Group #19155766 — subgroups

Subgroups — C₂₃×#8140 #19155766

Order 5888.

330 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 174 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×#1726	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1452	C₂	1	2	yes	maximal
3	C₂₃×#1275	C₂	1	2	yes	maximal
4	C₂₃×#1274	C₂	1	2	yes	maximal
5	C₂₃×#431	C₂²	1	4	yes	—
6	C₂×C₂₃×#97	C₄	1	2	yes	—
7	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×#376	C₂²	1	1	yes	characteristic
11	C₂₃×#322	C₄	1	8	yes	complemented
12	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
17	C₂₃×C₄⋊C₈	—	2	4	no	—
18	C₂₃×C₂²⋊C₈	—	2	4	no	—
19	C₂₃×#97	—	2	4	no	—
20	C₂₃×#97	—	2	2	no	—
21	C₄×C₈×C₂₃	—	2	4	no	—
22	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
23	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
24	C₂₃×#95	—	2	2	no	—
25	C₂²×C₄×C₂₃	—	2	2	no	—
26	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₃	#41	1	2	yes	—
28	C₂×C₈×C₂₃	—	2	2	no	—
29	C₂×C₈×C₂₃	—	2	2	no	—
30	C₂×C₈×C₂₃	—	2	2	no	—
31	C₂×C₈×C₂₃	—	2	2	no	—
32	C₂×C₈×C₂₃	—	2	8	no	—
33	C₂×C₈×C₂₃	C₄²	1	2	yes	—
34	C₂×C₈×C₂₃	C₄²	1	2	yes	—
35	C₂×C₈×C₂₃	M₁₆	1	4	yes	—
36	C₄²×C₂₃	—	2	2	no	—
37	C₄²×C₂₃	M₁₆	1	4	yes	—
38	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
39	C₄²×C₂₃	#41	1	2	yes	—
40	#8140	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂³×C₂₃	#117	1	1	yes	characteristic
42	C₂×C₄×C₂₃	—	2	4	no	—
43	C₂×C₄×C₂₃	—	4	2	no	—
44	C₂×C₄×C₂₃	—	4	4	no	—
45	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
46	C₂×C₄×C₂₃	#117	1	4	yes	—
47	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
48	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
49	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
50	C₈×C₂₃	—	2	2	no	—
51	C₈×C₂₃	—	2	2	no	—
52	C₈×C₂₃	—	2	4	no	—
53	C₈×C₂₃	—	4	8	no	—
54	#1726	C₂×C₂₃	1	1	yes	characteristic
55	#1452	C₂×C₂₃	1	2	yes	—
56	#1275	C₂×C₂₃	1	2	yes	—
57	#1274	C₂×C₂₃	1	2	yes	—
58	C₂²×C₂₃	—	2	2	no	—
59	C₂²×C₂₃	#376	1	1	yes	characteristic
60	C₂²×C₂₃	#432	1	2	yes	—
61	C₄×C₂₃	—	2	4	no	—
62	C₄×C₂₃	—	4	4	no	—
63	C₄×C₂₃	C₄×M₁₆	1	1	yes	characteristic
64	C₄×C₂₃	C₄×M₁₆	1	1	yes	characteristic
65	C₄×C₂₃	#432	1	2	yes	—
66	#431	C₂²×C₂₃	1	4	yes	—
67	C₂×#97	C₄×C₂₃	1	2	yes	—
68	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
69	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	—
70	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
71	#376	C₂²×C₂₃	1	1	yes	characteristic
72	#322	C₄×C₂₃	1	8	yes	complemented
73	C₂×C₂₃	—	2	2	no	—
74	C₂×C₂₃	#1745	1	1	yes	characteristic
75	C₂×C₂₃	#1747	1	2	yes	—
76	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
77	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
78	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
79	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
80	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
81	C₄⋊C₈	—	2	4	no	—
82	C₂²⋊C₈	—	2	4	no	—
83	#97	—	2	4	no	—
84	#97	—	2	2	no	—
85	C₄×C₈	—	2	4	no	—
86	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
87	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
88	#95	—	2	2	no	—
89	C₂₃	#8140	1	1	yes	characteristic, direct product factor, Sylow
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
92	C₂²×C₄	#41×C₂₃	1	2	yes	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	8	no	—
98	C₂×C₈	C₄²×C₂₃	1	2	yes	—
99	C₂×C₈	C₄²×C₂₃	1	2	yes	—
100	C₂×C₈	M₁₆×C₂₃	1	4	yes	—
101	C₄²	—	2	2	no	—
102	C₄²	M₁₆×C₂₃	1	4	yes	—
103	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
104	C₄²	#41×C₂₃	1	2	yes	—
105	C₂³	C₂₃×#117	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
110	C₂×C₄	C₂₃×#117	1	4	yes	—
111	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
112	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
113	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	2	2	no	—
116	C₈	—	2	4	no	—
117	C₈	—	4	8	no	—
118	C₂²	—	2	2	no	—
119	C₂²	C₂₃×#376	1	1	yes	characteristic
120	C₂²	C₂₃×#432	1	2	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	C₄×M₁₆×C₂₃	1	1	yes	characteristic
124	C₄	C₄×M₁₆×C₂₃	1	1	yes	characteristic
125	C₄	C₂₃×#432	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	C₂₃×#1745	1	1	yes	characteristic
128	C₂	C₂₃×#1747	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#1726

C₂

yes

maximal, characteristic

C₂₃×#1452

C₂

yes

maximal

C₂₃×#1275

C₂

yes

maximal

C₂₃×#1274

C₂

yes

maximal

C₂₃×#431

C₂²

yes

—

C₂×C₂₃×#97

C₄

yes

—

C₂×C₂₃×#97

C₂²

yes

characteristic

C₂×C₄×C₈×C₂₃

C₄

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#376

C₂²

yes

characteristic

C₂₃×#322

C₄

yes

complemented

C₂²×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₄²×C₂₃

C₂×C₄

yes

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂₃×C₂²⋊C₈

—

C₂₃×#97

—

C₂₃×#97

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

C₂×C₄

yes

—

C₄×C₈×C₂₃

C₂×C₄

yes

—

C₂₃×#95

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₂₃

#41

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₄²

yes

—

C₂×C₈×C₂₃

C₄²

yes

—

C₂×C₈×C₂₃

M₁₆

yes

—

C₄²×C₂₃

—

C₄²×C₂₃

M₁₆

yes

—

C₄²×C₂₃

C₂²×C₄

yes

—

C₄²×C₂₃

#41

yes

—

#8140

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#117

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₂₃

#117

yes

—

C₂×C₄×C₂₃

#117

yes

characteristic

C₂×C₄×C₂₃

C₂×M₁₆

yes

—

C₂×C₄×C₂₃

C₂×M₁₆

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

#1726

C₂×C₂₃

yes

characteristic

#1452

C₂×C₂₃

yes

—

#1275

C₂×C₂₃

yes

—

#1274

C₂×C₂₃

yes

—

C₂²×C₂₃

—

C₂²×C₂₃

#376

yes

characteristic

C₂²×C₂₃

#432

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×M₁₆

yes

characteristic

C₄×C₂₃

C₄×M₁₆

yes

characteristic

C₄×C₂₃

#432

yes

—

#431

C₂²×C₂₃

yes

—

C₂×#97

C₄×C₂₃

yes

—

C₂×#97

C₂²×C₂₃

yes

characteristic

C₂×C₄×C₈

C₄×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

#376

C₂²×C₂₃

yes

characteristic

#322

C₄×C₂₃

yes

complemented

C₂×C₂₃

—

C₂×C₂₃

#1745

yes

characteristic

C₂×C₂₃

#1747

yes

—

C₂²×C₈

C₂×C₄×C₂₃

yes

characteristic

C₂²×C₈

C₂×C₄×C₂₃

yes

characteristic

C₂²×C₈

C₂×C₄×C₂₃

yes

—

C₂×C₄²

C₂×C₄×C₂₃

yes

—

C₂×C₄²

C₂³×C₂₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×C₄×C₂₃

yes

—

C₄×C₈

C₂×C₄×C₂₃

yes

—

#95

—

C₂₃

#8140

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×C₄×C₂₃

yes

characteristic

C₂²×C₄

#41×C₂₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²×C₂₃

yes

—

C₂×C₈

C₄²×C₂₃

yes

—

100

C₂×C₈

M₁₆×C₂₃

yes

—

101

C₄²

—

102

C₄²

M₁₆×C₂₃

yes

—

103

C₄²

C₂²×C₄×C₂₃

yes

—

104

C₄²

#41×C₂₃

yes

—

105

C₂³

C₂₃×#117

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×C₄²×C₂₃

yes

characteristic

110

C₂×C₄

C₂₃×#117

yes

—

111

C₂×C₄

C₂₃×#117

yes

characteristic

112

C₂×C₄

C₂×M₁₆×C₂₃

yes

—

113

C₂×C₄

C₂×M₁₆×C₂₃

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₂²

—

119

C₂²

C₂₃×#376

yes

characteristic

120

C₂²

C₂₃×#432

yes

—

121

C₄

—

122

C₄

—

123

C₄

C₄×M₁₆×C₂₃

yes

characteristic

124

C₄

C₄×M₁₆×C₂₃

yes

characteristic

125

C₄

C₂₃×#432

yes

—

126

C₂

—

127

C₂

C₂₃×#1745

yes

characteristic

128

C₂

C₂₃×#1747

yes

—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#376
prime radical	C₂³×C₂₃	#117
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₂₃

C₂×C₄²

commutator

C₂×C₄

C₂×C₄²×C₂₃

Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

C₂²×C₂₃

C₂³×C₂₃

G ▷ C₂×C₄ ▷ C₁

C₁ ◁ C₂×C₄×C₂₃ ◁ G

N	H ≅ G/N	type
11. C₂₃×#322	C₄	semidirect
40. #8140	C₂₃	direct (N × H)
72. #322	C₄×C₂₃	semidirect
89. C₂₃	#8140	direct (N × H)

H ≅ G/N

type

11. C₂₃×#322

C₄

semidirect

40. #8140

C₂₃

direct (N × H)

72. #322

C₄×C₂₃

semidirect

89. C₂₃

#8140

direct (N × H)

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

subgroup	index
C₂₃×#1726	2
C₂₃×#1452	2
C₂₃×#1275	2
C₂₃×#1274	2
#8140	23

subgroup

index

C₂₃×#1726

C₂₃×#1452

C₂₃×#1275

C₂₃×#1274

#8140

Subgroups — C₂₃×#8140 #19155766

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#8140 #19155766

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8140 #19155766