Group #19212058 — subgroups

Subgroups — #19212058

Order 5888.

164 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 27 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	#15671564	C₂	1	1	yes	maximal, characteristic, complemented
2	#15671562	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1721	C₂	1	1	yes	maximal, characteristic
4	#13092216	—	2	1	no	—
5	#13092210	—	2	1	no	—
6	C₂₃×#493	—	2	1	no	—
7	C₂₃×#493	C₄	1	1	yes	characteristic, complemented
8	C₂₃×#492	C₂²	1	1	yes	characteristic
9	C₂₃×#486	—	2	1	no	—
10	C₂₃×#322	C₄	1	1	yes	characteristic, complemented
11	C₂×SD₁₆×C₂₃	—	4	1	no	—
12	C₂×D₈×C₂₃	—	4	1	no	—
13	C₂×D₈×C₂₃	—	4	2	no	—
14	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
15	C₂₃×#127	—	2	1	no	—
16	C₂₃×#127	D₄	1	1	yes	characteristic
17	C₂₃×#102	—	4	2	no	—
18	C₄×C₈×C₂₃	—	2	1	no	—
19	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
20	#10625018	—	4	1	no	—
21	#10625016	—	4	1	no	—
22	C₂×Q₈×C₂₃	—	2	1	no	—
23	C₂×D₄×C₂₃	—	2	1	no	—
24	C₂×D₄×C₂₃	—	4	1	no	—
25	C₂×D₄×C₂₃	—	4	1	no	—
26	C₂×D₄×C₂₃	—	4	1	no	—
27	C₂×D₄×C₂₃	—	4	1	no	—
28	SD₁₆×C₂₃	—	8	2	no	—
29	D₈×C₂₃	—	8	2	no	—
30	D₈×C₂₃	—	8	4	no	—
31	C₂×C₈×C₂₃	—	2	1	no	—
32	C₂×C₈×C₂₃	—	2	2	no	—
33	C₄⋊C₄×C₂₃	—	4	1	no	—
34	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²⋊Dic₂₃	—	8	1	no	—
36	#67267	—	8	1	no	—
37	#7527	—	23	1	no	maximal, Sylow
38	C₂³×C₂₃	—	4	1	no	—
39	C₂³×C₂₃	—	8	1	no	—
40	Q₈×C₂₃	—	8	1	no	—
41	D₄×C₂₃	—	8	1	no	—
42	D₄×C₂₃	—	8	2	no	—
43	D₄×C₂₃	—	8	2	no	—
44	D₄×C₂₃	—	8	2	no	—
45	D₄×C₂₃	—	8	2	no	—
46	C₂×C₄×C₂₃	—	2	1	no	—
47	C₂×C₄×C₂₃	—	4	1	no	—
48	C₂×C₄×C₂₃	#99	1	1	yes	characteristic
49	C₂×Dic₂₃	—	16	1	no	—
50	C₈×C₂₃	—	2	2	no	—
51	C₈×C₂₃	—	4	2	no	—
52	C₂₃⋊C₈	—	16	1	no	—
53	#1721	D₂₃	1	1	yes	characteristic
54	#1410	—	23	1	no	—
55	#1409	—	23	1	no	—
56	C₂²×C₂₃	—	8	1	no	—
57	C₂²×C₂₃	—	16	1	no	—
58	C₂²×C₂₃	—	16	1	no	—
59	C₂²×C₂₃	—	16	1	no	—
60	C₂²×C₂₃	—	16	1	no	—
61	C₂²×C₂₃	#353	1	1	yes	characteristic
62	C₄×C₂₃	—	2	2	no	—
63	C₄×C₂₃	—	2	1	no	—
64	C₄×C₂₃	—	8	1	no	—
65	Dic₂₃	—	32	1	no	—
66	#493	—	2	1	no	—
67	#493	Dic₂₃	1	1	yes	characteristic, complemented
68	#492	C₂×D₂₃	1	1	yes	characteristic
69	#486	—	2	1	no	—
70	#356	—	46	1	no	—
71	#353	—	46	1	no	—
72	#322	Dic₂₃	1	1	yes	characteristic, complemented
73	C₂×C₂₃	—	2	1	no	—
74	C₂×C₂₃	—	16	1	no	—
75	C₂×C₂₃	—	32	1	no	—
76	C₂×C₂₃	#1408	1	1	yes	characteristic
77	C₂×SD₁₆	—	4	1	no	—
78	C₂×D₈	—	4	1	no	—
79	C₂×D₈	—	4	2	no	—
80	C₄⋊Q₈	C₂²⋊D₂₃	1	1	yes	characteristic
81	#127	—	2	1	no	—
82	#127	C₂²⋊D₂₃	1	1	yes	characteristic
83	#102	—	4	2	no	—
84	#101	—	92	1	no	—
85	#99	—	92	1	no	—
86	C₄×C₈	—	2	1	no	—
87	C₄×C₈	C₂×Dic₂₃	1	1	yes	characteristic
88	C₂₃	#7527	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₈	—	2	1	no	—
90	C₂×D₄	—	2	1	no	—
91	C₂×D₄	—	4	1	no	—
92	C₂×D₄	—	4	1	no	—
93	C₂×D₄	—	4	1	no	—
94	C₂×D₄	—	4	1	no	—
95	SD₁₆	—	8	2	no	—
96	D₈	—	8	2	no	—
97	D₈	—	8	4	no	—
98	M₁₆	—	184	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	2	no	—
101	C₄⋊C₄	—	4	1	no	—
102	C₂²⋊C₄	—	184	1	no	—
103	C₄²	C₂²⋊Dic₂₃	1	1	yes	characteristic
104	C₂³	—	4	1	no	—
105	C₂³	—	8	1	no	—
106	Q₈	—	8	1	no	—
107	D₄	—	8	1	no	—
108	D₄	—	8	2	no	—
109	D₄	—	8	2	no	—
110	D₄	—	8	2	no	—
111	D₄	—	8	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	368	1	no	—
115	C₂×C₄	#10625016	1	1	yes	characteristic
116	C₈	—	2	2	no	—
117	C₈	—	4	2	no	—
118	C₈	—	368	1	no	—
119	C₂²	—	8	1	no	—
120	C₂²	—	16	1	no	—
121	C₂²	—	16	1	no	—
122	C₂²	—	16	1	no	—
123	C₂²	—	16	1	no	—
124	C₂²	#13092210	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	8	1	no	—
128	C₄	—	736	1	no	—
129	C₂	—	2	1	no	—
130	C₂	—	16	1	no	—
131	C₂	—	32	1	no	—
132	C₂	#15671560	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15671564

C₂

yes

maximal, characteristic, complemented

#15671562

C₂

yes

maximal, characteristic, complemented

C₂₃×#1721

C₂

yes

maximal, characteristic

#13092216

—

#13092210

—

C₂₃×#493

—

C₂₃×#493

C₄

yes

characteristic, complemented

C₂₃×#492

C₂²

yes

characteristic

C₂₃×#486

—

C₂₃×#322

C₄

yes

characteristic, complemented

C₂×SD₁₆×C₂₃

—

C₂×D₈×C₂₃

—

C₂×D₈×C₂₃

—

C₂₃×C₄⋊Q₈

D₄

yes

characteristic

C₂₃×#127

—

C₂₃×#127

D₄

yes

characteristic

C₂₃×#102

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

C₂×C₄

yes

characteristic

#10625018

—

#10625016

—

C₂×Q₈×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

SD₁₆×C₂₃

—

D₈×C₂₃

—

D₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂²⋊C₄

yes

characteristic

C₂²⋊Dic₂₃

—

#67267

—

#7527

—

maximal, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#99

yes

characteristic

C₂×Dic₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1721

D₂₃

yes

characteristic

#1410

—

#1409

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#353

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

#493

—

#493

Dic₂₃

yes

characteristic, complemented

#492

C₂×D₂₃

yes

—

—

—

Dic₂₃

yes

characteristic, complemented

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#1408

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

C₂²⋊D₂₃

yes

characteristic

#127

—

#127

C₂²⋊D₂₃

yes

—

—

—

C₄×C₈

—

C₄×C₈

C₂×Dic₂₃

yes

characteristic

C₂₃

#7527

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

D₈

—

M₁₆

—

184

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

184

—

103

C₄²

C₂²⋊Dic₂₃

yes

characteristic

104

C₂³

—

105

C₂³

—

106

Q₈

—

107

D₄

—

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

368

—

115

C₂×C₄

#10625016

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

—

368

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

#13092210

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

736

—

129

C₂

—

130

C₂

—

131

C₂

—

132

C₂

#15671560

yes

characteristic

	subgroup	G/H
center	C₂	#15671560
commutator	C₄×C₈×C₂₃	C₂×C₄
Frattini	#492	C₂×D₂₃
Fitting	C₂₃×#1721	C₂
socle	C₂×C₂₃	#1408
hypercenter	#1721	D₂₃
hypocenter	C₂₃	#7527
LCS	G ▷ C₄×C₈×C₂₃ ▷ C₄²×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #492 ◁ #1721

subgroup

G/H

center

C₂

#15671560

commutator

C₄×C₈×C₂₃

C₂×C₄

Frattini

#492

C₂×D₂₃

Fitting

C₂₃×#1721

C₂

C₂×C₂₃

D₂₃

C₂₃

G ▷ C₄×C₈×C₂₃ ▷ C₄²×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #492 ◁ #1721

N	H ≅ G/N	type
1. #15671564	C₂	semidirect
2. #15671562	C₂	semidirect
7. C₂₃×#493	C₄	semidirect
10. C₂₃×#322	C₄	semidirect
67. #493	Dic₂₃	semidirect
72. #322	Dic₂₃	semidirect
88. C₂₃	#7527	semidirect

H ≅ G/N

type

1. #15671564

C₂

semidirect

2. #15671562

C₂

semidirect

7. C₂₃×#493

C₄

semidirect

10. C₂₃×#322

C₄

semidirect

67. #493

Dic₂₃

semidirect

72. #322

Dic₂₃

semidirect

88. C₂₃

#7527

semidirect

Subgroups — #19212058

Conjugacy classes of subgroups

Complements