Group #24905752 — subgroups

Subgroups — #24905752

Order 10624.

208 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₄×#13050574	C₂	1	1	yes	maximal, characteristic
3	#19127423	C₂	1	2	yes	maximal, complemented
4	#19127422	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	C₄	1	2	yes	—
7	C₄⋊(C₈₃⋊C₈)	C₄	1	2	yes	—
8	C₂²⋊(C₈₃⋊C₈)	C₄	1	2	yes	—
9	#14581703	C₄	1	2	yes	—
10	C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
11	C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₈×C₈₃	C₄	1	2	yes	—
13	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₈₃	C₄	1	2	yes	—
15	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#13050574	—	2	4	no	—
18	C₂×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
19	C₂×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₈₃	—	2	4	no	—
24	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
26	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₈₃	C₂³	1	1	yes	characteristic
30	C₂³×C₈₃	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₈₃	—	2	2	no	—
32	C₂×C₄×C₈₃	—	2	2	no	—
33	C₂×C₄×C₈₃	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₈₃	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₈₃	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₈₃	M₁₆	1	4	yes	—
37	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₈₃	—	2	2	no	—
41	C₈×C₈₃	M₁₆	1	4	yes	complemented
42	C₈₃⋊C₈	—	2	4	no	—
43	C₈₃⋊C₈	—	2	4	no	—
44	C₂²×C₈₃	—	2	1	no	—
45	C₂²×C₈₃	—	2	1	no	—
46	C₂²×C₈₃	#97	1	2	yes	—
47	C₂²×C₈₃	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₈₃	—	2	1	no	—
49	C₄×C₈₃	—	2	1	no	—
50	C₄×C₈₃	#97	1	2	yes	—
51	C₄×C₈₃	C₂×M₁₆	1	2	yes	—
52	C₄×C₈₃	C₂×M₁₆	1	2	yes	—
53	C₄×C₈₃	#131	1	2	yes	—
54	C₂×C₈₃	—	2	2	no	—
55	C₂×C₈₃	C₂×#97	1	1	yes	characteristic
56	C₂×C₈₃	C₄×M₁₆	1	1	yes	characteristic
57	C₂×C₈₃	#405	1	1	yes	characteristic
58	#1451	—	83	1	no	maximal, Sylow
59	C₈₃	#1451	1	1	yes	characteristic, complemented, Sylow
60	#431	—	83	1	no	—
61	C₄×M₁₆	—	83	1	no	—
62	C₂×C₄×C₈	D₈₃	1	1	yes	characteristic
63	#322	—	83	2	no	—
64	#322	—	83	2	no	—
65	C₂×M₁₆	—	83	2	no	—
66	C₂²×C₈	Dic₈₃	1	2	yes	—
67	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
68	C₄⋊C₈	—	83	2	no	—
69	C₂²⋊C₈	—	83	2	no	—
70	#97	—	83	2	no	—
71	C₄×C₈	—	83	2	no	—
72	C₄×C₈	—	83	2	no	—
73	C₄×C₈	Dic₈₃	1	2	yes	—
74	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
75	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
76	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
77	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
79	M₁₆	—	166	4	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	—	83	4	no	—
82	C₂×C₈	—	83	4	no	—
83	C₂×C₈	C₂×Dic₈₃	1	2	yes	—
84	C₂×C₈	C₂×Dic₈₃	1	2	yes	—
85	C₄²	C₄×D₈₃	1	1	yes	characteristic
86	C₄²	C₄×D₈₃	1	1	yes	characteristic
87	C₄²	C₂×Dic₈₃	1	1	yes	characteristic
88	C₄²	C₂²×D₈₃	1	1	yes	characteristic
89	C₂³	C₄×Dic₈₃	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	C₄×Dic₈₃	1	1	yes	characteristic
93	C₂×C₄	C₄×Dic₈₃	1	1	yes	characteristic
94	C₂×C₄	C₄×Dic₈₃	1	1	yes	characteristic
95	C₂×C₄	#13050573	1	4	yes	—
96	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
97	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
98	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	166	4	no	—
101	C₈	—	166	4	no	—
102	C₈	#13050574	1	4	yes	complemented
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	#14581702	1	2	yes	—
106	C₂²	C₂×C₄×Dic₈₃	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	#14581702	1	2	yes	—
110	C₄	C₂×#13050573	1	2	yes	—
111	C₄	C₂×#13050574	1	2	yes	—
112	C₄	#14581798	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	C₂×#14581702	1	1	yes	characteristic
115	C₂	C₄×#13050574	1	1	yes	characteristic
116	C₂	#19127656	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127754

C₂

yes

maximal, characteristic

C₄×#13050574

C₂

yes

maximal, characteristic

#19127423

C₂

yes

maximal, complemented

#19127422

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₄

yes

—

C₄⋊(C₈₃⋊C₈)

C₄

yes

—

C₂²⋊(C₈₃⋊C₈)

C₄

yes

—

#14581703

C₄

yes

—

C₄×C₈₃⋊C₈

C₂²

yes

—

C₄×C₈₃⋊C₈

C₂²

yes

—

C₂²×C₈×C₈₃

C₄

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄×C₈×C₈₃

C₄

yes

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₄×C₈×C₈₃

C₂²

yes

characteristic

#13050574

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂³

yes

characteristic

C₂³×C₈₃

C₄²

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄²

yes

characteristic

C₂×C₄×C₈₃

C₄²

yes

characteristic

C₂×C₄×C₈₃

C₄²

yes

characteristic

C₂×C₄×C₈₃

M₁₆

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

M₁₆

yes

complemented

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#97

yes

—

C₂²×C₈₃

C₂×C₄²

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#97

yes

—

C₄×C₈₃

C₂×M₁₆

yes

—

C₄×C₈₃

C₂×M₁₆

yes

—

C₄×C₈₃

#131

yes

—

C₂×C₈₃

—

C₂×C₈₃

C₂×#97

yes

characteristic

C₂×C₈₃

C₄×M₁₆

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

D₈₃

yes

characteristic

#322

—

#322

—

C₂×M₁₆

—

C₂²×C₈

Dic₈₃

yes

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₈₃

yes

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

C₂×Dic₈₃

yes

characteristic

M₁₆

—

166

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₈₃

yes

—

C₂×C₈

C₂×Dic₈₃

yes

—

C₄²

C₄×D₈₃

yes

characteristic

C₄²

C₄×D₈₃

yes

characteristic

C₄²

C₂×Dic₈₃

yes

characteristic

C₄²

C₂²×D₈₃

yes

characteristic

C₂³

C₄×Dic₈₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₈₃

yes

characteristic

C₂×C₄

C₄×Dic₈₃

yes

characteristic

C₂×C₄

C₄×Dic₈₃

yes

characteristic

C₂×C₄

#13050573

yes

—

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

C₈

—

100

C₈

—

166

—

101

C₈

—

166

—

102

C₈

#13050574

yes

complemented

103

C₂²

—

104

C₂²

—

105

C₂²

#14581702

yes

—

106

C₂²

C₂×C₄×Dic₈₃

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

#14581702

yes

—

110

C₄

C₂×#13050573

yes

—

111

C₄

C₂×#13050574

yes

—

112

C₄

#14581798

yes

—

113

C₂

—

114

C₂

C₂×#14581702

yes

characteristic

115

C₂

C₄×#13050574

yes

characteristic

116

C₂

#19127656

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₈₃
commutator, socle	C₂²×C₈₃	C₂×C₄²
Fitting	C₂×C₄×C₈×C₈₃	C₂
prime radical	C₂³×C₈₃	C₄²
cosocle	C₄²×C₈₃	C₂³
hypercenter	C₂×C₄×C₈	D₈₃
hypocenter	C₈₃	#1451
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₄²

C₂²×D₈₃

commutator, socle

C₂²×C₈₃

C₂×C₄²

Fitting

C₂×C₄×C₈×C₈₃

C₂

prime radical

C₂³×C₈₃

C₄²

cosocle

C₄²×C₈₃

C₂³

hypercenter

C₂×C₄×C₈

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₈₃

C₁ ◁ C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
3. #19127423	C₂	semidirect
4. #19127422	C₂	semidirect
41. C₈×C₈₃	M₁₆	semidirect
59. C₈₃	#1451	semidirect
102. C₈	#13050574	semidirect

H ≅ G/N

type

3. #19127423

C₂

semidirect

4. #19127422

C₂

semidirect

41. C₈×C₈₃

M₁₆

semidirect

59. C₈₃

#1451

semidirect

102. C₈

#13050574

semidirect

Subgroups — #24905752

Conjugacy classes of subgroups

Complements