Group #19137318 — subgroups

Subgroups — #19137318

Order 5504.

740 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 197 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×#13063322	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063287	C₂	1	2	yes	maximal, complemented
3	#14582800	C₂	1	2	yes	maximal, complemented
4	#14582777	C₂	1	8	yes	maximal, complemented
5	#14582745	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₁₇₂	C₂²	1	2	yes	—
8	C₂²×D₁₇₂	C₂²	1	4	yes	—
9	C₂²×C₄×D₄₃	C₂²	1	4	yes	—
10	#13063322	C₄	1	8	yes	complemented
11	#13063287	—	2	8	no	—
12	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	4	yes	—
13	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	8	yes	—
14	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
16	C₄₃⋊#95	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
19	#95×C₄₃	C₂²	1	4	yes	—
20	C₂³×D₄₃	C₂³	1	4	yes	—
21	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
22	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
23	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₄₃	C₂³	1	2	yes	—
25	C₂²×Dic₄₃	C₂³	1	2	yes	—
26	C₂×D₁₇₂	—	2	4	no	—
27	C₂×D₁₇₂	—	2	4	no	—
28	C₂×D₁₇₂	—	4	8	no	—
29	C₂×D₁₇₂	C₂×C₄	1	8	yes	complemented
30	C₂×D₁₇₂	C₂×C₄	1	16	yes	complemented
31	C₂×C₄×D₄₃	—	2	8	no	—
32	C₂×C₄×D₄₃	—	4	8	no	—
33	C₄⋊C₄×C₄₃	—	2	4	no	—
34	C₄²×C₄₃	C₂×C₄	1	4	yes	complemented
35	(C₂×D₄₃)⋊C₄	—	2	8	no	—
36	(C₂×D₄₃)⋊C₄	—	4	8	no	—
37	C₄×Dic₄₃	—	2	2	no	—
38	C₄×Dic₄₃	D₄	1	4	yes	complemented
39	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
40	C₂²×D₄₃	—	2	4	no	—
41	C₂²×D₄₃	—	2	4	no	—
42	C₂²×D₄₃	—	2	8	no	—
43	C₂²×D₄₃	—	2	8	no	—
44	C₂²×D₄₃	C₂²×C₄	1	8	yes	—
45	C₂×C₄×C₄₃	—	2	4	no	—
46	C₂×C₄×C₄₃	—	2	4	no	—
47	C₂×C₄×C₄₃	—	2	8	no	—
48	C₂×C₄×C₄₃	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₄₃	#41	1	2	yes	—
52	C₂×Dic₄₃	—	2	4	no	—
53	C₂×Dic₄₃	—	2	2	no	—
54	C₂×Dic₄₃	—	2	4	no	—
55	C₂×Dic₄₃	C₂×D₄	1	4	yes	—
56	D₁₇₂	—	2	16	no	—
57	D₁₇₂	—	4	16	no	—
58	C₄×D₄₃	—	4	16	no	—
59	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
60	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
61	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
62	C₂²×C₄₃	H(5,𝔽₂)	1	4	yes	—
63	C₂×D₄₃	—	2	8	no	—
64	C₂×D₄₃	—	2	16	no	—
65	C₂×D₄₃	—	4	4	no	—
66	C₂×D₄₃	—	4	4	no	—
67	C₂×D₄₃	—	4	8	no	—
68	C₄×C₄₃	—	2	4	no	—
69	C₄×C₄₃	—	4	4	no	—
70	C₄×C₄₃	C₄×D₄	1	4	yes	—
71	Dic₄₃	—	2	4	no	—
72	Dic₄₃	—	4	2	no	—
73	#2328	—	43	1	no	maximal, Sylow
74	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
75	C₂×C₄₃	#518	1	2	yes	—
76	C₂×C₄₃	#535	1	2	yes	—
77	C₂×C₄₃	#540	1	2	yes	—
78	D₄₃	—	4	8	no	—
79	C₂×#127	—	43	1	no	—
80	C₂×C₄×D₄	—	43	2	no	—
81	#390	—	43	2	no	—
82	#386	—	43	8	no	—
83	#382	D₄₃	1	1	yes	characteristic, complemented
84	#376	—	43	1	no	—
85	C₄₃	#2328	1	1	yes	characteristic, complemented, Sylow
86	C₂²×D₄	—	43	2	no	—
87	C₂²×D₄	—	43	4	no	—
88	C₂³×C₄	—	43	4	no	—
89	#127	—	43	8	no	—
90	C₄×D₄	—	86	8	no	—
91	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
92	C₂×C₂²⋊C₄	—	43	4	no	—
93	C₂×C₂²⋊C₄	—	43	8	no	—
94	C₂×C₄²	—	43	1	no	—
95	C₂×C₄²	—	43	1	no	—
96	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
97	#95	—	43	4	no	—
98	#95	C₂×D₄₃	1	4	yes	—
99	C₂⁴	—	43	4	no	—
100	C₂×D₄	—	43	8	no	—
101	C₂×D₄	—	43	16	no	—
102	C₂×D₄	—	86	4	no	—
103	C₂×D₄	—	86	4	no	—
104	C₂×D₄	—	172	8	no	—
105	C₂²×C₄	—	43	2	no	—
106	C₂²×C₄	—	43	2	no	—
107	C₂²×C₄	—	86	8	no	—
108	C₂²×C₄	—	172	8	no	—
109	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
110	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
112	C₄⋊C₄	—	2	4	no	—
113	C₂²⋊C₄	—	86	8	no	—
114	C₂²⋊C₄	—	172	8	no	—
115	C₄²	—	43	4	no	—
116	C₄²	—	86	2	no	—
117	C₄²	C₄×D₄₃	1	4	yes	complemented
118	C₂³	—	43	8	no	—
119	C₂³	—	86	4	no	—
120	C₂³	—	86	4	no	—
121	C₂³	—	86	8	no	—
122	C₂³	—	86	8	no	—
123	C₂³	C₂³×D₄₃	1	1	yes	characteristic
124	D₄	—	86	16	no	—
125	D₄	—	172	16	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	8	no	—
129	C₂×C₄	—	43	4	no	—
130	C₂×C₄	—	86	4	no	—
131	C₂×C₄	—	86	2	no	—
132	C₂×C₄	—	86	4	no	—
133	C₂×C₄	—	172	16	no	—
134	C₂×C₄	C₂×C₄×D₄₃	1	4	yes	—
135	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
136	C₂×C₄	D₄×D₄₃	1	2	yes	—
137	C₂×C₄	#10621902	1	2	yes	—
138	C₂²	—	86	8	no	—
139	C₂²	—	86	16	no	—
140	C₂²	—	172	4	no	—
141	C₂²	—	172	4	no	—
142	C₂²	—	172	8	no	—
143	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
144	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
145	C₂²	C₂×#10621902	1	1	yes	characteristic
146	C₂²	#13063371	1	4	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	4	4	no	—
149	C₄	—	86	4	no	—
150	C₄	—	172	2	no	—
151	C₄	#13063287	1	4	yes	—
152	C₂	—	172	8	no	—
153	C₂	C₂×#13063287	1	1	yes	characteristic
154	C₂	#14583373	1	2	yes	—
155	C₂	#14583467	1	2	yes	—
156	C₂	#14583505	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063322

C₂

yes

maximal, characteristic

C₂×#13063287

C₂

yes

maximal, complemented

#14582800

C₂

yes

maximal, complemented

#14582777

C₂

yes

maximal, complemented

#14582745

C₂

yes

maximal, characteristic, complemented

C₄₃×#382

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₇₂

C₂²

yes

—

C₂²×D₁₇₂

C₂²

yes

—

C₂²×C₄×D₄₃

C₂²

yes

—

#13063322

C₄

yes

complemented

#13063287

—

C₂×(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

characteristic

C₂×C₄×Dic₄₃

C₂²

yes

characteristic

C₄₃⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#95×C₄₃

C₂²

yes

—

C₂³×D₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂×C₄

yes

complemented

C₂×D₁₇₂

C₂×C₄

yes

complemented

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

complemented

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

D₄

yes

complemented

C₂³×C₄₃

C₂⁴

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×D₄

yes

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₂²×C₄₃

C₂³×C₄

yes

characteristic

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂×#41

yes

characteristic

C₂²×C₄₃

H(5,𝔽₂)

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₄×D₄

yes

—

Dic₄₃

—

Dic₄₃

—

#2328

—

maximal, Sylow

C₂×C₄₃

C₂×C₄×D₄

yes

characteristic

C₂×C₄₃

#518

yes

—

C₂×C₄₃

#535

yes

—

C₂×C₄₃

#540

yes

—

D₄₃

—

C₂×#127

—

C₂×C₄×D₄

—

#390

—

#386

—

#382

D₄₃

yes

characteristic, complemented

#376

—

C₄₃

#2328

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂³×C₄

—

#127

—

C₄×D₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

#95

—

#95

C₂×D₄₃

yes

—

C₂⁴

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

172

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

172

—

109

C₂²×C₄

C₂²×D₄₃

yes

—

110

C₂²×C₄

C₂²×D₄₃

yes

—

111

C₂²×C₄

C₂²×D₄₃

yes

characteristic

112

C₄⋊C₄

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

172

—

115

C₄²

—

116

C₄²

—

117

C₄²

C₄×D₄₃

yes

complemented

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

C₂³×D₄₃

yes

characteristic

124

D₄

—

125

D₄

—

172

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

172

—

134

C₂×C₄

C₂×C₄×D₄₃

yes

—

135

C₂×C₄

C₂×C₄×D₄₃

yes

—

136

C₂×C₄

D₄×D₄₃

yes

—

137

C₂×C₄

#10621902

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

172

—

141

C₂²

—

172

—

142

C₂²

—

172

—

143

C₂²

C₂²×C₄×D₄₃

yes

characteristic

144

C₂²

C₂×D₄×D₄₃

yes

characteristic

145

C₂²

C₂×#10621902

yes

characteristic

146

C₂²

#13063371

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

172

—

151

C₄

#13063287

yes

—

152

C₂

—

172

—

153

C₂

C₂×#13063287

yes

characteristic

154

C₂

#14583373

yes

—

155

C₂

#14583467

yes

—

156

C₂

#14583505

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₄
Fitting	C₄₃×#382	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	#382	D₄₃
hypocenter	C₄₃	#2328
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator

C₂²×C₄₃

C₂³×C₄

Fitting

C₄₃×#382

C₂

socle

C₂³×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
2. C₂×#13063287	C₂	semidirect
3. #14582800	C₂	semidirect
4. #14582777	C₂	semidirect
5. #14582745	C₂	semidirect
6. C₄₃×#382	C₂	semidirect
10. #13063322	C₄	semidirect
29. C₂×D₁₇₂	C₂×C₄	semidirect
30. C₂×D₁₇₂	C₂×C₄	semidirect
34. C₄²×C₄₃	C₂×C₄	semidirect
38. C₄×Dic₄₃	D₄	semidirect
83. #382	D₄₃	semidirect
85. C₄₃	#2328	semidirect
117. C₄²	C₄×D₄₃	semidirect

H ≅ G/N

type

2. C₂×#13063287

C₂

semidirect

3. #14582800

C₂

semidirect

4. #14582777

C₂

semidirect

5. #14582745

C₂

semidirect

6. C₄₃×#382

C₂

semidirect

10. #13063322

C₄

semidirect

29. C₂×D₁₇₂

C₂×C₄

semidirect

30. C₂×D₁₇₂

C₂×C₄

semidirect

34. C₄²×C₄₃

C₂×C₄

semidirect

38. C₄×Dic₄₃

D₄

semidirect

83. #382

D₄₃

semidirect

85. C₄₃

#2328

semidirect

117. C₄²

C₄×D₄₃

semidirect

Subgroups — #19137318

Conjugacy classes of subgroups

Complements