Group #19135319 — subgroups

Subgroups — #19135319

Order 5504.

248 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 121 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×C₄⋊(C₄₃⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#13063236	C₂	1	1	yes	maximal, characteristic
4	#14582654	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
10	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
11	#13063236	—	2	2	no	—
12	C₂×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
13	C₂×C₄²×C₄₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
15	#95×C₄₃	C₄	1	1	yes	characteristic
16	#95×C₄₃	C₄	1	1	yes	characteristic
17	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₄₃	—	2	4	no	—
23	C₄²×C₄₃	—	2	2	no	—
24	C₄²×C₄₃	—	2	2	no	—
25	C₄²×C₄₃	D₄	1	2	yes	—
26	C₄²×C₄₃	Q₈	1	2	yes	—
27	C₂×C₄₃⋊C₈	—	2	1	no	—
28	C₂×C₄₃⋊C₈	—	2	1	no	—
29	C₂×C₄₃⋊C₈	—	2	1	no	—
30	C₂×C₄₃⋊C₈	—	2	1	no	—
31	C₂×C₄₃⋊C₈	—	2	2	no	—
32	C₂×C₄₃⋊C₈	—	2	2	no	—
33	C₂×C₄₃⋊C₈	—	2	2	no	—
34	C₂×C₄₃⋊C₈	D₄	1	2	yes	—
35	C₂×C₄₃⋊C₈	Q₈	1	2	yes	—
36	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₄₃	—	2	2	no	—
38	C₂×C₄×C₄₃	—	2	2	no	—
39	C₂×C₄×C₄₃	—	2	2	no	—
40	C₂×C₄×C₄₃	—	2	2	no	—
41	C₂×C₄×C₄₃	—	2	2	no	—
42	C₂×C₄×C₄₃	—	2	2	no	—
43	C₂×C₄×C₄₃	—	2	2	no	—
44	C₂×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₄₃	M₁₆	1	4	yes	—
47	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
52	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
53	C₄₃⋊C₈	—	2	2	no	—
54	C₄₃⋊C₈	—	2	2	no	—
55	C₄₃⋊C₈	—	4	2	no	—
56	C₂²×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄₃	C₄×D₄	1	1	yes	characteristic
58	C₂²×C₄₃	C₄×Q₈	1	1	yes	characteristic
59	C₂²×C₄₃	C₂×M₁₆	1	2	yes	—
60	C₂²×C₄₃	#131	1	2	yes	—
61	C₄×C₄₃	—	2	2	no	—
62	C₄×C₄₃	—	2	2	no	—
63	C₄×C₄₃	—	2	4	no	—
64	C₄×C₄₃	—	4	2	no	—
65	C₄×C₄₃	#121	1	1	yes	characteristic
66	C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
67	C₄×C₄₃	#125	1	1	yes	characteristic
68	C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
69	#1940	—	43	1	no	maximal, Sylow
70	C₂×C₄₃	#389	1	1	yes	characteristic
71	C₂×C₄₃	#423	1	1	yes	characteristic
72	C₂×C₄₃	#424	1	1	yes	characteristic
73	C₂×C₄₃	#435	1	2	yes	—
74	C₂×C₄₃	#446	1	2	yes	—
75	C₂×C₄⋊C₈	—	43	2	no	—
76	C₂×C₄⋊C₈	—	43	1	no	—
77	C₂×#97	—	43	1	no	—
78	C₄×C₄⋊C₄	D₄₃	1	1	yes	characteristic
79	#336	—	43	2	no	—
80	C₄₃	#1940	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	43	1	no	—
82	C₂²×C₈	—	43	1	no	—
83	C₂²×C₈	—	43	2	no	—
84	C₂×C₄⋊C₄	Dic₄₃	1	2	yes	—
85	C₂×C₄²	C₂×D₄₃	1	2	yes	—
86	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
87	C₄⋊C₈	—	86	4	no	—
88	C₄⋊C₈	—	86	2	no	—
89	#97	—	86	2	no	—
90	#95	Dic₄₃	1	1	yes	characteristic
91	#95	Dic₄₃	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₄₃	1	2	yes	—
93	C₂²×C₄	C₂×Dic₄₃	1	2	yes	—
94	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
95	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
97	C₂×C₈	—	43	2	no	—
98	C₂×C₈	—	43	2	no	—
99	C₂×C₈	—	86	1	no	—
100	C₂×C₈	—	86	1	no	—
101	C₂×C₈	—	86	1	no	—
102	C₂×C₈	—	86	1	no	—
103	C₂×C₈	—	86	2	no	—
104	C₂×C₈	—	86	2	no	—
105	C₂×C₈	—	86	2	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	Dic₈₆	1	2	yes	—
110	C₄²	D₁₇₂	1	2	yes	—
111	C₂³	C₂²×Dic₄₃	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	#10621877	1	4	yes	—
120	C₂×C₄	C₄⋊Dic₄₃	1	2	yes	—
121	C₂×C₄	C₄⋊Dic₄₃	1	2	yes	—
122	C₂×C₄	C₂×Dic₈₆	1	1	yes	characteristic
123	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
124	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
125	C₂×C₄	#10621900	1	1	yes	characteristic
126	C₂×C₄	Q₈×D₄₃	1	1	yes	characteristic
127	C₂×C₄	#10621902	1	1	yes	characteristic
128	C₈	—	86	2	no	—
129	C₈	—	86	2	no	—
130	C₈	—	172	2	no	—
131	C₂²	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
132	C₂²	D₄×Dic₄₃	1	1	yes	characteristic
133	C₂²	Q₈×Dic₄₃	1	1	yes	characteristic
134	C₂²	C₂×#10621877	1	2	yes	—
135	C₂²	#13063332	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	4	no	—
139	C₄	—	4	2	no	—
140	C₄	#13063300	1	1	yes	characteristic
141	C₄	#13063305	1	1	yes	characteristic
142	C₄	#13063316	1	1	yes	characteristic
143	C₄	#13063324	1	1	yes	characteristic
144	C₂	#14582798	1	1	yes	characteristic
145	C₂	#14582923	1	1	yes	characteristic
146	C₂	#14582926	1	1	yes	characteristic
147	C₂	#14582971	1	2	yes	—
148	C₂	#14583030	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₃⋊C₈)

C₂

yes

maximal

C₂×C₄⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic

C₂×#13063236

C₂

yes

maximal, characteristic

#14582654

C₂

yes

maximal

C₄×C₄⋊C₄×C₄₃

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

—

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

#13063236

—

C₂×C₄⋊C₄×C₄₃

C₄

yes

—

C₂×C₄²×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#95×C₄₃

C₄

yes

characteristic

#95×C₄₃

C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

D₄

yes

—

C₄²×C₄₃

Q₈

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

D₄

yes

—

C₂×C₄₃⋊C₈

Q₈

yes

—

C₂³×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄⋊C₄

yes

—

C₂×C₄×C₄₃

C₄⋊C₄

yes

—

C₂×C₄×C₄₃

M₁₆

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄₃

C₄×D₄

yes

characteristic

C₂²×C₄₃

C₄×Q₈

yes

characteristic

C₂²×C₄₃

C₂×M₁₆

yes

—

C₂²×C₄₃

#131

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#121

yes

characteristic

C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₄×C₄₃

#125

yes

characteristic

C₄×C₄₃

C₄⋊Q₈

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#446

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₄×C₄⋊C₄

D₄₃

yes

characteristic

#336

—

C₄₃

#1940

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

Dic₄₃

yes

—

C₂×C₄²

C₂×D₄₃

yes

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#95

Dic₄₃

yes

characteristic

#95

Dic₄₃

yes

characteristic

C₂²×C₄

C₂×Dic₄₃

yes

—

C₂²×C₄

C₂×Dic₄₃

yes

—

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

C₂²×C₄

C₂²×D₄₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

Dic₈₆

yes

—

110

C₄²

D₁₇₂

yes

—

111

C₂³

C₂²×Dic₄₃

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#10621877

yes

—

120

C₂×C₄

C₄⋊Dic₄₃

yes

—

121

C₂×C₄

C₄⋊Dic₄₃

yes

—

122

C₂×C₄

C₂×Dic₈₆

yes

characteristic

123

C₂×C₄

C₂×D₁₇₂

yes

characteristic

124

C₂×C₄

D₄×D₄₃

yes

characteristic

125

C₂×C₄

#10621900

yes

characteristic

126

C₂×C₄

Q₈×D₄₃

yes

characteristic

127

C₂×C₄

#10621902

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

172

—

131

C₂²

C₂×C₄⋊Dic₄₃

yes

characteristic

132

C₂²

D₄×Dic₄₃

yes

characteristic

133

C₂²

Q₈×Dic₄₃

yes

characteristic

134

C₂²

C₂×#10621877

yes

—

135

C₂²

#13063332

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#13063300

yes

characteristic

141

C₄

#13063305

yes

characteristic

142

C₄

#13063316

yes

characteristic

143

C₄

#13063324

yes

characteristic

144

C₂

#14582798

yes

characteristic

145

C₂

#14582923

yes

characteristic

146

C₂

#14582926

yes

characteristic

147

C₂

#14582971

yes

—

148

C₂

#14583030

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₄₃
commutator, socle	C₂³×C₄₃	C₂²×C₄
Fitting	C₄×C₄⋊C₄×C₄₃	C₂
hypercenter	C₄×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#1940
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₄₃

commutator, socle

C₂³×C₄₃

C₂²×C₄

Fitting

C₄×C₄⋊C₄×C₄₃

C₂

hypercenter

C₄×C₄⋊C₄

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
80. C₄₃	#1940	semidirect

H ≅ G/N

type

80. C₄₃

#1940

semidirect

Subgroups — #19135319

Conjugacy classes of subgroups

Complements