Subgroups — C₁₃×#19001 #15705340

Order 3328.

960 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#562	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3213	C₂	1	2	yes	maximal, complemented
3	C₁₃×#3142	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1662	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1656	C₂	1	4	yes	maximal, complemented
6	C₁₃×#1534	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1527	C₂	1	2	yes	maximal, complemented
8	C₁₃×#562	—	2	2	no	—
9	C₁₃×#562	—	2	4	no	—
10	C₁₃×#562	C₂²	1	4	yes	—
11	C₁₃×#532	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#124	C₂²	1	2	yes	—
14	C₂×C₁₃×#123	—	2	2	no	—
15	C₂×C₁₃×C₂²⋊Q₈	—	2	2	no	—
16	C₂×C₁₃×#121	—	2	2	no	—
17	C₁₃×#496	C₂²	1	4	yes	complemented
18	C₁₃×#491	C₂²	1	2	yes	complemented
19	C₁₃×#490	C₂²	1	2	yes	complemented
20	C₁₃×#489	C₂²	1	4	yes	complemented
21	C₁₃×#488	—	2	2	no	—
22	C₁₃×#463	—	2	4	no	—
23	C₁₃×#448	—	2	2	no	—
24	C₁₃×#447	—	2	2	no	—
25	C₁₃×#432	C₂²	1	2	yes	—
26	C₁₃×#417	—	2	2	no	—
27	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
28	C₁₃×#330	C₂²	1	4	yes	complemented
29	C₁₃×#329	C₂²	1	8	yes	complemented
30	C₂×C₁₃×#41	—	2	1	no	—
31	C₂²×Q₈×C₁₃	D₄	1	2	yes	—
32	C₂²×D₄×C₁₃	—	2	2	no	—
33	C₂²×D₄×C₁₃	D₄	1	2	yes	—
34	C₂³×C₄×C₁₃	—	2	2	no	—
35	C₂×C₁₃×SD₁₆	—	2	4	no	—
36	C₂×C₁₃×D₈	—	2	4	no	—
37	C₂×C₁₃×M₁₆	—	2	1	no	—
38	C₁₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
39	C₁₃×#127	C₂³	1	1	yes	characteristic
40	C₁₃×#125	—	2	2	no	—
41	C₁₃×#125	D₄	1	2	yes	—
42	C₁₃×#125	C₂³	1	2	yes	—
43	C₁₃×#124	—	2	2	no	—
44	C₁₃×#124	—	2	2	no	—
45	C₁₃×#124	—	2	1	no	—
46	C₁₃×#124	D₄	1	4	yes	—
47	C₁₃×#124	C₂³	1	4	yes	—
48	C₁₃×#123	—	2	8	no	—
49	C₁₃×#123	—	2	8	no	—
50	C₁₃×C₂²⋊Q₈	—	2	8	no	—
51	C₁₃×C₂²⋊Q₈	—	2	8	no	—
52	C₁₃×#121	—	2	8	no	—
53	C₁₃×#121	—	2	8	no	—
54	C₁₃×#121	—	2	2	no	—
55	C₄×D₄×C₁₃	—	2	2	no	—
56	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
58	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
59	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
60	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
61	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
62	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
63	C₁₃×C₄⋊C₈	—	2	2	no	—
64	C₁₃×#103	—	2	4	no	—
65	C₁₃×#103	—	2	4	no	—
66	C₁₃×#102	—	2	4	no	—
67	C₁₃×#102	—	2	4	no	—
68	C₁₃×C₂²⋊C₈	—	2	2	no	—
69	C₁₃×#97	C₂³	1	2	yes	—
70	C₁₃×#97	C₂³	1	4	yes	—
71	C₄×C₈×C₁₃	—	2	1	no	—
72	#19001	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂⁴×C₁₃	—	2	2	no	—
74	C₁₃×#41	—	4	4	no	—
75	C₂×Q₈×C₁₃	—	2	2	no	—
76	C₂×Q₈×C₁₃	—	2	1	no	—
77	C₂×Q₈×C₁₃	—	8	2	no	—
78	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
79	C₂×D₄×C₁₃	—	2	8	no	—
80	C₂×D₄×C₁₃	—	2	2	no	—
81	C₂×D₄×C₁₃	—	2	1	no	—
82	C₂×D₄×C₁₃	—	2	2	no	—
83	C₂×D₄×C₁₃	—	4	1	no	—
84	C₂×D₄×C₁₃	—	8	2	no	—
85	C₂×D₄×C₁₃	—	8	4	no	—
86	C₂×D₄×C₁₃	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₁₃	—	2	2	no	—
88	C₂²×C₄×C₁₃	—	2	2	no	—
89	C₂²×C₄×C₁₃	—	2	8	no	—
90	C₂²×C₄×C₁₃	—	4	1	no	—
91	C₂²×C₄×C₁₃	—	8	4	no	—
92	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
94	C₁₃×SD₁₆	—	8	4	no	—
95	C₁₃×D₈	—	8	4	no	—
96	C₁₃×M₁₆	—	4	2	no	—
97	C₂×C₈×C₁₃	—	2	2	no	—
98	C₂×C₈×C₁₃	—	2	4	no	—
99	C₁₃×C₄⋊C₄	—	2	8	no	—
100	C₁₃×C₄⋊C₄	—	2	4	no	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	2	4	no	—
103	C₁₃×C₄⋊C₄	—	2	2	no	—
104	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
105	C₁₃×C₂²⋊C₄	—	2	8	no	—
106	C₁₃×C₂²⋊C₄	—	2	4	no	—
107	C₁₃×C₂²⋊C₄	—	2	4	no	—
108	C₁₃×C₂²⋊C₄	—	2	4	no	—
109	C₁₃×C₂²⋊C₄	—	2	4	no	—
110	C₁₃×C₂²⋊C₄	—	4	2	no	—
111	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₃	C₂×D₄	1	2	yes	—
113	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
114	C₂×#562	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#3213	C₂×C₁₃	1	2	yes	complemented
116	#3142	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#1662	C₂×C₁₃	1	4	yes	complemented
118	#1656	C₂×C₁₃	1	4	yes	complemented
119	#1534	C₂×C₁₃	1	1	yes	characteristic, complemented
120	#1527	C₂×C₁₃	1	2	yes	complemented
121	C₂³×C₁₃	—	2	4	no	—
122	C₂³×C₁₃	—	4	4	no	—
123	C₂³×C₁₃	—	4	4	no	—
124	C₂³×C₁₃	—	4	4	no	—
125	C₂³×C₁₃	—	4	1	no	—
126	C₂³×C₁₃	#120	1	1	yes	characteristic
127	Q₈×C₁₃	—	4	4	no	—
128	Q₈×C₁₃	—	4	2	no	—
129	Q₈×C₁₃	—	8	2	no	—
130	D₄×C₁₃	—	4	4	no	—
131	D₄×C₁₃	—	4	4	no	—
132	D₄×C₁₃	—	4	2	no	—
133	D₄×C₁₃	—	8	2	no	—
134	D₄×C₁₃	—	8	8	no	—
135	D₄×C₁₃	—	8	2	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	4	no	—
138	C₂×C₄×C₁₃	—	2	4	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	4	no	—
141	C₂×C₄×C₁₃	—	4	4	no	—
142	C₂×C₄×C₁₃	—	4	1	no	—
143	C₂×C₄×C₁₃	—	8	2	no	—
144	C₂×C₄×C₁₃	—	8	8	no	—
145	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
146	C₂×C₄×C₁₃	#120	1	2	yes	—
147	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
149	C₈×C₁₃	—	4	2	no	—
150	C₈×C₁₃	—	4	4	no	—
151	#562	—	2	2	no	—
152	#562	—	2	4	no	—
153	#562	C₂²×C₁₃	1	4	yes	—
154	#532	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×#125	C₂²×C₁₃	1	1	yes	characteristic
156	C₂×#124	C₂²×C₁₃	1	2	yes	—
157	C₂×#123	—	2	2	no	—
158	C₂×C₂²⋊Q₈	—	2	2	no	—
159	C₂×#121	—	2	2	no	—
160	#496	C₂²×C₁₃	1	4	yes	complemented
161	#491	C₂²×C₁₃	1	2	yes	complemented
162	#490	C₂²×C₁₃	1	2	yes	complemented
163	#489	C₂²×C₁₃	1	4	yes	complemented
164	#488	—	2	2	no	—
165	#463	—	2	4	no	—
166	#448	—	2	2	no	—
167	#447	—	2	2	no	—
168	#432	C₂²×C₁₃	1	2	yes	—
169	#417	—	2	2	no	—
170	C₄×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
171	#330	C₂²×C₁₃	1	4	yes	complemented
172	#329	C₂²×C₁₃	1	8	yes	complemented
173	C₂²×C₁₃	—	2	2	no	—
174	C₂²×C₁₃	—	4	4	no	—
175	C₂²×C₁₃	—	4	4	no	—
176	C₂²×C₁₃	—	4	4	no	—
177	C₂²×C₁₃	—	8	8	no	—
178	C₂²×C₁₃	—	8	1	no	—
179	C₂²×C₁₃	—	8	2	no	—
180	C₂²×C₁₃	#454	1	2	yes	—
181	C₂²×C₁₃	C₂×#120	1	1	yes	characteristic
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	2	2	no	—
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	—	4	4	no	—
187	C₄×C₁₃	—	4	4	no	—
188	C₄×C₁₃	—	8	1	no	—
189	C₂×#41	—	2	1	no	—
190	C₂²×Q₈	D₄×C₁₃	1	2	yes	—
191	C₂²×D₄	—	2	2	no	—
192	C₂²×D₄	D₄×C₁₃	1	2	yes	—
193	C₂³×C₄	—	2	2	no	—
194	C₂×SD₁₆	—	2	4	no	—
195	C₂×D₈	—	2	4	no	—
196	C₂×M₁₆	—	2	1	no	—
197	C₄⋊Q₈	C₂³×C₁₃	1	1	yes	characteristic
198	#127	C₂³×C₁₃	1	1	yes	characteristic
199	#125	—	2	2	no	—
200	#125	D₄×C₁₃	1	2	yes	—
201	#125	C₂³×C₁₃	1	2	yes	—
202	#124	—	2	2	no	—
203	#124	—	2	2	no	—
204	#124	—	2	1	no	—
205	#124	D₄×C₁₃	1	4	yes	—
206	#124	C₂³×C₁₃	1	4	yes	—
207	#123	—	2	8	no	—
208	#123	—	2	8	no	—
209	C₂²⋊Q₈	—	2	8	no	—
210	C₂²⋊Q₈	—	2	8	no	—
211	#121	—	2	8	no	—
212	#121	—	2	8	no	—
213	#121	—	2	2	no	—
214	C₄×D₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
218	C₂×C₂²⋊C₄	—	2	2	no	—
219	C₂×C₂²⋊C₄	—	2	2	no	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
222	C₄⋊C₈	—	2	2	no	—
223	#103	—	2	4	no	—
224	#103	—	2	4	no	—
225	#102	—	2	4	no	—
226	#102	—	2	4	no	—
227	C₂²⋊C₈	—	2	2	no	—
228	#97	C₂³×C₁₃	1	2	yes	—
229	#97	C₂³×C₁₃	1	4	yes	—
230	C₄×C₈	—	2	1	no	—
231	C₂×C₁₃	—	2	2	no	—
232	C₂×C₁₃	—	8	4	no	—
233	C₂×C₁₃	—	16	1	no	—
234	C₂×C₁₃	C₂×#454	1	1	yes	characteristic
235	C₂×C₁₃	#3019	1	2	yes	—
236	C₂⁴	—	2	2	no	—
237	#41	—	4	4	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	8	2	no	—
241	C₂×Q₈	C₂×D₄×C₁₃	1	4	yes	—
242	C₂×D₄	—	2	8	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	8	2	no	—
248	C₂×D₄	—	8	4	no	—
249	C₂×D₄	C₂×D₄×C₁₃	1	4	yes	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	8	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	8	4	no	—
255	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
257	SD₁₆	—	8	4	no	—
258	D₈	—	8	4	no	—
259	M₁₆	—	4	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	4	no	—
262	C₄⋊C₄	—	2	8	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
268	C₂²⋊C₄	—	2	8	no	—
269	C₂²⋊C₄	—	2	4	no	—
270	C₂²⋊C₄	—	2	4	no	—
271	C₂²⋊C₄	—	2	4	no	—
272	C₂²⋊C₄	—	2	4	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
275	C₄²	C₂×D₄×C₁₃	1	2	yes	—
276	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
277	C₁₃	#19001	1	1	yes	characteristic, direct product factor, Sylow
278	C₂³	—	2	4	no	—
279	C₂³	—	4	4	no	—
280	C₂³	—	4	4	no	—
281	C₂³	—	4	4	no	—
282	C₂³	—	4	1	no	—
283	C₂³	C₁₃×#120	1	1	yes	characteristic
284	Q₈	—	4	4	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	8	2	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	4	no	—
289	D₄	—	4	2	no	—
290	D₄	—	8	2	no	—
291	D₄	—	8	8	no	—
292	D₄	—	8	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	8	8	no	—
302	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
303	C₂×C₄	C₁₃×#120	1	2	yes	—
304	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
306	C₈	—	4	2	no	—
307	C₈	—	4	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	—	8	8	no	—
313	C₂²	—	8	1	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₁₃×#454	1	2	yes	—
316	C₂²	C₂×C₁₃×#120	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	4	no	—
322	C₄	—	4	4	no	—
323	C₄	—	8	1	no	—
324	C₂	—	2	2	no	—
325	C₂	—	8	4	no	—
326	C₂	—	16	1	no	—
327	C₂	C₂×C₁₃×#454	1	1	yes	characteristic
328	C₂	C₁₃×#3019	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃
LCS	G ▷ C₄² ▷ C₂² ▷ 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. C₂×C₁₃×#562	C₂	semidirect
2. C₁₃×#3213	C₂	semidirect
3. C₁₃×#3142	C₂	semidirect
4. C₁₃×#1662	C₂	semidirect
5. C₁₃×#1656	C₂	semidirect
6. C₁₃×#1534	C₂	semidirect
7. C₁₃×#1527	C₂	semidirect
17. C₁₃×#496	C₂²	semidirect
18. C₁₃×#491	C₂²	semidirect
19. C₁₃×#490	C₂²	semidirect
20. C₁₃×#489	C₂²	semidirect
28. C₁₃×#330	C₂²	semidirect
29. C₁₃×#329	C₂²	semidirect
72. #19001	C₁₃	direct (N × H)
114. C₂×#562	C₂×C₁₃	semidirect
115. #3213	C₂×C₁₃	semidirect
116. #3142	C₂×C₁₃	semidirect
117. #1662	C₂×C₁₃	semidirect
118. #1656	C₂×C₁₃	semidirect
119. #1534	C₂×C₁₃	semidirect
120. #1527	C₂×C₁₃	semidirect
160. #496	C₂²×C₁₃	semidirect
161. #491	C₂²×C₁₃	semidirect
162. #490	C₂²×C₁₃	semidirect
163. #489	C₂²×C₁₃	semidirect
171. #330	C₂²×C₁₃	semidirect
172. #329	C₂²×C₁₃	semidirect
277. C₁₃	#19001	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×#562	2
C₁₃×#3213	2
C₁₃×#3142	2
C₁₃×#1662	2
C₁₃×#1656	2
C₁₃×#1534	2
C₁₃×#1527	2
#19001	13

Subgroups — C13×#19001 #15705340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#19001 #15705340