Group #15720315 — subgroups

Subgroups — C₂²×C₁₃×SD₆₄ #15720315

Order 3328.

502 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 222 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	C₂²×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×SD₆₄	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₁₃×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₁₃×Q₁₆	—	2	1	no	—
6	C₂²×C₁₃×D₈	—	2	1	no	—
7	C₂×C₁₃×Q₃₂	—	2	3	no	—
8	C₂×C₁₃×Q₃₂	C₂²	1	6	yes	complemented
9	C₂×C₁₃×D₁₆	—	2	3	no	—
10	C₂×C₁₃×D₁₆	C₂²	1	6	yes	—
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₁₃×SD₆₄	C₂²	1	16	yes	direct product factor
13	C₂×C₁₃×C₃₂	C₂²	1	6	yes	complemented
14	C₂²×Q₈×C₁₃	—	4	1	no	—
15	C₂²×D₄×C₁₃	—	4	1	no	—
16	C₂×C₁₃×Q₁₆	—	2	6	no	—
17	C₂×C₁₃×Q₁₆	—	4	3	no	—
18	C₂×C₁₃×D₈	—	2	6	no	—
19	C₂×C₁₃×D₈	—	4	3	no	—
20	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₁₃×Q₃₂	—	2	6	no	—
22	C₁₃×Q₃₂	C₂³	1	4	yes	complemented
23	C₁₃×D₁₆	—	2	6	no	—
24	C₁₃×D₁₆	C₂³	1	4	yes	—
25	C₂×C₁₃×C₁₆	D₄	1	3	yes	—
26	C₂×C₁₃×C₁₆	C₂³	1	3	yes	—
27	C₁₃×C₃₂	C₂³	1	4	yes	complemented
28	C₂²×SD₆₄	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂⁴×C₁₃	—	8	1	no	—
30	C₂×Q₈×C₁₃	—	4	6	no	—
31	C₂×Q₈×C₁₃	—	8	3	no	—
32	C₂×D₄×C₁₃	—	4	6	no	—
33	C₂×D₄×C₁₃	—	8	3	no	—
34	C₂²×C₄×C₁₃	—	8	1	no	—
35	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
36	C₁₃×Q₁₆	—	2	4	no	—
37	C₁₃×Q₁₆	—	4	6	no	—
38	C₁₃×D₈	—	2	4	no	—
39	C₁₃×D₈	—	4	6	no	—
40	C₂×C₈×C₁₃	D₈	1	3	yes	—
41	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
42	C₁₃×C₁₆	C₂×D₄	1	3	yes	—
43	C₁₃×C₁₆	C₂⁴	1	1	yes	characteristic
44	C₂²×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
45	C₂²×D₁₆	C₂×C₁₃	1	1	yes	characteristic
46	C₂×SD₆₄	C₂×C₁₃	1	12	yes	direct product factor
47	C₂²×C₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
48	C₂³×C₁₃	—	8	6	no	—
49	C₂³×C₁₃	—	16	4	no	—
50	C₂³×C₁₃	D₁₆	1	1	yes	characteristic
51	Q₈×C₁₃	—	4	4	no	—
52	Q₈×C₁₃	—	8	6	no	—
53	D₄×C₁₃	—	4	4	no	—
54	D₄×C₁₃	—	8	6	no	—
55	C₂×C₄×C₁₃	—	8	6	no	—
56	C₂×C₄×C₁₃	D₁₆	1	3	yes	—
57	C₂×C₄×C₁₃	C₂×D₈	1	3	yes	—
58	C₈×C₁₃	C₂×D₈	1	3	yes	—
59	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
60	C₂²×Q₁₆	—	2	1	no	—
61	C₂²×D₈	—	2	1	no	—
62	C₂×Q₃₂	—	2	3	no	—
63	C₂×Q₃₂	C₂²×C₁₃	1	6	yes	complemented
64	C₂×D₁₆	—	2	3	no	—
65	C₂×D₁₆	C₂²×C₁₃	1	6	yes	—
66	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
67	SD₆₄	C₂²×C₁₃	1	16	yes	direct product factor
68	C₂×C₃₂	C₂²×C₁₃	1	6	yes	complemented
69	C₂²×C₁₃	—	8	4	no	—
70	C₂²×C₁₃	—	16	12	no	—
71	C₂²×C₁₃	SD₆₄	1	4	yes	direct product factor
72	C₂²×C₁₃	C₂×D₁₆	1	3	yes	—
73	C₄×C₁₃	—	8	4	no	—
74	C₄×C₁₃	C₂×D₁₆	1	3	yes	—
75	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
76	C₂²×Q₈	—	4	1	no	—
77	C₂²×D₄	—	4	1	no	—
78	C₂×Q₁₆	—	2	6	no	—
79	C₂×Q₁₆	—	4	3	no	—
80	C₂×D₈	—	2	6	no	—
81	C₂×D₈	—	4	3	no	—
82	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
83	Q₃₂	—	2	6	no	—
84	Q₃₂	C₂³×C₁₃	1	4	yes	complemented
85	D₁₆	—	2	6	no	—
86	D₁₆	C₂³×C₁₃	1	4	yes	—
87	C₂×C₁₆	D₄×C₁₃	1	3	yes	—
88	C₂×C₁₆	C₂³×C₁₃	1	3	yes	—
89	C₃₂	C₂³×C₁₃	1	4	yes	complemented
90	C₂×C₁₃	—	16	4	no	—
91	C₂×C₁₃	C₂×SD₆₄	1	6	yes	direct product factor
92	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
93	C₂⁴	—	8	1	no	—
94	C₂×Q₈	—	4	6	no	—
95	C₂×Q₈	—	8	3	no	—
96	C₂×D₄	—	4	6	no	—
97	C₂×D₄	—	8	3	no	—
98	C₂²×C₄	—	8	1	no	—
99	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
100	Q₁₆	—	2	4	no	—
101	Q₁₆	—	4	6	no	—
102	D₈	—	2	4	no	—
103	D₈	—	4	6	no	—
104	C₂×C₈	C₁₃×D₈	1	3	yes	—
105	C₂×C₈	C₂×D₄×C₁₃	1	3	yes	—
106	C₁₆	C₂×D₄×C₁₃	1	3	yes	—
107	C₁₆	C₂⁴×C₁₃	1	1	yes	characteristic
108	C₁₃	C₂²×SD₆₄	1	1	yes	characteristic, direct product factor, Sylow
109	C₂³	—	8	6	no	—
110	C₂³	—	16	4	no	—
111	C₂³	C₁₃×D₁₆	1	1	yes	characteristic
112	Q₈	—	4	4	no	—
113	Q₈	—	8	6	no	—
114	D₄	—	4	4	no	—
115	D₄	—	8	6	no	—
116	C₂×C₄	—	8	6	no	—
117	C₂×C₄	C₁₃×D₁₆	1	3	yes	—
118	C₂×C₄	C₂×C₁₃×D₈	1	3	yes	—
119	C₈	C₂×C₁₃×D₈	1	3	yes	—
120	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
121	C₂²	—	8	4	no	—
122	C₂²	—	16	12	no	—
123	C₂²	C₁₃×SD₆₄	1	4	yes	direct product factor
124	C₂²	C₂×C₁₃×D₁₆	1	3	yes	—
125	C₄	—	8	4	no	—
126	C₄	C₂×C₁₃×D₁₆	1	3	yes	—
127	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
128	C₂	—	16	4	no	—
129	C₂	C₂×C₁₃×SD₆₄	1	6	yes	direct product factor
130	C₂	C₂²×C₁₃×D₁₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₃×Q₃₂

C₂

yes

maximal, characteristic, complemented

C₂²×C₁₃×D₁₆

C₂

yes

maximal, characteristic

C₂×C₁₃×SD₆₄

C₂

yes

maximal, direct product factor

C₂²×C₁₃×C₃₂

C₂

yes

maximal, characteristic, complemented

C₂²×C₁₃×Q₁₆

—

C₂²×C₁₃×D₈

—

C₂×C₁₃×Q₃₂

—

C₂×C₁₃×Q₃₂

C₂²

yes

complemented

C₂×C₁₃×D₁₆

—

C₂×C₁₃×D₁₆

C₂²

yes

—

C₂²×C₁₃×C₁₆

C₂²

yes

characteristic

C₁₃×SD₆₄

C₂²

yes

direct product factor

C₂×C₁₃×C₃₂

C₂²

yes

complemented

C₂²×Q₈×C₁₃

—

C₂²×D₄×C₁₃

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

—

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₁₃×Q₃₂

—

C₁₃×Q₃₂

C₂³

yes

complemented

C₁₃×D₁₆

—

C₁₃×D₁₆

C₂³

yes

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₂×C₁₃×C₁₆

C₂³

yes

—

C₁₃×C₃₂

C₂³

yes

complemented

C₂²×SD₆₄

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂⁴×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₈

yes

characteristic

C₁₃×Q₁₆

—

C₁₃×Q₁₆

—

C₁₃×D₈

—

C₁₃×D₈

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₁₃×C₁₆

C₂×D₄

yes

—

C₁₃×C₁₆

C₂⁴

yes

characteristic

C₂²×Q₃₂

C₂×C₁₃

yes

characteristic, complemented

C₂²×D₁₆

C₂×C₁₃

yes

characteristic

C₂×SD₆₄

C₂×C₁₃

yes

direct product factor

C₂²×C₃₂

C₂×C₁₃

yes

characteristic, complemented

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

D₁₆

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

D₁₆

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₈×C₁₃

C₂×D₈

yes

—

C₈×C₁₃

C₂²×D₄

yes

characteristic

C₂²×Q₁₆

—

C₂²×D₈

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂²×C₁₃

yes

complemented

C₂×D₁₆

—

C₂×D₁₆

C₂²×C₁₃

yes

—

C₂²×C₁₆

C₂²×C₁₃

yes

characteristic

SD₆₄

C₂²×C₁₃

yes

direct product factor

C₂×C₃₂

C₂²×C₁₃

yes

complemented

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

SD₆₄

yes

direct product factor

C₂²×C₁₃

C₂×D₁₆

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂×D₁₆

yes

—

C₄×C₁₃

C₂²×D₈

yes

characteristic

C₂²×Q₈

—

C₂²×D₄

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

D₄×C₁₃

yes

characteristic

Q₃₂

—

Q₃₂

C₂³×C₁₃

yes

complemented

D₁₆

—

D₁₆

C₂³×C₁₃

yes

—

C₂×C₁₆

D₄×C₁₃

yes

—

C₂×C₁₆

C₂³×C₁₃

yes

—

C₃₂

C₂³×C₁₃

yes

complemented

C₂×C₁₃

—

C₂×C₁₃

C₂×SD₆₄

yes

direct product factor

C₂×C₁₃

C₂²×D₁₆

yes

characteristic

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₁₃×D₈

yes

characteristic

100

Q₁₆

—

101

Q₁₆

—

102

D₈

—

103

D₈

—

104

C₂×C₈

C₁₃×D₈

yes

—

105

C₂×C₈

C₂×D₄×C₁₃

yes

—

106

C₁₆

C₂×D₄×C₁₃

yes

—

107

C₁₆

C₂⁴×C₁₃

yes

characteristic

108

C₁₃

C₂²×SD₆₄

yes

characteristic, direct product factor, Sylow

109

C₂³

—

110

C₂³

—

111

C₂³

C₁₃×D₁₆

yes

characteristic

112

Q₈

—

113

Q₈

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

C₁₃×D₁₆

yes

—

118

C₂×C₄

C₂×C₁₃×D₈

yes

—

119

C₈

C₂×C₁₃×D₈

yes

—

120

C₈

C₂²×D₄×C₁₃

yes

characteristic

121

C₂²

—

122

C₂²

—

123

C₂²

C₁₃×SD₆₄

yes

direct product factor

124

C₂²

C₂×C₁₃×D₁₆

yes

—

125

C₄

—

126

C₄

C₂×C₁₃×D₁₆

yes

—

127

C₄

C₂²×C₁₃×D₈

yes

characteristic

128

C₂

—

129

C₂

C₂×C₁₃×SD₆₄

yes

direct product factor

130

C₂

C₂²×C₁₃×D₁₆

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴×C₁₃
prime radical	C₂²×C₁₃×D₁₆	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ C₂²×C₁₃×C₁₆ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₃

D₁₆

commutator, Frattini, cosocle

C₁₆

C₂⁴×C₁₃

prime radical

C₂²×C₁₃×D₁₆

C₂

LCS

G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ C₂²×C₁₃×C₁₆ ◁ G

N	H ≅ G/N	type
1. C₂²×C₁₃×Q₃₂	C₂	semidirect
3. C₂×C₁₃×SD₆₄	C₂	direct (N × H)
4. C₂²×C₁₃×C₃₂	C₂	semidirect
8. C₂×C₁₃×Q₃₂	C₂²	semidirect
12. C₁₃×SD₆₄	C₂²	direct (N × H)
13. C₂×C₁₃×C₃₂	C₂²	semidirect
22. C₁₃×Q₃₂	C₂³	semidirect
27. C₁₃×C₃₂	C₂³	semidirect
28. C₂²×SD₆₄	C₁₃	direct (N × H)
44. C₂²×Q₃₂	C₂×C₁₃	semidirect
46. C₂×SD₆₄	C₂×C₁₃	direct (N × H)
47. C₂²×C₃₂	C₂×C₁₃	semidirect
63. C₂×Q₃₂	C₂²×C₁₃	semidirect
67. SD₆₄	C₂²×C₁₃	direct (N × H)
68. C₂×C₃₂	C₂²×C₁₃	semidirect
71. C₂²×C₁₃	SD₆₄	direct (N × H)
84. Q₃₂	C₂³×C₁₃	semidirect
89. C₃₂	C₂³×C₁₃	semidirect
91. C₂×C₁₃	C₂×SD₆₄	direct (N × H)
108. C₁₃	C₂²×SD₆₄	direct (N × H)
123. C₂²	C₁₃×SD₆₄	direct (N × H)
129. C₂	C₂×C₁₃×SD₆₄	direct (N × H)

H ≅ G/N

type

1. C₂²×C₁₃×Q₃₂

C₂

semidirect

3. C₂×C₁₃×SD₆₄

C₂

direct (N × H)

4. C₂²×C₁₃×C₃₂

C₂

semidirect

8. C₂×C₁₃×Q₃₂

C₂²

semidirect

12. C₁₃×SD₆₄

C₂²

direct (N × H)

13. C₂×C₁₃×C₃₂

C₂²

semidirect

22. C₁₃×Q₃₂

C₂³

semidirect

27. C₁₃×C₃₂

C₂³

semidirect

28. C₂²×SD₆₄

C₁₃

direct (N × H)

44. C₂²×Q₃₂

C₂×C₁₃

semidirect

46. C₂×SD₆₄

C₂×C₁₃

direct (N × H)

47. C₂²×C₃₂

C₂×C₁₃

semidirect

63. C₂×Q₃₂

C₂²×C₁₃

semidirect

67. SD₆₄

C₂²×C₁₃

direct (N × H)

68. C₂×C₃₂

C₂²×C₁₃

semidirect

71. C₂²×C₁₃

SD₆₄

direct (N × H)

84. Q₃₂

C₂³×C₁₃

semidirect

89. C₃₂

C₂³×C₁₃

semidirect

91. C₂×C₁₃

C₂×SD₆₄

direct (N × H)

108. C₁₃

C₂²×SD₆₄

direct (N × H)

123. C₂²

C₁₃×SD₆₄

direct (N × H)

129. C₂

C₂×C₁₃×SD₆₄

direct (N × H)

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

subgroup	index
C₂²×C₁₃×Q₃₂	2
C₂²×C₁₃×D₁₆	2
C₂×C₁₃×SD₆₄	2
C₂²×C₁₃×C₃₂	2
C₂²×SD₆₄	13

subgroup

index

C₂²×C₁₃×Q₃₂

C₂²×C₁₃×D₁₆

C₂×C₁₃×SD₆₄

C₂²×C₁₃×C₃₂

C₂²×SD₆₄

Subgroups — C₂²×C₁₃×SD₆₄ #15720315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×C13×SD64 #15720315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×SD₆₄ #15720315