Group #19212075 — subgroups

Subgroups — #19212075

Order 5888.

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

normal subgroups only

This group has 128 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	#15671566	C₂	1	1	yes	maximal, characteristic
2	#15671566	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1716	C₂	1	1	yes	maximal, characteristic
4	#13092216	—	2	1	no	—
5	#13092216	—	2	1	no	—
6	C₂₃×#498	—	2	1	no	—
7	C₂₃×#498	C₄	1	1	yes	characteristic
8	C₂₃×#494	C₂²	1	1	yes	characteristic
9	C₂₃×#487	—	2	1	no	—
10	C₂₃×#322	C₄	1	1	yes	characteristic
11	C₂×Q₁₆×C₂₃	—	4	1	no	—
12	C₂₃×C₄⋊Q₈	—	2	1	no	—
13	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
14	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
15	C₂₃×#106	—	4	2	no	—
16	C₂₃×#106	—	4	1	no	—
17	C₂₃×#103	—	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	#10625018	—	4	1	no	—
22	C₂×Q₈×C₂₃	—	2	1	no	—
23	C₂×Q₈×C₂₃	—	2	1	no	—
24	C₂×Q₈×C₂₃	—	4	1	no	—
25	Q₁₆×C₂₃	—	8	2	no	—
26	C₂×C₈×C₂₃	—	2	1	no	—
27	C₂×C₈×C₂₃	—	2	2	no	—
28	C₄⋊C₄×C₂₃	—	4	1	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	#67267	—	8	1	no	—
34	#67267	—	8	1	no	—
35	#7534	—	23	1	no	maximal, Sylow
36	Q₈×C₂₃	—	8	1	no	—
37	Q₈×C₂₃	—	8	1	no	—
38	Q₈×C₂₃	—	8	2	no	—
39	C₂×C₄×C₂₃	—	2	1	no	—
40	C₂×C₄×C₂₃	—	4	1	no	—
41	C₂×C₄×C₂₃	—	4	1	no	—
42	C₂×C₄×C₂₃	—	8	1	no	—
43	C₂×C₄×C₂₃	#99	1	1	yes	characteristic
44	C₈×C₂₃	—	2	2	no	—
45	C₈×C₂₃	—	4	2	no	—
46	C₂₃⋊C₈	—	16	1	no	—
47	C₂₃⋊C₈	—	16	1	no	—
48	#1716	D₂₃	1	1	yes	characteristic
49	#1411	—	23	1	no	—
50	#1411	—	23	1	no	—
51	C₂²×C₂₃	#353	1	1	yes	characteristic
52	C₄×C₂₃	—	2	2	no	—
53	C₄×C₂₃	—	2	1	no	—
54	C₄×C₂₃	—	8	1	no	—
55	C₄×C₂₃	—	8	1	no	—
56	C₄×C₂₃	—	16	1	no	—
57	#498	—	2	1	no	—
58	#498	Dic₂₃	1	1	yes	characteristic
59	#494	C₂×D₂₃	1	1	yes	characteristic
60	#487	—	2	1	no	—
61	#356	—	46	1	no	—
62	#356	—	46	1	no	—
63	#322	Dic₂₃	1	1	yes	characteristic
64	C₂×C₂₃	—	2	1	no	—
65	C₂×C₂₃	#1409	1	1	yes	characteristic
66	C₂×Q₁₆	—	4	1	no	—
67	C₄⋊Q₈	—	2	1	no	—
68	C₄⋊Q₈	C₂²⋊D₂₃	1	1	yes	characteristic
69	C₄⋊Q₈	C₂²⋊D₂₃	1	1	yes	characteristic
70	#106	—	4	2	no	—
71	#106	—	4	1	no	—
72	#103	—	4	2	no	—
73	#101	—	92	1	no	—
74	#101	—	92	1	no	—
75	C₄×C₈	—	2	1	no	—
76	C₄×C₈	C₂×Dic₂₃	1	1	yes	characteristic
77	C₂₃	#7534	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	4	1	no	—
81	Q₁₆	—	8	2	no	—
82	M₁₆	—	184	1	no	—
83	M₁₆	—	184	1	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	2	no	—
86	C₄⋊C₄	—	4	1	no	—
87	C₄⋊C₄	—	4	1	no	—
88	C₄⋊C₄	—	4	1	no	—
89	C₄⋊C₄	—	4	1	no	—
90	C₄²	C₂²⋊Dic₂₃	1	1	yes	characteristic
91	Q₈	—	8	1	no	—
92	Q₈	—	8	1	no	—
93	Q₈	—	8	2	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	—	8	1	no	—
98	C₂×C₄	#10625016	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	4	2	no	—
101	C₈	—	368	1	no	—
102	C₈	—	368	1	no	—
103	C₂²	#13092210	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	8	1	no	—
107	C₄	—	8	1	no	—
108	C₄	—	16	1	no	—
109	C₂	—	2	1	no	—
110	C₂	#15671562	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15671566

C₂

yes

maximal, characteristic

#15671566

C₂

yes

maximal, characteristic

C₂₃×#1716

C₂

yes

maximal, characteristic

#13092216

—

#13092216

—

C₂₃×#498

—

C₂₃×#498

C₄

yes

characteristic

C₂₃×#494

C₂²

yes

characteristic

C₂₃×#487

—

C₂₃×#322

C₄

yes

characteristic

C₂×Q₁₆×C₂₃

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

D₄

yes

characteristic

C₂₃×C₄⋊Q₈

D₄

yes

characteristic

C₂₃×#106

—

C₂₃×#106

—

C₂₃×#103

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

C₂×C₄

yes

characteristic

#10625018

—

#10625018

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

Q₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂²⋊C₄

yes

—

—

—

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#99

yes

characteristic

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1716

D₂₃

yes

characteristic

#1411

—

#1411

—

C₂²×C₂₃

#353

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

#498

—

#498

Dic₂₃

yes

characteristic

#494

C₂×D₂₃

yes

—

—

—

Dic₂₃

yes

characteristic

C₂×C₂₃

—

C₂×C₂₃

#1409

yes

characteristic

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²⋊D₂₃

yes

characteristic

C₄⋊Q₈

C₂²⋊D₂₃

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂×Dic₂₃

yes

characteristic

C₂₃

#7534

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

M₁₆

—

184

—

M₁₆

—

184

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊Dic₂₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#10625016

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

368

—

102

C₈

—

368

—

103

C₂²

#13092210

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₂

—

110

C₂

#15671562

yes

characteristic

	subgroup	G/H
center	C₂	#15671562
commutator	C₄×C₈×C₂₃	C₂×C₄
Frattini	#494	C₂×D₂₃
Fitting	C₂₃×#1716	C₂
socle	C₂×C₂₃	#1409
hypercenter	#1716	D₂₃
hypocenter	C₂₃	#7534
LCS	G ▷ C₄×C₈×C₂₃ ▷ C₄²×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #494 ◁ #1716

subgroup

G/H

center

C₂

#15671562

commutator

C₄×C₈×C₂₃

C₂×C₄

Frattini

#494

C₂×D₂₃

Fitting

C₂₃×#1716

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 ◁ #494 ◁ #1716

N	H ≅ G/N	type
77. C₂₃	#7534	semidirect

H ≅ G/N

type

77. C₂₃

#7534

semidirect

Subgroups — #19212075

Conjugacy classes of subgroups

Complements