Group #24983254 — subgroups

Subgroups — #24983254

Order 11008.

140 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₈×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
2	#19132513	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#1367	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₈₆	—	2	1	no	—
5	C₄×Dic₁₇₂	C₄	1	1	yes	characteristic
6	C₄×C₄₃⋊C₁₆	—	2	1	no	—
7	#14582652	C₄	1	1	yes	characteristic
8	#14582631	—	2	1	no	—
9	C₄₃×C₄⋊C₁₆	—	2	1	no	—
10	C₈²×C₄₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₇₂	C₈	1	1	yes	characteristic
12	C₄×Dic₈₆	—	2	1	no	—
13	C₂×C₄₃⋊C₁₆	—	2	2	no	—
14	#13063259	C₈	1	1	yes	characteristic
15	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
16	Dic₄₃⋊C₈	—	4	1	no	—
17	#13063250	—	2	1	no	—
18	C₈×Dic₄₃	—	4	1	no	—
19	C₂×C₁₆×C₄₃	—	4	1	no	—
20	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
22	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
23	C₂×Dic₈₆	—	2	1	no	—
24	C₂×C₈×C₄₃	—	2	1	no	—
25	C₂×C₈×C₄₃	C₂×C₈	1	1	yes	characteristic
26	C₂×C₈×C₄₃	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
28	C₂×C₈×C₄₃	SD₁₆	1	1	yes	characteristic
29	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₄⋊Dic₄₃	—	2	1	no	—
31	Dic₄₃⋊C₄	—	4	1	no	—
32	C₄×Dic₄₃	—	4	1	no	—
33	C₂×C₄₃⋊C₈	—	4	1	no	—
34	Dic₁₇₂	—	2	1	no	—
35	Dic₁₇₂	C₁₆	1	2	yes	complemented
36	C₁₆×C₄₃	—	8	1	no	—
37	C₄₃⋊C₁₆	—	4	2	no	—
38	C₂×C₄×C₄₃	C₂²⋊C₈	1	1	yes	characteristic
39	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
40	C₂×C₄×C₄₃	#104	1	1	yes	characteristic
41	C₂×Dic₄₃	—	4	1	no	—
42	C₂×Dic₄₃	—	4	1	no	—
43	Dic₈₆	—	2	2	no	—
44	Dic₈₆	—	4	1	no	—
45	C₈×C₄₃	—	2	2	no	—
46	C₈×C₄₃	—	2	1	no	—
47	C₈×C₄₃	—	2	1	no	—
48	C₈×C₄₃	—	2	1	no	—
49	C₈×C₄₃	C₂×C₁₆	1	1	yes	characteristic
50	C₈×C₄₃	M₃₂	1	1	yes	characteristic
51	C₄₃⋊C₈	—	8	1	no	—
52	#7071	—	43	1	no	maximal, Sylow
53	C₂²×C₄₃	D₄⋊C₈	1	1	yes	characteristic
54	C₄×C₄₃	—	2	1	no	—
55	C₄×C₄₃	C₂²⋊C₁₆	1	1	yes	characteristic
56	C₄×C₄₃	#350	1	1	yes	characteristic
57	C₄×C₄₃	#360	1	1	yes	characteristic
58	C₄×C₄₃	#362	1	1	yes	characteristic
59	Dic₄₃	—	4	2	no	—
60	Dic₄₃	—	8	1	no	—
61	C₈×Q₁₆	—	43	1	no	—
62	#1367	D₄₃	1	1	yes	characteristic
63	#1312	—	43	1	no	—
64	C₂×C₄₃	D₄⋊C₁₆	1	1	yes	characteristic
65	C₂×C₄₃	#1334	1	1	yes	characteristic
66	C₂×C₄₃	#1336	1	1	yes	characteristic
67	C₈×Q₈	—	86	1	no	—
68	C₄×Q₁₆	—	43	1	no	—
69	C₄⋊C₁₆	—	2	1	no	—
70	C₄×C₁₆	—	86	1	no	—
71	#335	—	43	1	no	—
72	Q₈⋊C₈	—	86	1	no	—
73	C₈²	C₂×D₄₃	1	1	yes	characteristic
74	C₄₃	#7071	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₁₆	—	43	1	no	—
76	C₄×Q₈	—	86	1	no	—
77	C₂×C₁₆	—	4	1	no	—
78	C₂×C₁₆	—	86	2	no	—
79	#107	—	43	1	no	—
80	C₄⋊C₈	—	86	1	no	—
81	C₄⋊C₈	—	172	1	no	—
82	#103	—	86	1	no	—
83	C₄×C₈	—	172	1	no	—
84	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
85	C₄×C₈	D₁₇₂	1	1	yes	characteristic
86	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
87	C₂×Q₈	—	86	1	no	—
88	Q₁₆	—	43	2	no	—
89	Q₁₆	—	86	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	172	1	no	—
92	C₂×C₈	C₈×D₄₃	1	1	yes	characteristic
93	C₂×C₈	#10621872	1	1	yes	characteristic
94	C₂×C₈	#10621882	1	1	yes	characteristic
95	C₂×C₈	#10621884	1	1	yes	characteristic
96	C₄⋊C₄	—	86	1	no	—
97	C₄⋊C₄	—	172	1	no	—
98	C₄²	—	172	1	no	—
99	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
100	C₁₆	—	8	1	no	—
101	C₁₆	—	172	2	no	—
102	Q₈	—	86	2	no	—
103	Q₈	—	172	1	no	—
104	C₂×C₄	—	172	1	no	—
105	C₂×C₄	—	172	1	no	—
106	C₂×C₄	(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
107	C₂×C₄	#13063245	1	1	yes	characteristic
108	C₂×C₄	#13063251	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	2	1	no	—
111	C₈	—	2	1	no	—
112	C₈	—	2	1	no	—
113	C₈	—	344	1	no	—
114	C₈	C₁₆×D₄₃	1	1	yes	characteristic
115	C₈	#13063264	1	1	yes	characteristic
116	C₂²	#14582626	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	344	1	no	—
119	C₄	—	172	2	no	—
120	C₄	(C₂×D₄₃)⋊C₁₆	1	1	yes	characteristic
121	C₄	#14582684	1	1	yes	characteristic
122	C₄	#14582708	1	1	yes	characteristic
123	C₄	#14582714	1	1	yes	characteristic
124	C₂	#19132549	1	1	yes	characteristic
125	C₂	#19132564	1	1	yes	characteristic
126	C₂	#19132570	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₁₇₂

C₂

yes

maximal, characteristic

#19132513

C₂

yes

maximal, characteristic

C₄₃×#1367

C₂

yes

maximal, characteristic

C₈×Dic₈₆

—

C₄×Dic₁₇₂

C₄

yes

characteristic

C₄×C₄₃⋊C₁₆

—

#14582652

C₄

yes

characteristic

#14582631

—

C₄₃×C₄⋊C₁₆

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

C₈

yes

characteristic

C₄×Dic₈₆

—

C₂×C₄₃⋊C₁₆

—

#13063259

C₈

yes

characteristic

C₄⋊(C₄₃⋊C₈)

—

Dic₄₃⋊C₈

—

#13063250

—

C₈×Dic₄₃

—

C₂×C₁₆×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂×C₈

yes

characteristic

C₂×C₈×C₄₃

M₁₆

yes

characteristic

C₂×C₈×C₄₃

D₈

yes

characteristic

C₂×C₈×C₄₃

SD₁₆

yes

characteristic

C₄²×C₄₃

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

Dic₁₇₂

C₁₆

yes

complemented

C₁₆×C₄₃

—

C₄₃⋊C₁₆

—

C₂×C₄×C₄₃

C₂²⋊C₈

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

C₂×C₁₆

yes

characteristic

C₈×C₄₃

M₃₂

yes

characteristic

C₄₃⋊C₈

—

#7071

—

maximal, Sylow

C₂²×C₄₃

D₄⋊C₈

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

C₂²⋊C₁₆

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

C₈×Q₁₆

—

#1367

D₄₃

yes

characteristic

#1312

—

C₂×C₄₃

D₄⋊C₁₆

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#7071

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

172

—

#103

—

C₄×C₈

—

172

—

C₄×C₈

C₄×D₄₃

yes

characteristic

C₄×C₈

D₁₇₂

yes

characteristic

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

172

—

C₂×C₈

C₈×D₄₃

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

172

—

C₄²

—

172

—

C₄²

(C₂×D₄₃)⋊C₄

yes

characteristic

100

C₁₆

—

101

C₁₆

—

172

—

102

Q₈

—

103

Q₈

—

172

—

104

C₂×C₄

—

172

—

105

C₂×C₄

—

172

—

106

C₂×C₄

(C₂×D₄₃)⋊C₈

yes

characteristic

107

C₂×C₄

#13063245

yes

characteristic

108

C₂×C₄

#13063251

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

344

—

114

C₈

C₁₆×D₄₃

yes

characteristic

115

C₈

#13063264

yes

characteristic

116

C₂²

#14582626

yes

characteristic

117

C₄

—

118

C₄

—

344

—

119

C₄

—

172

—

120

C₄

(C₂×D₄₃)⋊C₁₆

yes

characteristic

121

C₄

#14582684

yes

characteristic

122

C₄

#14582708

yes

characteristic

123

C₄

#14582714

yes

characteristic

124

C₂

#19132549

yes

characteristic

125

C₂

#19132564

yes

characteristic

126

C₂

#19132570

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13063245
commutator	C₈×C₄₃	C₂×C₁₆
Frattini	C₈²	C₂×D₄₃
Fitting	C₄₃×#1367	C₂
socle, prime radical	C₂²×C₄₃	D₄⋊C₈
cosocle	C₈²×C₄₃	C₂²
hypercenter	#1367	D₄₃
hypocenter	C₄₃	#7071
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1367

subgroup

G/H

center

C₂×C₄

#13063245

commutator

C₈×C₄₃

C₂×C₁₆

Frattini

C₈²

C₂×D₄₃

Fitting

C₄₃×#1367

C₂

socle, prime radical

C₂²×C₄₃

D₄⋊C₈

cosocle

C₈²×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1367

N	H ≅ G/N	type
35. Dic₁₇₂	C₁₆	semidirect
74. C₄₃	#7071	semidirect

H ≅ G/N

type

35. Dic₁₇₂

C₁₆

semidirect

74. C₄₃

#7071

semidirect

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

subgroup	index
C₈×Dic₁₇₂	2
#19132513	2
C₄₃×#1367	2
#7071	43

subgroup

index

C₈×Dic₁₇₂

#19132513

C₄₃×#1367

#7071

Subgroups — #24983254

Conjugacy classes of subgroups

Complements

Maximal subgroups