Group #24905396 — subgroups

Subgroups — #24905396

Order 10624.

134 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 37 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	#19127606	C₂	1	1	yes	maximal, characteristic
2	#19127427	C₂	1	1	yes	maximal, characteristic
3	#324×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₆₆	C₄	1	1	yes	characteristic
5	C₂×C₄×Dic₈₃	C₄	1	1	yes	characteristic
6	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
7	C₈₃⋊#95	—	2	1	no	—
8	C₈₃⋊#95	—	2	1	no	—
9	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
10	C₂²⋊C₈×C₈₃	—	2	1	no	—
11	C₂×Dic₁₆₆	—	2	1	no	—
12	C₂×Dic₁₆₆	—	4	1	no	—
13	C₂×Dic₁₆₆	—	4	1	no	—
14	C₂×Dic₁₆₆	C₈	1	2	yes	complemented
15	C₂²×Dic₈₃	—	2	1	no	—
16	C₂²×Dic₈₃	—	2	1	no	—
17	C₂×C₈₃⋊C₈	—	4	1	no	—
18	C₄×Dic₈₃	—	4	1	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	C₂×C₈×C₈₃	—	4	1	no	—
23	C₄⋊C₄×C₈₃	—	4	1	no	—
24	C₂³×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₈₃	—	2	1	no	—
26	C₂×C₄×C₈₃	—	2	1	no	—
27	C₂×C₄×C₈₃	—	2	1	no	—
28	C₂×C₄×C₈₃	—	2	1	no	—
29	C₂×C₄×C₈₃	—	4	1	no	—
30	C₂×C₄×C₈₃	—	4	1	no	—
31	C₂×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₈₃	M₁₆	1	1	yes	characteristic
33	C₂×Dic₈₃	—	2	2	no	—
34	C₂×Dic₈₃	—	2	2	no	—
35	C₂×Dic₈₃	—	4	1	no	—
36	C₂×Dic₈₃	—	4	1	no	—
37	C₂×Dic₈₃	—	4	1	no	—
38	C₂×Dic₈₃	—	4	1	no	—
39	Dic₁₆₆	—	4	2	no	—
40	Dic₁₆₆	—	4	2	no	—
41	C₈×C₈₃	—	8	1	no	—
42	C₈₃⋊C₈	—	8	1	no	—
43	C₂²×C₈₃	—	2	1	no	—
44	C₂²×C₈₃	—	2	1	no	—
45	C₂²×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
46	C₂²×C₈₃	#99	1	1	yes	characteristic
47	C₂²×C₈₃	#100	1	1	yes	characteristic
48	C₄×C₈₃	—	2	1	no	—
49	C₄×C₈₃	—	2	1	no	—
50	C₄×C₈₃	—	4	1	no	—
51	C₄×C₈₃	—	4	1	no	—
52	Dic₈₃	—	4	2	no	—
53	Dic₈₃	—	4	2	no	—
54	C₂×C₈₃	—	2	1	no	—
55	C₂×C₈₃	—	2	1	no	—
56	C₂×C₈₃	#323	1	1	yes	characteristic
57	C₂×C₈₃	#354	1	1	yes	characteristic
58	C₂×C₈₃	#356	1	1	yes	characteristic
59	#1319	—	83	1	no	maximal, Sylow
60	C₈₃	#1319	1	1	yes	characteristic, complemented, Sylow
61	#391	—	83	1	no	—
62	#324	—	83	1	no	—
63	#324	D₈₃	1	1	yes	characteristic
64	C₂²×Q₈	—	83	1	no	—
65	C₂×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
66	C₂×C₄²	—	83	1	no	—
67	C₂²⋊C₈	—	2	1	no	—
68	C₂²⋊C₈	—	166	1	no	—
69	#95	—	166	1	no	—
70	#95	—	166	1	no	—
71	C₂×Q₈	—	332	1	no	—
72	C₂×Q₈	—	332	1	no	—
73	C₂×Q₈	—	83	2	no	—
74	C₂×Q₈	—	166	1	no	—
75	C₂²×C₄	—	166	1	no	—
76	C₂²×C₄	—	166	1	no	—
77	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
78	C₂²×C₄	D₃₃₂	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
80	C₂×C₈	—	4	1	no	—
81	C₂×C₈	—	332	1	no	—
82	C₄⋊C₄	—	4	1	no	—
83	C₄²	—	332	1	no	—
84	C₂³	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
85	Q₈	—	332	2	no	—
86	Q₈	—	332	2	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	—	4	1	no	—
93	C₂×C₄	—	332	1	no	—
94	C₂×C₄	—	332	1	no	—
95	C₂×C₄	—	332	1	no	—
96	C₂×C₄	—	332	1	no	—
97	C₂×C₄	—	166	2	no	—
98	C₂×C₄	—	166	2	no	—
99	C₂×C₄	C₈×D₈₃	1	1	yes	characteristic
100	C₂×C₄	#13050573	1	1	yes	characteristic
101	C₈	—	8	1	no	—
102	C₈	—	664	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	(C₂×D₈₃)⋊C₈	1	1	yes	characteristic
106	C₂²	#14581706	1	1	yes	characteristic
107	C₂²	#14581708	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	4	1	no	—
111	C₄	—	4	1	no	—
112	C₄	—	332	2	no	—
113	C₄	—	332	2	no	—
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	#19127424	1	1	yes	characteristic
117	C₂	#19127495	1	1	yes	characteristic
118	C₂	#19127499	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127606

C₂

yes

maximal, characteristic

#19127427

C₂

yes

maximal, characteristic

#324×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

C₄

yes

characteristic

C₂×C₄×Dic₈₃

C₄

yes

characteristic

C₂²⋊(C₈₃⋊C₈)

—

C₈₃⋊#95

—

C₈₃⋊#95

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

characteristic

C₂²⋊C₈×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

C₈

yes

complemented

C₂²×Dic₈₃

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

—

C₄⋊C₄×C₈₃

—

C₂³×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₈

yes

characteristic

C₂×C₄×C₈₃

M₁₆

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂²⋊C₈

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#391

—

#324

—

#324

D₈₃

yes

characteristic

C₂²×Q₈

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

characteristic

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

—

166

—

#95

—

166

—

#95

—

166

—

C₂×Q₈

—

332

—

C₂×Q₈

—

332

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

D₃₃₂

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

332

—

C₄⋊C₄

—

C₄²

—

332

—

C₂³

(C₂×D₈₃)⋊C₄

yes

characteristic

Q₈

—

332

—

Q₈

—

332

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

332

—

C₂×C₄

—

332

—

C₂×C₄

—

332

—

C₂×C₄

—

332

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

C₈×D₈₃

yes

characteristic

100

C₂×C₄

#13050573

yes

characteristic

101

C₈

—

102

C₈

—

664

—

103

C₂²

—

104

C₂²

—

105

C₂²

(C₂×D₈₃)⋊C₈

yes

characteristic

106

C₂²

#14581706

yes

characteristic

107

C₂²

#14581708

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

332

—

113

C₄

—

332

—

114

C₂

—

115

C₂

—

116

C₂

#19127424

yes

characteristic

117

C₂

#19127495

yes

characteristic

118

C₂

#19127499

yes

characteristic

	subgroup	G/H
center	C₂²	#14581706
commutator	C₂×C₄×C₈₃	C₂×C₈
Frattini	C₂×C₄⋊C₄	C₂×D₈₃
Fitting	#324×C₈₃	C₂
socle	C₂²×C₈₃	#99
prime radical	C₂³×C₈₃	C₂²⋊C₄
cosocle	C₂×C₄⋊C₄×C₈₃	C₂²
hypercenter	#324	D₈₃
hypocenter	C₈₃	#1319
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #324

subgroup

G/H

center

C₂²

#14581706

commutator

C₂×C₄×C₈₃

C₂×C₈

Frattini

C₂×C₄⋊C₄

C₂×D₈₃

Fitting

#324×C₈₃

C₂

socle

C₂²×C₈₃

#99

prime radical

C₂³×C₈₃

C₂²⋊C₄

cosocle

C₂×C₄⋊C₄×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #324

N	H ≅ G/N	type
14. C₂×Dic₁₆₆	C₈	semidirect
60. C₈₃	#1319	semidirect

H ≅ G/N

type

14. C₂×Dic₁₆₆

C₈

semidirect

60. C₈₃

#1319

semidirect

Subgroups — #24905396

Conjugacy classes of subgroups

Complements