Group #19135568 — subgroups

Subgroups — #19135568

Order 5504.

276 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 109 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×#13063325	C₂	1	1	yes	maximal, characteristic
2	#14582765	C₂	1	1	yes	maximal, characteristic
3	#14582662	C₂	1	4	yes	maximal
4	C₂×#97×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₈₆	—	2	1	no	—
6	#13063325	—	2	2	no	—
7	#13063325	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₄₃	—	2	1	no	—
9	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
11	C₄₃⋊#95	—	2	1	no	—
12	C₄₃⋊#95	—	2	1	no	—
13	C₂²×C₈×C₄₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
15	#97×C₄₃	—	2	2	no	—
16	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₄₃	—	2	2	no	—
20	C₂²×Dic₄₃	—	2	2	no	—
21	C₂×Dic₈₆	—	2	4	no	—
22	C₂×Dic₈₆	—	4	4	no	—
23	C₂×C₈×C₄₃	—	2	2	no	—
24	C₂×C₈×C₄₃	—	2	2	no	—
25	C₂×C₈×C₄₃	Q₈	1	4	yes	complemented
26	C₄²×C₄₃	C₂×C₄	1	2	yes	—
27	C₄²×C₄₃	D₄	1	2	yes	—
28	C₄⋊Dic₄₃	—	2	4	no	—
29	C₄⋊Dic₄₃	—	2	2	no	—
30	C₄⋊Dic₄₃	—	2	4	no	—
31	C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
32	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₄₃	—	2	2	no	—
34	C₂×C₄×C₄₃	—	2	2	no	—
35	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₄₃	#41	1	2	yes	—
41	C₂×Dic₄₃	—	2	4	no	—
42	C₂×Dic₄₃	—	4	4	no	—
43	C₂×Dic₄₃	—	4	2	no	—
44	C₂×Dic₄₃	—	4	2	no	—
45	C₂×Dic₄₃	—	4	2	no	—
46	Dic₈₆	—	4	8	no	—
47	C₈×C₄₃	—	2	4	no	—
48	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄₃	#124	1	1	yes	characteristic
50	C₂²×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
52	C₂²×C₄₃	#137	1	2	yes	—
53	C₄×C₄₃	—	2	2	no	—
54	C₄×C₄₃	—	2	2	no	—
55	C₄×C₄₃	C₄×Q₈	1	2	yes	—
56	C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
57	Dic₄₃	—	4	4	no	—
58	Dic₄₃	—	8	2	no	—
59	#1993	—	43	1	no	maximal, Sylow
60	C₂×C₄₃	#391	1	1	yes	characteristic
61	C₂×C₄₃	#418	1	1	yes	characteristic
62	C₂×C₄₃	#419	1	1	yes	characteristic
63	C₂×C₄₃	#489	1	2	yes	—
64	C₂×C₄₃	#501	1	2	yes	—
65	C₂×C₄⋊Q₈	—	43	1	no	—
66	C₂×#97	D₄₃	1	1	yes	characteristic
67	#382	—	43	1	no	—
68	#340	—	43	4	no	—
69	C₄₃	#1993	1	1	yes	characteristic, complemented, Sylow
70	C₂²×Q₈	—	86	1	no	—
71	C₂²×C₈	C₂×D₄₃	1	2	yes	—
72	C₄⋊Q₈	—	43	4	no	—
73	C₄⋊Q₈	—	86	2	no	—
74	C₂×C₄⋊C₄	—	43	2	no	—
75	C₂×C₄⋊C₄	—	43	2	no	—
76	C₂×C₄⋊C₄	—	86	1	no	—
77	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
78	#97	—	2	2	no	—
79	#95	—	86	1	no	—
80	#95	—	86	1	no	—
81	C₂×Q₈	—	86	4	no	—
82	C₂×Q₈	—	172	4	no	—
83	C₂²×C₄	—	86	2	no	—
84	C₂²×C₄	—	86	2	no	—
85	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
86	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	Dic₈₆	1	4	yes	complemented
91	C₄⋊C₄	—	43	4	no	—
92	C₄⋊C₄	—	86	4	no	—
93	C₄⋊C₄	—	86	2	no	—
94	C₄⋊C₄	—	86	4	no	—
95	C₄²	C₄×D₄₃	1	2	yes	—
96	C₄²	C₂²⋊D₄₃	1	2	yes	—
97	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
98	Q₈	—	172	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	86	4	no	—
102	C₂×C₄	—	172	2	no	—
103	C₂×C₄	—	172	2	no	—
104	C₂×C₄	—	172	2	no	—
105	C₂×C₄	—	172	4	no	—
106	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
107	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
108	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
109	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
110	C₂×C₄	#10621898	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
112	C₈	—	2	4	no	—
113	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
114	C₂²	#13063313	1	1	yes	characteristic
115	C₂²	#13063325	1	1	yes	characteristic
116	C₂²	#13063353	1	2	yes	—
117	C₂²	#13063358	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	344	2	no	—
121	C₄	—	172	4	no	—
122	C₄	C₄×Dic₈₆	1	2	yes	—
123	C₄	#13063304	1	2	yes	—
124	C₂	#14582805	1	1	yes	characteristic
125	C₂	#14582905	1	1	yes	characteristic
126	C₂	#14582909	1	1	yes	characteristic
127	C₂	#14583259	1	2	yes	—
128	C₂	#14583308	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063325

C₂

yes

maximal, characteristic

#14582765

C₂

yes

maximal, characteristic

#14582662

C₂

yes

maximal

C₂×#97×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

—

#13063325

—

#13063325

C₄

yes

complemented

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#97×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

Q₈

yes

complemented

C₄²×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

D₄

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

#124

yes

characteristic

C₂²×C₄₃

C₄⋊Q₈

yes

characteristic

C₂²×C₄₃

AGL(1,ℤ/8)

yes

—

C₂²×C₄₃

#137

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₄×Q₈

yes

—

C₄×C₄₃

C₂²⋊Q₈

yes

—

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#501

yes

—

C₂×C₄⋊Q₈

—

C₂×#97

D₄₃

yes

characteristic

#382

—

#340

—

C₄₃

#1993

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

C₂×D₄₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄₃

yes

—

—

—

C₂×Q₈

—

C₂×Q₈

—

172

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₇₂

yes

characteristic

C₂²×C₄

D₁₇₂

yes

characteristic

C₂²×C₄

C₂²×D₄₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₈₆

yes

complemented

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₄₃

yes

—

C₄²

C₂²⋊D₄₃

yes

—

C₂³

C₂×D₁₇₂

yes

characteristic

Q₈

—

172

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

172

—

103

C₂×C₄

—

172

—

104

C₂×C₄

—

172

—

105

C₂×C₄

—

172

—

106

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

107

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

108

C₂×C₄

C₂×Dic₈₆

yes

—

109

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

110

C₂×C₄

#10621898

yes

—

111

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

112

C₈

—

113

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

114

C₂²

#13063313

yes

characteristic

115

C₂²

#13063325

yes

characteristic

116

C₂²

#13063353

yes

—

117

C₂²

#13063358

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

344

—

121

C₄

—

172

—

122

C₄

C₄×Dic₈₆

yes

—

123

C₄

#13063304

yes

—

124

C₂

#14582805

yes

characteristic

125

C₂

#14582905

yes

characteristic

126

C₂

#14582909

yes

characteristic

127

C₂

#14583259

yes

—

128

C₂

#14583308

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×#97×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×#97	D₄₃
hypocenter	C₄₃	#1993
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×#97×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

hypercenter

C₂×#97

D₄₃

hypocenter

C₄₃

#1993

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
7. #13063325	C₄	semidirect
25. C₂×C₈×C₄₃	Q₈	semidirect
69. C₄₃	#1993	semidirect
90. C₂×C₈	Dic₈₆	semidirect

H ≅ G/N

type

7. #13063325

C₄

semidirect

25. C₂×C₈×C₄₃

Q₈

semidirect

69. C₄₃

#1993

semidirect

90. C₂×C₈

Dic₈₆

semidirect

Subgroups — #19135568

Conjugacy classes of subgroups

Complements