Group #15669946 — subgroups

Subgroups — C₂₃×#2327 #15669946

Order 2944.

516 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 296 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₂₃	C₂	1	2	yes	maximal
3	C₂₃×#391	C₂	1	2	yes	maximal
4	C₂₃×#387	C₂	1	8	yes	maximal
5	C₂₃×#382	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#376	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
8	C₂₃×C₄⋊Q₈	C₄	1	8	yes	complemented
9	C₄×Q₈×C₂₃	—	2	8	no	—
10	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
13	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₂₃	C₂²	1	4	yes	—
15	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#95	C₂²	1	4	yes	—
18	C₂₃×#95	C₂²	1	4	yes	—
19	C₂₃×#95	C₂²	1	8	yes	—
20	C₂×Q₈×C₂₃	—	2	4	no	—
21	C₂×Q₈×C₂₃	—	2	4	no	—
22	C₂×Q₈×C₂₃	C₂×C₄	1	8	yes	—
23	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
24	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
28	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₂₃	—	2	8	no	—
30	C₄⋊C₄×C₂₃	—	2	4	no	—
31	C₄⋊C₄×C₂₃	C₂×C₄	1	16	yes	—
32	C₄²×C₂₃	—	2	8	no	—
33	C₄²×C₂₃	—	2	2	no	—
34	C₄²×C₂₃	C₂×C₄	1	4	yes	—
35	C₄²×C₂₃	D₄	1	4	yes	—
36	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
37	Q₈×C₂₃	—	2	16	no	—
38	C₂×C₄×C₂₃	—	2	4	no	—
39	C₂×C₄×C₂₃	—	2	4	no	—
40	C₂×C₄×C₂₃	—	2	4	no	—
41	C₂×C₄×C₂₃	—	2	4	no	—
42	C₂×C₄×C₂₃	—	2	4	no	—
43	C₂×C₄×C₂₃	—	2	2	no	—
44	C₂×C₄×C₂₃	—	2	8	no	—
45	C₂×C₄×C₂₃	—	2	4	no	—
46	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₂₃	C₂²×C₄	1	8	yes	—
48	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
50	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₂₃	#41	1	2	yes	—
52	#2327	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
54	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
55	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
56	C₂²×C₂₃	2₋¹⁺⁴	1	4	yes	—
57	C₄×C₂₃	—	2	8	no	—
58	C₄×C₂₃	—	2	4	no	—
59	C₄×C₂₃	—	2	4	no	—
60	C₄×C₂₃	—	4	4	no	—
61	C₄×C₂₃	—	4	2	no	—
62	C₄×C₂₃	C₄×D₄	1	4	yes	—
63	C₂×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
64	C₂×C₄×Q₈	C₂×C₂₃	1	2	yes	—
65	#391	C₂×C₂₃	1	2	yes	—
66	#387	C₂×C₂₃	1	8	yes	—
67	#382	C₂×C₂₃	1	1	yes	characteristic
68	#376	C₂×C₂₃	1	1	yes	characteristic
69	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
70	C₂×C₂₃	#519	1	2	yes	—
71	C₂×C₂₃	#536	1	2	yes	—
72	C₂×C₂₃	#541	1	2	yes	—
73	C₂²×Q₈	C₂²×C₂₃	1	2	yes	—
74	C₄⋊Q₈	C₄×C₂₃	1	8	yes	complemented
75	C₄×Q₈	—	2	8	no	—
76	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
77	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
78	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
79	C₂×C₄²	C₂²×C₂₃	1	4	yes	—
80	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
81	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
82	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
83	#95	C₂²×C₂₃	1	4	yes	—
84	#95	C₂²×C₂₃	1	4	yes	—
85	#95	C₂²×C₂₃	1	8	yes	—
86	C₂₃	#2327	1	1	yes	characteristic, direct product factor, Sylow
87	C₂×Q₈	—	2	4	no	—
88	C₂×Q₈	—	2	4	no	—
89	C₂×Q₈	C₂×C₄×C₂₃	1	8	yes	—
90	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
91	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
92	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
93	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
94	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
95	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
96	C₄⋊C₄	—	2	8	no	—
97	C₄⋊C₄	—	2	4	no	—
98	C₄⋊C₄	C₂×C₄×C₂₃	1	16	yes	—
99	C₄²	—	2	8	no	—
100	C₄²	—	2	2	no	—
101	C₄²	C₂×C₄×C₂₃	1	4	yes	—
102	C₄²	D₄×C₂₃	1	4	yes	—
103	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
104	Q₈	—	2	16	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	8	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
114	C₂×C₄	C₂²×C₄×C₂₃	1	8	yes	—
115	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
116	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
117	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
118	C₂×C₄	#41×C₂₃	1	2	yes	—
119	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
120	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
121	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
122	C₂²	C₂₃×2₋¹⁺⁴	1	4	yes	—
123	C₄	—	2	8	no	—
124	C₄	—	2	4	no	—
125	C₄	—	2	4	no	—
126	C₄	—	4	4	no	—
127	C₄	—	4	2	no	—
128	C₄	C₄×D₄×C₂₃	1	4	yes	—
129	C₂	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
130	C₂	C₂₃×#519	1	2	yes	—
131	C₂	C₂₃×#536	1	2	yes	—
132	C₂	C₂₃×#541	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×C₄×Q₈×C₂₃

C₂

yes

maximal

C₂₃×#391

C₂

yes

maximal

C₂₃×#387

C₂

yes

maximal

C₂₃×#382

C₂

yes

maximal, characteristic

C₂₃×#376

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₂²

yes

—

C₂₃×C₄⋊Q₈

C₄

yes

complemented

C₄×Q₈×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂₃×#95

C₂²

yes

—

C₂₃×#95

C₂²

yes

—

C₂₃×#95

C₂²

yes

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₂³×C₂₃

C₂⁴

yes

characteristic

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

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

#2327

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

C₂³×C₄

yes

characteristic

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₂²×C₂₃

C₂×#41

yes

characteristic

C₂²×C₂₃

2₋¹⁺⁴

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₂×C₄⋊Q₈

C₂×C₂₃

yes

characteristic

C₂×C₄×Q₈

C₂×C₂₃

yes

—

#391

C₂×C₂₃

yes

—

#387

C₂×C₂₃

yes

—

#382

C₂×C₂₃

yes

characteristic

#376

C₂×C₂₃

yes

characteristic

C₂×C₂₃

C₂×C₄×D₄

yes

characteristic

C₂×C₂₃

#519

yes

—

C₂×C₂₃

#536

yes

—

C₂×C₂₃

#541

yes

—

C₂²×Q₈

C₂²×C₂₃

yes

—

C₄⋊Q₈

C₄×C₂₃

yes

complemented

C₄×Q₈

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

characteristic

C₂×C₄²

C₂²×C₂₃

yes

characteristic

C₂×C₄²

C₂²×C₂₃

yes

characteristic

#95

C₂²×C₂₃

yes

—

#95

C₂²×C₂₃

yes

—

#95

C₂²×C₂₃

yes

—

C₂₃

#2327

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₄×C₂₃

yes

—

C₄²

—

100

C₄²

—

101

C₄²

C₂×C₄×C₂₃

yes

—

102

C₄²

D₄×C₂₃

yes

—

103

C₂³

C₂⁴×C₂₃

yes

characteristic

104

Q₈

—

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

C₂²×C₄×C₂₃

yes

—

114

C₂×C₄

C₂²×C₄×C₂₃

yes

—

115

C₂×C₄

C₂²×C₄×C₂₃

yes

—

116

C₂×C₄

C₂×D₄×C₂₃

yes

—

117

C₂×C₄

C₂×D₄×C₂₃

yes

—

118

C₂×C₄

#41×C₂₃

yes

—

119

C₂²

C₂³×C₄×C₂₃

yes

characteristic

120

C₂²

C₂²×D₄×C₂₃

yes

characteristic

121

C₂²

C₂×#41×C₂₃

yes

characteristic

122

C₂²

C₂₃×2₋¹⁺⁴

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

C₄×D₄×C₂₃

yes

—

129

C₂

C₂×C₄×D₄×C₂₃

yes

characteristic

130

C₂

C₂₃×#519

yes

—

131

C₂

C₂₃×#536

yes

—

132

C₂

C₂₃×#541

yes

—

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

subgroup

G/H

center, socle, prime radical

C₂³×C₂₃

C₂⁴

commutator

C₂²

C₂³×C₄×C₂₃

Frattini, cosocle

C₂³

C₂⁴×C₂₃

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ G

N	H ≅ G/N	type
8. C₂₃×C₄⋊Q₈	C₄	semidirect
52. #2327	C₂₃	direct (N × H)
74. C₄⋊Q₈	C₄×C₂₃	semidirect
86. C₂₃	#2327	direct (N × H)

H ≅ G/N

type

8. C₂₃×C₄⋊Q₈

C₄

semidirect

52. #2327

C₂₃

direct (N × H)

74. C₄⋊Q₈

C₄×C₂₃

semidirect

86. C₂₃

#2327

direct (N × H)

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

subgroup	index
C₂×C₂₃×C₄⋊Q₈	2
C₂×C₄×Q₈×C₂₃	2
C₂₃×#391	2
C₂₃×#387	2
C₂₃×#382	2
C₂₃×#376	2
#2327	23

subgroup

index

C₂×C₂₃×C₄⋊Q₈

C₂×C₄×Q₈×C₂₃

C₂₃×#391

C₂₃×#387

C₂₃×#382

C₂₃×#376

#2327

Subgroups — C₂₃×#2327 #15669946

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#2327 #15669946

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2327 #15669946