Group #24905360 — subgroups

Subgroups — #24905360

Order 10624.

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

normal subgroups only

This group has 148 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	C₂×C₄⋊C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13050574	—	2	1	no	—
5	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
6	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
7	C₄⋊(C₈₃⋊C₈)	C₄	1	2	yes	—
8	C₂²⋊(C₈₃⋊C₈)	C₄	1	2	yes	—
9	C₄×C₈₃⋊C₈	—	2	1	no	—
10	C₂²×C₈×C₈₃	—	2	1	no	—
11	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₈₃	—	2	1	no	—
13	C₄⋊C₈×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	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₈₃	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₈₃	—	2	2	no	—
22	C₂×C₈×C₈₃	—	4	1	no	—
23	C₂×C₈×C₈₃	—	4	1	no	—
24	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₈₃	D₄	1	1	yes	characteristic
27	C₄²×C₈₃	D₄	1	1	yes	characteristic
28	C₂³×C₈₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₈₃	—	2	2	no	—
30	C₂×C₄×C₈₃	—	4	1	no	—
31	C₂×C₄×C₈₃	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₈₃	D₈	1	1	yes	characteristic
38	C₂×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₈₃	Q₁₆	1	1	yes	characteristic
41	C₈×C₈₃	—	4	2	no	—
42	C₈₃⋊C₈	—	4	2	no	—
43	C₈₃⋊C₈	—	4	2	no	—
44	C₂²×C₈₃	—	2	1	no	—
45	C₂²×C₈₃	—	2	1	no	—
46	C₂²×C₈₃	#95	1	1	yes	characteristic
47	C₂²×C₈₃	#106	1	1	yes	characteristic
48	C₂²×C₈₃	#107	1	1	yes	characteristic
49	C₄×C₈₃	—	2	1	no	—
50	C₄×C₈₃	—	2	1	no	—
51	C₄×C₈₃	—	2	1	no	—
52	C₄×C₈₃	#102	1	1	yes	characteristic
53	C₄×C₈₃	#102	1	1	yes	characteristic
54	C₄×C₈₃	#103	1	1	yes	characteristic
55	C₄×C₈₃	#103	1	1	yes	characteristic
56	C₄×C₈₃	#108	1	2	yes	—
57	C₂×C₈₃	—	2	1	no	—
58	C₂×C₈₃	—	2	1	no	—
59	C₂×C₈₃	#340	1	1	yes	characteristic
60	C₂×C₈₃	#341	1	1	yes	characteristic
61	C₂×C₈₃	#344	1	1	yes	characteristic
62	#1301	—	83	1	no	maximal, Sylow
63	C₈₃	#1301	1	1	yes	characteristic, complemented, Sylow
64	#431	—	83	1	no	—
65	#423	—	83	1	no	—
66	C₂×C₄⋊C₈	D₈₃	1	1	yes	characteristic
67	C₂×M₁₆	—	166	1	no	—
68	C₂²×C₈	—	2	1	no	—
69	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	83	2	no	—
72	C₄⋊C₈	—	166	1	no	—
73	C₄⋊C₈	—	166	1	no	—
74	C₄⋊C₈	Dic₈₃	1	2	yes	—
75	C₂²⋊C₈	—	83	2	no	—
76	C₄×C₈	—	166	1	no	—
77	C₂²×C₄	Dic₁₆₆	1	1	yes	characteristic
78	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
80	M₁₆	—	332	1	no	—
81	M₁₆	—	332	1	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	166	2	no	—
86	C₂×C₈	—	166	2	no	—
87	C₄²	C₄×D₈₃	1	1	yes	characteristic
88	C₄²	D₃₃₂	1	1	yes	characteristic
89	C₄²	C₂×Dic₈₃	1	1	yes	characteristic
90	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
91	C₂³	Dic₈₃⋊C₄	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	C₄×Dic₈₃	1	1	yes	characteristic
95	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
96	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
97	C₂×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
98	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
99	C₂×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
100	C₂×C₄	#13050575	1	1	yes	characteristic
101	C₂×C₄	#13050577	1	1	yes	characteristic
102	C₂×C₄	#13050578	1	1	yes	characteristic
103	C₂×C₄	#13050580	1	1	yes	characteristic
104	C₈	—	4	2	no	—
105	C₈	—	332	2	no	—
106	C₈	—	332	2	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	C₈₃⋊#95	1	1	yes	characteristic
110	C₂²	#14581723	1	1	yes	characteristic
111	C₂²	#14581725	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	#14581712	1	1	yes	characteristic
116	C₄	#14581713	1	1	yes	characteristic
117	C₄	#14581715	1	1	yes	characteristic
118	C₄	#14581716	1	1	yes	characteristic
119	C₄	#14581728	1	2	yes	—
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#19127463	1	1	yes	characteristic
123	C₂	#19127466	1	1	yes	characteristic
124	C₂	#19127473	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127754

C₂

yes

maximal, characteristic

#19127725

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

C₄

yes

—

C₂²⋊(C₈₃⋊C₈)

C₄

yes

—

C₄×C₈₃⋊C₈

—

C₂²×C₈×C₈₃

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

C₄

yes

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

Q₈

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₂³×C₈₃

C₄⋊C₄

yes

characteristic

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

C₂²⋊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

C₈×C₈₃

—

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

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

—

C₂×C₄⋊C₈

D₈₃

yes

characteristic

C₂×M₁₆

—

166

—

C₂²×C₈

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₄⋊C₈

—

166

—

C₄⋊C₈

Dic₈₃

yes

—

C₂²⋊C₈

—

C₄×C₈

—

166

—

C₂²×C₄

Dic₁₆₆

yes

characteristic

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

M₁₆

—

332

—

M₁₆

—

332

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₄²

C₄×D₈₃

yes

characteristic

C₄²

D₃₃₂

yes

characteristic

C₄²

C₂×Dic₈₃

yes

characteristic

C₄²

C₂²⋊D₈₃

yes

characteristic

C₂³

Dic₈₃⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₈₃

yes

characteristic

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₈₃

yes

characteristic

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₈₃

yes

characteristic

100

C₂×C₄

#13050575

yes

characteristic

101

C₂×C₄

#13050577

yes

characteristic

102

C₂×C₄

#13050578

yes

characteristic

103

C₂×C₄

#13050580

yes

characteristic

104

C₈

—

105

C₈

—

332

—

106

C₈

—

332

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₈₃⋊#95

yes

characteristic

110

C₂²

#14581723

yes

characteristic

111

C₂²

#14581725

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#14581712

yes

characteristic

116

C₄

#14581713

yes

characteristic

117

C₄

#14581715

yes

characteristic

118

C₄

#14581716

yes

characteristic

119

C₄

#14581728

yes

—

120

C₂

—

121

C₂

—

122

C₂

#19127463

yes

characteristic

123

C₂

#19127466

yes

characteristic

124

C₂

#19127473

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₈₃)⋊C₄
commutator	C₂×C₄×C₈₃	C₄²
Frattini	C₂×C₄²	C₂×D₈₃
Fitting	C₂×C₄⋊C₈×C₈₃	C₂
socle	C₂²×C₈₃	#95
prime radical	C₂³×C₈₃	C₄⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂²
hypercenter	C₂×C₄⋊C₈	D₈₃
hypocenter	C₈₃	#1301
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

(C₂×D₈₃)⋊C₄

commutator

C₂×C₄×C₈₃

C₄²

Frattini

C₂×C₄²

C₂×D₈₃

Fitting

C₂×C₄⋊C₈×C₈₃

C₂

socle

C₂²×C₈₃

#95

prime radical

C₂³×C₈₃

C₄⋊C₄

cosocle

C₂×C₄²×C₈₃

C₂²

hypercenter

C₂×C₄⋊C₈

D₈₃

hypocenter

C₈₃

#1301

LCS

G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
63. C₈₃	#1301	semidirect

H ≅ G/N

type

63. C₈₃

#1301

semidirect

Subgroups — #24905360

Conjugacy classes of subgroups

Complements