Subgroups — C₁₃×#13205 #15699544

Order 3328.

522 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#495	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#358	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₃×#357	C₂	1	2	yes	maximal
4	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×SD₁₆	—	2	1	no	—
8	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#124	—	2	1	no	—
10	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
11	C₁₃×#495	—	2	2	no	—
12	C₁₃×#495	C₄	1	4	yes	complemented
13	C₂×C₁₃×#107	—	2	1	no	—
14	C₂×C₁₃×#107	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#384	—	2	1	no	—
17	C₁₃×#358	—	2	4	no	—
18	C₁₃×#357	—	2	4	no	—
19	C₁₃×#346	—	2	2	no	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂²×D₄×C₁₃	—	2	1	no	—
22	C₂×C₁₃×Q₁₆	—	2	2	no	—
23	C₂×C₁₃×Q₁₆	—	2	1	no	—
24	C₂×C₁₃×Q₁₆	—	2	2	no	—
25	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₁₃×SD₁₆	—	2	4	no	—
27	C₂×C₁₃×SD₁₆	—	4	2	no	—
28	C₂×C₁₃×SD₁₆	—	4	2	no	—
29	C₂×C₁₃×D₈	—	2	2	no	—
30	C₂×C₁₃×D₈	—	2	1	no	—
31	C₂×C₁₃×D₈	—	2	2	no	—
32	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×#124	—	2	4	no	—
36	C₁₃×#124	—	4	2	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
39	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
40	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
41	C₂×C₁₃×C₁₆	—	2	2	no	—
42	C₂×C₁₃×C₁₆	—	2	2	no	—
43	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
44	C₁₃×#107	—	2	2	no	—
45	C₁₃×#107	—	2	2	no	—
46	C₁₃×#107	—	4	2	no	—
47	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₁₃	D₄	1	2	yes	—
49	#13205	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂⁴×C₁₃	—	4	1	no	—
51	C₂×Q₈×C₁₃	—	2	2	no	—
52	C₂×Q₈×C₁₃	—	4	1	no	—
53	C₂×Q₈×C₁₃	—	4	1	no	—
54	C₂×Q₈×C₁₃	—	4	1	no	—
55	C₂×Q₈×C₁₃	—	4	1	no	—
56	C₂×Q₈×C₁₃	—	4	1	no	—
57	C₂×D₄×C₁₃	—	2	2	no	—
58	C₂×D₄×C₁₃	—	4	1	no	—
59	C₂×D₄×C₁₃	—	4	1	no	—
60	C₂×D₄×C₁₃	—	4	1	no	—
61	C₂×D₄×C₁₃	—	4	1	no	—
62	C₂×D₄×C₁₃	—	4	1	no	—
63	C₂²×C₄×C₁₃	—	4	1	no	—
64	C₂²×C₄×C₁₃	—	4	2	no	—
65	C₂²×C₄×C₁₃	D₈	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₁₃×Q₁₆	—	2	4	no	—
68	C₁₃×Q₁₆	—	4	2	no	—
69	C₁₃×SD₁₆	—	4	8	no	—
70	C₁₃×D₈	—	2	4	no	—
71	C₁₃×D₈	—	4	2	no	—
72	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₃	D₈	1	2	yes	—
74	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	#41	1	2	yes	—
79	C₁₃×C₄⋊C₄	—	4	2	no	—
80	C₁₃×C₄⋊C₄	—	4	2	no	—
81	C₁₃×C₄⋊C₄	—	8	4	no	—
82	C₁₃×C₂²⋊C₄	—	4	8	no	—
83	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₁₃	SD₁₆	1	2	yes	—
85	C₁₃×C₁₆	—	2	4	no	—
86	C₂×#495	C₂×C₁₃	1	1	yes	characteristic
87	C₂×#358	C₂×C₁₃	1	2	yes	complemented
88	C₂×#357	C₂×C₁₃	1	2	yes	—
89	C₂×#346	C₂×C₁₃	1	1	yes	characteristic, complemented
90	#1845	C₂×C₁₃	1	1	yes	characteristic, complemented
91	C₂³×C₁₃	—	4	2	no	—
92	C₂³×C₁₃	—	8	1	no	—
93	C₂³×C₁₃	—	8	1	no	—
94	C₂³×C₁₃	—	8	2	no	—
95	C₂³×C₁₃	—	8	2	no	—
96	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
97	Q₈×C₁₃	—	4	2	no	—
98	Q₈×C₁₃	—	4	2	no	—
99	Q₈×C₁₃	—	8	1	no	—
100	Q₈×C₁₃	—	8	1	no	—
101	D₄×C₁₃	—	4	2	no	—
102	D₄×C₁₃	—	4	2	no	—
103	D₄×C₁₃	—	8	1	no	—
104	D₄×C₁₃	—	8	1	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	—	8	1	no	—
109	C₂×C₄×C₁₃	—	8	1	no	—
110	C₂×C₄×C₁₃	—	8	2	no	—
111	C₂×C₄×C₁₃	—	8	4	no	—
112	C₂×C₄×C₁₃	#102	1	4	yes	—
113	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₈×C₁₃	#102	1	4	yes	—
120	C₈×C₁₃	C₄×D₄	1	2	yes	—
121	C₈×C₁₃	#121	1	2	yes	—
122	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
123	C₂²×SD₁₆	—	2	1	no	—
124	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
125	C₂×#124	—	2	1	no	—
126	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
127	#495	—	2	2	no	—
128	#495	C₄×C₁₃	1	4	yes	complemented
129	C₂×#107	—	2	1	no	—
130	C₂×#107	C₂²×C₁₃	1	2	yes	—
131	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
132	#384	—	2	1	no	—
133	#358	—	2	4	no	—
134	#357	—	2	4	no	—
135	#346	—	2	2	no	—
136	C₂²×C₁₃	—	8	4	no	—
137	C₂²×C₁₃	—	8	2	no	—
138	C₂²×C₁₃	—	16	2	no	—
139	C₂²×C₁₃	—	16	1	no	—
140	C₂²×C₁₃	—	16	1	no	—
141	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
142	C₂²×C₁₃	#486	1	1	yes	characteristic
143	C₂²×C₁₃	#493	1	1	yes	characteristic
144	C₂²×C₁₃	#509	1	2	yes	—
145	C₂²×C₁₃	#510	1	2	yes	—
146	C₄×C₁₃	—	2	4	no	—
147	C₄×C₁₃	—	8	2	no	—
148	C₄×C₁₃	—	16	2	no	—
149	C₄×C₁₃	#390	1	1	yes	characteristic
150	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
151	C₄×C₁₃	#486	1	1	yes	characteristic
152	C₄×C₁₃	#492	1	1	yes	characteristic
153	C₂²×Q₈	—	2	1	no	—
154	C₂²×D₄	—	2	1	no	—
155	C₂×Q₁₆	—	2	2	no	—
156	C₂×Q₁₆	—	2	1	no	—
157	C₂×Q₁₆	—	2	2	no	—
158	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
159	C₂×SD₁₆	—	2	4	no	—
160	C₂×SD₁₆	—	4	2	no	—
161	C₂×SD₁₆	—	4	2	no	—
162	C₂×D₈	—	2	2	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	—	2	2	no	—
165	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
166	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
168	#124	—	2	4	no	—
169	#124	—	4	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	4	2	no	—
172	C₂×C₂²⋊C₄	—	4	2	no	—
173	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	D₄×C₁₃	1	4	yes	—
177	#107	—	2	2	no	—
178	#107	—	2	2	no	—
179	#107	—	4	2	no	—
180	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
181	C₄×C₈	D₄×C₁₃	1	2	yes	—
182	C₂×C₁₃	—	16	2	no	—
183	C₂×C₁₃	#1957	1	1	yes	characteristic
184	C₂×C₁₃	#2139	1	2	yes	—
185	C₂×C₁₃	#2243	1	2	yes	—
186	C₂×C₁₃	#2250	1	1	yes	characteristic
187	C₂×C₁₃	#2251	1	1	yes	characteristic
188	C₂⁴	—	4	1	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	4	1	no	—
191	C₂×Q₈	—	4	1	no	—
192	C₂×Q₈	—	4	1	no	—
193	C₂×Q₈	—	4	1	no	—
194	C₂×Q₈	—	4	1	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	C₁₃×D₈	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
205	Q₁₆	—	2	4	no	—
206	Q₁₆	—	4	2	no	—
207	SD₁₆	—	4	8	no	—
208	D₈	—	2	4	no	—
209	D₈	—	4	2	no	—
210	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
211	C₂×C₈	C₁₃×D₈	1	2	yes	—
212	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
213	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
215	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
216	C₂×C₈	C₁₃×#41	1	2	yes	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	8	4	no	—
220	C₂²⋊C₄	—	4	8	no	—
221	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
222	C₄²	C₁₃×SD₁₆	1	2	yes	—
223	C₁₆	—	2	4	no	—
224	C₁₃	#13205	1	1	yes	characteristic, direct product factor, Sylow
225	C₂³	—	4	2	no	—
226	C₂³	—	8	1	no	—
227	C₂³	—	8	1	no	—
228	C₂³	—	8	2	no	—
229	C₂³	—	8	2	no	—
230	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
231	Q₈	—	4	2	no	—
232	Q₈	—	4	2	no	—
233	Q₈	—	8	1	no	—
234	Q₈	—	8	1	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	8	1	no	—
238	D₄	—	8	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	8	1	no	—
243	C₂×C₄	—	8	1	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	C₁₃×#102	1	4	yes	—
247	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
248	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
249	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
250	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
251	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
252	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
253	C₈	C₁₃×#102	1	4	yes	—
254	C₈	C₄×D₄×C₁₃	1	2	yes	—
255	C₈	C₁₃×#121	1	2	yes	—
256	C₂²	—	8	4	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	16	2	no	—
259	C₂²	—	16	1	no	—
260	C₂²	—	16	1	no	—
261	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
262	C₂²	C₁₃×#486	1	1	yes	characteristic
263	C₂²	C₁₃×#493	1	1	yes	characteristic
264	C₂²	C₁₃×#509	1	2	yes	—
265	C₂²	C₁₃×#510	1	2	yes	—
266	C₄	—	2	4	no	—
267	C₄	—	8	2	no	—
268	C₄	—	16	2	no	—
269	C₄	C₁₃×#390	1	1	yes	characteristic
270	C₄	C₂×C₁₃×#102	1	1	yes	characteristic
271	C₄	C₁₃×#486	1	1	yes	characteristic
272	C₄	C₁₃×#492	1	1	yes	characteristic
273	C₂	—	16	2	no	—
274	C₂	C₁₃×#1957	1	1	yes	characteristic
275	C₂	C₁₃×#2139	1	2	yes	—
276	C₂	C₁₃×#2243	1	2	yes	—
277	C₂	C₁₃×#2250	1	1	yes	characteristic
278	C₂	C₁₃×#2251	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#358	C₂	semidirect
4. C₂×C₁₃×#346	C₂	semidirect
5. C₁₃×#1845	C₂	semidirect
12. C₁₃×#495	C₄	semidirect
49. #13205	C₁₃	direct (N × H)
87. C₂×#358	C₂×C₁₃	semidirect
89. C₂×#346	C₂×C₁₃	semidirect
90. #1845	C₂×C₁₃	semidirect
128. #495	C₄×C₁₃	semidirect
224. C₁₃	#13205	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#495	2
C₂×C₁₃×#358	2
C₂×C₁₃×#357	2
C₂×C₁₃×#346	2
C₁₃×#1845	2
#13205	13

Subgroups — C13×#13205 #15699544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13205 #15699544