Group #24908439 — subgroups

Subgroups — #24908439

Order 10624.

284 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 97 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	#19128258	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	#19127462	C₂	1	4	yes	maximal
4	#423×C₈₃	C₂	1	1	yes	maximal, characteristic
5	#14581792	C₄	1	2	yes	complemented
6	#14581784	C₄	1	2	yes	complemented
7	#14581771	—	2	2	no	—
8	C₄×Dic₁₆₆	—	2	2	no	—
9	#14581751	C₂²	1	2	yes	—
10	C₂×C₄×Dic₈₃	—	2	2	no	—
11	C₂×C₄×Dic₈₃	C₂²	1	2	yes	—
12	C₂×M₁₆×C₈₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₈₃	—	2	2	no	—
15	C₂×Dic₁₆₆	—	2	2	no	—
16	C₂²×Dic₈₃	—	2	2	no	—
17	C₄⋊Dic₈₃	—	2	2	no	—
18	C₄⋊Dic₈₃	C₂×C₄	1	4	yes	—
19	Dic₈₃⋊C₄	—	2	4	no	—
20	C₂²⋊Dic₈₃	—	2	2	no	—
21	C₄×Dic₈₃	—	2	8	no	—
22	C₄×Dic₈₃	—	2	2	no	—
23	C₄×Dic₈₃	—	2	2	no	—
24	C₄×Dic₈₃	Q₈	1	4	yes	complemented
25	C₂²×C₄×C₈₃	D₄	1	2	yes	—
26	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
27	M₁₆×C₈₃	—	2	4	no	—
28	C₂×C₈×C₈₃	—	2	2	no	—
29	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₈₃	D₄	1	2	yes	—
32	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₈₃	—	2	4	no	—
34	C₂×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₈₃	#41	1	2	yes	—
40	C₂×Dic₈₃	—	2	8	no	—
41	C₂×Dic₈₃	—	2	4	no	—
42	C₂×Dic₈₃	—	2	4	no	—
43	Dic₁₆₆	—	4	4	no	—
44	C₈×C₈₃	—	4	2	no	—
45	C₂²×C₈₃	—	2	2	no	—
46	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₈₃	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₈₃	—	2	2	no	—
49	C₄×C₈₃	#104	1	4	yes	—
50	C₄×C₈₃	C₄×Q₈	1	2	yes	—
51	C₄×C₈₃	#124	1	1	yes	characteristic
52	C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₈₃	—	2	8	no	—
54	Dic₈₃	—	4	4	no	—
55	C₂×C₈₃	—	2	2	no	—
56	C₂×C₈₃	#391	1	1	yes	characteristic
57	C₂×C₈₃	C₂×#104	1	2	yes	—
58	#1994	—	83	1	no	maximal, Sylow
59	C₈₃	#1994	1	1	yes	characteristic, complemented, Sylow
60	#533	—	83	1	no	—
61	#423	D₈₃	1	1	yes	characteristic
62	C₄³	—	83	1	no	—
63	#339	—	83	4	no	—
64	C₂×M₁₆	C₂×D₈₃	1	2	yes	—
65	C₄⋊Q₈	—	83	2	no	—
66	#125	—	83	2	no	—
67	C₂²⋊Q₈	—	166	2	no	—
68	C₄×Q₈	—	166	2	no	—
69	#117	—	83	2	no	—
70	C₂×C₄²	—	83	2	no	—
71	C₂×C₄²	—	166	2	no	—
72	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
73	C₄⋊C₈	—	2	2	no	—
74	C₂×Q₈	—	166	2	no	—
75	C₂²×C₄	—	166	2	no	—
76	C₂²×C₄	D₃₃₂	1	2	yes	—
77	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
78	M₁₆	—	2	4	no	—
79	C₂×C₈	—	2	2	no	—
80	C₄⋊C₄	—	83	4	no	—
81	C₄⋊C₄	—	166	4	no	—
82	C₄⋊C₄	—	166	2	no	—
83	C₂²⋊C₄	—	166	2	no	—
84	C₄²	—	83	4	no	—
85	C₄²	—	166	8	no	—
86	C₄²	—	166	2	no	—
87	C₄²	—	166	2	no	—
88	C₄²	C₄×D₈₃	1	1	yes	characteristic
89	C₄²	C₄×D₈₃	1	1	yes	characteristic
90	C₄²	C₂²⋊D₈₃	1	2	yes	—
91	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
92	Q₈	—	332	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	166	4	no	—
95	C₂×C₄	—	166	8	no	—
96	C₂×C₄	—	166	4	no	—
97	C₂×C₄	(C₂×D₈₃)⋊C₄	1	2	yes	—
98	C₂×C₄	(C₂×D₈₃)⋊C₄	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
101	C₂×C₄	Q₈×D₈₃	1	2	yes	—
102	C₂×C₄	#13050589	1	2	yes	—
103	C₈	—	4	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
106	C₂²	#14581772	1	2	yes	—
107	C₄	—	2	2	no	—
108	C₄	—	332	4	no	—
109	C₄	—	166	8	no	—
110	C₄	#14581718	1	4	yes	—
111	C₄	#14581757	1	2	yes	—
112	C₄	#14581779	1	1	yes	characteristic
113	C₄	#14581790	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#19127606	1	1	yes	characteristic
116	C₂	C₂×#14581718	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19128258

C₂

yes

maximal, characteristic

C₄²×Dic₈₃

C₂

yes

maximal, characteristic

#19127462

C₂

yes

maximal

#423×C₈₃

C₂

yes

maximal, characteristic

#14581792

C₄

yes

complemented

#14581784

C₄

yes

complemented

#14581771

—

C₄×Dic₁₆₆

—

#14581751

C₂²

yes

—

C₂×C₄×Dic₈₃

—

C₂×C₄×Dic₈₃

C₂²

yes

—

C₂×M₁₆×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

Dic₈₃⋊C₄

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

Q₈

yes

complemented

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

D₄

yes

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

C₈×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₃

C₂²⋊Q₈

yes

—

C₄×C₈₃

—

C₄×C₈₃

#104

yes

—

C₄×C₈₃

C₄×Q₈

yes

—

C₄×C₈₃

#124

yes

characteristic

C₄×C₈₃

C₄⋊Q₈

yes

characteristic

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#391

yes

characteristic

C₂×C₈₃

C₂×#104

yes

—

#1994

—

maximal, Sylow

C₈₃

#1994

yes

characteristic, complemented, Sylow

#533

—

#423

D₈₃

yes

characteristic

C₄³

—

#339

—

C₂×M₁₆

C₂×D₈₃

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

166

—

C₄×Q₈

—

166

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

166

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₂×Q₈

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

D₃₃₂

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₂²⋊C₄

—

166

—

C₄²

—

C₄²

—

166

—

C₄²

—

166

—

C₄²

—

166

—

C₄²

C₄×D₈₃

yes

characteristic

C₄²

C₄×D₈₃

yes

characteristic

C₄²

C₂²⋊D₈₃

yes

—

C₂³

C₂×D₃₃₂

yes

characteristic

Q₈

—

332

—

C₂×C₄

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

—

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

101

C₂×C₄

Q₈×D₈₃

yes

—

102

C₂×C₄

#13050589

yes

—

103

C₈

—

104

C₂²

—

105

C₂²

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

106

C₂²

#14581772

yes

—

107

C₄

—

108

C₄

—

332

—

109

C₄

—

166

—

110

C₄

#14581718

yes

—

111

C₄

#14581757

yes

—

112

C₄

#14581779

yes

characteristic

113

C₄

#14581790

yes

characteristic

114

C₂

—

115

C₂

#19127606

yes

characteristic

116

C₂

C₂×#14581718

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#423×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₈₃	C₂×D₄
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#423	D₈₃
hypocenter	C₈₃	#1994
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#423×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₈₃

C₂×D₄

cosocle

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
5. #14581792	C₄	semidirect
6. #14581784	C₄	semidirect
24. C₄×Dic₈₃	Q₈	semidirect
59. C₈₃	#1994	semidirect

H ≅ G/N

type

5. #14581792

C₄

semidirect

6. #14581784

C₄

semidirect

24. C₄×Dic₈₃

Q₈

semidirect

59. C₈₃

#1994

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19128258	2
C₄²×Dic₈₃	2
#19127462	2
#423×C₈₃	2
#1994	83

subgroup

index

#19128258

C₄²×Dic₈₃

#19127462

#423×C₈₃

#1994

Subgroups — #24908439

Conjugacy classes of subgroups

Complements

Maximal subgroups