Group #24908236 — subgroups

Subgroups — #24908236

Order 10624.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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×#14581727	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄×#13050574	C₂	1	1	yes	maximal, characteristic
4	#19127465	C₂	1	2	yes	maximal
5	#423×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	C₂²	1	1	yes	characteristic
7	C₂×#13050574	C₂²	1	1	yes	characteristic
8	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
9	#14581727	—	2	4	no	—
10	C₄⋊(C₈₃⋊C₈)	C₄	1	2	yes	—
11	C₄⋊(C₈₃⋊C₈)	C₄	1	2	yes	—
12	#14581703	—	2	1	no	—
13	C₄×C₈₃⋊C₈	—	2	1	no	—
14	C₂×M₁₆×C₈₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₈₃	—	2	1	no	—
17	C₄⋊C₈×C₈₃	—	2	1	no	—
18	#13050574	—	2	1	no	—
19	#13050574	—	2	1	no	—
20	#13050574	D₄	1	2	yes	—
21	#13050574	Q₈	1	2	yes	—
22	C₂×C₈₃⋊C₈	—	2	2	no	—
23	C₂×C₈₃⋊C₈	—	2	1	no	—
24	C₂×C₈₃⋊C₈	—	2	1	no	—
25	C₂×C₈₃⋊C₈	—	2	2	no	—
26	C₂×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₈₃	Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₈₃	Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
30	M₁₆×C₈₃	—	2	2	no	—
31	M₁₆×C₈₃	—	2	2	no	—
32	C₂×C₈×C₈₃	—	2	2	no	—
33	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₈₃	D₄	1	1	yes	characteristic
36	C₄²×C₈₃	D₄	1	1	yes	characteristic
37	C₂³×C₈₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₈₃	—	2	2	no	—
39	C₂×C₄×C₈₃	—	4	1	no	—
40	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
49	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
50	C₈×C₈₃	—	4	2	no	—
51	C₈₃⋊C₈	—	2	4	no	—
52	C₈₃⋊C₈	—	2	2	no	—
53	C₈₃⋊C₈	—	2	2	no	—
54	C₂²×C₈₃	—	2	1	no	—
55	C₂²×C₈₃	—	2	1	no	—
56	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₈₃	#125	1	1	yes	characteristic
58	C₂²×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
59	C₄×C₈₃	—	2	1	no	—
60	C₄×C₈₃	—	2	1	no	—
61	C₄×C₈₃	—	2	1	no	—
62	C₄×C₈₃	#108	1	2	yes	complemented
63	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
64	C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
65	C₄×C₈₃	#121	1	1	yes	characteristic
66	C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₈₃	—	2	1	no	—
68	C₂×C₈₃	—	2	1	no	—
69	C₂×C₈₃	#389	1	1	yes	characteristic
70	C₂×C₈₃	C₂×#108	1	1	yes	characteristic
71	C₂×C₈₃	#430	1	1	yes	characteristic
72	#1950	—	83	1	no	maximal, Sylow
73	C₈₃	#1950	1	1	yes	characteristic, complemented, Sylow
74	C₂×#108	—	83	2	no	—
75	#423	D₈₃	1	1	yes	characteristic
76	C₂×C₄⋊C₈	—	83	1	no	—
77	C₄×M₁₆	—	83	1	no	—
78	#341	—	83	2	no	—
79	C₂×M₁₆	—	83	1	no	—
80	C₂×M₁₆	—	83	1	no	—
81	C₂×M₁₆	C₂×D₈₃	1	2	yes	—
82	C₂²×C₈	—	83	2	no	—
83	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
84	#108	—	166	4	no	—
85	C₄⋊C₈	—	2	1	no	—
86	C₄⋊C₈	—	2	1	no	—
87	C₄⋊C₈	—	83	2	no	—
88	C₄⋊C₈	—	83	2	no	—
89	#97	—	166	1	no	—
90	C₄×C₈	—	166	1	no	—
91	C₂²×C₄	Dic₁₆₆	1	1	yes	characteristic
92	C₂²×C₄	Dic₁₆₆	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
94	M₁₆	—	2	2	no	—
95	M₁₆	—	2	2	no	—
96	M₁₆	—	83	2	no	—
97	M₁₆	—	83	2	no	—
98	M₁₆	—	166	1	no	—
99	M₁₆	—	166	1	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	83	4	no	—
102	C₂×C₈	—	166	2	no	—
103	C₂×C₈	—	166	1	no	—
104	C₂×C₈	—	166	1	no	—
105	C₂×C₈	—	166	2	no	—
106	C₄²	C₄×D₈₃	1	1	yes	characteristic
107	C₄²	C₄×D₈₃	1	1	yes	characteristic
108	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
109	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
110	C₂³	C₂×Dic₁₆₆	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
114	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
115	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
116	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
118	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₈₃	1	1	yes	characteristic
121	C₂×C₄	#13050589	1	1	yes	characteristic
122	C₂×C₄	#13050591	1	1	yes	characteristic
123	C₈	—	4	2	no	—
124	C₈	—	166	2	no	—
125	C₈	—	166	2	no	—
126	C₈	—	166	4	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
130	C₂²	#14581783	1	1	yes	characteristic
131	C₂²	#14581791	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	#14581727	1	2	yes	complemented
136	C₄	#14581754	1	1	yes	characteristic
137	C₄	#14581757	1	1	yes	characteristic
138	C₄	#14581763	1	1	yes	characteristic
139	C₄	#14581768	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	2	1	no	—
142	C₂	#19127597	1	1	yes	characteristic
143	C₂	C₂×#14581727	1	1	yes	characteristic
144	C₂	#19127749	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581727

C₂

yes

maximal

C₂×C₄⋊(C₈₃⋊C₈)

C₂

yes

maximal, characteristic

C₄×#13050574

C₂

yes

maximal, characteristic

#19127465

C₂

yes

maximal

#423×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

characteristic

C₂×#13050574

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

—

#14581727

—

C₄⋊(C₈₃⋊C₈)

C₄

yes

—

C₄⋊(C₈₃⋊C₈)

C₄

yes

—

#14581703

—

C₄×C₈₃⋊C₈

—

C₂×M₁₆×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

—

#13050574

—

#13050574

—

#13050574

D₄

yes

—

#13050574

Q₈

yes

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₈₃

Q₈

yes

characteristic

C₂²×C₄×C₈₃

Q₈

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₂³×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₃

#125

yes

characteristic

C₂²×C₈₃

C₄⋊Q₈

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#108

yes

complemented

C₄×C₈₃

C₄×D₄

yes

characteristic

C₄×C₈₃

C₄×Q₈

yes

characteristic

C₄×C₈₃

#121

yes

characteristic

C₄×C₈₃

C₂²⋊Q₈

yes

characteristic

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×#108

—

#423

D₈₃

yes

characteristic

C₂×C₄⋊C₈

—

C₄×M₁₆

—

#341

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

—

C₂²×C₈

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#108

—

166

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

166

—

C₄×C₈

—

166

—

C₂²×C₄

Dic₁₆₆

yes

characteristic

C₂²×C₄

Dic₁₆₆

yes

characteristic

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

166

—

M₁₆

—

166

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

166

—

103

C₂×C₈

—

166

—

104

C₂×C₈

—

166

—

105

C₂×C₈

—

166

—

106

C₄²

C₄×D₈₃

yes

characteristic

107

C₄²

C₄×D₈₃

yes

characteristic

108

C₄²

C₂²⋊D₈₃

yes

characteristic

109

C₄²

C₂²⋊D₈₃

yes

characteristic

110

C₂³

C₂×Dic₁₆₆

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

114

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

115

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

116

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

117

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

118

C₂×C₄

D₄×D₈₃

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

120

C₂×C₄

Q₈×D₈₃

yes

characteristic

121

C₂×C₄

#13050589

yes

characteristic

122

C₂×C₄

#13050591

yes

characteristic

123

C₈

—

124

C₈

—

166

—

125

C₈

—

166

—

126

C₈

—

166

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×Dic₈₃⋊C₄

yes

characteristic

130

C₂²

#14581783

yes

characteristic

131

C₂²

#14581791

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#14581727

yes

complemented

136

C₄

#14581754

yes

characteristic

137

C₄

#14581757

yes

characteristic

138

C₄

#14581763

yes

characteristic

139

C₄

#14581768

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

#19127597

yes

characteristic

143

C₂

C₂×#14581727

yes

characteristic

144

C₂

#19127749

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#423×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₄⋊C₄
prime radical	C₂³×C₈₃	C₂×Q₈
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#423	D₈₃
hypocenter	C₈₃	#1950
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#423×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₄⋊C₄

prime radical

C₂³×C₈₃

C₂×Q₈

cosocle

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
62. C₄×C₈₃	#108	semidirect
73. C₈₃	#1950	semidirect
135. C₄	#14581727	semidirect

H ≅ G/N

type

62. C₄×C₈₃

#108

semidirect

73. C₈₃

#1950

semidirect

135. C₄

#14581727

semidirect

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

subgroup	index
C₂×#14581727	2
C₂×C₄⋊(C₈₃⋊C₈)	2
C₄×#13050574	2
#19127465	2
#423×C₈₃	2
#1950	83

subgroup

index

C₂×C₄⋊(C₈₃⋊C₈)

#423×C₈₃

Subgroups — #24908236

Conjugacy classes of subgroups

Complements

Maximal subgroups