Group #15674560 — subgroups

Subgroups — #15674560

Order 2944.

466 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 71 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	#13092272	C₂	1	3	yes	maximal
2	#13092184	C₂	1	3	yes	maximal, complemented
3	C₂₃×#543	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₂₃	C₂²	1	1	yes	characteristic
5	C₂₃×C₂²⋊Q₈	—	2	6	no	—
6	C₂×C₂²⋊C₄×C₂₃	C₂²	1	3	yes	—
7	C₂×C₂²⋊Dic₂₃	C₂²	1	3	yes	—
8	C₂×(C₂×D₂₃)⋊C₄	—	2	6	no	—
9	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
10	C₂³×D₂₃	—	2	3	no	—
11	C₂³×D₂₃	—	2	3	no	—
12	C₂³×D₂₃	—	4	4	no	—
13	C₂³×D₂₃	Q₈	1	2	yes	complemented
14	C₂×Q₈×C₂₃	—	2	2	no	—
15	C₂²×C₄×C₂₃	D₄	1	6	yes	—
16	C₂²×Dic₂₃	—	2	3	no	—
17	C₄⋊C₄×C₂₃	—	2	6	no	—
18	C₂²⋊C₄×C₂₃	—	2	6	no	—
19	C₂²⋊Dic₂₃	—	2	6	no	—
20	(C₂×D₂₃)⋊C₄	—	4	12	no	—
21	C₂³×C₂₃	—	4	2	no	—
22	C₂³×C₂₃	C₂×D₄	1	3	yes	—
23	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
24	C₂³×C₂₃	#41	1	3	yes	—
25	C₂²×D₂₃	—	2	12	no	—
26	C₂²×D₂₃	—	2	6	no	—
27	C₂²×D₂₃	—	4	4	no	—
28	C₂²×D₂₃	—	4	12	no	—
29	C₂²×D₂₃	—	4	6	no	—
30	C₂²×D₂₃	—	4	1	no	—
31	C₂²×D₂₃	—	4	3	no	—
32	Q₈×C₂₃	—	8	2	no	—
33	C₂×C₄×C₂₃	—	2	6	no	—
34	C₂×C₄×C₂₃	—	4	6	no	—
35	C₂×Dic₂₃	—	4	6	no	—
36	C₂×Dic₂₃	—	4	3	no	—
37	#2029	—	23	1	no	maximal, Sylow
38	C₂²×C₂₃	—	2	6	no	—
39	C₂²×C₂₃	—	4	1	no	—
40	C₂²×C₂₃	—	4	3	no	—
41	C₂²×C₂₃	#120	1	1	yes	characteristic
42	C₂²×C₂₃	C₂²⋊Q₈	1	3	yes	—
43	C₂²×C₂₃	#124	1	3	yes	—
44	C₂×D₂₃	—	2	12	no	—
45	C₂×D₂₃	—	4	12	no	—
46	C₂×D₂₃	—	4	6	no	—
47	C₂×D₂₃	—	4	2	no	—
48	C₂×D₂₃	—	4	4	no	—
49	C₄×C₂₃	—	4	6	no	—
50	Dic₂₃	—	8	3	no	—
51	#543	D₂₃	1	1	yes	characteristic, complemented
52	#379	—	23	3	no	—
53	#342	—	23	3	no	—
54	C₂×C₂₃	—	2	3	no	—
55	C₂×C₂₃	—	2	3	no	—
56	C₂×C₂₃	#393	1	1	yes	characteristic
57	C₂×C₂₃	B(4,𝔽₂)	1	2	yes	—
58	D₂₃	—	4	4	no	—
59	C₂⁵	—	23	1	no	—
60	C₂²⋊Q₈	—	2	6	no	—
61	C₂×C₂²⋊C₄	—	23	3	no	—
62	C₂×C₂²⋊C₄	—	46	6	no	—
63	C₂×C₂²⋊C₄	C₂×D₂₃	1	3	yes	—
64	C₂₃	#2029	1	1	yes	characteristic, complemented, Sylow
65	C₂⁴	—	23	2	no	—
66	C₂⁴	—	46	3	no	—
67	C₂⁴	—	46	3	no	—
68	C₂⁴	—	92	4	no	—
69	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
70	C₂×Q₈	—	2	2	no	—
71	C₂²×C₄	—	46	3	no	—
72	C₂²×C₄	C₂²⋊D₂₃	1	6	yes	—
73	C₄⋊C₄	—	2	6	no	—
74	C₂²⋊C₄	—	2	6	no	—
75	C₂²⋊C₄	—	46	6	no	—
76	C₂²⋊C₄	—	92	12	no	—
77	C₂³	—	4	2	no	—
78	C₂³	—	46	12	no	—
79	C₂³	—	46	6	no	—
80	C₂³	—	92	6	no	—
81	C₂³	—	92	4	no	—
82	C₂³	—	92	12	no	—
83	C₂³	—	92	3	no	—
84	C₂³	—	92	1	no	—
85	C₂³	Q₈×D₂₃	1	1	yes	characteristic
86	C₂³	#67292	1	3	yes	—
87	C₂³	C₂×C₂²⋊D₂₃	1	3	yes	—
88	Q₈	—	8	2	no	—
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	4	6	no	—
91	C₂×C₄	—	92	6	no	—
92	C₂×C₄	—	92	3	no	—
93	C₂²	—	2	6	no	—
94	C₂²	—	4	1	no	—
95	C₂²	—	4	3	no	—
96	C₂²	—	46	12	no	—
97	C₂²	—	92	12	no	—
98	C₂²	—	92	6	no	—
99	C₂²	—	92	2	no	—
100	C₂²	—	92	4	no	—
101	C₂²	#10625117	1	3	yes	—
102	C₂²	#10625118	1	3	yes	—
103	C₂²	#10625124	1	1	yes	characteristic
104	C₄	—	4	6	no	—
105	C₄	—	184	3	no	—
106	C₂	—	2	3	no	—
107	C₂	—	2	3	no	—
108	C₂	—	92	4	no	—
109	C₂	#13092328	1	1	yes	characteristic
110	C₂	#13092604	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092272

C₂

yes

maximal

#13092184

C₂

yes

maximal, complemented

C₂₃×#543

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₂₃

C₂²

yes

characteristic

C₂₃×C₂²⋊Q₈

—

C₂×C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂×C₂²⋊Dic₂₃

C₂²

yes

—

C₂×(C₂×D₂₃)⋊C₄

—

C₂⁴×C₂₃

C₂³

yes

characteristic

C₂³×D₂₃

—

C₂³×D₂₃

—

C₂³×D₂₃

—

C₂³×D₂₃

Q₈

yes

complemented

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×Dic₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊Dic₂₃

—

(C₂×D₂₃)⋊C₄

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×D₄

yes

—

C₂³×C₂₃

C₂×Q₈

yes

characteristic

C₂³×C₂₃

#41

yes

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

#2029

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#120

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

—

C₂²×C₂₃

#124

yes

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

Dic₂₃

—

#543

D₂₃

yes

characteristic, complemented

#379

—

#342

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#393

yes

characteristic

C₂×C₂₃

B(4,𝔽₂)

yes

—

D₂₃

—

C₂⁵

—

C₂²⋊Q₈

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₂₃

yes

—

C₂₃

#2029

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

C₂²×D₂₃

yes

characteristic

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

Q₈×D₂₃

yes

characteristic

C₂³

#67292

yes

—

C₂³

C₂×C₂²⋊D₂₃

yes

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

#10625117

yes

—

102

C₂²

#10625118

yes

—

103

C₂²

#10625124

yes

characteristic

104

C₄

—

105

C₄

—

184

—

106

C₂

—

107

C₂

—

108

C₂

—

109

C₂

#13092328

yes

characteristic

110

C₂

#13092604

yes

—

	subgroup	G/H
center	C₂²	#10625124
commutator, cosocle	C₂⁴×C₂₃	C₂³
Frattini	C₂⁴	C₂²×D₂₃
Fitting	C₂₃×#543	C₂
socle	C₂²×C₂₃	#120
prime radical	C₂⁴×D₂₃	C₂²
hypercenter	#543	D₂₃
hypocenter	C₂₃	#2029
LCS	G ▷ C₂⁴×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

subgroup

G/H

center

C₂²

#10625124

commutator, cosocle

C₂⁴×C₂₃

C₂³

Frattini

C₂⁴

C₂²×D₂₃

Fitting

C₂₃×#543

C₂

socle

C₂²×C₂₃

#120

prime radical

C₂⁴×D₂₃

C₂²

D₂₃

C₂₃

G ▷ C₂⁴×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

N	H ≅ G/N	type
2. #13092184	C₂	semidirect
3. C₂₃×#543	C₂	semidirect
13. C₂³×D₂₃	Q₈	semidirect
51. #543	D₂₃	semidirect
64. C₂₃	#2029	semidirect

H ≅ G/N

type

2. #13092184

C₂

semidirect

3. C₂₃×#543

C₂

semidirect

13. C₂³×D₂₃

Q₈

semidirect

51. #543

D₂₃

semidirect

64. C₂₃

#2029

semidirect

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

subgroup	index
#13092272	2
#13092184	2
C₂₃×#543	2
#2029	23

subgroup

index

#13092272

#13092184

C₂₃×#543

#2029

Subgroups — #15674560

Conjugacy classes of subgroups

Complements

Maximal subgroups