Group #15748965 — subgroups

Subgroups — C₂³×C₁₃×SD₃₂ #15748965

Order 3328.

2334 conjugacy classes of subgroups; only the 1054 normal subgroups are shown.

normal subgroups only

This group has 2334 conjugacy classes of subgroups — too many to draw. Only its 1054 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×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₃×SD₃₂	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×Q₈×C₁₃	—	2	1	no	—
6	C₂³×D₄×C₁₃	—	2	1	no	—
7	C₂²×C₁₃×Q₁₆	—	2	7	no	—
8	C₂²×C₁₃×Q₁₆	C₂²	1	14	yes	complemented
9	C₂²×C₁₃×D₈	—	2	7	no	—
10	C₂²×C₁₃×D₈	C₂²	1	14	yes	—
11	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×SD₃₂	C₂²	1	112	yes	direct product factor
13	C₂²×C₁₃×C₁₆	C₂²	1	14	yes	complemented
14	C₂⁵×C₁₃	—	4	1	no	—
15	C₂²×Q₈×C₁₃	—	2	14	no	—
16	C₂²×Q₈×C₁₃	—	4	7	no	—
17	C₂²×D₄×C₁₃	—	2	14	no	—
18	C₂²×D₄×C₁₃	—	4	7	no	—
19	C₂³×C₄×C₁₃	—	4	1	no	—
20	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
21	C₂×C₁₃×Q₁₆	—	2	42	no	—
22	C₂×C₁₃×Q₁₆	C₂³	1	28	yes	complemented
23	C₂×C₁₃×D₈	—	2	42	no	—
24	C₂×C₁₃×D₈	C₂³	1	28	yes	—
25	C₂²×C₈×C₁₃	D₄	1	7	yes	—
26	C₂²×C₈×C₁₃	C₂³	1	7	yes	—
27	C₁₃×SD₃₂	C₂³	1	64	yes	direct product factor
28	C₂×C₁₃×C₁₆	C₂³	1	28	yes	complemented
29	C₂³×SD₃₂	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂⁴×C₁₃	—	4	14	no	—
31	C₂⁴×C₁₃	—	8	8	no	—
32	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
33	C₂×Q₈×C₁₃	—	2	28	no	—
34	C₂×Q₈×C₁₃	—	4	42	no	—
35	C₂×D₄×C₁₃	—	2	28	no	—
36	C₂×D₄×C₁₃	—	4	42	no	—
37	C₂²×C₄×C₁₃	—	4	14	no	—
38	C₂²×C₄×C₁₃	D₈	1	7	yes	—
39	C₂²×C₄×C₁₃	C₂×D₄	1	7	yes	—
40	C₁₃×Q₁₆	—	2	28	no	—
41	C₁₃×Q₁₆	C₂⁴	1	8	yes	complemented
42	C₁₃×D₈	—	2	28	no	—
43	C₁₃×D₈	C₂⁴	1	8	yes	—
44	C₂×C₈×C₁₃	C₂×D₄	1	21	yes	—
45	C₂×C₈×C₁₃	C₂⁴	1	7	yes	—
46	C₁₃×C₁₆	C₂⁴	1	8	yes	complemented
47	C₂³×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
48	C₂³×D₈	C₂×C₁₃	1	1	yes	characteristic
49	C₂²×SD₃₂	C₂×C₁₃	1	28	yes	direct product factor
50	C₂³×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
51	C₂³×C₁₃	—	4	28	no	—
52	C₂³×C₁₃	—	8	56	no	—
53	C₂³×C₁₃	SD₃₂	1	8	yes	direct product factor
54	C₂³×C₁₃	C₂×D₈	1	7	yes	—
55	Q₈×C₁₃	—	2	8	no	—
56	Q₈×C₁₃	—	4	28	no	—
57	D₄×C₁₃	—	2	8	no	—
58	D₄×C₁₃	—	4	28	no	—
59	C₂×C₄×C₁₃	—	4	28	no	—
60	C₂×C₄×C₁₃	C₂×D₈	1	21	yes	—
61	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
62	C₈×C₁₃	C₂²×D₄	1	7	yes	—
63	C₈×C₁₃	C₂⁵	1	1	yes	characteristic
64	C₂³×Q₈	—	2	1	no	—
65	C₂³×D₄	—	2	1	no	—
66	C₂²×Q₁₆	—	2	7	no	—
67	C₂²×Q₁₆	C₂²×C₁₃	1	14	yes	complemented
68	C₂²×D₈	—	2	7	no	—
69	C₂²×D₈	C₂²×C₁₃	1	14	yes	—
70	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
71	C₂×SD₃₂	C₂²×C₁₃	1	112	yes	direct product factor
72	C₂²×C₁₆	C₂²×C₁₃	1	14	yes	complemented
73	C₂²×C₁₃	—	4	8	no	—
74	C₂²×C₁₃	—	8	56	no	—
75	C₂²×C₁₃	C₂×SD₃₂	1	28	yes	direct product factor
76	C₂²×C₁₃	C₂²×D₈	1	7	yes	—
77	C₄×C₁₃	—	4	8	no	—
78	C₄×C₁₃	C₂²×D₈	1	7	yes	—
79	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
80	C₂⁵	—	4	1	no	—
81	C₂²×Q₈	—	2	14	no	—
82	C₂²×Q₈	—	4	7	no	—
83	C₂²×D₄	—	2	14	no	—
84	C₂²×D₄	—	4	7	no	—
85	C₂³×C₄	—	4	1	no	—
86	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
87	C₂×Q₁₆	—	2	42	no	—
88	C₂×Q₁₆	C₂³×C₁₃	1	28	yes	complemented
89	C₂×D₈	—	2	42	no	—
90	C₂×D₈	C₂³×C₁₃	1	28	yes	—
91	C₂²×C₈	D₄×C₁₃	1	7	yes	—
92	C₂²×C₈	C₂³×C₁₃	1	7	yes	—
93	SD₃₂	C₂³×C₁₃	1	64	yes	direct product factor
94	C₂×C₁₆	C₂³×C₁₃	1	28	yes	complemented
95	C₂×C₁₃	—	8	8	no	—
96	C₂×C₁₃	C₂²×SD₃₂	1	14	yes	direct product factor
97	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
98	C₂⁴	—	4	14	no	—
99	C₂⁴	—	8	8	no	—
100	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
101	C₂×Q₈	—	2	28	no	—
102	C₂×Q₈	—	4	42	no	—
103	C₂×D₄	—	2	28	no	—
104	C₂×D₄	—	4	42	no	—
105	C₂²×C₄	—	4	14	no	—
106	C₂²×C₄	C₁₃×D₈	1	7	yes	—
107	C₂²×C₄	C₂×D₄×C₁₃	1	7	yes	—
108	Q₁₆	—	2	28	no	—
109	Q₁₆	C₂⁴×C₁₃	1	8	yes	complemented
110	D₈	—	2	28	no	—
111	D₈	C₂⁴×C₁₃	1	8	yes	—
112	C₂×C₈	C₂×D₄×C₁₃	1	21	yes	—
113	C₂×C₈	C₂⁴×C₁₃	1	7	yes	—
114	C₁₆	C₂⁴×C₁₃	1	8	yes	complemented
115	C₁₃	C₂³×SD₃₂	1	1	yes	characteristic, direct product factor, Sylow
116	C₂³	—	4	28	no	—
117	C₂³	—	8	56	no	—
118	C₂³	C₁₃×SD₃₂	1	8	yes	direct product factor
119	C₂³	C₂×C₁₃×D₈	1	7	yes	—
120	Q₈	—	2	8	no	—
121	Q₈	—	4	28	no	—
122	D₄	—	2	8	no	—
123	D₄	—	4	28	no	—
124	C₂×C₄	—	4	28	no	—
125	C₂×C₄	C₂×C₁₃×D₈	1	21	yes	—
126	C₂×C₄	C₂²×D₄×C₁₃	1	7	yes	—
127	C₈	C₂²×D₄×C₁₃	1	7	yes	—
128	C₈	C₂⁵×C₁₃	1	1	yes	characteristic
129	C₂²	—	4	8	no	—
130	C₂²	—	8	56	no	—
131	C₂²	C₂×C₁₃×SD₃₂	1	28	yes	direct product factor
132	C₂²	C₂²×C₁₃×D₈	1	7	yes	—
133	C₄	—	4	8	no	—
134	C₄	C₂²×C₁₃×D₈	1	7	yes	—
135	C₄	C₂³×D₄×C₁₃	1	1	yes	characteristic
136	C₂	—	8	8	no	—
137	C₂	C₂²×C₁₃×SD₃₂	1	14	yes	direct product factor
138	C₂	C₂³×C₁₃×D₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₁₃×Q₁₆

C₂

yes

maximal, characteristic, complemented

C₂³×C₁₃×D₈

C₂

yes

maximal, characteristic

C₂²×C₁₃×SD₃₂

C₂

yes

maximal, direct product factor

C₂³×C₁₃×C₁₆

C₂

yes

maximal, characteristic, complemented

C₂³×Q₈×C₁₃

—

C₂³×D₄×C₁₃

—

C₂²×C₁₃×Q₁₆

—

C₂²×C₁₃×Q₁₆

C₂²

yes

complemented

C₂²×C₁₃×D₈

—

C₂²×C₁₃×D₈

C₂²

yes

—

C₂³×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×SD₃₂

C₂²

112

yes

direct product factor

C₂²×C₁₃×C₁₆

C₂²

yes

complemented

C₂⁵×C₁₃

—

C₂²×Q₈×C₁₃

—

C₂²×Q₈×C₁₃

—

C₂²×D₄×C₁₃

—

C₂²×D₄×C₁₃

—

C₂³×C₄×C₁₃

—

C₂³×C₄×C₁₃

D₄

yes

characteristic

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

C₂³

yes

complemented

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

C₂³

yes

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

C₂³

yes

—

C₁₃×SD₃₂

C₂³

yes

direct product factor

C₂×C₁₃×C₁₆

C₂³

yes

complemented

C₂³×SD₃₂

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂⁴×C₁₃

—

C₂⁴×C₁₃

—

C₂⁴×C₁₃

D₈

yes

characteristic

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₈

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₁₃×Q₁₆

—

C₁₃×Q₁₆

C₂⁴

yes

complemented

C₁₃×D₈

—

C₁₃×D₈

C₂⁴

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂⁴

yes

—

C₁₃×C₁₆

C₂⁴

yes

complemented

C₂³×Q₁₆

C₂×C₁₃

yes

characteristic, complemented

C₂³×D₈

C₂×C₁₃

yes

characteristic

C₂²×SD₃₂

C₂×C₁₃

yes

direct product factor

C₂³×C₁₆

C₂×C₁₃

yes

characteristic, complemented

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

SD₃₂

yes

direct product factor

C₂³×C₁₃

C₂×D₈

yes

—

Q₈×C₁₃

—

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

C₂²×D₄

yes

—

C₈×C₁₃

C₂²×D₄

yes

—

C₈×C₁₃

C₂⁵

yes

characteristic

C₂³×Q₈

—

C₂³×D₄

—

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂²×C₁₃

yes

complemented

C₂²×D₈

—

C₂²×D₈

C₂²×C₁₃

yes

—

C₂³×C₈

C₂²×C₁₃

yes

characteristic

C₂×SD₃₂

C₂²×C₁₃

112

yes

direct product factor

C₂²×C₁₆

C₂²×C₁₃

yes

complemented

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×SD₃₂

yes

direct product factor

C₂²×C₁₃

C₂²×D₈

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂²×D₈

yes

—

C₄×C₁₃

C₂³×D₄

yes

characteristic

C₂⁵

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×D₄

—

C₂²×D₄

—

C₂³×C₄

—

C₂³×C₄

D₄×C₁₃

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

C₂³×C₁₃

yes

complemented

C₂×D₈

—

C₂×D₈

C₂³×C₁₃

yes

—

C₂²×C₈

D₄×C₁₃

yes

—

C₂²×C₈

C₂³×C₁₃

yes

—

SD₃₂

C₂³×C₁₃

yes

direct product factor

C₂×C₁₆

C₂³×C₁₃

yes

complemented

C₂×C₁₃

—

C₂×C₁₃

C₂²×SD₃₂

yes

direct product factor

C₂×C₁₃

C₂³×D₈

yes

characteristic

C₂⁴

—

C₂⁴

—

100

C₂⁴

C₁₃×D₈

yes

characteristic

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₁₃×D₈

yes

—

107

C₂²×C₄

C₂×D₄×C₁₃

yes

—

108

Q₁₆

—

109

Q₁₆

C₂⁴×C₁₃

yes

complemented

110

D₈

—

111

D₈

C₂⁴×C₁₃

yes

—

112

C₂×C₈

C₂×D₄×C₁₃

yes

—

113

C₂×C₈

C₂⁴×C₁₃

yes

—

114

C₁₆

C₂⁴×C₁₃

yes

complemented

115

C₁₃

C₂³×SD₃₂

yes

characteristic, direct product factor, Sylow

116

C₂³

—

117

C₂³

—

118

C₂³

C₁₃×SD₃₂

yes

direct product factor

119

C₂³

C₂×C₁₃×D₈

yes

—

120

Q₈

—

121

Q₈

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×C₁₃×D₈

yes

—

126

C₂×C₄

C₂²×D₄×C₁₃

yes

—

127

C₈

C₂²×D₄×C₁₃

yes

—

128

C₈

C₂⁵×C₁₃

yes

characteristic

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×C₁₃×SD₃₂

yes

direct product factor

132

C₂²

C₂²×C₁₃×D₈

yes

—

133

C₄

—

134

C₄

C₂²×C₁₃×D₈

yes

—

135

C₄

C₂³×D₄×C₁₃

yes

characteristic

136

C₂

—

137

C₂

C₂²×C₁₃×SD₃₂

yes

direct product factor

138

C₂

C₂³×C₁₃×D₈

yes

characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₃	D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₁₃
prime radical	C₂³×C₁₃×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

subgroup

G/H

center, socle

C₂⁴×C₁₃

D₈

commutator, Frattini, cosocle

C₈

C₂⁵×C₁₃

prime radical

C₂³×C₁₃×D₈

C₂

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂⁴×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₂³×C₁₃×Q₁₆	C₂	semidirect
3. C₂²×C₁₃×SD₃₂	C₂	direct (N × H)
4. C₂³×C₁₃×C₁₆	C₂	semidirect
8. C₂²×C₁₃×Q₁₆	C₂²	semidirect
12. C₂×C₁₃×SD₃₂	C₂²	direct (N × H)
13. C₂²×C₁₃×C₁₆	C₂²	semidirect
22. C₂×C₁₃×Q₁₆	C₂³	semidirect
27. C₁₃×SD₃₂	C₂³	direct (N × H)
28. C₂×C₁₃×C₁₆	C₂³	semidirect
29. C₂³×SD₃₂	C₁₃	direct (N × H)
41. C₁₃×Q₁₆	C₂⁴	semidirect
46. C₁₃×C₁₆	C₂⁴	semidirect
47. C₂³×Q₁₆	C₂×C₁₃	semidirect
49. C₂²×SD₃₂	C₂×C₁₃	direct (N × H)
50. C₂³×C₁₆	C₂×C₁₃	semidirect
53. C₂³×C₁₃	SD₃₂	direct (N × H)
67. C₂²×Q₁₆	C₂²×C₁₃	semidirect
71. C₂×SD₃₂	C₂²×C₁₃	direct (N × H)
72. C₂²×C₁₆	C₂²×C₁₃	semidirect
75. C₂²×C₁₃	C₂×SD₃₂	direct (N × H)
88. C₂×Q₁₆	C₂³×C₁₃	semidirect
93. SD₃₂	C₂³×C₁₃	direct (N × H)
94. C₂×C₁₆	C₂³×C₁₃	semidirect
96. C₂×C₁₃	C₂²×SD₃₂	direct (N × H)
109. Q₁₆	C₂⁴×C₁₃	semidirect
114. C₁₆	C₂⁴×C₁₃	semidirect
115. C₁₃	C₂³×SD₃₂	direct (N × H)
118. C₂³	C₁₃×SD₃₂	direct (N × H)
131. C₂²	C₂×C₁₃×SD₃₂	direct (N × H)
137. C₂	C₂²×C₁₃×SD₃₂	direct (N × H)

H ≅ G/N

type

1. C₂³×C₁₃×Q₁₆

C₂

semidirect

3. C₂²×C₁₃×SD₃₂

C₂

direct (N × H)

4. C₂³×C₁₃×C₁₆

C₂

semidirect

8. C₂²×C₁₃×Q₁₆

C₂²

semidirect

12. C₂×C₁₃×SD₃₂

C₂²

direct (N × H)

13. C₂²×C₁₃×C₁₆

C₂²

semidirect

22. C₂×C₁₃×Q₁₆

C₂³

semidirect

27. C₁₃×SD₃₂

C₂³

direct (N × H)

28. C₂×C₁₃×C₁₆

C₂³

semidirect

29. C₂³×SD₃₂

C₁₃

direct (N × H)

41. C₁₃×Q₁₆

C₂⁴

semidirect

46. C₁₃×C₁₆

C₂⁴

semidirect

47. C₂³×Q₁₆

C₂×C₁₃

semidirect

49. C₂²×SD₃₂

C₂×C₁₃

direct (N × H)

50. C₂³×C₁₆

C₂×C₁₃

semidirect

53. C₂³×C₁₃

SD₃₂

direct (N × H)

67. C₂²×Q₁₆

C₂²×C₁₃

semidirect

71. C₂×SD₃₂

C₂²×C₁₃

direct (N × H)

72. C₂²×C₁₆

C₂²×C₁₃

semidirect

75. C₂²×C₁₃

C₂×SD₃₂

direct (N × H)

88. C₂×Q₁₆

C₂³×C₁₃

semidirect

93. SD₃₂

C₂³×C₁₃

direct (N × H)

94. C₂×C₁₆

C₂³×C₁₃

semidirect

96. C₂×C₁₃

C₂²×SD₃₂

direct (N × H)

109. Q₁₆

C₂⁴×C₁₃

semidirect

114. C₁₆

C₂⁴×C₁₃

semidirect

115. C₁₃

C₂³×SD₃₂

direct (N × H)

118. C₂³

C₁₃×SD₃₂

direct (N × H)

131. C₂²

C₂×C₁₃×SD₃₂

direct (N × H)

137. C₂

C₂²×C₁₃×SD₃₂

direct (N × H)

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

subgroup	index
C₂³×C₁₃×Q₁₆	2
C₂³×C₁₃×D₈	2
C₂²×C₁₃×SD₃₂	2
C₂³×C₁₃×C₁₆	2
C₂³×SD₃₂	13

subgroup

index

C₂³×C₁₃×Q₁₆

C₂³×C₁₃×D₈

C₂²×C₁₃×SD₃₂

C₂³×C₁₃×C₁₆

C₂³×SD₃₂

Subgroups — C₂³×C₁₃×SD₃₂ #15748965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×C13×SD32 #15748965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₁₃×SD₃₂ #15748965