Group #24907210 — subgroups

Subgroups — #24907210

Order 10624.

272 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14581702	C₂	1	2	yes	maximal
2	C₄²×Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	#19127456	C₂	1	1	yes	maximal, characteristic
4	#19127455	C₂	1	2	yes	maximal
5	#336×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×Dic₈₃	C₄	1	2	yes	—
8	C₂×C₄×Dic₈₃	C₄	1	2	yes	—
9	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
11	#14581702	C₄	1	8	yes	complemented
12	C₂²×C₈×C₈₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₈₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₈₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₈₃	C₂×C₄	1	2	yes	—
17	C₂×C₈₃⋊C₈	—	2	2	no	—
18	C₂×C₈₃⋊C₈	—	2	2	no	—
19	C₂×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
20	C₄×Dic₈₃	—	2	4	no	—
21	C₄×Dic₈₃	—	2	1	no	—
22	C₄×Dic₈₃	—	2	1	no	—
23	C₄×Dic₈₃	—	2	4	no	—
24	C₄×Dic₈₃	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₈₃	—	2	2	no	—
28	C₂×C₈×C₈₃	—	2	2	no	—
29	C₂×C₈×C₈₃	C₂×C₄	1	4	yes	—
30	C₄²×C₈₃	—	2	2	no	—
31	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈₃	—	2	2	no	—
33	C₂×C₄×C₈₃	—	2	2	no	—
34	C₂×C₄×C₈₃	M₁₆	1	4	yes	—
35	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₈₃	#41	1	2	yes	—
37	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
38	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
39	C₂×Dic₈₃	—	2	2	no	—
40	C₂×Dic₈₃	—	2	2	no	—
41	C₂×Dic₈₃	—	2	2	no	—
42	C₂×Dic₈₃	—	2	2	no	—
43	C₂×Dic₈₃	C₄²	1	2	yes	—
44	C₂×Dic₈₃	C₄²	1	2	yes	—
45	C₂×Dic₈₃	M₁₆	1	4	yes	—
46	C₈×C₈₃	—	2	4	no	—
47	C₈₃⋊C₈	—	2	4	no	—
48	C₂²×C₈₃	C₂×C₄²	1	1	yes	characteristic
49	C₂²×C₈₃	#117	1	1	yes	characteristic
50	C₂²×C₈₃	#117	1	1	yes	characteristic
51	C₂²×C₈₃	C₂×M₁₆	1	2	yes	—
52	C₂²×C₈₃	C₂×M₁₆	1	2	yes	—
53	C₄×C₈₃	—	2	4	no	—
54	C₄×C₈₃	#117	1	2	yes	—
55	C₄×C₈₃	#117	1	2	yes	—
56	Dic₈₃	—	2	2	no	—
57	Dic₈₃	—	2	2	no	—
58	Dic₈₃	—	2	4	no	—
59	C₂×C₈₃	#376	1	1	yes	characteristic
60	C₂×C₈₃	C₄×M₁₆	1	1	yes	characteristic
61	C₂×C₈₃	C₄×M₁₆	1	1	yes	characteristic
62	C₂×C₈₃	#432	1	2	yes	—
63	C₂×C₈₃	#432	1	2	yes	—
64	#1745	—	83	1	no	maximal, Sylow
65	C₈₃	#1745	1	1	yes	characteristic, complemented, Sylow
66	C₂×#97	—	83	2	no	—
67	C₄³	—	83	1	no	—
68	#336	—	83	2	no	—
69	#336	—	83	1	no	—
70	#336	D₈₃	1	1	yes	characteristic
71	C₂²×C₈	—	83	2	no	—
72	C₂²×C₈	C₂×D₈₃	1	2	yes	—
73	C₂×C₄²	—	83	1	no	—
74	C₂×C₄²	—	83	2	no	—
75	C₂×C₄²	—	83	2	no	—
76	C₂×C₄²	—	83	1	no	—
77	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
78	#97	—	83	8	no	—
79	C₂²×C₄	—	83	2	no	—
80	C₂²×C₄	—	83	1	no	—
81	C₂²×C₄	—	83	1	no	—
82	C₂²×C₄	C₄×D₈₃	1	2	yes	—
83	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	83	4	no	—
87	C₂×C₈	—	166	2	no	—
88	C₂×C₈	—	166	2	no	—
89	C₂×C₈	C₄×D₈₃	1	4	yes	—
90	C₄²	—	2	2	no	—
91	C₄²	—	83	4	no	—
92	C₄²	—	166	4	no	—
93	C₄²	—	166	4	no	—
94	C₄²	—	166	1	no	—
95	C₄²	—	166	1	no	—
96	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	83	4	no	—
100	C₂×C₄	—	83	2	no	—
101	C₂×C₄	—	83	2	no	—
102	C₂×C₄	—	166	2	no	—
103	C₂×C₄	—	166	2	no	—
104	C₂×C₄	—	166	2	no	—
105	C₂×C₄	—	166	2	no	—
106	C₂×C₄	#13050573	1	4	yes	—
107	C₂×C₄	C₂×C₄×D₈₃	1	2	yes	—
108	C₂×C₄	#13050589	1	1	yes	characteristic
109	C₂×C₄	#13050589	1	2	yes	—
110	C₂×C₄	#13050591	1	1	yes	characteristic
111	C₈	—	2	4	no	—
112	C₈	—	166	4	no	—
113	C₂²	C₄²×D₈₃	1	1	yes	characteristic
114	C₂²	#14581748	1	1	yes	characteristic
115	C₂²	#14581750	1	1	yes	characteristic
116	C₂²	M₁₆×D₈₃	1	2	yes	—
117	C₂²	C₂×#13050573	1	2	yes	—
118	C₄	—	2	4	no	—
119	C₄	—	166	2	no	—
120	C₄	—	166	2	no	—
121	C₄	—	166	4	no	—
122	C₄	#14581748	1	2	yes	—
123	C₄	#14581750	1	2	yes	—
124	C₂	#19127546	1	1	yes	characteristic
125	C₂	C₄×#13050573	1	1	yes	characteristic
126	C₂	#19127653	1	1	yes	characteristic
127	C₂	#19127755	1	2	yes	—
128	C₂	#19127757	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581702

C₂

yes

maximal

C₄²×Dic₈₃

C₂

yes

maximal, characteristic

#19127456

C₂

yes

maximal, characteristic

#19127455

C₂

yes

maximal

#336×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

—

C₂×C₄×Dic₈₃

C₄

yes

—

C₂×C₄×Dic₈₃

C₄

yes

—

C₂×C₄×Dic₈₃

C₂²

yes

characteristic

C₂×C₄×Dic₈₃

C₂²

yes

characteristic

#14581702

C₄

yes

complemented

C₂²×C₈×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂²×Dic₈₃

C₂×C₄

yes

characteristic

C₂²×Dic₈₃

C₂×C₄

yes

characteristic

C₂²×Dic₈₃

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

M₁₆

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

C₄²

yes

—

C₂×Dic₈₃

C₄²

yes

—

C₂×Dic₈₃

M₁₆

yes

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₂×C₄²

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

C₂×M₁₆

yes

—

C₂²×C₈₃

C₂×M₁₆

yes

—

C₄×C₈₃

—

C₄×C₈₃

#117

yes

—

C₄×C₈₃

#117

yes

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

#376

yes

characteristic

C₂×C₈₃

C₄×M₁₆

yes

characteristic

C₂×C₈₃

C₄×M₁₆

yes

characteristic

C₂×C₈₃

#432

yes

—

C₂×C₈₃

#432

yes

—

#1745

—

maximal, Sylow

C₈₃

#1745

yes

characteristic, complemented, Sylow

C₂×#97

—

C₄³

—

#336

—

#336

—

#336

D₈₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₈₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#97

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₂×C₈

C₄×D₈₃

yes

—

C₄²

—

C₄²

—

C₄²

—

166

—

C₄²

—

166

—

C₄²

—

166

—

C₄²

—

166

—

C₂³

C₂×C₄×D₈₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

166

—

103

C₂×C₄

—

166

—

104

C₂×C₄

—

166

—

105

C₂×C₄

—

166

—

106

C₂×C₄

#13050573

yes

—

107

C₂×C₄

C₂×C₄×D₈₃

yes

—

108

C₂×C₄

#13050589

yes

characteristic

109

C₂×C₄

#13050589

yes

—

110

C₂×C₄

#13050591

yes

characteristic

111

C₈

—

112

C₈

—

166

—

113

C₂²

C₄²×D₈₃

yes

characteristic

114

C₂²

#14581748

yes

characteristic

115

C₂²

#14581750

yes

characteristic

116

C₂²

M₁₆×D₈₃

yes

—

117

C₂²

C₂×#13050573

yes

—

118

C₄

—

119

C₄

—

166

—

120

C₄

—

166

—

121

C₄

—

166

—

122

C₄

#14581748

yes

—

123

C₄

#14581750

yes

—

124

C₂

#19127546

yes

characteristic

125

C₂

C₄×#13050573

yes

characteristic

126

C₂

#19127653

yes

characteristic

127

C₂

#19127755

yes

—

128

C₂

#19127757

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₈₃
commutator	C₂²×C₈₃	C₂×C₄²
Fitting	#336×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂²×C₄
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#336	D₈₃
hypocenter	C₈₃	#1745
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₈₃

commutator

C₂²×C₈₃

C₂×C₄²

Fitting

#336×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂²×C₄

cosocle

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₈₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
11. #14581702	C₄	semidirect
65. C₈₃	#1745	semidirect

H ≅ G/N

type

11. #14581702

C₄

semidirect

65. C₈₃

#1745

semidirect

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

subgroup	index
C₂×#14581702	2
C₄²×Dic₈₃	2
#19127456	2
#19127455	2
#336×C₈₃	2
#1745	83

subgroup

index

C₄²×Dic₈₃

#336×C₈₃

Subgroups — #24907210

Conjugacy classes of subgroups

Complements

Maximal subgroups