Group #24907459 — subgroups

Subgroups — #24907459

Order 10624.

272 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 113 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	#19127725	C₂	1	2	yes	maximal
2	C₂×#14581711	C₂	1	2	yes	maximal
3	#19127470	C₂	1	2	yes	maximal
4	C₂×C₄×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13050574	C₂²	1	4	yes	—
6	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
7	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
8	#14581711	—	2	4	no	—
9	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₈₃	—	2	2	no	—
11	C₄×Q₈×C₈₃	—	2	2	no	—
12	C₂×C₄⋊C₄×C₈₃	C₄	1	2	yes	—
13	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₈₃	C₄	1	2	yes	—
15	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
16	#13050574	—	2	4	no	—
17	#13050574	—	2	4	no	—
18	C₂×C₈₃⋊C₈	—	2	4	no	—
19	C₂×Q₈×C₈₃	—	2	2	no	—
20	C₂×Q₈×C₈₃	—	2	2	no	—
21	C₂×Q₈×C₈₃	D₄	1	2	yes	—
22	C₂×Q₈×C₈₃	Q₈	1	2	yes	—
23	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₈₃	—	2	4	no	—
29	C₄⋊C₄×C₈₃	—	2	2	no	—
30	C₄²×C₈₃	—	2	2	no	—
31	C₄²×C₈₃	—	2	2	no	—
32	C₄²×C₈₃	D₄	1	2	yes	—
33	C₄²×C₈₃	D₄	1	2	yes	—
34	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₈₃	—	2	4	no	—
36	Q₈×C₈₃	—	2	4	no	—
37	C₂×C₄×C₈₃	—	2	2	no	—
38	C₂×C₄×C₈₃	—	2	2	no	—
39	C₂×C₄×C₈₃	—	2	2	no	—
40	C₂×C₄×C₈₃	—	2	2	no	—
41	C₂×C₄×C₈₃	—	4	1	no	—
42	C₂×C₄×C₈₃	—	4	2	no	—
43	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₈₃	C₄⋊C₄	1	4	yes	—
47	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₈₃	#41	1	2	yes	—
51	C₈₃⋊C₈	—	4	4	no	—
52	C₂²×C₈₃	—	2	1	no	—
53	C₂²×C₈₃	—	2	1	no	—
54	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₈₃	#117	1	1	yes	characteristic
57	C₄×C₈₃	—	2	4	no	—
58	C₄×C₈₃	—	2	1	no	—
59	C₄×C₈₃	—	2	2	no	—
60	C₄×C₈₃	—	4	2	no	—
61	C₄×C₈₃	#101	1	2	yes	complemented
62	C₄×C₈₃	#121	1	2	yes	—
63	C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₈₃	—	2	1	no	—
65	C₂×C₈₃	—	2	1	no	—
66	C₂×C₈₃	#380	1	1	yes	characteristic
67	C₂×C₈₃	C₂×#101	1	1	yes	characteristic
68	C₂×C₈₃	#413	1	1	yes	characteristic
69	#1801	—	83	1	no	maximal, Sylow
70	C₈₃	#1801	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄×Q₈	D₈₃	1	1	yes	characteristic
72	#423	—	83	2	no	—
73	C₂×#101	—	83	2	no	—
74	#343	—	83	2	no	—
75	C₂²×Q₈	C₂×D₈₃	1	1	yes	characteristic
76	C₂×M₁₆	—	83	4	no	—
77	C₄×Q₈	—	2	2	no	—
78	C₄×Q₈	—	2	2	no	—
79	C₂×C₄⋊C₄	Dic₈₃	1	2	yes	—
80	C₂×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
81	C₂×C₄²	Dic₈₃	1	2	yes	—
82	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
83	C₄⋊C₈	—	166	2	no	—
84	C₄⋊C₈	—	166	2	no	—
85	#101	—	166	4	no	—
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	Dic₁₆₆	1	2	yes	—
89	C₂×Q₈	D₃₃₂	1	2	yes	—
90	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
91	C₂²×C₄	C₂×Dic₈₃	1	2	yes	—
92	C₂²×C₄	C₂×Dic₈₃	1	2	yes	—
93	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
95	M₁₆	—	166	4	no	—
96	M₁₆	—	166	4	no	—
97	C₂×C₈	—	166	4	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	C₂²⋊D₈₃	1	2	yes	—
103	C₄²	C₂²⋊D₈₃	1	2	yes	—
104	C₂³	C₂²×Dic₈₃	1	1	yes	characteristic
105	Q₈	—	2	4	no	—
106	Q₈	—	2	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
114	C₂×C₄	C₂²⋊Dic₈₃	1	2	yes	—
115	C₂×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
116	C₂×C₄	C₄⋊Dic₈₃	1	4	yes	—
117	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
118	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
119	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
120	C₂×C₄	#13050590	1	2	yes	—
121	C₈	—	332	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
125	C₂²	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
126	C₂²	#14581751	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	2	no	—
131	C₄	#14581711	1	2	yes	complemented
132	C₄	#14581766	1	2	yes	—
133	C₄	#14581771	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#19127561	1	1	yes	characteristic
137	C₂	C₂×#14581711	1	1	yes	characteristic
138	C₂	#19127685	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127725

C₂

yes

maximal

C₂×#14581711

C₂

yes

maximal

#19127470

C₂

yes

maximal

C₂×C₄×Q₈×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

#14581711

—

C₂²×Q₈×C₈₃

C₂²

yes

characteristic

C₄×Q₈×C₈₃

—

C₄×Q₈×C₈₃

—

C₂×C₄⋊C₄×C₈₃

C₄

yes

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

characteristic

C₂×C₄²×C₈₃

C₄

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

D₄

yes

—

C₂×Q₈×C₈₃

Q₈

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

D₄

yes

—

C₄²×C₈₃

D₄

yes

—

C₂³×C₈₃

C₂²×C₄

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₃

#117

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#101

yes

complemented

C₄×C₈₃

#121

yes

—

C₄×C₈₃

C₂²⋊Q₈

yes

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₈₃

yes

—

—

—

C₂²×Q₈

C₂×D₈₃

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₈₃

yes

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

characteristic

C₂×C₄²

Dic₈₃

yes

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

166

—

C₄⋊C₈

—

166

—

#101

—

166

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₁₆₆

yes

—

C₂×Q₈

D₃₃₂

yes

—

C₂²×C₄

C₂×Dic₈₃

yes

characteristic

C₂²×C₄

C₂×Dic₈₃

yes

—

C₂²×C₄

C₂×Dic₈₃

yes

—

C₂²×C₄

C₂×Dic₈₃

yes

characteristic

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

166

—

M₁₆

—

166

—

C₂×C₈

—

166

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²⋊D₈₃

yes

—

103

C₄²

C₂²⋊D₈₃

yes

—

104

C₂³

C₂²×Dic₈₃

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²⋊Dic₈₃

yes

characteristic

114

C₂×C₄

C₂²⋊Dic₈₃

yes

—

115

C₂×C₄

C₂²⋊Dic₈₃

yes

characteristic

116

C₂×C₄

C₄⋊Dic₈₃

yes

—

117

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

118

C₂×C₄

C₂×D₃₃₂

yes

characteristic

119

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

120

C₂×C₄

#13050590

yes

—

121

C₈

—

332

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×C₂²⋊Dic₈₃

yes

characteristic

125

C₂²

C₂×C₄⋊Dic₈₃

yes

characteristic

126

C₂²

#14581751

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#14581711

yes

complemented

132

C₄

#14581766

yes

—

133

C₄

#14581771

yes

—

134

C₂

—

135

C₂

—

136

C₂

#19127561

yes

characteristic

137

C₂

C₂×#14581711

yes

characteristic

138

C₂

#19127685

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₈₃
commutator, prime radical	C₂³×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	C₂×C₄×Q₈×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	C₂×C₄×Q₈	D₈₃
hypocenter	C₈₃	#1801
LCS	G ▷ C₂³×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₈₃

commutator, prime radical

C₂³×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

C₂×C₄×Q₈×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₈₃

C₂³

hypercenter

C₂×C₄×Q₈

D₈₃

hypocenter

C₈₃

#1801

LCS

G ▷ C₂³×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
61. C₄×C₈₃	#101	semidirect
70. C₈₃	#1801	semidirect
131. C₄	#14581711	semidirect

H ≅ G/N

type

61. C₄×C₈₃

#101

semidirect

70. C₈₃

#1801

semidirect

131. C₄

#14581711

semidirect

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

subgroup	index
#19127725	2
C₂×#14581711	2
#19127470	2
C₂×C₄×Q₈×C₈₃	2
#1801	83

subgroup

index

C₂×C₄×Q₈×C₈₃

Subgroups — #24907459

Conjugacy classes of subgroups

Complements

Maximal subgroups