Group #15674973 — subgroups

Subgroups — #15674973

Order 2944.

158 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 59 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	#13092466	C₂	1	1	yes	maximal, characteristic
2	#13092440	C₂	1	1	yes	maximal, characteristic
3	#13092187	C₂	1	2	yes	maximal
4	#13092176	C₂	1	2	yes	maximal
5	C₂₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#117	C₂²	1	2	yes	—
8	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
9	C₂×#67267	C₂²	1	2	yes	—
10	C₂×#67267	C₂²	1	1	yes	characteristic
11	#10625001	—	2	2	no	—
12	Dic₂₃⋊C₈	—	2	2	no	—
13	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
14	C₂²×Dic₂₃	—	2	1	no	—
15	M₁₆×C₂₃	—	2	2	no	—
16	C₂×C₈×C₂₃	D₄	1	2	yes	—
17	C₄⋊C₄×C₂₃	—	2	2	no	—
18	C₂²⋊C₄×C₂₃	—	2	2	no	—
19	C₄²×C₂₃	—	2	2	no	—
20	C₄×Dic₂₃	—	2	1	no	—
21	C₄×Dic₂₃	Q₈	1	1	yes	characteristic
22	C₄×Dic₂₃	Q₈	1	1	yes	characteristic
23	#67267	—	2	2	no	—
24	#67267	—	2	2	no	—
25	#67267	—	2	2	no	—
26	C₂×C₂₃⋊C₈	—	2	1	no	—
27	C₂×C₂₃⋊C₈	—	2	2	no	—
28	C₂³×C₂₃	#41	1	1	yes	characteristic
29	C₂×C₄×C₂₃	—	4	2	no	—
30	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₂₃	#41	1	2	yes	—
33	C₂×C₄×C₂₃	#41	1	2	yes	—
34	C₂×Dic₂₃	—	2	1	no	—
35	C₂×Dic₂₃	—	2	1	no	—
36	C₂×Dic₂₃	—	4	2	no	—
37	C₈×C₂₃	—	2	2	no	—
38	C₂₃⋊C₈	—	4	1	no	—
39	C₂₃⋊C₈	—	4	2	no	—
40	#2102	—	23	1	no	maximal, Sylow
41	C₂²×C₂₃	—	4	1	no	—
42	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
43	C₂²×C₂₃	#126	1	2	yes	—
44	C₄×C₂₃	—	8	2	no	—
45	C₄×C₂₃	#123	1	1	yes	characteristic
46	C₄×C₂₃	#124	1	1	yes	characteristic
47	C₄×C₂₃	#125	1	2	yes	—
48	Dic₂₃	—	2	2	no	—
49	Dic₂₃	—	4	1	no	—
50	#430	—	23	1	no	—
51	#423	—	23	1	no	—
52	#344	—	23	2	no	—
53	#339	—	23	2	no	—
54	H(3,ℤ/4)	D₂₃	1	1	yes	characteristic
55	C₂×C₂₃	—	2	1	no	—
56	C₂×C₂₃	—	2	2	no	—
57	C₂×C₂₃	#400	1	1	yes	characteristic
58	C₂×M₁₆	—	23	2	no	—
59	C₂×M₁₆	—	23	1	no	—
60	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
61	#117	C₂×D₂₃	1	2	yes	—
62	C₂×C₄²	—	23	1	no	—
63	#108	—	46	2	no	—
64	C₄⋊C₈	—	46	2	no	—
65	C₂₃	#2102	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₄	—	46	1	no	—
67	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	46	2	no	—
70	M₁₆	—	46	2	no	—
71	M₁₆	—	46	2	no	—
72	C₂×C₈	—	46	1	no	—
73	C₂×C₈	—	46	2	no	—
74	C₂×C₈	D₉₂	1	2	yes	—
75	C₄⋊C₄	—	2	2	no	—
76	C₂²⋊C₄	—	2	2	no	—
77	C₄²	—	2	2	no	—
78	C₄²	—	23	1	no	—
79	C₄²	—	23	1	no	—
80	C₄²	—	46	1	no	—
81	C₂³	#67290	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	46	1	no	—
84	C₂×C₄	—	46	1	no	—
85	C₂×C₄	—	92	2	no	—
86	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
87	C₂×C₄	#67288	1	2	yes	—
88	C₂×C₄	#67290	1	2	yes	—
89	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	92	2	no	—
92	C₈	—	92	1	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#10625050	1	2	yes	—
95	C₂²	#10625068	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	—	46	2	no	—
98	C₄	—	92	1	no	—
99	C₄	#10625046	1	1	yes	characteristic
100	C₄	#10625057	1	1	yes	characteristic
101	C₄	#10625060	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	—	2	1	no	—
104	C₂	#13092357	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092466

C₂

yes

maximal, characteristic

#13092440

C₂

yes

maximal, characteristic

#13092187

C₂

yes

maximal

#13092176

C₂

yes

maximal

C₂₃×H(3,ℤ/4)

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂×#67267

C₂²

yes

characteristic

#10625001

—

Dic₂₃⋊C₈

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

Q₈

yes

characteristic

C₄×Dic₂₃

Q₈

yes

—

—

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

#41

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#2102

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₃

#126

yes

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

Dic₂₃

—

Dic₂₃

—

—

—

—

—

D₂₃

yes

characteristic

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#400

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

characteristic

#117

C₂×D₂₃

yes

—

C₂×C₄²

—

#108

—

C₄⋊C₈

—

C₂₃

#2102

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₉₂

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

#67290

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₉₂

yes

characteristic

C₂×C₄

#67288

yes

—

C₂×C₄

#67290

yes

—

C₂×C₄

Q₈×D₂₃

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

#10625050

yes

—

C₂²

#10625068

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#10625046

yes

characteristic

100

C₄

#10625057

yes

characteristic

101

C₄

#10625060

yes

—

102

C₂

—

103

C₂

—

104

C₂

#13092357

yes

characteristic

	subgroup	G/H
center	C₄	#10625046
commutator, cosocle	C₂²×C₄×C₂₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×H(3,ℤ/4)	C₂
socle	C₂×C₂₃	#400
prime radical	C₂³×C₂₃	#41
hypercenter	H(3,ℤ/4)	D₂₃
hypocenter	C₂₃	#2102
LCS	G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#10625046

commutator, cosocle

C₂²×C₄×C₂₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×H(3,ℤ/4)

C₂

C₂×C₂₃

C₂³×C₂₃

D₂₃

C₂₃

G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
65. C₂₃	#2102	semidirect

H ≅ G/N

type

65. C₂₃

#2102

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13092466	2
#13092440	2
#13092187	2
#13092176	2
C₂₃×H(3,ℤ/4)	2
#2102	23

subgroup

index

C₂₃×H(3,ℤ/4)

Subgroups — #15674973

Conjugacy classes of subgroups

Complements

Maximal subgroups