Group #15670748 — subgroups

Subgroups — C₂×C₂₃×#487 #15670748

Order 2944.

424 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 232 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×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#487	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×#103	C₂	1	4	yes	maximal
5	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₂₃	—	2	1	no	—
7	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
8	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
9	C₂₃×C₄⋊Q₈	—	2	2	no	—
10	C₂₃×C₄⋊Q₈	—	2	2	no	—
11	C₂₃×C₄⋊Q₈	C₂²	1	4	yes	—
12	C₂₃×C₄⋊Q₈	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
14	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
15	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#103	C₂²	1	16	yes	—
19	C₄×C₈×C₂₃	C₂²	1	4	yes	—
20	C₂×Q₈×C₂₃	—	2	2	no	—
21	C₂×Q₈×C₂₃	—	2	4	no	—
22	C₂×Q₈×C₂₃	—	2	8	no	—
23	C₂×Q₈×C₂₃	—	4	4	no	—
24	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
25	C₂²×C₄×C₂₃	—	2	2	no	—
26	C₂²×C₄×C₂₃	—	2	2	no	—
27	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₂₃	—	2	4	no	—
31	C₂×C₈×C₂₃	C₂³	1	4	yes	—
32	C₄⋊C₄×C₂₃	—	2	4	no	—
33	C₄⋊C₄×C₂₃	—	2	4	no	—
34	C₄⋊C₄×C₂₃	—	2	4	no	—
35	C₄⋊C₄×C₂₃	—	2	2	no	—
36	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
37	C₄²×C₂₃	D₄	1	2	yes	—
38	C₄²×C₂₃	C₂³	1	2	yes	—
39	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₂₃	—	2	8	no	—
41	Q₈×C₂₃	—	4	4	no	—
42	Q₈×C₂₃	—	4	8	no	—
43	C₂×C₄×C₂₃	—	2	4	no	—
44	C₂×C₄×C₂₃	—	2	4	no	—
45	C₂×C₄×C₂₃	—	4	4	no	—
46	C₂×C₄×C₂₃	—	4	4	no	—
47	C₂×C₄×C₂₃	SD₁₆	1	4	yes	—
48	C₂×C₄×C₂₃	Q₁₆	1	4	yes	—
49	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₂₃	#41	1	4	yes	—
53	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
54	C₈×C₂₃	—	2	4	no	—
55	C₂×#487	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂²×C₂₃	#124	1	2	yes	—
57	C₂²×C₂₃	C₂×SD₁₆	1	2	yes	—
58	C₂²×C₂₃	C₂×Q₁₆	1	2	yes	—
59	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₂₃	—	4	4	no	—
61	C₄×C₂₃	—	4	4	no	—
62	C₄×C₂₃	#124	1	2	yes	—
63	C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
64	C₄×C₂₃	C₂×Q₁₆	1	4	yes	—
65	C₄×C₂₃	C₂×#41	1	2	yes	—
66	C₂×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
67	C₂×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
68	#487	C₂×C₂₃	1	8	yes	direct product factor
69	C₂×#103	C₂×C₂₃	1	4	yes	—
70	C₂×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
71	C₂×C₂₃	#487	1	4	yes	direct product factor
72	C₂×C₂₃	C₂×#124	1	1	yes	characteristic
73	C₂×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
74	C₂×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×Q₈	C₂²×C₂₃	1	2	yes	—
77	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
78	C₄⋊Q₈	—	2	2	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	C₂²×C₂₃	1	4	yes	—
81	C₄⋊Q₈	C₂²×C₂₃	1	4	yes	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
86	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
87	#103	C₂²×C₂₃	1	16	yes	—
88	C₄×C₈	C₂²×C₂₃	1	4	yes	—
89	C₂₃	C₂×#487	1	1	yes	characteristic, direct product factor, Sylow
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	4	no	—
92	C₂×Q₈	—	2	8	no	—
93	C₂×Q₈	—	4	4	no	—
94	C₂×Q₈	C₂³×C₂₃	1	4	yes	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
98	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
99	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	C₂³×C₂₃	1	4	yes	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	4	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
107	C₄²	D₄×C₂₃	1	2	yes	—
108	C₄²	C₂³×C₂₃	1	2	yes	—
109	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	4	8	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	SD₁₆×C₂₃	1	4	yes	—
118	C₂×C₄	Q₁₆×C₂₃	1	4	yes	—
119	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
120	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
121	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
122	C₂×C₄	#41×C₂₃	1	4	yes	—
123	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
124	C₈	—	2	4	no	—
125	C₂²	C₂₃×#124	1	2	yes	—
126	C₂²	C₂×SD₁₆×C₂₃	1	2	yes	—
127	C₂²	C₂×Q₁₆×C₂₃	1	2	yes	—
128	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
129	C₄	—	4	4	no	—
130	C₄	—	4	4	no	—
131	C₄	C₂₃×#124	1	2	yes	—
132	C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
133	C₄	C₂×Q₁₆×C₂₃	1	4	yes	—
134	C₄	C₂×#41×C₂₃	1	2	yes	—
135	C₂	C₂₃×#487	1	4	yes	direct product factor
136	C₂	C₂×C₂₃×#124	1	1	yes	characteristic
137	C₂	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
138	C₂	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×C₂₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂₃×#487

C₂

yes

maximal, direct product factor

C₂×C₂₃×#103

C₂

yes

maximal

C₂×C₄×C₈×C₂₃

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

—

C₂²×Q₈×C₂₃

C₂²

yes

—

C₂²×C₈×C₂₃

C₂²

yes

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

C₂²

yes

—

C₂₃×C₄⋊Q₈

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂₃×#103

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂³

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂³

yes

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

Q₁₆

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₈×C₂₃

—

C₂×#487

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

#124

yes

—

C₂²×C₂₃

C₂×SD₁₆

yes

—

C₂²×C₂₃

C₂×Q₁₆

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#124

yes

—

C₄×C₂₃

C₂×SD₁₆

yes

—

C₄×C₂₃

C₂×Q₁₆

yes

—

C₄×C₂₃

C₂×#41

yes

—

C₂×C₄⋊Q₈

C₂×C₂₃

yes

characteristic

C₂×C₄⋊Q₈

C₂×C₂₃

yes

characteristic

#487

C₂×C₂₃

yes

direct product factor

C₂×#103

C₂×C₂₃

yes

—

C₂×C₄×C₈

C₂×C₂₃

yes

characteristic

C₂×C₂₃

#487

yes

direct product factor

C₂×C₂₃

C₂×#124

yes

characteristic

C₂×C₂₃

C₂²×SD₁₆

yes

characteristic

C₂×C₂₃

C₂²×Q₁₆

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

C₂²×C₂₃

yes

—

C₂²×C₈

C₂²×C₂₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²×C₂₃

yes

—

C₄⋊Q₈

C₂²×C₂₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

characteristic

#103

C₂²×C₂₃

yes

—

C₄×C₈

C₂²×C₂₃

yes

—

C₂₃

C₂×#487

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂³×C₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₃

yes

characteristic

C₂²×C₄

D₄×C₂₃

yes

characteristic

C₂²×C₄

C₂³×C₂₃

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

C₂³×C₂₃

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂³×C₂₃

yes

—

107

C₄²

D₄×C₂₃

yes

—

108

C₄²

C₂³×C₂₃

yes

—

109

C₂³

C₂×D₄×C₂₃

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

SD₁₆×C₂₃

yes

—

118

C₂×C₄

Q₁₆×C₂₃

yes

—

119

C₂×C₄

C₂×D₄×C₂₃

yes

characteristic

120

C₂×C₄

C₂×D₄×C₂₃

yes

—

121

C₂×C₄

C₂×D₄×C₂₃

yes

—

122

C₂×C₄

#41×C₂₃

yes

—

123

C₂×C₄

C₂⁴×C₂₃

yes

characteristic

124

C₈

—

125

C₂²

C₂₃×#124

yes

—

126

C₂²

C₂×SD₁₆×C₂₃

yes

—

127

C₂²

C₂×Q₁₆×C₂₃

yes

—

128

C₂²

C₂²×D₄×C₂₃

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

C₂₃×#124

yes

—

132

C₄

C₂×SD₁₆×C₂₃

yes

—

133

C₄

C₂×Q₁₆×C₂₃

yes

—

134

C₄

C₂×#41×C₂₃

yes

—

135

C₂

C₂₃×#487

yes

direct product factor

136

C₂

C₂×C₂₃×#124

yes

characteristic

137

C₂

C₂²×SD₁₆×C₂₃

yes

characteristic

138

C₂

C₂²×Q₁₆×C₂₃

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₃

C₂×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₂₃

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
3. C₂₃×#487	C₂	direct (N × H)
55. C₂×#487	C₂₃	direct (N × H)
68. #487	C₂×C₂₃	direct (N × H)
71. C₂×C₂₃	#487	direct (N × H)
89. C₂₃	C₂×#487	direct (N × H)
135. C₂	C₂₃×#487	direct (N × H)

H ≅ G/N

type

3. C₂₃×#487

C₂

direct (N × H)

55. C₂×#487

C₂₃

direct (N × H)

68. #487

C₂×C₂₃

direct (N × H)

71. C₂×C₂₃

#487

direct (N × H)

89. C₂₃

C₂×#487

direct (N × H)

135. C₂

C₂₃×#487

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
C₂₃×#487	2
C₂×C₂₃×#103	2
C₂×C₄×C₈×C₂₃	2
C₂×#487	23

subgroup

index

C₂×C₂₃×C₄⋊Q₈

C₂₃×#487

C₂×C₂₃×#103

C₂×C₄×C₈×C₂₃

C₂×#487

Subgroups — C₂×C₂₃×#487 #15670748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C23×#487 #15670748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#487 #15670748