Group #15669579 — subgroups

Subgroups — C₂₃×#1960 #15669579

Order 2944.

308 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 128 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	C₂×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#103	C₂	1	4	yes	maximal
3	C₂×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#382	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
6	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
7	C₂₃×C₄⋊Q₈	—	2	2	no	—
8	C₂₃×C₄⋊Q₈	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
10	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#103	—	2	8	no	—
13	C₂₃×#97	—	2	2	no	—
14	C₂₃×#95	—	2	2	no	—
15	C₂×Q₈×C₂₃	—	2	2	no	—
16	C₂×Q₈×C₂₃	—	2	4	no	—
17	C₂×Q₈×C₂₃	—	2	4	no	—
18	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₂₃	—	2	2	no	—
20	C₂²×C₄×C₂₃	—	2	2	no	—
21	C₂²×C₄×C₂₃	D₄	1	2	yes	—
22	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₂₃	—	2	2	no	—
24	C₂×C₈×C₂₃	—	2	2	no	—
25	C₂×C₈×C₂₃	D₄	1	4	yes	—
26	C₄⋊C₄×C₂₃	—	2	8	no	—
27	C₄⋊C₄×C₂₃	—	2	4	no	—
28	C₄²×C₂₃	C₂×C₄	1	2	yes	—
29	C₄²×C₂₃	D₄	1	2	yes	—
30	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₂₃	—	2	8	no	—
32	Q₈×C₂₃	—	4	4	no	—
33	C₂×C₄×C₂₃	—	2	4	no	—
34	C₂×C₄×C₂₃	—	2	4	no	—
35	C₂×C₄×C₂₃	—	4	2	no	—
36	C₂×C₄×C₂₃	—	4	2	no	—
37	C₂×C₄×C₂₃	—	4	2	no	—
38	C₂×C₄×C₂₃	—	4	4	no	—
39	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
40	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₃	#41	1	2	yes	—
44	C₈×C₂₃	—	2	4	no	—
45	#1960	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₂₃	#124	1	1	yes	characteristic
48	C₂²×C₂₃	#127	1	1	yes	characteristic
49	C₂²×C₂₃	#137	1	4	yes	—
50	C₄×C₂₃	—	2	4	no	—
51	C₄×C₂₃	—	4	4	no	—
52	C₄×C₂₃	—	8	2	no	—
53	C₄×C₂₃	C₄×D₄	1	2	yes	—
54	C₄×C₂₃	#121	1	2	yes	—
55	C₂×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
56	C₂×#103	C₂×C₂₃	1	4	yes	—
57	C₂×#97	C₂×C₂₃	1	1	yes	characteristic
58	#382	C₂×C₂₃	1	1	yes	characteristic
59	C₂×C₂₃	#390	1	1	yes	characteristic
60	C₂×C₂₃	#419	1	2	yes	—
61	C₂×C₂₃	#491	1	2	yes	—
62	C₂×C₂₃	#497	1	2	yes	—
63	C₂²×Q₈	C₂²×C₂₃	1	2	yes	—
64	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
65	C₄⋊Q₈	—	2	2	no	—
66	C₄⋊Q₈	C₄×C₂₃	1	4	yes	complemented
67	C₂×C₄⋊C₄	—	2	2	no	—
68	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
69	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
70	#103	—	2	8	no	—
71	#97	—	2	2	no	—
72	#95	—	2	2	no	—
73	C₂₃	#1960	1	1	yes	characteristic, direct product factor, Sylow
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×C₄×C₂₃	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	D₄×C₂₃	1	2	yes	—
81	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	D₄×C₂₃	1	4	yes	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	2	4	no	—
87	C₄²	C₂×C₄×C₂₃	1	2	yes	—
88	C₄²	D₄×C₂₃	1	2	yes	—
89	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
99	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
100	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
101	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
102	C₂×C₄	#41×C₂₃	1	2	yes	—
103	C₈	—	2	4	no	—
104	C₂²	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
105	C₂²	C₂₃×#124	1	1	yes	characteristic
106	C₂²	C₂₃×#127	1	1	yes	characteristic
107	C₂²	C₂₃×#137	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	8	2	no	—
111	C₄	C₄×D₄×C₂₃	1	2	yes	—
112	C₄	C₂₃×#121	1	2	yes	—
113	C₂	C₂₃×#390	1	1	yes	characteristic
114	C₂	C₂₃×#419	1	2	yes	—
115	C₂	C₂₃×#491	1	2	yes	—
116	C₂	C₂₃×#497	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×C₂₃×#103

C₂

yes

maximal

C₂×C₂₃×#97

C₂

yes

maximal, characteristic

C₂₃×#382

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₂²

yes

—

C₂²×C₈×C₂₃

C₂²

yes

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂₃×#103

—

C₂₃×#97

—

C₂₃×#95

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₈×C₂₃

—

#1960

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

#121

yes

—

C₂×C₄⋊Q₈

C₂×C₂₃

yes

characteristic

C₂×#103

C₂×C₂₃

yes

—

C₂×#97

C₂×C₂₃

yes

characteristic

#382

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#491

yes

—

C₂×C₂₃

#497

yes

—

C₂²×Q₈

C₂²×C₂₃

yes

—

C₂²×C₈

C₂²×C₂₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₄×C₂₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

—

—

—

C₂₃

#1960

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₄×C₂₃

yes

—

C₄²

D₄×C₂₃

yes

—

C₂³

C₂×D₄×C₂₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₂₃

yes

—

C₂×C₄

C₂²×C₄×C₂₃

yes

characteristic

100

C₂×C₄

C₂×D₄×C₂₃

yes

—

101

C₂×C₄

C₂×D₄×C₂₃

yes

characteristic

102

C₂×C₄

#41×C₂₃

yes

—

103

C₈

—

104

C₂²

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

105

C₂²

C₂₃×#124

yes

characteristic

106

C₂²

C₂₃×#127

yes

characteristic

107

C₂²

C₂₃×#137

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

C₄×D₄×C₂₃

yes

—

112

C₄

C₂₃×#121

yes

—

113

C₂

C₂₃×#390

yes

characteristic

114

C₂

C₂₃×#419

yes

—

115

C₂

C₂₃×#491

yes

—

116

C₂

C₂₃×#497

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂³×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₃

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₂₃

Frattini, cosocle

C₂²×C₄

C₂³×C₂₃

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
8. C₂₃×C₄⋊Q₈	C₄	semidirect
45. #1960	C₂₃	direct (N × H)
66. C₄⋊Q₈	C₄×C₂₃	semidirect
73. C₂₃	#1960	direct (N × H)

H ≅ G/N

type

8. C₂₃×C₄⋊Q₈

C₄

semidirect

45. #1960

C₂₃

direct (N × H)

66. C₄⋊Q₈

C₄×C₂₃

semidirect

73. C₂₃

#1960

direct (N × H)

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

subgroup	index
C₂×C₂₃×C₄⋊Q₈	2
C₂×C₂₃×#103	2
C₂×C₂₃×#97	2
C₂₃×#382	2
#1960	23

subgroup

index

C₂×C₂₃×C₄⋊Q₈

C₂×C₂₃×#103

C₂×C₂₃×#97

C₂₃×#382

#1960

Subgroups — C₂₃×#1960 #15669579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#1960 #15669579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1960 #15669579