Group #15671681 — subgroups

Subgroups — #15671681

Order 2944.

212 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 133 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	#13092470	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#13092139	C₂	1	2	yes	maximal, complemented
4	C₈×C₂₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂₃×#431	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₂₃⋊C₈	C₄	1	2	yes	—
8	C₂₃×C₄⋊C₈	C₄	1	2	yes	—
9	C₂₃×C₂²⋊C₈	C₄	1	2	yes	—
10	C₄×C₈×C₂₃	C₂²	1	2	yes	—
11	C₂²⋊(C₂₃⋊C₈)	C₄	1	2	yes	—
12	C₄⋊(C₂₃⋊C₈)	C₄	1	2	yes	—
13	C₄×C₂₃⋊C₈	C₄	1	2	yes	—
14	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
17	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₂₃	C₂×C₄	1	4	yes	—
21	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₂₃	C₂³	1	1	yes	characteristic
25	C₂×C₂₃⋊C₈	C₈	1	8	yes	complemented
26	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
27	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
28	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
29	C₂³×C₂₃	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₂₃	—	2	2	no	—
31	C₂×C₄×C₂₃	—	2	2	no	—
32	C₂×C₄×C₂₃	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₂₃	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₂×C₈	1	4	yes	—
36	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₈×C₂₃	—	2	4	no	—
40	C₂₃⋊C₈	—	2	4	no	—
41	C₂₃⋊C₈	C₂×C₈	1	4	yes	—
42	C₂₃⋊C₈	C₂×C₈	1	4	yes	—
43	#1450	—	23	1	no	maximal, Sylow
44	C₂²×C₂₃	—	2	1	no	—
45	C₂²×C₂₃	—	2	1	no	—
46	C₂²×C₂₃	C₄×C₈	1	2	yes	—
47	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₂₃	—	2	1	no	—
49	C₄×C₂₃	—	2	1	no	—
50	C₄×C₂₃	C₄×C₈	1	2	yes	—
51	C₄×C₂₃	C₂²×C₈	1	2	yes	—
52	C₄×C₂₃	#131	1	2	yes	—
53	C₄×C₂₃	#131	1	2	yes	—
54	#431	—	23	1	no	—
55	#431	D₂₃	1	1	yes	characteristic
56	C₂×C₄×C₈	—	23	1	no	—
57	#322	—	23	2	no	—
58	C₈²	—	23	2	no	—
59	C₂×C₂₃	—	2	2	no	—
60	C₂×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
61	C₂×C₂₃	#405	1	1	yes	characteristic
62	C₂×C₂₃	#405	1	1	yes	characteristic
63	C₂²×C₈	—	23	2	no	—
64	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
65	C₄⋊C₈	—	23	2	no	—
66	C₄⋊C₈	Dic₂₃	1	2	yes	—
67	C₂²⋊C₈	—	23	2	no	—
68	C₂²⋊C₈	Dic₂₃	1	2	yes	—
69	C₄×C₈	—	23	2	no	—
70	C₄×C₈	—	23	1	no	—
71	C₄×C₈	—	23	2	no	—
72	C₄×C₈	—	23	1	no	—
73	C₄×C₈	C₂×D₂₃	1	2	yes	—
74	C₂₃	#1450	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
76	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
77	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
78	C₂×C₈	—	23	8	no	—
79	C₂×C₈	—	23	2	no	—
80	C₂×C₈	—	23	2	no	—
81	C₂×C₈	—	23	4	no	—
82	C₂×C₈	C₂×Dic₂₃	1	4	yes	—
83	C₄²	C₄×D₂₃	1	1	yes	characteristic
84	C₄²	C₄×D₂₃	1	1	yes	characteristic
85	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
86	C₄²	C₂²×D₂₃	1	1	yes	characteristic
87	C₂³	C₄×Dic₂₃	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	C₈×D₂₃	1	4	yes	—
91	C₂×C₄	C₄×Dic₂₃	1	1	yes	characteristic
92	C₂×C₄	C₄×Dic₂₃	1	1	yes	characteristic
93	C₂×C₄	C₄×Dic₂₃	1	1	yes	characteristic
94	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
95	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
96	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
97	C₈	—	2	4	no	—
98	C₈	—	23	4	no	—
99	C₈	—	23	4	no	—
100	C₈	—	46	4	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	C₈×Dic₂₃	1	2	yes	—
104	C₂²	C₂×C₄×Dic₂₃	1	1	yes	characteristic
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	C₈×Dic₂₃	1	2	yes	—
108	C₄	C₂×C₈×D₂₃	1	2	yes	—
109	C₄	#10625078	1	2	yes	—
110	C₄	#10625119	1	2	yes	—
111	C₂	—	2	2	no	—
112	C₂	C₂×C₈×Dic₂₃	1	1	yes	characteristic
113	C₂	#13092373	1	1	yes	characteristic
114	C₂	#13092375	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092470

C₂

yes

maximal, characteristic

C₂×C₄×C₂₃⋊C₈

C₂

yes

maximal, characteristic

#13092139

C₂

yes

maximal, complemented

C₈×C₂₃⋊C₈

C₂

yes

maximal, complemented

C₂₃×#431

C₂

yes

maximal, characteristic

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₄

yes

—

C₂₃×C₄⋊C₈

C₄

yes

—

C₂₃×C₂²⋊C₈

C₄

yes

—

C₄×C₈×C₂₃

C₂²

yes

—

C₂²⋊(C₂₃⋊C₈)

C₄

yes

—

C₄⋊(C₂₃⋊C₈)

C₄

yes

—

C₄×C₂₃⋊C₈

C₄

yes

—

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₄×C₂₃⋊C₈

C₂²

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

C₂×C₂₃⋊C₈

C₈

yes

complemented

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂³×C₂₃

C₄²

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄²

yes

characteristic

C₂×C₄×C₂₃

C₄²

yes

characteristic

C₂×C₄×C₂₃

C₄²

yes

characteristic

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

C₂×C₈

yes

—

C₂₃⋊C₈

C₂×C₈

yes

—

#1450

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₄×C₈

yes

—

C₂²×C₂₃

C₂×C₄²

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×C₈

yes

—

C₄×C₂₃

C₂²×C₈

yes

—

C₄×C₂₃

#131

yes

—

C₄×C₂₃

#131

yes

—

#431

—

#431

D₂₃

yes

characteristic

C₂×C₄×C₈

—

#322

—

C₈²

—

C₂×C₂₃

—

C₂×C₂₃

C₂×C₄×C₈

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×C₈

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

Dic₂₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

Dic₂₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₂₃

yes

—

C₂₃

#1450

yes

characteristic, complemented, Sylow

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₂₃

yes

—

C₄²

C₄×D₂₃

yes

characteristic

C₄²

C₄×D₂₃

yes

characteristic

C₄²

C₂×Dic₂₃

yes

characteristic

C₄²

C₂²×D₂₃

yes

characteristic

C₂³

C₄×Dic₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₂₃

yes

—

C₂×C₄

C₄×Dic₂₃

yes

characteristic

C₂×C₄

C₄×Dic₂₃

yes

characteristic

C₂×C₄

C₄×Dic₂₃

yes

characteristic

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₈×Dic₂₃

yes

—

104

C₂²

C₂×C₄×Dic₂₃

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

C₈×Dic₂₃

yes

—

108

C₄

C₂×C₈×D₂₃

yes

—

109

C₄

#10625078

yes

—

110

C₄

#10625119

yes

—

111

C₂

—

112

C₂

C₂×C₈×Dic₂₃

yes

characteristic

113

C₂

#13092373

yes

characteristic

114

C₂

#13092375

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₂₃
commutator	C₂×C₂₃	C₂×C₄×C₈
Fitting	C₂₃×#431	C₂
socle	C₂²×C₂₃	C₂×C₄²
prime radical	C₂³×C₂₃	C₄²
cosocle	C₄²×C₂₃	C₂³
hypercenter	#431	D₂₃
hypocenter	C₂₃	#1450
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄² ◁ #431

subgroup

G/H

center, Frattini

C₄²

C₂²×D₂₃

commutator

C₂×C₂₃

C₂×C₄×C₈

Fitting

C₂₃×#431

C₂

socle

C₂²×C₂₃

C₂×C₄²

prime radical

C₂³×C₂₃

C₄²

cosocle

C₄²×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₂₃ ▷ C₂₃

C₁ ◁ C₄² ◁ #431

N	H ≅ G/N	type
3. #13092139	C₂	semidirect
4. C₈×C₂₃⋊C₈	C₂	semidirect
25. C₂×C₂₃⋊C₈	C₈	semidirect
74. C₂₃	#1450	semidirect

H ≅ G/N

type

3. #13092139

C₂

semidirect

4. C₈×C₂₃⋊C₈

C₂

semidirect

25. C₂×C₂₃⋊C₈

C₈

semidirect

74. C₂₃

#1450

semidirect

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

subgroup	index
#13092470	2
C₂×C₄×C₂₃⋊C₈	2
#13092139	2
C₈×C₂₃⋊C₈	2
C₂₃×#431	2
#1450	23

subgroup

index

#13092470

C₂×C₄×C₂₃⋊C₈

#13092139

C₈×C₂₃⋊C₈

C₂₃×#431

#1450

Subgroups — #15671681

Conjugacy classes of subgroups

Complements

Maximal subgroups