Group #24987759 — subgroups

Subgroups — #24987759

Order 11008.

390 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 179 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	#19133343	C₂	1	2	yes	maximal
2	#19132887	C₂	1	2	yes	maximal
3	#19132428	C₂	1	2	yes	maximal
4	C₄²×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
5	#14582952	C₂²	1	4	yes	—
6	#14582652	C₄	1	4	yes	complemented
7	#14582650	C₄	1	4	yes	complemented
8	#14582621	—	2	4	no	—
9	C₂×C₄×C₈×C₄₃	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₄₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
13	C₄³×C₄₃	C₂²	1	1	yes	characteristic
14	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	—
15	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
16	C₄×C₄₃⋊C₈	—	2	4	no	—
17	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₄₃	—	2	2	no	—
21	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₄₃	—	2	4	no	—
24	C₄×C₈×C₄₃	—	2	4	no	—
25	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₄₃	D₄	1	2	yes	—
28	C₄×C₈×C₄₃	Q₈	1	2	yes	—
29	C₂²×C₄×C₄₃	—	2	2	no	—
30	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₃	#41	1	2	yes	—
32	C₂×C₈×C₄₃	—	2	2	no	—
33	C₂×C₈×C₄₃	—	2	2	no	—
34	C₂×C₈×C₄₃	—	2	2	no	—
35	C₂×C₈×C₄₃	C₄²	1	2	yes	—
36	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
39	C₂×C₈×C₄₃	M₁₆	1	4	yes	—
40	C₄²×C₄₃	—	2	8	no	—
41	C₄²×C₄₃	—	2	2	no	—
42	C₄²×C₄₃	M₁₆	1	4	yes	—
43	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
44	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄₃⋊C₈	—	2	8	no	—
47	C₂³×C₄₃	#117	1	1	yes	characteristic
48	C₂×C₄×C₄₃	—	2	8	no	—
49	C₂×C₄×C₄₃	—	2	4	no	—
50	C₂×C₄×C₄₃	—	2	4	no	—
51	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
54	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
55	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
56	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
57	C₈×C₄₃	—	2	2	no	—
58	C₈×C₄₃	—	2	4	no	—
59	C₈×C₄₃	#97	1	2	yes	—
60	C₈×C₄₃	#97	1	2	yes	—
61	C₄₃⋊C₈	—	4	8	no	—
62	#8267	—	43	1	no	maximal, Sylow
63	C₂²×C₄₃	—	2	2	no	—
64	C₂²×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄₃	#432	1	2	yes	—
66	C₄×C₄₃	—	2	8	no	—
67	C₄×C₄₃	—	2	2	no	—
68	C₄×C₄₃	C₂×#97	1	1	yes	characteristic
69	C₄×C₄₃	C₄×M₁₆	1	1	yes	characteristic
70	C₄×C₄₃	#423	1	2	yes	—
71	C₄×C₄₃	#443	1	4	yes	—
72	C₄²×C₈	D₄₃	1	1	yes	characteristic
73	#1564	—	43	2	no	—
74	#1452	—	43	2	no	—
75	#1276	—	43	2	no	—
76	C₂×C₄₃	—	2	2	no	—
77	C₂×C₄₃	#1767	1	1	yes	characteristic
78	C₂×C₄₃	#1772	1	2	yes	—
79	#431	—	43	4	no	—
80	C₂×C₄×C₈	Dic₄₃	1	2	yes	—
81	C₂×C₄×C₈	Dic₄₃	1	2	yes	—
82	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
84	C₄³	C₂×D₄₃	1	1	yes	characteristic
85	#335	—	43	4	no	—
86	#334	—	43	4	no	—
87	#322	—	86	4	no	—
88	C₄₃	#8267	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
90	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
91	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
92	C₂×C₄²	—	2	2	no	—
93	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
94	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
95	C₄⋊C₈	—	43	8	no	—
96	C₂²⋊C₈	—	86	4	no	—
97	C₄×C₈	—	2	4	no	—
98	C₄×C₈	—	2	4	no	—
99	C₄×C₈	—	86	4	no	—
100	C₄×C₈	Dic₈₆	1	2	yes	—
101	C₄×C₈	C₄×D₄₃	1	2	yes	—
102	C₄×C₈	C₄×D₄₃	1	2	yes	—
103	C₄×C₈	D₁₇₂	1	2	yes	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	#10621898	1	2	yes	—
106	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	86	8	no	—
111	C₂×C₈	#10621877	1	4	yes	—
112	C₂×C₈	C₄×Dic₄₃	1	2	yes	—
113	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
114	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
115	C₂×C₈	C₄⋊Dic₄₃	1	4	yes	—
116	C₄²	—	2	8	no	—
117	C₄²	—	2	2	no	—
118	C₄²	#10621877	1	4	yes	—
119	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
120	C₄²	C₂×C₄×D₄₃	1	2	yes	—
121	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
122	C₂³	#13063284	1	1	yes	characteristic
123	C₂×C₄	—	2	8	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
128	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
129	C₂×C₄	C₄×Dic₈₆	1	2	yes	—
130	C₂×C₄	C₂×#10621877	1	2	yes	—
131	C₂×C₄	C₂×#10621877	1	2	yes	—
132	C₈	—	2	2	no	—
133	C₈	—	2	4	no	—
134	C₈	—	172	8	no	—
135	C₈	#13063236	1	2	yes	—
136	C₈	#13063236	1	2	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	C₄×C₄⋊Dic₄₃	1	1	yes	characteristic
139	C₂²	#14582956	1	2	yes	—
140	C₄	—	2	8	no	—
141	C₄	—	2	2	no	—
142	C₄	C₂×#13063236	1	1	yes	characteristic
143	C₄	C₄×#10621877	1	1	yes	characteristic
144	C₄	#14582923	1	2	yes	—
145	C₄	#14583017	1	4	yes	—
146	C₂	—	2	2	no	—
147	C₂	#19134448	1	1	yes	characteristic
148	C₂	#19134474	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19133343

C₂

yes

maximal

#19132887

C₂

yes

maximal

#19132428

C₂

yes

maximal

C₄²×C₈×C₄₃

C₂

yes

maximal, characteristic

#14582952

C₂²

yes

—

#14582652

C₄

yes

complemented

#14582650

C₄

yes

complemented

#14582621

—

C₂×C₄×C₈×C₄₃

C₄

yes

—

C₂×C₄×C₈×C₄₃

C₄

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄³×C₄₃

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂×C₄

yes

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₂²×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₃

C₂×C₄

yes

—

C₂²×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₄²

yes

—

C₂×C₈×C₄₃

C₄²

yes

characteristic

C₂×C₈×C₄₃

C₄²

yes

characteristic

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₂×C₈×C₄₃

M₁₆

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

M₁₆

yes

—

C₄²×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#117

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

C₄×Q₈

yes

—

C₂×C₄×C₄₃

C₂×M₁₆

yes

—

C₂×C₄×C₄₃

C₂×M₁₆

yes

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#97

yes

—

C₈×C₄₃

#97

yes

—

C₄₃⋊C₈

—

#8267

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₄₃

#432

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×#97

yes

characteristic

C₄×C₄₃

C₄×M₁₆

yes

characteristic

C₄×C₄₃

#423

yes

—

C₄×C₄₃

#443

yes

—

C₄²×C₈

D₄₃

yes

—

—

—

C₂×C₄₃

—

C₂×C₄₃

#1767

yes

characteristic

C₂×C₄₃

#1772

yes

—

#431

—

C₂×C₄×C₈

Dic₄₃

yes

—

C₂×C₄×C₈

Dic₄₃

yes

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

C₄³

C₂×D₄₃

yes

—

—

—

C₄₃

#8267

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₄₃

yes

characteristic

C₂²×C₈

C₂×Dic₄₃

yes

—

C₂²×C₈

C₂×Dic₄₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×Dic₄₃

yes

—

C₂×C₄²

C₂²×D₄₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

Dic₈₆

yes

—

101

C₄×C₈

C₄×D₄₃

yes

—

102

C₄×C₈

C₄×D₄₃

yes

—

103

C₄×C₈

D₁₇₂

yes

—

104

C₂²×C₄

—

105

C₂²×C₄

#10621898

yes

—

106

C₂²×C₄

C₂²×Dic₄₃

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

#10621877

yes

—

112

C₂×C₈

C₄×Dic₄₃

yes

—

113

C₂×C₈

C₄×Dic₄₃

yes

characteristic

114

C₂×C₈

C₄×Dic₄₃

yes

characteristic

115

C₂×C₈

C₄⋊Dic₄₃

yes

—

116

C₄²

—

117

C₄²

—

118

C₄²

#10621877

yes

—

119

C₄²

C₂×Dic₈₆

yes

characteristic

120

C₄²

C₂×C₄×D₄₃

yes

—

121

C₄²

C₂×D₁₇₂

yes

characteristic

122

C₂³

#13063284

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂×C₄×Dic₄₃

yes

characteristic

127

C₂×C₄

C₂×C₄⋊Dic₄₃

yes

characteristic

128

C₂×C₄

C₄×D₁₇₂

yes

—

129

C₂×C₄

C₄×Dic₈₆

yes

—

130

C₂×C₄

C₂×#10621877

yes

—

131

C₂×C₄

C₂×#10621877

yes

—

132

C₈

—

133

C₈

—

134

C₈

—

172

—

135

C₈

#13063236

yes

—

136

C₈

#13063236

yes

—

137

C₂²

—

138

C₂²

C₄×C₄⋊Dic₄₃

yes

characteristic

139

C₂²

#14582956

yes

—

140

C₄

—

141

C₄

—

142

C₄

C₂×#13063236

yes

characteristic

143

C₄

C₄×#10621877

yes

characteristic

144

C₄

#14582923

yes

—

145

C₄

#14583017

yes

—

146

C₂

—

147

C₂

#19134448

yes

characteristic

148

C₂

#19134474

yes

—

	subgroup	G/H
center	C₄²	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄²×C₈×C₄₃	C₂
socle	C₂²×C₄₃	C₄×C₄⋊C₄
prime radical	C₂³×C₄₃	#117
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄²×C₈	D₄₃
hypocenter	C₄₃	#8267
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄²×C₈

subgroup

G/H

center

C₄²

C₂×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄²×C₈×C₄₃

C₂

socle

C₂²×C₄₃

C₄×C₄⋊C₄

prime radical

C₂³×C₄₃

#117

cosocle

C₂×C₄²×C₄₃

C₂³

hypercenter

C₄²×C₈

D₄₃

hypocenter

C₄₃

#8267

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄²×C₈

N	H ≅ G/N	type
6. #14582652	C₄	semidirect
7. #14582650	C₄	semidirect
88. C₄₃	#8267	semidirect

H ≅ G/N

type

6. #14582652

C₄

semidirect

7. #14582650

C₄

semidirect

88. C₄₃

#8267

semidirect

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

subgroup	index
#19133343	2
#19132887	2
#19132428	2
C₄²×C₈×C₄₃	2
#8267	43

subgroup

index

C₄²×C₈×C₄₃

Subgroups — #24987759

Conjugacy classes of subgroups

Complements

Maximal subgroups