Group #19180941 — subgroups

Subgroups — C₂₃×#33315 #19180941

Order 5888.

806 conjugacy classes of subgroups; only the 382 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 382 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×#499	C₂	1	4	yes	maximal
2	C₂₃×#2859	C₂	1	2	yes	maximal
3	C₂₃×#1847	C₂	1	8	yes	maximal
4	C₄×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×#125	C₂²	1	4	yes	—
6	C₂₃×#499	—	2	16	no	—
7	C₂×C₂₃×#107	C₂²	1	8	yes	—
8	C₂×C₂₃×#106	C₂²	1	8	yes	—
9	C₂×C₂₃×#97	C₂²	1	4	yes	—
10	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
11	C₂₃×#389	—	2	8	no	—
12	C₂₃×#384	C₂²	1	4	yes	—
13	C₂₃×#382	C₂²	1	4	yes	—
14	C₄³×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
16	C₂₃×#125	—	2	16	no	—
17	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
18	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
19	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
20	C₂×C₄²×C₂₃	D₄	1	4	yes	—
21	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
23	C₂₃×#107	—	2	16	no	—
24	C₂₃×#106	—	2	16	no	—
25	C₂₃×#97	Q₈	1	16	yes	—
26	C₄×C₈×C₂₃	D₄	1	8	yes	—
27	C₂₃×#95	—	2	8	no	—
28	C₂²×C₄×C₂₃	—	2	8	no	—
29	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
30	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
31	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
32	C₂×C₈×C₂₃	C₂×D₄	1	8	yes	—
33	C₂×C₈×C₂₃	C₂×Q₈	1	16	yes	—
34	C₄⋊C₄×C₂₃	—	2	16	no	—
35	C₄⋊C₄×C₂₃	—	4	16	no	—
36	C₄⋊C₄×C₂₃	—	4	16	no	—
37	C₄²×C₂₃	—	4	4	no	—
38	C₄²×C₂₃	C₂×D₄	1	4	yes	—
39	C₄²×C₂₃	C₂×Q₈	1	8	yes	—
40	#33315	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	—	2	8	no	—
43	C₂×C₄×C₂₃	—	2	4	no	—
44	C₂×C₄×C₂₃	—	4	8	no	—
45	C₂×C₄×C₂₃	—	4	16	no	—
46	C₂×C₄×C₂₃	#127	1	4	yes	—
47	C₂×C₄×C₂₃	C₄⋊Q₈	1	8	yes	—
48	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
49	C₂×C₄×C₂₃	C₂²×Q₈	1	4	yes	—
50	C₈×C₂₃	C₄⋊Q₈	1	16	yes	—
51	C₂×#499	C₂×C₂₃	1	4	yes	—
52	#2859	C₂×C₂₃	1	2	yes	—
53	#1847	C₂×C₂₃	1	8	yes	—
54	C₄×#97	C₂×C₂₃	1	1	yes	characteristic
55	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
56	C₂²×C₂₃	C₂×C₄⋊Q₈	1	2	yes	—
57	C₂²×C₂₃	#575	1	4	yes	—
58	C₄×C₂₃	—	2	4	no	—
59	C₄×C₂₃	—	2	8	no	—
60	C₄×C₂₃	—	8	8	no	—
61	C₄×C₂₃	C₂×C₄⋊Q₈	1	4	yes	—
62	C₂×#125	C₂²×C₂₃	1	4	yes	—
63	#499	—	2	16	no	—
64	C₂×#107	C₂²×C₂₃	1	8	yes	—
65	C₂×#106	C₂²×C₂₃	1	8	yes	—
66	C₂×#97	C₂²×C₂₃	1	4	yes	—
67	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
68	#389	—	2	8	no	—
69	#384	C₂²×C₂₃	1	4	yes	—
70	#382	C₂²×C₂₃	1	4	yes	—
71	C₄³	C₂²×C₂₃	1	1	yes	characteristic
72	C₂×C₂₃	#2868	1	1	yes	characteristic
73	C₂×C₂₃	#3150	1	2	yes	—
74	C₂×C₂₃	#3166	1	4	yes	—
75	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
76	#125	—	2	16	no	—
77	C₂×C₄⋊C₄	—	2	8	no	—
78	C₂×C₄⋊C₄	—	2	8	no	—
79	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
80	C₂×C₄²	D₄×C₂₃	1	4	yes	—
81	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
82	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
83	#107	—	2	16	no	—
84	#106	—	2	16	no	—
85	#97	Q₈×C₂₃	1	16	yes	—
86	C₄×C₈	D₄×C₂₃	1	8	yes	—
87	#95	—	2	8	no	—
88	C₂₃	#33315	1	1	yes	characteristic, direct product factor, Sylow
89	C₂²×C₄	—	2	8	no	—
90	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
91	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
92	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
93	C₂×C₈	C₂×D₄×C₂₃	1	8	yes	—
94	C₂×C₈	C₂×Q₈×C₂₃	1	16	yes	—
95	C₄⋊C₄	—	2	16	no	—
96	C₄⋊C₄	—	4	16	no	—
97	C₄⋊C₄	—	4	16	no	—
98	C₄²	—	4	4	no	—
99	C₄²	C₂×D₄×C₂₃	1	4	yes	—
100	C₄²	C₂×Q₈×C₂₃	1	8	yes	—
101	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	16	no	—
105	C₂×C₄	—	4	8	no	—
106	C₂×C₄	C₂₃×#127	1	4	yes	—
107	C₂×C₄	C₂₃×C₄⋊Q₈	1	8	yes	—
108	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
109	C₂×C₄	C₂²×Q₈×C₂₃	1	4	yes	—
110	C₈	C₂₃×C₄⋊Q₈	1	16	yes	—
111	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
112	C₂²	C₂×C₂₃×C₄⋊Q₈	1	2	yes	—
113	C₂²	C₂₃×#575	1	4	yes	—
114	C₄	—	2	8	no	—
115	C₄	—	2	4	no	—
116	C₄	—	8	8	no	—
117	C₄	C₂×C₂₃×C₄⋊Q₈	1	4	yes	—
118	C₂	C₂₃×#2868	1	1	yes	characteristic
119	C₂	C₂₃×#3150	1	2	yes	—
120	C₂	C₂₃×#3166	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×#499

C₂

yes

maximal

C₂₃×#2859

C₂

yes

maximal

C₂₃×#1847

C₂

yes

maximal

C₄×C₂₃×#97

C₂

yes

maximal, characteristic

C₂×C₂₃×#125

C₂²

yes

—

C₂₃×#499

—

C₂×C₂₃×#107

C₂²

yes

—

C₂×C₂₃×#106

C₂²

yes

—

C₂×C₂₃×#97

C₂²

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

—

C₂₃×#389

—

C₂₃×#384

C₂²

yes

—

C₂₃×#382

C₂²

yes

—

C₄³×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂³

yes

—

C₂₃×#125

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂³

yes

—

C₂×C₄²×C₂₃

D₄

yes

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂×C₄²×C₂₃

C₂³

yes

—

C₂₃×#107

—

C₂₃×#106

—

C₂₃×#97

Q₈

yes

—

C₄×C₈×C₂₃

D₄

yes

—

C₂₃×#95

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×D₄

yes

—

C₂²×C₄×C₂₃

C₂×D₄

yes

—

C₂²×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×C₈×C₂₃

C₂×D₄

yes

—

C₂×C₈×C₂₃

C₂×Q₈

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂×D₄

yes

—

C₄²×C₂₃

C₂×Q₈

yes

—

#33315

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#127

yes

—

C₂×C₄×C₂₃

C₄⋊Q₈

yes

—

C₂×C₄×C₂₃

C₂²×D₄

yes

—

C₂×C₄×C₂₃

C₂²×Q₈

yes

—

C₈×C₂₃

C₄⋊Q₈

yes

—

C₂×#499

C₂×C₂₃

yes

—

#2859

C₂×C₂₃

yes

—

#1847

C₂×C₂₃

yes

—

C₄×#97

C₂×C₂₃

yes

characteristic

C₂²×C₂₃

C₂×#127

yes

characteristic

C₂²×C₂₃

C₂×C₄⋊Q₈

yes

—

C₂²×C₂₃

#575

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×C₄⋊Q₈

yes

—

C₂×#125

C₂²×C₂₃

yes

—

#499

—

C₂×#107

C₂²×C₂₃

yes

—

C₂×#106

C₂²×C₂₃

yes

—

C₂×#97

C₂²×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

—

#389

—

#384

C₂²×C₂₃

yes

—

#382

C₂²×C₂₃

yes

—

C₄³

C₂²×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#3166

yes

—

C₂²×C₈

C₂³×C₂₃

yes

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂³×C₂₃

yes

—

C₂×C₄²

D₄×C₂₃

yes

—

C₂×C₄²

C₂³×C₂₃

yes

characteristic

C₂×C₄²

C₂³×C₂₃

yes

—

#107

—

#106

—

#97

Q₈×C₂₃

yes

—

C₄×C₈

D₄×C₂₃

yes

—

#95

—

C₂₃

#33315

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₂₃

yes

—

C₂²×C₄

C₂×D₄×C₂₃

yes

—

C₂²×C₄

C₂⁴×C₂₃

yes

characteristic

C₂×C₈

C₂×D₄×C₂₃

yes

—

C₂×C₈

C₂×Q₈×C₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂×D₄×C₂₃

yes

—

100

C₄²

C₂×Q₈×C₂₃

yes

—

101

C₂³

C₂²×D₄×C₂₃

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂₃×#127

yes

—

107

C₂×C₄

C₂₃×C₄⋊Q₈

yes

—

108

C₂×C₄

C₂²×D₄×C₂₃

yes

—

109

C₂×C₄

C₂²×Q₈×C₂₃

yes

—

110

C₈

C₂₃×C₄⋊Q₈

yes

—

111

C₂²

C₂×C₂₃×#127

yes

characteristic

112

C₂²

C₂×C₂₃×C₄⋊Q₈

yes

—

113

C₂²

C₂₃×#575

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

C₂×C₂₃×C₄⋊Q₈

yes

—

118

C₂

C₂₃×#2868

yes

characteristic

119

C₂

C₂₃×#3150

yes

—

120

C₂

C₂₃×#3166

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₃

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₂₃

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

N	H ≅ G/N	type
40. #33315	C₂₃	direct (N × H)
88. C₂₃	#33315	direct (N × H)

H ≅ G/N

type

40. #33315

C₂₃

direct (N × H)

88. C₂₃

#33315

direct (N × H)

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

subgroup	index
C₂×C₂₃×#499	2
C₂₃×#2859	2
C₂₃×#1847	2
C₄×C₂₃×#97	2
#33315	23

subgroup

index

C₂×C₂₃×#499

C₂₃×#2859

C₂₃×#1847

C₄×C₂₃×#97

#33315

Subgroups — C₂₃×#33315 #19180941

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#33315 #19180941

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33315 #19180941