Subgroups — C₁₃×#33651 #15719990

Order 3328.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3059	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2913	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2237	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2237	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2236	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2235	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2159	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2141	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#358	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2113	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
14	C₁₃×#536	—	2	1	no	—
15	C₁₃×#520	—	2	1	no	—
16	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
18	C₁₃×#470	—	2	2	no	—
19	C₁₃×#467	C₂²	1	2	yes	—
20	C₁₃×#467	C₂²	1	2	yes	—
21	C₁₃×#428	—	2	1	no	—
22	C₁₃×#428	—	2	2	no	—
23	C₁₃×#427	C₂²	1	1	yes	characteristic
24	C₁₃×#427	C₂²	1	1	yes	characteristic
25	C₁₃×#427	C₂²	1	2	yes	—
26	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
27	C₂×C₁₃×#107	C₂²	1	2	yes	—
28	C₁₃×#419	—	2	1	no	—
29	C₁₃×#367	C₂²	1	1	yes	characteristic
30	C₁₃×#367	C₂²	1	1	yes	characteristic
31	C₁₃×#367	C₂²	1	2	yes	—
32	C₁₃×#366	C₂²	1	1	yes	characteristic
33	C₁₃×#366	C₂²	1	1	yes	characteristic
34	C₁₃×#366	C₂²	1	2	yes	—
35	C₁₃×#358	C₂²	1	2	yes	—
36	C₁₃×#358	C₂²	1	2	yes	—
37	C₁₃×#358	C₂²	1	2	yes	—
38	C₁₃×#358	C₂²	1	2	yes	—
39	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
41	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
42	C₂²×Q₈×C₁₃	—	2	1	no	—
43	C₂×C₁₃×Q₁₆	—	2	1	no	—
44	C₂×C₁₃×Q₁₆	—	2	2	no	—
45	C₂×C₁₃×Q₁₆	—	2	2	no	—
46	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
47	C₁₃×#131	—	2	2	no	—
48	C₁₃×#131	—	2	2	no	—
49	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
50	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
51	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×C₄⋊Q₈	—	4	1	no	—
54	C₁₃×#124	—	4	1	no	—
55	C₁₃×#123	—	4	2	no	—
56	C₁₃×C₂²⋊Q₈	—	2	2	no	—
57	C₁₃×C₂²⋊Q₈	—	2	2	no	—
58	C₄×Q₈×C₁₃	—	4	1	no	—
59	C₄×D₄×C₁₃	—	4	2	no	—
60	C₄×D₄×C₁₃	—	4	1	no	—
61	C₁₃×#117	—	2	1	no	—
62	C₁₃×#117	—	2	2	no	—
63	C₁₃×#117	—	2	1	no	—
64	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
65	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
66	C₁₃×M₃₂	—	2	2	no	—
67	C₂×C₁₃×C₁₆	—	2	2	no	—
68	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
70	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
71	C₁₃×#107	—	2	2	no	—
72	C₁₃×#107	—	2	1	no	—
73	C₁₃×#107	C₂³	1	2	yes	—
74	C₁₃×#107	C₂³	1	1	yes	characteristic
75	C₁₃×#107	C₂³	1	1	yes	characteristic
76	C₁₃×#107	C₂³	1	1	yes	characteristic
77	C₁₃×#107	C₂³	1	1	yes	characteristic
78	C₁₃×#106	—	2	2	no	—
79	C₁₃×#106	—	2	1	no	—
80	C₁₃×#106	—	2	1	no	—
81	C₁₃×#103	—	2	2	no	—
82	C₁₃×#103	—	2	2	no	—
83	#33651	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₁₃×#41	—	2	2	no	—
85	C₁₃×#41	—	2	2	no	—
86	C₂×Q₈×C₁₃	—	2	2	no	—
87	C₂×Q₈×C₁₃	—	4	1	no	—
88	C₂×Q₈×C₁₃	—	4	2	no	—
89	C₂×Q₈×C₁₃	—	4	2	no	—
90	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
91	C₂×D₄×C₁₃	D₈	1	2	yes	—
92	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₁₃	—	4	1	no	—
94	C₂²×C₄×C₁₃	—	4	2	no	—
95	C₂²×C₄×C₁₃	—	4	1	no	—
96	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₁₃×Q₁₆	—	2	2	no	—
99	C₁₃×Q₁₆	—	2	2	no	—
100	C₁₃×Q₁₆	—	4	2	no	—
101	C₁₃×M₁₆	—	2	2	no	—
102	C₁₃×M₁₆	—	2	4	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	#41	1	2	yes	—
108	C₂×C₈×C₁₃	#41	1	2	yes	—
109	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
110	C₁₃×C₄⋊C₄	—	2	1	no	—
111	C₁₃×C₄⋊C₄	—	2	1	no	—
112	C₁₃×C₄⋊C₄	—	2	1	no	—
113	C₁₃×C₄⋊C₄	—	2	2	no	—
114	C₁₃×C₄⋊C₄	—	2	1	no	—
115	C₁₃×C₄⋊C₄	—	4	2	no	—
116	C₁₃×C₄⋊C₄	—	4	1	no	—
117	C₁₃×C₄⋊C₄	—	4	2	no	—
118	C₁₃×C₄⋊C₄	—	4	2	no	—
119	C₁₃×C₂²⋊C₄	—	4	2	no	—
120	C₁₃×C₂²⋊C₄	—	4	1	no	—
121	C₁₃×C₂²⋊C₄	—	4	1	no	—
122	C₁₃×C₂²⋊C₄	—	4	2	no	—
123	C₁₃×C₂²⋊C₄	—	4	2	no	—
124	C₄²×C₁₃	—	4	2	no	—
125	C₄²×C₁₃	—	4	1	no	—
126	C₄²×C₁₃	—	4	1	no	—
127	C₁₃×C₁₆	—	2	1	no	—
128	C₁₃×C₁₆	—	2	1	no	—
129	C₁₃×C₁₆	—	2	2	no	—
130	#3059	C₂×C₁₃	1	1	yes	characteristic
131	#2913	C₂×C₁₃	1	1	yes	characteristic
132	#2237	C₂×C₁₃	1	2	yes	complemented
133	#2237	C₂×C₁₃	1	2	yes	complemented
134	#2236	C₂×C₁₃	1	2	yes	complemented
135	#2235	C₂×C₁₃	1	2	yes	complemented
136	#2159	C₂×C₁₃	1	1	yes	characteristic, complemented
137	#2158	C₂×C₁₃	1	1	yes	characteristic, complemented
138	#2141	C₂×C₁₃	1	1	yes	characteristic, complemented
139	C₂×#358	C₂×C₁₃	1	1	yes	characteristic, complemented
140	#2113	C₂×C₁₃	1	1	yes	characteristic
141	C₂³×C₁₃	C₂×D₈	1	2	yes	—
142	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
143	Q₈×C₁₃	—	2	2	no	—
144	Q₈×C₁₃	—	4	2	no	—
145	Q₈×C₁₃	—	4	2	no	—
146	Q₈×C₁₃	—	8	2	no	—
147	D₄×C₁₃	—	2	4	no	—
148	D₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	—	4	1	no	—
152	C₂×C₄×C₁₃	—	4	1	no	—
153	C₂×C₄×C₁₃	—	4	1	no	—
154	C₂×C₄×C₁₃	—	4	2	no	—
155	C₂×C₄×C₁₃	—	4	1	no	—
156	C₂×C₄×C₁₃	—	4	2	no	—
157	C₂×C₄×C₁₃	—	8	2	no	—
158	C₂×C₄×C₁₃	—	16	1	no	—
159	C₂×C₄×C₁₃	—	16	2	no	—
160	C₂×C₄×C₁₃	#123	1	2	yes	—
161	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
162	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₁₃	#137	1	2	yes	—
164	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
165	C₈×C₁₃	—	2	2	no	—
166	C₈×C₁₃	#123	1	2	yes	—
167	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
168	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
169	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
170	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
171	#536	—	2	1	no	—
172	#520	—	2	1	no	—
173	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
174	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
175	#470	—	2	2	no	—
176	#467	C₂²×C₁₃	1	2	yes	—
177	#467	C₂²×C₁₃	1	2	yes	—
178	#428	—	2	1	no	—
179	#428	—	2	2	no	—
180	#427	C₂²×C₁₃	1	1	yes	characteristic
181	#427	C₂²×C₁₃	1	2	yes	—
182	#427	C₂²×C₁₃	1	1	yes	characteristic
183	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
184	C₂×#107	C₂²×C₁₃	1	2	yes	—
185	#419	—	2	1	no	—
186	#367	C₂²×C₁₃	1	1	yes	characteristic
187	#367	C₂²×C₁₃	1	1	yes	characteristic
188	#367	C₂²×C₁₃	1	2	yes	—
189	#366	C₂²×C₁₃	1	1	yes	characteristic
190	#366	C₂²×C₁₃	1	1	yes	characteristic
191	#366	C₂²×C₁₃	1	2	yes	—
192	#358	C₂²×C₁₃	1	2	yes	—
193	#358	C₂²×C₁₃	1	2	yes	—
194	#358	C₂²×C₁₃	1	2	yes	—
195	#358	C₂²×C₁₃	1	2	yes	—
196	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
197	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
198	C₂²×C₁₃	—	2	1	no	—
199	C₂²×C₁₃	—	2	1	no	—
200	C₂²×C₁₃	—	2	2	no	—
201	C₂²×C₁₃	—	2	2	no	—
202	C₂²×C₁₃	—	2	2	no	—
203	C₂²×C₁₃	#480	1	2	yes	—
204	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
205	C₄×C₁₃	—	2	2	no	—
206	C₄×C₁₃	—	8	2	no	—
207	C₄×C₁₃	—	8	1	no	—
208	C₄×C₁₃	—	8	2	no	—
209	C₄×C₁₃	—	8	1	no	—
210	C₄×C₁₃	—	8	1	no	—
211	C₄×C₁₃	—	8	1	no	—
212	C₄×C₁₃	#480	1	2	yes	—
213	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
214	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
215	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
216	C₂²×Q₈	—	2	1	no	—
217	C₂×Q₁₆	—	2	1	no	—
218	C₂×Q₁₆	—	2	2	no	—
219	C₂×Q₁₆	—	2	2	no	—
220	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
221	#131	—	2	2	no	—
222	#131	—	2	2	no	—
223	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
224	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
225	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
226	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
227	C₄⋊Q₈	—	4	1	no	—
228	#124	—	4	1	no	—
229	#123	—	4	2	no	—
230	C₂²⋊Q₈	—	2	2	no	—
231	C₂²⋊Q₈	—	2	2	no	—
232	C₄×Q₈	—	4	1	no	—
233	C₄×D₄	—	4	2	no	—
234	C₄×D₄	—	4	1	no	—
235	#117	—	2	2	no	—
236	#117	—	2	1	no	—
237	#117	—	2	1	no	—
238	C₂×C₄⋊C₄	—	2	2	no	—
239	C₂×C₄⋊C₄	—	2	1	no	—
240	M₃₂	—	2	2	no	—
241	C₂×C₁₆	—	2	2	no	—
242	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
243	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
244	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
245	#107	—	2	2	no	—
246	#107	—	2	1	no	—
247	#107	C₂³×C₁₃	1	2	yes	—
248	#107	C₂³×C₁₃	1	1	yes	characteristic
249	#107	C₂³×C₁₃	1	1	yes	characteristic
250	#107	C₂³×C₁₃	1	1	yes	characteristic
251	#107	C₂³×C₁₃	1	1	yes	characteristic
252	#106	—	2	2	no	—
253	#106	—	2	1	no	—
254	#106	—	2	1	no	—
255	#103	—	2	2	no	—
256	#103	—	2	2	no	—
257	C₂×C₁₃	—	2	2	no	—
258	C₂×C₁₃	—	4	2	no	—
259	C₂×C₁₃	C₂×#480	1	1	yes	characteristic
260	C₂×C₁₃	#3416	1	1	yes	characteristic
261	C₂×C₁₃	#3417	1	1	yes	characteristic
262	#41	—	2	2	no	—
263	#41	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	4	2	no	—
267	C₂×Q₈	—	4	2	no	—
268	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
269	C₂×D₄	C₁₃×D₈	1	2	yes	—
270	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
271	C₂²×C₄	—	4	1	no	—
272	C₂²×C₄	—	4	2	no	—
273	C₂²×C₄	—	4	1	no	—
274	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
275	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
276	Q₁₆	—	2	2	no	—
277	Q₁₆	—	2	2	no	—
278	Q₁₆	—	4	2	no	—
279	M₁₆	—	2	2	no	—
280	M₁₆	—	2	4	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
284	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
285	C₂×C₈	C₁₃×#41	1	2	yes	—
286	C₂×C₈	C₁₃×#41	1	2	yes	—
287	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	2	no	—
296	C₄⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	4	1	no	—
299	C₂²⋊C₄	—	4	1	no	—
300	C₂²⋊C₄	—	4	2	no	—
301	C₂²⋊C₄	—	4	2	no	—
302	C₄²	—	4	2	no	—
303	C₄²	—	4	1	no	—
304	C₄²	—	4	1	no	—
305	C₁₆	—	2	1	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	2	no	—
308	C₁₃	#33651	1	1	yes	characteristic, direct product factor, Sylow
309	C₂³	C₂×C₁₃×D₈	1	2	yes	—
310	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
311	Q₈	—	2	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	8	2	no	—
315	D₄	—	2	4	no	—
316	D₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	8	2	no	—
326	C₂×C₄	—	16	1	no	—
327	C₂×C₄	—	16	2	no	—
328	C₂×C₄	C₁₃×#123	1	2	yes	—
329	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
330	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
331	C₂×C₄	C₁₃×#137	1	2	yes	—
332	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	C₁₃×#123	1	2	yes	—
335	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
336	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	C₁₃×#480	1	2	yes	—
343	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	2	no	—
348	C₄	—	8	1	no	—
349	C₄	—	8	1	no	—
350	C₄	—	8	1	no	—
351	C₄	C₁₃×#480	1	2	yes	—
352	C₄	C₂×C₁₃×#123	1	1	yes	characteristic
353	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
354	C₂	—	2	2	no	—
355	C₂	—	4	2	no	—
356	C₂	C₂×C₁₃×#480	1	1	yes	characteristic
357	C₂	C₁₃×#3416	1	1	yes	characteristic
358	C₂	C₁₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₁₃×#41 ◁ C₂×C₁₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#2237	C₂	semidirect
4. C₁₃×#2237	C₂	semidirect
5. C₁₃×#2236	C₂	semidirect
6. C₁₃×#2235	C₂	semidirect
7. C₁₃×#2159	C₂	semidirect
8. C₁₃×#2158	C₂	semidirect
9. C₁₃×#2141	C₂	semidirect
10. C₂×C₁₃×#358	C₂	semidirect
83. #33651	C₁₃	direct (N × H)
132. #2237	C₂×C₁₃	semidirect
133. #2237	C₂×C₁₃	semidirect
134. #2236	C₂×C₁₃	semidirect
135. #2235	C₂×C₁₃	semidirect
136. #2159	C₂×C₁₃	semidirect
137. #2158	C₂×C₁₃	semidirect
138. #2141	C₂×C₁₃	semidirect
139. C₂×#358	C₂×C₁₃	semidirect
308. C₁₃	#33651	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×#3059	2
C₁₃×#2913	2
C₁₃×#2237	2
C₁₃×#2236	2
C₁₃×#2235	2
C₁₃×#2159	2
C₁₃×#2158	2
C₁₃×#2141	2
C₂×C₁₃×#358	2
C₁₃×#2113	2
#33651	13

Subgroups — C13×#33651 #15719990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33651 #15719990