Group #19212078 — subgroups

Subgroups — #19212078

Order 5888.

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

normal subgroups only

This group has 150 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×#1705	C₂	1	1	yes	maximal, characteristic
4	#13092216	—	2	1	no	—
5	#13092210	—	2	1	no	—
6	C₂₃×#498	—	2	1	no	—
7	C₂₃×#498	C₄	1	1	yes	characteristic, complemented
8	C₂₃×#492	—	2	1	no	—
9	C₂₃×#492	C₂²	1	1	yes	characteristic
10	C₈²×C₂₃	C₄	1	1	yes	characteristic, complemented
11	C₂×SD₁₆×C₂₃	—	4	1	no	—
12	C₂×SD₁₆×C₂₃	—	4	2	no	—
13	C₂₃×C₄⋊Q₈	—	2	1	no	—
14	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
15	C₂₃×#127	D₄	1	1	yes	characteristic
16	C₂₃×#106	—	4	2	no	—
17	C₂₃×#106	—	4	1	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	SD₁₆×C₂₃	—	8	4	no	—
28	C₂×C₈×C₂₃	—	2	2	no	—
29	C₂×C₈×C₂₃	—	2	1	no	—
30	C₄⋊C₄×C₂₃	—	4	1	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	#67267	—	8	1	no	—
36	#7535	—	23	1	no	maximal, Sylow
37	C₂³×C₂₃	—	4	1	no	—
38	Q₈×C₂₃	—	8	1	no	—
39	Q₈×C₂₃	—	8	2	no	—
40	D₄×C₂₃	—	8	1	no	—
41	D₄×C₂₃	—	8	2	no	—
42	C₂×C₄×C₂₃	—	2	1	no	—
43	C₂×C₄×C₂₃	—	4	1	no	—
44	C₂×C₄×C₂₃	—	8	1	no	—
45	C₂×C₄×C₂₃	#99	1	1	yes	characteristic
46	C₂×Dic₂₃	—	16	1	no	—
47	C₈×C₂₃	—	2	4	no	—
48	C₈×C₂₃	—	2	2	no	—
49	C₂₃⋊C₈	—	16	1	no	—
50	#1705	D₂₃	1	1	yes	characteristic
51	#1410	—	23	1	no	—
52	#1409	—	23	1	no	—
53	C₂²×C₂₃	—	8	1	no	—
54	C₂²×C₂₃	—	16	1	no	—
55	C₂²×C₂₃	#353	1	1	yes	characteristic
56	C₄×C₂₃	—	2	2	no	—
57	C₄×C₂₃	—	2	1	no	—
58	C₄×C₂₃	—	8	1	no	—
59	C₄×C₂₃	—	16	1	no	—
60	Dic₂₃	—	32	1	no	—
61	#498	—	2	1	no	—
62	#498	Dic₂₃	1	1	yes	characteristic, complemented
63	#492	—	2	1	no	—
64	#492	C₂×D₂₃	1	1	yes	characteristic
65	#356	—	46	1	no	—
66	#353	—	46	1	no	—
67	C₈²	Dic₂₃	1	1	yes	characteristic, complemented
68	C₂×C₂₃	—	2	1	no	—
69	C₂×C₂₃	—	16	1	no	—
70	C₂×C₂₃	#1409	1	1	yes	characteristic
71	C₂×SD₁₆	—	4	1	no	—
72	C₂×SD₁₆	—	4	2	no	—
73	C₄⋊Q₈	—	2	1	no	—
74	C₄⋊Q₈	C₂²⋊D₂₃	1	1	yes	characteristic
75	#127	C₂²⋊D₂₃	1	1	yes	characteristic
76	#106	—	4	2	no	—
77	#106	—	4	1	no	—
78	#101	—	92	1	no	—
79	#99	—	92	1	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	C₂×Dic₂₃	1	1	yes	characteristic
82	C₂₃	#7535	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	4	1	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	4	1	no	—
87	SD₁₆	—	8	2	no	—
88	SD₁₆	—	8	4	no	—
89	M₁₆	—	184	1	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	1	no	—
92	C₄⋊C₄	—	4	1	no	—
93	C₄⋊C₄	—	4	1	no	—
94	C₄⋊C₄	—	4	1	no	—
95	C₂²⋊C₄	—	184	1	no	—
96	C₄²	C₂²⋊Dic₂₃	1	1	yes	characteristic
97	C₂³	—	4	1	no	—
98	Q₈	—	8	1	no	—
99	Q₈	—	8	2	no	—
100	D₄	—	8	1	no	—
101	D₄	—	8	2	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	8	1	no	—
105	C₂×C₄	—	368	1	no	—
106	C₂×C₄	#10625016	1	1	yes	characteristic
107	C₈	—	2	4	no	—
108	C₈	—	2	2	no	—
109	C₈	—	368	1	no	—
110	C₂²	—	8	1	no	—
111	C₂²	—	16	1	no	—
112	C₂²	#13092210	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	2	1	no	—
115	C₄	—	8	1	no	—
116	C₄	—	16	1	no	—
117	C₄	—	736	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	16	1	no	—
120	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×#1705

C₂

yes

maximal, characteristic

#13092216

—

#13092210

—

C₂₃×#498

—

C₂₃×#498

C₄

yes

characteristic, complemented

C₂₃×#492

—

C₂₃×#492

C₂²

yes

characteristic

C₈²×C₂₃

C₄

yes

characteristic, complemented

C₂×SD₁₆×C₂₃

—

C₂×SD₁₆×C₂₃

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

D₄

yes

characteristic

C₂₃×#127

D₄

yes

characteristic

C₂₃×#106

—

C₂₃×#106

—

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

—

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

—

#7535

—

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

—

#1705

D₂₃

yes

characteristic

#1410

—

#1409

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#353

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

#498

—

#498

Dic₂₃

yes

characteristic, complemented

#492

—

#492

C₂×D₂₃

yes

characteristic

#356

—

#353

—

C₈²

Dic₂₃

yes

characteristic, complemented

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#1409

yes

characteristic

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²⋊D₂₃

yes

characteristic

#127

C₂²⋊D₂₃

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂×Dic₂₃

yes

characteristic

C₂₃

#7535

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

M₁₆

—

184

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

184

—

C₄²

C₂²⋊Dic₂₃

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

368

—

106

C₂×C₄

#10625016

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

368

—

110

C₂²

—

111

C₂²

—

112

C₂²

#13092210

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

736

—

118

C₂

—

119

C₂

—

120

C₂

#15671562

yes

characteristic

	subgroup	G/H
center	C₂	#15671562
commutator	C₄×C₈×C₂₃	C₂×C₄
Frattini	#492	C₂×D₂₃
Fitting	C₂₃×#1705	C₂
socle	C₂×C₂₃	#1409
hypercenter	#1705	D₂₃
hypocenter	C₂₃	#7535
LCS	G ▷ C₄×C₈×C₂₃ ▷ C₄²×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #492 ◁ #1705

subgroup

G/H

center

C₂

#15671562

commutator

C₄×C₈×C₂₃

C₂×C₄

Frattini

#492

C₂×D₂₃

Fitting

C₂₃×#1705

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 ◁ #1705

N	H ≅ G/N	type
7. C₂₃×#498	C₄	semidirect
10. C₈²×C₂₃	C₄	semidirect
62. #498	Dic₂₃	semidirect
67. C₈²	Dic₂₃	semidirect
82. C₂₃	#7535	semidirect

H ≅ G/N

type

7. C₂₃×#498

C₄

semidirect

10. C₈²×C₂₃

C₄

semidirect

62. #498

Dic₂₃

semidirect

67. C₈²

Dic₂₃

semidirect

82. C₂₃

#7535

semidirect

Subgroups — #19212078

Conjugacy classes of subgroups

Complements