Group #15670227 — subgroups

Subgroups — C₂₃×#2608 #15670227

Order 2944.

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

normal subgroups only

This group has 428 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	C₂₃×#396	C₂	1	4	yes	maximal
2	C₂₃×#389	C₂	1	4	yes	maximal, complemented
3	C₂₃×#387	C₂	1	4	yes	maximal, complemented
4	C₂₃×#381	C₂	1	1	yes	maximal, characteristic
5	C₄×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
6	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
8	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
9	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
10	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
11	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#95	C₂²	1	4	yes	—
15	C₂₃×#95	C₂²	1	4	yes	—
16	C₂₃×#95	C₂²	1	4	yes	—
17	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	—	4	2	no	—
19	C₂²×C₄×C₂₃	Q₈	1	4	yes	—
20	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
21	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
22	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
23	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
24	C₄⋊C₄×C₂₃	—	2	8	no	—
25	C₄⋊C₄×C₂₃	—	2	8	no	—
26	C₄⋊C₄×C₂₃	—	4	4	no	—
27	C₂²⋊C₄×C₂₃	—	2	8	no	—
28	C₂²⋊C₄×C₂₃	—	4	2	no	—
29	C₄²×C₂₃	—	4	2	no	—
30	C₄²×C₂₃	—	4	2	no	—
31	C₂³×C₂₃	—	2	4	no	—
32	C₂³×C₂₃	—	2	2	no	—
33	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
34	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
35	C₂×C₄×C₂₃	—	2	4	no	—
36	C₂×C₄×C₂₃	—	2	4	no	—
37	C₂×C₄×C₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	—	2	2	no	—
39	C₂×C₄×C₂₃	—	2	4	no	—
40	C₂×C₄×C₂₃	—	2	4	no	—
41	C₂×C₄×C₂₃	—	2	4	no	—
42	C₂×C₄×C₂₃	—	2	4	no	—
43	C₂×C₄×C₂₃	—	2	4	no	—
44	C₂×C₄×C₂₃	—	2	4	no	—
45	C₂×C₄×C₂₃	—	4	4	no	—
46	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
47	C₂×C₄×C₂₃	#41	1	8	yes	—
48	#2608	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂²×C₂₃	—	2	4	no	—
50	C₂²×C₂₃	—	2	2	no	—
51	C₂²×C₂₃	—	4	4	no	—
52	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
53	C₂²×C₂₃	C₂×#41	1	4	yes	—
54	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
55	C₄×C₂₃	—	2	4	no	—
56	C₄×C₂₃	—	2	8	no	—
57	C₄×C₂₃	—	4	4	no	—
58	C₄×C₂₃	—	4	4	no	—
59	#396	C₂×C₂₃	1	4	yes	—
60	#389	C₂×C₂₃	1	4	yes	complemented
61	#387	C₂×C₂₃	1	4	yes	complemented
62	#381	C₂×C₂₃	1	1	yes	characteristic
63	C₄×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
64	C₂×C₂₃	—	4	2	no	—
65	C₂×C₂₃	#533	1	2	yes	—
66	C₂×C₂₃	#543	1	1	yes	characteristic
67	C₂×C₂₃	#553	1	4	yes	—
68	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
69	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
70	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
71	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
72	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
73	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
74	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
75	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
76	#95	C₂²×C₂₃	1	4	yes	—
77	#95	C₂²×C₂₃	1	4	yes	—
78	#95	C₂²×C₂₃	1	4	yes	—
79	C₂₃	#2608	1	1	yes	characteristic, direct product factor, Sylow
80	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
81	C₂²×C₄	—	4	2	no	—
82	C₂²×C₄	Q₈×C₂₃	1	4	yes	—
83	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
84	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
85	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
86	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
87	C₄⋊C₄	—	2	8	no	—
88	C₄⋊C₄	—	2	8	no	—
89	C₄⋊C₄	—	4	4	no	—
90	C₂²⋊C₄	—	2	8	no	—
91	C₂²⋊C₄	—	4	2	no	—
92	C₄²	—	4	2	no	—
93	C₄²	—	4	2	no	—
94	C₂³	—	2	4	no	—
95	C₂³	—	2	2	no	—
96	C₂³	C₂×Q₈×C₂₃	1	2	yes	—
97	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	C₂×Q₈×C₂₃	1	4	yes	—
110	C₂×C₄	#41×C₂₃	1	8	yes	—
111	C₂²	—	2	4	no	—
112	C₂²	—	2	2	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
115	C₂²	C₂×#41×C₂₃	1	4	yes	—
116	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
117	C₄	—	2	8	no	—
118	C₄	—	2	4	no	—
119	C₄	—	4	4	no	—
120	C₄	—	4	4	no	—
121	C₂	—	4	2	no	—
122	C₂	C₂₃×#533	1	2	yes	—
123	C₂	C₂₃×#543	1	1	yes	characteristic
124	C₂	C₂₃×#553	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#396

C₂

yes

maximal

C₂₃×#389

C₂

yes

maximal, complemented

C₂₃×#387

C₂

yes

maximal, complemented

C₂₃×#381

C₂

yes

maximal, characteristic

C₄×C₂²⋊C₄×C₂₃

C₂

yes

maximal

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

—

C₂₃×#95

C₂²

yes

—

C₂₃×#95

C₂²

yes

—

C₂₃×#95

C₂²

yes

—

C₂⁴×C₂₃

C₂³

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

Q₈

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×Q₈

yes

—

C₂³×C₂₃

C₂⁴

yes

characteristic

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

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

#41

yes

—

#2608

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂²×Q₈

yes

characteristic

C₂²×C₂₃

C₂×#41

yes

—

C₂²×C₂₃

H(5,𝔽₂)

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

#396

C₂×C₂₃

yes

—

#389

C₂×C₂₃

yes

complemented

#387

C₂×C₂₃

yes

complemented

#381

C₂×C₂₃

yes

characteristic

C₄×C₂²⋊C₄

C₂×C₂₃

yes

—

C₂×C₂₃

—

C₂×C₂₃

#533

yes

—

C₂×C₂₃

#543

yes

characteristic

C₂×C₂₃

#553

yes

—

C₂³×C₄

C₂²×C₂₃

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₂₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

—

#95

C₂²×C₂₃

yes

—

#95

C₂²×C₂₃

yes

—

#95

C₂²×C₂₃

yes

—

C₂₃

#2608

yes

characteristic, direct product factor, Sylow

C₂⁴

C₂³×C₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

Q₈×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₂³

—

C₂³

—

C₂³

C₂×Q₈×C₂₃

yes

—

C₂³

C₂⁴×C₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×Q₈×C₂₃

yes

—

110

C₂×C₄

#41×C₂₃

yes

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂²×Q₈×C₂₃

yes

characteristic

115

C₂²

C₂×#41×C₂₃

yes

—

116

C₂²

C₂₃×H(5,𝔽₂)

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₂

—

122

C₂

C₂₃×#533

yes

—

123

C₂

C₂₃×#543

yes

characteristic

124

C₂

C₂₃×#553

yes

—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂³
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

C₂⁴×C₂₃

C₂³

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ G

N	H ≅ G/N	type
2. C₂₃×#389	C₂	semidirect
3. C₂₃×#387	C₂	semidirect
48. #2608	C₂₃	direct (N × H)
60. #389	C₂×C₂₃	semidirect
61. #387	C₂×C₂₃	semidirect
79. C₂₃	#2608	direct (N × H)

H ≅ G/N

type

2. C₂₃×#389

C₂

semidirect

3. C₂₃×#387

C₂

semidirect

48. #2608

C₂₃

direct (N × H)

60. #389

C₂×C₂₃

semidirect

61. #387

C₂×C₂₃

semidirect

79. C₂₃

#2608

direct (N × H)

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

subgroup	index
C₂₃×#396	2
C₂₃×#389	2
C₂₃×#387	2
C₂₃×#381	2
C₄×C₂²⋊C₄×C₂₃	2
#2608	23

subgroup

index

C₂₃×#396

C₂₃×#389

C₂₃×#387

C₂₃×#381

C₄×C₂²⋊C₄×C₂₃

#2608

Subgroups — C₂₃×#2608 #15670227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#2608 #15670227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2608 #15670227