Group #13120045 — subgroups

Subgroups — #13120045

Order 1664.

500 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 189 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic
3	#11718890	C₂	1	4	yes	maximal
4	#11718637	C₂	1	2	yes	maximal
5	#11718635	C₂	1	2	yes	maximal
6	#11718634	C₂	1	4	yes	maximal
7	C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
9	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
10	C₁₃×C₄⋊Q₈	—	2	4	no	—
11	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
16	C₂×#6101	C₂²	1	2	yes	—
17	C₂×#6101	C₂²	1	8	yes	—
18	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
19	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
20	C₄×Dic₂₆	—	2	4	no	—
21	#88071	C₂²	1	2	yes	—
22	#88071	C₂²	1	8	yes	—
23	C₂×Q₈×C₁₃	—	2	4	no	—
24	C₂×Q₈×C₁₃	—	4	4	no	—
25	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₁₃	C₂³	1	2	yes	—
30	C₂²×Dic₁₃	C₂³	1	2	yes	—
31	C₂²×Dic₁₃	C₂³	1	4	yes	—
32	C₂×Dic₂₆	—	2	2	no	—
33	C₂×Dic₂₆	—	2	2	no	—
34	C₂×Dic₂₆	Q₈	1	4	yes	complemented
35	C₁₃×C₄⋊C₄	—	2	4	no	—
36	C₁₃×C₄⋊C₄	—	2	4	no	—
37	C₄²×C₁₃	D₄	1	4	yes	—
38	C₄⋊Dic₁₃	—	2	2	no	—
39	C₄⋊Dic₁₃	Q₈	1	8	yes	complemented
40	#6101	—	2	4	no	—
41	#6101	—	4	8	no	—
42	C₄×Dic₁₃	—	2	4	no	—
43	C₄×Dic₁₃	—	2	4	no	—
44	#2567	—	13	1	no	maximal, Sylow
45	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
46	Q₈×C₁₃	—	4	8	no	—
47	C₂×C₄×C₁₃	—	2	2	no	—
48	C₂×C₄×C₁₃	—	2	4	no	—
49	C₂×C₄×C₁₃	—	2	8	no	—
50	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
55	C₂×C₄×C₁₃	#41	1	2	yes	—
56	C₂×Dic₁₃	—	2	2	no	—
57	C₂×Dic₁₃	—	2	4	no	—
58	C₂×Dic₁₃	—	2	2	no	—
59	C₂×Dic₁₃	—	2	4	no	—
60	C₂×Dic₁₃	—	2	2	no	—
61	C₂×Dic₁₃	—	2	4	no	—
62	C₂×Dic₁₃	C₂×Q₈	1	8	yes	—
63	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
64	Dic₂₆	—	2	8	no	—
65	C₂×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
66	C₂×C₄×Q₈	—	13	1	no	—
67	#395	—	13	4	no	—
68	#391	—	13	2	no	—
69	#389	—	13	4	no	—
70	#384	—	13	2	no	—
71	C₄×C₄⋊C₄	—	13	1	no	—
72	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
74	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
75	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
76	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
77	C₄×C₁₃	—	2	2	no	—
78	C₄×C₁₃	—	4	4	no	—
79	C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
80	C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
81	Dic₁₃	—	2	4	no	—
82	Dic₁₃	—	2	8	no	—
83	Dic₁₃	—	4	2	no	—
84	C₂²×Q₈	—	13	1	no	—
85	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
86	C₄⋊Q₈	—	2	4	no	—
87	C₄×Q₈	—	26	4	no	—
88	C₂×C₄⋊C₄	—	13	2	no	—
89	C₂×C₄⋊C₄	—	13	2	no	—
90	C₂×C₄⋊C₄	—	13	1	no	—
91	C₂×C₄⋊C₄	—	13	8	no	—
92	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
93	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
94	C₂×C₄²	—	13	2	no	—
95	C₂×C₄²	—	13	2	no	—
96	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
97	#95	—	13	2	no	—
98	#95	—	13	8	no	—
99	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₁₃	#535	1	1	yes	characteristic
102	C₂×C₁₃	#554	1	2	yes	—
103	C₂×C₁₃	Q₈²	1	2	yes	—
104	C₂×Q₈	—	2	4	no	—
105	C₂×Q₈	—	4	4	no	—
106	C₂×Q₈	—	13	4	no	—
107	C₂×Q₈	—	26	2	no	—
108	C₂×Q₈	—	26	2	no	—
109	C₂²×C₄	—	13	2	no	—
110	C₂²×C₄	—	13	4	no	—
111	C₂²×C₄	—	13	2	no	—
112	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
113	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	13	8	no	—
119	C₄⋊C₄	—	26	4	no	—
120	C₄⋊C₄	—	26	2	no	—
121	C₄⋊C₄	—	52	8	no	—
122	C₄²	—	26	4	no	—
123	C₄²	—	26	4	no	—
124	C₄²	C₂²⋊D₁₃	1	4	yes	—
125	C₁₃	#2567	1	1	yes	characteristic, complemented, Sylow
126	C₂³	C₂³×D₁₃	1	1	yes	characteristic
127	Q₈	—	4	8	no	—
128	Q₈	—	26	8	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	13	8	no	—
133	C₂×C₄	—	13	4	no	—
134	C₂×C₄	—	26	4	no	—
135	C₂×C₄	—	26	4	no	—
136	C₂×C₄	—	26	2	no	—
137	C₂×C₄	—	26	2	no	—
138	C₂×C₄	—	26	4	no	—
139	C₂×C₄	—	26	2	no	—
140	C₂×C₄	Q₈×D₁₃	1	4	yes	—
141	C₂×C₄	Q₈×D₁₃	1	2	yes	—
142	C₂×C₄	#6131	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
144	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
145	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
146	C₂²	#88251	1	2	yes	—
147	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
148	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
149	C₂²	C₂×#6131	1	1	yes	characteristic
150	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	4	4	no	—
153	C₄	—	26	8	no	—
154	C₄	—	26	4	no	—
155	C₄	—	52	2	no	—
156	C₄	#88198	1	4	yes	—
157	C₄	#88200	1	4	yes	—
158	C₂	#11719427	1	2	yes	—
159	C₂	#11719435	1	2	yes	—
160	C₂	#11719510	1	1	yes	characteristic
161	C₂	C₂×#88198	1	1	yes	characteristic
162	C₂	C₂×#88200	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×C₄×Dic₂₆

C₂

yes

maximal, characteristic

#11718890

C₂

yes

maximal

#11718637

C₂

yes

maximal

#11718635

C₂

yes

maximal

#11718634

C₂

yes

maximal

C₄×C₄⋊Dic₁₃

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₃

C₂²

yes

—

C₂²×Dic₂₆

C₂²

yes

characteristic

C₁₃×C₄⋊Q₈

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₄×Dic₂₆

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

Q₈

yes

complemented

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

Q₈

yes

complemented

#6101

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

#2567

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

C₂×Q₈

yes

—

Dic₂₆

—

C₂×C₄⋊Q₈

D₁₃

yes

characteristic

C₂×C₄×Q₈

—

—

—

—

—

C₄×C₄⋊C₄

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

—

C₂²×C₁₃

C₂×#41

yes

characteristic

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂²⋊Q₈

yes

—

C₄×C₁₃

C₄⋊Q₈

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₁₃

yes

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#95

—

#95

—

C₂×C₁₃

C₂×C₂²⋊Q₈

yes

characteristic

100

C₂×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

101

C₂×C₁₃

#535

yes

characteristic

102

C₂×C₁₃

#554

yes

—

103

C₂×C₁₃

Q₈²

yes

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂²×D₁₃

yes

—

113

C₂²×C₄

C₂²×D₁₃

yes

characteristic

114

C₂²×C₄

C₂²×D₁₃

yes

characteristic

115

C₂²×C₄

C₂²×D₁₃

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄²

—

123

C₄²

—

124

C₄²

C₂²⋊D₁₃

yes

—

125

C₁₃

#2567

yes

characteristic, complemented, Sylow

126

C₂³

C₂³×D₁₃

yes

characteristic

127

Q₈

—

128

Q₈

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

Q₈×D₁₃

yes

—

141

C₂×C₄

Q₈×D₁₃

yes

—

142

C₂×C₄

#6131

yes

—

143

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

144

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

145

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

146

C₂²

#88251

yes

—

147

C₂²

C₂×Q₈×D₁₃

yes

characteristic

148

C₂²

C₂×Q₈×D₁₃

yes

—

149

C₂²

C₂×#6131

yes

characteristic

150

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#88198

yes

—

157

C₄

#88200

yes

—

158

C₂

#11719427

yes

—

159

C₂

#11719435

yes

—

160

C₂

#11719510

yes

characteristic

161

C₂

C₂×#88198

yes

characteristic

162

C₂

C₂×#88200

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×C₄⋊Q₈	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#2567
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂×C₁₃×C₄⋊Q₈

C₂

hypercenter

C₂×C₄⋊Q₈

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
34. C₂×Dic₂₆	Q₈	semidirect
39. C₄⋊Dic₁₃	Q₈	semidirect
125. C₁₃	#2567	semidirect

H ≅ G/N

type

34. C₂×Dic₂₆

Q₈

semidirect

39. C₄⋊Dic₁₃

Q₈

semidirect

125. C₁₃

#2567

semidirect

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

subgroup	index
C₂×C₁₃×C₄⋊Q₈	2
C₂×C₄×Dic₂₆	2
#11718890	2
#11718637	2
#11718635	2
#11718634	2
C₄×C₄⋊Dic₁₃	2
#2567	13

subgroup

index

C₂×C₁₃×C₄⋊Q₈

C₂×C₄×Dic₂₆

C₄×C₄⋊Dic₁₃

Subgroups — #13120045

Conjugacy classes of subgroups

Complements

Maximal subgroups