Group #19145940 — subgroups

Subgroups — #19145940

Order 5504.

348 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 137 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	#14583456	C₂	1	2	yes	maximal
2	#14583195	C₂	1	4	yes	maximal, complemented
3	#14583140	C₂	1	4	yes	maximal
4	#14583110	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₇₂	C₂²	1	4	yes	—
7	#13063325	C₂²	1	2	yes	—
8	#13063325	C₂²	1	2	yes	—
9	#13063317	—	2	2	no	—
10	#13063304	—	2	2	no	—
11	#13063304	C₂²	1	4	yes	—
12	C₄×Dic₈₆	—	2	2	no	—
13	C₄×Dic₈₆	C₂²	1	4	yes	—
14	#13063284	—	2	2	no	—
15	#13063259	C₂²	1	4	yes	—
16	#13063250	C₂²	1	8	yes	—
17	C₂²×C₈×C₄₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₄₃	C₂²	1	4	yes	—
20	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
23	C₂×Dic₈₆	—	2	2	no	—
24	C₂×Dic₈₆	C₂³	1	4	yes	—
25	C₂×C₈×C₄₃	—	4	2	no	—
26	C₂×C₈×C₄₃	C₂³	1	4	yes	—
27	C₄²×C₄₃	D₄	1	2	yes	—
28	C₄²×C₄₃	C₂³	1	2	yes	—
29	C₂²⋊Dic₄₃	—	2	4	no	—
30	C₄⋊Dic₄₃	—	2	2	no	—
31	C₄⋊Dic₄₃	—	2	2	no	—
32	C₄⋊Dic₄₃	—	2	2	no	—
33	C₄⋊Dic₄₃	C₂³	1	4	yes	—
34	Dic₄₃⋊C₄	—	2	4	no	—
35	Dic₄₃⋊C₄	—	2	4	no	—
36	C₄×Dic₄₃	—	2	4	no	—
37	Dic₁₇₂	—	4	4	no	—
38	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₄₃	—	2	1	no	—
40	C₂×C₄×C₄₃	—	2	1	no	—
41	C₂×C₄×C₄₃	—	4	1	no	—
42	C₂×C₄×C₄₃	Q₁₆	1	4	yes	complemented
43	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₄₃	—	2	4	no	—
48	C₂×Dic₄₃	—	2	4	no	—
49	Dic₈₆	—	2	4	no	—
50	Dic₈₆	—	2	4	no	—
51	Dic₈₆	—	4	4	no	—
52	C₈×C₄₃	—	2	4	no	—
53	C₂²×C₄₃	—	2	1	no	—
54	C₂²×C₄₃	—	2	1	no	—
55	C₂²×C₄₃	C₂×Q₁₆	1	2	yes	—
56	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₄₃	—	2	1	no	—
58	C₄×C₄₃	—	2	2	no	—
59	C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
60	C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
61	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
62	Dic₄₃	—	4	4	no	—
63	Dic₄₃	—	4	4	no	—
64	#3249	—	43	1	no	maximal, Sylow
65	C₂×C₄₃	—	2	2	no	—
66	C₂×C₄₃	#537	1	1	yes	characteristic
67	C₂×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
68	C₂×C₄₃	#575	1	1	yes	characteristic
69	#533	—	43	2	no	—
70	#477	—	43	4	no	—
71	#467	—	43	4	no	—
72	#462	—	43	4	no	—
73	C₂×C₄⋊C₈	D₄₃	1	1	yes	characteristic
74	C₄₃	#3249	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₁₆	—	43	4	no	—
76	C₂²×C₈	C₂×D₄₃	1	2	yes	—
77	C₄⋊Q₈	—	43	2	no	—
78	C₄⋊Q₈	—	43	2	no	—
79	#125	—	86	2	no	—
80	C₂²⋊Q₈	—	43	4	no	—
81	C₂²⋊Q₈	—	86	2	no	—
82	C₄×Q₈	—	43	4	no	—
83	C₄×Q₈	—	86	2	no	—
84	#117	—	86	2	no	—
85	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
86	#107	—	43	4	no	—
87	C₄⋊C₈	C₂×D₄₃	1	4	yes	—
88	#103	—	43	8	no	—
89	C₂×Q₈	—	43	4	no	—
90	C₂×Q₈	—	86	2	no	—
91	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
92	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
94	Q₁₆	—	172	4	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	C₂²×D₄₃	1	4	yes	—
97	C₄⋊C₄	—	43	4	no	—
98	C₄⋊C₄	—	86	2	no	—
99	C₄⋊C₄	—	86	2	no	—
100	C₄⋊C₄	—	86	4	no	—
101	C₄⋊C₄	—	86	4	no	—
102	C₄⋊C₄	—	86	2	no	—
103	C₂²⋊C₄	—	86	4	no	—
104	C₄²	—	86	4	no	—
105	C₄²	D₁₇₂	1	2	yes	—
106	C₄²	C₂²×D₄₃	1	2	yes	—
107	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
108	Q₈	—	86	4	no	—
109	Q₈	—	86	4	no	—
110	Q₈	—	172	4	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	86	4	no	—
115	C₂×C₄	—	86	4	no	—
116	C₂×C₄	Dic₁₇₂	1	4	yes	complemented
117	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
118	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
119	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
120	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×Dic₁₇₂	1	2	yes	—
125	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	1	no	—
128	C₄	—	172	4	no	—
129	C₄	—	172	4	no	—
130	C₄	C₂×Dic₁₇₂	1	4	yes	—
131	C₄	#13063370	1	1	yes	characteristic
132	C₄	#13063373	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	#14583480	1	1	yes	characteristic
135	C₂	C₂²×Dic₁₇₂	1	1	yes	characteristic
136	C₂	#14583699	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583456

C₂

yes

maximal

#14583195

C₂

yes

maximal, complemented

#14583140

C₂

yes

maximal

#14583110

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₄₃

C₂

yes

maximal, characteristic

C₂×Dic₁₇₂

C₂²

yes

—

#13063325

C₂²

yes

—

#13063325

C₂²

yes

—

#13063317

—

#13063304

—

#13063304

C₂²

yes

—

C₄×Dic₈₆

—

C₄×Dic₈₆

C₂²

yes

—

#13063284

—

#13063259

C₂²

yes

—

#13063250

C₂²

yes

—

C₂²×C₈×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂³

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂³

yes

—

C₄²×C₄₃

D₄

yes

—

C₄²×C₄₃

C₂³

yes

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂³

yes

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

Dic₁₇₂

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

Q₁₆

yes

complemented

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×Q₁₆

yes

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×Q₁₆

yes

—

C₄×C₄₃

H(5,𝔽₂)

yes

characteristic

C₄×C₄₃

2₋¹⁺⁴

yes

characteristic

Dic₄₃

—

Dic₄₃

—

#3249

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

#537

yes

characteristic

C₂×C₄₃

C₂²×Q₁₆

yes

C₂×C₄₃

yes

—

—

—

—

C₂×C₄⋊C₈

D₄₃

yes

characteristic

C₄₃

#3249

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂²×C₈

C₂×D₄₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

#107

—

C₄⋊C₈

C₂×D₄₃

yes

—

#103

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

D₁₇₂

yes

characteristic

C₂²×C₄

D₁₇₂

yes

characteristic

C₂²×C₄

C₂²×D₄₃

yes

characteristic

Q₁₆

—

172

—

C₂×C₈

—

C₂×C₈

C₂²×D₄₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₄²

D₁₇₂

yes

—

106

C₄²

C₂²×D₄₃

yes

—

107

C₂³

C₂×D₁₇₂

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

172

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

Dic₁₇₂

yes

complemented

117

C₂×C₄

C₂×D₁₇₂

yes

—

118

C₂×C₄

C₂×D₁₇₂

yes

characteristic

119

C₂×C₄

C₂×D₁₇₂

yes

—

120

C₂×C₄

C₂³×D₄₃

yes

characteristic

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×Dic₁₇₂

yes

—

125

C₂²

C₂²×D₁₇₂

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

172

—

129

C₄

—

172

—

130

C₄

C₂×Dic₁₇₂

yes

—

131

C₄

#13063370

yes

characteristic

132

C₄

#13063373

yes

characteristic

133

C₂

—

134

C₂

#14583480

yes

characteristic

135

C₂

C₂²×Dic₁₇₂

yes

characteristic

136

C₂

#14583699

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₂×C₄⋊C₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₂×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#3249
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²

C₂²×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₂×C₄⋊C₈×C₄₃

C₂

socle

C₂²×C₄₃

C₂²×D₄

hypercenter

C₂×C₄⋊C₈

D₄₃

hypocenter

C₄₃

#3249

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
2. #14583195	C₂	semidirect
4. #14583110	C₂	semidirect
42. C₂×C₄×C₄₃	Q₁₆	semidirect
74. C₄₃	#3249	semidirect
116. C₂×C₄	Dic₁₇₂	semidirect

H ≅ G/N

type

2. #14583195

C₂

semidirect

4. #14583110

C₂

semidirect

42. C₂×C₄×C₄₃

Q₁₆

semidirect

74. C₄₃

#3249

semidirect

116. C₂×C₄

Dic₁₇₂

semidirect

Subgroups — #19145940

Conjugacy classes of subgroups

Complements