Group #19212063 — subgroups

Subgroups — #19212063

Order 5888.

186 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 25 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	#15673011	C₂	1	1	yes	maximal, characteristic
2	#15671560	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1408	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092794	C₄	1	1	yes	characteristic, complemented
5	#13092793	—	2	1	no	—
6	#13092793	—	2	1	no	—
7	#13092210	—	2	1	no	—
8	C₈×C₂₃⋊C₈	C₄	1	1	yes	characteristic, complemented
9	C₂₃×#493	C₂²	1	1	yes	characteristic
10	C₂₃×#353	—	2	1	no	—
11	C₂×D₈×C₂₃	—	4	1	no	—
12	C₂₃×#127	D₄	1	1	yes	characteristic
13	C₂₃×#127	D₄	1	1	yes	characteristic
14	C₂×#67272	—	4	2	no	—
15	C₂×#67272	—	4	2	no	—
16	C₂×D₁₈₄	—	4	1	no	—
17	#10625045	—	2	1	no	—
18	C₂₃×#99	—	4	1	no	—
19	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
20	#10625016	—	4	1	no	—
21	C₄×C₂₃⋊C₈	—	2	1	no	—
22	C₂×D₄×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×D₉₂	—	4	1	no	—
27	C₂×D₉₂	—	4	1	no	—
28	C₂×D₉₂	—	4	1	no	—
29	D₈×C₂₃	—	8	2	no	—
30	C₂×C₈×C₂₃	—	2	1	no	—
31	C₂²⋊C₄×C₂₃	—	8	1	no	—
32	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²⋊Dic₂₃	—	8	1	no	—
34	#67272	—	8	4	no	—
35	#67272	—	8	4	no	—
36	C₂×C₂₃⋊C₈	—	2	2	no	—
37	D₁₈₄	—	8	2	no	—
38	#7529	—	23	1	no	maximal, Sylow
39	C₂³×C₂₃	—	4	1	no	—
40	C₂³×C₂₃	—	4	1	no	—
41	C₂²×D₂₃	—	8	1	no	—
42	D₄×C₂₃	—	8	1	no	—
43	D₄×C₂₃	—	8	1	no	—
44	D₄×C₂₃	—	8	2	no	—
45	D₄×C₂₃	—	8	2	no	—
46	C₂×C₄×C₂₃	—	2	1	no	—
47	C₂×C₄×C₂₃	—	16	1	no	—
48	C₂×C₄×C₂₃	#99	1	1	yes	characteristic
49	C₂×Dic₂₃	—	16	1	no	—
50	D₉₂	—	8	2	no	—
51	D₉₂	—	8	2	no	—
52	D₉₂	—	8	2	no	—
53	C₈×C₂₃	—	2	2	no	—
54	C₂₃⋊C₈	—	2	4	no	—
55	#1712	—	23	1	no	—
56	#1408	—	23	1	no	—
57	#1408	D₂₃	1	1	yes	characteristic, complemented
58	C₂²×C₂₃	—	8	1	no	—
59	C₂²×C₂₃	—	8	1	no	—
60	C₂²×C₂₃	—	16	1	no	—
61	C₂²×C₂₃	—	16	1	no	—
62	C₂²×C₂₃	#353	1	1	yes	characteristic
63	C₂×D₂₃	—	16	1	no	—
64	C₂×D₂₃	—	16	1	no	—
65	C₂×D₂₃	—	16	1	no	—
66	C₄×C₂₃	—	2	2	no	—
67	C₄×C₂₃	—	2	1	no	—
68	C₄×C₂₃	—	32	1	no	—
69	Dic₂₃	—	32	1	no	—
70	#493	—	23	1	no	—
71	#493	—	46	1	no	—
72	#493	—	46	1	no	—
73	#493	C₂×D₂₃	1	1	yes	characteristic
74	#353	—	2	1	no	—
75	#353	—	46	1	no	—
76	C₈²	—	23	1	no	—
77	C₂×C₂₃	—	2	1	no	—
78	C₂×C₂₃	—	16	1	no	—
79	C₂×C₂₃	—	16	1	no	—
80	C₂×C₂₃	#1408	1	1	yes	characteristic
81	D₂₃	—	32	1	no	—
82	C₂×D₈	—	4	1	no	—
83	C₂×D₈	—	92	1	no	—
84	C₂×D₈	—	92	2	no	—
85	C₂×D₈	—	92	2	no	—
86	#127	—	46	1	no	—
87	#127	D₉₂	1	1	yes	characteristic
88	#127	C₂²⋊D₂₃	1	1	yes	characteristic
89	#99	—	4	1	no	—
90	#99	—	92	1	no	—
91	C₄×C₈	—	46	1	no	—
92	C₄×C₈	C₄×D₂₃	1	1	yes	characteristic
93	C₂₃	#7529	1	1	yes	characteristic, complemented, Sylow
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	4	1	no	—
97	C₂×D₄	—	4	1	no	—
98	C₂×D₄	—	92	1	no	—
99	C₂×D₄	—	92	1	no	—
100	C₂×D₄	—	92	1	no	—
101	D₈	—	8	2	no	—
102	D₈	—	184	2	no	—
103	D₈	—	184	4	no	—
104	D₈	—	184	4	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	46	2	no	—
107	C₂²⋊C₄	—	8	1	no	—
108	C₂²⋊C₄	—	184	1	no	—
109	C₄²	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
110	C₂³	—	4	1	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	184	1	no	—
113	D₄	—	8	1	no	—
114	D₄	—	8	1	no	—
115	D₄	—	8	2	no	—
116	D₄	—	8	2	no	—
117	D₄	—	184	2	no	—
118	D₄	—	184	2	no	—
119	D₄	—	184	2	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	16	1	no	—
122	C₂×C₄	—	368	1	no	—
123	C₂×C₄	#10624988	1	1	yes	characteristic
124	C₈	—	2	2	no	—
125	C₈	—	46	4	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	—	16	1	no	—
129	C₂²	—	16	1	no	—
130	C₂²	—	368	1	no	—
131	C₂²	—	368	1	no	—
132	C₂²	—	368	1	no	—
133	C₂²	#13092209	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	2	1	no	—
136	C₄	—	32	1	no	—
137	C₄	—	736	1	no	—
138	C₂	—	2	1	no	—
139	C₂	—	16	1	no	—
140	C₂	—	16	1	no	—
141	C₂	—	736	1	no	—
142	C₂	#15671559	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15673011

C₂

yes

maximal, characteristic

#15671560

C₂

yes

maximal, characteristic, complemented

C₂₃×#1408

C₂

yes

maximal, characteristic, complemented

#13092794

C₄

yes

characteristic, complemented

#13092793

—

#13092793

—

#13092210

—

C₈×C₂₃⋊C₈

C₄

yes

characteristic, complemented

C₂₃×#493

C₂²

yes

characteristic

C₂₃×#353

—

C₂×D₈×C₂₃

—

C₂₃×#127

D₄

yes

characteristic

C₂₃×#127

D₄

yes

characteristic

C₂×#67272

—

C₂×#67272

—

C₂×D₁₈₄

—

#10625045

—

C₂₃×#99

—

C₄×C₈×C₂₃

C₂×C₄

yes

characteristic

#10625016

—

C₄×C₂₃⋊C₈

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

D₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

C₂²⋊C₄

yes

characteristic

C₂²⋊Dic₂₃

—

#67272

—

#67272

—

C₂×C₂₃⋊C₈

—

D₁₈₄

—

#7529

—

maximal, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂²×D₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#99

yes

characteristic

C₂×Dic₂₃

—

D₉₂

—

D₉₂

—

D₉₂

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1712

—

#1408

—

#1408

D₂₃

yes

characteristic, complemented

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#353

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

—

—

—

C₂×D₂₃

yes

characteristic

#353

—

#353

—

C₈²

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#1408

yes

characteristic

D₂₃

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

#127

—

#127

D₉₂

yes

characteristic

#127

C₂²⋊D₂₃

yes

characteristic

#99

—

#99

—

C₄×C₈

—

C₄×C₈

C₄×D₂₃

yes

characteristic

C₂₃

#7529

yes

characteristic, complemented, Sylow

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

D₈

—

102

D₈

—

184

—

103

D₈

—

184

—

104

D₈

—

184

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

184

—

109

C₄²

(C₂×D₂₃)⋊C₄

yes

characteristic

110

C₂³

—

111

C₂³

—

112

C₂³

—

184

—

113

D₄

—

114

D₄

—

115

D₄

—

116

D₄

—

117

D₄

—

184

—

118

D₄

—

184

—

119

D₄

—

184

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

368

—

123

C₂×C₄

#10624988

yes

characteristic

124

C₈

—

125

C₈

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

368

—

131

C₂²

—

368

—

132

C₂²

—

368

—

133

C₂²

#13092209

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

736

—

138

C₂

—

139

C₂

—

140

C₂

—

141

C₂

—

736

—

142

C₂

#15671559

yes

characteristic

	subgroup	G/H
center	C₂	#15671559
commutator	C₄×C₈×C₂₃	C₂×C₄
Frattini	#493	C₂×D₂₃
Fitting	C₂₃×#1408	C₂
socle	C₂×C₂₃	#1408
hypercenter	#1408	D₂₃
hypocenter	C₂₃	#7529
LCS	G ▷ C₄×C₈×C₂₃ ▷ C₄²×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #493 ◁ #1408

subgroup

G/H

center

C₂

#15671559

commutator

C₄×C₈×C₂₃

C₂×C₄

Frattini

#493

C₂×D₂₃

Fitting

C₂₃×#1408

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 ◁ #493 ◁ #1408

N	H ≅ G/N	type
2. #15671560	C₂	semidirect
3. C₂₃×#1408	C₂	semidirect
4. #13092794	C₄	semidirect
8. C₈×C₂₃⋊C₈	C₄	semidirect
57. #1408	D₂₃	semidirect
93. C₂₃	#7529	semidirect

H ≅ G/N

type

2. #15671560

C₂

semidirect

3. C₂₃×#1408

C₂

semidirect

4. #13092794

C₄

semidirect

8. C₈×C₂₃⋊C₈

C₄

semidirect

57. #1408

D₂₃

semidirect

93. C₂₃

#7529

semidirect

Subgroups — #19212063

Conjugacy classes of subgroups

Complements