Group #24990413 — subgroups

Subgroups — #24990413

Order 11008.

306 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 131 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	#19134732	C₂	1	2	yes	maximal
2	#19133353	C₂	1	2	yes	maximal
3	#19133351	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582621	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1566	C₂	1	1	yes	maximal, characteristic
6	#14582923	C₂²	1	2	yes	—
7	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	#14582663	—	2	4	no	—
10	#14582651	—	2	4	no	—
11	#14582650	—	2	2	no	—
12	#14582621	C₄	1	4	yes	—
13	C₄₃×#423	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#334	—	2	2	no	—
16	C₂×#10621877	—	2	2	no	—
17	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
18	C₂²×C₄₃⋊C₈	Q₈	1	2	yes	—
19	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
20	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
21	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
22	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
24	C₄×C₄₃⋊C₈	Q₈	1	2	yes	—
25	C₂×M₁₆×C₄₃	—	2	2	no	—
26	C₂²×C₈×C₄₃	Q₈	1	2	yes	—
27	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₄₃	—	2	2	no	—
29	C₄⋊C₈×C₄₃	—	2	2	no	—
30	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₄₃	D₄	1	2	yes	—
32	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
35	M₁₆×C₄₃	—	4	4	no	—
36	C₂×C₈×C₄₃	—	4	2	no	—
37	C₂×C₈×C₄₃	—	4	2	no	—
38	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
39	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
41	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
43	#10621877	—	4	4	no	—
44	C₂×C₄₃⋊C₈	—	2	4	no	—
45	C₂×C₄₃⋊C₈	—	4	2	no	—
46	C₂×C₄₃⋊C₈	—	4	2	no	—
47	C₂×C₄₃⋊C₈	C₄⋊C₄	1	4	yes	—
48	C₂×C₄₃⋊C₈	C₄⋊C₄	1	4	yes	—
49	C₂³×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₄×C₄₃	—	2	2	no	—
51	C₂×C₄×C₄₃	—	2	2	no	—
52	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
56	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₄₃	#135	1	4	yes	—
59	C₈×C₄₃	—	2	4	no	—
60	C₈×C₄₃	—	8	2	no	—
61	C₄₃⋊C₈	—	2	4	no	—
62	C₄₃⋊C₈	—	2	4	no	—
63	C₄₃⋊C₈	—	8	2	no	—
64	#8827	—	43	1	no	maximal, Sylow
65	C₂²×C₄₃	—	2	1	no	—
66	C₂²×C₄₃	—	2	1	no	—
67	C₂²×C₄₃	#384	1	1	yes	characteristic
68	C₂²×C₄₃	#499	1	2	yes	—
69	C₄×C₄₃	—	2	1	no	—
70	C₄×C₄₃	—	2	1	no	—
71	C₄×C₄₃	#427	1	2	yes	—
72	C₄×C₄₃	#430	1	2	yes	—
73	C₄×C₄₃	#430	1	2	yes	—
74	C₄×C₄₃	#495	1	2	yes	—
75	#1837	—	43	2	no	—
76	#1567	—	43	2	no	—
77	#1566	—	43	1	no	—
78	#1566	D₄₃	1	1	yes	characteristic
79	C₂×#322	—	43	1	no	—
80	C₂×C₄₃	—	2	2	no	—
81	C₂×C₄₃	#1846	1	1	yes	characteristic
82	C₂×C₄₃	#1849	1	1	yes	characteristic
83	C₂×C₄₃	#1849	1	1	yes	characteristic
84	#423	—	43	2	no	—
85	#423	C₂×D₄₃	1	2	yes	—
86	C₂×C₄×C₈	—	43	1	no	—
87	C₂×C₄×C₈	—	43	1	no	—
88	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
89	#341	—	86	4	no	—
90	#335	—	86	4	no	—
91	#334	—	2	2	no	—
92	#334	—	86	2	no	—
93	#322	—	43	4	no	—
94	C₄₃	#8827	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	86	2	no	—
97	C₂²×C₈	—	43	2	no	—
98	C₂²×C₈	—	43	2	no	—
99	C₂²×C₈	Dic₈₆	1	2	yes	—
100	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	86	2	no	—
104	C₄⋊C₈	—	86	2	no	—
105	C₄×C₈	—	43	2	no	—
106	C₄×C₈	—	43	2	no	—
107	C₄×C₈	—	43	2	no	—
108	C₄×C₈	—	43	2	no	—
109	C₄×C₈	C₄×D₄₃	1	2	yes	—
110	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
111	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
112	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
113	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
114	M₁₆	—	4	4	no	—
115	M₁₆	—	172	4	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	43	4	no	—
119	C₂×C₈	—	43	4	no	—
120	C₂×C₈	—	86	4	no	—
121	C₂×C₈	—	172	2	no	—
122	C₂×C₈	—	172	2	no	—
123	C₂×C₈	Dic₄₃⋊C₄	1	4	yes	—
124	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
125	C₄²	D₄×D₄₃	1	1	yes	characteristic
126	C₄²	Q₈×D₄₃	1	1	yes	characteristic
127	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
128	C₂³	#13063324	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
132	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
133	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
134	C₂×C₄	#13063321	1	1	yes	characteristic
135	C₂×C₄	#13063323	1	1	yes	characteristic
136	C₂×C₄	#13063324	1	1	yes	characteristic
137	C₂×C₄	#13063346	1	4	yes	—
138	C₈	—	2	4	no	—
139	C₈	—	8	2	no	—
140	C₈	—	86	4	no	—
141	C₈	—	86	4	no	—
142	C₈	—	344	2	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	#14582768	1	1	yes	characteristic
146	C₂²	#14583299	1	2	yes	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	#14582934	1	2	yes	—
150	C₄	#14582946	1	2	yes	—
151	C₄	#14582947	1	2	yes	—
152	C₄	#14583280	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	#19134756	1	1	yes	characteristic
155	C₂	#19134768	1	1	yes	characteristic
156	C₂	#19134769	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19134732

C₂

yes

maximal

#19133353

C₂

yes

maximal

#19133351

C₂

yes

maximal, characteristic

C₂×#14582621

C₂

yes

maximal, characteristic

C₄₃×#1566

C₂

yes

maximal, characteristic

#14582923

C₂²

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

—

—

C₄

yes

—

C₄₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#334

—

C₂×#10621877

—

C₂²×C₄₃⋊C₈

D₄

yes

—

C₂²×C₄₃⋊C₈

Q₈

yes

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

C₂×C₄

yes

—

C₄×C₄₃⋊C₈

C₂×C₄

yes

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₄×C₄₃⋊C₈

Q₈

yes

—

C₂×M₁₆×C₄₃

—

C₂²×C₈×C₄₃

Q₈

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₄₃

C₂×Q₈

yes

characteristic

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×Q₈

yes

characteristic

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₄⋊C₄

yes

—

C₂×C₄₃⋊C₈

C₄⋊C₄

yes

—

C₂³×C₄₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

#135

yes

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#8827

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#384

yes

characteristic

C₂²×C₄₃

#499

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#427

yes

—

C₄×C₄₃

#430

yes

—

C₄×C₄₃

#430

yes

—

C₄×C₄₃

#495

yes

—

—

—

—

D₄₃

yes

characteristic

C₂×#322

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×D₄₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₃

yes

—

—

—

—

—

C₄₃

#8827

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

Dic₈₆

yes

—

100

C₂×C₄²

C₂²×D₄₃

yes

characteristic

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

C₄×D₄₃

yes

—

110

C₄×C₈

C₂²⋊D₄₃

yes

—

111

C₂²×C₄

C₂×Dic₈₆

yes

characteristic

112

C₂²×C₄

D₄×D₄₃

yes

characteristic

113

C₂²×C₄

Q₈×D₄₃

yes

characteristic

114

M₁₆

—

115

M₁₆

—

172

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

172

—

122

C₂×C₈

—

172

—

123

C₂×C₈

Dic₄₃⋊C₄

yes

—

124

C₄²

C₂×C₄×D₄₃

yes

characteristic

125

C₄²

D₄×D₄₃

yes

characteristic

126

C₄²

Q₈×D₄₃

yes

characteristic

127

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

128

C₂³

#13063324

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₄⋊C₄×D₄₃

yes

characteristic

132

C₂×C₄

C₄⋊C₄×D₄₃

yes

characteristic

133

C₂×C₄

C₂×Dic₄₃⋊C₄

yes

characteristic

134

C₂×C₄

#13063321

yes

characteristic

135

C₂×C₄

#13063323

yes

characteristic

136

C₂×C₄

#13063324

yes

characteristic

137

C₂×C₄

#13063346

yes

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

344

—

143

C₂²

—

144

C₂²

—

145

C₂²

#14582768

yes

characteristic

146

C₂²

#14583299

yes

—

147

C₄

—

148

C₄

—

149

C₄

#14582934

yes

—

150

C₄

#14582946

yes

—

151

C₄

#14582947

yes

—

152

C₄

#14583280

yes

—

153

C₂

—

154

C₂

#19134756

yes

characteristic

155

C₂

#19134768

yes

characteristic

156

C₂

#19134769

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13063321
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1566	C₂
socle	C₂²×C₄₃	#384
prime radical	C₂³×C₄₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1566	D₄₃
hypocenter	C₄₃	#8827
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#13063321

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1566

C₂

socle

C₂²×C₄₃

#384

prime radical

C₂³×C₄₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
94. C₄₃	#8827	semidirect

H ≅ G/N

type

94. C₄₃

#8827

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19134732	2
#19133353	2
#19133351	2
C₂×#14582621	2
C₄₃×#1566	2
#8827	43

subgroup

index

C₄₃×#1566

Subgroups — #24990413

Conjugacy classes of subgroups

Complements

Maximal subgroups