Group #19212081 — subgroups

Subgroups — #19212081

Order 5888.

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

normal subgroups only

This group has 138 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
2	#15671562	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1708	C₂	1	1	yes	maximal, characteristic
4	#13092216	—	2	1	no	—
5	#13092210	—	2	1	no	—
6	C₂₃×#498	C₄	1	1	yes	characteristic, complemented
7	C₂₃×#492	C₂²	1	1	yes	characteristic
8	C₂₃×#487	—	2	1	no	—
9	C₂₃×#486	—	2	1	no	—
10	C₈²×C₂₃	C₄	1	1	yes	characteristic, complemented
11	C₂×SD₁₆×C₂₃	—	4	1	no	—
12	C₂₃×C₄⋊Q₈	—	2	1	no	—
13	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
14	C₂₃×#127	D₄	1	1	yes	characteristic
15	C₂₃×#106	—	4	1	no	—
16	C₂₃×#103	—	4	2	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	#10625018	—	4	1	no	—
21	#10625016	—	4	1	no	—
22	C₂×Q₈×C₂₃	—	2	1	no	—
23	C₂×Q₈×C₂₃	—	4	1	no	—
24	C₂×D₄×C₂₃	—	2	1	no	—
25	C₂×D₄×C₂₃	—	4	1	no	—
26	SD₁₆×C₂₃	—	8	2	no	—
27	C₂×C₈×C₂₃	—	2	1	no	—
28	C₂×C₈×C₂₃	—	2	2	no	—
29	C₄⋊C₄×C₂₃	—	4	1	no	—
30	C₄⋊C₄×C₂₃	—	4	1	no	—
31	C₄⋊C₄×C₂₃	—	4	1	no	—
32	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²⋊Dic₂₃	—	8	1	no	—
34	#67267	—	8	1	no	—
35	#7536	—	23	1	no	maximal, Sylow
36	C₂³×C₂₃	—	4	1	no	—
37	Q₈×C₂₃	—	8	1	no	—
38	Q₈×C₂₃	—	8	2	no	—
39	D₄×C₂₃	—	8	1	no	—
40	D₄×C₂₃	—	8	2	no	—
41	C₂×C₄×C₂₃	—	2	1	no	—
42	C₂×C₄×C₂₃	—	4	1	no	—
43	C₂×C₄×C₂₃	—	8	1	no	—
44	C₂×C₄×C₂₃	#99	1	1	yes	characteristic
45	C₂×Dic₂₃	—	16	1	no	—
46	C₈×C₂₃	—	2	2	no	—
47	C₈×C₂₃	—	4	2	no	—
48	C₂₃⋊C₈	—	16	1	no	—
49	#1708	D₂₃	1	1	yes	characteristic
50	#1410	—	23	1	no	—
51	#1409	—	23	1	no	—
52	C₂²×C₂₃	—	8	1	no	—
53	C₂²×C₂₃	—	16	1	no	—
54	C₂²×C₂₃	#353	1	1	yes	characteristic
55	C₄×C₂₃	—	2	2	no	—
56	C₄×C₂₃	—	2	1	no	—
57	C₄×C₂₃	—	8	1	no	—
58	C₄×C₂₃	—	16	1	no	—
59	Dic₂₃	—	32	1	no	—
60	#498	Dic₂₃	1	1	yes	characteristic, complemented
61	#492	C₂×D₂₃	1	1	yes	characteristic
62	#487	—	2	1	no	—
63	#486	—	2	1	no	—
64	#356	—	46	1	no	—
65	#353	—	46	1	no	—
66	C₈²	Dic₂₃	1	1	yes	characteristic, complemented
67	C₂×C₂₃	—	2	1	no	—
68	C₂×C₂₃	—	16	1	no	—
69	C₂×C₂₃	#1409	1	1	yes	characteristic
70	C₂×SD₁₆	—	4	1	no	—
71	C₄⋊Q₈	—	2	1	no	—
72	C₄⋊Q₈	C₂²⋊D₂₃	1	1	yes	characteristic
73	#127	C₂²⋊D₂₃	1	1	yes	characteristic
74	#106	—	4	1	no	—
75	#103	—	4	2	no	—
76	#102	—	4	2	no	—
77	#101	—	92	1	no	—
78	#99	—	92	1	no	—
79	C₄×C₈	—	2	1	no	—
80	C₄×C₈	C₂×Dic₂₃	1	1	yes	characteristic
81	C₂₃	#7536	1	1	yes	characteristic, complemented, Sylow
82	C₂×Q₈	—	2	1	no	—
83	C₂×Q₈	—	4	1	no	—
84	C₂×D₄	—	2	1	no	—
85	C₂×D₄	—	4	1	no	—
86	SD₁₆	—	8	2	no	—
87	M₁₆	—	184	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	2	no	—
90	C₄⋊C₄	—	4	1	no	—
91	C₄⋊C₄	—	4	1	no	—
92	C₄⋊C₄	—	4	1	no	—
93	C₂²⋊C₄	—	184	1	no	—
94	C₄²	C₂²⋊Dic₂₃	1	1	yes	characteristic
95	C₂³	—	4	1	no	—
96	Q₈	—	8	1	no	—
97	Q₈	—	8	2	no	—
98	D₄	—	8	1	no	—
99	D₄	—	8	2	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	8	1	no	—
103	C₂×C₄	—	368	1	no	—
104	C₂×C₄	#10625016	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	4	2	no	—
107	C₈	—	368	1	no	—
108	C₂²	—	8	1	no	—
109	C₂²	—	16	1	no	—
110	C₂²	#13092210	1	1	yes	characteristic
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	—	8	1	no	—
114	C₄	—	16	1	no	—
115	C₄	—	736	1	no	—
116	C₂	—	2	1	no	—
117	C₂	—	16	1	no	—
118	C₂	#15671562	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15671564

C₂

yes

maximal, characteristic

#15671562

C₂

yes

maximal, characteristic

C₂₃×#1708

C₂

yes

maximal, characteristic

#13092216

—

#13092210

—

C₂₃×#498

C₄

yes

characteristic, complemented

C₂₃×#492

C₂²

yes

characteristic

C₂₃×#487

—

C₂₃×#486

—

C₈²×C₂₃

C₄

yes

characteristic, complemented

C₂×SD₁₆×C₂₃

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

D₄

yes

characteristic

C₂₃×#127

D₄

yes

characteristic

C₂₃×#106

—

C₂₃×#103

—

C₂₃×#102

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

C₂×C₄

yes

characteristic

#10625018

—

#10625016

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

SD₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂²⋊C₄

yes

characteristic

C₂²⋊Dic₂₃

—

#67267

—

#7536

—

maximal, Sylow

C₂³×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#99

yes

characteristic

C₂×Dic₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1708

D₂₃

yes

characteristic

#1410

—

#1409

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#353

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

#498

Dic₂₃

yes

characteristic, complemented

#492

C₂×D₂₃

yes

—

—

—

—

C₈²

Dic₂₃

yes

characteristic, complemented

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#1409

yes

characteristic

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²⋊D₂₃

yes

characteristic

#127

C₂²⋊D₂₃

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂×Dic₂₃

yes

characteristic

C₂₃

#7536

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

M₁₆

—

184

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

184

—

C₄²

C₂²⋊Dic₂₃

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

368

—

104

C₂×C₄

#10625016

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

368

—

108

C₂²

—

109

C₂²

—

110

C₂²

#13092210

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

736

—

116

C₂

—

117

C₂

—

118

C₂

#15671562

yes

characteristic

	subgroup	G/H
center	C₂	#15671562
commutator	C₄×C₈×C₂₃	C₂×C₄
Frattini	#492	C₂×D₂₃
Fitting	C₂₃×#1708	C₂
socle	C₂×C₂₃	#1409
hypercenter	#1708	D₂₃
hypocenter	C₂₃	#7536
LCS	G ▷ C₄×C₈×C₂₃ ▷ C₄²×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #492 ◁ #1708

subgroup

G/H

center

C₂

#15671562

commutator

C₄×C₈×C₂₃

C₂×C₄

Frattini

#492

C₂×D₂₃

Fitting

C₂₃×#1708

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 ◁ #1708

N	H ≅ G/N	type
6. C₂₃×#498	C₄	semidirect
10. C₈²×C₂₃	C₄	semidirect
60. #498	Dic₂₃	semidirect
66. C₈²	Dic₂₃	semidirect
81. C₂₃	#7536	semidirect

H ≅ G/N

type

6. C₂₃×#498

C₄

semidirect

10. C₈²×C₂₃

C₄

semidirect

60. #498

Dic₂₃

semidirect

66. C₈²

Dic₂₃

semidirect

81. C₂₃

#7536

semidirect

Subgroups — #19212081

Conjugacy classes of subgroups

Complements