Group #19137079 — subgroups

Subgroups — #19137079

Order 5504.

656 conjugacy classes of subgroups; only the 317 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 317 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×#13063284	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	#14582798	C₂	1	4	yes	maximal, complemented
4	#14582774	C₂	1	4	yes	maximal
5	C₄×C₄⋊Dic₄₃	C₂	1	4	yes	maximal, complemented
6	C₂×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₄₃	C₂²	1	2	yes	—
8	#13063284	—	2	4	no	—
9	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₄₃	C₂²	1	8	yes	—
11	C₂×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
12	C₂×C₄×Dic₄₃	—	2	4	no	—
13	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
14	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
15	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
16	C₄₃⋊#95	C₂²	1	8	yes	—
17	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
18	#117×C₄₃	C₄	1	8	yes	complemented
19	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₄₃	C₂²	1	2	yes	—
22	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₄₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₄₃	D₄	1	4	yes	—
25	C₂²×C₄×C₄₃	Q₈	1	4	yes	—
26	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
27	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₄₃	—	2	8	no	—
30	C₂²×Dic₄₃	—	2	4	no	—
31	C₂²×Dic₄₃	C₂³	1	4	yes	—
32	C₂²×Dic₄₃	C₂³	1	4	yes	—
33	C₄⋊C₄×C₄₃	C₂×C₄	1	8	yes	complemented
34	C₂²⋊C₄×C₄₃	C₂×C₄	1	8	yes	—
35	C₄²×C₄₃	C₂×C₄	1	8	yes	complemented
36	C₂²⋊Dic₄₃	—	2	4	no	—
37	C₄⋊Dic₄₃	—	2	4	no	—
38	C₄⋊Dic₄₃	—	2	16	no	—
39	C₄×Dic₄₃	—	2	4	no	—
40	C₄×Dic₄₃	—	2	4	no	—
41	C₄×Dic₄₃	—	2	4	no	—
42	C₂³×C₄₃	—	2	4	no	—
43	C₂³×C₄₃	C₂²×C₄	1	2	yes	—
44	C₂³×C₄₃	C₂×D₄	1	2	yes	—
45	C₂³×C₄₃	C₂×Q₈	1	2	yes	—
46	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
47	C₂×C₄×C₄₃	—	2	8	no	—
48	C₂×C₄×C₄₃	C₄⋊C₄	1	16	yes	complemented
49	C₂×C₄×C₄₃	C₂²×C₄	1	8	yes	—
50	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
56	C₂×Dic₄₃	—	2	16	no	—
57	C₂×Dic₄₃	—	2	4	no	—
58	C₂×Dic₄₃	—	2	4	no	—
59	C₂×Dic₄₃	—	2	4	no	—
60	C₂×Dic₄₃	—	2	8	no	—
61	C₂×Dic₄₃	#41	1	8	yes	—
62	C₂²×C₄₃	—	2	4	no	—
63	C₂²×C₄₃	—	2	4	no	—
64	C₂²×C₄₃	—	2	4	no	—
65	C₂²×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
66	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
67	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
68	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
69	C₂²×C₄₃	C₂×#41	1	4	yes	—
70	C₄×C₄₃	—	2	8	no	—
71	C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
72	C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
73	Dic₄₃	—	2	8	no	—
74	Dic₄₃	—	4	4	no	—
75	#2296	—	43	1	no	maximal, Sylow
76	C₂×C₄₃	—	2	4	no	—
77	C₂×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄₃	C₄×#41	1	2	yes	—
79	C₂×C₄₃	#525	1	2	yes	—
80	C₂×C₄₃	#533	1	2	yes	—
81	C₂×#117	—	43	1	no	—
82	C₂×#117	D₄₃	1	1	yes	characteristic
83	C₂²×C₄²	—	43	1	no	—
84	#389	—	43	4	no	—
85	#385	—	43	4	no	—
86	C₄×C₄⋊C₄	—	43	4	no	—
87	C₄₃	#2296	1	1	yes	characteristic, complemented, Sylow
88	C₂³×C₄	—	43	2	no	—
89	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
90	#117	—	86	4	no	—
91	#117	Dic₄₃	1	8	yes	complemented
92	C₂×C₄⋊C₄	—	43	2	no	—
93	C₂×C₄⋊C₄	—	43	8	no	—
94	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
95	C₂×C₂²⋊C₄	—	43	2	no	—
96	C₂×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
97	C₂×C₄²	—	43	2	no	—
98	C₂×C₄²	—	43	2	no	—
99	C₂×C₄²	—	43	2	no	—
100	C₂×C₄²	—	86	4	no	—
101	C₂×C₄²	C₂×D₄₃	1	2	yes	—
102	#95	—	43	8	no	—
103	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
104	C₂²×C₄	—	43	4	no	—
105	C₂²×C₄	—	43	4	no	—
106	C₂²×C₄	—	86	8	no	—
107	C₂²×C₄	—	86	4	no	—
108	C₂²×C₄	Dic₈₆	1	4	yes	—
109	C₂²×C₄	D₁₇₂	1	4	yes	—
110	C₂²×C₄	C₂×Dic₄₃	1	4	yes	—
111	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
112	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
114	C₄⋊C₄	—	86	16	no	—
115	C₄⋊C₄	—	86	4	no	—
116	C₄⋊C₄	C₂×Dic₄₃	1	8	yes	complemented
117	C₂²⋊C₄	—	86	4	no	—
118	C₂²⋊C₄	C₂×Dic₄₃	1	8	yes	—
119	C₄²	—	86	4	no	—
120	C₄²	—	86	4	no	—
121	C₄²	—	86	4	no	—
122	C₄²	C₂×Dic₄₃	1	8	yes	complemented
123	C₂³	—	2	4	no	—
124	C₂³	C₂×Dic₈₆	1	2	yes	—
125	C₂³	C₂×D₁₇₂	1	2	yes	—
126	C₂³	C₂²×Dic₄₃	1	2	yes	—
127	C₂³	C₂³×D₄₃	1	1	yes	characteristic
128	C₂×C₄	—	2	8	no	—
129	C₂×C₄	—	43	8	no	—
130	C₂×C₄	—	86	16	no	—
131	C₂×C₄	—	86	4	no	—
132	C₂×C₄	—	86	4	no	—
133	C₂×C₄	—	86	4	no	—
134	C₂×C₄	—	86	8	no	—
135	C₂×C₄	C₄⋊Dic₄₃	1	16	yes	complemented
136	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
137	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
138	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
139	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
140	C₂×C₄	C₂²×Dic₄₃	1	8	yes	—
141	C₂×C₄	C₂²×Dic₄₃	1	2	yes	—
142	C₂×C₄	C₂²×Dic₄₃	1	2	yes	—
143	C₂²	—	2	4	no	—
144	C₂²	—	2	4	no	—
145	C₂²	—	2	4	no	—
146	C₂²	C₂×C₄⋊Dic₄₃	1	4	yes	—
147	C₂²	C₂³×Dic₄₃	1	1	yes	characteristic
148	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
149	C₂²	C₂²×Dic₈₆	1	1	yes	characteristic
150	C₂²	#41×D₄₃	1	4	yes	—
151	C₄	—	2	8	no	—
152	C₄	—	86	8	no	—
153	C₄	—	172	4	no	—
154	C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
155	C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
156	C₂	—	2	4	no	—
157	C₂	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
158	C₂	#41×Dic₄₃	1	2	yes	—
159	C₂	#14583411	1	2	yes	—
160	C₂	#14583452	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063284

C₂

yes

maximal, characteristic

C₂²×C₄×Dic₄₃

C₂

yes

maximal, characteristic

#14582798

C₂

yes

maximal, complemented

#14582774

C₂

yes

maximal

C₄×C₄⋊Dic₄₃

C₂

yes

maximal, complemented

C₂×#117×C₄₃

C₂

yes

maximal, characteristic

C₂³×Dic₄₃

C₂²

yes

—

#13063284

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

C₂²

yes

—

C₂×C₄×Dic₄₃

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₂³×C₄×C₄₃

C₂²

yes

characteristic

#117×C₄₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

—

C₂⁴×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₄⋊C₄×C₄₃

C₂×C₄

yes

complemented

C₂²⋊C₄×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

C₂×C₄

yes

complemented

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂²×C₄

yes

—

C₂³×C₄₃

C₂×D₄

yes

—

C₂³×C₄₃

C₂×Q₈

yes

—

C₂³×C₄₃

C₂⁴

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄⋊C₄

yes

complemented

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

#41

yes

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₄⋊C₄

yes

—

C₂²×C₄₃

C₂³×C₄

yes

characteristic

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂²×Q₈

yes

characteristic

C₂²×C₄₃

C₂×#41

yes

—

C₄×C₄₃

—

C₄×C₄₃

C₂×C₄⋊C₄

yes

—

C₄×C₄₃

C₂×C₄⋊C₄

yes

—

Dic₄₃

—

Dic₄₃

—

#2296

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₄₃

C₄×#41

yes

—

C₂×C₄₃

#525

yes

—

C₂×C₄₃

#533

yes

—

C₂×#117

—

C₂×#117

D₄₃

yes

characteristic

C₂²×C₄²

—

#389

—

#385

—

C₄×C₄⋊C₄

—

C₄₃

#2296

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

C₂×D₄₃

yes

characteristic

#117

—

#117

Dic₄₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₄₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

C₂×D₄₃

yes

—

102

#95

—

103

C₂⁴

C₂²×D₄₃

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

Dic₈₆

yes

—

109

C₂²×C₄

D₁₇₂

yes

—

110

C₂²×C₄

C₂×Dic₄₃

yes

—

111

C₂²×C₄

C₂²×D₄₃

yes

—

112

C₂²×C₄

C₂²×D₄₃

yes

characteristic

113

C₂²×C₄

C₂²×D₄₃

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

C₂×Dic₄₃

yes

complemented

117

C₂²⋊C₄

—

118

C₂²⋊C₄

C₂×Dic₄₃

yes

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₂×Dic₄₃

yes

complemented

123

C₂³

—

124

C₂³

C₂×Dic₈₆

yes

—

125

C₂³

C₂×D₁₇₂

yes

—

126

C₂³

C₂²×Dic₄₃

yes

—

127

C₂³

C₂³×D₄₃

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₄⋊Dic₄₃

yes

complemented

136

C₂×C₄

C₂×Dic₈₆

yes

—

137

C₂×C₄

C₂×Dic₈₆

yes

—

138

C₂×C₄

C₂×D₁₇₂

yes

—

139

C₂×C₄

C₂×D₁₇₂

yes

—

140

C₂×C₄

C₂²×Dic₄₃

yes

—

141

C₂×C₄

C₂²×Dic₄₃

yes

—

142

C₂×C₄

C₂²×Dic₄₃

yes

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×C₄⋊Dic₄₃

yes

—

147

C₂²

C₂³×Dic₄₃

yes

characteristic

148

C₂²

C₂²×D₁₇₂

yes

characteristic

149

C₂²

C₂²×Dic₈₆

yes

characteristic

150

C₂²

#41×D₄₃

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

172

—

154

C₄

C₂×C₄⋊Dic₄₃

yes

—

155

C₄

C₂×C₄⋊Dic₄₃

yes

—

156

C₂

—

157

C₂

C₂²×C₄⋊Dic₄₃

yes

characteristic

158

C₂

#41×Dic₄₃

yes

—

159

C₂

#14583411

yes

—

160

C₂

#14583452

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂³×C₄
Frattini	C₂³	C₂³×D₄₃
Fitting	C₂×#117×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₂×#117	D₄₃
hypocenter	C₄₃	#2296
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

subgroup

G/H

center

C₂²×C₄

C₂²×D₄₃

commutator

C₂²×C₄₃

C₂³×C₄

Frattini

C₂³

C₂³×D₄₃

Fitting

C₂×#117×C₄₃

C₂

socle

C₂³×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂²×C₄ ◁ C₂×#117

N	H ≅ G/N	type
3. #14582798	C₂	semidirect
5. C₄×C₄⋊Dic₄₃	C₂	semidirect
18. #117×C₄₃	C₄	semidirect
33. C₄⋊C₄×C₄₃	C₂×C₄	semidirect
35. C₄²×C₄₃	C₂×C₄	semidirect
48. C₂×C₄×C₄₃	C₄⋊C₄	semidirect
87. C₄₃	#2296	semidirect
91. #117	Dic₄₃	semidirect
116. C₄⋊C₄	C₂×Dic₄₃	semidirect
122. C₄²	C₂×Dic₄₃	semidirect
135. C₂×C₄	C₄⋊Dic₄₃	semidirect

H ≅ G/N

type

3. #14582798

C₂

semidirect

5. C₄×C₄⋊Dic₄₃

C₂

semidirect

18. #117×C₄₃

C₄

semidirect

33. C₄⋊C₄×C₄₃

C₂×C₄

semidirect

35. C₄²×C₄₃

C₂×C₄

semidirect

48. C₂×C₄×C₄₃

C₄⋊C₄

semidirect

87. C₄₃

#2296

semidirect

91. #117

Dic₄₃

semidirect

116. C₄⋊C₄

C₂×Dic₄₃

semidirect

122. C₄²

C₂×Dic₄₃

semidirect

135. C₂×C₄

C₄⋊Dic₄₃

semidirect

Subgroups — #19137079

Conjugacy classes of subgroups

Complements