Group #24908951 — subgroups

Subgroups — #24908951

Order 10624.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14581726	C₂	1	1	yes	maximal, characteristic
2	#19127599	C₂	1	2	yes	maximal
3	#19127464	C₂	1	2	yes	maximal
4	#19127464	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
8	C₂×Dic₈₃⋊C₄	—	2	2	no	—
9	#14581726	—	2	2	no	—
10	C₈₃⋊#95	—	2	2	no	—
11	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₈₃	—	2	2	no	—
15	C₂²×Dic₈₃	—	2	2	no	—
16	C₂²×Dic₈₃	—	2	2	no	—
17	C₄⋊Dic₈₃	—	2	2	no	—
18	C₄⋊Dic₈₃	—	2	2	no	—
19	C₄⋊Dic₈₃	—	2	2	no	—
20	C₄⋊Dic₈₃	Q₈	1	4	yes	—
21	Dic₈₃⋊C₄	—	4	4	no	—
22	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₈₃	—	2	2	no	—
26	C₂×C₈×C₈₃	—	2	1	no	—
27	C₂×C₈×C₈₃	—	2	2	no	—
28	C₂×C₈×C₈₃	D₄	1	2	yes	—
29	C₄²×C₈₃	—	2	1	no	—
30	C₄²×C₈₃	—	2	1	no	—
31	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₈₃	—	2	1	no	—
33	C₂×C₄×C₈₃	—	2	1	no	—
34	C₂×C₄×C₈₃	—	2	1	no	—
35	C₂×C₄×C₈₃	—	2	2	no	—
36	C₂×C₄×C₈₃	D₈	1	2	yes	—
37	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₈₃	#41	1	2	yes	—
40	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
41	C₂×Dic₈₃	—	2	4	no	—
42	C₂×Dic₈₃	—	4	2	no	—
43	C₂×Dic₈₃	—	4	2	no	—
44	C₂×Dic₈₃	—	4	2	no	—
45	C₂×Dic₈₃	—	4	2	no	—
46	C₂×Dic₈₃	—	4	2	no	—
47	C₈×C₈₃	—	2	2	no	—
48	C₈×C₈₃	—	4	1	no	—
49	C₂²×C₈₃	#121	1	1	yes	characteristic
50	C₂²×C₈₃	C₂²⋊Q₈	1	2	yes	—
51	C₂²×C₈₃	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₈₃	#135	1	1	yes	characteristic
53	C₂²×C₈₃	#137	1	2	yes	—
54	C₄×C₈₃	—	2	2	no	—
55	C₄×C₈₃	—	4	1	no	—
56	C₄×C₈₃	#123	1	1	yes	characteristic
57	C₄×C₈₃	#125	1	2	yes	—
58	C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
59	Dic₈₃	—	4	4	no	—
60	Dic₈₃	—	8	2	no	—
61	C₂×C₈₃	#398	1	1	yes	characteristic
62	C₂×C₈₃	#466	1	1	yes	characteristic
63	C₂×C₈₃	#470	1	1	yes	characteristic
64	C₂×C₈₃	#474	1	2	yes	—
65	C₂×C₈₃	#479	1	2	yes	—
66	#2086	—	83	1	no	maximal, Sylow
67	C₈₃	#2086	1	1	yes	characteristic, complemented, Sylow
68	C₂×#107	—	83	1	no	—
69	C₂×C₄⋊C₈	D₈₃	1	1	yes	characteristic
70	#389	—	83	2	no	—
71	#340	—	83	1	no	—
72	#340	—	83	2	no	—
73	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	83	2	no	—
76	C₂×C₄⋊C₄	—	83	2	no	—
77	C₂×C₄⋊C₄	—	166	2	no	—
78	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
79	#107	—	166	2	no	—
80	C₄⋊C₈	—	2	2	no	—
81	#95	—	166	2	no	—
82	C₂²×C₄	—	166	2	no	—
83	C₂²×C₄	—	166	2	no	—
84	C₂²×C₄	D₃₃₂	1	1	yes	characteristic
85	C₂²×C₄	D₃₃₂	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	C₂²⋊D₈₃	1	2	yes	—
91	C₄⋊C₄	—	332	4	no	—
92	C₄⋊C₄	—	83	4	no	—
93	C₄⋊C₄	—	166	2	no	—
94	C₄⋊C₄	—	166	2	no	—
95	C₄⋊C₄	—	166	2	no	—
96	C₄²	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	332	2	no	—
104	C₂×C₄	—	332	2	no	—
105	C₂×C₄	—	332	2	no	—
106	C₂×C₄	—	332	2	no	—
107	C₂×C₄	—	332	2	no	—
108	C₂×C₄	—	166	4	no	—
109	C₂×C₄	D₆₆₄	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₈₃	1	2	yes	—
112	C₂×C₄	#13050589	1	2	yes	—
113	C₂×C₄	#13050590	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	4	1	no	—
116	C₂²	#14581766	1	1	yes	characteristic
117	C₂²	#14581772	1	2	yes	—
118	C₂²	C₂×D₆₆₄	1	1	yes	characteristic
119	C₂²	#14581814	1	1	yes	characteristic
120	C₂²	#14581825	1	2	yes	—
121	C₄	—	2	2	no	—
122	C₄	—	4	1	no	—
123	C₄	—	332	4	no	—
124	C₄	—	664	2	no	—
125	C₄	#14581773	1	1	yes	characteristic
126	C₄	#14581782	1	2	yes	—
127	C₄	#14581790	1	1	yes	characteristic
128	C₂	#19127631	1	1	yes	characteristic
129	C₂	#19127937	1	1	yes	characteristic
130	C₂	#19127959	1	1	yes	characteristic
131	C₂	#19127982	1	2	yes	—
132	C₂	#19128009	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581726

C₂

yes

maximal, characteristic

#19127599

C₂

yes

maximal

#19127464

C₂

yes

maximal

#19127464

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₈₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₈₃

C₂²

yes

—

C₂×C₄⋊Dic₈₃

C₂²

yes

—

C₂×Dic₈₃⋊C₄

—

#14581726

—

C₈₃⋊#95

—

C₂²×C₈×C₈₃

C₂²

yes

characteristic

C₂²×C₈×C₈₃

C₂²

yes

characteristic

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

—

C₂²×Dic₈₃

—

C₂²×Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

Q₈

yes

—

Dic₈₃⋊C₄

—

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₄

yes

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

D₈

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₂²×C₈₃

#121

yes

characteristic

C₂²×C₈₃

C₂²⋊Q₈

yes

—

C₂²×C₈₃

C₂×D₈

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#123

yes

characteristic

C₄×C₈₃

#125

yes

—

C₄×C₈₃

C₄⋊Q₈

yes

characteristic

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₂×C₈₃

#479

yes

—

#2086

—

maximal, Sylow

C₈₃

#2086

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×C₄⋊C₈

D₈₃

yes

—

—

—

C₂²×C₈

C₂×D₈₃

yes

characteristic

C₂²×C₈

C₂×D₈₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#107

—

166

—

C₄⋊C₈

—

#95

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

D₃₃₂

yes

characteristic

C₂²×C₄

D₃₃₂

yes

characteristic

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₈₃

yes

—

C₄⋊C₄

—

332

—

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₄²

—

C₄²

—

C₂³

C₂×D₃₃₂

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

332

—

104

C₂×C₄

—

332

—

105

C₂×C₄

—

332

—

106

C₂×C₄

—

332

—

107

C₂×C₄

—

332

—

108

C₂×C₄

—

166

—

109

C₂×C₄

D₆₆₄

yes

—

110

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

111

C₂×C₄

Q₈×D₈₃

yes

—

112

C₂×C₄

#13050589

yes

—

113

C₂×C₄

#13050590

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

#14581766

yes

characteristic

117

C₂²

#14581772

yes

—

118

C₂²

C₂×D₆₆₄

yes

characteristic

119

C₂²

#14581814

yes

characteristic

120

C₂²

#14581825

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

332

—

124

C₄

—

664

—

125

C₄

#14581773

yes

characteristic

126

C₄

#14581782

yes

—

127

C₄

#14581790

yes

characteristic

128

C₂

#19127631

yes

characteristic

129

C₂

#19127937

yes

characteristic

130

C₂

#19127959

yes

characteristic

131

C₂

#19127982

yes

—

132

C₂

#19128009

yes

—

	subgroup	G/H
center	C₂³	C₂×D₃₃₂
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	C₂×C₄⋊C₈×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₈₃
hypocenter	C₈₃	#2086
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₃₃₂

commutator, cosocle

C₂²×C₄×C₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

C₂×C₄⋊C₈×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₈₃

hypocenter

C₈₃

#2086

LCS

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
67. C₈₃	#2086	semidirect

H ≅ G/N

type

67. C₈₃

#2086

semidirect

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

subgroup	index
C₂×#14581726	2
#19127599	2
#19127464	2
C₂×C₄⋊C₈×C₈₃	2
#2086	83

subgroup

index

C₂×C₄⋊C₈×C₈₃

Subgroups — #24908951

Conjugacy classes of subgroups

Complements

Maximal subgroups