Group #19136631 — subgroups

Subgroups — #19136631

Order 5504.

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

normal subgroups only

This group has 186 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×D₆₈₈	C₂	1	2	yes	maximal, complemented
2	#14583276	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582701	C₂	1	2	yes	maximal, complemented
5	C₄₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₃₄₄	—	2	1	no	—
7	C₂×D₃₄₄	C₂²	1	2	yes	—
8	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
9	#13063322	—	2	1	no	—
10	C₄×D₁₇₂	—	2	1	no	—
11	D₆₈₈	—	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×D₁₇₂	—	2	2	no	—
18	C₂×D₁₇₂	—	4	1	no	—
19	C₂×D₁₇₂	—	4	1	no	—
20	C₂×C₄×D₄₃	—	4	1	no	—
21	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
23	C₄²×C₄₃	D₄	1	1	yes	characteristic
24	(C₂×D₄₃)⋊C₄	—	4	1	no	—
25	C₄⋊Dic₄₃	—	2	1	no	—
26	D₃₄₄	—	2	2	no	—
27	D₃₄₄	—	2	1	no	—
28	D₃₄₄	—	2	2	no	—
29	D₃₄₄	—	4	2	no	—
30	D₃₄₄	D₄	1	2	yes	complemented
31	C₁₆×C₄₃	—	2	2	no	—
32	C₂²×D₄₃	—	4	1	no	—
33	C₂²×D₄₃	—	4	2	no	—
34	C₂×C₄×C₄₃	D₈	1	1	yes	characteristic
35	C₂×C₄×C₄₃	D₈	1	1	yes	characteristic
36	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₄₃	—	4	1	no	—
38	D₁₇₂	—	2	2	no	—
39	D₁₇₂	—	4	2	no	—
40	D₁₇₂	—	4	1	no	—
41	D₁₇₂	—	4	2	no	—
42	D₁₇₂	—	8	2	no	—
43	D₁₇₂	—	8	2	no	—
44	C₄×D₄₃	—	8	2	no	—
45	C₈×C₄₃	—	2	1	no	—
46	C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₈×C₄₃	#41	1	1	yes	characteristic
48	C₂²×C₄₃	C₂×D₈	1	1	yes	characteristic
49	C₂×D₄₃	—	4	2	no	—
50	C₂×D₄₃	—	8	1	no	—
51	C₂×D₄₃	—	8	1	no	—
52	C₂×D₄₃	—	8	2	no	—
53	C₂×D₄₃	—	8	2	no	—
54	C₂×D₄₃	—	8	2	no	—
55	C₄×C₄₃	—	2	1	no	—
56	C₄×C₄₃	D₁₆	1	2	yes	complemented
57	C₄×C₄₃	#121	1	1	yes	characteristic
58	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
59	Dic₄₃	—	8	1	no	—
60	#2206	—	43	1	no	maximal, Sylow
61	C₂×C₄₃	#459	1	1	yes	characteristic
62	C₂×C₄₃	C₂×D₁₆	1	1	yes	characteristic
63	C₂×C₄₃	#509	1	1	yes	characteristic
64	D₄₃	—	8	2	no	—
65	D₄₃	—	16	2	no	—
66	C₂×D₁₆	—	43	2	no	—
67	#493	—	43	1	no	—
68	C₄×D₈	—	43	1	no	—
69	C₄⋊C₁₆	D₄₃	1	1	yes	characteristic, complemented
70	#357	—	43	2	no	—
71	C₄₃	#2206	1	1	yes	characteristic, complemented, Sylow
72	C₂×D₈	—	43	1	no	—
73	C₂×D₈	—	43	2	no	—
74	C₂×D₈	—	86	1	no	—
75	#127	—	86	1	no	—
76	C₄×D₄	—	86	1	no	—
77	D₁₆	—	86	4	no	—
78	C₂×C₁₆	C₂×D₄₃	1	2	yes	—
79	#107	—	43	1	no	—
80	#102	—	86	1	no	—
81	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
82	C₂×D₄	—	86	1	no	—
83	C₂×D₄	—	86	2	no	—
84	C₂×D₄	—	172	1	no	—
85	C₂×D₄	—	172	1	no	—
86	C₂²×C₄	—	172	1	no	—
87	D₈	—	43	2	no	—
88	D₈	—	86	2	no	—
89	D₈	—	86	1	no	—
90	D₈	—	86	2	no	—
91	D₈	—	172	2	no	—
92	C₂×C₈	D₁₇₂	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
94	C₄⋊C₄	—	86	1	no	—
95	C₂²⋊C₄	—	172	1	no	—
96	C₄²	D₁₇₂	1	1	yes	characteristic
97	C₁₆	—	2	2	no	—
98	C₂³	—	172	2	no	—
99	C₂³	—	172	1	no	—
100	D₄	—	86	2	no	—
101	D₄	—	344	2	no	—
102	D₄	—	344	2	no	—
103	D₄	—	172	2	no	—
104	D₄	—	172	2	no	—
105	D₄	—	172	1	no	—
106	C₂×C₄	—	344	2	no	—
107	C₂×C₄	—	172	1	no	—
108	C₂×C₄	D₃₄₄	1	1	yes	characteristic
109	C₂×C₄	D₃₄₄	1	1	yes	characteristic
110	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
111	C₈	—	2	1	no	—
112	C₈	D₄×D₄₃	1	1	yes	characteristic
113	C₈	#10621902	1	1	yes	characteristic
114	C₂²	—	344	1	no	—
115	C₂²	—	344	1	no	—
116	C₂²	—	344	2	no	—
117	C₂²	—	344	2	no	—
118	C₂²	—	344	2	no	—
119	C₂²	—	172	2	no	—
120	C₂²	C₂×D₃₄₄	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	344	1	no	—
123	C₄	D₆₈₈	1	2	yes	complemented
124	C₄	#13063300	1	1	yes	characteristic
125	C₄	#13063353	1	1	yes	characteristic
126	C₂	—	344	2	no	—
127	C₂	—	688	2	no	—
128	C₂	#14583095	1	1	yes	characteristic
129	C₂	C₂×D₆₈₈	1	1	yes	characteristic
130	C₂	#14583337	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₆₈₈

C₂

yes

maximal, complemented

#14583276

C₂

yes

maximal, characteristic, complemented

C₄×D₃₄₄

C₂

yes

maximal, characteristic, complemented

#14582701

C₂

yes

maximal, complemented

C₄₃×C₄⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×D₃₄₄

—

C₂×D₃₄₄

C₂²

yes

—

C₂×D₃₄₄

C₂²

yes

characteristic

#13063322

—

C₄×D₁₇₂

—

D₆₈₈

—

#13063259

C₂²

yes

characteristic

#13063247

—

C₂×C₁₆×C₄₃

C₂²

yes

—

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×C₄×D₄₃

—

C₂×C₈×C₄₃

D₄

yes

characteristic

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

D₃₄₄

—

D₃₄₄

—

D₃₄₄

—

D₃₄₄

—

D₃₄₄

D₄

yes

complemented

C₁₆×C₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

D₈

yes

characteristic

C₂×C₄×C₄₃

D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

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

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

D₁₆

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

C₂×D₁₆

yes

characteristic

C₂×C₄₃

#509

yes

characteristic

D₄₃

—

D₄₃

—

C₂×D₁₆

—

#493

—

C₄×D₈

—

C₄⋊C₁₆

D₄₃

yes

characteristic, complemented

#357

—

C₄₃

#2206

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

#127

—

C₄×D₄

—

D₁₆

—

C₂×C₁₆

C₂×D₄₃

yes

—

#107

—

#102

—

C₄×C₈

C₂×D₄₃

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

172

—

C₂×D₄

—

172

—

C₂²×C₄

—

172

—

D₈

—

D₈

—

D₈

—

D₈

—

D₈

—

172

—

C₂×C₈

D₁₇₂

yes

characteristic

C₂×C₈

C₂²×D₄₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

172

—

C₄²

D₁₇₂

yes

characteristic

C₁₆

—

C₂³

—

172

—

C₂³

—

172

—

100

D₄

—

101

D₄

—

344

—

102

D₄

—

344

—

103

D₄

—

172

—

104

D₄

—

172

—

105

D₄

—

172

—

106

C₂×C₄

—

344

—

107

C₂×C₄

—

172

—

108

C₂×C₄

D₃₄₄

yes

characteristic

109

C₂×C₄

D₃₄₄

yes

characteristic

110

C₂×C₄

C₂×D₁₇₂

yes

characteristic

111

C₈

—

112

C₈

D₄×D₄₃

yes

characteristic

113

C₈

#10621902

yes

characteristic

114

C₂²

—

344

—

115

C₂²

—

344

—

116

C₂²

—

344

—

117

C₂²

—

344

—

118

C₂²

—

344

—

119

C₂²

—

172

—

120

C₂²

C₂×D₃₄₄

yes

characteristic

121

C₄

—

122

C₄

—

344

—

123

C₄

D₆₈₈

yes

complemented

124

C₄

#13063300

yes

characteristic

125

C₄

#13063353

yes

characteristic

126

C₂

—

344

—

127

C₂

—

688

—

128

C₂

#14583095

yes

characteristic

129

C₂

C₂×D₆₈₈

yes

characteristic

130

C₂

#14583337

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	#2206
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

#2206

LCS

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
1. C₂×D₆₈₈	C₂	semidirect
2. #14583276	C₂	semidirect
3. C₄×D₃₄₄	C₂	semidirect
4. #14582701	C₂	semidirect
5. C₄₃×C₄⋊C₁₆	C₂	semidirect
30. D₃₄₄	D₄	semidirect
56. C₄×C₄₃	D₁₆	semidirect
69. C₄⋊C₁₆	D₄₃	semidirect
71. C₄₃	#2206	semidirect
123. C₄	D₆₈₈	semidirect

H ≅ G/N

type

1. C₂×D₆₈₈

C₂

semidirect

2. #14583276

C₂

semidirect

3. C₄×D₃₄₄

C₂

semidirect

4. #14582701

C₂

semidirect

5. C₄₃×C₄⋊C₁₆

C₂

semidirect

30. D₃₄₄

D₄

semidirect

56. C₄×C₄₃

D₁₆

semidirect

69. C₄⋊C₁₆

D₄₃

semidirect

71. C₄₃

#2206

semidirect

123. C₄

D₆₈₈

semidirect

Subgroups — #19136631

Conjugacy classes of subgroups

Complements