Group #15681992 — subgroups

Subgroups — #15681992

Order 2944.

476 conjugacy classes of subgroups; only the 293 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 293 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	#13092900	C₂	1	2	yes	maximal
2	#13092459	C₂	1	6	yes	maximal, complemented
3	#13092455	C₂	1	3	yes	maximal, complemented
4	C₂×#10624999	C₂	1	3	yes	maximal, complemented
5	C₂×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#131	C₄	1	8	yes	complemented
8	C₂×M₁₆×C₂₃	C₂²	1	3	yes	—
9	C₂²×C₈×C₂₃	C₂²	1	3	yes	—
10	Q₈×Dic₂₃	—	2	2	no	—
11	D₄×Dic₂₃	—	2	6	no	—
12	#10625097	C₂²	1	6	yes	—
13	C₂×C₄⋊Dic₂₃	C₂²	1	6	yes	—
14	#10625000	C₂²	1	6	yes	—
15	#10624999	C₂²	1	3	yes	—
16	#10624999	C₂²	1	6	yes	—
17	#10624999	C₂²	1	1	yes	characteristic
18	#41×C₂₃	C₂×C₄	1	4	yes	—
19	#41×C₂₃	Q₈	1	4	yes	—
20	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
21	C₂×D₄×C₂₃	D₄	1	3	yes	—
22	C₂²×C₄×C₂₃	C₂³	1	3	yes	—
23	C₂²×Dic₂₃	—	2	6	no	—
24	M₁₆×C₂₃	C₂×C₄	1	12	yes	complemented
25	C₂×C₈×C₂₃	C₂×C₄	1	12	yes	complemented
26	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₂₃	C₂³	1	3	yes	—
28	C₂²⋊Dic₂₃	—	2	6	no	—
29	C₄⋊Dic₂₃	—	2	6	no	—
30	C₄⋊Dic₂₃	C₂³	1	6	yes	—
31	C₄⋊Dic₂₃	C₂³	1	2	yes	—
32	C₄×Dic₂₃	—	2	6	no	—
33	C₂³×C₂₃	C₂×D₄	1	3	yes	—
34	Q₈×C₂₃	C₄⋊C₄	1	4	yes	—
35	D₄×C₂₃	C₄⋊C₄	1	12	yes	—
36	C₂×C₄×C₂₃	C₂²×C₄	1	6	yes	—
37	C₂×C₄×C₂₃	C₂×D₄	1	3	yes	—
38	C₂×C₄×C₂₃	C₂×Q₈	1	6	yes	—
39	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
40	C₂×Dic₂₃	—	2	2	no	—
41	C₂×Dic₂₃	—	2	6	no	—
42	C₂×Dic₂₃	—	8	6	no	—
43	C₈×C₂₃	C₂²×C₄	1	2	yes	—
44	C₈×C₂₃	C₂²×C₄	1	6	yes	—
45	#2912	—	23	1	no	maximal, Sylow
46	C₂²×C₂₃	—	2	6	no	—
47	C₂²×C₂₃	C₂×C₄⋊C₄	1	6	yes	—
48	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₂₃	C₂×C₄⋊C₄	1	6	yes	—
50	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
51	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
52	Dic₂₃	—	4	2	no	—
53	Dic₂₃	—	4	6	no	—
54	C₂×#131	D₂₃	1	1	yes	characteristic
55	#520	—	23	2	no	—
56	#428	—	23	6	no	—
57	#427	—	23	3	no	—
58	C₂×#106	—	23	3	no	—
59	C₂×C₂₃	—	2	6	no	—
60	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₂₃	#577	1	2	yes	—
62	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
63	#131	Dic₂₃	1	8	yes	complemented
64	C₂×M₁₆	C₂×D₂₃	1	3	yes	—
65	C₂²×C₈	C₂×D₂₃	1	3	yes	—
66	C₄×Q₈	—	46	2	no	—
67	C₄×D₄	—	46	6	no	—
68	#117	—	23	6	no	—
69	C₂×C₄⋊C₄	—	23	6	no	—
70	#107	—	23	6	no	—
71	#106	—	23	3	no	—
72	#106	—	23	6	no	—
73	#106	—	23	1	no	—
74	C₂₃	#2912	1	1	yes	characteristic, complemented, Sylow
75	#41	Dic₄₆	1	4	yes	—
76	#41	C₂×Dic₂₃	1	4	yes	—
77	C₂×Q₈	D₉₂	1	1	yes	characteristic
78	C₂×D₄	D₉₂	1	3	yes	—
79	C₂²×C₄	—	46	6	no	—
80	C₂²×C₄	C₂²×D₂₃	1	3	yes	—
81	M₁₆	C₂×Dic₂₃	1	12	yes	complemented
82	C₂×C₈	C₂×Dic₂₃	1	12	yes	complemented
83	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
84	C₂×C₈	C₂²×D₂₃	1	3	yes	—
85	C₄⋊C₄	—	23	6	no	—
86	C₄⋊C₄	—	23	2	no	—
87	C₄⋊C₄	—	46	6	no	—
88	C₂²⋊C₄	—	46	6	no	—
89	C₄²	—	46	6	no	—
90	C₂³	C₂×D₉₂	1	3	yes	—
91	Q₈	C₄⋊Dic₂₃	1	4	yes	—
92	D₄	C₄⋊Dic₂₃	1	12	yes	—
93	C₂×C₄	—	46	2	no	—
94	C₂×C₄	—	46	6	no	—
95	C₂×C₄	—	184	6	no	—
96	C₂×C₄	C₂×Dic₄₆	1	6	yes	—
97	C₂×C₄	C₂×D₉₂	1	3	yes	—
98	C₂×C₄	C₂²×Dic₂₃	1	6	yes	—
99	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
100	C₈	C₂²×Dic₂₃	1	2	yes	—
101	C₈	C₂²×Dic₂₃	1	6	yes	—
102	C₂²	—	2	6	no	—
103	C₂²	C₂×C₄⋊Dic₂₃	1	6	yes	—
104	C₂²	C₂²×D₉₂	1	1	yes	characteristic
105	C₄	—	92	6	no	—
106	C₄	—	92	2	no	—
107	C₄	C₂×C₄⋊Dic₂₃	1	6	yes	—
108	C₄	C₂²×Dic₄₆	1	1	yes	characteristic
109	C₄	C₂³×Dic₂₃	1	1	yes	characteristic
110	C₂	—	2	6	no	—
111	C₂	C₂²×C₄⋊Dic₂₃	1	1	yes	characteristic
112	C₂	#13093229	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092900

C₂

yes

maximal

#13092459

C₂

yes

maximal, complemented

#13092455

C₂

yes

maximal, complemented

C₂×#10624999

C₂

yes

maximal, complemented

C₂×C₂₃×#131

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

C₂²

yes

characteristic

C₂₃×#131

C₄

yes

complemented

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂²×C₈×C₂₃

C₂²

yes

—

Q₈×Dic₂₃

—

D₄×Dic₂₃

—

#10625097

C₂²

yes

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

#10625000

C₂²

yes

—

#10624999

C₂²

yes

—

#10624999

C₂²

yes

—

#10624999

C₂²

yes

characteristic

#41×C₂₃

C₂×C₄

yes

—

#41×C₂₃

Q₈

yes

—

C₂×Q₈×C₂₃

D₄

yes

characteristic

C₂×D₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×Dic₂₃

—

M₁₆×C₂₃

C₂×C₄

yes

complemented

C₂×C₈×C₂₃

C₂×C₄

yes

complemented

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₃

C₂³

yes

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂³

yes

—

C₄⋊Dic₂₃

C₂³

yes

—

C₄×Dic₂₃

—

C₂³×C₂₃

C₂×D₄

yes

—

Q₈×C₂₃

C₄⋊C₄

yes

—

D₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

C₂²×C₄

yes

—

C₈×C₂₃

C₂²×C₄

yes

—

#2912

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

C₂×C₄⋊C₄

yes

—

C₄×C₂₃

C₂³×C₄

yes

characteristic

C₄×C₂₃

C₂²×Q₈

yes

characteristic

Dic₂₃

—

Dic₂₃

—

C₂×#131

D₂₃

yes

—

—

—

—

C₂×C₂₃

—

C₂×C₂₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₂₃

#577

yes

—

C₂×#41

C₂×D₂₃

yes

characteristic

#131

Dic₂₃

yes

complemented

C₂×M₁₆

C₂×D₂₃

yes

—

C₂²×C₈

C₂×D₂₃

yes

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×C₄⋊C₄

—

—

—

—

—

C₂₃

#2912

yes

characteristic, complemented, Sylow

#41

Dic₄₆

yes

—

#41

C₂×Dic₂₃

yes

—

C₂×Q₈

D₉₂

yes

characteristic

C₂×D₄

D₉₂

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

—

M₁₆

C₂×Dic₂₃

yes

complemented

C₂×C₈

C₂×Dic₂₃

yes

complemented

C₂×C₈

C₂²×D₂₃

yes

characteristic

C₂×C₈

C₂²×D₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

C₂×D₉₂

yes

—

Q₈

C₄⋊Dic₂₃

yes

—

D₄

C₄⋊Dic₂₃

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

184

—

C₂×C₄

C₂×Dic₄₆

yes

—

C₂×C₄

C₂×D₉₂

yes

—

C₂×C₄

C₂²×Dic₂₃

yes

—

C₂×C₄

C₂³×D₂₃

yes

characteristic

100

C₈

C₂²×Dic₂₃

yes

—

101

C₈

C₂²×Dic₂₃

yes

—

102

C₂²

—

103

C₂²

C₂×C₄⋊Dic₂₃

yes

—

104

C₂²

C₂²×D₉₂

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

C₂×C₄⋊Dic₂₃

yes

—

108

C₄

C₂²×Dic₄₆

yes

characteristic

109

C₄

C₂³×Dic₂₃

yes

characteristic

110

C₂

—

111

C₂

C₂²×C₄⋊Dic₂₃

yes

characteristic

112

C₂

#13093229

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₉₂
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₂₃×#131	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#41×C₂₃	C₂²
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂×#131	D₂₃
hypocenter	C₂₃	#2912
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

subgroup

G/H

center

C₂²

C₂²×D₉₂

commutator

C₄×C₂₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂×C₂₃×#131

C₂

socle

C₂²×C₂₃

C₂²×D₄

prime radical

C₂×#41×C₂₃

C₂²

cosocle

C₂×C₄×C₂₃

C₂⁴

D₂₃

C₂₃

G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

N	H ≅ G/N	type
2. #13092459	C₂	semidirect
3. #13092455	C₂	semidirect
4. C₂×#10624999	C₂	semidirect
7. C₂₃×#131	C₄	semidirect
24. M₁₆×C₂₃	C₂×C₄	semidirect
25. C₂×C₈×C₂₃	C₂×C₄	semidirect
63. #131	Dic₂₃	semidirect
74. C₂₃	#2912	semidirect
81. M₁₆	C₂×Dic₂₃	semidirect
82. C₂×C₈	C₂×Dic₂₃	semidirect

H ≅ G/N

type

2. #13092459

C₂

semidirect

3. #13092455

C₂

semidirect

4. C₂×#10624999

C₂

semidirect

7. C₂₃×#131

C₄

semidirect

24. M₁₆×C₂₃

C₂×C₄

semidirect

25. C₂×C₈×C₂₃

C₂×C₄

semidirect

63. #131

Dic₂₃

semidirect

74. C₂₃

#2912

semidirect

81. M₁₆

C₂×Dic₂₃

semidirect

82. C₂×C₈

C₂×Dic₂₃

semidirect

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

subgroup	index
#13092900	2
#13092459	2
#13092455	2
C₂×#10624999	2
C₂×C₂₃×#131	2
#2912	23

subgroup

index

C₂×C₂₃×#131

Subgroups — #15681992

Conjugacy classes of subgroups

Complements

Maximal subgroups