Group #15670260 — subgroups

Subgroups — C₂₃×#2641 #15670260

Order 2944.

584 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 200 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×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#394	C₂	1	8	yes	maximal, complemented
4	C₂₃×#390	C₂	1	4	yes	maximal, complemented
5	C₂₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
7	C₂²×D₄×C₂₃	C₂²	1	4	yes	—
8	C₂₃×#127	—	2	4	no	—
9	C₂₃×#125	—	2	4	no	—
10	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₄×C₂₃	C₂²	1	16	yes	—
13	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#95	C₂²	1	4	yes	—
16	C₂⁴×C₂₃	C₂³	1	4	yes	—
17	C₂×D₄×C₂₃	—	2	8	no	—
18	C₂×D₄×C₂₃	—	2	4	no	—
19	C₂×D₄×C₂₃	—	2	8	no	—
20	C₂×D₄×C₂₃	—	4	8	no	—
21	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
22	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
23	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
24	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₂₃	—	2	8	no	—
26	C₄⋊C₄×C₂₃	—	2	4	no	—
27	C₂²⋊C₄×C₂₃	—	4	16	no	—
28	C₄²×C₂₃	—	2	4	no	—
29	C₄²×C₂₃	D₄	1	4	yes	—
30	C₂³×C₂₃	—	2	4	no	—
31	C₂³×C₂₃	—	2	8	no	—
32	C₂³×C₂₃	—	2	16	no	—
33	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
34	D₄×C₂₃	—	4	16	no	—
35	D₄×C₂₃	—	4	8	no	—
36	C₂×C₄×C₂₃	—	2	4	no	—
37	C₂×C₄×C₂₃	—	2	4	no	—
38	C₂×C₄×C₂₃	—	2	4	no	—
39	C₂×C₄×C₂₃	—	2	8	no	—
40	C₂×C₄×C₂₃	—	2	8	no	—
41	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₃	#41	1	4	yes	—
44	#2641	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×C₂₃	—	4	4	no	—
46	C₂²×C₂₃	—	4	8	no	—
47	C₂²×C₂₃	—	4	16	no	—
48	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
49	C₂²×C₂₃	C₂×#41	1	2	yes	—
50	C₂²×C₂₃	H(5,𝔽₂)	1	4	yes	—
51	C₄×C₂₃	—	2	4	no	—
52	C₄×C₂₃	—	4	4	no	—
53	C₄×C₂₃	—	4	4	no	—
54	C₄×C₂₃	#124	1	4	yes	—
55	C₂×#127	C₂×C₂₃	1	1	yes	characteristic
56	C₂×#125	C₂×C₂₃	1	1	yes	characteristic, complemented
57	#394	C₂×C₂₃	1	8	yes	complemented
58	#390	C₂×C₂₃	1	4	yes	complemented
59	#376	C₂×C₂₃	1	1	yes	characteristic, complemented
60	C₂×C₂₃	—	8	4	no	—
61	C₂×C₂₃	C₂×#124	1	1	yes	characteristic
62	C₂×C₂₃	#535	1	2	yes	—
63	C₂×C₂₃	#540	1	4	yes	—
64	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
65	C₂²×D₄	C₂²×C₂₃	1	4	yes	—
66	#127	—	2	4	no	—
67	#125	—	2	4	no	—
68	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
69	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
70	C₂×C₂²⋊C₄	C₂²×C₂₃	1	16	yes	—
71	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
72	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
73	#95	C₂²×C₂₃	1	4	yes	—
74	C₂₃	#2641	1	1	yes	characteristic, direct product factor, Sylow
75	C₂⁴	C₂³×C₂₃	1	4	yes	—
76	C₂×D₄	—	2	8	no	—
77	C₂×D₄	—	2	4	no	—
78	C₂×D₄	—	2	8	no	—
79	C₂×D₄	—	4	8	no	—
80	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
81	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
82	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
83	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
84	C₄⋊C₄	—	2	8	no	—
85	C₄⋊C₄	—	2	4	no	—
86	C₂²⋊C₄	—	4	16	no	—
87	C₄²	—	2	4	no	—
88	C₄²	D₄×C₂₃	1	4	yes	—
89	C₂³	—	2	4	no	—
90	C₂³	—	2	8	no	—
91	C₂³	—	2	16	no	—
92	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
93	D₄	—	4	16	no	—
94	D₄	—	4	8	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	8	no	—
99	C₂×C₄	—	2	8	no	—
100	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
101	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
102	C₂×C₄	#41×C₂₃	1	4	yes	—
103	C₂²	—	4	4	no	—
104	C₂²	—	4	8	no	—
105	C₂²	—	4	16	no	—
106	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
107	C₂²	C₂×#41×C₂₃	1	2	yes	—
108	C₂²	C₂₃×H(5,𝔽₂)	1	4	yes	—
109	C₄	—	2	4	no	—
110	C₄	—	4	4	no	—
111	C₄	—	4	4	no	—
112	C₄	C₂₃×#124	1	4	yes	—
113	C₂	—	8	4	no	—
114	C₂	C₂×C₂₃×#124	1	1	yes	characteristic
115	C₂	C₂₃×#535	1	2	yes	—
116	C₂	C₂₃×#540	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×#127

C₂

yes

maximal, characteristic

C₂×C₂₃×#125

C₂

yes

maximal, characteristic, complemented

C₂₃×#394

C₂

yes

maximal, complemented

C₂₃×#390

C₂

yes

maximal, complemented

C₂₃×#376

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₂₃

C₂²

yes

—

C₂²×D₄×C₂₃

C₂²

yes

—

C₂₃×#127

—

C₂₃×#125

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂₃×#95

C₂²

yes

—

C₂⁴×C₂₃

C₂³

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

D₄

yes

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂⁴

yes

characteristic

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

#2641

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₂²×C₂₃

C₂×#41

yes

—

C₂²×C₂₃

H(5,𝔽₂)

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#124

yes

—

C₂×#127

C₂×C₂₃

yes

characteristic

C₂×#125

C₂×C₂₃

yes

characteristic, complemented

#394

C₂×C₂₃

yes

complemented

#390

C₂×C₂₃

yes

complemented

#376

C₂×C₂₃

yes

characteristic, complemented

C₂×C₂₃

—

C₂×C₂₃

C₂×#124

yes

characteristic

C₂×C₂₃

#535

yes

—

C₂×C₂₃

#540

yes

—

C₂²×D₄

C₂²×C₂₃

yes

—

C₂²×D₄

C₂²×C₂₃

yes

—

#127

—

#125

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

characteristic

#95

C₂²×C₂₃

yes

—

C₂₃

#2641

yes

characteristic, direct product factor, Sylow

C₂⁴

C₂³×C₂₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

D₄×C₂₃

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂⁴×C₂₃

yes

characteristic

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂×D₄×C₂₃

yes

—

101

C₂×C₄

C₂×D₄×C₂₃

yes

—

102

C₂×C₄

#41×C₂₃

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂²×D₄×C₂₃

yes

characteristic

107

C₂²

C₂×#41×C₂₃

yes

—

108

C₂²

C₂₃×H(5,𝔽₂)

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

C₂₃×#124

yes

—

113

C₂

—

114

C₂

C₂×C₂₃×#124

yes

characteristic

115

C₂

C₂₃×#535

yes

—

116

C₂

C₂₃×#540

yes

—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₂₃

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₂₃

prime radical

C₂×C₂₃×#127

C₂

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ G

N	H ≅ G/N	type
2. C₂×C₂₃×#125	C₂	semidirect
3. C₂₃×#394	C₂	semidirect
4. C₂₃×#390	C₂	semidirect
5. C₂₃×#376	C₂	semidirect
44. #2641	C₂₃	direct (N × H)
56. C₂×#125	C₂×C₂₃	semidirect
57. #394	C₂×C₂₃	semidirect
58. #390	C₂×C₂₃	semidirect
59. #376	C₂×C₂₃	semidirect
74. C₂₃	#2641	direct (N × H)

H ≅ G/N

type

2. C₂×C₂₃×#125

C₂

semidirect

3. C₂₃×#394

C₂

semidirect

4. C₂₃×#390

C₂

semidirect

5. C₂₃×#376

C₂

semidirect

44. #2641

C₂₃

direct (N × H)

56. C₂×#125

C₂×C₂₃

semidirect

57. #394

C₂×C₂₃

semidirect

58. #390

C₂×C₂₃

semidirect

59. #376

C₂×C₂₃

semidirect

74. C₂₃

#2641

direct (N × H)

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

subgroup	index
C₂×C₂₃×#127	2
C₂×C₂₃×#125	2
C₂₃×#394	2
C₂₃×#390	2
C₂₃×#376	2
#2641	23

subgroup

index

C₂×C₂₃×#127

C₂×C₂₃×#125

C₂₃×#394

C₂₃×#390

C₂₃×#376

#2641

Subgroups — C₂₃×#2641 #15670260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#2641 #15670260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2641 #15670260