Group #19160133 — subgroups

Subgroups — C₂₃×#12507 #19160133

Order 5888.

258 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2254	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	C₂₃×#1311	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×#367	—	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	2	yes	—
28	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₂₃	—	4	2	no	—
30	C₂×Q₈×C₂₃	—	4	2	no	—
31	Q₁₆×C₂₃	—	4	4	no	—
32	Q₁₆×C₂₃	—	4	4	no	—
33	C₂×C₈×C₂₃	D₈	1	4	yes	—
34	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₂₃	—	4	2	no	—
37	C₄⋊C₄×C₂₃	—	4	2	no	—
38	C₄⋊C₄×C₂₃	—	4	2	no	—
39	C₄⋊C₄×C₂₃	—	4	2	no	—
40	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₂₃	—	2	2	no	—
42	C₁₆×C₂₃	SD₁₆	1	4	yes	—
43	#12507	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	Q₈×C₂₃	—	8	2	no	—
45	Q₈×C₂₃	—	8	2	no	—
46	Q₈×C₂₃	—	8	4	no	—
47	C₂×C₄×C₂₃	—	8	2	no	—
48	C₂×C₄×C₂₃	—	8	2	no	—
49	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
50	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
51	C₈×C₂₃	—	2	4	no	—
52	C₈×C₂₃	C₂×SD₁₆	1	2	yes	—
53	C₈×C₂₃	#135	1	2	yes	—
54	#2254	C₂×C₂₃	1	1	yes	characteristic
55	#2247	C₂×C₂₃	1	1	yes	characteristic
56	#2241	C₂×C₂₃	1	2	yes	—
57	#1707	C₂×C₂₃	1	2	yes	—
58	#1311	C₂×C₂₃	1	1	yes	characteristic
59	C₂²×C₂₃	#493	1	1	yes	characteristic
60	C₄×C₂₃	—	16	2	no	—
61	C₄×C₂₃	—	16	2	no	—
62	C₄×C₂₃	#492	1	1	yes	characteristic
63	C₄×C₂₃	#495	1	1	yes	characteristic
64	C₄×C₂₃	#510	1	4	yes	—
65	C₂×Q₃₂	—	2	2	no	—
66	#500	C₂²×C₂₃	1	2	yes	—
67	#498	—	2	2	no	—
68	#494	C₂²×C₂₃	1	2	yes	—
69	#487	—	2	2	no	—
70	#367	—	2	2	no	—
71	#358	—	2	4	no	—
72	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
73	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
74	C₈²	C₂²×C₂₃	1	1	yes	characteristic
75	C₂×C₂₃	#1706	1	1	yes	characteristic
76	C₂×C₂₃	#2251	1	2	yes	—
77	C₂×Q₁₆	—	2	2	no	—
78	C₂×Q₁₆	—	2	2	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	—	2	2	no	—
81	Q₃₂	—	4	4	no	—
82	C₂×C₁₆	D₄×C₂₃	1	2	yes	—
83	C₂×C₁₆	D₄×C₂₃	1	2	yes	—
84	#107	—	2	2	no	—
85	#107	—	2	2	no	—
86	#106	—	4	4	no	—
87	#103	—	4	4	no	—
88	C₄×C₈	D₄×C₂₃	1	2	yes	—
89	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
90	C₂₃	#12507	1	1	yes	characteristic, direct product factor, Sylow
91	C₂×Q₈	—	4	2	no	—
92	C₂×Q₈	—	4	2	no	—
93	Q₁₆	—	4	4	no	—
94	Q₁₆	—	4	4	no	—
95	C₂×C₈	D₈×C₂₃	1	4	yes	—
96	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
97	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	SD₁₆×C₂₃	1	4	yes	—
105	Q₈	—	8	2	no	—
106	Q₈	—	8	2	no	—
107	Q₈	—	8	4	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	—	8	2	no	—
110	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
111	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	C₂×SD₁₆×C₂₃	1	2	yes	—
114	C₈	C₂₃×#135	1	2	yes	—
115	C₂²	C₂₃×#493	1	1	yes	characteristic
116	C₄	—	16	2	no	—
117	C₄	—	16	2	no	—
118	C₄	C₂₃×#492	1	1	yes	characteristic
119	C₄	C₂₃×#495	1	1	yes	characteristic
120	C₄	C₂₃×#510	1	4	yes	—
121	C₂	C₂₃×#1706	1	1	yes	characteristic
122	C₂	C₂₃×#2251	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2254

C₂

yes

maximal, characteristic

C₂₃×#2247

C₂

yes

maximal, characteristic

C₂₃×#2241

C₂

yes

maximal

C₂₃×#1707

C₂

yes

maximal

C₂₃×#1311

C₂

yes

maximal, characteristic

C₂×C₂₃×Q₃₂

—

C₂₃×#500

C₂²

yes

—

C₂₃×#498

—

C₂₃×#494

C₂²

yes

—

C₂₃×#487

—

C₂₃×#367

—

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

—

C₄×C₈×C₂₃

C₂³

yes

characteristic

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

Q₁₆×C₂₃

—

Q₁₆×C₂₃

—

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

—

#12507

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

—

C₈×C₂₃

—

C₈×C₂₃

C₂×SD₁₆

yes

—

C₈×C₂₃

#135

yes

—

#2254

C₂×C₂₃

yes

characteristic

#2247

C₂×C₂₃

yes

characteristic

#2241

C₂×C₂₃

yes

—

#1707

C₂×C₂₃

yes

—

#1311

C₂×C₂₃

yes

characteristic

C₂²×C₂₃

#493

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

C₂×Q₃₂

—

#500

C₂²×C₂₃

yes

—

#498

—

#494

C₂²×C₂₃

yes

—

#487

—

#367

—

#358

—

C₄×C₁₆

C₂²×C₂₃

yes

characteristic

C₄×C₁₆

C₂²×C₂₃

yes

characteristic

C₈²

C₂²×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

—

C₄×C₈

C₂³×C₂₃

yes

characteristic

C₂₃

#12507

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

D₈×C₂₃

yes

—

C₂×C₈

C₂×D₄×C₂₃

yes

characteristic

C₂×C₈

C₂×D₄×C₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

C₂×D₄×C₂₃

yes

characteristic

103

C₁₆

—

104

C₁₆

SD₁₆×C₂₃

yes

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂₃×#127

yes

characteristic

111

C₂×C₄

C₂×D₈×C₂₃

yes

—

112

C₈

—

113

C₈

C₂×SD₁₆×C₂₃

yes

—

114

C₈

C₂₃×#135

yes

—

115

C₂²

C₂₃×#493

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

C₂₃×#492

yes

characteristic

119

C₄

C₂₃×#495

yes

characteristic

120

C₄

C₂₃×#510

yes

—

121

C₂

C₂₃×#1706

yes

characteristic

122

C₂

C₂₃×#2251

yes

—

	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
43. #12507	C₂₃	direct (N × H)
90. C₂₃	#12507	direct (N × H)

H ≅ G/N

type

43. #12507

C₂₃

direct (N × H)

90. C₂₃

#12507

direct (N × H)

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

subgroup	index
C₂₃×#2254	2
C₂₃×#2247	2
C₂₃×#2241	2
C₂₃×#1707	2
C₂₃×#1311	2
#12507	23

subgroup

index

C₂₃×#2254

C₂₃×#2247

C₂₃×#2241

C₂₃×#1707

C₂₃×#1311

#12507

Subgroups — C₂₃×#12507 #19160133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#12507 #19160133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12507 #19160133