Group #24908882 — subgroups

Subgroups — #24908882

Order 10624.

222 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 79 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×#14581715	C₂	1	2	yes	maximal
2	#19127607	C₂	1	1	yes	maximal, characteristic
3	#19127463	C₂	1	2	yes	maximal
4	#19127456	C₂	1	1	yes	maximal, characteristic
5	#382×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₆₆	C₂²	1	1	yes	characteristic
7	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
9	#14581715	—	2	4	no	—
10	C₈₃⋊#95	—	2	2	no	—
11	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
13	#95×C₈₃	—	2	2	no	—
14	C₂×Dic₁₆₆	—	2	2	no	—
15	C₂×Dic₁₆₆	—	2	2	no	—
16	C₂×Dic₁₆₆	—	2	1	no	—
17	C₂×Dic₁₆₆	D₄	1	2	yes	—
18	C₂²×Dic₈₃	—	2	1	no	—
19	C₂²×Dic₈₃	—	2	1	no	—
20	C₂×C₈₃⋊C₈	—	2	2	no	—
21	C₂×C₈₃⋊C₈	—	2	2	no	—
22	C₂×C₈₃⋊C₈	—	2	2	no	—
23	C₄⋊Dic₈₃	—	2	2	no	—
24	C₂²×C₄×C₈₃	—	2	2	no	—
25	C₂²×C₄×C₈₃	D₄	1	2	yes	—
26	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₈₃	—	2	2	no	—
28	C₄⋊C₄×C₈₃	—	2	2	no	—
29	C₄²×C₈₃	—	2	2	no	—
30	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₈₃	—	2	2	no	—
32	C₂×C₄×C₈₃	—	2	2	no	—
33	C₂×C₄×C₈₃	—	2	2	no	—
34	C₂×C₄×C₈₃	—	4	2	no	—
35	C₂×C₄×C₈₃	—	4	2	no	—
36	C₂×C₄×C₈₃	—	4	2	no	—
37	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
39	C₂×C₄×C₈₃	#41	1	2	yes	—
40	C₂×C₄×C₈₃	#41	1	2	yes	—
41	C₂×Dic₈₃	—	2	2	no	—
42	C₂×Dic₈₃	—	4	1	no	—
43	C₂×Dic₈₃	—	4	1	no	—
44	C₂×Dic₈₃	—	4	1	no	—
45	C₂×Dic₈₃	—	4	2	no	—
46	Dic₁₆₆	—	2	4	no	—
47	Dic₁₆₆	—	4	2	no	—
48	C₈₃⋊C₈	—	4	2	no	—
49	C₂²×C₈₃	#121	1	1	yes	characteristic
50	C₂²×C₈₃	#123	1	1	yes	characteristic
51	C₂²×C₈₃	#124	1	1	yes	characteristic
52	C₂²×C₈₃	#135	1	2	yes	—
53	C₂²×C₈₃	#137	1	2	yes	—
54	C₄×C₈₃	—	4	2	no	—
55	C₄×C₈₃	—	8	2	no	—
56	C₄×C₈₃	#123	1	2	yes	—
57	C₄×C₈₃	#126	1	2	yes	—
58	Dic₈₃	—	4	2	no	—
59	Dic₈₃	—	8	1	no	—
60	C₂×C₈₃	#397	1	1	yes	characteristic
61	C₂×C₈₃	#464	1	2	yes	—
62	C₂×C₈₃	#485	1	2	yes	—
63	C₂×C₈₃	#488	1	1	yes	characteristic
64	C₂×C₈₃	#491	1	1	yes	characteristic
65	#2074	—	83	1	no	maximal, Sylow
66	C₈₃	#2074	1	1	yes	characteristic, complemented, Sylow
67	C₂×#103	—	83	2	no	—
68	#391	—	83	1	no	—
69	#382	D₈₃	1	1	yes	characteristic
70	#340	—	83	2	no	—
71	#336	—	83	1	no	—
72	C₂²×Q₈	—	83	1	no	—
73	C₂²×C₈	—	83	2	no	—
74	C₂×C₄⋊C₄	—	83	1	no	—
75	C₂×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
76	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
77	#103	—	166	4	no	—
78	#95	—	2	2	no	—
79	#95	—	166	2	no	—
80	C₂×Q₈	—	83	2	no	—
81	C₂×Q₈	—	166	2	no	—
82	C₂×Q₈	—	166	2	no	—
83	C₂×Q₈	—	166	1	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	166	1	no	—
86	C₂²×C₄	—	166	1	no	—
87	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
88	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
89	C₂×C₈	—	166	2	no	—
90	C₂×C₈	—	166	2	no	—
91	C₂×C₈	—	166	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	166	2	no	—
95	C₄²	—	2	2	no	—
96	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
97	Q₈	—	332	2	no	—
98	Q₈	—	166	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	332	1	no	—
106	C₂×C₄	—	332	1	no	—
107	C₂×C₄	—	332	1	no	—
108	C₂×C₄	—	332	2	no	—
109	C₂×C₄	—	166	2	no	—
110	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
111	C₂×C₄	#13050589	1	1	yes	characteristic
112	C₂×C₄	#13050590	1	2	yes	—
113	C₂×C₄	#13050591	1	2	yes	—
114	C₈	—	332	2	no	—
115	C₂²	#14581763	1	1	yes	characteristic
116	C₂²	#14581775	1	1	yes	characteristic
117	C₂²	#14581781	1	1	yes	characteristic
118	C₂²	#14581812	1	2	yes	—
119	C₂²	#14581822	1	2	yes	—
120	C₄	—	4	2	no	—
121	C₄	—	8	2	no	—
122	C₄	—	332	2	no	—
123	C₄	—	664	1	no	—
124	C₄	#14581777	1	2	yes	—
125	C₄	#14581787	1	2	yes	—
126	C₂	#19127627	1	1	yes	characteristic
127	C₂	#19127920	1	2	yes	—
128	C₂	#19128042	1	2	yes	—
129	C₂	#19128058	1	1	yes	characteristic
130	C₂	#19128071	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581715

C₂

yes

maximal

#19127607

C₂

yes

maximal, characteristic

#19127463

C₂

yes

maximal

#19127456

C₂

yes

maximal, characteristic

#382×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₈₃

C₂²

yes

characteristic

#14581715

—

C₈₃⋊#95

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

#95×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

D₄

yes

—

C₂²×Dic₈₃

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈₃⋊C₈

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

—

C₂²×C₈₃

#137

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#123

yes

—

C₄×C₈₃

#126

yes

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

#397

yes

characteristic

C₂×C₈₃

#464

yes

—

C₂×C₈₃

#485

yes

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×#103

—

#391

—

#382

D₈₃

yes

characteristic

#340

—

#336

—

C₂²×Q₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#103

—

166

—

#95

—

#95

—

166

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂²×C₄

—

C₂²×C₄

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

C₂²⋊D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₄²

—

C₂³

C₂×C₂²⋊D₈₃

yes

characteristic

Q₈

—

332

—

Q₈

—

166

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

332

—

106

C₂×C₄

—

332

—

107

C₂×C₄

—

332

—

108

C₂×C₄

—

332

—

109

C₂×C₄

—

166

—

110

C₂×C₄

D₄×D₈₃

yes

characteristic

111

C₂×C₄

#13050589

yes

characteristic

112

C₂×C₄

#13050590

yes

—

113

C₂×C₄

#13050591

yes

—

114

C₈

—

332

—

115

C₂²

#14581763

yes

characteristic

116

C₂²

#14581775

yes

characteristic

117

C₂²

#14581781

yes

characteristic

118

C₂²

#14581812

yes

—

119

C₂²

#14581822

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

332

—

123

C₄

—

664

—

124

C₄

#14581777

yes

—

125

C₄

#14581787

yes

—

126

C₂

#19127627

yes

characteristic

127

C₂

#19127920

yes

—

128

C₂

#19128042

yes

—

129

C₂

#19128058

yes

characteristic

130

C₂

#19128071

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₃
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#382×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₄
hypercenter	#382	D₈₃
hypocenter	C₈₃	#2074
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₃

commutator, cosocle

C₂²×C₄×C₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#382×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂×D₄

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
66. C₈₃	#2074	semidirect

H ≅ G/N

type

66. C₈₃

#2074

semidirect

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

subgroup	index
C₂×#14581715	2
#19127607	2
#19127463	2
#19127456	2
#382×C₈₃	2
#2074	83

subgroup

index

#382×C₈₃

Subgroups — #24908882

Conjugacy classes of subgroups

Complements

Maximal subgroups