Group #24999666 — subgroups

Subgroups — #24999666

Order 11008.

484 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 93 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	#19135418	C₂	1	3	yes	maximal
2	#19132476	C₂	1	3	yes	maximal, complemented
3	C₄₃×#2844	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13063322	C₂²	1	1	yes	characteristic
5	C₂×#13063245	—	2	3	no	—
6	C₂×#13063245	—	2	3	no	—
7	C₂×#13063236	C₂²	1	3	yes	—
8	C₄₃×#400	—	2	3	no	—
9	C₄₃×#395	—	2	3	no	—
10	C₄₃×#382	C₂²	1	3	yes	—
11	C₂²×D₁₇₂	—	2	3	no	—
12	C₂²×C₄₃⋊C₈	—	2	3	no	—
13	#13063322	—	2	3	no	—
14	#13063322	Q₈	1	2	yes	complemented
15	#13063245	—	4	6	no	—
16	#13063245	—	4	6	no	—
17	#13063236	—	2	6	no	—
18	C₂²×Q₈×C₄₃	—	2	1	no	—
19	C₂×C₄⋊C₄×C₄₃	—	2	3	no	—
20	C₂×C₄⋊C₄×C₄₃	D₄	1	3	yes	—
21	C₂×C₄⋊C₄×C₄₃	D₄	1	3	yes	—
22	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
23	#95×C₄₃	—	2	3	no	—
24	#95×C₄₃	—	2	1	no	—
25	#95×C₄₃	—	2	6	no	—
26	C₂³×D₄₃	—	4	1	no	—
27	C₂×Q₈×C₄₃	—	8	3	no	—
28	C₂²×C₄×C₄₃	—	2	3	no	—
29	C₂²×C₄×C₄₃	—	2	3	no	—
30	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
31	C₂×D₁₇₂	—	2	6	no	—
32	C₂×D₁₇₂	—	4	3	no	—
33	C₂×D₁₇₂	—	4	3	no	—
34	C₂×D₁₇₂	—	4	3	no	—
35	C₂×D₁₇₂	—	4	3	no	—
36	C₂×D₁₇₂	—	4	3	no	—
37	C₄⋊C₄×C₄₃	—	4	3	no	—
38	C₄⋊C₄×C₄₃	—	4	3	no	—
39	C₄⋊C₄×C₄₃	—	8	3	no	—
40	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
41	C₄²×C₄₃	#41	1	3	yes	—
42	C₂×C₄₃⋊C₈	—	2	6	no	—
43	C₂×C₄₃⋊C₈	—	4	3	no	—
44	C₂×C₄₃⋊C₈	—	4	3	no	—
45	C₂³×C₄₃	#120	1	1	yes	characteristic
46	C₂²×D₄₃	—	4	2	no	—
47	C₂²×D₄₃	—	8	3	no	—
48	C₂²×D₄₃	—	8	3	no	—
49	Q₈×C₄₃	—	8	4	no	—
50	C₂×C₄×C₄₃	—	2	3	no	—
51	C₂×C₄×C₄₃	—	2	3	no	—
52	C₂×C₄×C₄₃	—	4	3	no	—
53	C₂×C₄×C₄₃	—	4	3	no	—
54	C₂×C₄×C₄₃	—	4	6	no	—
55	C₂×C₄×C₄₃	—	4	6	no	—
56	C₂×C₄×C₄₃	C₂²⋊Q₈	1	3	yes	—
57	C₂×C₄×C₄₃	#124	1	3	yes	—
58	D₁₇₂	—	4	6	no	—
59	D₁₇₂	—	4	6	no	—
60	D₁₇₂	—	8	3	no	—
61	D₁₇₂	—	8	3	no	—
62	C₄₃⋊C₈	—	4	6	no	—
63	#10358	—	43	1	no	maximal, Sylow
64	C₂²×C₄₃	#393	1	1	yes	characteristic
65	C₂²×C₄₃	#453	1	3	yes	—
66	C₂²×C₄₃	#454	1	3	yes	—
67	C₂×D₄₃	—	8	6	no	—
68	C₂×D₄₃	—	16	1	no	—
69	C₂×D₄₃	—	16	3	no	—
70	C₄×C₄₃	—	2	3	no	—
71	C₄×C₄₃	—	2	3	no	—
72	C₄×C₄₃	—	8	3	no	—
73	C₄×C₄₃	—	8	3	no	—
74	#2844	D₄₃	1	1	yes	characteristic, complemented
75	#1961	—	43	3	no	—
76	#1295	—	43	3	no	—
77	C₂×C₄₃	#1655	1	1	yes	characteristic
78	C₂×C₄₃	#1657	1	3	yes	—
79	C₂×C₄₃	#2027	1	3	yes	—
80	D₄₃	—	16	2	no	—
81	C₂×#127	—	43	1	no	—
82	C₂×#102	—	86	3	no	—
83	C₂×#102	—	86	3	no	—
84	C₂×#97	—	43	3	no	—
85	#400	—	2	3	no	—
86	#395	—	2	3	no	—
87	#382	C₂×D₄₃	1	3	yes	—
88	C₄₃	#10358	1	1	yes	characteristic, complemented, Sylow
89	C₂²×Q₈	—	2	1	no	—
90	C₂²×D₄	—	86	3	no	—
91	C₂²×C₈	—	86	3	no	—
92	#127	—	43	2	no	—
93	#127	—	86	3	no	—
94	C₂×C₄⋊C₄	—	2	3	no	—
95	C₂×C₄⋊C₄	C₂²⋊D₄₃	1	3	yes	—
96	C₂×C₄⋊C₄	C₂²⋊D₄₃	1	3	yes	—
97	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
98	#102	—	172	6	no	—
99	#102	—	172	6	no	—
100	#97	—	86	6	no	—
101	#95	—	2	3	no	—
102	#95	—	2	1	no	—
103	#95	—	2	6	no	—
104	C₂⁴	—	172	1	no	—
105	C₂×Q₈	—	8	3	no	—
106	C₂×D₄	—	86	6	no	—
107	C₂×D₄	—	172	3	no	—
108	C₂×D₄	—	172	3	no	—
109	C₂×D₄	—	172	3	no	—
110	C₂×D₄	—	172	3	no	—
111	C₂×D₄	—	172	3	no	—
112	C₂²×C₄	—	2	3	no	—
113	C₂²×C₄	—	2	3	no	—
114	C₂²×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
115	C₂×C₈	—	86	6	no	—
116	C₂×C₈	—	172	3	no	—
117	C₂×C₈	—	172	3	no	—
118	C₄⋊C₄	—	4	3	no	—
119	C₄⋊C₄	—	4	3	no	—
120	C₄⋊C₄	—	8	3	no	—
121	C₄²	Q₈×D₄₃	1	1	yes	characteristic
122	C₄²	#10621902	1	3	yes	—
123	C₂³	—	344	3	no	—
124	C₂³	—	344	3	no	—
125	C₂³	—	172	2	no	—
126	C₂³	#13063294	1	1	yes	characteristic
127	Q₈	—	8	4	no	—
128	D₄	—	344	3	no	—
129	D₄	—	344	3	no	—
130	D₄	—	172	6	no	—
131	D₄	—	172	6	no	—
132	C₂×C₄	—	2	3	no	—
133	C₂×C₄	—	2	3	no	—
134	C₂×C₄	—	4	3	no	—
135	C₂×C₄	—	4	3	no	—
136	C₂×C₄	—	4	6	no	—
137	C₂×C₄	—	4	6	no	—
138	C₂×C₄	#13063301	1	3	yes	—
139	C₂×C₄	#13063314	1	3	yes	—
140	C₈	—	172	6	no	—
141	C₂²	—	344	6	no	—
142	C₂²	—	688	1	no	—
143	C₂²	—	688	3	no	—
144	C₂²	#14582810	1	1	yes	characteristic
145	C₂²	#14583067	1	3	yes	—
146	C₂²	#14583072	1	3	yes	—
147	C₄	—	2	3	no	—
148	C₄	—	2	3	no	—
149	C₄	—	8	3	no	—
150	C₄	—	8	3	no	—
151	C₂	—	688	2	no	—
152	C₂	#19133884	1	1	yes	characteristic
153	C₂	#19133893	1	3	yes	—
154	C₂	#19135748	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19135418

C₂

yes

maximal

#19132476

C₂

yes

maximal, complemented

C₄₃×#2844

C₂

yes

maximal, characteristic, complemented

C₂×#13063322

C₂²

yes

—

—

C₂²

yes

—

C₄₃×#400

—

C₄₃×#395

—

C₄₃×#382

C₂²

yes

—

C₂²×D₁₇₂

—

C₂²×C₄₃⋊C₈

—

#13063322

—

#13063322

Q₈

yes

—

—

—

C₂²×Q₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

D₄

yes

—

C₂×C₄⋊C₄×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#95×C₄₃

—

#95×C₄₃

—

#95×C₄₃

—

C₂³×D₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₄²×C₄₃

#41

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#120

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₃

#124

yes

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#454

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

#2844

D₄₃

yes

characteristic, complemented

#1961

—

#1295

—

C₂×C₄₃

#1655

yes

characteristic

C₂×C₄₃

#1657

yes

—

C₂×C₄₃

#2027

yes

—

D₄₃

—

C₂×#127

—

C₂×#102

—

C₂×#102

—

C₂×#97

—

#400

—

#395

—

#382

C₂×D₄₃

yes

—

C₄₃

#10358

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

C₂²×C₈

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₄₃

yes

—

C₂×C₄⋊C₄

C₂²⋊D₄₃

yes

—

C₂×C₄²

C₂²×D₄₃

yes

characteristic

#102

—

172

—

#102

—

172

—

100

#97

—

101

#95

—

102

#95

—

103

#95

—

104

C₂⁴

—

172

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

172

—

108

C₂×D₄

—

172

—

109

C₂×D₄

—

172

—

110

C₂×D₄

—

172

—

111

C₂×D₄

—

172

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

—

115

C₂×C₈

—

116

C₂×C₈

—

172

—

117

C₂×C₈

—

172

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄²

Q₈×D₄₃

yes

characteristic

122

C₄²

#10621902

yes

—

123

C₂³

—

344

—

124

C₂³

—

344

—

125

C₂³

—

172

—

126

C₂³

#13063294

yes

characteristic

127

Q₈

—

128

D₄

—

344

—

129

D₄

—

344

—

130

D₄

—

172

—

131

D₄

—

172

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#13063301

yes

—

139

C₂×C₄

#13063314

yes

—

140

C₈

—

172

—

141

C₂²

—

344

—

142

C₂²

—

688

—

143

C₂²

—

688

—

144

C₂²

#14582810

yes

characteristic

145

C₂²

#14583067

yes

—

146

C₂²

#14583072

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₂

—

688

—

152

C₂

#19133884

yes

characteristic

153

C₂

#19133893

yes

—

154

C₂

#19135748

yes

—

	subgroup	G/H
center	C₂³	#13063294
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#2844	C₂
socle	C₂³×C₄₃	#120
prime radical	C₂×#13063322	C₂²
hypercenter	#2844	D₄₃
hypocenter	C₄₃	#10358
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

subgroup

G/H

center

C₂³

#13063294

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#2844

C₂

C₂³×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

N	H ≅ G/N	type
2. #19132476	C₂	semidirect
3. C₄₃×#2844	C₂	semidirect
14. #13063322	Q₈	semidirect
74. #2844	D₄₃	semidirect
88. C₄₃	#10358	semidirect

H ≅ G/N

type

2. #19132476

C₂

semidirect

3. C₄₃×#2844

C₂

semidirect

14. #13063322

Q₈

semidirect

74. #2844

D₄₃

semidirect

88. C₄₃

#10358

semidirect

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

subgroup	index
#19135418	2
#19132476	2
C₄₃×#2844	2
#10358	43

subgroup

index

#19135418

#19132476

C₄₃×#2844

#10358

Subgroups — #24999666

Conjugacy classes of subgroups

Complements

Maximal subgroups