Group #14583509 — subgroups

Subgroups — #14583509

Order 2752.

192 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 95 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	#13063323	C₂	1	1	yes	maximal, characteristic, complemented
2	#13063318	C₂	1	2	yes	maximal
3	#13063318	C₂	1	2	yes	maximal
4	#13063316	C₂	1	1	yes	maximal, characteristic, complemented
5	#13063316	C₂	1	2	yes	maximal, complemented
6	#13063315	C₂	1	2	yes	maximal, complemented
7	#13063304	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#13063290	C₂	1	1	yes	maximal, characteristic, complemented
10	#13063284	C₂	1	1	yes	maximal, characteristic
11	C₂²⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₈×C₄₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₈₆	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
17	C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
18	C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
19	C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
20	C₂²⋊Dic₄₃	C₂²	1	2	yes	—
21	C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
23	C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
24	C₄⋊Dic₄₃	C₂²	1	2	yes	—
25	C₄⋊Dic₄₃	C₂²	1	2	yes	—
26	Dic₄₃⋊C₄	C₂²	1	2	yes	—
27	Dic₄₃⋊C₄	C₂²	1	2	yes	—
28	Dic₄₃⋊C₄	C₂²	1	2	yes	—
29	Dic₄₃⋊C₄	C₂²	1	2	yes	—
30	Dic₄₃⋊C₄	C₂²	1	2	yes	—
31	C₄×Dic₄₃	C₂²	1	1	yes	characteristic
32	C₄×Dic₄₃	C₂²	1	1	yes	characteristic
33	C₄×Dic₄₃	C₂²	1	2	yes	—
34	C₄×Dic₄₃	C₂²	1	2	yes	—
35	C₂³×C₄₃	C₂³	1	1	yes	characteristic
36	Q₈×C₄₃	—	2	2	no	—
37	C₂×C₄×C₄₃	—	2	2	no	—
38	C₂×C₄×C₄₃	C₂³	1	2	yes	—
39	C₂×C₄×C₄₃	C₂³	1	2	yes	—
40	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
42	C₂×Dic₄₃	C₂³	1	1	yes	characteristic
43	C₂×Dic₄₃	C₂³	1	1	yes	characteristic
44	C₂×Dic₄₃	C₂³	1	1	yes	characteristic
45	C₂×Dic₄₃	C₂³	1	2	yes	—
46	C₂×Dic₄₃	C₂³	1	1	yes	characteristic
47	C₂×Dic₄₃	C₂³	1	2	yes	—
48	Dic₈₆	—	2	2	no	—
49	C₂²×C₄₃	—	2	1	no	—
50	C₂²×C₄₃	—	2	1	no	—
51	C₂²×C₄₃	—	2	1	no	—
52	C₂²×C₄₃	C₂⁴	1	1	yes	characteristic
53	C₄×C₄₃	—	2	1	no	—
54	C₄×C₄₃	—	2	2	no	—
55	C₄×C₄₃	—	2	2	no	—
56	C₄×C₄₃	#41	1	2	yes	—
57	Dic₄₃	—	2	2	no	—
58	Dic₄₃	—	2	1	no	—
59	Dic₄₃	—	2	1	no	—
60	Dic₄₃	—	2	1	no	—
61	Dic₄₃	—	2	1	no	—
62	Dic₄₃	—	2	2	no	—
63	C₂×C₄₃	—	4	1	no	—
64	C₂×C₄₃	C₂×#41	1	1	yes	characteristic
65	C₂×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
66	C₂×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
67	#541	—	43	1	no	maximal, Sylow
68	C₄₃	#541	1	1	yes	characteristic, complemented, Sylow
69	C₄⋊Q₈	—	43	1	no	—
70	#126	—	43	2	no	—
71	#126	—	43	2	no	—
72	#125	—	43	2	no	—
73	#125	—	43	1	no	—
74	#125	—	43	2	no	—
75	C₂²⋊Q₈	—	43	1	no	—
76	C₂²⋊Q₈	D₄₃	1	1	yes	characteristic
77	C₄×Q₈	—	43	1	no	—
78	C₄×Q₈	—	43	1	no	—
79	#117	—	43	1	no	—
80	C₂×Q₈	—	43	1	no	—
81	C₂×Q₈	C₂×D₄₃	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₄₃	1	1	yes	characteristic
83	C₄⋊C₄	—	43	1	no	—
84	C₄⋊C₄	—	43	1	no	—
85	C₄⋊C₄	—	43	1	no	—
86	C₄⋊C₄	—	43	2	no	—
87	C₄⋊C₄	—	43	2	no	—
88	C₄⋊C₄	—	43	2	no	—
89	C₄⋊C₄	—	43	2	no	—
90	C₄⋊C₄	—	43	2	no	—
91	C₄⋊C₄	—	43	2	no	—
92	C₄⋊C₄	—	43	2	no	—
93	C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
94	C₄⋊C₄	C₂×D₄₃	1	2	yes	—
95	C₂²⋊C₄	—	43	1	no	—
96	C₂²⋊C₄	—	43	1	no	—
97	C₂²⋊C₄	—	43	2	no	—
98	C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
99	C₄²	—	43	1	no	—
100	C₄²	—	43	1	no	—
101	C₄²	—	43	2	no	—
102	C₄²	—	43	2	no	—
103	C₂³	C₂²×D₄₃	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	Q₈	—	86	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	43	1	no	—
108	C₂×C₄	—	43	1	no	—
109	C₂×C₄	—	43	1	no	—
110	C₂×C₄	—	43	1	no	—
111	C₂×C₄	—	43	2	no	—
112	C₂×C₄	—	43	2	no	—
113	C₂×C₄	C₂²×D₄₃	1	2	yes	—
114	C₂×C₄	C₂²×D₄₃	1	2	yes	—
115	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂³×D₄₃	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	86	2	no	—
125	C₄	—	86	1	no	—
126	C₄	—	86	1	no	—
127	C₄	—	86	1	no	—
128	C₄	—	86	1	no	—
129	C₄	—	86	2	no	—
130	C₄	#10621900	1	2	yes	—
131	C₂	—	4	1	no	—
132	C₂	C₂×#10621900	1	1	yes	characteristic
133	C₂	#13063372	1	1	yes	characteristic
134	C₂	#13063373	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13063323

C₂

yes

maximal, characteristic, complemented

#13063318

C₂

yes

maximal

#13063318

C₂

yes

maximal

#13063316

C₂

yes

maximal, characteristic, complemented

#13063316

C₂

yes

maximal, complemented

#13063315

C₂

yes

maximal, complemented

#13063304

C₂

yes

maximal, characteristic

Q₈×Dic₄₃

C₂

yes

maximal, characteristic, complemented

#13063290

C₂

yes

maximal, characteristic, complemented

#13063284

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₄₃

C₂

yes

maximal, characteristic

C₂×Q₈×C₄₃

C₂²

yes

characteristic

C₂²×C₄×C₄₃

C₂²

yes

characteristic

C₂×Dic₈₆

C₂²

yes

characteristic

C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₄⋊C₄×C₄₃

C₂²

yes

—

C₂²⋊C₄×C₄₃

C₂²

yes

—

C₂²⋊Dic₄₃

C₂²

yes

characteristic

C₂²⋊Dic₄₃

C₂²

yes

characteristic

C₂²⋊Dic₄₃

C₂²

yes

—

C₄⋊Dic₄₃

C₂²

yes

characteristic

C₄⋊Dic₄₃

C₂²

yes

characteristic

C₄⋊Dic₄₃

C₂²

yes

characteristic

C₄⋊Dic₄₃

C₂²

yes

—

C₄⋊Dic₄₃

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

C₄×Dic₄₃

C₂²

yes

characteristic

C₄×Dic₄₃

C₂²

yes

characteristic

C₄×Dic₄₃

C₂²

yes

—

C₄×Dic₄₃

C₂²

yes

—

C₂³×C₄₃

C₂³

yes

characteristic

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂³

yes

—

C₂×C₄×C₄₃

C₂³

yes

—

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂×Dic₄₃

C₂³

yes

characteristic

C₂×Dic₄₃

C₂³

yes

characteristic

C₂×Dic₄₃

C₂³

yes

characteristic

C₂×Dic₄₃

C₂³

yes

—

C₂×Dic₄₃

C₂³

yes

characteristic

C₂×Dic₄₃

C₂³

yes

—

Dic₈₆

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂⁴

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#41

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

C₂×C₄₃

—

C₂×C₄₃

C₂×#41

yes

characteristic

C₂×C₄₃

2₋¹⁺⁴

yes

characteristic

C₂×C₄₃

2₋¹⁺⁴

yes

—

C₄₃

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₄₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄₃

yes

characteristic

C₂²×C₄

C₂×D₄₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄₃

yes

characteristic

C₄⋊C₄

C₂×D₄₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₄₃

yes

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₂³

C₂²×D₄₃

yes

characteristic

104

Q₈

—

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²×D₄₃

yes

—

114

C₂×C₄

C₂²×D₄₃

yes

—

115

C₂×C₄

C₂²×D₄₃

yes

characteristic

116

C₂×C₄

C₂²×D₄₃

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂³×D₄₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#10621900

yes

—

131

C₂

—

132

C₂

C₂×#10621900

yes

characteristic

133

C₂

#13063372

yes

characteristic

134

C₂

#13063373

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₃
commutator, socle, cosocle	C₂²×C₄₃	C₂⁴
Fitting	C₂²⋊Q₈×C₄₃	C₂
prime radical	C₂³×C₄₃	C₂³
hypercenter	C₂²⋊Q₈	D₄₃
hypocenter	C₄₃	#541
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₃

commutator, socle, cosocle

C₂²×C₄₃

C₂⁴

Fitting

C₂²⋊Q₈×C₄₃

C₂

prime radical

C₂³×C₄₃

C₂³

hypercenter

C₂²⋊Q₈

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #13063323	C₂	semidirect
4. #13063316	C₂	semidirect
5. #13063316	C₂	semidirect
6. #13063315	C₂	semidirect
8. Q₈×Dic₄₃	C₂	semidirect
9. #13063290	C₂	semidirect
68. C₄₃	#541	semidirect

H ≅ G/N

type

1. #13063323

C₂

semidirect

4. #13063316

C₂

semidirect

5. #13063316

C₂

semidirect

6. #13063315

C₂

semidirect

8. Q₈×Dic₄₃

C₂

semidirect

9. #13063290

C₂

semidirect

68. C₄₃

#541

semidirect

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

subgroup	index
#13063323	2
#13063318	2
#13063316	2
#13063315	2
#13063304	2
Q₈×Dic₄₃	2
#13063290	2
#13063284	2
C₂²⋊Q₈×C₄₃	2
#541	43

subgroup

index

Q₈×Dic₄₃

C₂²⋊Q₈×C₄₃

Subgroups — #14583509

Conjugacy classes of subgroups

Complements

Maximal subgroups