Group #24905366 — subgroups

Subgroups — #24905366

Order 10624.

176 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 77 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	#19127754	C₂	1	1	yes	maximal, characteristic
2	#19127725	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₄×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13050574	—	2	1	no	—
5	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
6	C₄⋊(C₈₃⋊C₈)	C₄	1	2	yes	complemented
7	C₄⋊(C₈₃⋊C₈)	C₄	1	2	yes	complemented
8	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
9	C₄×C₈₃⋊C₈	—	2	1	no	—
10	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
11	C₂×C₄²×C₈₃	—	2	1	no	—
12	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
13	#95×C₈₃	C₄	1	2	yes	—
14	#13050574	—	4	1	no	—
15	#13050574	—	4	1	no	—
16	C₂×C₈₃⋊C₈	—	2	2	no	—
17	C₂×C₈₃⋊C₈	—	2	2	no	—
18	C₂²×C₄×C₈₃	—	2	2	no	—
19	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
22	C₄⋊C₄×C₈₃	—	2	4	no	—
23	C₄²×C₈₃	—	4	1	no	—
24	C₄²×C₈₃	—	4	1	no	—
25	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₈₃	D₄	1	1	yes	characteristic
28	C₄²×C₈₃	Q₈	1	1	yes	characteristic
29	C₂³×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₈₃	—	2	2	no	—
31	C₂×C₄×C₈₃	—	2	4	no	—
32	C₂×C₄×C₈₃	—	4	1	no	—
33	C₂×C₄×C₈₃	—	4	2	no	—
34	C₂×C₄×C₈₃	—	4	2	no	—
35	C₂×C₄×C₈₃	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₈₃	D₈	1	1	yes	characteristic
42	C₂×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
43	C₂×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
44	C₂×C₄×C₈₃	Q₁₆	1	1	yes	characteristic
45	C₈₃⋊C₈	—	4	2	no	—
46	C₈₃⋊C₈	—	4	2	no	—
47	C₂²×C₈₃	—	2	1	no	—
48	C₂²×C₈₃	—	2	1	no	—
49	C₂²×C₈₃	#95	1	1	yes	characteristic
50	C₂²×C₈₃	#102	1	1	yes	characteristic
51	C₂²×C₈₃	#103	1	1	yes	characteristic
52	C₄×C₈₃	—	2	1	no	—
53	C₄×C₈₃	—	2	1	no	—
54	C₄×C₈₃	—	2	1	no	—
55	C₄×C₈₃	—	4	4	no	—
56	C₄×C₈₃	#102	1	1	yes	characteristic
57	C₄×C₈₃	#103	1	1	yes	characteristic
58	C₄×C₈₃	#104	1	2	yes	—
59	C₄×C₈₃	#106	1	1	yes	characteristic
60	C₄×C₈₃	#107	1	1	yes	characteristic
61	C₂×C₈₃	—	2	1	no	—
62	C₂×C₈₃	—	2	1	no	—
63	C₂×C₈₃	#339	1	1	yes	characteristic
64	C₂×C₈₃	#340	1	1	yes	characteristic
65	C₂×C₈₃	#344	1	1	yes	characteristic
66	#1303	—	83	1	no	maximal, Sylow
67	C₈₃	#1303	1	1	yes	characteristic, complemented, Sylow
68	#431	—	83	1	no	—
69	#423	—	83	1	no	—
70	C₄×C₄⋊C₄	D₈₃	1	1	yes	characteristic
71	C₂×M₁₆	—	166	1	no	—
72	C₂×C₄⋊C₄	—	2	1	no	—
73	C₂×C₄²	—	2	1	no	—
74	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
75	C₄⋊C₈	—	83	2	no	—
76	C₄⋊C₈	—	83	2	no	—
77	C₄⋊C₈	—	166	1	no	—
78	C₂²⋊C₈	—	166	1	no	—
79	C₄×C₈	—	166	1	no	—
80	#95	Dic₈₃	1	2	yes	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
83	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
84	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
85	M₁₆	—	332	1	no	—
86	M₁₆	—	332	1	no	—
87	C₂×C₈	—	166	2	no	—
88	C₂×C₈	—	166	2	no	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄²	—	4	1	no	—
91	C₄²	—	4	1	no	—
92	C₄²	Dic₁₆₆	1	1	yes	characteristic
93	C₄²	C₄×D₈₃	1	1	yes	characteristic
94	C₄²	C₄×D₈₃	1	1	yes	characteristic
95	C₄²	D₃₃₂	1	1	yes	characteristic
96	C₂³	C₂²⋊Dic₈₃	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	C₄×Dic₈₃	1	1	yes	characteristic
103	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
104	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
105	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
106	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
107	C₂×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
108	C₂×C₄	#13050575	1	1	yes	characteristic
109	C₂×C₄	#13050577	1	1	yes	characteristic
110	C₂×C₄	#13050578	1	1	yes	characteristic
111	C₂×C₄	#13050580	1	1	yes	characteristic
112	C₈	—	332	2	no	—
113	C₈	—	332	2	no	—
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	C₈₃⋊#95	1	1	yes	characteristic
117	C₂²	#14581713	1	1	yes	characteristic
118	C₂²	#14581716	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	2	1	no	—
121	C₄	—	2	1	no	—
122	C₄	—	4	4	no	—
123	C₄	#14581712	1	1	yes	characteristic
124	C₄	#14581715	1	1	yes	characteristic
125	C₄	#14581720	1	2	yes	—
126	C₄	#14581723	1	1	yes	characteristic
127	C₄	#14581725	1	1	yes	characteristic
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#19127461	1	1	yes	characteristic
131	C₂	#19127463	1	1	yes	characteristic
132	C₂	#19127471	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127754

C₂

yes

maximal, characteristic

#19127725

C₂

yes

maximal, characteristic

C₄×C₄⋊C₄×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

C₄

yes

complemented

C₄⋊(C₈₃⋊C₈)

C₄

yes

complemented

C₂²⋊(C₈₃⋊C₈)

—

C₄×C₈₃⋊C₈

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

#95×C₈₃

C₄

yes

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

Q₈

yes

characteristic

C₂³×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄²

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

D₈

yes

characteristic

C₂×C₄×C₈₃

SD₁₆

yes

characteristic

C₂×C₄×C₈₃

SD₁₆

yes

characteristic

C₂×C₄×C₈₃

Q₁₆

yes

characteristic

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#431

—

#423

—

C₄×C₄⋊C₄

D₈₃

yes

characteristic

C₂×M₁₆

—

166

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₂²⋊C₈

—

166

—

C₄×C₈

—

166

—

#95

Dic₈₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₈₃

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

M₁₆

—

332

—

M₁₆

—

332

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

Dic₁₆₆

yes

characteristic

C₄²

C₄×D₈₃

yes

characteristic

C₄²

C₄×D₈₃

yes

characteristic

C₄²

D₃₃₂

yes

characteristic

C₂³

C₂²⋊Dic₈₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₄×Dic₈₃

yes

characteristic

103

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

104

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

105

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

106

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

107

C₂×C₄

C₄⋊Dic₈₃

yes

characteristic

108

C₂×C₄

#13050575

yes

characteristic

109

C₂×C₄

#13050577

yes

characteristic

110

C₂×C₄

#13050578

yes

characteristic

111

C₂×C₄

#13050580

yes

characteristic

112

C₈

—

332

—

113

C₈

—

332

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₈₃⋊#95

yes

characteristic

117

C₂²

#14581713

yes

characteristic

118

C₂²

#14581716

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#14581712

yes

characteristic

124

C₄

#14581715

yes

characteristic

125

C₄

#14581720

yes

—

126

C₄

#14581723

yes

characteristic

127

C₄

#14581725

yes

characteristic

128

C₂

—

129

C₂

—

130

C₂

#19127461

yes

characteristic

131

C₂

#19127463

yes

characteristic

132

C₂

#19127471

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₈₃)⋊C₄
commutator	C₂×C₄×C₈₃	C₄²
Frattini	C₂×C₄²	C₂×D₈₃
Fitting	C₄×C₄⋊C₄×C₈₃	C₂
socle	C₂²×C₈₃	#95
prime radical	C₂³×C₈₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂²
hypercenter	C₄×C₄⋊C₄	D₈₃
hypocenter	C₈₃	#1303
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂×C₄

(C₂×D₈₃)⋊C₄

commutator

C₂×C₄×C₈₃

C₄²

Frattini

C₂×C₄²

C₂×D₈₃

Fitting

C₄×C₄⋊C₄×C₈₃

C₂

socle

C₂²×C₈₃

#95

prime radical

C₂³×C₈₃

C₂²⋊C₄

cosocle

C₂×C₄²×C₈₃

C₂²

hypercenter

C₄×C₄⋊C₄

D₈₃

hypocenter

C₈₃

#1303

LCS

G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
6. C₄⋊(C₈₃⋊C₈)	C₄	semidirect
7. C₄⋊(C₈₃⋊C₈)	C₄	semidirect
67. C₈₃	#1303	semidirect

H ≅ G/N

type

6. C₄⋊(C₈₃⋊C₈)

C₄

semidirect

7. C₄⋊(C₈₃⋊C₈)

C₄

semidirect

67. C₈₃

#1303

semidirect

Subgroups — #24905366

Conjugacy classes of subgroups

Complements