Group #15675116 — subgroups

Subgroups — #15675116

Order 2944.

220 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×Dic₉₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#10624994	C₂	1	1	yes	maximal, characteristic
3	#13092220	C₂	1	4	yes	maximal, complemented
4	C₂₃×#427	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₆	—	2	1	no	—
6	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#117	—	2	1	no	—
8	C₂×Dic₉₂	—	2	2	no	—
9	C₂×Dic₉₂	—	2	1	no	—
10	C₂×Dic₉₂	C₄	1	4	yes	complemented
11	C₂×Dic₉₂	C₂²	1	2	yes	—
12	C₂₃×#107	C₂²	1	2	yes	—
13	C₂₃×#106	—	2	1	no	—
14	#10624994	—	2	2	no	—
15	C₂×C₂₃⋊C₁₆	C₂²	1	2	yes	—
16	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
17	C₂²×Dic₂₃	—	4	1	no	—
18	C₂×Dic₄₆	—	2	4	no	—
19	C₂×Dic₄₆	—	2	2	no	—
20	C₂×Dic₄₆	—	4	1	no	—
21	C₂×Dic₄₆	—	4	2	no	—
22	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₂₃	D₄	1	2	yes	—
24	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
25	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₂²⋊C₄×C₂₃	—	4	1	no	—
28	C₄²×C₂₃	—	4	1	no	—
29	Dic₉₂	—	2	4	no	—
30	Dic₉₂	—	2	2	no	—
31	Dic₉₂	C₂×C₄	1	4	yes	—
32	C₂₃⋊C₁₆	—	2	2	no	—
33	C₂³×C₂₃	D₈	1	1	yes	characteristic
34	C₂×C₄×C₂₃	—	4	2	no	—
35	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
37	C₂×C₄×C₂₃	SD₁₆	1	2	yes	—
38	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₂₃	—	4	4	no	—
40	C₂×Dic₂₃	—	4	2	no	—
41	Dic₄₆	—	2	4	no	—
42	Dic₄₆	—	4	4	no	—
43	Dic₄₆	—	4	2	no	—
44	C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
45	C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
47	#2141	—	23	1	no	maximal, Sylow
48	C₂²×C₂₃	—	2	2	no	—
49	C₂²×C₂₃	#102	1	2	yes	—
50	C₂²×C₂₃	C₂×D₈	1	1	yes	characteristic
51	C₄×C₂₃	—	8	2	no	—
52	C₄×C₂₃	#102	1	2	yes	—
53	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
55	Dic₂₃	—	4	4	no	—
56	C₂²×Q₁₆	—	23	1	no	—
57	C₂×M₃₂	—	23	1	no	—
58	#427	D₂₃	1	1	yes	characteristic
59	#358	—	23	4	no	—
60	C₂×C₂₃	—	2	2	no	—
61	C₂×C₂₃	C₂×#102	1	1	yes	characteristic
62	C₂×C₂₃	#510	1	2	yes	—
63	C₂²×Q₈	—	46	1	no	—
64	C₂×Q₁₆	—	23	4	no	—
65	C₂×Q₁₆	—	23	2	no	—
66	C₂×Q₁₆	—	46	2	no	—
67	C₂×Q₁₆	—	46	1	no	—
68	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
69	#117	—	2	1	no	—
70	M₃₂	—	46	2	no	—
71	C₂×C₁₆	—	23	2	no	—
72	#107	C₂×D₂₃	1	2	yes	—
73	#106	—	2	1	no	—
74	C₂₃	#2141	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₈	—	46	4	no	—
76	C₂×Q₈	—	46	2	no	—
77	C₂×Q₈	—	92	1	no	—
78	C₂×Q₈	—	92	2	no	—
79	C₂²×C₄	—	92	1	no	—
80	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
81	Q₁₆	—	23	4	no	—
82	Q₁₆	—	46	4	no	—
83	Q₁₆	—	46	2	no	—
84	C₂×C₈	C₄×D₂₃	1	2	yes	—
85	C₂×C₈	D₉₂	1	2	yes	—
86	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
87	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
88	C₄⋊C₄	—	2	2	no	—
89	C₂²⋊C₄	—	4	1	no	—
90	C₄²	—	4	1	no	—
91	C₁₆	—	46	2	no	—
92	C₂³	#67272	1	1	yes	characteristic
93	Q₈	—	46	4	no	—
94	Q₈	—	92	2	no	—
95	Q₈	—	92	4	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	92	2	no	—
98	C₂×C₄	—	92	4	no	—
99	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
100	C₂×C₄	#67272	1	1	yes	characteristic
101	C₂×C₄	#67274	1	2	yes	—
102	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
103	C₈	(C₂×D₂₃)⋊C₄	1	2	yes	—
104	C₈	C₂×C₄×D₂₃	1	1	yes	characteristic
105	C₈	C₂×D₉₂	1	1	yes	characteristic
106	C₂²	—	2	2	no	—
107	C₂²	#10624991	1	2	yes	—
108	C₂²	C₂×#67272	1	1	yes	characteristic
109	C₄	—	8	2	no	—
110	C₄	—	92	4	no	—
111	C₄	#10624991	1	2	yes	—
112	C₄	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
113	C₄	C₂×#67274	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	C₂×#10624991	1	1	yes	characteristic
116	C₂	#13092857	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₉₂

C₂

yes

maximal, characteristic

C₂×#10624994

C₂

yes

maximal, characteristic

#13092220

C₂

yes

maximal, complemented

C₂₃×#427

C₂

yes

maximal, characteristic

C₂²×Dic₄₆

—

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#117

—

C₂×Dic₉₂

—

C₂×Dic₉₂

—

C₂×Dic₉₂

C₄

yes

complemented

C₂×Dic₉₂

C₂²

yes

—

C₂₃×#107

C₂²

yes

—

C₂₃×#106

—

#10624994

—

C₂×C₂₃⋊C₁₆

C₂²

yes

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

Dic₉₂

—

Dic₉₂

—

Dic₉₂

C₂×C₄

yes

—

C₂₃⋊C₁₆

—

C₂³×C₂₃

D₈

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

D₈

yes

characteristic

C₂×C₄×C₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

C₂²⋊C₄

yes

—

C₈×C₂₃

C₂²×C₄

yes

characteristic

C₈×C₂₃

C₂×D₄

yes

characteristic

#2141

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

#102

yes

—

C₂²×C₂₃

C₂×D₈

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

#102

yes

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

C₂×SD₁₆

yes

characteristic

Dic₂₃

—

C₂²×Q₁₆

—

C₂×M₃₂

—

#427

D₂₃

yes

characteristic

#358

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#102

yes

characteristic

C₂×C₂₃

#510

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

—

#107

C₂×D₂₃

yes

—

#106

—

C₂₃

#2141

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

C₄×D₂₃

yes

—

C₂×C₈

D₉₂

yes

—

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

C₂×C₈

C₂²×D₂₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₂³

#67272

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

100

C₂×C₄

#67272

yes

characteristic

101

C₂×C₄

#67274

yes

—

102

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

103

C₈

(C₂×D₂₃)⋊C₄

yes

—

104

C₈

C₂×C₄×D₂₃

yes

characteristic

105

C₈

C₂×D₉₂

yes

characteristic

106

C₂²

—

107

C₂²

#10624991

yes

—

108

C₂²

C₂×#67272

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

#10624991

yes

—

112

C₄

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

113

C₄

C₂×#67274

yes

characteristic

114

C₂

—

115

C₂

C₂×#10624991

yes

characteristic

116

C₂

#13092857

yes

—

	subgroup	G/H
center	C₂²	C₂×#67272
commutator	C₈×C₂₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₃
Fitting	C₂₃×#427	C₂
socle	C₂²×C₂₃	C₂×D₈
prime radical	C₂³×C₂₃	D₈
cosocle	C₂×C₈×C₂₃	C₂³
hypercenter	#427	D₂₃
hypocenter	C₂₃	#2141
LCS	G ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₂²

C₂×#67272

commutator

C₈×C₂₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₂₃

Fitting

C₂₃×#427

C₂

socle

C₂²×C₂₃

C₂×D₈

prime radical

C₂³×C₂₃

D₈

cosocle

C₂×C₈×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
3. #13092220	C₂	semidirect
10. C₂×Dic₉₂	C₄	semidirect
74. C₂₃	#2141	semidirect

H ≅ G/N

type

3. #13092220

C₂

semidirect

10. C₂×Dic₉₂

C₄

semidirect

74. C₂₃

#2141

semidirect

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

subgroup	index
C₂²×Dic₉₂	2
C₂×#10624994	2
#13092220	2
C₂₃×#427	2
#2141	23

subgroup

index

C₂²×Dic₉₂

C₂×#10624994

#13092220

C₂₃×#427

#2141

Subgroups — #15675116

Conjugacy classes of subgroups

Complements

Maximal subgroups