Subgroups — Dic₁₃×#346 #15777739

Order 3328.

510 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718053	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718094	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₁₃	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂²×#6090	C₂²	1	2	yes	—
13	C₄²×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₄×#1055	C₄	1	1	yes	characteristic
15	C₁₃×#346	C₄	1	4	yes	complemented
16	#11718094	C₄	1	8	yes	complemented
17	C₁₆×Dic₁₃	C₄	1	8	yes	complemented
18	#11718053	C₄	1	4	yes	complemented
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃	C₈	1	4	yes	—
23	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
28	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
29	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₈×Dic₁₃	C₈	1	4	yes	—
33	C₈×Dic₁₃	C₈	1	4	yes	—
34	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
35	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
36	C₂×#6090	C₂×C₄	1	4	yes	—
37	C₂×#6090	C₂×C₄	1	4	yes	—
38	C₂×#6090	C₂×C₄	1	4	yes	—
39	C₄×#1055	C₈	1	4	yes	—
40	C₄×#346	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
44	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₁₃	C₄²	1	2	yes	—
46	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
51	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₁₃	C₄²	1	2	yes	—
53	C₄²×C₁₃	C₂×C₈	1	2	yes	—
54	C₄×Dic₁₃	—	2	8	no	—
55	C₄×Dic₁₃	C₄²	1	2	yes	—
56	C₄×Dic₁₃	C₄²	1	2	yes	—
57	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
58	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
59	C₂×#1055	C₄²	1	2	yes	—
60	C₂×#1055	C₄²	1	2	yes	—
61	C₂×#1055	C₂×C₈	1	2	yes	—
62	C₂×#1055	C₂×C₈	1	2	yes	—
63	C₂×#1055	C₂×C₈	1	2	yes	—
64	C₂×#1055	C₂×C₈	1	2	yes	—
65	C₁₃×C₁₆	C₄²	1	8	yes	complemented
66	#6090	C₄²	1	8	yes	complemented
67	C₂×#346	—	13	2	no	—
68	C₂×#346	—	13	1	no	—
69	C₂×#346	D₁₃	1	1	yes	characteristic
70	C₂×C₄×C₁₆	—	13	2	no	—
71	C₄²×C₈	—	13	1	no	—
72	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃	—	2	4	no	—
74	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
75	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×Dic₁₃	—	2	4	no	—
82	C₂×Dic₁₃	C₄×C₈	1	4	yes	—
83	C₂×Dic₁₃	C₄×C₈	1	4	yes	—
84	C₂×Dic₁₃	M₃₂	1	8	yes	—
85	C₈×C₁₃	C₄×C₈	1	4	yes	—
86	C₈×C₁₃	C₂×C₄²	1	2	yes	—
87	C₈×C₁₃	C₂×C₄²	1	2	yes	—
88	#1055	C₄×C₈	1	4	yes	—
89	#1055	C₄×C₈	1	4	yes	—
90	C₂²×C₁₆	—	13	2	no	—
91	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
92	C₂×C₄×C₈	—	13	1	no	—
93	C₂×C₄×C₈	—	13	1	no	—
94	C₂×C₄×C₈	—	13	1	no	—
95	C₂×C₄×C₈	—	13	1	no	—
96	C₂×C₄×C₈	—	13	1	no	—
97	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
98	C₄³	—	13	1	no	—
99	#346	—	13	8	no	—
100	#346	—	13	4	no	—
101	#346	Dic₁₃	1	4	yes	direct product factor
102	C₄×C₁₆	—	13	8	no	—
103	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
104	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
105	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
106	C₂²×C₁₃	C₂×M₃₂	1	4	yes	—
107	C₄×C₁₃	—	2	4	no	—
108	C₄×C₁₃	C₄³	1	1	yes	characteristic
109	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
110	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
111	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
112	Dic₁₃	—	2	4	no	—
113	Dic₁₃	#346	1	8	yes	direct product factor
114	C₂²×C₈	—	13	1	no	—
115	C₂²×C₈	—	13	1	no	—
116	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
118	C₂×C₄²	—	13	1	no	—
119	C₂×C₄²	—	13	1	no	—
120	C₂×C₄²	—	13	4	no	—
121	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
122	C₂×C₁₆	—	13	4	no	—
123	C₂×C₁₆	—	13	4	no	—
124	C₂×C₁₆	—	13	4	no	—
125	C₂×C₁₆	C₄×D₁₃	1	4	yes	—
126	C₂×C₁₆	C₄×D₁₃	1	4	yes	—
127	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
128	C₄×C₈	—	13	4	no	—
129	C₄×C₈	—	13	4	no	—
130	C₄×C₈	—	13	4	no	—
131	C₄×C₈	—	13	4	no	—
132	C₄×C₈	—	13	4	no	—
133	C₄×C₈	C₄×D₁₃	1	2	yes	—
134	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
135	C₂×C₁₃	C₄²×C₈	1	1	yes	characteristic
136	C₂×C₁₃	C₂×#346	1	2	yes	—
137	C₂×C₁₃	C₄×M₃₂	1	2	yes	—
138	C₂×C₁₃	C₄×M₃₂	1	2	yes	—
139	C₂²×C₄	—	13	2	no	—
140	C₂²×C₄	—	13	2	no	—
141	C₂²×C₄	C₈×D₁₃	1	2	yes	—
142	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
143	C₂×C₈	—	13	2	no	—
144	C₂×C₈	—	13	2	no	—
145	C₂×C₈	—	13	2	no	—
146	C₂×C₈	—	13	2	no	—
147	C₂×C₈	—	13	2	no	—
148	C₂×C₈	—	13	2	no	—
149	C₂×C₈	C₈×D₁₃	1	2	yes	—
150	C₂×C₈	C₈×D₁₃	1	2	yes	—
151	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
152	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
153	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
154	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
155	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
156	C₄²	—	13	2	no	—
157	C₄²	—	13	2	no	—
158	C₄²	—	13	2	no	—
159	C₄²	—	13	2	no	—
160	C₄²	—	26	8	no	—
161	C₄²	C₈×D₁₃	1	2	yes	—
162	C₄²	C₄×Dic₁₃	1	2	yes	—
163	C₁₆	—	13	8	no	—
164	C₁₆	C₄×Dic₁₃	1	8	yes	complemented
165	C₁₃	C₄×#346	1	1	yes	characteristic, complemented, Sylow
166	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	13	8	no	—
169	C₂×C₄	—	13	4	no	—
170	C₂×C₄	—	13	4	no	—
171	C₂×C₄	—	26	4	no	—
172	C₂×C₄	C₈×Dic₁₃	1	4	yes	—
173	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
174	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
175	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
176	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
177	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
179	C₈	—	13	4	no	—
180	C₈	—	13	4	no	—
181	C₈	C₈×Dic₁₃	1	4	yes	—
182	C₈	C₄²×D₁₃	1	2	yes	—
183	C₈	C₂×C₄×Dic₁₃	1	2	yes	—
184	C₂²	C₄×C₈×D₁₃	1	1	yes	characteristic
185	C₂²	C₄×C₈×D₁₃	1	1	yes	characteristic
186	C₂²	D₁₃×M₃₂	1	4	yes	—
187	C₂²	C₂×C₈×Dic₁₃	1	1	yes	characteristic
188	C₄	—	2	4	no	—
189	C₄	—	13	8	no	—
190	C₄	—	26	4	no	—
191	C₄	C₄×C₈×D₁₃	1	1	yes	characteristic
192	C₄	C₄×C₈×D₁₃	1	1	yes	characteristic
193	C₄	C₄²×Dic₁₃	1	1	yes	characteristic
194	C₄	C₂×C₈×Dic₁₃	1	1	yes	characteristic
195	C₂	C₄×C₈×Dic₁₃	1	1	yes	characteristic
196	C₂	D₁₃×#346	1	2	yes	—
197	C₂	M₃₂×Dic₁₃	1	2	yes	—
198	C₂	#13116429	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₃
commutator	C₂×C₁₃	C₄²×C₈
Fitting	C₂×C₁₃×#346	C₂
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	C₄×#346
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
15. C₁₃×#346	C₄	semidirect
16. #11718094	C₄	semidirect
17. C₁₆×Dic₁₃	C₄	semidirect
18. #11718053	C₄	semidirect
65. C₁₃×C₁₆	C₄²	semidirect
66. #6090	C₄²	semidirect
101. #346	Dic₁₃	direct (N × H)
113. Dic₁₃	#346	direct (N × H)
164. C₁₆	C₄×Dic₁₃	semidirect
165. C₁₃	C₄×#346	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718053	2
C₂×#11718094	2
C₂×C₁₆×Dic₁₃	2
C₄×C₈×Dic₁₃	2
C₂×C₁₃×#346	2
C₄×#346	13

Subgroups — Dic13×#346 #15777739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₁₃×#346 #15777739