Group #25104997 — subgroups

Subgroups — #25104997

Order 11008.

1266 conjugacy classes of subgroups; only the 307 normal subgroups are shown.

normal subgroups only

This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 307 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×C₈×D₁₇₂	C₂	1	6	yes	maximal, complemented
2	#19137876	C₂	1	1	yes	maximal, characteristic
3	#19135393	C₂	1	6	yes	maximal, complemented
4	#19134749	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₄₃	C₂²	1	4	yes	—
7	C₂×C₄×D₁₇₂	C₂²	1	6	yes	—
8	C₈×D₁₇₂	—	2	24	no	—
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	6	yes	—
10	C₂×(C₂×D₄₃)⋊C₈	C₂²	1	12	yes	—
11	#14582836	C₄	1	4	yes	—
12	#14582809	C₄	1	4	yes	—
13	C₂×#97×C₄₃	C₂²	1	3	yes	—
14	C₂×C₄×C₈×C₄₃	C₂²	1	3	yes	—
15	C₄₃×#376	C₂²	1	1	yes	characteristic
16	C₂²×D₁₇₂	C₂×C₄	1	6	yes	—
17	C₂²×C₄×D₄₃	C₂³	1	4	yes	—
18	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
19	C₂×C₈×D₄₃	—	2	24	no	—
20	C₄×D₁₇₂	—	2	12	no	—
21	C₄×D₁₇₂	—	2	12	no	—
22	C₂×C₄⋊Dic₄₃	C₂×C₄	1	6	yes	—
23	C₂×(C₂×D₄₃)⋊C₄	C₂×C₄	1	12	yes	—
24	C₄⋊(C₄₃⋊C₈)	—	2	12	no	—
25	(C₂×D₄₃)⋊C₈	—	2	24	no	—
26	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
27	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
29	#97×C₄₃	—	2	6	no	—
30	C₄×C₈×C₄₃	D₄	1	12	yes	complemented
31	#95×C₄₃	C₂×C₄	1	4	yes	—
32	C₂³×D₄₃	C₂²×C₄	1	4	yes	—
33	C₂²×C₄×C₄₃	C₂²×C₄	1	6	yes	—
34	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
35	C₂²×Dic₄₃	C₂²×C₄	1	4	yes	—
36	C₂×D₁₇₂	—	2	12	no	—
37	C₂×D₁₇₂	—	2	24	no	—
38	C₂×C₄×D₄₃	—	2	12	no	—
39	C₂×C₄×D₄₃	—	2	12	no	—
40	C₂×C₈×C₄₃	—	2	6	no	—
41	C₂×C₈×C₄₃	C₂×D₄	1	6	yes	—
42	C₂×C₈×C₄₃	C₂×D₄	1	6	yes	—
43	C₄²×C₄₃	C₂×D₄	1	6	yes	—
44	C₄²×C₄₃	#41	1	6	yes	—
45	(C₂×D₄₃)⋊C₄	—	2	24	no	—
46	C₄⋊Dic₄₃	—	2	12	no	—
47	C₂×C₄₃⋊C₈	—	2	12	no	—
48	C₈×D₄₃	—	4	16	no	—
49	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
50	C₂²×D₄₃	—	2	12	no	—
51	C₂²×D₄₃	—	2	12	no	—
52	C₂²×D₄₃	#131	1	8	yes	—
53	C₂×C₄×C₄₃	—	2	12	no	—
54	C₂×C₄×C₄₃	C₄×D₄	1	12	yes	—
55	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
56	C₂×C₄×C₄₃	C₂×#41	1	3	yes	—
57	C₂×Dic₄₃	—	2	12	no	—
58	D₁₇₂	—	2	48	no	—
59	C₄×D₄₃	—	4	4	no	—
60	C₄×D₄₃	—	4	12	no	—
61	C₈×C₄₃	—	2	6	no	—
62	C₈×C₄₃	#127	1	4	yes	—
63	C₄₃⋊C₈	—	4	4	no	—
64	#21683	—	43	1	no	maximal, Sylow
65	C₂²×C₄₃	C₂×C₄×D₄	1	3	yes	—
66	C₂²×C₄₃	C₂×#131	1	4	yes	—
67	C₂×D₄₃	—	2	24	no	—
68	C₂×D₄₃	—	4	4	no	—
69	C₂×D₄₃	—	4	12	no	—
70	C₄×C₄₃	—	2	12	no	—
71	C₄×C₄₃	C₂×#127	1	1	yes	characteristic
72	C₄×C₄₃	#532	1	3	yes	—
73	Dic₄₃	—	4	4	no	—
74	C₂×C₈×D₄	—	43	6	no	—
75	#2397	—	43	1	no	—
76	#1955	—	43	6	no	—
77	#1843	—	43	1	no	—
78	#1726	D₄₃	1	1	yes	characteristic, complemented
79	C₂×C₄₃	C₄×#127	1	1	yes	characteristic
80	C₂×C₄₃	#2929	1	6	yes	—
81	D₄₃	—	4	8	no	—
82	C₂³×C₈	—	43	4	no	—
83	C₂×C₄×D₄	—	43	6	no	—
84	C₈×D₄	—	86	24	no	—
85	C₂×C₄⋊C₈	—	43	6	no	—
86	C₂×C₂²⋊C₈	—	43	12	no	—
87	C₂×#97	C₂×D₄₃	1	3	yes	—
88	C₂×C₄×C₈	C₂×D₄₃	1	3	yes	—
89	#399	—	43	4	no	—
90	#392	—	43	4	no	—
91	#376	C₂×D₄₃	1	1	yes	characteristic
92	C₄₃	#21683	1	1	yes	characteristic, complemented, Sylow
93	C₂²×D₄	—	43	6	no	—
94	C₂³×C₄	—	43	4	no	—
95	C₂²×C₈	—	43	4	no	—
96	C₂²×C₈	—	86	24	no	—
97	C₂²×C₈	C₂²×D₄₃	1	3	yes	—
98	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
99	C₄×D₄	—	86	12	no	—
100	C₄×D₄	—	86	12	no	—
101	C₂×C₄⋊C₄	—	43	6	no	—
102	C₂×C₂²⋊C₄	—	43	12	no	—
103	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
104	C₄⋊C₈	—	86	12	no	—
105	C₂²⋊C₈	—	86	24	no	—
106	#97	—	2	6	no	—
107	C₄×C₈	D₁₇₂	1	12	yes	complemented
108	#95	C₄×D₄₃	1	4	yes	—
109	C₂⁴	—	43	4	no	—
110	C₂×D₄	—	86	24	no	—
111	C₂×D₄	—	86	12	no	—
112	C₂²×C₄	—	43	4	no	—
113	C₂²×C₄	—	86	12	no	—
114	C₂²×C₄	—	86	12	no	—
115	C₂²×C₄	C₂×C₄×D₄₃	1	6	yes	—
116	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
117	C₂×C₈	—	2	6	no	—
118	C₂×C₈	—	86	12	no	—
119	C₂×C₈	—	172	16	no	—
120	C₂×C₈	C₂×D₁₇₂	1	6	yes	—
121	C₂×C₈	C₂×D₁₇₂	1	6	yes	—
122	C₄⋊C₄	—	86	12	no	—
123	C₂²⋊C₄	—	86	24	no	—
124	C₄²	C₂×D₁₇₂	1	6	yes	—
125	C₄²	#10621898	1	6	yes	—
126	C₂³	—	43	8	no	—
127	C₂³	—	86	12	no	—
128	C₂³	—	86	12	no	—
129	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
130	D₄	—	86	48	no	—
131	C₂×C₄	—	2	12	no	—
132	C₂×C₄	—	86	12	no	—
133	C₂×C₄	—	172	4	no	—
134	C₂×C₄	—	172	12	no	—
135	C₂×C₄	C₄×D₁₇₂	1	12	yes	—
136	C₂×C₄	C₂²×D₁₇₂	1	3	yes	—
137	C₂×C₄	C₂×#10621898	1	3	yes	—
138	C₈	—	2	6	no	—
139	C₈	—	172	4	no	—
140	C₈	#13063322	1	4	yes	—
141	C₂²	—	86	24	no	—
142	C₂²	—	172	12	no	—
143	C₂²	—	172	4	no	—
144	C₂²	C₂×C₄×D₁₇₂	1	3	yes	—
145	C₂²	#131×D₄₃	1	4	yes	—
146	C₄	—	2	12	no	—
147	C₄	—	172	4	no	—
148	C₄	C₂×#13063322	1	1	yes	characteristic
149	C₄	#14583449	1	3	yes	—
150	C₂	—	172	8	no	—
151	C₂	C₄×#13063322	1	1	yes	characteristic
152	C₂	#19143291	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×D₁₇₂

C₂

yes

maximal, complemented

#19137876

C₂

yes

maximal, characteristic

#19135393

C₂

yes

maximal, complemented

#19134749

C₂

yes

maximal, characteristic, complemented

C₄₃×#1726

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₄₃

C₂²

yes

—

C₂×C₄×D₁₇₂

C₂²

yes

—

C₈×D₁₇₂

—

C₂×C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

C₂×(C₂×D₄₃)⋊C₈

C₂²

yes

—

#14582836

C₄

yes

—

#14582809

C₄

yes

—

C₂×#97×C₄₃

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

—

C₄₃×#376

C₂²

yes

characteristic

C₂²×D₁₇₂

C₂×C₄

yes

—

C₂²×C₄×D₄₃

C₂³

yes

—

C₂²×C₄₃⋊C₈

C₂³

yes

—

C₂×C₈×D₄₃

—

C₄×D₁₇₂

—

C₄×D₁₇₂

—

C₂×C₄⋊Dic₄₃

C₂×C₄

yes

—

C₂×(C₂×D₄₃)⋊C₄

C₂×C₄

yes

—

C₄⋊(C₄₃⋊C₈)

—

(C₂×D₄₃)⋊C₈

—

C₂²×C₈×C₄₃

C₂³

yes

—

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄²×C₄₃

C₂³

yes

—

#97×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

complemented

#95×C₄₃

C₂×C₄

yes

—

C₂³×D₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²×Dic₄₃

C₂²×C₄

yes

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₈×D₄₃

—

C₂³×C₄₃

C₂³×C₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

#131

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

C₂²×D₄

yes

—

C₂×C₄×C₄₃

C₂×#41

yes

—

C₂×Dic₄₃

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#127

yes

—

C₄₃⋊C₈

—

#21683

—

maximal, Sylow

C₂²×C₄₃

C₂×C₄×D₄

yes

—

C₂²×C₄₃

C₂×#131

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×#127

yes

characteristic

C₄×C₄₃

#532

yes

—

Dic₄₃

—

C₂×C₈×D₄

—

—

—

—

D₄₃

yes

characteristic, complemented

C₂×C₄₃

C₄×#127

yes

characteristic

C₂×C₄₃

#2929

yes

—

D₄₃

—

C₂³×C₈

—

C₂×C₄×D₄

—

C₈×D₄

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×#97

C₂×D₄₃

yes

—

C₂×C₄×C₈

C₂×D₄₃

yes

—

#399

—

#392

—

#376

C₂×D₄₃

yes

characteristic

C₄₃

#21683

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₄₃

yes

—

C₂²×C₈

C₂²×D₄₃

yes

characteristic

C₄×D₄

—

100

C₄×D₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₂²⋊C₄

—

103

C₂×C₄²

C₂²×D₄₃

yes

—

104

C₄⋊C₈

—

105

C₂²⋊C₈

—

106

#97

—

107

C₄×C₈

D₁₇₂

yes

complemented

108

#95

C₄×D₄₃

yes

—

109

C₂⁴

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

C₂×C₄×D₄₃

yes

—

116

C₂²×C₄

C₂³×D₄₃

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

172

—

120

C₂×C₈

C₂×D₁₇₂

yes

—

121

C₂×C₈

C₂×D₁₇₂

yes

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₄²

C₂×D₁₇₂

yes

—

125

C₄²

#10621898

yes

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

C₂²×C₄×D₄₃

yes

characteristic

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

172

—

134

C₂×C₄

—

172

—

135

C₂×C₄

C₄×D₁₇₂

yes

—

136

C₂×C₄

C₂²×D₁₇₂

yes

—

137

C₂×C₄

C₂×#10621898

yes

—

138

C₈

—

139

C₈

—

172

—

140

C₈

#13063322

yes

—

141

C₂²

—

142

C₂²

—

172

—

143

C₂²

—

172

—

144

C₂²

C₂×C₄×D₁₇₂

yes

—

145

C₂²

#131×D₄₃

yes

—

146

C₄

—

147

C₄

—

172

—

148

C₄

C₂×#13063322

yes

characteristic

149

C₄

#14583449

yes

—

150

C₂

—

172

—

151

C₂

C₄×#13063322

yes

characteristic

152

C₂

#19143291

yes

—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₄₃
commutator, socle	C₂³×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1726	C₂
prime radical	#19137876	C₂
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1726	D₄₃
hypocenter	C₄₃	#21683
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

subgroup

G/H

center

C₂²×C₈

C₂²×D₄₃

commutator, socle

C₂³×C₄₃

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₄₃

Fitting

C₄₃×#1726

C₂

prime radical

#19137876

C₂

cosocle

C₂²×C₄×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂²×C₈ ◁ #1726

N	H ≅ G/N	type
1. C₂×C₈×D₁₇₂	C₂	semidirect
3. #19135393	C₂	semidirect
4. #19134749	C₂	semidirect
5. C₄₃×#1726	C₂	semidirect
30. C₄×C₈×C₄₃	D₄	semidirect
78. #1726	D₄₃	semidirect
92. C₄₃	#21683	semidirect
107. C₄×C₈	D₁₇₂	semidirect

H ≅ G/N

type

1. C₂×C₈×D₁₇₂

C₂

semidirect

3. #19135393

C₂

semidirect

4. #19134749

C₂

semidirect

5. C₄₃×#1726

C₂

semidirect

30. C₄×C₈×C₄₃

D₄

semidirect

78. #1726

D₄₃

semidirect

92. C₄₃

#21683

semidirect

107. C₄×C₈

D₁₇₂

semidirect

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

subgroup	index
C₂×C₈×D₁₇₂	2
#19137876	2
#19135393	2
#19134749	2
C₄₃×#1726	2
#21683	43

subgroup

index

C₂×C₈×D₁₇₂

C₄₃×#1726

Subgroups — #25104997

Conjugacy classes of subgroups

Complements

Maximal subgroups