄

C₂²⋊C₄×D₇₃

yes

characteristic

143

C₄

#14556547

yes

characteristic

144

C₄

#14556568

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#17988445

yes

characteristic

	subgroup	G/H
center	C₄	#14556547
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#410×C₇₃	C₂
socle	C₂×C₇₃	#380
hypercenter	#410	D₇₃
hypocenter	C₇₃	#1812
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

#14556547

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#410×C₇₃

C₂

C₂×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
2. #17988623	C₂	semidirect
4. #17988343	C₂	semidirect
8. C₂×#11924012	C₄	semidirect
10. C₂²×#10596831	C₄	semidirect
75. C₇₃	#1812	semidirect

H ≅ G/N

type

2. #17988623

C₂

semidirect

4. #17988343

C₂

semidirect

8. C₂×#11924012

C₄

semidirect

10. C₂²×#10596831

C₄

semidirect

75. C₇₃

#1812

semidirect

Subgroups — #22609975

Conjugacy classes of subgroups

Complements