Group #25012012 — subgroups

Subgroups — #25012012

Order 11008.

306 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 135 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×#14582728	C₂	1	2	yes	maximal
2	C₂×#14582674	C₂	1	1	yes	maximal, characteristic
3	#108×Dic₄₃	C₂	1	2	yes	maximal
4	#19132689	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#13063261	C₂²	1	2	yes	—
8	M₁₆×Dic₄₃	—	2	2	no	—
9	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
10	#14582728	C₄	1	8	yes	complemented
11	#14582674	C₄	1	4	yes	—
12	#14582663	—	2	2	no	—
13	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×#108×C₄₃	C₂²	1	2	yes	—
15	C₂×#10621877	—	2	2	no	—
16	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
18	C₂×C₄₃⋊C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	—
20	#13063261	—	2	2	no	—
21	#13063261	C₂×C₄	1	4	yes	—
22	#13063235	—	4	2	no	—
23	C₈×Dic₄₃	—	4	2	no	—
24	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
25	C₈×Dic₄₃	Q₈	1	2	yes	—
26	C₂×M₁₆×C₄₃	—	2	2	no	—
27	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₄₃	—	2	2	no	—
29	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
30	#108×C₄₃	—	2	2	no	—
31	#108×C₄₃	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₄₃	Q₁₆	1	2	yes	—
34	M₁₆×C₄₃	—	2	4	no	—
35	M₁₆×C₄₃	—	4	2	no	—
36	C₂×C₈×C₄₃	—	4	2	no	—
37	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
39	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₄₃	#41	1	2	yes	—
41	C₄×Dic₄₃	C₄⋊C₄	1	2	yes	—
42	C₄×Dic₄₃	D₈	1	2	yes	—
43	#10621877	—	2	4	no	—
44	#10621877	—	4	2	no	—
45	C₂×C₄₃⋊C₈	—	2	2	no	—
46	C₂×C₄₃⋊C₈	—	4	2	no	—
47	C₂×C₄₃⋊C₈	C₄⋊C₄	1	2	yes	—
48	C₁₆×C₄₃	—	2	2	no	—
49	C₄₃⋊C₁₆	C₄²	1	4	yes	—
50	C₂³×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
53	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₄₃	#135	1	2	yes	—
55	C₂×Dic₄₃	—	4	2	no	—
56	C₂×Dic₄₃	#107	1	4	yes	—
57	C₈×C₄₃	—	4	4	no	—
58	C₈×C₄₃	C₂×C₄²	1	1	yes	characteristic
59	C₈×C₄₃	#117	1	1	yes	characteristic
60	C₈×C₄₃	C₄×Q₈	1	2	yes	—
61	C₄₃⋊C₈	—	2	2	no	—
62	C₄₃⋊C₈	—	4	4	no	—
63	#12676	—	43	1	no	maximal, Sylow
64	C₂²×C₄₃	—	2	2	no	—
65	C₂²×C₄₃	C₂×#107	1	1	yes	characteristic
66	C₂²×C₄₃	C₄×Q₁₆	1	2	yes	—
67	C₄×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
68	C₄×C₄₃	#427	1	1	yes	characteristic
69	C₄×C₄₃	C₄×D₈	1	2	yes	—
70	Dic₄₃	—	2	4	no	—
71	C₂×#368	—	43	2	no	—
72	C₂×#346	—	43	1	no	—
73	C₄×#108	—	43	2	no	—
74	#1381	—	43	1	no	—
75	#1381	D₄₃	1	1	yes	characteristic
76	C₂×C₄₃	—	2	2	no	—
77	C₂×C₄₃	C₄×#107	1	1	yes	characteristic
78	C₂×C₄₃	#2161	1	2	yes	—
79	C₂²×C₁₆	—	43	1	no	—
80	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
81	C₂×#108	—	43	2	no	—
82	C₂×#108	C₂×D₄₃	1	2	yes	—
83	C₄×M₁₆	—	86	2	no	—
84	C₂×C₄×C₈	—	43	1	no	—
85	#368	—	43	8	no	—
86	#346	—	43	4	no	—
87	#341	—	86	2	no	—
88	C₄₃	#12676	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	—	86	2	no	—
91	C₂²×C₈	—	43	1	no	—
92	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
93	C₂×C₄²	—	43	1	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	—	43	2	no	—
96	C₂×C₁₆	—	43	4	no	—
97	C₂×C₁₆	C₄×D₄₃	1	2	yes	—
98	#108	—	2	2	no	—
99	#108	—	43	4	no	—
100	#108	—	86	2	no	—
101	#108	C₄×D₄₃	1	4	yes	—
102	#97	—	172	2	no	—
103	C₄×C₈	—	43	2	no	—
104	C₄×C₈	—	43	2	no	—
105	C₄×C₈	—	172	2	no	—
106	C₂²×C₄	—	43	2	no	—
107	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
108	M₁₆	—	2	4	no	—
109	M₁₆	—	4	2	no	—
110	M₁₆	—	86	4	no	—
111	M₁₆	—	172	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	43	2	no	—
114	C₂×C₈	—	86	2	no	—
115	C₂×C₈	—	172	2	no	—
116	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
117	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
118	C₂×C₈	#10621900	1	2	yes	—
119	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
120	C₄²	—	43	2	no	—
121	C₄²	—	43	2	no	—
122	C₁₆	—	2	2	no	—
123	C₁₆	—	43	4	no	—
124	C₂³	Q₁₆×D₄₃	1	1	yes	characteristic
125	C₂×C₄	—	43	4	no	—
126	C₂×C₄	—	172	2	no	—
127	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
128	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
129	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
130	C₂×C₄	#13063346	1	2	yes	—
131	C₈	—	4	4	no	—
132	C₈	—	86	2	no	—
133	C₈	—	172	4	no	—
134	C₈	C₄²×D₄₃	1	1	yes	characteristic
135	C₈	#13063281	1	1	yes	characteristic
136	C₈	#13063290	1	2	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	#107×D₄₃	1	1	yes	characteristic
139	C₂²	#14582991	1	2	yes	—
140	C₄	—	86	4	no	—
141	C₄	#14582750	1	1	yes	characteristic
142	C₄	#14582934	1	1	yes	characteristic
143	C₄	#14582979	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	#19134487	1	1	yes	characteristic
146	C₂	#19136388	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582728

C₂

yes

maximal

C₂×#14582674

C₂

yes

maximal, characteristic

#108×Dic₄₃

C₂

yes

maximal

#19132689

C₂

yes

maximal, characteristic

C₄₃×#1381

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₁₆

C₂²

yes

characteristic

C₂×#13063261

C₂²

yes

—

M₁₆×Dic₄₃

—

C₂×C₈×Dic₄₃

C₂²

yes

characteristic

#14582728

C₄

yes

complemented

#14582674

C₄

yes

—

#14582663

—

C₂²×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×#108×C₄₃

C₂²

yes

—

C₂×#10621877

—

C₂²×C₄₃⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₄₃

D₄

yes

characteristic

C₂×C₄₃⋊C₁₆

C₂×C₄

yes

—

C₂×C₄₃⋊C₁₆

C₂×C₄

yes

—

#13063261

—

#13063261

C₂×C₄

yes

—

#13063235

—

C₈×Dic₄₃

—

C₈×Dic₄₃

C₂×C₄

yes

—

C₈×Dic₄₃

Q₈

yes

—

C₂×M₁₆×C₄₃

—

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

C₂×C₄

yes

—

#108×C₄₃

—

#108×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

Q₁₆

yes

—

M₁₆×C₄₃

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂²×C₄

yes

—

C₂×C₈×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₃

#41

yes

—

C₄×Dic₄₃

C₄⋊C₄

yes

—

C₄×Dic₄₃

D₈

yes

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₄⋊C₄

yes

—

C₁₆×C₄₃

—

C₄₃⋊C₁₆

C₄²

yes

—

C₂³×C₄₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

#135

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

#107

yes

—

C₈×C₄₃

—

C₈×C₄₃

C₂×C₄²

yes

characteristic

C₈×C₄₃

#117

yes

characteristic

C₈×C₄₃

C₄×Q₈

yes

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#12676

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

C₂×#107

yes

characteristic

C₂²×C₄₃

C₄×Q₁₆

yes

—

C₄×C₄₃

C₄×C₄⋊C₄

yes

characteristic

C₄×C₄₃

#427

yes

characteristic

C₄×C₄₃

C₄×D₈

yes

—

Dic₄₃

—

—

—

—

—

D₄₃

yes

characteristic

C₂×C₄₃

—

C₂×C₄₃

C₄×#107

yes

characteristic

C₂×C₄₃

#2161

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₄₃

yes

characteristic

C₂×#108

—

C₂×#108

C₂×D₄₃

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#368

—

#346

—

#341

—

C₄₃

#12676

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₄₃

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₄₃

yes

—

#108

—

#108

—

100

#108

—

101

#108

C₄×D₄₃

yes

—

102

#97

—

172

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

172

—

106

C₂²×C₄

—

107

C₂²×C₄

D₄×D₄₃

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

M₁₆

—

172

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

172

—

116

C₂×C₈

C₂×C₄×D₄₃

yes

characteristic

117

C₂×C₈

C₂×C₄×D₄₃

yes

—

118

C₂×C₈

#10621900

yes

—

119

C₂×C₈

Q₈×D₄₃

yes

characteristic

120

C₄²

—

121

C₄²

—

122

C₁₆

—

123

C₁₆

—

124

C₂³

Q₁₆×D₄₃

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

172

—

127

C₂×C₄

C₄⋊C₄×D₄₃

yes

characteristic

128

C₂×C₄

(C₂²⋊D₄₃)⋊C₄

yes

—

129

C₂×C₄

D₈×D₄₃

yes

characteristic

130

C₂×C₄

#13063346

yes

—

131

C₈

—

132

C₈

—

133

C₈

—

172

—

134

C₈

C₄²×D₄₃

yes

characteristic

135

C₈

#13063281

yes

characteristic

136

C₈

#13063290

yes

—

137

C₂²

—

138

C₂²

#107×D₄₃

yes

characteristic

139

C₂²

#14582991

yes

—

140

C₄

—

141

C₄

#14582750

yes

characteristic

142

C₄

#14582934

yes

characteristic

143

C₄

#14582979

yes

—

144

C₂

—

145

C₂

#19134487

yes

characteristic

146

C₂

#19136388

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₃
commutator	C₈×C₄₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1381	C₂
socle	C₂²×C₄₃	C₂×#107
prime radical	C₂³×C₄₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1381	D₄₃
hypocenter	C₄₃	#12676
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₄₃

commutator

C₈×C₄₃

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄₃×#1381

C₂

socle

C₂²×C₄₃

C₂×#107

prime radical

C₂³×C₄₃

C₂×Q₁₆

cosocle

C₂²×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
10. #14582728	C₄	semidirect
88. C₄₃	#12676	semidirect

H ≅ G/N

type

10. #14582728

C₄

semidirect

88. C₄₃

#12676

semidirect

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

subgroup	index
C₂×#14582728	2
C₂×#14582674	2
#108×Dic₄₃	2
#19132689	2
C₄₃×#1381	2
#12676	43

subgroup

index

#108×Dic₄₃

C₄₃×#1381

Subgroups — #25012012

Conjugacy classes of subgroups

Complements

Maximal subgroups