Group #15670753 — subgroups

Subgroups — C₂×C₂₃×#491 #15670753

Order 2944.

384 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 200 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×#125	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#491	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×#103	C₂	1	4	yes	maximal
5	C₂×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
7	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
8	C₂₃×C₄⋊Q₈	—	2	2	no	—
9	C₂₃×C₄⋊Q₈	C₂²	1	4	yes	—
10	C₂₃×#125	—	2	2	no	—
11	C₂₃×#125	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
13	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#103	C₂²	1	16	yes	—
17	C₂₃×#97	C₂²	1	4	yes	—
18	C₂×Q₈×C₂₃	—	2	2	no	—
19	C₂×Q₈×C₂₃	—	2	8	no	—
20	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
21	C₂²×C₄×C₂₃	—	2	2	no	—
22	C₂²×C₄×C₂₃	—	2	2	no	—
23	C₂²×C₄×C₂₃	D₄	1	2	yes	—
24	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₂₃	—	2	4	no	—
26	C₂×C₈×C₂₃	C₂³	1	4	yes	—
27	C₄⋊C₄×C₂₃	—	2	8	no	—
28	C₄⋊C₄×C₂₃	—	2	8	no	—
29	C₄⋊C₄×C₂₃	—	2	2	no	—
30	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
31	C₄²×C₂₃	D₄	1	2	yes	—
32	C₄²×C₂₃	C₂³	1	2	yes	—
33	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₂₃	—	2	8	no	—
35	Q₈×C₂₃	—	4	4	no	—
36	C₂×C₄×C₂₃	—	2	4	no	—
37	C₂×C₄×C₂₃	—	2	4	no	—
38	C₂×C₄×C₂₃	—	2	4	no	—
39	C₂×C₄×C₂₃	—	4	4	no	—
40	C₂×C₄×C₂₃	—	4	4	no	—
41	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₃	#41	1	4	yes	—
44	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
45	C₈×C₂₃	—	2	4	no	—
46	C₂×#491	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂²×C₂₃	#124	1	2	yes	—
48	C₂²×C₂₃	#137	1	4	yes	—
49	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
50	C₄×C₂₃	—	2	4	no	—
51	C₄×C₂₃	—	4	4	no	—
52	C₄×C₂₃	—	4	4	no	—
53	C₄×C₂₃	#124	1	2	yes	—
54	C₄×C₂₃	C₂×#41	1	2	yes	—
55	C₂×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
56	C₂×#125	C₂×C₂₃	1	1	yes	characteristic
57	#491	C₂×C₂₃	1	8	yes	direct product factor
58	C₂×#103	C₂×C₂₃	1	4	yes	—
59	C₂×#97	C₂×C₂₃	1	1	yes	characteristic
60	C₂×C₂₃	#491	1	4	yes	direct product factor
61	C₂×C₂₃	C₂×#124	1	1	yes	characteristic
62	C₂×C₂₃	C₂×#137	1	2	yes	—
63	C₂²×Q₈	C₂²×C₂₃	1	2	yes	—
64	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
65	C₄⋊Q₈	—	2	2	no	—
66	C₄⋊Q₈	C₂²×C₂₃	1	4	yes	—
67	#125	—	2	2	no	—
68	#125	C₂²×C₂₃	1	4	yes	—
69	C₂×C₄⋊C₄	—	2	2	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
72	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
73	#103	C₂²×C₂₃	1	16	yes	—
74	#97	C₂²×C₂₃	1	4	yes	—
75	C₂₃	C₂×#491	1	1	yes	characteristic, direct product factor, Sylow
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	2	8	no	—
78	C₂×Q₈	C₂³×C₂₃	1	4	yes	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	D₄×C₂₃	1	2	yes	—
82	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	C₂³×C₂₃	1	4	yes	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	2	8	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
89	C₄²	D₄×C₂₃	1	2	yes	—
90	C₄²	C₂³×C₂₃	1	2	yes	—
91	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
92	Q₈	—	2	8	no	—
93	Q₈	—	4	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
100	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
101	C₂×C₄	#41×C₂₃	1	4	yes	—
102	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₂²	C₂₃×#124	1	2	yes	—
105	C₂²	C₂₃×#137	1	4	yes	—
106	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
107	C₄	—	2	4	no	—
108	C₄	—	4	4	no	—
109	C₄	—	4	4	no	—
110	C₄	C₂₃×#124	1	2	yes	—
111	C₄	C₂×#41×C₂₃	1	2	yes	—
112	C₂	C₂₃×#491	1	4	yes	direct product factor
113	C₂	C₂×C₂₃×#124	1	1	yes	characteristic
114	C₂	C₂×C₂₃×#137	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×C₂₃×#125

C₂

yes

maximal, characteristic

C₂₃×#491

C₂

yes

maximal, direct product factor

C₂×C₂₃×#103

C₂

yes

maximal

C₂×C₂₃×#97

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₂²

yes

—

C₂²×C₈×C₂₃

C₂²

yes

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

C₂²

yes

—

C₂₃×#125

—

C₂₃×#125

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂₃×#103

C₂²

yes

—

C₂₃×#97

C₂²

yes

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

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

C₂³

yes

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₈×C₂₃

—

C₂×#491

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

#124

yes

—

C₂²×C₂₃

#137

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#124

yes

—

C₄×C₂₃

C₂×#41

yes

—

C₂×C₄⋊Q₈

C₂×C₂₃

yes

characteristic

C₂×#125

C₂×C₂₃

yes

characteristic

#491

C₂×C₂₃

yes

direct product factor

C₂×#103

C₂×C₂₃

yes

—

C₂×#97

C₂×C₂₃

yes

characteristic

C₂×C₂₃

#491

yes

direct product factor

C₂×C₂₃

C₂×#124

yes

characteristic

C₂×C₂₃

C₂×#137

yes

—

C₂²×Q₈

C₂²×C₂₃

yes

—

C₂²×C₈

C₂²×C₂₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²×C₂₃

yes

—

#125

—

#125

C₂²×C₂₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

characteristic

#103

C₂²×C₂₃

yes

—

#97

C₂²×C₂₃

yes

—

C₂₃

C₂×#491

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂³×C₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂³×C₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂³×C₂₃

yes

—

C₄²

D₄×C₂₃

yes

—

C₄²

C₂³×C₂₃

yes

—

C₂³

C₂×D₄×C₂₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₂₃

yes

—

100

C₂×C₄

C₂×D₄×C₂₃

yes

characteristic

101

C₂×C₄

#41×C₂₃

yes

—

102

C₂×C₄

C₂⁴×C₂₃

yes

characteristic

103

C₈

—

104

C₂²

C₂₃×#124

yes

—

105

C₂²

C₂₃×#137

yes

—

106

C₂²

C₂²×D₄×C₂₃

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

C₂₃×#124

yes

—

111

C₄

C₂×#41×C₂₃

yes

—

112

C₂

C₂₃×#491

yes

direct product factor

113

C₂

C₂×C₂₃×#124

yes

characteristic

114

C₂

C₂×C₂₃×#137

yes

—

	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×#491	C₂	direct (N × H)
46. C₂×#491	C₂₃	direct (N × H)
57. #491	C₂×C₂₃	direct (N × H)
60. C₂×C₂₃	#491	direct (N × H)
75. C₂₃	C₂×#491	direct (N × H)
112. C₂	C₂₃×#491	direct (N × H)

H ≅ G/N

type

3. C₂₃×#491

C₂

direct (N × H)

46. C₂×#491

C₂₃

direct (N × H)

57. #491

C₂×C₂₃

direct (N × H)

60. C₂×C₂₃

#491

direct (N × H)

75. C₂₃

C₂×#491

direct (N × H)

112. C₂

C₂₃×#491

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×C₂₃×#125	2
C₂₃×#491	2
C₂×C₂₃×#103	2
C₂×C₂₃×#97	2
C₂×#491	23

subgroup

index

C₂×C₂₃×C₄⋊Q₈

C₂×C₂₃×#125

C₂₃×#491

C₂×C₂₃×#103

C₂×C₂₃×#97

C₂×#491

Subgroups — C₂×C₂₃×#491 #15670753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C23×#491 #15670753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#491 #15670753