Group #19136641 — subgroups

Subgroups — #19136641

Order 5504.

166 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 55 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×#13063270	C₂	1	2	yes	maximal
2	#14583279	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic
4	#14582704	C₂	1	2	yes	maximal, complemented
5	C₄₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₁₇₂	—	2	1	no	—
7	C₂×Dic₁₇₂	C₂²	1	2	yes	—
8	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
9	#13063325	—	2	1	no	—
10	C₄×D₁₇₂	—	2	1	no	—
11	#13063270	—	2	4	no	—
12	#13063259	C₂²	1	1	yes	characteristic
13	#13063247	—	2	1	no	—
14	C₂×C₁₆×C₄₃	C₂²	1	2	yes	—
15	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×D₁₇₂	—	2	1	no	—
17	C₂×C₄×D₄₃	—	4	1	no	—
18	C₂×Dic₈₆	—	2	2	no	—
19	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
20	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
21	C₄²×C₄₃	D₄	1	1	yes	characteristic
22	(C₂×D₄₃)⋊C₄	—	4	1	no	—
23	C₄⋊Dic₄₃	—	2	1	no	—
24	C₄⋊Dic₄₃	—	4	1	no	—
25	C₄⋊Dic₄₃	—	4	1	no	—
26	Dic₁₇₂	—	2	2	no	—
27	Dic₁₇₂	—	2	2	no	—
28	Dic₁₇₂	—	4	2	no	—
29	D₃₄₄	—	2	1	no	—
30	D₃₄₄	D₄	1	2	yes	—
31	C₁₆×C₄₃	—	2	2	no	—
32	C₂²×D₄₃	—	4	1	no	—
33	C₂×C₄×C₄₃	D₈	1	1	yes	characteristic
34	C₂×C₄×C₄₃	D₈	1	1	yes	characteristic
35	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
36	C₂×Dic₄₃	—	4	1	no	—
37	C₂×Dic₄₃	—	4	2	no	—
38	D₁₇₂	—	2	2	no	—
39	D₁₇₂	—	4	1	no	—
40	C₄×D₄₃	—	8	2	no	—
41	Dic₈₆	—	4	2	no	—
42	Dic₈₆	—	4	2	no	—
43	C₈×C₄₃	—	2	1	no	—
44	C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
45	C₈×C₄₃	#41	1	1	yes	characteristic
46	C₂²×C₄₃	C₂×D₈	1	1	yes	characteristic
47	C₂×D₄₃	—	4	2	no	—
48	C₂×D₄₃	—	8	1	no	—
49	C₂×D₄₃	—	8	1	no	—
50	C₄×C₄₃	—	2	1	no	—
51	C₄×C₄₃	SD₃₂	1	2	yes	complemented
52	C₄×C₄₃	#121	1	1	yes	characteristic
53	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
54	Dic₄₃	—	8	2	no	—
55	Dic₄₃	—	8	1	no	—
56	#2208	—	43	1	no	maximal, Sylow
57	C₂×C₄₃	#459	1	1	yes	characteristic
58	C₂×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
59	C₂×C₄₃	#510	1	1	yes	characteristic
60	D₄₃	—	8	2	no	—
61	C₂×SD₃₂	—	43	2	no	—
62	#494	—	43	1	no	—
63	C₄×D₈	—	43	1	no	—
64	C₄⋊C₁₆	D₄₃	1	1	yes	characteristic, complemented
65	#358	—	43	2	no	—
66	C₄₃	#2208	1	1	yes	characteristic, complemented, Sylow
67	C₂×Q₁₆	—	43	2	no	—
68	C₂×Q₁₆	—	86	1	no	—
69	C₂×D₈	—	43	1	no	—
70	C₄⋊Q₈	—	86	1	no	—
71	C₄×D₄	—	86	1	no	—
72	SD₃₂	—	86	4	no	—
73	C₂×C₁₆	C₂×D₄₃	1	2	yes	—
74	#107	—	43	1	no	—
75	#102	—	86	1	no	—
76	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
77	C₂×Q₈	—	86	2	no	—
78	C₂×D₄	—	86	1	no	—
79	C₂²×C₄	—	172	1	no	—
80	Q₁₆	—	86	2	no	—
81	Q₁₆	—	86	2	no	—
82	Q₁₆	—	172	2	no	—
83	D₈	—	43	2	no	—
84	D₈	—	86	1	no	—
85	C₂×C₈	D₁₇₂	1	1	yes	characteristic
86	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
87	C₄⋊C₄	—	86	1	no	—
88	C₄⋊C₄	—	172	1	no	—
89	C₄⋊C₄	—	172	1	no	—
90	C₂²⋊C₄	—	172	1	no	—
91	C₄²	D₁₇₂	1	1	yes	characteristic
92	C₁₆	—	2	2	no	—
93	C₂³	—	172	1	no	—
94	Q₈	—	172	2	no	—
95	Q₈	—	172	2	no	—
96	D₄	—	86	2	no	—
97	D₄	—	172	1	no	—
98	C₂×C₄	—	344	2	no	—
99	C₂×C₄	—	172	1	no	—
100	C₂×C₄	—	172	2	no	—
101	C₂×C₄	D₃₄₄	1	1	yes	characteristic
102	C₂×C₄	D₃₄₄	1	1	yes	characteristic
103	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
104	C₈	—	2	1	no	—
105	C₈	D₄×D₄₃	1	1	yes	characteristic
106	C₈	#10621902	1	1	yes	characteristic
107	C₂²	—	344	1	no	—
108	C₂²	—	344	1	no	—
109	C₂²	—	172	2	no	—
110	C₂²	C₂×D₃₄₄	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	344	1	no	—
113	C₄	—	344	2	no	—
114	C₄	#13063270	1	2	yes	complemented
115	C₄	#13063300	1	1	yes	characteristic
116	C₄	#13063353	1	1	yes	characteristic
117	C₂	—	344	2	no	—
118	C₂	#14583095	1	1	yes	characteristic
119	C₂	C₂×#13063270	1	1	yes	characteristic
120	C₂	#14583342	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063270

C₂

yes

maximal

#14583279

C₂

yes

maximal, characteristic, complemented

C₄×D₃₄₄

C₂

yes

maximal, characteristic

#14582704

C₂

yes

maximal, complemented

C₄₃×C₄⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

C₂²

yes

—

C₂×D₃₄₄

C₂²

yes

characteristic

#13063325

—

C₄×D₁₇₂

—

#13063270

—

#13063259

C₂²

yes

characteristic

#13063247

—

C₂×C₁₆×C₄₃

C₂²

yes

—

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×D₁₇₂

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

D₄

yes

characteristic

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

D₃₄₄

—

D₃₄₄

D₄

yes

—

C₁₆×C₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

D₈

yes

characteristic

C₂×C₄×C₄₃

D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂×D₄

yes

characteristic

C₈×C₄₃

#41

yes

characteristic

C₂²×C₄₃

C₂×D₈

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

SD₃₂

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

C₂×SD₃₂

yes

characteristic

C₂×C₄₃

#510

yes

characteristic

D₄₃

—

C₂×SD₃₂

—

#494

—

C₄×D₈

—

C₄⋊C₁₆

D₄₃

yes

characteristic, complemented

#358

—

C₄₃

#2208

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄×D₄

—

SD₃₂

—

C₂×C₁₆

C₂×D₄₃

yes

—

#107

—

#102

—

C₄×C₈

C₂×D₄₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

172

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

172

—

D₈

—

D₈

—

C₂×C₈

D₁₇₂

yes

characteristic

C₂×C₈

C₂²×D₄₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

172

—

C₄⋊C₄

—

172

—

C₂²⋊C₄

—

172

—

C₄²

D₁₇₂

yes

characteristic

C₁₆

—

C₂³

—

172

—

Q₈

—

172

—

Q₈

—

172

—

D₄

—

D₄

—

172

—

C₂×C₄

—

344

—

C₂×C₄

—

172

—

100

C₂×C₄

—

172

—

101

C₂×C₄

D₃₄₄

yes

characteristic

102

C₂×C₄

D₃₄₄

yes

characteristic

103

C₂×C₄

C₂×D₁₇₂

yes

characteristic

104

C₈

—

105

C₈

D₄×D₄₃

yes

characteristic

106

C₈

#10621902

yes

characteristic

107

C₂²

—

344

—

108

C₂²

—

344

—

109

C₂²

—

172

—

110

C₂²

C₂×D₃₄₄

yes

characteristic

111

C₄

—

112

C₄

—

344

—

113

C₄

—

344

—

114

C₄

#13063270

yes

complemented

115

C₄

#13063300

yes

characteristic

116

C₄

#13063353

yes

characteristic

117

C₂

—

344

—

118

C₂

#14583095

yes

characteristic

119

C₂

C₂×#13063270

yes

characteristic

120

C₂

#14583342

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₃₄₄
commutator	C₂×C₈×C₄₃	C₂³
Frattini	C₂×C₈	C₂²×D₄₃
Fitting	C₄₃×C₄⋊C₁₆	C₂
socle	C₂²×C₄₃	C₂×D₈
hypercenter	C₄⋊C₁₆	D₄₃
hypocenter	C₄₃	#2208
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₃₄₄

commutator

C₂×C₈×C₄₃

C₂³

Frattini

C₂×C₈

C₂²×D₄₃

Fitting

C₄₃×C₄⋊C₁₆

C₂

socle

C₂²×C₄₃

C₂×D₈

hypercenter

C₄⋊C₁₆

D₄₃

hypocenter

C₄₃

#2208

LCS

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
2. #14583279	C₂	semidirect
4. #14582704	C₂	semidirect
5. C₄₃×C₄⋊C₁₆	C₂	semidirect
51. C₄×C₄₃	SD₃₂	semidirect
64. C₄⋊C₁₆	D₄₃	semidirect
66. C₄₃	#2208	semidirect
114. C₄	#13063270	semidirect

H ≅ G/N

type

2. #14583279

C₂

semidirect

4. #14582704

C₂

semidirect

5. C₄₃×C₄⋊C₁₆

C₂

semidirect

51. C₄×C₄₃

SD₃₂

semidirect

64. C₄⋊C₁₆

D₄₃

semidirect

66. C₄₃

#2208

semidirect

114. C₄

#13063270

semidirect

Subgroups — #19136641

Conjugacy classes of subgroups

Complements