Group #25123151 — subgroups

Subgroups — #25123151

Order 11008.

912 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 215 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	C₄×#13063322	C₂	1	1	yes	maximal, characteristic
2	#19134891	C₂	1	12	yes	maximal, complemented
3	#19134738	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₃×#1839	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#10621872	C₂²	1	4	yes	—
6	C₂×#13063322	C₄	1	1	yes	characteristic
7	C₂×C₄×D₁₇₂	C₂²	1	6	yes	—
8	C₂×(C₂×D₄₃)⋊C₈	C₂²	1	12	yes	—
9	#14582801	C₄	1	6	yes	—
10	#14582769	C₄	1	1	yes	characteristic
11	#14582654	C₂²	1	6	yes	—
12	C₄³×C₄₃	C₂²	1	1	yes	characteristic
13	C₄₃×#336	C₂²	1	6	yes	—
14	C₂²×D₁₇₂	C₂×C₄	1	6	yes	—
15	C₂²×C₄×D₄₃	C₂³	1	4	yes	—
16	C₂×#10621872	—	4	12	no	—
17	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
18	#13063322	—	2	4	no	—
19	C₄×D₁₇₂	—	4	6	no	—
20	C₄×D₁₇₂	—	4	6	no	—
21	C₂×C₄⋊Dic₄₃	C₂×C₄	1	6	yes	—
22	C₂×(C₂×D₄₃)⋊C₄	C₂×C₄	1	12	yes	—
23	(C₂×D₄₃)⋊C₈	—	4	12	no	—
24	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
25	C₂×C₄²×C₄₃	C₂×C₄	1	3	yes	—
26	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
28	C₂³×D₄₃	C₂²×C₄	1	4	yes	—
29	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
30	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
31	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
32	C₂²×Dic₄₃	C₂²×C₄	1	4	yes	—
33	C₂×D₁₇₂	—	2	12	no	—
34	C₂×D₁₇₂	—	4	6	no	—
35	C₂×D₁₇₂	—	4	6	no	—
36	C₂×C₄×D₄₃	—	2	12	no	—
37	C₂×C₄×D₄₃	—	2	12	no	—
38	C₂×C₈×C₄₃	—	2	12	no	—
39	C₄²×C₄₃	—	2	6	no	—
40	C₄²×C₄₃	—	4	3	no	—
41	C₄²×C₄₃	M₁₆	1	4	yes	complemented
42	(C₂×D₄₃)⋊C₄	—	4	12	no	—
43	C₄⋊Dic₄₃	—	4	6	no	—
44	C₂×C₄₃⋊C₈	—	2	12	no	—
45	#10621872	—	4	16	no	—
46	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
47	C₂²×D₄₃	—	2	12	no	—
48	C₂²×D₄₃	—	2	4	no	—
49	C₂²×D₄₃	—	2	12	no	—
50	C₂×C₄×C₄₃	—	2	3	no	—
51	C₂×C₄×C₄₃	—	2	3	no	—
52	C₂×C₄×C₄₃	—	2	3	no	—
53	C₂×C₄×C₄₃	—	2	6	no	—
54	C₂×C₄×C₄₃	C₂×M₁₆	1	6	yes	—
55	C₂×C₄×C₄₃	H(5,𝔽₂)	1	6	yes	—
56	C₂×Dic₄₃	—	2	12	no	—
57	D₁₇₂	—	4	24	no	—
58	C₄×D₄₃	—	4	4	no	—
59	C₄×D₄₃	—	4	12	no	—
60	C₈×C₄₃	—	4	4	no	—
61	C₄₃⋊C₈	—	4	4	no	—
62	#23357	—	43	1	no	maximal, Sylow
63	C₂²×C₄₃	#518	1	3	yes	—
64	C₂²×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
65	C₂²×C₄₃	#568	1	3	yes	—
66	C₂×D₄₃	—	4	4	no	—
67	C₂×D₄₃	—	4	12	no	—
68	C₂×D₄₃	—	4	12	no	—
69	C₄×C₄₃	—	2	6	no	—
70	C₄×C₄₃	—	4	3	no	—
71	C₄×C₄₃	#560	1	4	yes	—
72	Dic₄₃	—	4	4	no	—
73	C₄×#127	—	43	1	no	—
74	#1870	—	43	12	no	—
75	#1839	—	43	1	no	—
76	#1839	D₄₃	1	1	yes	characteristic, complemented
77	C₂×C₄₃	#2380	1	1	yes	characteristic
78	C₂×C₄₃	#2975	1	3	yes	—
79	C₂×C₄₃	#2980	1	3	yes	—
80	D₄₃	—	8	4	no	—
81	C₂²×M₁₆	—	43	4	no	—
82	C₂×#127	—	43	1	no	—
83	C₂×C₄×D₄	—	43	6	no	—
84	C₂×C₂²⋊C₈	—	43	12	no	—
85	#390	—	43	6	no	—
86	#384	—	43	1	no	—
87	C₄³	C₂×D₄₃	1	1	yes	characteristic
88	#336	—	43	6	no	—
89	#336	C₂×D₄₃	1	6	yes	—
90	C₄₃	#23357	1	1	yes	characteristic, complemented, Sylow
91	C₂²×D₄	—	43	6	no	—
92	C₂³×C₄	—	43	4	no	—
93	C₂×M₁₆	—	172	12	no	—
94	C₂²×C₈	—	43	4	no	—
95	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
96	#127	—	86	4	no	—
97	C₄×D₄	—	172	6	no	—
98	C₄×D₄	—	172	6	no	—
99	C₂×C₄⋊C₄	—	43	6	no	—
100	C₂×C₂²⋊C₄	—	43	12	no	—
101	C₂×C₄²	C₄×D₄₃	1	3	yes	—
102	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
103	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
104	C₂²⋊C₈	—	172	12	no	—
105	C₂⁴	—	43	4	no	—
106	C₂×D₄	—	86	12	no	—
107	C₂×D₄	—	172	6	no	—
108	C₂×D₄	—	172	6	no	—
109	C₂²×C₄	—	43	4	no	—
110	C₂²×C₄	—	86	12	no	—
111	C₂²×C₄	—	86	12	no	—
112	C₂²×C₄	C₂×C₄×D₄₃	1	3	yes	—
113	C₂²×C₄	C₂×C₄×D₄₃	1	3	yes	—
114	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
115	M₁₆	—	172	16	no	—
116	C₂×C₈	—	2	12	no	—
117	C₂×C₈	—	86	12	no	—
118	C₄⋊C₄	—	172	6	no	—
119	C₂²⋊C₄	—	172	12	no	—
120	C₄²	—	2	6	no	—
121	C₄²	—	4	3	no	—
122	C₄²	#10621872	1	4	yes	complemented
123	C₂³	—	86	12	no	—
124	C₂³	—	86	4	no	—
125	C₂³	—	86	12	no	—
126	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
127	D₄	—	172	24	no	—
128	C₂×C₄	—	2	3	no	—
129	C₂×C₄	—	2	3	no	—
130	C₂×C₄	—	2	3	no	—
131	C₂×C₄	—	2	6	no	—
132	C₂×C₄	—	86	12	no	—
133	C₂×C₄	—	172	4	no	—
134	C₂×C₄	—	172	12	no	—
135	C₂×C₄	C₂×#10621872	1	6	yes	—
136	C₂×C₄	#13063371	1	6	yes	—
137	C₈	—	4	4	no	—
138	C₈	—	172	4	no	—
139	C₂²	—	172	12	no	—
140	C₂²	—	172	4	no	—
141	C₂²	—	172	12	no	—
142	C₂²	#14583373	1	3	yes	—
143	C₂²	C₂²×#10621872	1	1	yes	characteristic
144	C₂²	#14583665	1	3	yes	—
145	C₄	—	2	6	no	—
146	C₄	—	4	3	no	—
147	C₄	—	172	4	no	—
148	C₄	#14583640	1	4	yes	—
149	C₂	—	344	4	no	—
150	C₂	#19137771	1	1	yes	characteristic
151	C₂	#19143635	1	3	yes	—
152	C₂	#19143665	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13063322

C₂

yes

maximal, characteristic

#19134891

C₂

yes

maximal, complemented

#19134738

C₂

yes

maximal, characteristic, complemented

C₄₃×#1839

C₂

yes

maximal, characteristic, complemented

C₂²×#10621872

C₂²

yes

—

C₂×#13063322

C₄

yes

characteristic

C₂×C₄×D₁₇₂

C₂²

yes

—

C₂×(C₂×D₄₃)⋊C₈

C₂²

yes

—

#14582801

C₄

yes

—

#14582769

C₄

yes

characteristic

#14582654

C₂²

yes

—

C₄³×C₄₃

C₂²

yes

characteristic

C₄₃×#336

C₂²

yes

—

C₂²×D₁₇₂

C₂×C₄

yes

—

C₂²×C₄×D₄₃

C₂³

yes

—

C₂×#10621872

—

C₂²×C₄₃⋊C₈

C₂³

yes

—

#13063322

—

C₄×D₁₇₂

—

C₄×D₁₇₂

—

C₂×C₄⋊Dic₄₃

C₂×C₄

yes

—

C₂×(C₂×D₄₃)⋊C₄

C₂×C₄

yes

—

(C₂×D₄₃)⋊C₈

—

C₂²×C₈×C₄₃

C₂³

yes

—

C₂×C₄²×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₃

C₂³

yes

—

C₂³×D₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

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×D₁₇₂

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

M₁₆

yes

complemented

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

#10621872

—

C₂³×C₄₃

C₂³×C₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×M₁₆

yes

—

C₂×C₄×C₄₃

H(5,𝔽₂)

yes

—

C₂×Dic₄₃

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#23357

—

maximal, Sylow

C₂²×C₄₃

#518

yes

—

C₂²×C₄₃

C₂²×M₁₆

yes

characteristic

C₂²×C₄₃

#568

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#560

yes

—

Dic₄₃

—

—

—

—

D₄₃

yes

characteristic, complemented

C₂×C₄₃

#2380

yes

characteristic

C₂×C₄₃

#2975

yes

—

C₂×C₄₃

#2980

yes

—

D₄₃

—

C₂²×M₁₆

—

C₂×#127

—

C₂×C₄×D₄

—

C₂×C₂²⋊C₈

—

#390

—

#384

—

C₄³

C₂×D₄₃

yes

characteristic

#336

—

#336

C₂×D₄₃

yes

—

C₄₃

#23357

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂³×C₄

—

C₂×M₁₆

—

172

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₄₃

yes

—

#127

—

C₄×D₄

—

172

—

C₄×D₄

—

172

—

C₂×C₄⋊C₄

—

100

C₂×C₂²⋊C₄

—

101

C₂×C₄²

C₄×D₄₃

yes

—

102

C₂×C₄²

C₄×D₄₃

yes

characteristic

103

C₂×C₄²

C₂²×D₄₃

yes

—

104

C₂²⋊C₈

—

172

—

105

C₂⁴

—

106

C₂×D₄

—

107

C₂×D₄

—

172

—

108

C₂×D₄

—

172

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂×C₄×D₄₃

yes

—

113

C₂²×C₄

C₂×C₄×D₄₃

yes

—

114

C₂²×C₄

C₂³×D₄₃

yes

characteristic

115

M₁₆

—

172

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

172

—

119

C₂²⋊C₄

—

172

—

120

C₄²

—

121

C₄²

—

122

C₄²

#10621872

yes

complemented

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

C₂²×C₄×D₄₃

yes

characteristic

127

D₄

—

172

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

172

—

134

C₂×C₄

—

172

—

135

C₂×C₄

C₂×#10621872

yes

—

136

C₂×C₄

#13063371

yes

—

137

C₈

—

138

C₈

—

172

—

139

C₂²

—

172

—

140

C₂²

—

172

—

141

C₂²

—

172

—

142

C₂²

#14583373

yes

—

143

C₂²

C₂²×#10621872

yes

characteristic

144

C₂²

#14583665

yes

—

145

C₄

—

146

C₄

—

147

C₄

—

172

—

148

C₄

#14583640

yes

—

149

C₂

—

344

—

150

C₂

#19137771

yes

characteristic

151

C₂

#19143635

yes

—

152

C₂

#19143665

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂³×C₄
Fitting	C₄₃×#1839	C₂
prime radical	C₂×#13063322	C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1839	D₄₃
hypocenter	C₄₃	#23357
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1839

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₄₃

commutator, socle

C₂³×C₄₃

C₂³×C₄

Fitting

C₄₃×#1839

C₂

prime radical

C₂×#13063322

C₄

cosocle

C₂²×C₄×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂²×C₄ ◁ #1839

N	H ≅ G/N	type
2. #19134891	C₂	semidirect
3. #19134738	C₂	semidirect
4. C₄₃×#1839	C₂	semidirect
41. C₄²×C₄₃	M₁₆	semidirect
76. #1839	D₄₃	semidirect
90. C₄₃	#23357	semidirect
122. C₄²	#10621872	semidirect

H ≅ G/N

type

2. #19134891

C₂

semidirect

3. #19134738

C₂

semidirect

4. C₄₃×#1839

C₂

semidirect

41. C₄²×C₄₃

M₁₆

semidirect

76. #1839

D₄₃

semidirect

90. C₄₃

#23357

semidirect

122. C₄²

#10621872

semidirect

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

subgroup	index
C₄×#13063322	2
#19134891	2
#19134738	2
C₄₃×#1839	2
#23357	43

subgroup

index

C₄₃×#1839

Subgroups — #25123151

Conjugacy classes of subgroups

Complements

Maximal subgroups