Subgroups — C₁₃×#9105 #15695444

Order 3328.

602 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2975	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2328	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1478	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1460	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1274	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#127	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#436	—	2	4	no	—
10	C₁₃×#431	C₂²	1	2	yes	—
11	C₁₃×#431	C₂²	1	2	yes	—
12	C₁₃×#407	—	2	2	no	—
13	C₁₃×#390	C₄	1	1	yes	characteristic
14	C₁₃×#390	C₄	1	1	yes	characteristic
15	C₁₃×#386	—	2	2	no	—
16	C₁₃×#386	—	2	2	no	—
17	C₁₃×#382	C₄	1	1	yes	characteristic
18	C₁₃×#376	C₂²	1	1	yes	characteristic
19	C₁₃×D₄⋊C₈	—	2	4	no	—
20	C₁₃×#323	—	2	2	no	—
21	C₂²×D₄×C₁₃	—	2	2	no	—
22	C₂²×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	—	2	1	no	—
25	C₂³×C₄×C₁₃	D₄	1	2	yes	—
26	C₂×C₁₃×M₁₆	—	2	4	no	—
27	C₁₃×#127	—	2	2	no	—
28	C₁₃×#127	—	2	2	no	—
29	C₄×D₄×C₁₃	—	2	2	no	—
30	C₄×D₄×C₁₃	—	2	2	no	—
31	C₄×D₄×C₁₃	—	2	2	no	—
32	C₄×D₄×C₁₃	—	2	2	no	—
33	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
36	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×C₄⋊C₈	—	2	2	no	—
43	C₁₃×C₄⋊C₈	—	2	2	no	—
44	C₁₃×C₂²⋊C₈	—	2	4	no	—
45	C₁₃×C₂²⋊C₈	—	2	2	no	—
46	C₁₃×C₂²⋊C₈	—	2	2	no	—
47	C₄×C₈×C₁₃	—	2	2	no	—
48	C₄×C₈×C₁₃	D₄	1	4	yes	complemented
49	C₁₃×#95	—	2	2	no	—
50	C₁₃×#95	—	2	1	no	—
51	C₁₃×#95	—	2	1	no	—
52	#9105	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂⁴×C₁₃	—	2	1	no	—
54	C₂⁴×C₁₃	C₂²⋊C₄	1	2	yes	—
55	C₂×D₄×C₁₃	—	2	8	no	—
56	C₂×D₄×C₁₃	—	2	2	no	—
57	C₂×D₄×C₁₃	—	2	2	no	—
58	C₂×D₄×C₁₃	—	2	1	no	—
59	C₂×D₄×C₁₃	—	2	1	no	—
60	C₂×D₄×C₁₃	—	2	1	no	—
61	C₂×D₄×C₁₃	—	2	2	no	—
62	C₂×D₄×C₁₃	—	2	2	no	—
63	C₂×D₄×C₁₃	—	8	2	no	—
64	C₂×D₄×C₁₃	—	8	2	no	—
65	C₂×D₄×C₁₃	M₁₆	1	2	yes	complemented
66	C₂²×C₄×C₁₃	—	2	1	no	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	—	2	1	no	—
70	C₂²×C₄×C₁₃	—	2	1	no	—
71	C₂²×C₄×C₁₃	—	4	2	no	—
72	C₂²×C₄×C₁₃	—	8	2	no	—
73	C₂²×C₄×C₁₃	—	8	2	no	—
74	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₁₃×M₁₆	—	8	4	no	—
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	—	4	2	no	—
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₄⋊C₄	—	2	2	no	—
83	C₁₃×C₂²⋊C₄	—	2	4	no	—
84	C₁₃×C₂²⋊C₄	—	4	1	no	—
85	C₁₃×C₂²⋊C₄	—	4	1	no	—
86	C₁₃×C₂²⋊C₄	—	8	2	no	—
87	C₁₃×C₂²⋊C₄	—	8	2	no	—
88	C₄²×C₁₃	—	2	1	no	—
89	C₄²×C₁₃	—	2	2	no	—
90	C₄²×C₁₃	M₁₆	1	2	yes	complemented
91	C₄²×C₁₃	C₂×D₄	1	2	yes	—
92	C₄²×C₁₃	#41	1	2	yes	—
93	#2975	C₂×C₁₃	1	1	yes	characteristic, complemented
94	#2328	C₂×C₁₃	1	1	yes	characteristic
95	#1478	C₂×C₁₃	1	2	yes	complemented
96	#1460	C₂×C₁₃	1	1	yes	characteristic, complemented
97	#1274	C₂×C₁₃	1	2	yes	complemented
98	C₂³×C₁₃	—	2	2	no	—
99	C₂³×C₁₃	—	2	2	no	—
100	C₂³×C₁₃	—	2	2	no	—
101	C₂³×C₁₃	—	2	2	no	—
102	C₂³×C₁₃	—	4	2	no	—
103	C₂³×C₁₃	—	4	2	no	—
104	C₂³×C₁₃	—	4	2	no	—
105	C₂³×C₁₃	—	4	2	no	—
106	C₂³×C₁₃	—	4	1	no	—
107	C₂³×C₁₃	—	4	1	no	—
108	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	D₄×C₁₃	—	2	4	no	—
110	D₄×C₁₃	—	4	2	no	—
111	D₄×C₁₃	—	4	2	no	—
112	D₄×C₁₃	—	4	2	no	—
113	D₄×C₁₃	—	8	8	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	—	4	1	no	—
120	C₂×C₄×C₁₃	—	4	1	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	1	no	—
123	C₂×C₄×C₁₃	—	4	1	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	1	no	—
126	C₂×C₄×C₁₃	—	4	1	no	—
127	C₂×C₄×C₁₃	—	8	2	no	—
128	C₂×C₄×C₁₃	—	8	4	no	—
129	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#121	1	2	yes	—
132	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#131	1	2	yes	—
136	C₈×C₁₃	—	4	4	no	—
137	C₈×C₁₃	—	8	2	no	—
138	C₂×#127	C₄×C₁₃	1	1	yes	characteristic
139	C₂×C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
140	C₂×C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
141	#436	—	2	4	no	—
142	#431	C₂²×C₁₃	1	2	yes	—
143	#431	C₂²×C₁₃	1	2	yes	—
144	#407	—	2	2	no	—
145	#390	C₄×C₁₃	1	1	yes	characteristic
146	#390	C₄×C₁₃	1	1	yes	characteristic
147	#386	—	2	2	no	—
148	#386	—	2	2	no	—
149	#382	C₄×C₁₃	1	1	yes	characteristic
150	#376	C₂²×C₁₃	1	1	yes	characteristic
151	D₄⋊C₈	—	2	4	no	—
152	#323	—	2	2	no	—
153	C₂²×C₁₃	—	2	1	no	—
154	C₂²×C₁₃	—	2	1	no	—
155	C₂²×C₁₃	—	4	4	no	—
156	C₂²×C₁₃	—	4	4	no	—
157	C₂²×C₁₃	—	4	2	no	—
158	C₂²×C₁₃	—	4	2	no	—
159	C₂²×C₁₃	—	4	2	no	—
160	C₂²×C₁₃	—	4	2	no	—
161	C₂²×C₁₃	—	8	2	no	—
162	C₂²×C₁₃	—	8	2	no	—
163	C₂²×C₁₃	—	8	1	no	—
164	C₂²×C₁₃	—	8	1	no	—
165	C₂²×C₁₃	#386	1	1	yes	characteristic
166	C₂²×C₁₃	#407	1	1	yes	characteristic
167	C₂²×C₁₃	#408	1	1	yes	characteristic
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	2	1	no	—
170	C₄×C₁₃	—	2	1	no	—
171	C₄×C₁₃	—	4	2	no	—
172	C₄×C₁₃	—	4	2	no	—
173	C₄×C₁₃	—	8	1	no	—
174	C₄×C₁₃	—	8	1	no	—
175	C₄×C₁₃	#435	1	2	yes	—
176	C₄×C₁₃	#436	1	2	yes	—
177	C₂²×D₄	—	2	2	no	—
178	C₂²×D₄	C₂×C₄×C₁₃	1	1	yes	characteristic
179	C₂²×D₄	C₂×C₄×C₁₃	1	1	yes	characteristic
180	C₂³×C₄	—	2	1	no	—
181	C₂³×C₄	D₄×C₁₃	1	2	yes	—
182	C₂×M₁₆	—	2	4	no	—
183	#127	—	2	2	no	—
184	#127	—	2	2	no	—
185	C₄×D₄	—	2	2	no	—
186	C₄×D₄	—	2	2	no	—
187	C₄×D₄	—	2	2	no	—
188	C₄×D₄	—	2	2	no	—
189	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
191	C₂×C₂²⋊C₄	—	2	1	no	—
192	C₂×C₂²⋊C₄	—	2	2	no	—
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
195	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
196	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
197	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
198	C₄⋊C₈	—	2	2	no	—
199	C₄⋊C₈	—	2	2	no	—
200	C₂²⋊C₈	—	2	4	no	—
201	C₂²⋊C₈	—	2	2	no	—
202	C₂²⋊C₈	—	2	2	no	—
203	C₄×C₈	—	2	2	no	—
204	C₄×C₈	D₄×C₁₃	1	4	yes	complemented
205	#95	—	2	2	no	—
206	#95	—	2	1	no	—
207	#95	—	2	1	no	—
208	C₂×C₁₃	—	2	1	no	—
209	C₂×C₁₃	—	2	1	no	—
210	C₂×C₁₃	—	8	2	no	—
211	C₂×C₁₃	—	8	2	no	—
212	C₂×C₁₃	#1870	1	1	yes	characteristic
213	C₂×C₁₃	#1910	1	1	yes	characteristic
214	C₂×C₁₃	#1911	1	1	yes	characteristic
215	C₂⁴	—	2	1	no	—
216	C₂⁴	C₁₃×C₂²⋊C₄	1	2	yes	—
217	C₂×D₄	—	2	8	no	—
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	2	1	no	—
221	C₂×D₄	—	2	1	no	—
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	8	2	no	—
226	C₂×D₄	—	8	2	no	—
227	C₂×D₄	C₁₃×M₁₆	1	2	yes	complemented
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	8	2	no	—
235	C₂²×C₄	—	8	2	no	—
236	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
237	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
240	M₁₆	—	8	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	4	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	—	4	1	no	—
247	C₂²⋊C₄	—	4	1	no	—
248	C₂²⋊C₄	—	8	2	no	—
249	C₂²⋊C₄	—	8	2	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	2	2	no	—
252	C₄²	C₁₃×M₁₆	1	2	yes	complemented
253	C₄²	C₂×D₄×C₁₃	1	2	yes	—
254	C₄²	C₁₃×#41	1	2	yes	—
255	C₁₃	#9105	1	1	yes	characteristic, direct product factor, Sylow
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	2	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	1	no	—
265	C₂³	—	4	1	no	—
266	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
267	D₄	—	2	4	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	2	no	—
270	D₄	—	4	2	no	—
271	D₄	—	8	8	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	8	2	no	—
286	C₂×C₄	—	8	4	no	—
287	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
288	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
289	C₂×C₄	C₁₃×#121	1	2	yes	—
290	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
291	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
292	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
293	C₂×C₄	C₁₃×#131	1	2	yes	—
294	C₈	—	4	4	no	—
295	C₈	—	8	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	4	4	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	1	no	—
307	C₂²	—	8	1	no	—
308	C₂²	C₁₃×#386	1	1	yes	characteristic
309	C₂²	C₁₃×#407	1	1	yes	characteristic
310	C₂²	C₁₃×#408	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	2	no	—
316	C₄	—	8	1	no	—
317	C₄	—	8	1	no	—
318	C₄	C₁₃×#435	1	2	yes	—
319	C₄	C₁₃×#436	1	2	yes	—
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	8	2	no	—
323	C₂	—	8	2	no	—
324	C₂	C₁₃×#1870	1	1	yes	characteristic
325	C₂	C₁₃×#1910	1	1	yes	characteristic
326	C₂	C₁₃×#1911	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#386
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₂×C₁₃×#127	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#2975	C₂	semidirect
3. C₁₃×#1478	C₂	semidirect
4. C₁₃×#1460	C₂	semidirect
5. C₁₃×#1274	C₂	semidirect
48. C₄×C₈×C₁₃	D₄	semidirect
52. #9105	C₁₃	direct (N × H)
65. C₂×D₄×C₁₃	M₁₆	semidirect
90. C₄²×C₁₃	M₁₆	semidirect
93. #2975	C₂×C₁₃	semidirect
95. #1478	C₂×C₁₃	semidirect
96. #1460	C₂×C₁₃	semidirect
97. #1274	C₂×C₁₃	semidirect
204. C₄×C₈	D₄×C₁₃	semidirect
227. C₂×D₄	C₁₃×M₁₆	semidirect
252. C₄²	C₁₃×M₁₆	semidirect
255. C₁₃	#9105	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#2975	2
C₁₃×#2328	2
C₁₃×#1478	2
C₁₃×#1460	2
C₁₃×#1274	2
#9105	13

Subgroups — C13×#9105 #15695444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9105 #15695444