Group #19145181 — subgroups

Subgroups — #19145181

Order 5504.

368 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 149 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	#14583456	C₂	1	2	yes	maximal
2	#14583304	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583279	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583250	C₂	1	2	yes	maximal, complemented
5	#14583140	C₂	1	4	yes	maximal
6	C₄×Dic₁₇₂	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₇₂	C₂²	1	4	yes	—
9	#13063325	C₂²	1	2	yes	—
10	#13063325	C₂²	1	2	yes	—
11	#13063317	—	2	2	no	—
12	#13063304	—	2	2	no	—
13	#13063304	C₂²	1	4	yes	—
14	C₄×Dic₈₆	—	2	2	no	—
15	C₄×Dic₈₆	C₂²	1	4	yes	—
16	#13063284	—	2	2	no	—
17	#13063259	C₂²	1	4	yes	—
18	#13063250	C₂²	1	8	yes	—
19	C₂²×C₈×C₄₃	C₂²	1	2	yes	—
20	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄×C₈×C₄₃	C₂²	1	2	yes	—
24	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₈₆	—	2	2	no	—
28	C₂×Dic₈₆	C₂³	1	4	yes	—
29	C₂×C₈×C₄₃	—	2	4	no	—
30	C₂×C₈×C₄₃	C₂³	1	2	yes	—
31	C₂×C₈×C₄₃	C₂³	1	2	yes	—
32	C₄²×C₄₃	D₄	1	1	yes	characteristic
33	C₄²×C₄₃	D₄	1	1	yes	characteristic
34	C₄²×C₄₃	C₂³	1	1	yes	characteristic
35	C₄²×C₄₃	C₂³	1	1	yes	characteristic
36	C₂²⋊Dic₄₃	—	2	4	no	—
37	C₄⋊Dic₄₃	—	2	2	no	—
38	C₄⋊Dic₄₃	—	2	2	no	—
39	C₄⋊Dic₄₃	—	2	2	no	—
40	C₄⋊Dic₄₃	C₂³	1	4	yes	—
41	Dic₄₃⋊C₄	—	2	4	no	—
42	Dic₄₃⋊C₄	—	2	4	no	—
43	C₄×Dic₄₃	—	2	4	no	—
44	Dic₁₇₂	—	2	8	no	—
45	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₃	—	2	2	no	—
47	C₂×C₄×C₄₃	—	2	2	no	—
48	C₂×C₄×C₄₃	Q₁₆	1	4	yes	complemented
49	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₄₃	—	2	4	no	—
56	C₂×Dic₄₃	—	2	4	no	—
57	Dic₈₆	—	2	8	no	—
58	Dic₈₆	—	4	4	no	—
59	C₈×C₄₃	—	2	2	no	—
60	C₈×C₄₃	#41	1	4	yes	—
61	C₂²×C₄₃	—	2	1	no	—
62	C₂²×C₄₃	—	2	1	no	—
63	C₂²×C₄₃	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₄₃	—	2	1	no	—
66	C₄×C₄₃	—	2	1	no	—
67	C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
68	C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
69	C₄×C₄₃	#135	1	2	yes	—
70	C₄×C₄₃	C₂×#41	1	2	yes	—
71	Dic₄₃	—	4	4	no	—
72	Dic₄₃	—	4	4	no	—
73	#3170	—	43	1	no	maximal, Sylow
74	C₂×C₄₃	—	2	2	no	—
75	C₂×C₄₃	#532	1	1	yes	characteristic
76	C₂×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
77	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
78	#533	—	43	2	no	—
79	#500	—	43	1	no	—
80	#494	—	43	1	no	—
81	#487	—	43	2	no	—
82	#467	—	43	4	no	—
83	C₄×Q₁₆	—	43	4	no	—
84	C₂×C₄×C₈	D₄₃	1	1	yes	characteristic
85	C₄₃	#3170	1	1	yes	characteristic, complemented, Sylow
86	C₂×Q₁₆	—	43	4	no	—
87	C₂²×C₈	C₂×D₄₃	1	2	yes	—
88	C₄⋊Q₈	—	43	2	no	—
89	C₄⋊Q₈	—	43	2	no	—
90	#125	—	86	2	no	—
91	C₂²⋊Q₈	—	43	4	no	—
92	C₂²⋊Q₈	—	86	2	no	—
93	C₄×Q₈	—	43	4	no	—
94	C₄×Q₈	—	86	2	no	—
95	#117	—	86	2	no	—
96	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
97	#107	—	43	4	no	—
98	#103	—	43	8	no	—
99	C₄×C₈	C₂×D₄₃	1	2	yes	—
100	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
101	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
102	C₂×Q₈	—	43	4	no	—
103	C₂×Q₈	—	86	2	no	—
104	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
105	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
107	Q₁₆	—	86	8	no	—
108	C₂×C₈	—	2	4	no	—
109	C₂×C₈	C₂²×D₄₃	1	2	yes	—
110	C₂×C₈	C₂²×D₄₃	1	2	yes	—
111	C₄⋊C₄	—	43	4	no	—
112	C₄⋊C₄	—	86	2	no	—
113	C₄⋊C₄	—	86	2	no	—
114	C₄⋊C₄	—	86	4	no	—
115	C₄⋊C₄	—	86	4	no	—
116	C₄⋊C₄	—	86	2	no	—
117	C₂²⋊C₄	—	86	4	no	—
118	C₄²	—	86	4	no	—
119	C₄²	D₁₇₂	1	1	yes	characteristic
120	C₄²	D₁₇₂	1	1	yes	characteristic
121	C₄²	C₂²×D₄₃	1	1	yes	characteristic
122	C₄²	C₂²×D₄₃	1	1	yes	characteristic
123	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
124	Q₈	—	86	8	no	—
125	Q₈	—	172	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	86	4	no	—
129	C₂×C₄	—	86	4	no	—
130	C₂×C₄	Dic₁₇₂	1	4	yes	complemented
131	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
132	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
133	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
134	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
135	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
136	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	#10621898	1	4	yes	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₂×Dic₁₇₂	1	2	yes	—
142	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	172	4	no	—
146	C₄	—	172	4	no	—
147	C₄	C₂×Dic₁₇₂	1	2	yes	—
148	C₄	C₂×Dic₁₇₂	1	2	yes	—
149	C₄	#13063347	1	2	yes	—
150	C₄	C₂×#10621898	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	#14583449	1	1	yes	characteristic
153	C₂	C₂²×Dic₁₇₂	1	1	yes	characteristic
154	C₂	C₂×#13063347	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583456

C₂

yes

maximal

#14583304

C₂

yes

maximal, characteristic, complemented

#14583279

C₂

yes

maximal, characteristic, complemented

#14583250

C₂

yes

maximal, complemented

#14583140

C₂

yes

maximal

C₄×Dic₁₇₂

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₄₃

C₂

yes

maximal, characteristic

C₂×Dic₁₇₂

C₂²

yes

—

#13063325

C₂²

yes

—

#13063325

C₂²

yes

—

#13063317

—

#13063304

—

#13063304

C₂²

yes

—

C₄×Dic₈₆

—

C₄×Dic₈₆

C₂²

yes

—

#13063284

—

#13063259

C₂²

yes

—

#13063250

C₂²

yes

—

C₂²×C₈×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄×C₈×C₄₃

C₂²

yes

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂³

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂³

yes

—

C₂×C₈×C₄₃

C₂³

yes

—

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂³

yes

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

Dic₁₇₂

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

Q₁₆

yes

complemented

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

#41

yes

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×Q₁₆

yes

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×Q₁₆

yes

—

C₄×C₄₃

C₂×Q₁₆

yes

—

C₄×C₄₃

#135

yes

—

C₄×C₄₃

C₂×#41

yes

—

Dic₄₃

—

Dic₄₃

—

#3170

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

#532

yes

characteristic

C₂×C₄₃

C₂²×Q₁₆

yes

C₂×C₄₃

yes

—

—

—

—

—

C₄×Q₁₆

—

C₂×C₄×C₈

D₄₃

yes

characteristic

C₄₃

#3170

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂²×C₈

C₂×D₄₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

#107

—

#103

—

C₄×C₈

C₂×D₄₃

yes

—

100

C₄×C₈

C₂×D₄₃

yes

characteristic

101

C₄×C₈

C₂×D₄₃

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂²×C₄

D₁₇₂

yes

characteristic

105

C₂²×C₄

D₁₇₂

yes

characteristic

106

C₂²×C₄

C₂²×D₄₃

yes

characteristic

107

Q₁₆

—

108

C₂×C₈

—

109

C₂×C₈

C₂²×D₄₃

yes

—

110

C₂×C₈

C₂²×D₄₃

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

D₁₇₂

yes

characteristic

120

C₄²

D₁₇₂

yes

characteristic

121

C₄²

C₂²×D₄₃

yes

characteristic

122

C₄²

C₂²×D₄₃

yes

characteristic

123

C₂³

C₂×D₁₇₂

yes

characteristic

124

Q₈

—

125

Q₈

—

172

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

Dic₁₇₂

yes

complemented

131

C₂×C₄

C₂×D₁₇₂

yes

characteristic

132

C₂×C₄

C₂×D₁₇₂

yes

characteristic

133

C₂×C₄

C₂×D₁₇₂

yes

characteristic

134

C₂×C₄

C₂×D₁₇₂

yes

characteristic

135

C₂×C₄

C₂×D₁₇₂

yes

characteristic

136

C₂×C₄

C₂³×D₄₃

yes

characteristic

137

C₈

—

138

C₈

#10621898

yes

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×Dic₁₇₂

yes

—

142

C₂²

C₂²×D₁₇₂

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

172

—

146

C₄

—

172

—

147

C₄

C₂×Dic₁₇₂

yes

—

148

C₄

C₂×Dic₁₇₂

yes

—

149

C₄

#13063347

yes

—

150

C₄

C₂×#10621898

yes

—

151

C₂

—

152

C₂

#14583449

yes

characteristic

153

C₂

C₂²×Dic₁₇₂

yes

characteristic

154

C₂

C₂×#13063347

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₂×C₄×C₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₂×C₄×C₈	D₄₃
hypocenter	C₄₃	#3170
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₂×C₄×C₈×C₄₃

C₂

socle

C₂²×C₄₃

C₂²×D₄

hypercenter

C₂×C₄×C₈

D₄₃

hypocenter

C₄₃

#3170

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #14583304	C₂	semidirect
3. #14583279	C₂	semidirect
4. #14583250	C₂	semidirect
6. C₄×Dic₁₇₂	C₂	semidirect
48. C₂×C₄×C₄₃	Q₁₆	semidirect
85. C₄₃	#3170	semidirect
130. C₂×C₄	Dic₁₇₂	semidirect

H ≅ G/N

type

2. #14583304

C₂

semidirect

3. #14583279

C₂

semidirect

4. #14583250

C₂

semidirect

6. C₄×Dic₁₇₂

C₂

semidirect

48. C₂×C₄×C₄₃

Q₁₆

semidirect

85. C₄₃

#3170

semidirect

130. C₂×C₄

Dic₁₇₂

semidirect

Subgroups — #19145181

Conjugacy classes of subgroups

Complements