Group #15669681 — subgroups

Subgroups — C₂₃×#2062 #15669681

Order 2944.

314 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 92 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	C₂₃×#543	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#385	C₂	1	3	yes	maximal
3	C₂₃×#342	C₂	1	3	yes	maximal
4	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
5	C₂₃×C₂²⋊Q₈	—	2	3	no	—
6	C₂₃×C₂²⋊Q₈	—	2	3	no	—
7	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
8	C₂×C₂²⋊C₄×C₂₃	C₂²	1	3	yes	—
9	C₂×C₂²⋊C₄×C₂₃	C₂²	1	3	yes	—
10	C₂₃×#95	—	2	3	no	—
11	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
12	C₂×Q₈×C₂₃	—	2	1	no	—
13	C₂×Q₈×C₂₃	—	2	1	no	—
14	C₂²×C₄×C₂₃	—	2	3	no	—
15	C₂²×C₄×C₂₃	—	2	3	no	—
16	C₂²×C₄×C₂₃	—	4	2	no	—
17	C₂²×C₄×C₂₃	D₄	1	3	yes	—
18	C₂²×C₄×C₂₃	D₄	1	3	yes	—
19	C₄⋊C₄×C₂₃	—	2	3	no	—
20	C₄⋊C₄×C₂₃	—	2	3	no	—
21	C₄⋊C₄×C₂₃	—	4	6	no	—
22	C₂²⋊C₄×C₂₃	—	2	6	no	—
23	C₂²⋊C₄×C₂₃	—	2	3	no	—
24	C₂²⋊C₄×C₂₃	Q₈	1	6	yes	complemented
25	C₂³×C₂₃	—	4	1	no	—
26	C₂³×C₂₃	—	4	1	no	—
27	C₂³×C₂₃	C₂×D₄	1	3	yes	—
28	C₂³×C₂₃	C₂×Q₈	1	3	yes	—
29	C₂³×C₂₃	#41	1	1	yes	characteristic
30	Q₈×C₂₃	—	8	1	no	—
31	Q₈×C₂₃	—	8	1	no	—
32	C₂×C₄×C₂₃	—	2	6	no	—
33	C₂×C₄×C₂₃	—	2	3	no	—
34	C₂×C₄×C₂₃	—	2	3	no	—
35	C₂×C₄×C₂₃	—	2	6	no	—
36	C₂×C₄×C₂₃	—	4	3	no	—
37	C₂×C₄×C₂₃	—	4	3	no	—
38	C₂×C₄×C₂₃	—	4	3	no	—
39	C₂×C₄×C₂₃	—	4	1	no	—
40	C₂×C₄×C₂₃	—	4	3	no	—
41	C₂×C₄×C₂₃	—	4	3	no	—
42	#2062	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₂₃	—	2	3	no	—
44	C₂²×C₂₃	—	2	3	no	—
45	C₂²×C₂₃	—	4	3	no	—
46	C₂²×C₂₃	—	4	1	no	—
47	C₂²×C₂₃	#120	1	1	yes	characteristic
48	C₂²×C₂₃	C₂²⋊Q₈	1	3	yes	—
49	C₂²×C₂₃	C₄⋊Q₈	1	3	yes	—
50	C₄×C₂₃	—	4	6	no	—
51	C₄×C₂₃	—	4	2	no	—
52	C₄×C₂₃	—	4	3	no	—
53	C₄×C₂₃	—	4	3	no	—
54	#543	C₂×C₂₃	1	1	yes	characteristic
55	#385	C₂×C₂₃	1	3	yes	—
56	#342	C₂×C₂₃	1	3	yes	—
57	C₂×C₂₃	—	2	3	no	—
58	C₂×C₂₃	—	2	3	no	—
59	C₂×C₂₃	#395	1	1	yes	characteristic
60	C₂×C₂₃	B(4,𝔽₂)	1	1	yes	characteristic
61	C₂×C₂₃	#458	1	1	yes	characteristic
62	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
63	C₂²⋊Q₈	—	2	3	no	—
64	C₂²⋊Q₈	—	2	3	no	—
65	C₂×C₄⋊C₄	—	2	3	no	—
66	C₂×C₂²⋊C₄	C₂²×C₂₃	1	3	yes	—
67	C₂×C₂²⋊C₄	C₂²×C₂₃	1	3	yes	—
68	#95	—	2	3	no	—
69	C₂₃	#2062	1	1	yes	characteristic, direct product factor, Sylow
70	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
71	C₂×Q₈	—	2	1	no	—
72	C₂×Q₈	—	2	1	no	—
73	C₂²×C₄	—	2	3	no	—
74	C₂²×C₄	—	2	3	no	—
75	C₂²×C₄	—	4	2	no	—
76	C₂²×C₄	D₄×C₂₃	1	3	yes	—
77	C₂²×C₄	D₄×C₂₃	1	3	yes	—
78	C₄⋊C₄	—	2	3	no	—
79	C₄⋊C₄	—	2	3	no	—
80	C₄⋊C₄	—	4	6	no	—
81	C₂²⋊C₄	—	2	6	no	—
82	C₂²⋊C₄	—	2	3	no	—
83	C₂²⋊C₄	Q₈×C₂₃	1	6	yes	complemented
84	C₂³	—	4	1	no	—
85	C₂³	—	4	1	no	—
86	C₂³	C₂×D₄×C₂₃	1	3	yes	—
87	C₂³	C₂×Q₈×C₂₃	1	3	yes	—
88	C₂³	#41×C₂₃	1	1	yes	characteristic
89	Q₈	—	8	1	no	—
90	Q₈	—	8	1	no	—
91	C₂×C₄	—	2	6	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	2	3	no	—
94	C₂×C₄	—	2	6	no	—
95	C₂×C₄	—	4	3	no	—
96	C₂×C₄	—	4	3	no	—
97	C₂×C₄	—	4	3	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	4	3	no	—
100	C₂×C₄	—	4	3	no	—
101	C₂²	—	2	3	no	—
102	C₂²	—	2	3	no	—
103	C₂²	—	4	3	no	—
104	C₂²	—	4	1	no	—
105	C₂²	C₂₃×#120	1	1	yes	characteristic
106	C₂²	C₂₃×C₂²⋊Q₈	1	3	yes	—
107	C₂²	C₂₃×C₄⋊Q₈	1	3	yes	—
108	C₄	—	4	6	no	—
109	C₄	—	4	2	no	—
110	C₄	—	4	3	no	—
111	C₄	—	4	3	no	—
112	C₂	—	2	3	no	—
113	C₂	—	2	3	no	—
114	C₂	C₂₃×#395	1	1	yes	characteristic
115	C₂	C₂₃×B(4,𝔽₂)	1	1	yes	characteristic
116	C₂	C₂₃×#458	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#543

C₂

yes

maximal, characteristic

C₂₃×#385

C₂

yes

maximal

C₂₃×#342

C₂

yes

maximal

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂₃×C₂²⋊Q₈

—

C₂₃×C₂²⋊Q₈

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂₃×#95

—

C₂⁴×C₂₃

C₂³

yes

characteristic

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

Q₈

yes

complemented

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×D₄

yes

—

C₂³×C₂₃

C₂×Q₈

yes

—

C₂³×C₂₃

#41

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

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

#2062

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#120

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

—

C₂²×C₂₃

C₄⋊Q₈

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

#543

C₂×C₂₃

yes

characteristic

#385

C₂×C₂₃

yes

—

#342

C₂×C₂₃

yes

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂³×C₄

C₂²×C₂₃

yes

characteristic

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

C₂²×C₂₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₂₃

yes

—

#95

—

C₂₃

#2062

yes

characteristic, direct product factor, Sylow

C₂⁴

C₂³×C₂₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₃

yes

—

C₂²×C₄

D₄×C₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

Q₈×C₂₃

yes

complemented

C₂³

—

C₂³

—

C₂³

C₂×D₄×C₂₃

yes

—

C₂³

C₂×Q₈×C₂₃

yes

—

C₂³

#41×C₂₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

C₂₃×#120

yes

characteristic

106

C₂²

C₂₃×C₂²⋊Q₈

yes

—

107

C₂²

C₂₃×C₄⋊Q₈

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₂

—

113

C₂

—

114

C₂

C₂₃×#395

yes

characteristic

115

C₂

C₂₃×B(4,𝔽₂)

yes

characteristic

116

C₂

C₂₃×#458

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#120
commutator, Frattini, cosocle	C₂⁴	C₂³×C₂₃
prime radical	C₂⁴×C₂₃	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂⁴×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#120

commutator, Frattini, cosocle

C₂⁴

C₂³×C₂₃

prime radical

C₂⁴×C₂₃

C₂³

LCS

G ▷ C₂⁴ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂⁴×C₂₃ ◁ G

N	H ≅ G/N	type
24. C₂²⋊C₄×C₂₃	Q₈	semidirect
42. #2062	C₂₃	direct (N × H)
69. C₂₃	#2062	direct (N × H)
83. C₂²⋊C₄	Q₈×C₂₃	semidirect

H ≅ G/N

type

24. C₂²⋊C₄×C₂₃

Q₈

semidirect

42. #2062

C₂₃

direct (N × H)

69. C₂₃

#2062

direct (N × H)

83. C₂²⋊C₄

Q₈×C₂₃

semidirect

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

subgroup	index
C₂₃×#543	2
C₂₃×#385	2
C₂₃×#342	2
#2062	23

subgroup

index

C₂₃×#543

C₂₃×#385

C₂₃×#342

#2062

Subgroups — C₂₃×#2062 #15669681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#2062 #15669681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2062 #15669681