Group #24905331 — subgroups

Subgroups — #24905331

Order 10624.

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

normal subgroups only

This group has 228 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	#19127561	C₂	1	2	yes	maximal
2	C₂×C₂²⋊C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊Dic₈₃	—	2	2	no	—
4	C₂×C₂²⋊Dic₈₃	—	2	2	no	—
5	C₈₃⋊#95	C₄	1	4	yes	complemented
6	C₂³×C₄×C₈₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₈₃	C₄	1	2	yes	complemented
8	C₂²⋊C₈×C₈₃	—	2	2	no	—
9	C₂²×Dic₈₃	—	2	4	no	—
10	C₄⋊Dic₈₃	—	4	4	no	—
11	C₂²⋊Dic₈₃	—	4	2	no	—
12	C₂²⋊Dic₈₃	—	4	2	no	—
13	C₂⁴×C₈₃	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₈₃	—	4	2	no	—
15	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₈₃	D₄	1	2	yes	—
18	C₂²×C₄×C₈₃	Q₈	1	1	yes	characteristic
19	C₂×C₈×C₈₃	—	2	2	no	—
20	C₂×C₈×C₈₃	—	4	2	no	—
21	C₂³×C₈₃	—	4	1	no	—
22	C₂³×C₈₃	—	4	1	no	—
23	C₂³×C₈₃	C₂²⋊C₄	1	2	yes	—
24	C₂³×C₈₃	C₄⋊C₄	1	1	yes	characteristic
25	C₂³×C₈₃	D₈	1	1	yes	characteristic
26	C₂³×C₈₃	SD₁₆	1	2	yes	—
27	C₂³×C₈₃	Q₁₆	1	1	yes	characteristic
28	C₂×C₄×C₈₃	—	2	2	no	—
29	C₂×C₄×C₈₃	—	2	4	no	—
30	C₂×C₄×C₈₃	—	4	2	no	—
31	C₂×C₄×C₈₃	—	4	1	no	—
32	C₂×C₄×C₈₃	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
35	C₂×Dic₈₃	—	4	4	no	—
36	C₂×Dic₈₃	—	4	4	no	—
37	C₂×Dic₈₃	—	4	4	no	—
38	C₈×C₈₃	—	4	2	no	—
39	C₂²×C₈₃	—	2	2	no	—
40	C₂²×C₈₃	—	2	1	no	—
41	C₂²×C₈₃	—	2	2	no	—
42	C₂²×C₈₃	—	2	1	no	—
43	C₂²×C₈₃	—	4	1	no	—
44	C₂²×C₈₃	—	4	2	no	—
45	C₂²×C₈₃	—	4	1	no	—
46	C₂²×C₈₃	#95	1	1	yes	characteristic
47	C₂²×C₈₃	#102	1	2	yes	—
48	C₂²×C₈₃	#103	1	2	yes	—
49	C₂²×C₈₃	#106	1	1	yes	characteristic
50	C₂²×C₈₃	#107	1	1	yes	characteristic
51	C₄×C₈₃	—	2	1	no	—
52	C₄×C₈₃	—	2	2	no	—
53	C₄×C₈₃	#99	1	2	yes	—
54	Dic₈₃	—	8	4	no	—
55	C₂×C₈₃	—	2	2	no	—
56	C₂×C₈₃	—	2	2	no	—
57	C₂×C₈₃	—	2	1	no	—
58	C₂×C₈₃	—	2	1	no	—
59	C₂×C₈₃	H(3,ℤ/4)	1	1	yes	characteristic
60	C₂×C₈₃	#340	1	1	yes	characteristic
61	C₂×C₈₃	#342	1	1	yes	characteristic
62	#1289	—	83	1	no	maximal, Sylow
63	C₈₃	#1289	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₂²⋊C₈	D₈₃	1	1	yes	characteristic
65	#380	—	83	2	no	—
66	C₂³×C₄	C₂×D₈₃	1	1	yes	characteristic
67	C₂²×C₈	Dic₈₃	1	2	yes	complemented
68	C₂×C₄⋊C₄	—	166	2	no	—
69	C₂×C₂²⋊C₄	—	166	2	no	—
70	C₂²⋊C₈	—	2	2	no	—
71	#95	—	83	4	no	—
72	C₂⁴	D₃₃₂	1	1	yes	characteristic
73	C₂²×C₄	—	4	2	no	—
74	C₂²×C₄	—	166	4	no	—
75	C₂²×C₄	Dic₁₆₆	1	1	yes	characteristic
76	C₂²×C₄	C₄×D₈₃	1	2	yes	—
77	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
78	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	4	2	no	—
81	C₄⋊C₄	—	332	4	no	—
82	C₂²⋊C₄	—	332	2	no	—
83	C₂²⋊C₄	—	332	2	no	—
84	C₂³	—	4	1	no	—
85	C₂³	—	4	1	no	—
86	C₂³	(C₂×D₈₃)⋊C₄	1	2	yes	—
87	C₂³	C₄⋊Dic₈₃	1	1	yes	characteristic
88	C₂³	D₆₆₄	1	1	yes	characteristic
89	C₂³	#13050579	1	2	yes	—
90	C₂³	Dic₃₃₂	1	1	yes	characteristic
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	332	4	no	—
96	C₂×C₄	—	332	4	no	—
97	C₂×C₄	—	332	4	no	—
98	C₂×C₄	C₄×Dic₈₃	1	1	yes	characteristic
99	C₂×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
100	C₂×C₄	Dic₈₃⋊C₄	1	2	yes	—
101	C₈	—	4	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	4	1	no	—
107	C₂²	—	4	2	no	—
108	C₂²	—	4	1	no	—
109	C₂²	C₈₃⋊#95	1	1	yes	characteristic
110	C₂²	#14581714	1	2	yes	—
111	C₂²	#14581717	1	2	yes	—
112	C₂²	#14581724	1	1	yes	characteristic
113	C₂²	#14581726	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	2	2	no	—
116	C₄	—	664	4	no	—
117	C₄	#14581707	1	2	yes	—
118	C₂	—	2	2	no	—
119	C₂	—	2	2	no	—
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#19127458	1	1	yes	characteristic
123	C₂	#19127464	1	1	yes	characteristic
124	C₂	#19127468	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127561

C₂

yes

maximal

C₂×C₂²⋊C₈×C₈₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₈₃

—

C₂×C₂²⋊Dic₈₃

—

C₈₃⋊#95

C₄

yes

complemented

C₂³×C₄×C₈₃

C₂²

yes

characteristic

C₂²×C₈×C₈₃

C₄

yes

complemented

C₂²⋊C₈×C₈₃

—

C₂²×Dic₈₃

—

C₄⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₂⁴×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

Q₈

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

C₂²⋊C₄

yes

—

C₂³×C₈₃

C₄⋊C₄

yes

characteristic

C₂³×C₈₃

D₈

yes

characteristic

C₂³×C₈₃

SD₁₆

yes

—

C₂³×C₈₃

Q₁₆

yes

characteristic

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

C₄⋊C₄

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#95

yes

characteristic

C₂²×C₈₃

#102

yes

—

C₂²×C₈₃

#103

yes

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#99

yes

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×C₂²⋊C₈

D₈₃

yes

characteristic

#380

—

C₂³×C₄

C₂×D₈₃

yes

characteristic

C₂²×C₈

Dic₈₃

yes

complemented

C₂×C₄⋊C₄

—

166

—

C₂×C₂²⋊C₄

—

166

—

C₂²⋊C₈

—

#95

—

C₂⁴

D₃₃₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

166

—

C₂²×C₄

Dic₁₆₆

yes

characteristic

C₂²×C₄

C₄×D₈₃

yes

—

C₂²×C₄

C₂×Dic₈₃

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

332

—

C₂²⋊C₄

—

332

—

C₂²⋊C₄

—

332

—

C₂³

—

C₂³

—

C₂³

(C₂×D₈₃)⋊C₄

yes

—

C₂³

C₄⋊Dic₈₃

yes

characteristic

C₂³

D₆₆₄

yes

characteristic

C₂³

#13050579

yes

—

C₂³

Dic₃₃₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

332

—

C₂×C₄

—

332

—

C₂×C₄

—

332

—

C₂×C₄

C₄×Dic₈₃

yes

characteristic

C₂×C₄

C₂²⋊Dic₈₃

yes

characteristic

100

C₂×C₄

Dic₈₃⋊C₄

yes

—

101

C₈

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₈₃⋊#95

yes

characteristic

110

C₂²

#14581714

yes

—

111

C₂²

#14581717

yes

—

112

C₂²

#14581724

yes

characteristic

113

C₂²

#14581726

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

664

—

117

C₄

#14581707

yes

—

118

C₂

—

119

C₂

—

120

C₂

—

121

C₂

—

122

C₂

#19127458

yes

characteristic

123

C₂

#19127464

yes

characteristic

124

C₂

#19127468

yes

characteristic

	subgroup	G/H
center	C₂²	C₈₃⋊#95
commutator	C₂×C₄×C₈₃	C₄²
Frattini	C₂³×C₄	C₂×D₈₃
Fitting	C₂×C₂²⋊C₈×C₈₃	C₂
socle	C₂²×C₈₃	#95
prime radical	C₂⁴×C₈₃	D₄
cosocle	C₂³×C₄×C₈₃	C₂²
hypercenter	C₂×C₂²⋊C₈	D₈₃
hypocenter	C₈₃	#1289
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂²

C₈₃⋊#95

commutator

C₂×C₄×C₈₃

C₄²

Frattini

C₂³×C₄

C₂×D₈₃

Fitting

C₂×C₂²⋊C₈×C₈₃

C₂

socle

C₂²×C₈₃

#95

prime radical

C₂⁴×C₈₃

D₄

cosocle

C₂³×C₄×C₈₃

C₂²

hypercenter

C₂×C₂²⋊C₈

D₈₃

hypocenter

C₈₃

#1289

LCS

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
5. C₈₃⋊#95	C₄	semidirect
7. C₂²×C₈×C₈₃	C₄	semidirect
63. C₈₃	#1289	semidirect
67. C₂²×C₈	Dic₈₃	semidirect

H ≅ G/N

type

5. C₈₃⋊#95

C₄

semidirect

7. C₂²×C₈×C₈₃

C₄

semidirect

63. C₈₃

#1289

semidirect

67. C₂²×C₈

Dic₈₃

semidirect

Subgroups — #24905331

Conjugacy classes of subgroups

Complements