Group #13117562 — subgroups

Subgroups — #13117562

Order 1664.

740 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 237 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×C₁₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×D₄×Dic₁₃	C₂	1	2	yes	maximal, complemented
3	#11718882	C₂	1	2	yes	maximal, complemented
4	#11718881	C₂	1	8	yes	maximal, complemented
5	#11718597	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718593	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
8	C₂²×D₄×C₁₃	C₂²	1	4	yes	—
9	C₂³×Dic₁₃	C₂²	1	4	yes	—
10	C₁₃×#127	C₄	1	8	yes	complemented
11	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Dic₁₃	C₂²	1	4	yes	—
13	C₂×C₂²⋊Dic₁₃	C₂²	1	8	yes	—
14	D₄×Dic₁₃	—	2	8	no	—
15	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
18	#88071	C₂²	1	4	yes	—
19	#88071	C₂²	1	4	yes	—
20	C₂⁴×C₁₃	C₂³	1	4	yes	—
21	C₂×D₄×C₁₃	—	2	4	no	—
22	C₂×D₄×C₁₃	—	2	4	no	—
23	C₂×D₄×C₁₃	—	4	8	no	—
24	C₂×D₄×C₁₃	C₂×C₄	1	8	yes	complemented
25	C₂×D₄×C₁₃	C₂×C₄	1	16	yes	complemented
26	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂²×Dic₁₃	—	2	8	no	—
29	C₂²×Dic₁₃	—	4	8	no	—
30	C₂²×Dic₁₃	C₂³	1	2	yes	—
31	C₂²×Dic₁₃	C₂³	1	4	yes	—
32	C₂²×Dic₁₃	C₂³	1	2	yes	—
33	C₄²×C₁₃	C₂×C₄	1	4	yes	complemented
34	C₂²⋊Dic₁₃	—	2	8	no	—
35	C₂²⋊Dic₁₃	—	4	8	no	—
36	C₄⋊Dic₁₃	—	2	4	no	—
37	C₄×Dic₁₃	—	2	2	no	—
38	C₄×Dic₁₃	D₄	1	4	yes	complemented
39	#2328	—	13	1	no	maximal, Sylow
40	C₂³×C₁₃	—	2	4	no	—
41	C₂³×C₁₃	—	2	4	no	—
42	C₂³×C₁₃	—	2	8	no	—
43	C₂³×C₁₃	—	2	8	no	—
44	C₂³×C₁₃	C₂²×C₄	1	8	yes	—
45	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
46	D₄×C₁₃	—	2	16	no	—
47	D₄×C₁₃	—	4	16	no	—
48	C₂×C₄×C₁₃	—	2	4	no	—
49	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₃	#41	1	2	yes	—
53	C₂×Dic₁₃	—	2	4	no	—
54	C₂×Dic₁₃	—	2	4	no	—
55	C₂×Dic₁₃	—	2	2	no	—
56	C₂×Dic₁₃	—	2	4	no	—
57	C₂×Dic₁₃	—	2	8	no	—
58	C₂×Dic₁₃	—	4	16	no	—
59	C₂×Dic₁₃	C₂×D₄	1	4	yes	—
60	C₂×#127	D₁₃	1	1	yes	characteristic
61	C₂×C₄×D₄	—	13	2	no	—
62	#390	—	13	2	no	—
63	#386	—	13	8	no	—
64	#382	—	13	1	no	—
65	#376	—	13	1	no	—
66	C₂²×C₁₃	—	2	8	no	—
67	C₂²×C₁₃	—	2	16	no	—
68	C₂²×C₁₃	—	4	4	no	—
69	C₂²×C₁₃	—	4	4	no	—
70	C₂²×C₁₃	—	4	8	no	—
71	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
72	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
74	C₂²×C₁₃	H(5,𝔽₂)	1	4	yes	—
75	C₄×C₁₃	—	2	4	no	—
76	C₄×C₁₃	C₄×D₄	1	4	yes	—
77	Dic₁₃	—	2	4	no	—
78	Dic₁₃	—	4	4	no	—
79	Dic₁₃	—	4	2	no	—
80	C₂²×D₄	C₂×D₁₃	1	2	yes	—
81	C₂²×D₄	C₂×D₁₃	1	4	yes	—
82	C₂³×C₄	—	13	4	no	—
83	#127	Dic₁₃	1	8	yes	complemented
84	C₄×D₄	—	26	8	no	—
85	C₂×C₄⋊C₄	—	13	2	no	—
86	C₂×C₂²⋊C₄	—	13	4	no	—
87	C₂×C₂²⋊C₄	—	13	8	no	—
88	C₂×C₄²	—	13	1	no	—
89	C₂×C₄²	—	13	1	no	—
90	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
91	#95	—	13	4	no	—
92	#95	—	13	4	no	—
93	C₂×C₁₃	—	4	8	no	—
94	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
95	C₂×C₁₃	#518	1	2	yes	—
96	C₂×C₁₃	#535	1	2	yes	—
97	C₂×C₁₃	#540	1	2	yes	—
98	C₂⁴	C₂²×D₁₃	1	4	yes	—
99	C₂×D₄	—	2	4	no	—
100	C₂×D₄	—	2	4	no	—
101	C₂×D₄	—	4	8	no	—
102	C₂×D₄	C₂×Dic₁₃	1	8	yes	complemented
103	C₂×D₄	C₂×Dic₁₃	1	16	yes	complemented
104	C₂²×C₄	—	13	2	no	—
105	C₂²×C₄	—	13	2	no	—
106	C₂²×C₄	—	13	4	no	—
107	C₂²×C₄	—	26	8	no	—
108	C₂²×C₄	—	52	8	no	—
109	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
110	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
111	C₄⋊C₄	—	26	4	no	—
112	C₂²⋊C₄	—	26	8	no	—
113	C₂²⋊C₄	—	52	8	no	—
114	C₄²	—	13	4	no	—
115	C₄²	—	26	2	no	—
116	C₄²	C₂×Dic₁₃	1	4	yes	complemented
117	C₁₃	#2328	1	1	yes	characteristic, complemented, Sylow
118	C₂³	—	2	4	no	—
119	C₂³	—	2	4	no	—
120	C₂³	—	2	8	no	—
121	C₂³	—	2	8	no	—
122	C₂³	C₂²×Dic₁₃	1	8	yes	—
123	C₂³	C₂³×D₁₃	1	1	yes	characteristic
124	D₄	—	2	16	no	—
125	D₄	—	4	16	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	13	4	no	—
128	C₂×C₄	—	26	4	no	—
129	C₂×C₄	—	26	4	no	—
130	C₂×C₄	—	26	2	no	—
131	C₂×C₄	—	26	4	no	—
132	C₂×C₄	—	26	8	no	—
133	C₂×C₄	—	52	16	no	—
134	C₂×C₄	D₄×D₁₃	1	2	yes	—
135	C₂×C₄	#6129	1	2	yes	—
136	C₂×C₄	C₂²×Dic₁₃	1	4	yes	—
137	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
138	C₂²	—	2	8	no	—
139	C₂²	—	2	16	no	—
140	C₂²	—	4	8	no	—
141	C₂²	—	4	4	no	—
142	C₂²	—	4	4	no	—
143	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
144	C₂²	C₂×#6129	1	1	yes	characteristic
145	C₂²	#88251	1	4	yes	—
146	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
147	C₄	—	2	4	no	—
148	C₄	—	26	4	no	—
149	C₄	—	52	2	no	—
150	C₄	—	52	4	no	—
151	C₄	D₄×Dic₁₃	1	4	yes	—
152	C₂	—	4	8	no	—
153	C₂	#11719420	1	2	yes	—
154	C₂	#11719424	1	2	yes	—
155	C₂	C₂×D₄×Dic₁₃	1	1	yes	characteristic
156	C₂	#11719503	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#127

C₂

yes

maximal, characteristic

C₂×D₄×Dic₁₃

C₂

yes

maximal, complemented

#11718882

C₂

yes

maximal, complemented

#11718881

C₂

yes

maximal, complemented

#11718597

C₂

yes

maximal, characteristic, complemented

#11718593

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₃

C₂²

yes

—

C₂²×D₄×C₁₃

C₂²

yes

—

C₂³×Dic₁₃

C₂²

yes

—

C₁₃×#127

C₄

yes

complemented

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

D₄×Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₂×C₄

yes

complemented

C₂×D₄×C₁₃

C₂×C₄

yes

complemented

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₄²×C₁₃

C₂×C₄

yes

complemented

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

D₄

yes

complemented

#2328

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂²×C₄

yes

—

C₂³×C₁₃

C₂⁴

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

C₂×#127

D₁₃

yes

characteristic

C₂×C₄×D₄

—

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

characteristic

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₄×D₄

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×D₄

C₂×D₁₃

yes

—

C₂²×D₄

C₂×D₁₃

yes

—

C₂³×C₄

—

#127

Dic₁₃

yes

complemented

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#95

—

#95

—

C₂×C₁₃

—

C₂×C₁₃

C₂×C₄×D₄

yes

characteristic

C₂×C₁₃

#518

yes

—

C₂×C₁₃

#535

yes

—

C₂×C₁₃

#540

yes

—

C₂⁴

C₂²×D₁₃

yes

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

C₂×Dic₁₃

yes

complemented

103

C₂×D₄

C₂×Dic₁₃

yes

complemented

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂²×D₁₃

yes

—

110

C₂²×C₄

C₂²×D₁₃

yes

characteristic

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₂×Dic₁₃

yes

complemented

117

C₁₃

#2328

yes

characteristic, complemented, Sylow

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

C₂²×Dic₁₃

yes

—

123

C₂³

C₂³×D₁₃

yes

characteristic

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

D₄×D₁₃

yes

—

135

C₂×C₄

#6129

yes

—

136

C₂×C₄

C₂²×Dic₁₃

yes

—

137

C₂×C₄

C₂²×Dic₁₃

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×D₄×D₁₃

yes

characteristic

144

C₂²

C₂×#6129

yes

characteristic

145

C₂²

#88251

yes

—

146

C₂²

C₂³×Dic₁₃

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

D₄×Dic₁₃

yes

—

152

C₂

—

153

C₂

#11719420

yes

—

154

C₂

#11719424

yes

—

155

C₂

C₂×D₄×Dic₁₃

yes

characteristic

156

C₂

#11719503

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Fitting, prime radical	C₂×C₁₃×#127	C₂
socle, cosocle	C₂³×C₁₃	C₂⁴
hypercenter	C₂×#127	D₁₃
hypocenter	C₁₃	#2328
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator

C₂²×C₁₃

C₂³×C₄

Fitting, prime radical

C₂×C₁₃×#127

C₂

socle, cosocle

C₂³×C₁₃

C₂⁴

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
2. C₂×D₄×Dic₁₃	C₂	semidirect
3. #11718882	C₂	semidirect
4. #11718881	C₂	semidirect
5. #11718597	C₂	semidirect
6. #11718593	C₂	semidirect
10. C₁₃×#127	C₄	semidirect
24. C₂×D₄×C₁₃	C₂×C₄	semidirect
25. C₂×D₄×C₁₃	C₂×C₄	semidirect
33. C₄²×C₁₃	C₂×C₄	semidirect
38. C₄×Dic₁₃	D₄	semidirect
83. #127	Dic₁₃	semidirect
102. C₂×D₄	C₂×Dic₁₃	semidirect
103. C₂×D₄	C₂×Dic₁₃	semidirect
116. C₄²	C₂×Dic₁₃	semidirect
117. C₁₃	#2328	semidirect

H ≅ G/N

type

2. C₂×D₄×Dic₁₃

C₂

semidirect

3. #11718882

C₂

semidirect

4. #11718881

C₂

semidirect

5. #11718597

C₂

semidirect

6. #11718593

C₂

semidirect

10. C₁₃×#127

C₄

semidirect

24. C₂×D₄×C₁₃

C₂×C₄

semidirect

25. C₂×D₄×C₁₃

C₂×C₄

semidirect

33. C₄²×C₁₃

C₂×C₄

semidirect

38. C₄×Dic₁₃

D₄

semidirect

83. #127

Dic₁₃

semidirect

102. C₂×D₄

C₂×Dic₁₃

semidirect

103. C₂×D₄

C₂×Dic₁₃

semidirect

116. C₄²

C₂×Dic₁₃

semidirect

117. C₁₃

#2328

semidirect

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

subgroup	index
C₂×C₁₃×#127	2
C₂×D₄×Dic₁₃	2
#11718882	2
#11718881	2
#11718597	2
#11718593	2
#2328	13

subgroup

index

C₂×C₁₃×#127

C₂×D₄×Dic₁₃

Subgroups — #13117562

Conjugacy classes of subgroups

Complements

Maximal subgroups