Group #19145126 — subgroups

Subgroups — #19145126

Order 5504.

340 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 165 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	#14583527	C₂	1	2	yes	maximal
2	#14583308	C₂	1	4	yes	maximal, complemented
3	#14583301	C₂	1	2	yes	maximal, complemented
4	#14583297	C₂	1	2	yes	maximal, complemented
5	#14582941	C₂	1	2	yes	maximal
6	#14582937	C₂	1	1	yes	maximal, characteristic
7	C₂×#13063257	C₂	1	1	yes	maximal, characteristic
8	C₄₃×#405	C₂	1	1	yes	maximal, characteristic
9	#13063325	C₂²	1	4	yes	—
10	#13063324	—	2	2	no	—
11	#13063317	C₂²	1	4	yes	—
12	#13063316	—	2	2	no	—
13	#13063302	—	2	4	no	—
14	#13063284	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
16	#13063259	C₂²	1	4	yes	—
17	#13063259	C₂²	1	2	yes	—
18	#13063257	C₂²	1	1	yes	characteristic
19	#13063257	C₂²	1	2	yes	—
20	#13063257	C₂²	1	4	yes	—
21	#13063257	C₂²	1	1	yes	characteristic
22	#13063257	C₂²	1	2	yes	—
23	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	#117×C₄₃	C₂²	1	1	yes	characteristic
26	#97×C₄₃	C₂²	1	2	yes	—
27	C₄×C₈×C₄₃	C₂²	1	2	yes	—
28	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₄₃	—	2	2	no	—
30	C₂×Dic₈₆	—	2	4	no	—
31	M₁₆×C₄₃	Q₈	1	4	yes	complemented
32	C₂×C₈×C₄₃	Q₈	1	4	yes	complemented
33	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₄₃	C₂³	1	2	yes	—
35	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₄₃	D₄	1	2	yes	—
37	C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
38	C₄²×C₄₃	C₂³	1	2	yes	—
39	C₂²⋊Dic₄₃	—	2	2	no	—
40	C₄⋊Dic₄₃	—	2	2	no	—
41	C₄⋊Dic₄₃	—	2	4	no	—
42	C₄⋊Dic₄₃	C₂³	1	4	yes	—
43	C₄⋊Dic₄₃	C₂³	1	2	yes	—
44	C₄⋊Dic₄₃	C₂³	1	2	yes	—
45	Dic₄₃⋊C₄	—	2	4	no	—
46	Dic₄₃⋊C₄	—	2	4	no	—
47	C₄×Dic₄₃	—	2	2	no	—
48	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂×D₄	1	4	yes	—
50	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₄₃	—	2	2	no	—
55	C₂×Dic₄₃	—	2	2	no	—
56	C₂×Dic₄₃	—	2	4	no	—
57	C₂×Dic₄₃	—	8	2	no	—
58	Dic₈₆	—	8	4	no	—
59	C₈×C₄₃	C₂×Q₈	1	2	yes	—
60	C₈×C₄₃	C₂×Q₈	1	2	yes	—
61	C₈×C₄₃	C₂×Q₈	1	4	yes	—
62	C₂²×C₄₃	—	2	2	no	—
63	C₂²×C₄₃	C₄⋊Q₈	1	2	yes	—
64	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₄₃	—	2	4	no	—
66	C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
67	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
68	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
69	Dic₄₃	—	4	2	no	—
70	Dic₄₃	—	4	4	no	—
71	Dic₄₃	—	4	2	no	—
72	#3163	—	43	1	no	maximal, Sylow
73	C₂×C₄₃	—	2	2	no	—
74	C₂×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄₃	#577	1	2	yes	—
76	#544	—	43	2	no	—
77	#501	—	43	4	no	—
78	#499	—	43	2	no	—
79	#498	—	43	2	no	—
80	#428	—	43	2	no	—
81	#427	—	43	1	no	—
82	C₂×#106	—	43	1	no	—
83	#405	D₄₃	1	1	yes	characteristic
84	C₄₃	#3163	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
87	C₄⋊Q₈	—	43	4	no	—
88	C₄⋊Q₈	—	86	2	no	—
89	#125	—	43	4	no	—
90	#125	—	86	2	no	—
91	C₂²⋊Q₈	—	86	4	no	—
92	#117	—	43	2	no	—
93	#117	C₂×D₄₃	1	1	yes	characteristic
94	C₂×C₄⋊C₄	—	43	2	no	—
95	#107	—	43	2	no	—
96	#107	—	43	4	no	—
97	#106	—	43	2	no	—
98	#106	—	43	2	no	—
99	#106	—	43	1	no	—
100	#106	—	43	1	no	—
101	#106	—	43	4	no	—
102	#97	C₂×D₄₃	1	2	yes	—
103	C₄×C₈	C₂×D₄₃	1	2	yes	—
104	C₂×Q₈	—	86	4	no	—
105	C₂²×C₄	—	86	2	no	—
106	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
107	M₁₆	Dic₈₆	1	4	yes	complemented
108	C₂×C₈	Dic₈₆	1	4	yes	complemented
109	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
110	C₂×C₈	C₂²×D₄₃	1	2	yes	—
111	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
112	C₄⋊C₄	—	43	4	no	—
113	C₄⋊C₄	—	43	2	no	—
114	C₄⋊C₄	—	43	2	no	—
115	C₄⋊C₄	—	86	2	no	—
116	C₄⋊C₄	—	86	4	no	—
117	C₄⋊C₄	—	86	4	no	—
118	C₄⋊C₄	—	86	4	no	—
119	C₄⋊C₄	D₁₇₂	1	2	yes	—
120	C₂²⋊C₄	—	86	2	no	—
121	C₂²⋊C₄	D₁₇₂	1	2	yes	—
122	C₄²	—	86	2	no	—
123	C₄²	C₂²×D₄₃	1	2	yes	—
124	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
125	Q₈	—	344	4	no	—
126	C₂×C₄	—	86	2	no	—
127	C₂×C₄	—	86	2	no	—
128	C₂×C₄	—	86	4	no	—
129	C₂×C₄	—	344	2	no	—
130	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
131	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
132	C₂×C₄	C₂×D₁₇₂	1	4	yes	—
133	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
134	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
135	C₈	C₂×Dic₈₆	1	2	yes	—
136	C₈	C₂×Dic₈₆	1	2	yes	—
137	C₈	C₂×Dic₈₆	1	4	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	#13063325	1	2	yes	—
140	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	172	4	no	—
143	C₄	—	172	2	no	—
144	C₄	—	172	2	no	—
145	C₄	#13063325	1	2	yes	—
146	C₄	C₂²×Dic₈₆	1	1	yes	characteristic
147	C₄	C₂²×Dic₈₆	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	C₂×#13063325	1	1	yes	characteristic
150	C₂	#14583711	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14583527

C₂

yes

maximal

#14583308

C₂

yes

maximal, complemented

#14583301

C₂

yes

maximal, complemented

#14583297

C₂

yes

maximal, complemented

#14582941

C₂

yes

maximal

#14582937

C₂

yes

maximal, characteristic

C₂×#13063257

C₂

yes

maximal, characteristic

C₄₃×#405

C₂

yes

maximal, characteristic

#13063325

C₂²

yes

—

#13063324

—

#13063317

C₂²

yes

—

#13063316

—

#13063302

—

#13063284

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

#13063259

C₂²

yes

—

#13063259

C₂²

yes

—

#13063257

C₂²

yes

characteristic

#13063257

C₂²

yes

—

#13063257

C₂²

yes

—

#13063257

C₂²

yes

characteristic

#13063257

C₂²

yes

—

C₂×M₁₆×C₄₃

C₂²

yes

characteristic

C₂²×C₈×C₄₃

C₂²

yes

characteristic

#117×C₄₃

C₂²

yes

characteristic

#97×C₄₃

C₂²

yes

—

C₄×C₈×C₄₃

C₂²

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

M₁₆×C₄₃

Q₈

yes

complemented

C₂×C₈×C₄₃

Q₈

yes

complemented

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₈×C₄₃

C₂³

yes

—

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₄⋊C₄×C₄₃

D₄

yes

—

C₂²⋊C₄×C₄₃

D₄

yes

—

C₄²×C₄₃

C₂³

yes

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂³

yes

—

C₄⋊Dic₄₃

C₂³

yes

—

C₄⋊Dic₄₃

C₂³

yes

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

C₂×Q₈

yes

—

C₈×C₄₃

C₂×Q₈

yes

—

C₈×C₄₃

C₂×Q₈

yes

—

C₂²×C₄₃

—

C₂²×C₄₃

C₄⋊Q₈

yes

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

C₄⋊Q₈

yes

—

C₄×C₄₃

C₂²×Q₈

yes

characteristic

C₄×C₄₃

C₂²×Q₈

yes

characteristic

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

#3163

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₄₃

#577

yes

—

—

—

—

—

—

—

—

D₄₃

yes

characteristic

C₄₃

#3163

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₂×D₄₃

yes

characteristic

C₂²×C₈

C₂×D₄₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

#117

—

#117

C₂×D₄₃

yes

characteristic

C₂×C₄⋊C₄

—

—

—

—

—

—

100

#106

—

101

#106

—

102

#97

C₂×D₄₃

yes

—

103

C₄×C₈

C₂×D₄₃

yes

—

104

C₂×Q₈

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂²×D₄₃

yes

characteristic

107

M₁₆

Dic₈₆

yes

complemented

108

C₂×C₈

Dic₈₆

yes

complemented

109

C₂×C₈

C₂²×D₄₃

yes

characteristic

110

C₂×C₈

C₂²×D₄₃

yes

—

111

C₂×C₈

C₂²×D₄₃

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

D₁₇₂

yes

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

D₁₇₂

yes

—

122

C₄²

—

123

C₄²

C₂²×D₄₃

yes

—

124

C₂³

C₂×D₁₇₂

yes

characteristic

125

Q₈

—

344

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

344

—

130

C₂×C₄

C₂×Dic₈₆

yes

—

131

C₂×C₄

C₂×Dic₈₆

yes

—

132

C₂×C₄

C₂×D₁₇₂

yes

—

133

C₂×C₄

C₂×D₁₇₂

yes

characteristic

134

C₂×C₄

C₂³×D₄₃

yes

characteristic

135

C₈

C₂×Dic₈₆

yes

—

136

C₈

C₂×Dic₈₆

yes

—

137

C₈

C₂×Dic₈₆

yes

—

138

C₂²

—

139

C₂²

#13063325

yes

—

140

C₂²

C₂²×D₁₇₂

yes

characteristic

141

C₄

—

142

C₄

—

172

—

143

C₄

—

172

—

144

C₄

—

172

—

145

C₄

#13063325

yes

—

146

C₄

C₂²×Dic₈₆

yes

characteristic

147

C₄

C₂²×Dic₈₆

yes

characteristic

148

C₂

—

149

C₂

C₂×#13063325

yes

characteristic

150

C₂

#14583711

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#405	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#405	D₄₃
hypocenter	C₄₃	#3163
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

subgroup

G/H

center

C₂²

C₂²×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₄₃×#405

C₂

socle

C₂²×C₄₃

C₂²×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405

N	H ≅ G/N	type
2. #14583308	C₂	semidirect
3. #14583301	C₂	semidirect
4. #14583297	C₂	semidirect
31. M₁₆×C₄₃	Q₈	semidirect
32. C₂×C₈×C₄₃	Q₈	semidirect
84. C₄₃	#3163	semidirect
107. M₁₆	Dic₈₆	semidirect
108. C₂×C₈	Dic₈₆	semidirect

H ≅ G/N

type

2. #14583308

C₂

semidirect

3. #14583301

C₂

semidirect

4. #14583297

C₂

semidirect

31. M₁₆×C₄₃

Q₈

semidirect

32. C₂×C₈×C₄₃

Q₈

semidirect

84. C₄₃

#3163

semidirect

107. M₁₆

Dic₈₆

semidirect

108. C₂×C₈

Dic₈₆

semidirect

Subgroups — #19145126

Conjugacy classes of subgroups

Complements