Group #19212072 — subgroups

Subgroups — #19212072

Order 5888.

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

normal subgroups only

This group has 156 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	#15671560	C₂	1	1	yes	maximal, characteristic
2	#15671560	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1715	C₂	1	1	yes	maximal, characteristic
4	#13092210	—	2	1	no	—
5	#13092210	—	2	1	no	—
6	C₂₃×#498	C₄	1	1	yes	characteristic, complemented
7	C₂₃×#493	C₂²	1	1	yes	characteristic
8	C₂₃×#492	—	2	1	no	—
9	C₂₃×#486	—	2	1	no	—
10	C₂₃×#322	C₄	1	1	yes	characteristic, complemented
11	C₂×SD₁₆×C₂₃	—	4	2	no	—
12	C₂×D₈×C₂₃	—	4	1	no	—
13	C₂₃×C₄⋊Q₈	—	2	1	no	—
14	C₂₃×#127	D₄	1	1	yes	characteristic
15	C₂₃×#127	D₄	1	1	yes	characteristic
16	C₂₃×#106	—	4	1	no	—
17	C₂₃×#102	—	4	2	no	—
18	C₄×C₈×C₂₃	—	2	1	no	—
19	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
20	#10625016	—	4	1	no	—
21	#10625016	—	4	1	no	—
22	C₂×Q₈×C₂₃	—	4	1	no	—
23	C₂×D₄×C₂₃	—	2	1	no	—
24	C₂×D₄×C₂₃	—	2	1	no	—
25	C₂×D₄×C₂₃	—	4	1	no	—
26	C₂×D₄×C₂₃	—	4	1	no	—
27	SD₁₆×C₂₃	—	8	4	no	—
28	D₈×C₂₃	—	8	2	no	—
29	C₂×C₈×C₂₃	—	2	1	no	—
30	C₂×C₈×C₂₃	—	2	2	no	—
31	C₄⋊C₄×C₂₃	—	4	1	no	—
32	C₄⋊C₄×C₂₃	—	4	1	no	—
33	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²⋊Dic₂₃	—	8	1	no	—
35	C₂²⋊Dic₂₃	—	8	1	no	—
36	#7533	—	23	1	no	maximal, Sylow
37	C₂³×C₂₃	—	4	1	no	—
38	C₂³×C₂₃	—	4	1	no	—
39	Q₈×C₂₃	—	8	2	no	—
40	D₄×C₂₃	—	8	1	no	—
41	D₄×C₂₃	—	8	1	no	—
42	D₄×C₂₃	—	8	2	no	—
43	D₄×C₂₃	—	8	2	no	—
44	C₂×C₄×C₂₃	—	2	1	no	—
45	C₂×C₄×C₂₃	—	8	1	no	—
46	C₂×C₄×C₂₃	#99	1	1	yes	characteristic
47	C₂×Dic₂₃	—	16	1	no	—
48	C₂×Dic₂₃	—	16	1	no	—
49	C₈×C₂₃	—	2	2	no	—
50	C₈×C₂₃	—	4	2	no	—
51	#1715	D₂₃	1	1	yes	characteristic
52	#1408	—	23	1	no	—
53	#1408	—	23	1	no	—
54	C₂²×C₂₃	—	8	1	no	—
55	C₂²×C₂₃	—	8	1	no	—
56	C₂²×C₂₃	—	16	1	no	—
57	C₂²×C₂₃	—	16	1	no	—
58	C₂²×C₂₃	#353	1	1	yes	characteristic
59	C₄×C₂₃	—	2	2	no	—
60	C₄×C₂₃	—	2	1	no	—
61	C₄×C₂₃	—	16	1	no	—
62	Dic₂₃	—	32	1	no	—
63	Dic₂₃	—	32	1	no	—
64	#498	Dic₂₃	1	1	yes	characteristic, complemented
65	#493	C₂×D₂₃	1	1	yes	characteristic
66	#492	—	2	1	no	—
67	#486	—	2	1	no	—
68	#353	—	46	1	no	—
69	#353	—	46	1	no	—
70	#322	Dic₂₃	1	1	yes	characteristic, complemented
71	C₂×C₂₃	—	2	1	no	—
72	C₂×C₂₃	—	16	1	no	—
73	C₂×C₂₃	—	16	1	no	—
74	C₂×C₂₃	#1409	1	1	yes	characteristic
75	C₂×SD₁₆	—	4	2	no	—
76	C₂×D₈	—	4	1	no	—
77	C₄⋊Q₈	—	2	1	no	—
78	#127	C₂²⋊D₂₃	1	1	yes	characteristic
79	#127	C₂²⋊D₂₃	1	1	yes	characteristic
80	#106	—	4	1	no	—
81	#102	—	4	2	no	—
82	#99	—	92	1	no	—
83	#99	—	92	1	no	—
84	C₄×C₈	—	2	1	no	—
85	C₄×C₈	C₂×Dic₂₃	1	1	yes	characteristic
86	C₂₃	#7533	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₈	—	4	1	no	—
88	C₂×D₄	—	2	1	no	—
89	C₂×D₄	—	2	1	no	—
90	C₂×D₄	—	4	1	no	—
91	C₂×D₄	—	4	1	no	—
92	SD₁₆	—	8	4	no	—
93	D₈	—	8	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	2	no	—
96	C₄⋊C₄	—	4	1	no	—
97	C₄⋊C₄	—	4	1	no	—
98	C₂²⋊C₄	—	184	1	no	—
99	C₂²⋊C₄	—	184	1	no	—
100	C₄²	C₂²⋊Dic₂₃	1	1	yes	characteristic
101	C₂³	—	4	1	no	—
102	C₂³	—	4	1	no	—
103	Q₈	—	8	2	no	—
104	D₄	—	8	1	no	—
105	D₄	—	8	1	no	—
106	D₄	—	8	2	no	—
107	D₄	—	8	2	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	8	1	no	—
110	C₂×C₄	—	368	1	no	—
111	C₂×C₄	—	368	1	no	—
112	C₂×C₄	#10625016	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	4	2	no	—
115	C₂²	—	8	1	no	—
116	C₂²	—	8	1	no	—
117	C₂²	—	16	1	no	—
118	C₂²	—	16	1	no	—
119	C₂²	#13092210	1	1	yes	characteristic
120	C₄	—	2	2	no	—
121	C₄	—	2	1	no	—
122	C₄	—	16	1	no	—
123	C₄	—	736	1	no	—
124	C₄	—	736	1	no	—
125	C₂	—	2	1	no	—
126	C₂	—	16	1	no	—
127	C₂	—	16	1	no	—
128	C₂	#15671562	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15671560

C₂

yes

maximal, characteristic

#15671560

C₂

yes

maximal, characteristic

C₂₃×#1715

C₂

yes

maximal, characteristic

#13092210

—

#13092210

—

C₂₃×#498

C₄

yes

characteristic, complemented

C₂₃×#493

C₂²

yes

characteristic

C₂₃×#492

—

C₂₃×#486

—

C₂₃×#322

C₄

yes

characteristic, complemented

C₂×SD₁₆×C₂₃

—

C₂×D₈×C₂₃

—

C₂₃×C₄⋊Q₈

—

C₂₃×#127

D₄

yes

characteristic

C₂₃×#127

D₄

yes

characteristic

C₂₃×#106

—

C₂₃×#102

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

C₂×C₄

yes

characteristic

#10625016

—

#10625016

—

C₂×Q₈×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

SD₁₆×C₂₃

—

D₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂²⋊C₄

yes

characteristic

C₂²⋊Dic₂₃

—

C₂²⋊Dic₂₃

—

#7533

—

maximal, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#99

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

#1715

D₂₃

yes

characteristic

#1408

—

#1408

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#353

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

Dic₂₃

—

#498

Dic₂₃

yes

characteristic, complemented

#493

C₂×D₂₃

yes

—

—

—

—

Dic₂₃

yes

characteristic, complemented

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#1409

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

#127

C₂²⋊D₂₃

yes

characteristic

#127

C₂²⋊D₂₃

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂×Dic₂₃

yes

characteristic

C₂₃

#7533

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

184

—

C₂²⋊C₄

—

184

—

100

C₄²

C₂²⋊Dic₂₃

yes

characteristic

101

C₂³

—

102

C₂³

—

103

Q₈

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

368

—

111

C₂×C₄

—

368

—

112

C₂×C₄

#10625016

yes

characteristic

113

C₈

—

114

C₈

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#13092210

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

736

—

124

C₄

—

736

—

125

C₂

—

126

C₂

—

127

C₂

—

128

C₂

#15671562

yes

characteristic

	subgroup	G/H
center	C₂	#15671562
commutator	C₄×C₈×C₂₃	C₂×C₄
Frattini	#493	C₂×D₂₃
Fitting	C₂₃×#1715	C₂
socle	C₂×C₂₃	#1409
hypercenter	#1715	D₂₃
hypocenter	C₂₃	#7533
LCS	G ▷ C₄×C₈×C₂₃ ▷ C₄²×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #493 ◁ #1715

subgroup

G/H

center

C₂

#15671562

commutator

C₄×C₈×C₂₃

C₂×C₄

Frattini

#493

C₂×D₂₃

Fitting

C₂₃×#1715

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 ◁ #1715

N	H ≅ G/N	type
6. C₂₃×#498	C₄	semidirect
10. C₂₃×#322	C₄	semidirect
64. #498	Dic₂₃	semidirect
70. #322	Dic₂₃	semidirect
86. C₂₃	#7533	semidirect

H ≅ G/N

type

6. C₂₃×#498

C₄

semidirect

10. C₂₃×#322

C₄

semidirect

64. #498

Dic₂₃

semidirect

70. #322

Dic₂₃

semidirect

86. C₂₃

#7533

semidirect

Subgroups — #19212072

Conjugacy classes of subgroups

Complements