Group #24907614 — subgroups

Subgroups — #24907614

Order 10624.

300 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 153 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×C₄⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₈₃⋊C₈	C₂	1	4	yes	maximal
3	C₄²×Dic₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×Dic₈₃	C₄	1	2	yes	—
7	C₂×C₄×Dic₈₃	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₈₃	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₈₃	C₂²	1	2	yes	—
10	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
11	Dic₈₃⋊C₈	—	2	8	no	—
12	C₂²×C₈×C₈₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₈₃	—	2	2	no	—
15	C₂²×Dic₈₃	C₂×C₄	1	2	yes	—
16	C₂²×Dic₈₃	C₂×C₄	1	2	yes	—
17	C₂×C₈₃⋊C₈	—	2	4	no	—
18	C₂×C₈₃⋊C₈	—	2	2	no	—
19	C₄×Dic₈₃	—	2	2	no	—
20	C₄×Dic₈₃	—	2	4	no	—
21	C₄×Dic₈₃	C₈	1	4	yes	complemented
22	C₄×Dic₈₃	D₄	1	4	yes	—
23	C₄×Dic₈₃	Q₈	1	4	yes	—
24	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₈₃	—	2	2	no	—
28	C₂×C₈×C₈₃	—	2	4	no	—
29	C₄²×C₈₃	D₄	1	2	yes	—
30	C₄²×C₈₃	Q₈	1	2	yes	—
31	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈₃	—	2	2	no	—
33	C₂×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₈₃	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₈₃	M₁₆	1	2	yes	—
37	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₈₃	—	2	2	no	—
42	C₂×Dic₈₃	—	2	2	no	—
43	C₂×Dic₈₃	C₄⋊C₄	1	4	yes	—
44	C₂×Dic₈₃	C₄⋊C₄	1	4	yes	—
45	C₂×Dic₈₃	C₂×C₈	1	4	yes	—
46	C₂×Dic₈₃	M₁₆	1	4	yes	—
47	C₈×C₈₃	—	4	2	no	—
48	C₈₃⋊C₈	—	4	2	no	—
49	C₂²×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
50	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₈₃	C₂²×C₈	1	1	yes	characteristic
52	C₂²×C₈₃	C₂×M₁₆	1	2	yes	—
53	C₂²×C₈₃	C₂×M₁₆	1	1	yes	characteristic
54	C₄×C₈₃	—	2	2	no	—
55	C₄×C₈₃	C₄⋊C₈	1	4	yes	complemented
56	C₄×C₈₃	#127	1	1	yes	characteristic
57	C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₄×C₈₃	C₄⋊Q₈	1	2	yes	—
59	Dic₈₃	—	2	4	no	—
60	Dic₈₃	C₄⋊C₈	1	8	yes	complemented
61	C₂×C₈₃	#384	1	1	yes	characteristic
62	C₂×C₈₃	C₂×C₄⋊C₈	1	2	yes	—
63	C₂×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₈₃	#423	1	2	yes	—
65	C₂×C₈₃	#423	1	1	yes	characteristic
66	#1842	—	83	1	no	maximal, Sylow
67	C₈₃	#1842	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄⋊C₈	—	83	4	no	—
69	C₂×C₄⋊C₈	—	83	1	no	—
70	C₂×C₄⋊C₈	D₈₃	1	1	yes	characteristic
71	C₄³	—	83	1	no	—
72	C₂²×C₈	—	83	2	no	—
73	C₂²×C₈	C₂×D₈₃	1	2	yes	—
74	C₂×C₄²	—	83	1	no	—
75	C₂×C₄²	—	83	2	no	—
76	C₂×C₄²	—	83	2	no	—
77	C₂×C₄²	—	83	1	no	—
78	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
79	C₄⋊C₈	—	2	2	no	—
80	C₄⋊C₈	—	166	8	no	—
81	C₄⋊C₈	—	166	2	no	—
82	C₂²×C₄	—	83	2	no	—
83	C₂²×C₄	—	83	2	no	—
84	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
85	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
87	C₂×C₈	—	2	4	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	166	4	no	—
90	C₂×C₈	—	166	2	no	—
91	C₄²	—	83	4	no	—
92	C₄²	—	83	4	no	—
93	C₄²	—	83	4	no	—
94	C₄²	—	166	4	no	—
95	C₄²	—	166	2	no	—
96	C₄²	Dic₁₆₆	1	2	yes	—
97	C₄²	C₂²⋊D₈₃	1	2	yes	—
98	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	83	4	no	—
101	C₂×C₄	—	83	4	no	—
102	C₂×C₄	—	83	4	no	—
103	C₂×C₄	—	83	4	no	—
104	C₂×C₄	—	166	2	no	—
105	C₂×C₄	—	166	2	no	—
106	C₂×C₄	Dic₈₃⋊C₄	1	2	yes	—
107	C₂×C₄	Dic₈₃⋊C₄	1	2	yes	—
108	C₂×C₄	C₈×D₈₃	1	2	yes	—
109	C₂×C₄	#13050573	1	2	yes	—
110	C₂×C₄	D₄×D₈₃	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
112	C₂×C₄	Q₈×D₈₃	1	2	yes	—
113	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₈	—	332	2	no	—
116	C₂²	C₄⋊C₄×D₈₃	1	2	yes	—
117	C₂²	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
118	C₂²	C₂×C₈×D₈₃	1	1	yes	characteristic
119	C₂²	M₁₆×D₈₃	1	2	yes	—
120	C₂²	C₂×#13050573	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	83	8	no	—
123	C₄	—	166	4	no	—
124	C₄	Dic₈₃⋊C₈	1	4	yes	complemented
125	C₄	#14581788	1	1	yes	characteristic
126	C₄	#14581790	1	1	yes	characteristic
127	C₄	#14581791	1	2	yes	—
128	C₂	#19127570	1	1	yes	characteristic
129	C₂	C₄⋊C₈×D₈₃	1	2	yes	—
130	C₂	C₂×Dic₈₃⋊C₈	1	1	yes	characteristic
131	C₂	#19127723	1	2	yes	—
132	C₂	#19127724	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₈₃⋊C₈)

C₂

yes

maximal, characteristic

C₂×Dic₈₃⋊C₈

C₂

yes

maximal

C₄²×Dic₈₃

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

—

C₂×C₄×Dic₈₃

C₄

yes

—

C₂×C₄×Dic₈₃

C₄

yes

characteristic

C₂×C₄×Dic₈₃

C₄

yes

characteristic

C₂×C₄×Dic₈₃

C₂²

yes

—

C₄⋊(C₈₃⋊C₈)

—

Dic₈₃⋊C₈

—

C₂²×C₈×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

—

C₂²×Dic₈₃

C₂×C₄

yes

—

C₂²×Dic₈₃

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

C₈

yes

complemented

C₄×Dic₈₃

D₄

yes

—

C₄×Dic₈₃

Q₈

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₄²×C₈₃

D₄

yes

—

C₄²×C₈₃

Q₈

yes

—

C₂³×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄⋊C₄

yes

—

C₂×C₄×C₈₃

C₄⋊C₄

yes

—

C₂×C₄×C₈₃

C₂×C₈

yes

—

C₂×C₄×C₈₃

M₁₆

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

C₄⋊C₄

yes

—

C₂×Dic₈₃

C₄⋊C₄

yes

—

C₂×Dic₈₃

C₂×C₈

yes

—

C₂×Dic₈₃

M₁₆

yes

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₂×C₄⋊C₄

yes

—

C₂²×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₃

C₂²×C₈

yes

characteristic

C₂²×C₈₃

C₂×M₁₆

yes

—

C₂²×C₈₃

C₂×M₁₆

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

C₄⋊C₈

yes

complemented

C₄×C₈₃

#127

yes

characteristic

C₄×C₈₃

C₄⋊Q₈

yes

characteristic

C₄×C₈₃

C₄⋊Q₈

yes

—

Dic₈₃

—

Dic₈₃

C₄⋊C₈

yes

complemented

C₂×C₈₃

#384

yes

characteristic

C₂×C₈₃

C₂×C₄⋊C₈

yes

—

C₂×C₈₃

C₂×C₄⋊C₈

yes

characteristic

C₂×C₈₃

#423

yes

—

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₈₃

yes

characteristic

C₄³

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₄⋊C₈

—

166

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

166

—

C₄²

—

166

—

C₄²

Dic₁₆₆

yes

—

C₄²

C₂²⋊D₈₃

yes

—

C₂³

C₂×C₄×D₈₃

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

166

—

105

C₂×C₄

—

166

—

106

C₂×C₄

Dic₈₃⋊C₄

yes

—

107

C₂×C₄

Dic₈₃⋊C₄

yes

—

108

C₂×C₄

C₈×D₈₃

yes

—

109

C₂×C₄

#13050573

yes

—

110

C₂×C₄

D₄×D₈₃

yes

—

111

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

112

C₂×C₄

Q₈×D₈₃

yes

—

113

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

114

C₈

—

115

C₈

—

332

—

116

C₂²

C₄⋊C₄×D₈₃

yes

—

117

C₂²

C₂×Dic₈₃⋊C₄

yes

characteristic

118

C₂²

C₂×C₈×D₈₃

yes

characteristic

119

C₂²

M₁₆×D₈₃

yes

—

120

C₂²

C₂×#13050573

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

166

—

124

C₄

Dic₈₃⋊C₈

yes

complemented

125

C₄

#14581788

yes

characteristic

126

C₄

#14581790

yes

characteristic

127

C₄

#14581791

yes

—

128

C₂

#19127570

yes

characteristic

129

C₂

C₄⋊C₈×D₈₃

yes

—

130

C₂

C₂×Dic₈₃⋊C₈

yes

characteristic

131

C₂

#19127723

yes

—

132

C₂

#19127724

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₈₃
commutator	C₂²×C₈₃	C₂²×C₈
Fitting	C₂×C₄⋊C₈×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂²×C₄
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	C₂×C₄⋊C₈	D₈₃
hypocenter	C₈₃	#1842
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₈₃

commutator

C₂²×C₈₃

C₂²×C₈

Fitting

C₂×C₄⋊C₈×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂²×C₄

cosocle

C₂²×C₄×C₈₃

C₂³

hypercenter

C₂×C₄⋊C₈

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₈₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
21. C₄×Dic₈₃	C₈	semidirect
55. C₄×C₈₃	C₄⋊C₈	semidirect
60. Dic₈₃	C₄⋊C₈	semidirect
67. C₈₃	#1842	semidirect
124. C₄	Dic₈₃⋊C₈	semidirect

H ≅ G/N

type

21. C₄×Dic₈₃

C₈

semidirect

55. C₄×C₈₃

C₄⋊C₈

semidirect

60. Dic₈₃

C₄⋊C₈

semidirect

67. C₈₃

#1842

semidirect

124. C₄

Dic₈₃⋊C₈

semidirect

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

subgroup	index
C₂×C₄⋊(C₈₃⋊C₈)	2
C₂×Dic₈₃⋊C₈	2
C₄²×Dic₈₃	2
C₂×C₄⋊C₈×C₈₃	2
#1842	83

subgroup

index

C₂×C₄⋊(C₈₃⋊C₈)

C₂×Dic₈₃⋊C₈

C₄²×Dic₈₃

C₂×C₄⋊C₈×C₈₃

#1842

Subgroups — #24907614

Conjugacy classes of subgroups

Complements

Maximal subgroups