Group #19160510 — subgroups

Subgroups — C₂₃×#12884 #19160510

Order 5888.

302 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 166 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×#2150	C₂	1	2	yes	maximal
2	C₂₃×#1726	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1376	C₂	1	4	yes	maximal
4	C₂×C₂₃×M₃₂	C₂²	1	4	yes	—
5	C₂×C₂₃×#97	C₄	1	2	yes	—
6	C₂×C₂₃×#97	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#376	C₄	1	1	yes	characteristic
10	C₂₃×C₄⋊C₁₆	—	2	4	no	—
11	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂₃×M₃₂	—	2	8	no	—
17	C₂×C₁₆×C₂₃	—	2	4	no	—
18	C₂₃×#97	—	2	4	no	—
19	C₂₃×#97	C₈	1	4	yes	—
20	C₄×C₈×C₂₃	—	2	4	no	—
21	C₄×C₈×C₂₃	D₄	1	2	yes	—
22	C₄×C₈×C₂₃	Q₈	1	2	yes	—
23	C₂₃×#95	—	2	2	no	—
24	C₂²×C₄×C₂₃	—	2	2	no	—
25	C₂²×C₄×C₂₃	M₁₆	1	2	yes	—
26	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₂₃	—	2	2	no	—
28	C₂×C₈×C₂₃	—	2	2	no	—
29	C₂×C₈×C₂₃	—	2	2	no	—
30	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₂₃	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₂₃	#41	1	4	yes	—
35	C₄²×C₂₃	—	2	2	no	—
36	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
37	C₄²×C₂₃	C₂×C₈	1	2	yes	—
38	C₄²×C₂₃	M₁₆	1	4	yes	—
39	C₁₆×C₂₃	—	4	4	no	—
40	#12884	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
42	C₂×C₄×C₂₃	—	2	4	no	—
43	C₂×C₄×C₂₃	—	4	2	no	—
44	C₂×C₄×C₂₃	—	4	4	no	—
45	C₂×C₄×C₂₃	C₄⋊C₈	1	4	yes	—
46	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃	#117	1	2	yes	—
48	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
50	C₈×C₂₃	—	2	2	no	—
51	C₈×C₂₃	—	2	4	no	—
52	C₈×C₂₃	#124	1	2	yes	—
53	C₈×C₂₃	#125	1	2	yes	—
54	#2150	C₂×C₂₃	1	2	yes	—
55	#1726	C₂×C₂₃	1	1	yes	characteristic
56	#1376	C₂×C₂₃	1	4	yes	—
57	C₂²×C₂₃	—	2	2	no	—
58	C₂²×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
59	C₂²×C₂₃	#432	1	2	yes	—
60	C₄×C₂₃	—	2	4	no	—
61	C₄×C₂₃	—	4	4	no	—
62	C₄×C₂₃	#382	1	1	yes	characteristic
63	C₄×C₂₃	#423	1	1	yes	characteristic
64	C₄×C₂₃	#431	1	2	yes	—
65	C₂×M₃₂	C₂²×C₂₃	1	4	yes	—
66	C₂×#97	C₄×C₂₃	1	2	yes	—
67	C₂×#97	C₄×C₂₃	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
69	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
70	#376	C₄×C₂₃	1	1	yes	characteristic
71	C₄⋊C₁₆	—	2	4	no	—
72	C₂×C₂₃	—	2	2	no	—
73	C₂×C₂₃	#1831	1	1	yes	characteristic
74	C₂×C₂₃	#2153	1	2	yes	—
75	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
76	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
77	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
78	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
79	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
80	M₃₂	—	2	8	no	—
81	C₂×C₁₆	—	2	4	no	—
82	#97	—	2	4	no	—
83	#97	C₈×C₂₃	1	4	yes	—
84	C₄×C₈	—	2	4	no	—
85	C₄×C₈	D₄×C₂₃	1	2	yes	—
86	C₄×C₈	Q₈×C₂₃	1	2	yes	—
87	#95	—	2	2	no	—
88	C₂₃	#12884	1	1	yes	characteristic, direct product factor, Sylow
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	M₁₆×C₂₃	1	2	yes	—
91	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
96	C₂×C₈	C₂×C₈×C₂₃	1	4	yes	—
97	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
98	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
99	C₂×C₈	#41×C₂₃	1	4	yes	—
100	C₄²	—	2	2	no	—
101	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
102	C₄²	C₂×C₈×C₂₃	1	2	yes	—
103	C₄²	M₁₆×C₂₃	1	4	yes	—
104	C₁₆	—	4	4	no	—
105	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	C₂₃×C₄⋊C₈	1	4	yes	—
110	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
111	C₂×C₄	C₂₃×#117	1	2	yes	—
112	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
113	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	2	4	no	—
116	C₈	C₂₃×#124	1	2	yes	—
117	C₈	C₂₃×#125	1	2	yes	—
118	C₂²	—	2	2	no	—
119	C₂²	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
120	C₂²	C₂₃×#432	1	2	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	C₂₃×#382	1	1	yes	characteristic
124	C₄	C₂₃×#423	1	1	yes	characteristic
125	C₄	C₂₃×#431	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	C₂₃×#1831	1	1	yes	characteristic
128	C₂	C₂₃×#2153	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2150

C₂

yes

maximal

C₂₃×#1726

C₂

yes

maximal, characteristic

C₂₃×#1376

C₂

yes

maximal

C₂×C₂₃×M₃₂

C₂²

yes

—

C₂×C₂₃×#97

C₄

yes

—

C₂×C₂₃×#97

C₄

yes

characteristic

C₂×C₄×C₈×C₂₃

C₂²

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#376

C₄

yes

characteristic

C₂₃×C₄⋊C₁₆

—

C₂²×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₂₃

C₂×C₄

yes

—

C₂²×C₈×C₂₃

C₂³

yes

characteristic

C₂×C₄²×C₂₃

C₂×C₄

yes

—

C₂×C₄²×C₂₃

C₂×C₄

yes

characteristic

C₂₃×M₃₂

—

C₂×C₁₆×C₂₃

—

C₂₃×#97

—

C₂₃×#97

C₈

yes

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

D₄

yes

—

C₄×C₈×C₂₃

Q₈

yes

—

C₂₃×#95

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

M₁₆

yes

—

C₂²×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₄⋊C₄

yes

—

C₂×C₈×C₂₃

C₂×C₈

yes

—

C₂×C₈×C₂₃

C₂×D₄

yes

characteristic

C₂×C₈×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₂₃

#41

yes

—

C₄²×C₂₃

—

C₄²×C₂₃

C₄⋊C₄

yes

—

C₄²×C₂₃

C₂×C₈

yes

—

C₄²×C₂₃

M₁₆

yes

—

C₁₆×C₂₃

—

#12884

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₈

yes

—

C₂×C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

#117

yes

—

C₂×C₄×C₂₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₂₃

C₂×M₁₆

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

#124

yes

—

C₈×C₂₃

#125

yes

—

#2150

C₂×C₂₃

yes

—

#1726

C₂×C₂₃

yes

characteristic

#1376

C₂×C₂₃

yes

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₂₃

#432

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

C₂×M₃₂

C₂²×C₂₃

yes

—

C₂×#97

C₄×C₂₃

yes

—

C₂×#97

C₄×C₂₃

yes

characteristic

C₂×C₄×C₈

C₂²×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

#376

C₄×C₂₃

yes

characteristic

C₄⋊C₁₆

—

C₂×C₂₃

—

C₂×C₂₃

#1831

yes

characteristic

C₂×C₂₃

#2153

yes

—

C₂²×C₈

C₂×C₄×C₂₃

yes

characteristic

C₂²×C₈

C₂×C₄×C₂₃

yes

—

C₂²×C₈

C₂³×C₂₃

yes

characteristic

C₂×C₄²

C₂×C₄×C₂₃

yes

—

C₂×C₄²

C₂×C₄×C₂₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

#97

—

#97

C₈×C₂₃

yes

—

C₄×C₈

—

C₄×C₈

D₄×C₂₃

yes

—

C₄×C₈

Q₈×C₂₃

yes

—

#95

—

C₂₃

#12884

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

M₁₆×C₂₃

yes

—

C₂²×C₄

C₂²×C₄×C₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₂₃

yes

—

C₂×C₈

C₂×C₈×C₂₃

yes

—

C₂×C₈

C₂×D₄×C₂₃

yes

characteristic

C₂×C₈

C₂×Q₈×C₂₃

yes

characteristic

C₂×C₈

#41×C₂₃

yes

—

100

C₄²

—

101

C₄²

C₄⋊C₄×C₂₃

yes

—

102

C₄²

C₂×C₈×C₂₃

yes

—

103

C₄²

M₁₆×C₂₃

yes

—

104

C₁₆

—

105

C₂³

C₂×M₁₆×C₂₃

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂₃×C₄⋊C₈

yes

—

110

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

characteristic

111

C₂×C₄

C₂₃×#117

yes

—

112

C₂×C₄

C₂²×C₈×C₂₃

yes

characteristic

113

C₂×C₄

C₂×M₁₆×C₂₃

yes

—

114

C₈

—

115

C₈

—

116

C₈

C₂₃×#124

yes

—

117

C₈

C₂₃×#125

yes

—

118

C₂²

—

119

C₂²

C₂×C₂₃×C₄⋊C₈

yes

characteristic

120

C₂²

C₂₃×#432

yes

—

121

C₄

—

122

C₄

—

123

C₄

C₂₃×#382

yes

characteristic

124

C₄

C₂₃×#423

yes

characteristic

125

C₄

C₂₃×#431

yes

—

126

C₂

—

127

C₂

C₂₃×#1831

yes

characteristic

128

C₂

C₂₃×#2153

yes

—

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₄⋊C₈
prime radical	C₂³×C₂₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

subgroup

G/H

center

C₂×C₈×C₂₃

C₂×D₄

commutator

C₂×C₄

C₂²×C₈×C₂₃

Frattini, cosocle

C₂²×C₈

C₂³×C₂₃

socle

C₂²×C₂₃

C₂×C₄⋊C₈

prime radical

C₂³×C₂₃

C₂×M₁₆

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

N	H ≅ G/N	type
40. #12884	C₂₃	direct (N × H)
88. C₂₃	#12884	direct (N × H)

H ≅ G/N

type

40. #12884

C₂₃

direct (N × H)

88. C₂₃

#12884

direct (N × H)

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

subgroup	index
C₂₃×#2150	2
C₂₃×#1726	2
C₂₃×#1376	2
#12884	23

subgroup

index

C₂₃×#2150

C₂₃×#1726

C₂₃×#1376

#12884

Subgroups — C₂₃×#12884 #19160510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#12884 #19160510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12884 #19160510