Group #19157837 — subgroups

Subgroups — C₂₃×#10211 #19157837

Order 5888.

406 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3237	C₂	1	3	yes	maximal
2	C₂₃×#1515	C₂	1	3	yes	maximal
3	C₂₃×#1308	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#533	C₂²	1	3	yes	—
5	C₂₃×#501	—	2	3	no	—
6	C₂₃×#491	—	2	3	no	—
7	C₂₃×#485	—	2	6	no	—
8	C₂₃×#477	—	2	3	no	—
9	C₂₃×#475	—	2	3	no	—
10	C₂₃×#432	C₂²	1	3	yes	—
11	C₂₃×#383	C₂²	1	1	yes	characteristic
12	C₂₃×#328	—	2	6	no	—
13	C₂₃×C₄⋊Q₈	D₄	1	3	yes	—
14	C₂₃×C₄⋊Q₈	D₄	1	3	yes	—
15	C₂₃×#125	—	2	3	no	—
16	C₂₃×C₂²⋊Q₈	—	2	3	no	—
17	C₂₃×C₂²⋊Q₈	—	2	3	no	—
18	C₄×Q₈×C₂₃	—	2	3	no	—
19	C₄×Q₈×C₂₃	—	2	3	no	—
20	C₂₃×#117	—	2	3	no	—
21	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
22	C₂₃×#107	—	2	6	no	—
23	C₂₃×#106	—	2	6	no	—
24	C₂₃×C₄⋊C₈	—	2	3	no	—
25	C₂₃×#103	—	2	6	no	—
26	C₂₃×#103	—	2	6	no	—
27	C₂₃×C₂²⋊C₈	D₄	1	6	yes	—
28	C₂₃×#97	—	2	3	no	—
29	C₂₃×#95	—	2	3	no	—
30	C₂×Q₈×C₂₃	—	2	3	no	—
31	C₂×Q₈×C₂₃	—	2	3	no	—
32	C₂²×C₄×C₂₃	—	4	1	no	—
33	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
34	C₂×C₈×C₂₃	—	2	6	no	—
35	C₄⋊C₄×C₂₃	—	2	3	no	—
36	C₄⋊C₄×C₂₃	—	2	3	no	—
37	C₄⋊C₄×C₂₃	—	2	3	no	—
38	C₄⋊C₄×C₂₃	—	2	3	no	—
39	C₄⋊C₄×C₂₃	—	4	3	no	—
40	C₄⋊C₄×C₂₃	—	4	3	no	—
41	C₂²⋊C₄×C₂₃	—	4	3	no	—
42	C₄²×C₂₃	—	4	3	no	—
43	C₄²×C₂₃	C₂×D₄	1	3	yes	—
44	C₄²×C₂₃	#41	1	1	yes	characteristic
45	#10211	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂³×C₂₃	#127	1	1	yes	characteristic
47	Q₈×C₂₃	—	4	6	no	—
48	Q₈×C₂₃	—	8	3	no	—
49	C₂×C₄×C₂₃	—	4	3	no	—
50	C₂×C₄×C₂₃	—	4	3	no	—
51	C₂×C₄×C₂₃	—	4	3	no	—
52	C₂×C₄×C₂₃	—	8	3	no	—
53	C₂×C₄×C₂₃	#120	1	3	yes	—
54	C₂×C₄×C₂₃	#121	1	3	yes	—
55	C₈×C₂₃	—	4	6	no	—
56	#3237	C₂×C₂₃	1	3	yes	—
57	#1515	C₂×C₂₃	1	3	yes	—
58	#1308	C₂×C₂₃	1	1	yes	characteristic
59	C₂²×C₂₃	—	2	3	no	—
60	C₂²×C₂₃	#392	1	1	yes	characteristic
61	C₄×C₂₃	—	2	3	no	—
62	C₄×C₂₃	—	2	3	no	—
63	C₄×C₂₃	—	8	3	no	—
64	C₄×C₂₃	—	8	3	no	—
65	C₄×C₂₃	—	16	1	no	—
66	#533	C₂²×C₂₃	1	3	yes	—
67	#501	—	2	3	no	—
68	#491	—	2	3	no	—
69	#485	—	2	6	no	—
70	#477	—	2	3	no	—
71	#475	—	2	3	no	—
72	#432	C₂²×C₂₃	1	3	yes	—
73	#383	C₂²×C₂₃	1	1	yes	characteristic
74	#328	—	2	6	no	—
75	C₂×C₂₃	—	4	1	no	—
76	C₂×C₂₃	#2009	1	3	yes	—
77	C₄⋊Q₈	D₄×C₂₃	1	3	yes	—
78	C₄⋊Q₈	D₄×C₂₃	1	3	yes	—
79	#125	—	2	3	no	—
80	C₂²⋊Q₈	—	2	3	no	—
81	C₂²⋊Q₈	—	2	3	no	—
82	C₄×Q₈	—	2	3	no	—
83	C₄×Q₈	—	2	3	no	—
84	#117	—	2	3	no	—
85	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
86	#107	—	2	6	no	—
87	#106	—	2	6	no	—
88	C₄⋊C₈	—	2	3	no	—
89	#103	—	2	6	no	—
90	#103	—	2	6	no	—
91	C₂²⋊C₈	D₄×C₂₃	1	6	yes	—
92	#97	—	2	3	no	—
93	#95	—	2	3	no	—
94	C₂₃	#10211	1	1	yes	characteristic, direct product factor, Sylow
95	C₂×Q₈	—	2	3	no	—
96	C₂×Q₈	—	2	3	no	—
97	C₂²×C₄	—	4	1	no	—
98	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
99	C₂×C₈	—	2	6	no	—
100	C₄⋊C₄	—	2	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	2	3	no	—
104	C₄⋊C₄	—	4	3	no	—
105	C₄⋊C₄	—	4	3	no	—
106	C₂²⋊C₄	—	4	3	no	—
107	C₄²	—	4	3	no	—
108	C₄²	C₂×D₄×C₂₃	1	3	yes	—
109	C₄²	#41×C₂₃	1	1	yes	characteristic
110	C₂³	C₂₃×#127	1	1	yes	characteristic
111	Q₈	—	4	6	no	—
112	Q₈	—	8	3	no	—
113	C₂×C₄	—	4	3	no	—
114	C₂×C₄	—	4	3	no	—
115	C₂×C₄	—	4	3	no	—
116	C₂×C₄	—	8	3	no	—
117	C₂×C₄	C₂₃×#120	1	3	yes	—
118	C₂×C₄	C₂₃×#121	1	3	yes	—
119	C₈	—	4	6	no	—
120	C₂²	—	2	3	no	—
121	C₂²	C₂₃×#392	1	1	yes	characteristic
122	C₄	—	2	3	no	—
123	C₄	—	2	3	no	—
124	C₄	—	8	3	no	—
125	C₄	—	8	3	no	—
126	C₄	—	16	1	no	—
127	C₂	—	4	1	no	—
128	C₂	C₂₃×#2009	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#3237

C₂

yes

maximal

C₂₃×#1515

C₂

yes

maximal

C₂₃×#1308

C₂

yes

maximal, characteristic

C₂₃×#533

C₂²

yes

—

C₂₃×#501

—

C₂₃×#491

—

C₂₃×#485

—

C₂₃×#477

—

C₂₃×#475

—

C₂₃×#432

C₂²

yes

—

C₂₃×#383

C₂²

yes

characteristic

C₂₃×#328

—

C₂₃×C₄⋊Q₈

D₄

yes

—

C₂₃×C₄⋊Q₈

D₄

yes

—

C₂₃×#125

—

C₂₃×C₂²⋊Q₈

—

C₂₃×C₂²⋊Q₈

—

C₄×Q₈×C₂₃

—

C₄×Q₈×C₂₃

—

C₂₃×#117

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×#107

—

C₂₃×#106

—

C₂₃×C₄⋊C₈

—

C₂₃×#103

—

C₂₃×#103

—

C₂₃×C₂²⋊C₈

D₄

yes

—

C₂₃×#97

—

C₂₃×#95

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂×D₄

yes

—

C₄²×C₂₃

#41

yes

characteristic

#10211

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#127

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#120

yes

—

C₂×C₄×C₂₃

#121

yes

—

C₈×C₂₃

—

#3237

C₂×C₂₃

yes

—

#1515

C₂×C₂₃

yes

—

#1308

C₂×C₂₃

yes

characteristic

C₂²×C₂₃

—

C₂²×C₂₃

#392

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

#533

C₂²×C₂₃

yes

—

—

—

—

—

—

C₂²×C₂₃

yes

—

#383

C₂²×C₂₃

yes

characteristic

#328

—

C₂×C₂₃

—

C₂×C₂₃

#2009

yes

—

C₄⋊Q₈

D₄×C₂₃

yes

—

C₄⋊Q₈

D₄×C₂₃

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂³×C₂₃

yes

characteristic

#107

—

#106

—

C₄⋊C₈

—

#103

—

#103

—

C₂²⋊C₈

D₄×C₂₃

yes

—

#97

—

#95

—

C₂₃

#10211

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₂₃

yes

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₄²

C₂×D₄×C₂₃

yes

—

109

C₄²

#41×C₂₃

yes

characteristic

110

C₂³

C₂₃×#127

yes

characteristic

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂₃×#120

yes

—

118

C₂×C₄

C₂₃×#121

yes

—

119

C₈

—

120

C₂²

—

121

C₂²

C₂₃×#392

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₂

—

128

C₂

C₂₃×#2009

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂³×C₂₃	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#392

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

prime radical

C₂³×C₂₃

#127

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
45. #10211	C₂₃	direct (N × H)
94. C₂₃	#10211	direct (N × H)

H ≅ G/N

type

45. #10211

C₂₃

direct (N × H)

94. C₂₃

#10211

direct (N × H)

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

subgroup	index
C₂₃×#3237	2
C₂₃×#1515	2
C₂₃×#1308	2
#10211	23

subgroup

index

C₂₃×#3237

C₂₃×#1515

C₂₃×#1308

#10211

Subgroups — C₂₃×#10211 #19157837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#10211 #19157837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10211 #19157837