Group #15673026 — subgroups

Subgroups — #15673026

Order 2944.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	#13092794	C₂	1	1	yes	maximal, characteristic
2	#13092791	C₂	1	2	yes	maximal, complemented
3	#13092765	C₂	1	2	yes	maximal, complemented
4	#13092138	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×#67274	—	2	4	no	—
8	C₂×D₁₈₄	—	2	2	no	—
9	#10625045	C₂²	1	1	yes	characteristic
10	#10625045	C₂²	1	1	yes	characteristic
11	C₂₃×#106	—	2	2	no	—
12	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	#10624991	—	2	4	no	—
14	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
15	C₂×Q₈×C₂₃	—	2	2	no	—
16	C₂×D₉₂	—	2	1	no	—
17	C₂×D₉₂	—	2	1	no	—
18	C₂×D₉₂	—	2	2	no	—
19	C₂×D₉₂	—	2	2	no	—
20	C₂×C₈×C₂₃	D₄	1	2	yes	—
21	C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₄²×C₂₃	C₂³	1	1	yes	characteristic
24	#67274	—	4	8	no	—
25	C₂×C₂₃⋊C₈	D₄	1	4	yes	—
26	D₁₈₄	—	4	4	no	—
27	C₂²×D₂₃	—	4	1	no	—
28	C₂²×D₂₃	—	4	1	no	—
29	Q₈×C₂₃	—	4	4	no	—
30	C₂×C₄×C₂₃	—	4	2	no	—
31	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
33	D₉₂	—	4	2	no	—
34	D₉₂	—	4	2	no	—
35	D₉₂	—	4	4	no	—
36	D₉₂	—	4	4	no	—
37	C₈×C₂₃	SD₁₆	1	4	yes	complemented
38	C₂₃⋊C₈	—	2	4	no	—
39	#1715	—	23	1	no	maximal, Sylow
40	C₂²×C₂₃	#127	1	1	yes	characteristic
41	C₂×D₂₃	—	8	1	no	—
42	C₂×D₂₃	—	8	1	no	—
43	C₂×D₂₃	—	8	2	no	—
44	C₂×D₂₃	—	8	2	no	—
45	C₄×C₂₃	—	8	2	no	—
46	C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
47	C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
48	#498	D₂₃	1	1	yes	characteristic, complemented
49	#493	—	23	1	no	—
50	#492	—	23	2	no	—
51	#486	—	23	2	no	—
52	#322	—	23	1	no	—
53	C₂×C₂₃	#492	1	1	yes	characteristic
54	C₂×C₂₃	#496	1	2	yes	—
55	D₂₃	—	16	1	no	—
56	D₂₃	—	16	1	no	—
57	C₂×SD₁₆	—	46	4	no	—
58	C₂×D₈	—	46	2	no	—
59	C₄⋊Q₈	C₂×D₂₃	1	2	yes	—
60	#127	—	23	1	no	—
61	#127	—	23	1	no	—
62	#106	—	2	2	no	—
63	#102	—	46	4	no	—
64	C₄×C₈	—	23	2	no	—
65	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
66	C₂₃	#1715	1	1	yes	characteristic, complemented, Sylow
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	46	2	no	—
69	C₂×D₄	—	46	2	no	—
70	C₂×D₄	—	46	1	no	—
71	C₂×D₄	—	46	1	no	—
72	SD₁₆	—	92	8	no	—
73	D₈	—	92	4	no	—
74	C₂×C₈	—	23	4	no	—
75	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂²×D₂₃	1	1	yes	characteristic
79	C₂³	—	92	1	no	—
80	C₂³	—	92	1	no	—
81	Q₈	—	4	4	no	—
82	D₄	—	92	4	no	—
83	D₄	—	92	4	no	—
84	D₄	—	92	2	no	—
85	D₄	—	92	2	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	D₄×D₂₃	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
89	C₈	—	46	4	no	—
90	C₈	#67274	1	4	yes	complemented
91	C₂²	—	184	1	no	—
92	C₂²	—	184	1	no	—
93	C₂²	—	184	2	no	—
94	C₂²	—	184	2	no	—
95	C₂²	#10625111	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	#10625085	1	4	yes	—
98	C₄	C₂×#67274	1	2	yes	—
99	C₂	—	368	1	no	—
100	C₂	—	368	1	no	—
101	C₂	#13092791	1	1	yes	characteristic
102	C₂	#13092803	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092794

C₂

yes

maximal, characteristic

#13092791

C₂

yes

maximal, complemented

#13092765

C₂

yes

maximal, complemented

#13092138

C₂

yes

maximal, characteristic, complemented

C₂₃×#498

C₂

yes

maximal, characteristic, complemented

C₂₃×C₄⋊Q₈

C₂²

yes

—

C₂×#67274

—

C₂×D₁₈₄

—

#10625045

C₂²

yes

characteristic

#10625045

C₂²

yes

characteristic

C₂₃×#106

—

C₄×C₈×C₂₃

C₂²

yes

characteristic

#10624991

—

C₄×C₂₃⋊C₈

C₂²

yes

—

C₂×Q₈×C₂₃

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×C₈×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

#67274

—

C₂×C₂₃⋊C₈

D₄

yes

—

D₁₈₄

—

C₂²×D₂₃

—

C₂²×D₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

D₉₂

—

D₉₂

—

D₉₂

—

D₉₂

—

C₈×C₂₃

SD₁₆

yes

complemented

C₂₃⋊C₈

—

—

C₂²×C₂₃

yes

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×SD₁₆

yes

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

#498

D₂₃

yes

characteristic, complemented

—

—

—

—

C₂×C₂₃

#492

yes

characteristic

C₂×C₂₃

#496

yes

—

D₂₃

—

D₂₃

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂×D₂₃

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#1715

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₂₃

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

D₄×D₂₃

yes

—

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

C₈

—

C₈

#67274

yes

complemented

C₂²

—

184

—

C₂²

—

184

—

C₂²

—

184

—

C₂²

—

184

—

C₂²

#10625111

yes

characteristic

C₄

—

C₄

#10625085

yes

—

C₄

C₂×#67274

yes

—

C₂

—

368

—

100

C₂

—

368

—

101

C₂

#13092791

yes

characteristic

102

C₂

#13092803

yes

—

	subgroup	G/H
center	C₂²	#10625111
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#498	C₂
socle	C₂²×C₂₃	#127
prime radical	#13092794	C₂
hypercenter	#498	D₂₃
hypocenter	C₂₃	#1715
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#10625111

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#498

C₂

C₂²×C₂₃

C₂

D₂₃

C₂₃

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #13092791	C₂	semidirect
3. #13092765	C₂	semidirect
4. #13092138	C₂	semidirect
5. C₂₃×#498	C₂	semidirect
37. C₈×C₂₃	SD₁₆	semidirect
48. #498	D₂₃	semidirect
66. C₂₃	#1715	semidirect
90. C₈	#67274	semidirect

H ≅ G/N

type

2. #13092791

C₂

semidirect

3. #13092765

C₂

semidirect

4. #13092138

C₂

semidirect

5. C₂₃×#498

C₂

semidirect

37. C₈×C₂₃

SD₁₆

semidirect

48. #498

D₂₃

semidirect

66. C₂₃

#1715

semidirect

90. C₈

#67274

semidirect

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

subgroup	index
#13092794	2
#13092791	2
#13092765	2
#13092138	2
C₂₃×#498	2
#1715	23

subgroup

index

C₂₃×#498

Subgroups — #15673026

Conjugacy classes of subgroups

Complements

Maximal subgroups