Group #19160130 — subgroups

Subgroups — C₂₃×#12504 #19160130

Order 5888.

262 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2252	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2247	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2241	C₂	1	2	yes	maximal
4	C₂₃×#1707	C₂	1	2	yes	maximal
5	C₈×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×Q₃₂	—	2	2	no	—
7	C₂₃×#500	C₂²	1	2	yes	—
8	C₂₃×#498	—	2	2	no	—
9	C₂₃×#494	C₂²	1	2	yes	—
10	C₂₃×#487	—	2	2	no	—
11	C₂₃×#366	—	2	2	no	—
12	C₂₃×#358	—	2	4	no	—
13	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₈²×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₂₃	—	2	2	no	—
17	C₂×Q₁₆×C₂₃	—	2	2	no	—
18	C₂₃×C₄⋊Q₈	—	2	2	no	—
19	C₂₃×C₄⋊Q₈	—	2	2	no	—
20	C₂₃×Q₃₂	—	4	4	no	—
21	C₂×C₁₆×C₂₃	D₄	1	2	yes	—
22	C₂×C₁₆×C₂₃	D₄	1	2	yes	—
23	C₂₃×#107	—	2	2	no	—
24	C₂₃×#107	—	2	2	no	—
25	C₂₃×#106	—	4	4	no	—
26	C₂₃×#103	—	4	4	no	—
27	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₂₃	—	4	2	no	—
31	C₂×Q₈×C₂₃	—	4	2	no	—
32	Q₁₆×C₂₃	—	4	4	no	—
33	Q₁₆×C₂₃	—	4	4	no	—
34	C₂×C₈×C₂₃	D₈	1	2	yes	—
35	C₂×C₈×C₂₃	D₈	1	2	yes	—
36	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₂₃	—	4	2	no	—
39	C₄⋊C₄×C₂₃	—	4	2	no	—
40	C₄⋊C₄×C₂₃	—	4	2	no	—
41	C₄⋊C₄×C₂₃	—	4	2	no	—
42	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₂₃	—	2	2	no	—
44	C₁₆×C₂₃	SD₁₆	1	4	yes	—
45	#12504	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	Q₈×C₂₃	—	8	2	no	—
47	Q₈×C₂₃	—	8	2	no	—
48	Q₈×C₂₃	—	8	4	no	—
49	C₂×C₄×C₂₃	—	8	2	no	—
50	C₂×C₄×C₂₃	—	8	2	no	—
51	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
52	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
54	C₈×C₂₃	—	2	2	no	—
55	C₈×C₂₃	Q₃₂	1	4	yes	complemented
56	C₈×C₂₃	C₂×SD₁₆	1	2	yes	—
57	C₈×C₂₃	#135	1	2	yes	—
58	#2252	C₂×C₂₃	1	1	yes	characteristic
59	#2247	C₂×C₂₃	1	1	yes	characteristic
60	#2241	C₂×C₂₃	1	2	yes	—
61	#1707	C₂×C₂₃	1	2	yes	—
62	C₈×C₁₆	C₂×C₂₃	1	1	yes	characteristic
63	C₂²×C₂₃	#493	1	1	yes	characteristic
64	C₄×C₂₃	—	16	2	no	—
65	C₄×C₂₃	—	16	2	no	—
66	C₄×C₂₃	#492	1	1	yes	characteristic
67	C₄×C₂₃	#495	1	1	yes	characteristic
68	C₄×C₂₃	C₂×Q₃₂	1	2	yes	—
69	C₄×C₂₃	#508	1	2	yes	—
70	C₂×Q₃₂	—	2	2	no	—
71	#500	C₂²×C₂₃	1	2	yes	—
72	#498	—	2	2	no	—
73	#494	C₂²×C₂₃	1	2	yes	—
74	#487	—	2	2	no	—
75	#366	—	2	2	no	—
76	#358	—	2	4	no	—
77	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
78	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
79	C₈²	C₂²×C₂₃	1	1	yes	characteristic
80	C₂×C₂₃	#1706	1	1	yes	characteristic
81	C₂×C₂₃	#2247	1	1	yes	characteristic
82	C₂×C₂₃	#2249	1	1	yes	characteristic
83	C₂×Q₁₆	—	2	2	no	—
84	C₂×Q₁₆	—	2	2	no	—
85	C₄⋊Q₈	—	2	2	no	—
86	C₄⋊Q₈	—	2	2	no	—
87	Q₃₂	—	4	4	no	—
88	C₂×C₁₆	D₄×C₂₃	1	2	yes	—
89	C₂×C₁₆	D₄×C₂₃	1	2	yes	—
90	#107	—	2	2	no	—
91	#107	—	2	2	no	—
92	#106	—	4	4	no	—
93	#103	—	4	4	no	—
94	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
95	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
96	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
97	C₂₃	#12504	1	1	yes	characteristic, direct product factor, Sylow
98	C₂×Q₈	—	4	2	no	—
99	C₂×Q₈	—	4	2	no	—
100	Q₁₆	—	4	4	no	—
101	Q₁₆	—	4	4	no	—
102	C₂×C₈	D₈×C₂₃	1	2	yes	—
103	C₂×C₈	D₈×C₂₃	1	2	yes	—
104	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
105	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	SD₁₆×C₂₃	1	4	yes	—
113	Q₈	—	8	2	no	—
114	Q₈	—	8	2	no	—
115	Q₈	—	8	4	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	—	8	2	no	—
118	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
119	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
120	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	C₂₃×Q₃₂	1	4	yes	complemented
123	C₈	C₂×SD₁₆×C₂₃	1	2	yes	—
124	C₈	C₂₃×#135	1	2	yes	—
125	C₂²	C₂₃×#493	1	1	yes	characteristic
126	C₄	—	16	2	no	—
127	C₄	—	16	2	no	—
128	C₄	C₂₃×#492	1	1	yes	characteristic
129	C₄	C₂₃×#495	1	1	yes	characteristic
130	C₄	C₂×C₂₃×Q₃₂	1	2	yes	—
131	C₄	C₂₃×#508	1	2	yes	—
132	C₂	C₂₃×#1706	1	1	yes	characteristic
133	C₂	C₂₃×#2247	1	1	yes	characteristic
134	C₂	C₂₃×#2249	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2252

C₂

yes

maximal, characteristic

C₂₃×#2247

C₂

yes

maximal, characteristic

C₂₃×#2241

C₂

yes

maximal

C₂₃×#1707

C₂

yes

maximal

C₈×C₁₆×C₂₃

C₂

yes

maximal, characteristic

C₂×C₂₃×Q₃₂

—

C₂₃×#500

C₂²

yes

—

C₂₃×#498

—

C₂₃×#494

C₂²

yes

—

C₂₃×#487

—

C₂₃×#366

—

C₂₃×#358

—

C₄×C₁₆×C₂₃

C₂²

yes

characteristic

C₄×C₁₆×C₂₃

C₂²

yes

characteristic

C₈²×C₂₃

C₂²

yes

characteristic

C₂×Q₁₆×C₂₃

—

C₂×Q₁₆×C₂₃

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

—

C₂₃×Q₃₂

—

C₂×C₁₆×C₂₃

D₄

yes

—

C₂×C₁₆×C₂₃

D₄

yes

—

C₂₃×#107

—

C₂₃×#107

—

C₂₃×#106

—

C₂₃×#103

—

C₄×C₈×C₂₃

D₄

yes

characteristic

C₄×C₈×C₂₃

D₄

yes

characteristic

C₄×C₈×C₂₃

C₂³

yes

characteristic

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

Q₁₆×C₂₃

—

Q₁₆×C₂₃

—

C₂×C₈×C₂₃

D₈

yes

—

C₂×C₈×C₂₃

D₈

yes

—

C₂×C₈×C₂₃

C₂×D₄

yes

characteristic

C₂×C₈×C₂₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×D₄

yes

characteristic

C₁₆×C₂₃

—

C₁₆×C₂₃

SD₁₆

yes

—

#12504

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#127

yes

characteristic

C₂×C₄×C₂₃

C₂×D₈

yes

characteristic

C₂×C₄×C₂₃

C₂×D₈

yes

characteristic

C₈×C₂₃

—

C₈×C₂₃

Q₃₂

yes

complemented

C₈×C₂₃

C₂×SD₁₆

yes

—

C₈×C₂₃

#135

yes

—

#2252

C₂×C₂₃

yes

characteristic

#2247

C₂×C₂₃

yes

characteristic

#2241

C₂×C₂₃

yes

—

#1707

C₂×C₂₃

yes

—

C₈×C₁₆

C₂×C₂₃

yes

characteristic

C₂²×C₂₃

#493

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

C₂×Q₃₂

yes

—

C₄×C₂₃

#508

yes

—

C₂×Q₃₂

—

#500

C₂²×C₂₃

yes

—

#498

—

#494

C₂²×C₂₃

yes

—

#487

—

#366

—

#358

—

C₄×C₁₆

C₂²×C₂₃

yes

characteristic

C₄×C₁₆

C₂²×C₂₃

yes

characteristic

C₈²

C₂²×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

D₄×C₂₃

yes

—

C₂×C₁₆

D₄×C₂₃

yes

—

—

—

—

—

C₄×C₈

D₄×C₂₃

yes

characteristic

C₄×C₈

D₄×C₂₃

yes

characteristic

C₄×C₈

C₂³×C₂₃

yes

characteristic

C₂₃

#12504

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

—

102

C₂×C₈

D₈×C₂₃

yes

—

103

C₂×C₈

D₈×C₂₃

yes

—

104

C₂×C₈

C₂×D₄×C₂₃

yes

characteristic

105

C₂×C₈

C₂×D₄×C₂₃

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄²

C₂×D₄×C₂₃

yes

characteristic

111

C₁₆

—

112

C₁₆

SD₁₆×C₂₃

yes

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂₃×#127

yes

characteristic

119

C₂×C₄

C₂×D₈×C₂₃

yes

characteristic

120

C₂×C₄

C₂×D₈×C₂₃

yes

characteristic

121

C₈

—

122

C₈

C₂₃×Q₃₂

yes

complemented

123

C₈

C₂×SD₁₆×C₂₃

yes

—

124

C₈

C₂₃×#135

yes

—

125

C₂²

C₂₃×#493

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

C₂₃×#492

yes

characteristic

129

C₄

C₂₃×#495

yes

characteristic

130

C₄

C₂×C₂₃×Q₃₂

yes

—

131

C₄

C₂₃×#508

yes

—

132

C₂

C₂₃×#1706

yes

characteristic

133

C₂

C₂₃×#2247

yes

characteristic

134

C₂

C₂₃×#2249

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₃	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₂₃
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ C₈²×C₂₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₂₃

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₂₃

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ C₈²×C₂₃ ◁ G

N	H ≅ G/N	type
45. #12504	C₂₃	direct (N × H)
55. C₈×C₂₃	Q₃₂	semidirect
97. C₂₃	#12504	direct (N × H)
122. C₈	C₂₃×Q₃₂	semidirect

H ≅ G/N

type

45. #12504

C₂₃

direct (N × H)

55. C₈×C₂₃

Q₃₂

semidirect

97. C₂₃

#12504

direct (N × H)

122. C₈

C₂₃×Q₃₂

semidirect

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

subgroup	index
C₂₃×#2252	2
C₂₃×#2247	2
C₂₃×#2241	2
C₂₃×#1707	2
C₈×C₁₆×C₂₃	2
#12504	23

subgroup

index

C₂₃×#2252

C₂₃×#2247

C₂₃×#2241

C₂₃×#1707

C₈×C₁₆×C₂₃

#12504

Subgroups — C₂₃×#12504 #19160130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#12504 #19160130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12504 #19160130