Group #15676702 — subgroups

Subgroups — #15676702

Order 2944.

400 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 157 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	#13092359	C₂	1	4	yes	maximal
2	#13092351	C₂	1	4	yes	maximal
3	C₄×Dic₂₃⋊C₄	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₂₃	C₂	1	2	yes	maximal
5	C₂₃×#376	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₂₃	C₂²	1	4	yes	—
9	C₂×Dic₂₃⋊C₄	C₂²	1	4	yes	—
10	C₂×Dic₂₃⋊C₄	C₂²	1	4	yes	—
11	C₂×C₄×Dic₂₃	C₂²	1	4	yes	—
12	C₂₃×#95	C₂²	1	4	yes	—
13	C₂₃⋊#95	C₂²	1	2	yes	—
14	C₂₃⋊#95	C₂²	1	4	yes	—
15	C₂₃⋊#95	C₂²	1	4	yes	—
16	C₂₃⋊#95	C₂²	1	2	yes	—
17	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
18	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
19	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₂₃	C₂³	1	4	yes	—
21	C₂²×Dic₂₃	C₂³	1	4	yes	—
22	C₄²×C₂₃	—	2	4	no	—
23	C₄²×C₂₃	Q₈	1	4	yes	—
24	C₄⋊Dic₂₃	—	2	8	no	—
25	Dic₂₃⋊C₄	—	2	8	no	—
26	Dic₂₃⋊C₄	—	2	8	no	—
27	C₄×Dic₂₃	—	2	8	no	—
28	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
29	C₂×C₄×C₂₃	—	2	4	no	—
30	C₂×C₄×C₂₃	—	2	4	no	—
31	C₂×C₄×C₂₃	—	2	4	no	—
32	C₂×C₄×C₂₃	—	2	4	no	—
33	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
34	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₂₃	#41	1	4	yes	—
36	C₂×Dic₂₃	—	2	4	no	—
37	C₂×Dic₂₃	—	2	4	no	—
38	C₂×Dic₂₃	—	2	4	no	—
39	C₂×Dic₂₃	—	2	4	no	—
40	C₂×Dic₂₃	—	2	4	no	—
41	C₂×Dic₂₃	#41	1	8	yes	—
42	#2402	—	23	1	no	maximal, Sylow
43	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
44	C₂²×C₂₃	C₂×#41	1	2	yes	—
45	C₂²×C₂₃	C₂×#41	1	4	yes	—
46	C₄×C₂₃	—	2	4	no	—
47	C₄×C₂₃	—	4	4	no	—
48	C₄×C₂₃	#125	1	4	yes	—
49	Dic₂₃	—	2	8	no	—
50	Dic₂₃	—	4	4	no	—
51	#400	—	23	4	no	—
52	#398	—	23	4	no	—
53	C₄×C₄⋊C₄	—	23	2	no	—
54	C₄×C₄⋊C₄	—	23	4	no	—
55	#376	D₂₃	1	1	yes	characteristic
56	C₂×C₂₃	C₂×#125	1	1	yes	characteristic
57	C₂×C₂₃	#529	1	4	yes	—
58	C₂×C₂₃	#533	1	2	yes	—
59	C₂×C₄⋊C₄	—	23	4	no	—
60	C₂×C₄⋊C₄	—	23	4	no	—
61	C₂×C₄⋊C₄	—	23	4	no	—
62	C₂×C₄²	—	23	4	no	—
63	C₂×C₄²	C₂×D₂₃	1	2	yes	—
64	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
65	#95	—	23	2	no	—
66	#95	—	23	4	no	—
67	#95	—	23	4	no	—
68	#95	—	23	2	no	—
69	#95	C₂×D₂₃	1	4	yes	—
70	C₂₃	#2402	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₄	—	23	4	no	—
72	C₂²×C₄	—	23	4	no	—
73	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
74	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
75	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
76	C₄⋊C₄	—	46	8	no	—
77	C₄⋊C₄	—	46	8	no	—
78	C₄⋊C₄	—	46	8	no	—
79	C₄²	—	2	4	no	—
80	C₄²	—	46	8	no	—
81	C₄²	Dic₄₆	1	4	yes	—
82	C₂³	C₂³×D₂₃	1	1	yes	characteristic
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	23	8	no	—
88	C₂×C₄	—	46	4	no	—
89	C₂×C₄	—	46	4	no	—
90	C₂×C₄	—	46	4	no	—
91	C₂×C₄	—	46	4	no	—
92	C₂×C₄	—	46	4	no	—
93	C₂×C₄	C₂×Dic₄₆	1	4	yes	—
94	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
95	C₂×C₄	#67288	1	4	yes	—
96	C₂²	C₂²×Dic₄₆	1	1	yes	characteristic
97	C₂²	C₂×#67288	1	2	yes	—
98	C₂²	#41×D₂₃	1	4	yes	—
99	C₄	—	2	4	no	—
100	C₄	—	4	4	no	—
101	C₄	—	46	8	no	—
102	C₄	—	92	4	no	—
103	C₄	#10625041	1	4	yes	—
104	C₂	C₂×#10625041	1	1	yes	characteristic
105	C₂	#13092945	1	4	yes	—
106	C₂	#13092970	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092359

C₂

yes

maximal

#13092351

C₂

yes

maximal

C₄×Dic₂₃⋊C₄

C₂

yes

maximal

C₄×C₄⋊Dic₂₃

C₂

yes

maximal

C₂₃×#376

C₂

yes

maximal, characteristic

C₂×C₄²×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×Dic₂₃⋊C₄

C₂²

yes

—

C₂×Dic₂₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

—

C₂₃×#95

C₂²

yes

—

C₂₃⋊#95

C₂²

yes

—

C₂₃⋊#95

C₂²

yes

—

C₂₃⋊#95

C₂²

yes

—

C₂₃⋊#95

C₂²

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂³

yes

—

C₂²×Dic₂₃

C₂³

yes

—

C₄²×C₂₃

—

C₄²×C₂₃

Q₈

yes

—

C₄⋊Dic₂₃

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₂³×C₂₃

C₂⁴

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

#41

yes

—

#2402

—

maximal, Sylow

C₂²×C₂₃

C₂²×Q₈

yes

characteristic

C₂²×C₂₃

C₂×#41

yes

—

C₂²×C₂₃

C₂×#41

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#125

yes

—

Dic₂₃

—

Dic₂₃

—

#400

—

#398

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

#376

D₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#533

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

—

—

—

—

C₂×D₂₃

yes

—

C₂₃

#2402

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

Dic₄₆

yes

—

C₂³

C₂³×D₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₄₆

yes

—

C₂×C₄

C₂×Dic₄₆

yes

—

C₂×C₄

#67288

yes

—

C₂²

C₂²×Dic₄₆

yes

characteristic

C₂²

C₂×#67288

yes

—

C₂²

#41×D₂₃

yes

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

#10625041

yes

—

104

C₂

C₂×#10625041

yes

characteristic

105

C₂

#13092945

yes

—

106

C₂

#13092970

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₃
commutator, socle, prime radical, cosocle	C₂³×C₂₃	C₂⁴
Frattini	C₂³	C₂³×D₂₃
Fitting	C₂₃×#376	C₂
hypercenter	#376	D₂₃
hypocenter	C₂₃	#2402
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ #376

subgroup

G/H

center

C₂²×C₄

C₂²×D₂₃

commutator, socle, prime radical, cosocle

C₂³×C₂₃

C₂⁴

Frattini

C₂³

C₂³×D₂₃

Fitting

C₂₃×#376

C₂

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ #376

N	H ≅ G/N	type
70. C₂₃	#2402	semidirect

H ≅ G/N

type

70. C₂₃

#2402

semidirect

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

subgroup	index
#13092359	2
#13092351	2
C₄×Dic₂₃⋊C₄	2
C₄×C₄⋊Dic₂₃	2
C₂₃×#376	2
#2402	23

subgroup

index

#13092359

#13092351

C₄×Dic₂₃⋊C₄

C₄×C₄⋊Dic₂₃

C₂₃×#376

#2402

Subgroups — #15676702

Conjugacy classes of subgroups

Complements

Maximal subgroups