Group #15669645 — subgroups

Subgroups — C₂₃×#2026 #15669645

Order 2944.

258 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 100 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×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#103	C₂	1	2	yes	maximal
3	C₂₃×#391	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	C₂₃×#125	—	2	4	no	—
9	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
10	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#103	—	2	4	no	—
14	C₂₃×#95	—	2	2	no	—
15	C₂×Q₈×C₂₃	—	2	1	no	—
16	C₂×Q₈×C₂₃	—	2	2	no	—
17	C₂×Q₈×C₂₃	—	2	2	no	—
18	C₂×Q₈×C₂₃	Q₈	1	2	yes	—
19	C₂²×C₄×C₂₃	—	2	2	no	—
20	C₂²×C₄×C₂₃	—	2	1	no	—
21	C₂²×C₄×C₂₃	—	2	1	no	—
22	C₂²×C₄×C₂₃	D₄	1	2	yes	—
23	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₂₃	—	2	2	no	—
25	C₂×C₈×C₂₃	—	2	2	no	—
26	C₂×C₈×C₂₃	—	2	2	no	—
27	C₄⋊C₄×C₂₃	—	2	8	no	—
28	C₄⋊C₄×C₂₃	—	2	2	no	—
29	C₄⋊C₄×C₂₃	—	2	2	no	—
30	C₄⋊C₄×C₂₃	D₄	1	4	yes	—
31	C₄²×C₂₃	—	2	2	no	—
32	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
33	Q₈×C₂₃	—	2	4	no	—
34	Q₈×C₂₃	—	4	2	no	—
35	C₂×C₄×C₂₃	—	2	2	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	2	no	—
40	C₂×C₄×C₂₃	—	4	1	no	—
41	C₂×C₄×C₂₃	—	4	1	no	—
42	C₂×C₄×C₂₃	—	4	2	no	—
43	C₂×C₄×C₂₃	—	4	2	no	—
44	C₂×C₄×C₂₃	—	4	1	no	—
45	C₂×C₄×C₂₃	—	4	2	no	—
46	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₂₃	#41	1	2	yes	—
49	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
50	C₈×C₂₃	—	4	2	no	—
51	#2026	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂²×C₂₃	#120	1	1	yes	characteristic
53	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₂₃	#124	1	1	yes	characteristic
55	C₂²×C₂₃	#135	1	2	yes	—
56	C₂²×C₂₃	#137	1	2	yes	—
57	C₄×C₂₃	—	4	2	no	—
58	C₄×C₂₃	—	4	2	no	—
59	C₄×C₂₃	—	4	4	no	—
60	C₄×C₂₃	—	8	1	no	—
61	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
62	C₄×C₂₃	#124	1	2	yes	—
63	C₂×#125	C₂×C₂₃	1	1	yes	characteristic
64	C₂×#103	C₂×C₂₃	1	2	yes	—
65	#391	C₂×C₂₃	1	1	yes	characteristic
66	#340	C₂×C₂₃	1	2	yes	—
67	#336	C₂×C₂₃	1	1	yes	characteristic
68	C₂×C₂₃	#393	1	1	yes	characteristic
69	C₂×C₂₃	#452	1	2	yes	—
70	C₂×C₂₃	#479	1	2	yes	—
71	C₂×C₂₃	#488	1	1	yes	characteristic
72	C₂×C₂₃	#491	1	1	yes	characteristic
73	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
74	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
75	#125	—	2	4	no	—
76	C₂×C₄⋊C₄	—	2	2	no	—
77	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
78	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
79	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
80	#103	—	2	4	no	—
81	#95	—	2	2	no	—
82	C₂₃	#2026	1	1	yes	characteristic, direct product factor, Sylow
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	—	2	2	no	—
86	C₂×Q₈	Q₈×C₂₃	1	2	yes	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	D₄×C₂₃	1	2	yes	—
91	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	D₄×C₂₃	1	4	yes	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
101	Q₈	—	2	4	no	—
102	Q₈	—	4	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
115	C₂×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
116	C₂×C₄	#41×C₂₃	1	2	yes	—
117	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
118	C₈	—	4	2	no	—
119	C₂²	C₂₃×#120	1	1	yes	characteristic
120	C₂²	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
121	C₂²	C₂₃×#124	1	1	yes	characteristic
122	C₂²	C₂₃×#135	1	2	yes	—
123	C₂²	C₂₃×#137	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	4	2	no	—
126	C₄	—	4	4	no	—
127	C₄	—	8	1	no	—
128	C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
129	C₄	C₂₃×#124	1	2	yes	—
130	C₂	C₂₃×#393	1	1	yes	characteristic
131	C₂	C₂₃×#452	1	2	yes	—
132	C₂	C₂₃×#479	1	2	yes	—
133	C₂	C₂₃×#488	1	1	yes	characteristic
134	C₂	C₂₃×#491	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×#125

C₂

yes

maximal, characteristic

C₂×C₂₃×#103

C₂

yes

maximal

C₂₃×#391

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

—

C₂₃×#125

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂₃×#103

—

C₂₃×#95

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

Q₈

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

—

C₄⋊C₄×C₂₃

D₄

yes

—

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

—

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

characteristic

C₈×C₂₃

—

#2026

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

#120

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

#137

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

#124

yes

—

C₂×#125

C₂×C₂₃

yes

characteristic

C₂×#103

C₂×C₂₃

yes

—

#391

C₂×C₂₃

yes

characteristic

#340

C₂×C₂₃

yes

—

#336

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#479

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×Q₈

C₂²×C₂₃

yes

characteristic

C₂²×C₈

C₂²×C₂₃

yes

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

characteristic

#103

—

#95

—

C₂₃

#2026

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Q₈×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

—

C₄⋊C₄

D₄×C₂₃

yes

—

C₄²

—

100

C₂³

C₂×D₄×C₂₃

yes

characteristic

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×D₄×C₂₃

yes

—

115

C₂×C₄

C₂×Q₈×C₂₃

yes

characteristic

116

C₂×C₄

#41×C₂₃

yes

—

117

C₂×C₄

#41×C₂₃

yes

characteristic

118

C₈

—

119

C₂²

C₂₃×#120

yes

characteristic

120

C₂²

C₂₃×C₂²⋊Q₈

yes

characteristic

121

C₂²

C₂₃×#124

yes

characteristic

122

C₂²

C₂₃×#135

yes

—

123

C₂²

C₂₃×#137

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

C₂₃×C₂²⋊Q₈

yes

—

129

C₄

C₂₃×#124

yes

—

130

C₂

C₂₃×#393

yes

characteristic

131

C₂

C₂₃×#452

yes

—

132

C₂

C₂₃×#479

yes

—

133

C₂

C₂₃×#488

yes

characteristic

134

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
51. #2026	C₂₃	direct (N × H)
82. C₂₃	#2026	direct (N × H)

H ≅ G/N

type

51. #2026

C₂₃

direct (N × H)

82. C₂₃

#2026

direct (N × H)

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

subgroup	index
C₂×C₂₃×#125	2
C₂×C₂₃×#103	2
C₂₃×#391	2
C₂₃×#340	2
C₂₃×#336	2
#2026	23

subgroup

index

C₂×C₂₃×#125

C₂×C₂₃×#103

C₂₃×#391

C₂₃×#340

C₂₃×#336

#2026

Subgroups — C₂₃×#2026 #15669645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#2026 #15669645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2026 #15669645