Group #25027644 — subgroups

Subgroups — #25027644

Order 11008.

1462 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1462 conjugacy classes of subgroups — too many to draw. Only its 411 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×#14582767	C₂	1	2	yes	maximal
2	#19132866	C₂	1	4	yes	maximal
3	#19132863	C₂	1	8	yes	maximal
4	C₂²×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
5	#14582767	—	2	8	no	—
6	C₂×C₄₃⋊#95	C₂²	1	8	yes	—
7	C₂×C₄₃⋊#95	C₂²	1	4	yes	—
8	C₂×C₄₃⋊#95	C₂²	1	16	yes	—
9	C₂³×Q₈×C₄₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Q₈×C₄₃	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₄₃	C₄	1	8	yes	complemented
12	C₂²×C₄⋊C₄×C₄₃	C₂²	1	4	yes	—
13	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
14	C₂³×Dic₄₃	C₂³	1	8	yes	—
15	C₄₃⋊#95	—	2	16	no	—
16	C₄₃⋊#95	—	2	8	no	—
17	C₄₃⋊#95	—	4	16	no	—
18	C₂²×Q₈×C₄₃	—	2	8	no	—
19	C₂²×Q₈×C₄₃	—	4	8	no	—
20	C₂²×Q₈×C₄₃	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₄₃	C₂³	1	4	yes	—
22	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
23	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
24	C₄⋊Q₈×C₄₃	—	2	32	no	—
25	C₂×C₄⋊C₄×C₄₃	—	2	16	no	—
26	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	16	yes	—
27	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₄₃	D₄	1	8	yes	—
29	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
30	C₂×Q₈×C₄₃	—	2	16	no	—
31	C₂×Q₈×C₄₃	—	4	16	no	—
32	C₂×Q₈×C₄₃	—	4	32	no	—
33	C₂²×C₄×C₄₃	—	2	8	no	—
34	C₂²×C₄×C₄₃	—	2	4	no	—
35	C₂²×C₄×C₄₃	—	2	16	no	—
36	C₂²×C₄×C₄₃	—	2	8	no	—
37	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
38	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
39	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₄₃	C₂×D₄	1	8	yes	—
41	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
42	C₂²×Dic₄₃	—	2	8	no	—
43	C₂²×Dic₄₃	—	2	16	no	—
44	C₂²×Dic₄₃	—	2	32	no	—
45	C₄⋊C₄×C₄₃	—	2	32	no	—
46	C₄²×C₄₃	C₂²⋊C₄	1	16	yes	—
47	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
48	C₂³×C₄₃	C₂²×D₄	1	2	yes	—
49	C₂³×C₄₃	H(5,𝔽₂)	1	4	yes	—
50	C₂³×C₄₃	2₋¹⁺⁴	1	8	yes	—
51	Q₈×C₄₃	—	4	32	no	—
52	C₂×C₄×C₄₃	—	2	4	no	—
53	C₂×C₄×C₄₃	—	2	8	no	—
54	C₂×C₄×C₄₃	—	2	8	no	—
55	C₂×C₄×C₄₃	—	2	16	no	—
56	C₂×C₄×C₄₃	—	2	8	no	—
57	C₂×C₄×C₄₃	—	2	16	no	—
58	C₂×C₄×C₄₃	—	4	16	no	—
59	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	8	yes	—
60	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	4	yes	—
61	C₂×Dic₄₃	—	4	8	no	—
62	C₂×Dic₄₃	—	4	16	no	—
63	C₂×Dic₄₃	—	4	32	no	—
64	#14663	—	43	1	no	maximal, Sylow
65	C₂²×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄₃	#518	1	2	yes	—
67	C₂²×C₄₃	#519	1	4	yes	—
68	C₂²×C₄₃	#535	1	4	yes	—
69	C₂²×C₄₃	#536	1	8	yes	—
70	C₂²×C₄₃	#563	1	16	yes	—
71	C₄×C₄₃	—	2	4	no	—
72	C₄×C₄₃	—	2	8	no	—
73	C₄×C₄₃	—	4	8	no	—
74	Dic₄₃	—	8	8	no	—
75	C₂²×C₄⋊Q₈	D₄₃	1	1	yes	characteristic
76	C₂×#383	—	43	2	no	—
77	#1446	—	43	4	no	—
78	#1445	—	43	8	no	—
79	C₂×C₄₃	#2309	1	1	yes	characteristic
80	C₂×C₄₃	#2310	1	2	yes	—
81	C₂×C₄₃	#2381	1	4	yes	—
82	C₂×C₄₃	#2821	1	8	yes	—
83	C₂³×Q₈	C₂×D₄₃	1	2	yes	—
84	C₂×C₄⋊Q₈	—	2	8	no	—
85	C₂×C₄⋊Q₈	Dic₄₃	1	8	yes	complemented
86	C₂²×C₄⋊C₄	C₂×D₄₃	1	4	yes	—
87	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
88	#383	—	86	8	no	—
89	C₂×#95	—	43	8	no	—
90	C₂×#95	—	43	4	no	—
91	C₂×#95	—	43	16	no	—
92	C₄₃	#14663	1	1	yes	characteristic, complemented, Sylow
93	C₂²×Q₈	—	2	8	no	—
94	C₂²×Q₈	—	4	8	no	—
95	C₂²×Q₈	C₂×Dic₄₃	1	8	yes	—
96	C₂³×C₄	—	43	8	no	—
97	C₂³×C₄	C₂²×D₄₃	1	4	yes	—
98	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
99	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
100	C₄⋊Q₈	—	2	32	no	—
101	C₂×C₄⋊C₄	—	2	16	no	—
102	C₂×C₄⋊C₄	C₂×Dic₄₃	1	16	yes	—
103	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
104	C₂×C₄²	C₂²⋊D₄₃	1	8	yes	—
105	#95	—	86	16	no	—
106	#95	—	86	8	no	—
107	#95	—	172	16	no	—
108	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
109	C₂×Q₈	—	2	16	no	—
110	C₂×Q₈	—	4	16	no	—
111	C₂×Q₈	—	4	32	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	—	2	8	no	—
115	C₂²×C₄	—	2	16	no	—
116	C₂²×C₄	—	86	8	no	—
117	C₂²×C₄	—	86	16	no	—
118	C₂²×C₄	—	86	32	no	—
119	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
120	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
121	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₄₃	1	8	yes	—
123	C₂²×C₄	C₂×C₂²⋊D₄₃	1	4	yes	—
124	C₄⋊C₄	—	2	32	no	—
125	C₄²	C₂²⋊Dic₄₃	1	16	yes	—
126	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
127	C₂³	C₂²×C₂²⋊D₄₃	1	2	yes	—
128	C₂³	#13063370	1	4	yes	—
129	C₂³	#13063373	1	8	yes	—
130	Q₈	—	4	32	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	8	no	—
133	C₂×C₄	—	2	8	no	—
134	C₂×C₄	—	2	16	no	—
135	C₂×C₄	—	2	8	no	—
136	C₂×C₄	—	2	16	no	—
137	C₂×C₄	—	4	16	no	—
138	C₂×C₄	—	172	32	no	—
139	C₂×C₄	—	172	8	no	—
140	C₂×C₄	—	172	16	no	—
141	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	8	yes	—
142	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	4	yes	—
143	C₂²	C₂²×C₂²⋊Dic₄₃	1	1	yes	characteristic
144	C₂²	#14583374	1	2	yes	—
145	C₂²	#14583377	1	4	yes	—
146	C₂²	#14583466	1	4	yes	—
147	C₂²	#14583473	1	8	yes	—
148	C₂²	#14583648	1	16	yes	—
149	C₄	—	2	4	no	—
150	C₄	—	2	8	no	—
151	C₄	—	4	8	no	—
152	C₄	—	344	8	no	—
153	C₂	#19137177	1	1	yes	characteristic
154	C₂	#19137181	1	2	yes	—
155	C₂	#19137776	1	4	yes	—
156	C₂	#19142654	1	8	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582767

C₂

yes

maximal

#19132866

C₂

yes

maximal

#19132863

C₂

yes

maximal

C₂²×C₄⋊Q₈×C₄₃

C₂

yes

maximal, characteristic

#14582767

—

C₂×C₄₃⋊#95

C₂²

yes

—

C₂×C₄₃⋊#95

C₂²

yes

—

C₂×C₄₃⋊#95

C₂²

yes

—

C₂³×Q₈×C₄₃

C₂²

yes

—

C₂×C₄⋊Q₈×C₄₃

—

C₂×C₄⋊Q₈×C₄₃

C₄

yes

complemented

C₂²×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂²×C₄²×C₄₃

C₂²

yes

characteristic

C₂³×Dic₄₃

C₂³

yes

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₂²×Q₈×C₄₃

—

C₂²×Q₈×C₄₃

—

C₂²×Q₈×C₄₃

C₂×C₄

yes

—

C₂³×C₄×C₄₃

C₂³

yes

—

C₂³×C₄×C₄₃

C₂³

yes

—

C₂³×C₄×C₄₃

C₂³

yes

characteristic

C₄⋊Q₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂⁴×C₄₃

C₂⁴

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂²⋊C₄

yes

—

C₂³×C₄₃

C₂³×C₄

yes

characteristic

C₂³×C₄₃

C₂²×D₄

yes

—

C₂³×C₄₃

H(5,𝔽₂)

yes

—

C₂³×C₄₃

2₋¹⁺⁴

yes

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

#14663

—

maximal, Sylow

C₂²×C₄₃

C₂²×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

#518

yes

—

C₂²×C₄₃

#519

yes

—

C₂²×C₄₃

#535

yes

—

C₂²×C₄₃

#536

yes

—

C₂²×C₄₃

#563

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

Dic₄₃

—

C₂²×C₄⋊Q₈

D₄₃

yes

—

—

—

C₂×C₄₃

#2309

yes

characteristic

C₂×C₄₃

#2310

yes

—

C₂×C₄₃

#2381

yes

—

C₂×C₄₃

#2821

yes

—

C₂³×Q₈

C₂×D₄₃

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

Dic₄₃

yes

complemented

C₂²×C₄⋊C₄

C₂×D₄₃

yes

—

C₂²×C₄²

C₂×D₄₃

yes

characteristic

#383

—

C₂×#95

—

C₂×#95

—

C₂×#95

—

C₄₃

#14663

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×Dic₄₃

yes

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₄₃

yes

—

C₂³×C₄

C₂²×D₄₃

yes

—

C₂³×C₄

C₂²×D₄₃

yes

characteristic

100

C₄⋊Q₈

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

C₂×Dic₄₃

yes

—

103

C₂×C₄²

C₂×Dic₄₃

yes

—

104

C₂×C₄²

C₂²⋊D₄₃

yes

—

105

#95

—

106

#95

—

107

#95

—

172

—

108

C₂⁴

C₂³×D₄₃

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂²×Dic₄₃

yes

—

120

C₂²×C₄

C₂²×Dic₄₃

yes

—

121

C₂²×C₄

C₂²×Dic₄₃

yes

—

122

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

—

123

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

—

124

C₄⋊C₄

—

125

C₄²

C₂²⋊Dic₄₃

yes

—

126

C₂³

C₂³×Dic₄₃

yes

characteristic

127

C₂³

C₂²×C₂²⋊D₄₃

yes

—

128

C₂³

#13063370

yes

—

129

C₂³

#13063373

yes

—

130

Q₈

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

172

—

139

C₂×C₄

—

172

—

140

C₂×C₄

—

172

—

141

C₂×C₄

C₂×C₂²⋊Dic₄₃

yes

—

142

C₂×C₄

C₂×C₂²⋊Dic₄₃

yes

—

143

C₂²

C₂²×C₂²⋊Dic₄₃

yes

characteristic

144

C₂²

#14583374

yes

—

145

C₂²

#14583377

yes

—

146

C₂²

#14583466

yes

—

147

C₂²

#14583473

yes

—

148

C₂²

#14583648

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

344

—

153

C₂

#19137177

yes

characteristic

154

C₂

#19137181

yes

—

155

C₂

#19137776

yes

—

156

C₂

#19142654

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₃
commutator	C₂³×C₄₃	C₂³×C₄
Fitting	C₂²×C₄⋊Q₈×C₄₃	C₂
socle, prime radical, cosocle	C₂⁴×C₄₃	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	#14663
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₄₃

commutator

C₂³×C₄₃

C₂³×C₄

Fitting

C₂²×C₄⋊Q₈×C₄₃

C₂

socle, prime radical, cosocle

C₂⁴×C₄₃

C₂⁴

hypercenter

C₂²×C₄⋊Q₈

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

N	H ≅ G/N	type
11. C₂×C₄⋊Q₈×C₄₃	C₄	semidirect
85. C₂×C₄⋊Q₈	Dic₄₃	semidirect
92. C₄₃	#14663	semidirect

H ≅ G/N

type

11. C₂×C₄⋊Q₈×C₄₃

C₄

semidirect

85. C₂×C₄⋊Q₈

Dic₄₃

semidirect

92. C₄₃

#14663

semidirect

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

subgroup	index
C₂×#14582767	2
#19132866	2
#19132863	2
C₂²×C₄⋊Q₈×C₄₃	2
#14663	43

subgroup

index

C₂²×C₄⋊Q₈×C₄₃

Subgroups — #25027644

Conjugacy classes of subgroups

Complements

Maximal subgroups