Group #22626049 — subgroups

Subgroups — #22626049

Order 9344.

196 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 68 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	#17989598	C₂	1	2	yes	maximal
2	C₂×#14556660	C₂	1	1	yes	maximal, characteristic
3	#17989550	C₂	1	1	yes	maximal, characteristic
4	#17989466	C₂	1	2	yes	maximal
5	#108×D₇₃	C₂	1	1	yes	maximal, characteristic
6	D₇₃⋊M₁₆	C₂²	1	1	yes	characteristic
7	#14556670	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#64408	C₂²	1	2	yes	—
9	#14556660	—	2	2	no	—
10	#14556660	—	2	1	no	—
11	#14556660	—	2	1	no	—
12	#14556659	—	2	2	no	—
13	#14556645	—	2	1	no	—
14	C₈×#64408	—	2	1	no	—
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	—	2	2	no	—
21	#11924038	—	2	1	no	—
22	#11924037	Q₈	1	2	yes	—
23	#11924036	—	2	1	no	—
24	C₄⋊#64408	—	2	2	no	—
25	C₄⋊#64408	—	2	2	no	—
26	C₄⋊#64408	D₄	1	1	yes	characteristic
27	C₄⋊#64408	D₄	1	1	yes	characteristic
28	C₂²⋊#64408	—	2	1	no	—
29	C₄×#64408	—	2	1	no	—
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	D₄	1	1	yes	characteristic
36	C₈×D₇₃	D₄	1	1	yes	characteristic
37	C₈×D₇₃	Q₈	1	1	yes	characteristic
38	C₈×D₇₃	Q₈	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	#41	1	1	yes	characteristic
42	C₂×#64408	—	2	1	no	—
43	C₂×#64408	—	2	1	no	—
44	C₂×#64408	—	4	2	no	—
45	C₂×#64408	—	8	2	no	—
46	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
48	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
49	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
50	C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
51	C₄×D₇₃	#41	1	1	yes	characteristic
52	#10596833	—	2	2	no	—
53	C₈×C₇₃	—	2	2	no	—
54	C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
55	C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
56	#10596831	—	2	2	no	—
57	#10596831	C₄⋊C₄	1	1	yes	characteristic
58	#10596831	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₇₃	C₄×Q₈	1	1	yes	characteristic
60	C₂×D₇₃	—	2	1	no	—
61	C₂×D₇₃	—	2	1	no	—
62	C₂×D₇₃	#121	1	1	yes	characteristic
63	C₂×D₇₃	#125	1	1	yes	characteristic
64	#64408	—	4	1	no	—
65	#64408	—	4	1	no	—
66	#64408	—	8	2	no	—
67	C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
69	Dic₇₃	C₂²⋊Q₈	1	1	yes	characteristic
70	Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₇₃	—	2	1	no	—
72	C₂×C₇₃	#389	1	1	yes	characteristic
73	D₇₃	—	2	1	no	—
74	D₇₃	#472	1	1	yes	characteristic
75	D₇₃	#473	1	1	yes	characteristic
76	#1952	—	73	1	no	maximal, Sylow
77	C₇₃	#1952	1	1	yes	characteristic, complemented, Sylow
78	C₂×#108	—	73	1	no	—
79	#428	—	73	2	no	—
80	C₂×#107	—	73	1	no	—
81	#405	—	73	1	no	—
82	#343	—	73	2	no	—
83	C₂×M₁₆	—	73	1	no	—
84	C₂×M₁₆	—	73	2	no	—
85	C₂²×C₈	—	73	1	no	—
86	#117	—	73	1	no	—
87	C₂×C₄⋊C₄	—	73	2	no	—
88	#108	—	73	2	no	—
89	#108	—	73	1	no	—
90	#108	#64408	1	1	yes	characteristic, complemented
91	#107	—	146	2	no	—
92	#107	—	146	1	no	—
93	#107	—	146	1	no	—
94	#106	—	146	2	no	—
95	#97	—	146	1	no	—
96	C₄×C₈	—	146	1	no	—
97	C₂²×C₄	—	73	1	no	—
98	C₂²×C₄	—	146	2	no	—
99	M₁₆	—	73	2	no	—
100	M₁₆	—	73	2	no	—
101	M₁₆	—	146	2	no	—
102	M₁₆	—	146	1	no	—
103	M₁₆	C₂×#64408	1	2	yes	—
104	C₂×C₈	—	73	1	no	—
105	C₂×C₈	—	73	1	no	—
106	C₂×C₈	—	73	1	no	—
107	C₂×C₈	—	73	1	no	—
108	C₂×C₈	—	73	1	no	—
109	C₂×C₈	—	146	2	no	—
110	C₂×C₈	—	146	1	no	—
111	C₂×C₈	C₂×#64408	1	1	yes	characteristic
112	C₄⋊C₄	—	73	1	no	—
113	C₄⋊C₄	—	73	1	no	—
114	C₄⋊C₄	—	146	2	no	—
115	C₄⋊C₄	—	146	2	no	—
116	C₂²⋊C₄	—	146	1	no	—
117	C₄²	—	146	1	no	—
118	C₂³	—	73	1	no	—
119	C₂×C₄	—	292	2	no	—
120	C₂×C₄	—	584	2	no	—
121	C₂×C₄	—	73	1	no	—
122	C₂×C₄	—	73	1	no	—
123	C₂×C₄	—	73	1	no	—
124	C₂×C₄	—	73	1	no	—
125	C₂×C₄	—	73	1	no	—
126	C₂×C₄	—	146	1	no	—
127	C₂×C₄	—	146	1	no	—
128	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	73	1	no	—
131	C₈	—	73	1	no	—
132	C₈	—	146	2	no	—
133	C₈	—	146	2	no	—
134	C₈	C₄⋊#64408	1	1	yes	characteristic
135	C₈	C₄⋊#64408	1	1	yes	characteristic
136	C₂²	—	73	1	no	—
137	C₂²	—	73	1	no	—
138	C₂²	—	146	1	no	—
139	C₂²	—	146	1	no	—
140	C₂²	Q₈×#64408	1	1	yes	characteristic
141	C₄	—	292	1	no	—
142	C₄	—	292	1	no	—
143	C₄	—	584	2	no	—
144	C₄	—	73	1	no	—
145	C₄	—	73	1	no	—
146	C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
147	C₄	D₄×#64408	1	1	yes	characteristic
148	C₂	—	2	1	no	—
149	C₂	—	73	1	no	—
150	C₂	—	73	1	no	—
151	C₂	—	146	1	no	—
152	C₂	#17989538	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989598

C₂

yes

maximal

C₂×#14556660

C₂

yes

maximal, characteristic

#17989550

C₂

yes

maximal, characteristic

#17989466

C₂

yes

maximal

#108×D₇₃

C₂

yes

maximal, characteristic

D₇₃⋊M₁₆

C₂²

yes

characteristic

#14556670

C₂²

yes

characteristic

C₂×C₄⋊#64408

C₂²

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

—

#11924038

—

#11924037

Q₈

yes

—

—

—

—

D₄

yes

characteristic

C₄⋊#64408

D₄

yes

characteristic

C₂²⋊#64408

—

C₄×#64408

—

C₂×C₄×D₇₃

C₂³

yes

characteristic

#11924012

C₂×C₄

yes

—

#11924011

—

C₂×#10596831

C₂×C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

D₄

yes

characteristic

C₈×D₇₃

D₄

yes

characteristic

C₈×D₇₃

Q₈

yes

characteristic

C₈×D₇₃

Q₈

yes

characteristic

M₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

C₂×C₄

yes

C₂²×D₇₃

yes

—

—

—

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

C₂×Q₈

yes

characteristic

C₄×D₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

C₂×Q₈

yes

C₄×D₇₃

yes

—

C₈×C₇₃

—

C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₈×C₇₃

C₄⋊C₄

yes

characteristic

#10596831

—

#10596831

C₄⋊C₄

yes

characteristic

#10596831

C₄⋊C₄

yes

characteristic

C₂²×C₇₃

C₄×Q₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

yes

C₂×D₇₃

yes

—

—

—

C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₃

C₄×D₄

yes

characteristic

Dic₇₃

C₂²⋊Q₈

yes

characteristic

Dic₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

#389

yes

characteristic

D₇₃

—

D₇₃

yes

D₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

C₂×C₄⋊C₄

—

—

—

yes

characteristic, complemented

#107

—

146

—

#107

—

146

—

#107

—

146

—

#106

—

146

—

#97

—

146

—

C₄×C₈

—

146

—

C₂²×C₄

—

C₂²×C₄

—

146

—

M₁₆

—

100

M₁₆

—

101

M₁₆

—

146

—

102

M₁₆

—

146

—

103

M₁₆

C₂×#64408

yes

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

146

—

110

C₂×C₈

—

146

—

111

C₂×C₈

C₂×#64408

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

146

—

115

C₄⋊C₄

—

146

—

116

C₂²⋊C₄

—

146

—

117

C₄²

—

146

—

118

C₂³

—

119

C₂×C₄

—

292

—

120

C₂×C₄

—

584

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

146

—

127

C₂×C₄

—

146

—

128

C₂×C₄

C₂²×#64408

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

146

—

133

C₈

—

146

—

134

C₈

C₄⋊#64408

yes

characteristic

135

C₈

C₄⋊#64408

yes

characteristic

136

C₂²

—

137

C₂²

—

138

C₂²

—

146

—

139

C₂²

—

146

—

140

C₂²

Q₈×#64408

yes

characteristic

141

C₄

—

292

—

142

C₄

—

292

—

143

C₄

—

584

—

144

C₄

—

145

C₄

—

146

C₄

C₂×C₄⋊#64408

yes

characteristic

147

C₄

D₄×#64408

yes

characteristic

148

C₂

—

149

C₂

—

150

C₂

—

151

C₂

—

146

—

152

C₂

#17989538

yes

characteristic

	subgroup	G/H
center	C₂	#17989538
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	#108×C₇₃	C₄
socle	C₂×C₇₃	#389
hypercenter	#108	#64408
hypocenter	C₇₃	#1952
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₂

#17989538

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

#108×C₇₃

C₄

C₂×C₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
17. #14556527	C₄	semidirect
18. #14556526	C₄	semidirect
19. #108×C₇₃	C₄	semidirect
77. C₇₃	#1952	semidirect
90. #108	#64408	semidirect

H ≅ G/N

type

17. #14556527

C₄

semidirect

18. #14556526

C₄

semidirect

19. #108×C₇₃

C₄

semidirect

77. C₇₃

#1952

semidirect

90. #108

#64408

semidirect

Subgroups — #22626049

Conjugacy classes of subgroups

Complements