Group #15793966 — subgroups

Subgroups — #15793966

Order 3328.

1462 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1462 conjugacy classes of subgroups — too many to draw. Only its 411 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×#11718598	C₂	1	2	yes	maximal
2	#13113107	C₂	1	4	yes	maximal
3	#13113104	C₂	1	8	yes	maximal
4	C₂²×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
6	C₂×C₁₃×C₄⋊Q₈	—	2	8	no	—
7	C₂×C₁₃×C₄⋊Q₈	C₄	1	8	yes	complemented
8	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88071	C₂²	1	8	yes	—
11	C₂×#88071	C₂²	1	4	yes	—
12	C₂×#88071	C₂²	1	16	yes	—
13	#11718598	—	2	8	no	—
14	C₂²×Q₈×C₁₃	—	2	8	no	—
15	C₂²×Q₈×C₁₃	—	4	8	no	—
16	C₂²×Q₈×C₁₃	C₂×C₄	1	8	yes	—
17	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
18	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
19	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂³×Dic₁₃	C₂³	1	8	yes	—
21	C₁₃×C₄⋊Q₈	—	2	32	no	—
22	C₂×C₁₃×C₄⋊C₄	—	2	16	no	—
23	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	16	yes	—
24	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₁₃	D₄	1	8	yes	—
26	#88071	—	2	16	no	—
27	#88071	—	2	8	no	—
28	#88071	—	4	16	no	—
29	#14663	—	13	1	no	maximal, Sylow
30	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
31	C₂×Q₈×C₁₃	—	2	16	no	—
32	C₂×Q₈×C₁₃	—	4	16	no	—
33	C₂×Q₈×C₁₃	—	4	32	no	—
34	C₂²×C₄×C₁₃	—	2	4	no	—
35	C₂²×C₄×C₁₃	—	2	8	no	—
36	C₂²×C₄×C₁₃	—	2	8	no	—
37	C₂²×C₄×C₁₃	—	2	16	no	—
38	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
39	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
42	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
43	C₂²×Dic₁₃	—	2	8	no	—
44	C₂²×Dic₁₃	—	2	16	no	—
45	C₂²×Dic₁₃	—	2	32	no	—
46	C₁₃×C₄⋊C₄	—	2	32	no	—
47	C₄²×C₁₃	C₂²⋊C₄	1	16	yes	—
48	C₂²×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
49	C₂×#383	—	13	2	no	—
50	#1446	—	13	4	no	—
51	#1445	—	13	8	no	—
52	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
53	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
54	C₂³×C₁₃	H(5,𝔽₂)	1	4	yes	—
55	C₂³×C₁₃	2₋¹⁺⁴	1	8	yes	—
56	Q₈×C₁₃	—	4	32	no	—
57	C₂×C₄×C₁₃	—	2	4	no	—
58	C₂×C₄×C₁₃	—	2	8	no	—
59	C₂×C₄×C₁₃	—	2	8	no	—
60	C₂×C₄×C₁₃	—	2	16	no	—
61	C₂×C₄×C₁₃	—	2	8	no	—
62	C₂×C₄×C₁₃	—	2	16	no	—
63	C₂×C₄×C₁₃	—	4	16	no	—
64	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	8	yes	—
65	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
66	C₂×Dic₁₃	—	4	8	no	—
67	C₂×Dic₁₃	—	4	16	no	—
68	C₂×Dic₁₃	—	4	32	no	—
69	C₂³×Q₈	C₂×D₁₃	1	2	yes	—
70	C₂×C₄⋊Q₈	—	2	8	no	—
71	C₂×C₄⋊Q₈	Dic₁₃	1	8	yes	complemented
72	C₂²×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
73	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
74	#383	—	26	8	no	—
75	C₂×#95	—	13	8	no	—
76	C₂×#95	—	13	4	no	—
77	C₂×#95	—	13	16	no	—
78	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₁₃	#518	1	2	yes	—
80	C₂²×C₁₃	#519	1	4	yes	—
81	C₂²×C₁₃	#535	1	4	yes	—
82	C₂²×C₁₃	#536	1	8	yes	—
83	C₂²×C₁₃	#563	1	16	yes	—
84	C₄×C₁₃	—	2	4	no	—
85	C₄×C₁₃	—	2	8	no	—
86	C₄×C₁₃	—	4	8	no	—
87	Dic₁₃	—	8	8	no	—
88	C₂²×Q₈	—	2	8	no	—
89	C₂²×Q₈	—	4	8	no	—
90	C₂²×Q₈	C₂×Dic₁₃	1	8	yes	—
91	C₂³×C₄	—	13	8	no	—
92	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
93	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
94	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
95	C₄⋊Q₈	—	2	32	no	—
96	C₂×C₄⋊C₄	—	2	16	no	—
97	C₂×C₄⋊C₄	C₂×Dic₁₃	1	16	yes	—
98	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
99	C₂×C₄²	C₂²⋊D₁₃	1	8	yes	—
100	#95	—	26	16	no	—
101	#95	—	26	8	no	—
102	#95	—	52	16	no	—
103	C₂×C₁₃	#2309	1	1	yes	characteristic
104	C₂×C₁₃	#2310	1	2	yes	—
105	C₂×C₁₃	#2381	1	4	yes	—
106	C₂×C₁₃	#2821	1	8	yes	—
107	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
108	C₂×Q₈	—	2	16	no	—
109	C₂×Q₈	—	4	16	no	—
110	C₂×Q₈	—	4	32	no	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	2	8	no	—
113	C₂²×C₄	—	2	16	no	—
114	C₂²×C₄	—	2	8	no	—
115	C₂²×C₄	—	26	16	no	—
116	C₂²×C₄	—	26	32	no	—
117	C₂²×C₄	—	26	8	no	—
118	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
119	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
120	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
121	C₂²×C₄	C₂×C₂²⋊D₁₃	1	8	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
123	C₄⋊C₄	—	2	32	no	—
124	C₄²	C₂²⋊Dic₁₃	1	16	yes	—
125	C₁₃	#14663	1	1	yes	characteristic, complemented, Sylow
126	C₂³	#88251	1	4	yes	—
127	C₂³	#88254	1	8	yes	—
128	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
129	C₂³	C₂²×C₂²⋊D₁₃	1	2	yes	—
130	Q₈	—	4	32	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	8	no	—
133	C₂×C₄	—	2	8	no	—
134	C₂×C₄	—	2	16	no	—
135	C₂×C₄	—	2	16	no	—
136	C₂×C₄	—	2	8	no	—
137	C₂×C₄	—	4	16	no	—
138	C₂×C₄	—	52	32	no	—
139	C₂×C₄	—	52	8	no	—
140	C₂×C₄	—	52	16	no	—
141	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	8	yes	—
142	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	4	yes	—
143	C₂²	#11719440	1	16	yes	—
144	C₂²	#11719503	1	2	yes	—
145	C₂²	#11719510	1	4	yes	—
146	C₂²	#11719517	1	4	yes	—
147	C₂²	#11719521	1	8	yes	—
148	C₂²	C₂²×C₂²⋊Dic₁₃	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	2	8	no	—
151	C₄	—	4	8	no	—
152	C₄	—	104	8	no	—
153	C₂	#13117418	1	1	yes	characteristic
154	C₂	#13117422	1	2	yes	—
155	C₂	#13118017	1	4	yes	—
156	C₂	#13122895	1	8	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718598

C₂

yes

maximal

#13113107

C₂

yes

maximal

#13113104

C₂

yes

maximal

C₂²×C₁₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂³×Q₈×C₁₃

C₂²

yes

—

C₂×C₁₃×C₄⋊Q₈

—

C₂×C₁₃×C₄⋊Q₈

C₄

yes

complemented

C₂²×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂²×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#88071

C₂²

yes

—

C₂×#88071

C₂²

yes

—

C₂×#88071

C₂²

yes

—

#11718598

—

C₂²×Q₈×C₁₃

—

C₂²×Q₈×C₁₃

—

C₂²×Q₈×C₁₃

C₂×C₄

yes

—

C₂³×C₄×C₁₃

C₂³

yes

—

C₂³×C₄×C₁₃

C₂³

yes

—

C₂³×C₄×C₁₃

C₂³

yes

characteristic

C₂³×Dic₁₃

C₂³

yes

—

C₁₃×C₄⋊Q₈

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

—

—

—

—

—

C₂⁴×C₁₃

C₂⁴

yes

characteristic

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄⋊Q₈

D₁₃

yes

—

—

—

C₂³×C₁₃

C₂³×C₄

yes

characteristic

C₂³×C₁₃

C₂²×D₄

yes

—

C₂³×C₁₃

H(5,𝔽₂)

yes

—

C₂³×C₁₃

2₋¹⁺⁴

yes

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂³×Q₈

C₂×D₁₃

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

Dic₁₃

yes

complemented

C₂²×C₄⋊C₄

C₂×D₁₃

yes

—

C₂²×C₄²

C₂×D₁₃

yes

characteristic

#383

—

C₂×#95

—

C₂×#95

—

C₂×#95

—

C₂²×C₁₃

C₂²×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

#518

yes

—

C₂²×C₁₃

#519

yes

—

C₂²×C₁₃

#535

yes

—

C₂²×C₁₃

#536

yes

—

C₂²×C₁₃

#563

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×Dic₁₃

yes

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₁₃

yes

—

C₂³×C₄

C₂²×D₁₃

yes

—

C₂³×C₄

C₂²×D₁₃

yes

characteristic

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×Dic₁₃

yes

—

C₂×C₄²

C₂×Dic₁₃

yes

—

C₂×C₄²

C₂²⋊D₁₃

yes

—

100

#95

—

101

#95

—

102

#95

—

103

C₂×C₁₃

#2309

yes

characteristic

104

C₂×C₁₃

#2310

yes

—

105

C₂×C₁₃

#2381

yes

—

106

C₂×C₁₃

#2821

yes

—

107

C₂⁴

C₂³×D₁₃

yes

characteristic

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

C₂²×Dic₁₃

yes

—

119

C₂²×C₄

C₂²×Dic₁₃

yes

—

120

C₂²×C₄

C₂²×Dic₁₃

yes

—

121

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

122

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

123

C₄⋊C₄

—

124

C₄²

C₂²⋊Dic₁₃

yes

—

125

C₁₃

#14663

yes

characteristic, complemented, Sylow

126

C₂³

#88251

yes

—

127

C₂³

#88254

yes

—

128

C₂³

C₂³×Dic₁₃

yes

characteristic

129

C₂³

C₂²×C₂²⋊D₁₃

yes

—

130

Q₈

—

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

—

141

C₂×C₄

C₂×C₂²⋊Dic₁₃

yes

—

142

C₂×C₄

C₂×C₂²⋊Dic₁₃

yes

—

143

C₂²

#11719440

yes

—

144

C₂²

#11719503

yes

—

145

C₂²

#11719510

yes

—

146

C₂²

#11719517

yes

—

147

C₂²

#11719521

yes

—

148

C₂²

C₂²×C₂²⋊Dic₁₃

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

104

—

153

C₂

#13117418

yes

characteristic

154

C₂

#13117422

yes

—

155

C₂

#13118017

yes

—

156

C₂

#13122895

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₁₃
commutator	C₂³×C₁₃	C₂³×C₄
Fitting	C₂²×C₁₃×C₄⋊Q₈	C₂
socle, prime radical, cosocle	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#14663
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₁₃

commutator

C₂³×C₁₃

C₂³×C₄

Fitting

C₂²×C₁₃×C₄⋊Q₈

C₂

socle, prime radical, cosocle

C₂⁴×C₁₃

C₂⁴

hypercenter

C₂²×C₄⋊Q₈

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

N	H ≅ G/N	type
7. C₂×C₁₃×C₄⋊Q₈	C₄	semidirect
71. C₂×C₄⋊Q₈	Dic₁₃	semidirect
125. C₁₃	#14663	semidirect

H ≅ G/N

type

7. C₂×C₁₃×C₄⋊Q₈

C₄

semidirect

71. C₂×C₄⋊Q₈

Dic₁₃

semidirect

125. C₁₃

#14663

semidirect

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

subgroup	index
C₂×#11718598	2
#13113107	2
#13113104	2
C₂²×C₁₃×C₄⋊Q₈	2
#14663	13

subgroup

index

C₂²×C₁₃×C₄⋊Q₈

Subgroups — #15793966

Conjugacy classes of subgroups

Complements

Maximal subgroups