Group #22625750 — subgroups

Subgroups — #108×#64408 #22625750

Order 9344.

228 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 92 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	D₇₃⋊#108	C₂	1	1	yes	maximal, characteristic
2	M₁₆×#64408	C₂	1	2	yes	maximal
3	C₂×C₈×#64408	C₂	1	1	yes	maximal, characteristic
4	#17989462	C₂	1	2	yes	maximal
5	#108×D₇₃	C₂	1	1	yes	maximal, characteristic
6	D₇₃⋊M₁₆	C₂²	1	2	yes	—
7	C₂×#11924036	C₂²	1	1	yes	characteristic
8	C₂×C₄×#64408	C₂²	1	1	yes	characteristic
9	#14556662	C₄	1	2	yes	complemented
10	#14556661	C₄	1	2	yes	complemented
11	#14556645	—	2	2	no	—
12	C₈×#64408	—	2	2	no	—
13	C₈×#64408	C₄	1	2	yes	—
14	C₈×#64408	C₄	1	2	yes	—
15	M₁₆×D₇₃	C₂²	1	2	yes	—
16	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
17	#14556527	C₄	1	1	yes	characteristic, complemented
18	#14556526	C₄	1	2	yes	complemented
19	#108×C₇₃	C₄	1	1	yes	characteristic, complemented
20	C₂²×#64408	Q₈	1	2	yes	—
21	#11924038	C₂×C₄	1	4	yes	—
22	#11924037	—	2	2	no	—
23	#11924036	—	2	2	no	—
24	#11924036	—	2	1	no	—
25	#11924036	C₂×C₄	1	2	yes	—
26	C₂×#10596833	—	2	1	no	—
27	C₄×#64408	C₂×C₄	1	2	yes	—
28	C₄×#64408	D₄	1	1	yes	characteristic
29	C₄×#64408	D₄	1	1	yes	characteristic
30	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
31	#11924012	C₂×C₄	1	2	yes	—
32	#11924011	—	2	2	no	—
33	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
34	C₈×D₇₃	—	2	2	no	—
35	C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
36	C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
37	C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
38	C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
39	M₁₆×C₇₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×D₇₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×#64408	—	2	4	no	—
43	C₂×#64408	C₄⋊C₄	1	2	yes	—
44	C₂×#64408	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₇₃	#41	1	1	yes	characteristic
47	C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
50	C₄×D₇₃	#41	1	1	yes	characteristic
51	#10596833	—	2	2	no	—
52	#10596833	—	2	4	no	—
53	C₈×C₇₃	—	2	2	no	—
54	C₈×C₇₃	C₄²	1	1	yes	characteristic
55	C₈×C₇₃	C₄²	1	1	yes	characteristic
56	#10596831	—	2	2	no	—
57	#10596831	C₄²	1	1	yes	characteristic
58	#10596831	C₄²	1	1	yes	characteristic
59	C₂²×C₇₃	C₄×Q₈	1	1	yes	characteristic
60	C₂×D₇₃	—	2	2	no	—
61	C₂×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×D₇₃	C₄×Q₈	1	1	yes	characteristic
63	#64408	—	2	2	no	—
64	#64408	#108	1	4	yes	direct product factor
65	C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
66	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
67	Dic₇₃	#117	1	1	yes	characteristic
68	Dic₇₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₇₃	—	2	1	no	—
70	C₂×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
71	D₇₃	—	2	1	no	—
72	D₇₃	C₂×#108	1	2	yes	—
73	C₄×#108	—	73	1	no	maximal, Sylow
74	C₇₃	C₄×#108	1	1	yes	characteristic, complemented, Sylow
75	C₂×#108	—	73	1	no	—
76	C₂×#108	—	73	1	no	—
77	C₄×M₁₆	—	73	2	no	—
78	C₂×C₄×C₈	—	73	1	no	—
79	#341	—	73	2	no	—
80	C₂×M₁₆	—	73	2	no	—
81	C₂×M₁₆	—	73	2	no	—
82	C₂²×C₈	—	73	1	no	—
83	C₂²×C₈	—	73	1	no	—
84	C₂×C₄²	—	73	1	no	—
85	#108	—	73	2	no	—
86	#108	—	73	2	no	—
87	#108	—	73	2	no	—
88	#108	—	73	1	no	—
89	#108	#64408	1	1	yes	characteristic, direct product factor
90	#97	—	146	2	no	—
91	C₄×C₈	—	73	2	no	—
92	C₄×C₈	—	73	2	no	—
93	C₄×C₈	—	146	2	no	—
94	C₂²×C₄	—	73	1	no	—
95	C₂²×C₄	—	73	2	no	—
96	M₁₆	—	73	4	no	—
97	M₁₆	—	73	2	no	—
98	M₁₆	—	146	2	no	—
99	M₁₆	—	146	2	no	—
100	M₁₆	C₂×#64408	1	2	yes	—
101	C₂×C₈	—	73	1	no	—
102	C₂×C₈	—	73	1	no	—
103	C₂×C₈	—	73	1	no	—
104	C₂×C₈	—	73	1	no	—
105	C₂×C₈	—	73	1	no	—
106	C₂×C₈	—	73	2	no	—
107	C₂×C₈	—	146	2	no	—
108	C₂×C₈	—	146	1	no	—
109	C₂×C₈	—	146	2	no	—
110	C₂×C₈	—	146	1	no	—
111	C₂×C₈	C₂×#64408	1	1	yes	characteristic
112	C₄²	—	73	2	no	—
113	C₄²	—	73	1	no	—
114	C₄²	—	73	1	no	—
115	C₂³	—	73	1	no	—
116	C₂×C₄	—	73	2	no	—
117	C₂×C₄	—	73	1	no	—
118	C₂×C₄	—	73	1	no	—
119	C₂×C₄	—	73	1	no	—
120	C₂×C₄	—	73	1	no	—
121	C₂×C₄	—	73	1	no	—
122	C₂×C₄	—	73	2	no	—
123	C₂×C₄	—	146	4	no	—
124	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	73	1	no	—
127	C₈	—	73	1	no	—
128	C₈	—	146	2	no	—
129	C₈	—	146	4	no	—
130	C₈	—	146	2	no	—
131	C₈	C₄×#64408	1	1	yes	characteristic
132	C₈	C₄×#64408	1	1	yes	characteristic
133	C₂²	—	73	1	no	—
134	C₂²	—	73	1	no	—
135	C₂²	—	146	2	no	—
136	C₂²	Q₈×#64408	1	1	yes	characteristic
137	C₄	—	73	4	no	—
138	C₄	—	73	1	no	—
139	C₄	—	73	1	no	—
140	C₄	—	146	2	no	—
141	C₄	C₂×C₄×#64408	1	1	yes	characteristic
142	C₄	D₄×#64408	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	73	2	no	—
145	C₂	—	146	1	no	—
146	C₂	C₄⋊C₄×#64408	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₇₃⋊#108

C₂

yes

maximal, characteristic

M₁₆×#64408

C₂

yes

maximal

C₂×C₈×#64408

C₂

yes

maximal, characteristic

#17989462

C₂

yes

maximal

#108×D₇₃

C₂

yes

maximal, characteristic

D₇₃⋊M₁₆

C₂²

yes

—

C₂×#11924036

C₂²

yes

characteristic

C₂×C₄×#64408

C₂²

yes

characteristic

#14556662

C₄

yes

complemented

#14556661

C₄

yes

—

—

C₄

yes

—

C₈×#64408

C₄

yes

—

M₁₆×D₇₃

C₂²

yes

—

C₂×C₈×D₇₃

C₂²

yes

characteristic

#14556527

C₄

yes

characteristic, complemented

#14556526

C₄

yes

complemented

#108×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

Q₈

yes

—

#11924038

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#10596833

—

C₄×#64408

C₂×C₄

yes

—

C₄×#64408

D₄

yes

characteristic

C₄×#64408

D₄

yes

characteristic

C₂×C₄×D₇₃

C₂³

yes

characteristic

#11924012

C₂×C₄

yes

—

#11924011

—

C₂×#10596831

C₂×C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

C₂×C₄

yes

characteristic

C₈×D₇₃

C₂×C₄

yes

characteristic

C₈×D₇₃

C₂×C₄

yes

characteristic

C₈×D₇₃

C₂×C₄

yes

characteristic

M₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

C₂×Q₈

yes

characteristic

C₂×#64408

—

C₂×#64408

C₄⋊C₄

yes

—

C₂×#64408

C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

#41

yes

characteristic

C₄×D₇₃

C₂²×C₄

yes

characteristic

C₄×D₇₃

C₂²×C₄

yes

characteristic

C₄×D₇₃

C₂×D₄

yes

C₄×D₇₃

yes

—

—

C₈×C₇₃

—

C₈×C₇₃

C₄²

yes

characteristic

C₈×C₇₃

C₄²

yes

characteristic

#10596831

—

#10596831

C₄²

yes

characteristic

#10596831

C₄²

yes

characteristic

C₂²×C₇₃

C₄×Q₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₇₃

C₄×Q₈

yes

—

yes

direct product factor

C₄×C₇₃

C₂×C₄²

yes

characteristic

C₄×C₇₃

C₄×D₄

yes

characteristic

Dic₇₃

#117

yes

characteristic

Dic₇₃

C₄×D₄

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

C₄×C₄⋊C₄

yes

characteristic

D₇₃

—

D₇₃

C₂×#108

yes

—

C₄×#108

—

maximal, Sylow

C₇₃

C₄×#108

yes

characteristic, complemented, Sylow

C₂×#108

—

C₂×#108

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#341

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

—

—

—

—

yes

characteristic, direct product factor

#97

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

146

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

146

—

M₁₆

—

146

—

100

M₁₆

C₂×#64408

yes

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

146

—

108

C₂×C₈

—

146

—

109

C₂×C₈

—

146

—

110

C₂×C₈

—

146

—

111

C₂×C₈

C₂×#64408

yes

characteristic

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₂³

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

146

—

124

C₂×C₄

C₂²×#64408

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

146

—

129

C₈

—

146

—

130

C₈

—

146

—

131

C₈

C₄×#64408

yes

characteristic

132

C₈

C₄×#64408

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

146

—

136

C₂²

Q₈×#64408

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

146

—

141

C₄

C₂×C₄×#64408

yes

characteristic

142

C₄

D₄×#64408

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

146

—

146

C₂

C₄⋊C₄×#64408

yes

characteristic

	subgroup	G/H
center	C₄	D₄×#64408
commutator	C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄	C₂²×#64408
Fitting	#108×C₇₃	C₄
socle	C₂×C₇₃	C₄×C₄⋊C₄
hypercenter	#108	#64408
hypocenter	C₇₃	C₄×#108
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₄

D₄×#64408

commutator

C₄×C₇₃

C₂×C₄²

Frattini

C₂×C₄

C₂²×#64408

Fitting

#108×C₇₃

C₄

socle

C₂×C₇₃

C₄×C₄⋊C₄

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
9. #14556662	C₄	semidirect
10. #14556661	C₄	semidirect
17. #14556527	C₄	semidirect
18. #14556526	C₄	semidirect
19. #108×C₇₃	C₄	semidirect
64. #64408	#108	direct (N × H)
74. C₇₃	C₄×#108	semidirect
89. #108	#64408	direct (N × H)

H ≅ G/N

type

9. #14556662

C₄

semidirect

10. #14556661

C₄

semidirect

17. #14556527

C₄

semidirect

18. #14556526

C₄

semidirect

19. #108×C₇₃

C₄

semidirect

64. #64408

#108

direct (N × H)

74. C₇₃

C₄×#108

semidirect

89. #108

#64408

direct (N × H)

Subgroups — #108×#64408 #22625750

Conjugacy classes of subgroups

Complements