Group #22626226 — subgroups

Subgroups — #22626226

Order 9344.

230 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 54 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×#14556652	C₂	1	2	yes	maximal, complemented
2	#17989497	C₂	1	2	yes	maximal, complemented
3	#17989495	C₂	1	2	yes	maximal, complemented
4	#17989158	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924037	—	2	2	no	—
6	#14556652	—	2	2	no	—
7	#14556652	C₂²	1	4	yes	—
8	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
9	#14556589	C₄	1	1	yes	characteristic
10	#14556589	C₂²	1	1	yes	characteristic
11	#14556578	C₄	1	1	yes	characteristic
12	#14556578	C₂²	1	1	yes	characteristic
13	#14556572	—	2	2	no	—
14	#14556570	—	2	2	no	—
15	#14556550	C₄	1	1	yes	characteristic
16	C₂×#41×C₇₃	C₄	1	1	yes	characteristic
17	#11924037	—	2	4	no	—
18	#11924037	—	4	2	no	—
19	C₂×#10596833	—	4	2	no	—
20	C₂×Dic₁₄₆	—	4	2	no	—
21	C₂×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₁₄₆	D₄	1	2	yes	—
23	C₂×Dic₁₄₆	C₂³	1	1	yes	characteristic
24	C₂²×Dic₇₃	D₄	1	2	yes	—
25	C₄⋊Dic₇₃	—	2	1	no	—
26	#11924007	—	2	2	no	—
27	#11924007	—	2	2	no	—
28	C₂²⋊Dic₇₃	—	2	1	no	—
29	C₂²⋊Dic₇₃	—	2	2	no	—
30	C₂²⋊Dic₇₃	—	2	2	no	—
31	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
33	#41×C₇₃	—	4	2	no	—
34	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×D₄×C₇₃	—	2	1	no	—
36	C₂×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	—	2	1	no	—
38	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₇₃	—	2	1	no	—
40	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
41	Q₈×C₇₃	—	4	1	no	—
42	D₄×C₇₃	—	4	1	no	—
43	D₄×C₇₃	—	4	2	no	—
44	C₂×C₄×C₇₃	—	2	2	no	—
45	C₂×C₄×C₇₃	—	2	1	no	—
46	C₂×C₄×C₇₃	—	4	2	no	—
47	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₇₃	—	2	2	no	—
50	C₂×Dic₇₃	—	2	2	no	—
51	C₂×Dic₇₃	—	2	1	no	—
52	C₂×Dic₇₃	—	2	1	no	—
53	C₂×Dic₇₃	C₂²⋊C₄	1	2	yes	—
54	C₂×Dic₇₃	C₂×D₄	1	2	yes	—
55	Dic₁₄₆	—	4	1	no	—
56	Dic₁₄₆	—	4	2	no	—
57	Dic₁₄₆	—	4	1	no	—
58	#10596833	—	4	4	no	—
59	C₂²×C₇₃	—	4	1	no	—
60	C₂²×C₇₃	—	4	2	no	—
61	C₂²×C₇₃	—	4	1	no	—
62	C₂²×C₇₃	#99	1	2	yes	—
63	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₄×C₇₃	—	2	1	no	—
65	C₄×C₇₃	—	2	1	no	—
66	C₄×C₇₃	—	4	1	no	—
67	Dic₇₃	—	2	2	no	—
68	Dic₇₃	—	2	2	no	—
69	Dic₇₃	—	4	1	no	—
70	Dic₇₃	—	4	1	no	—
71	C₂×C₇₃	—	2	1	no	—
72	C₂×C₇₃	—	2	2	no	—
73	C₂×C₇₃	—	8	1	no	—
74	C₂×C₇₃	C₂×#99	1	1	yes	characteristic
75	#2133	—	73	1	no	maximal, Sylow
76	C₇₃	#2133	1	1	yes	characteristic, complemented, Sylow
77	#536	—	73	1	no	—
78	C₂×#101	—	73	2	no	—
79	#356	—	73	2	no	—
80	#355	—	73	2	no	—
81	C₂×#41	#64408	1	1	yes	characteristic
82	C₂²×Q₈	—	73	1	no	—
83	C₂×M₁₆	—	146	2	no	—
84	C₄⋊Q₈	—	73	1	no	—
85	C₄⋊Q₈	—	73	1	no	—
86	#124	—	73	1	no	—
87	#124	—	73	1	no	—
88	#123	—	146	2	no	—
89	C₂²⋊Q₈	—	146	2	no	—
90	#117	—	73	1	no	—
91	#101	—	73	4	no	—
92	#101	—	146	2	no	—
93	#41	—	4	2	no	—
94	C₂×Q₈	—	292	2	no	—
95	C₂×Q₈	—	73	2	no	—
96	C₂×Q₈	—	73	1	no	—
97	C₂×Q₈	—	73	1	no	—
98	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	C₂×#64408	1	1	yes	characteristic
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	73	2	no	—
103	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
104	M₁₆	—	292	2	no	—
105	M₁₆	—	146	4	no	—
106	C₂×C₈	—	292	2	no	—
107	C₄⋊C₄	—	146	2	no	—
108	C₄⋊C₄	—	146	2	no	—
109	C₄⋊C₄	—	146	1	no	—
110	C₂²⋊C₄	—	146	2	no	—
111	C₂²⋊C₄	—	146	2	no	—
112	C₂²⋊C₄	—	146	1	no	—
113	C₄²	—	73	1	no	—
114	C₄²	—	73	1	no	—
115	C₂³	—	2	1	no	—
116	C₂³	C₂²⋊#64408	1	1	yes	characteristic
117	Q₈	—	4	1	no	—
118	Q₈	—	292	2	no	—
119	Q₈	—	292	1	no	—
120	Q₈	—	292	1	no	—
121	D₄	—	4	1	no	—
122	D₄	—	4	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	73	2	no	—
127	C₂×C₄	—	73	2	no	—
128	C₂×C₄	—	146	1	no	—
129	C₂×C₄	—	146	1	no	—
130	C₂×C₄	—	146	2	no	—
131	C₂×C₄	—	146	2	no	—
132	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
133	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
134	C₈	—	292	4	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	2	no	—
137	C₂²	—	4	1	no	—
138	C₂²	#14556649	1	2	yes	—
139	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	4	1	no	—
143	C₄	—	292	1	no	—
144	C₄	—	292	1	no	—
145	C₄	—	146	2	no	—
146	C₄	—	146	2	no	—
147	C₂	—	2	1	no	—
148	C₂	—	2	2	no	—
149	C₂	—	8	1	no	—
150	C₂	C₂×#14556649	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556652

C₂

yes

maximal, complemented

#17989497

C₂

yes

maximal, complemented

#17989495

C₂

yes

maximal, complemented

#17989158

C₂

yes

maximal, characteristic

C₂×#11924037

—

#14556652

—

#14556652

C₂²

yes

—

C₂²×Dic₁₄₆

C₂²

yes

characteristic

#14556589

C₄

yes

characteristic

#14556589

C₂²

yes

characteristic

#14556578

C₄

yes

characteristic

#14556578

C₂²

yes

—

—

C₄

yes

characteristic

C₂×#41×C₇₃

C₄

yes

—

—

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂×C₄

yes

characteristic

C₂×Dic₁₄₆

D₄

yes

—

C₂×Dic₁₄₆

C₂³

yes

characteristic

C₂²×Dic₇₃

D₄

yes

—

C₄⋊Dic₇₃

—

#11924007

—

#11924007

—

C₂²⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

C₂×C₄

yes

characteristic

#41×C₇₃

—

C₂×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

C₂²⋊C₄

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

—

C₂×Dic₇₃

C₂×D₄

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#99

yes

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₂×#99

yes

—

C₇₃

yes

characteristic, complemented, Sylow

—

—

—

—

C₂×#41

#64408

yes

characteristic

C₂²×Q₈

—

C₂×M₁₆

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#124

—

#124

—

#123

—

146

—

C₂²⋊Q₈

—

146

—

#117

—

#101

—

#101

—

146

—

#41

—

C₂×Q₈

—

292

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×#64408

yes

characteristic

C₂×D₄

—

100

C₂×D₄

C₂×#64408

yes

characteristic

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂×#64408

yes

characteristic

104

M₁₆

—

292

—

105

M₁₆

—

146

—

106

C₂×C₈

—

292

—

107

C₄⋊C₄

—

146

—

108

C₄⋊C₄

—

146

—

109

C₄⋊C₄

—

146

—

110

C₂²⋊C₄

—

146

—

111

C₂²⋊C₄

—

146

—

112

C₂²⋊C₄

—

146

—

113

C₄²

—

114

C₄²

—

115

C₂³

—

116

C₂³

C₂²⋊#64408

yes

characteristic

117

Q₈

—

118

Q₈

—

292

—

119

Q₈

—

292

—

120

Q₈

—

292

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

146

—

129

C₂×C₄

—

146

—

130

C₂×C₄

—

146

—

131

C₂×C₄

—

146

—

132

C₂×C₄

C₂²⋊#64408

yes

characteristic

133

C₂×C₄

C₂²×#64408

yes

characteristic

134

C₈

—

292

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#14556649

yes

—

139

C₂²

C₂×C₂²⋊#64408

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

292

—

144

C₄

—

292

—

145

C₄

—

146

—

146

C₄

—

146

—

147

C₂

—

148

C₂

—

149

C₂

—

150

C₂

C₂×#14556649

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#14556649
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₂×#41×C₇₃	C₄
socle	C₂×C₇₃	C₂×#99
hypercenter	C₂×#41	#64408
hypocenter	C₇₃	#2133
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#41

subgroup

G/H

center

C₂

C₂×#14556649

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₂×#41×C₇₃

C₄

socle

C₂×C₇₃

C₂×#99

C₂×#41

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#41

N	H ≅ G/N	type
1. C₂×#14556652	C₂	semidirect
2. #17989497	C₂	semidirect
3. #17989495	C₂	semidirect
76. C₇₃	#2133	semidirect

H ≅ G/N

type

1. C₂×#14556652

C₂

semidirect

2. #17989497

C₂

semidirect

3. #17989495

C₂

semidirect

76. C₇₃

#2133

semidirect

Subgroups — #22626226

Conjugacy classes of subgroups

Complements