Group #15669693 — subgroups

Subgroups — C₂₃×#2074 #15669693

Order 2944.

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

normal subgroups only

This group has 222 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	C₂×C₂₃×#103	C₂	1	2	yes	maximal
2	C₂₃×#391	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#382	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#340	C₂	1	2	yes	maximal
5	C₂₃×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#103	—	2	4	no	—
12	C₂₃×#95	—	2	2	no	—
13	C₂₃×#95	—	2	2	no	—
14	C₂×Q₈×C₂₃	—	2	1	no	—
15	C₂×Q₈×C₂₃	—	2	2	no	—
16	C₂×Q₈×C₂₃	—	2	2	no	—
17	C₂×Q₈×C₂₃	D₄	1	2	yes	—
18	C₂²×C₄×C₂₃	—	2	1	no	—
19	C₂²×C₄×C₂₃	—	2	2	no	—
20	C₂²×C₄×C₂₃	—	2	1	no	—
21	C₂²×C₄×C₂₃	D₄	1	2	yes	—
22	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₂₃	—	2	2	no	—
24	C₂×C₈×C₂₃	—	2	2	no	—
25	C₂×C₈×C₂₃	—	2	2	no	—
26	C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₄⋊C₄×C₂₃	—	2	2	no	—
28	C₄⋊C₄×C₂₃	—	2	2	no	—
29	C₄²×C₂₃	—	2	2	no	—
30	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₂₃	—	2	4	no	—
32	Q₈×C₂₃	—	4	2	no	—
33	C₂×C₄×C₂₃	—	2	2	no	—
34	C₂×C₄×C₂₃	—	2	2	no	—
35	C₂×C₄×C₂₃	—	2	2	no	—
36	C₂×C₄×C₂₃	—	2	2	no	—
37	C₂×C₄×C₂₃	—	4	1	no	—
38	C₂×C₄×C₂₃	—	4	1	no	—
39	C₂×C₄×C₂₃	—	4	2	no	—
40	C₂×C₄×C₂₃	—	4	1	no	—
41	C₂×C₄×C₂₃	—	4	2	no	—
42	C₂×C₄×C₂₃	—	4	2	no	—
43	C₂×C₄×C₂₃	—	4	2	no	—
44	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	#41	1	2	yes	—
46	C₂×C₄×C₂₃	#41	1	2	yes	—
47	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
48	C₈×C₂₃	—	4	2	no	—
49	#2074	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂²×C₂₃	#121	1	1	yes	characteristic
51	C₂²×C₂₃	#123	1	1	yes	characteristic
52	C₂²×C₂₃	#124	1	1	yes	characteristic
53	C₂²×C₂₃	#135	1	2	yes	—
54	C₂²×C₂₃	#137	1	2	yes	—
55	C₄×C₂₃	—	4	2	no	—
56	C₄×C₂₃	—	4	2	no	—
57	C₄×C₂₃	—	8	2	no	—
58	C₄×C₂₃	—	8	1	no	—
59	C₄×C₂₃	#123	1	2	yes	—
60	C₄×C₂₃	#126	1	2	yes	—
61	C₂×#103	C₂×C₂₃	1	2	yes	—
62	#391	C₂×C₂₃	1	1	yes	characteristic
63	#382	C₂×C₂₃	1	1	yes	characteristic
64	#340	C₂×C₂₃	1	2	yes	—
65	#336	C₂×C₂₃	1	1	yes	characteristic
66	C₂×C₂₃	#397	1	1	yes	characteristic
67	C₂×C₂₃	#464	1	2	yes	—
68	C₂×C₂₃	#485	1	2	yes	—
69	C₂×C₂₃	#488	1	1	yes	characteristic
70	C₂×C₂₃	#491	1	1	yes	characteristic
71	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
72	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
73	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
74	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
75	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
76	#103	—	2	4	no	—
77	#95	—	2	2	no	—
78	#95	—	2	2	no	—
79	C₂₃	#2074	1	1	yes	characteristic, direct product factor, Sylow
80	C₂×Q₈	—	2	1	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	D₄×C₂₃	1	2	yes	—
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	D₄×C₂₃	1	2	yes	—
88	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄²	—	2	2	no	—
96	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
97	Q₈	—	2	4	no	—
98	Q₈	—	4	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
111	C₂×C₄	#41×C₂₃	1	2	yes	—
112	C₂×C₄	#41×C₂₃	1	2	yes	—
113	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₂²	C₂₃×#121	1	1	yes	characteristic
116	C₂²	C₂₃×#123	1	1	yes	characteristic
117	C₂²	C₂₃×#124	1	1	yes	characteristic
118	C₂²	C₂₃×#135	1	2	yes	—
119	C₂²	C₂₃×#137	1	2	yes	—
120	C₄	—	4	2	no	—
121	C₄	—	4	2	no	—
122	C₄	—	8	2	no	—
123	C₄	—	8	1	no	—
124	C₄	C₂₃×#123	1	2	yes	—
125	C₄	C₂₃×#126	1	2	yes	—
126	C₂	C₂₃×#397	1	1	yes	characteristic
127	C₂	C₂₃×#464	1	2	yes	—
128	C₂	C₂₃×#485	1	2	yes	—
129	C₂	C₂₃×#488	1	1	yes	characteristic
130	C₂	C₂₃×#491	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×#103

C₂

yes

maximal

C₂₃×#391

C₂

yes

maximal, characteristic

C₂₃×#382

C₂

yes

maximal, characteristic

C₂₃×#340

C₂

yes

maximal

C₂₃×#336

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂₃×#103

—

C₂₃×#95

—

C₂₃×#95

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₂³×C₂₃

C₂×D₄

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

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

characteristic

C₈×C₂₃

—

#2074

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

#137

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#123

yes

—

C₄×C₂₃

#126

yes

—

C₂×#103

C₂×C₂₃

yes

—

#391

C₂×C₂₃

yes

characteristic

#382

C₂×C₂₃

yes

characteristic

#340

C₂×C₂₃

yes

—

#336

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#485

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×Q₈

C₂²×C₂₃

yes

characteristic

C₂²×C₈

C₂²×C₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

—

—

—

C₂₃

#2074

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₄×C₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₂³

C₂×D₄×C₂₃

yes

characteristic

Q₈

—

Q₈

—

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

—

110

C₂×C₄

C₂×D₄×C₂₃

yes

characteristic

111

C₂×C₄

#41×C₂₃

yes

—

112

C₂×C₄

#41×C₂₃

yes

—

113

C₂×C₄

#41×C₂₃

yes

characteristic

114

C₈

—

115

C₂²

C₂₃×#121

yes

characteristic

116

C₂²

C₂₃×#123

yes

characteristic

117

C₂²

C₂₃×#124

yes

characteristic

118

C₂²

C₂₃×#135

yes

—

119

C₂²

C₂₃×#137

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

C₂₃×#123

yes

—

125

C₄

C₂₃×#126

yes

—

126

C₂

C₂₃×#397

yes

characteristic

127

C₂

C₂₃×#464

yes

—

128

C₂

C₂₃×#485

yes

—

129

C₂

C₂₃×#488

yes

characteristic

130

C₂

C₂₃×#491

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₃

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₂₃

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
49. #2074	C₂₃	direct (N × H)
79. C₂₃	#2074	direct (N × H)

H ≅ G/N

type

49. #2074

C₂₃

direct (N × H)

79. C₂₃

#2074

direct (N × H)

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

subgroup	index
C₂×C₂₃×#103	2
C₂₃×#391	2
C₂₃×#382	2
C₂₃×#340	2
C₂₃×#336	2
#2074	23

subgroup

index

C₂×C₂₃×#103

C₂₃×#391

C₂₃×#382

C₂₃×#340

C₂₃×#336

#2074

Subgroups — C₂₃×#2074 #15669693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#2074 #15669693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2074 #15669693