Group #24983425 — subgroups

Subgroups — #24983425

Order 11008.

306 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 207 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	C₂×C₄×C₄₃⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×C₈²×C₄₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄₃⋊C₁₆	C₄	1	4	yes	—
4	C₄×C₄₃⋊C₁₆	—	2	2	no	—
5	C₄×C₄₃⋊C₁₆	—	2	2	no	—
6	C₂×C₄×C₈×C₄₃	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
8	C₈²×C₄₃	—	2	2	no	—
9	C₂×C₄₃⋊C₁₆	—	2	4	no	—
10	C₂×C₄₃⋊C₁₆	—	2	4	no	—
11	C₂×C₄₃⋊C₁₆	C₈	1	8	yes	complemented
12	C₂²×C₈×C₄₃	C₈	1	4	yes	—
13	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₄₃	—	2	2	no	—
16	C₄×C₈×C₄₃	—	2	2	no	—
17	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
18	C₄×C₈×C₄₃	D₄	1	2	yes	—
19	C₄×C₈×C₄₃	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
22	C₂×C₈×C₄₃	—	2	4	no	—
23	C₂×C₈×C₄₃	—	2	4	no	—
24	C₂×C₈×C₄₃	C₁₆	1	8	yes	complemented
25	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
26	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₄₃	C₂×C₈	1	4	yes	—
28	C₂×C₈×C₄₃	M₁₆	1	4	yes	—
29	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₄²×C₄₃	C₄⋊C₄	1	1	yes	characteristic
31	C₄²×C₄₃	M₁₆	1	2	yes	—
32	C₄₃⋊C₁₆	—	2	8	no	—
33	C₂³×C₄₃	C₄×C₈	1	1	yes	characteristic
34	C₂×C₄×C₄₃	#95	1	1	yes	characteristic
35	C₂×C₄×C₄₃	C₄×C₈	1	2	yes	—
36	C₂×C₄×C₄₃	#97	1	2	yes	—
37	C₂×C₄×C₄₃	#97	1	1	yes	characteristic
38	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
39	C₂×C₄×C₄₃	C₄⋊C₈	1	2	yes	—
40	C₂×C₄×C₄₃	C₂×C₁₆	1	4	yes	—
41	C₂×C₄×C₄₃	M₃₂	1	4	yes	—
42	C₈×C₄₃	—	2	8	no	—
43	C₈×C₄₃	C₂²⋊C₈	1	4	yes	—
44	C₈×C₄₃	C₄⋊C₈	1	4	yes	—
45	#7136	—	43	1	no	maximal, Sylow
46	C₂²×C₄₃	C₈²	1	1	yes	characteristic
47	C₂²×C₄₃	#322	1	1	yes	characteristic
48	C₂²×C₄₃	#336	1	1	yes	characteristic
49	C₂²×C₄₃	C₄×C₁₆	1	2	yes	—
50	C₂²×C₄₃	#346	1	2	yes	—
51	C₄×C₄₃	#322	1	2	yes	—
52	C₄×C₄₃	#336	1	2	yes	—
53	C₄×C₄₃	C₂²⋊C₁₆	1	4	yes	—
54	C₄×C₄₃	C₄⋊C₁₆	1	4	yes	—
55	C₂×C₄×C₁₆	—	43	2	no	—
56	C₂×C₈²	D₄₃	1	1	yes	characteristic
57	C₂×C₄₃	#1273	1	1	yes	characteristic
58	C₂×C₄₃	C₈×C₁₆	1	1	yes	characteristic
59	C₂×C₄₃	#1311	1	1	yes	characteristic
60	C₂×C₄₃	#1312	1	2	yes	—
61	C₂×C₄₃	#1374	1	2	yes	—
62	C₂²×C₁₆	—	43	4	no	—
63	C₂×C₄×C₈	Dic₄₃	1	2	yes	—
64	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
65	C₄×C₁₆	—	86	2	no	—
66	C₄×C₁₆	—	86	2	no	—
67	C₈²	—	2	2	no	—
68	C₄₃	#7136	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	C₄₃⋊C₈	1	4	yes	—
70	C₂²×C₈	C₄×D₄₃	1	2	yes	—
71	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
72	C₂×C₁₆	—	43	8	no	—
73	C₂×C₁₆	—	86	4	no	—
74	C₂×C₁₆	—	86	4	no	—
75	C₄×C₈	—	2	2	no	—
76	C₄×C₈	—	2	2	no	—
77	C₄×C₈	Dic₈₆	1	1	yes	characteristic
78	C₄×C₈	D₁₇₂	1	1	yes	characteristic
79	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
80	C₂²×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
81	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	C₄₃⋊C₁₆	1	8	yes	complemented
85	C₂×C₈	C₈×D₄₃	1	4	yes	—
86	C₂×C₈	#10621872	1	4	yes	—
87	C₂×C₈	Dic₄₃⋊C₄	1	2	yes	—
88	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
89	C₄²	#10621877	1	2	yes	—
90	C₄²	C₄⋊Dic₄₃	1	1	yes	characteristic
91	C₄²	C₂²⋊Dic₄₃	1	1	yes	characteristic
92	C₁₆	—	86	8	no	—
93	C₂³	C₄×C₄₃⋊C₈	1	1	yes	characteristic
94	C₂×C₄	C₄₃⋊#95	1	1	yes	characteristic
95	C₂×C₄	C₈×Dic₄₃	1	2	yes	—
96	C₂×C₄	#13063235	1	2	yes	—
97	C₂×C₄	#13063236	1	1	yes	characteristic
98	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
99	C₂×C₄	C₄⋊(C₄₃⋊C₈)	1	2	yes	—
100	C₂×C₄	C₂×C₄₃⋊C₁₆	1	4	yes	—
101	C₂×C₄	#13063265	1	4	yes	—
102	C₈	—	2	8	no	—
103	C₈	(C₂×D₄₃)⋊C₈	1	4	yes	—
104	C₈	Dic₄₃⋊C₈	1	4	yes	—
105	C₂²	C₈×C₄₃⋊C₈	1	1	yes	characteristic
106	C₂²	#14582620	1	1	yes	characteristic
107	C₂²	#14582654	1	1	yes	characteristic
108	C₂²	C₄×C₄₃⋊C₁₆	1	2	yes	—
109	C₂²	#14582675	1	2	yes	—
110	C₄	#14582621	1	2	yes	—
111	C₄	#14582653	1	2	yes	—
112	C₄	C₂²⋊(C₄₃⋊C₁₆)	1	4	yes	—
113	C₄	C₄⋊(C₄₃⋊C₁₆)	1	4	yes	—
114	C₂	C₄₃⋊#1273	1	1	yes	characteristic
115	C₂	C₈×C₄₃⋊C₁₆	1	1	yes	characteristic
116	C₂	#19132512	1	1	yes	characteristic
117	C₂	#19132513	1	2	yes	—
118	C₂	#19132673	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₄₃⋊C₁₆

C₂

yes

maximal

C₂×C₈²×C₄₃

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₁₆

C₄

yes

—

C₄×C₄₃⋊C₁₆

—

C₄×C₄₃⋊C₁₆

—

C₂×C₄×C₈×C₄₃

C₄

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₈²×C₄₃

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

C₈

yes

complemented

C₂²×C₈×C₄₃

C₈

yes

—

C₂²×C₈×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

Q₈

yes

characteristic

C₂²×C₄×C₄₃

C₄²

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₈

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₁₆

yes

complemented

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₂×C₈×C₄₃

C₂×C₈

yes

—

C₂×C₈×C₄₃

M₁₆

yes

—

C₄²×C₄₃

C₂²⋊C₄

yes

characteristic

C₄²×C₄₃

C₄⋊C₄

yes

characteristic

C₄²×C₄₃

M₁₆

yes

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₄×C₈

yes

characteristic

C₂×C₄×C₄₃

#95

yes

characteristic

C₂×C₄×C₄₃

C₄×C₈

yes

—

C₂×C₄×C₄₃

#97

yes

—

C₂×C₄×C₄₃

#97

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₈

yes

—

C₂×C₄×C₄₃

C₄⋊C₈

yes

—

C₂×C₄×C₄₃

C₂×C₁₆

yes

—

C₂×C₄×C₄₃

M₃₂

yes

—

C₈×C₄₃

—

C₈×C₄₃

C₂²⋊C₈

yes

—

C₈×C₄₃

C₄⋊C₈

yes

—

#7136

—

maximal, Sylow

C₂²×C₄₃

C₈²

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

C₄×C₁₆

yes

—

C₂²×C₄₃

#346

yes

—

C₄×C₄₃

#322

yes

—

C₄×C₄₃

#336

yes

—

C₄×C₄₃

C₂²⋊C₁₆

yes

—

C₄×C₄₃

C₄⋊C₁₆

yes

—

C₂×C₄×C₁₆

—

C₂×C₈²

D₄₃

yes

characteristic

C₂×C₄₃

#1273

yes

characteristic

C₂×C₄₃

C₈×C₁₆

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#1374

yes

—

C₂²×C₁₆

—

C₂×C₄×C₈

Dic₄₃

yes

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

—

C₄₃

#7136

yes

characteristic, complemented, Sylow

C₂²×C₈

C₄₃⋊C₈

yes

—

C₂²×C₈

C₄×D₄₃

yes

—

C₂×C₄²

C₂×Dic₄₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₈₆

yes

characteristic

C₄×C₈

D₁₇₂

yes

characteristic

C₄×C₈

C₂²⋊D₄₃

yes

—

C₂²×C₄

C₂×C₄₃⋊C₈

yes

—

C₂²×C₄

C₄×Dic₄₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄₃⋊C₁₆

yes

complemented

C₂×C₈

C₈×D₄₃

yes

—

C₂×C₈

#10621872

yes

—

C₂×C₈

Dic₄₃⋊C₄

yes

—

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

C₄²

#10621877

yes

—

C₄²

C₄⋊Dic₄₃

yes

characteristic

C₄²

C₂²⋊Dic₄₃

yes

characteristic

C₁₆

—

C₂³

C₄×C₄₃⋊C₈

yes

characteristic

C₂×C₄

C₄₃⋊#95

yes

characteristic

C₂×C₄

C₈×Dic₄₃

yes

—

C₂×C₄

#13063235

yes

—

C₂×C₄

#13063236

yes

characteristic

C₂×C₄

C₂²⋊(C₄₃⋊C₈)

yes

—

C₂×C₄

C₄⋊(C₄₃⋊C₈)

yes

—

100

C₂×C₄

C₂×C₄₃⋊C₁₆

yes

—

101

C₂×C₄

#13063265

yes

—

102

C₈

—

103

C₈

(C₂×D₄₃)⋊C₈

yes

—

104

C₈

Dic₄₃⋊C₈

yes

—

105

C₂²

C₈×C₄₃⋊C₈

yes

characteristic

106

C₂²

#14582620

yes

characteristic

107

C₂²

#14582654

yes

characteristic

108

C₂²

C₄×C₄₃⋊C₁₆

yes

—

109

C₂²

#14582675

yes

—

110

C₄

#14582621

yes

—

111

C₄

#14582653

yes

—

112

C₄

C₂²⋊(C₄₃⋊C₁₆)

yes

—

113

C₄

C₄⋊(C₄₃⋊C₁₆)

yes

—

114

C₂

C₄₃⋊#1273

yes

characteristic

115

C₂

C₈×C₄₃⋊C₁₆

yes

characteristic

116

C₂

#19132512

yes

characteristic

117

C₂

#19132513

yes

—

118

C₂

#19132673

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄×C₈	C₂×D₄₃
commutator	C₂×C₄₃	C₈×C₁₆
Fitting	C₂×C₈²×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₄×C₈
cosocle	C₂×C₄×C₈×C₄₃	C₂²
hypercenter	C₂×C₈²	D₄₃
hypocenter	C₄₃	#7136
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

subgroup

G/H

center, Frattini

C₂×C₄×C₈

C₂×D₄₃

commutator

C₂×C₄₃

C₈×C₁₆

Fitting

C₂×C₈²×C₄₃

C₂

socle, prime radical

C₂³×C₄₃

C₄×C₈

cosocle

C₂×C₄×C₈×C₄₃

C₂²

hypercenter

C₂×C₈²

D₄₃

C₄₃

G ▷ C₂×C₄₃ ▷ C₄₃

C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

N	H ≅ G/N	type
11. C₂×C₄₃⋊C₁₆	C₈	semidirect
24. C₂×C₈×C₄₃	C₁₆	semidirect
68. C₄₃	#7136	semidirect
84. C₂×C₈	C₄₃⋊C₁₆	semidirect

H ≅ G/N

type

11. C₂×C₄₃⋊C₁₆

C₈

semidirect

24. C₂×C₈×C₄₃

C₁₆

semidirect

68. C₄₃

#7136

semidirect

84. C₂×C₈

C₄₃⋊C₁₆

semidirect

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

subgroup	index
C₂×C₄×C₄₃⋊C₁₆	2
C₂×C₈²×C₄₃	2
#7136	43

subgroup

index

C₂×C₄×C₄₃⋊C₁₆

C₂×C₈²×C₄₃

#7136

Subgroups — #24983425

Conjugacy classes of subgroups

Complements

Maximal subgroups