Subgroups — C₁₃×#9822 #15696161

Order 3328.

578 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#486	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#1957	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1957	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1956	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
10	C₁₃×#486	—	2	2	no	—
11	C₁₃×#486	C₄	1	4	yes	complemented
12	C₂×C₁₃×#103	—	2	2	no	—
13	C₂×C₁₃×#102	—	2	2	no	—
14	C₂×C₁₃×#102	—	2	2	no	—
15	C₂×C₁₃×#102	—	2	2	no	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#384	C₂²	1	1	yes	characteristic
20	C₁₃×#384	C₂²	1	1	yes	characteristic
21	C₁₃×#340	—	2	2	no	—
22	C₁₃×#340	—	2	2	no	—
23	C₈²×C₁₃	—	2	2	no	—
24	C₂²×Q₈×C₁₃	—	2	1	no	—
25	C₂²×D₄×C₁₃	—	2	1	no	—
26	C₂²×D₄×C₁₃	—	2	1	no	—
27	C₂²×D₄×C₁₃	—	2	1	no	—
28	C₂²×C₈×C₁₃	D₄	1	2	yes	—
29	C₂²×C₈×C₁₃	D₄	1	2	yes	—
30	C₂²×C₈×C₁₃	D₄	1	2	yes	—
31	C₁₃×C₄⋊Q₈	—	2	1	no	—
32	C₁₃×C₄⋊Q₈	—	2	1	no	—
33	C₁₃×C₄⋊Q₈	—	2	1	no	—
34	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₁₃×#127	—	2	1	no	—
36	C₁₃×#127	—	2	1	no	—
37	C₁₃×#127	—	2	1	no	—
38	C₁₃×#127	C₂×C₄	1	2	yes	—
39	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×#103	—	4	4	no	—
49	C₁₃×#102	—	2	8	no	—
50	C₁₃×#102	—	4	4	no	—
51	C₁₃×#102	—	4	4	no	—
52	C₄×C₈×C₁₃	—	2	1	no	—
53	C₄×C₈×C₁₃	—	2	1	no	—
54	C₄×C₈×C₁₃	—	2	1	no	—
55	C₄×C₈×C₁₃	—	2	1	no	—
56	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₁₃	D₄	1	2	yes	—
58	#9822	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂⁴×C₁₃	—	4	1	no	—
60	C₂×Q₈×C₁₃	—	2	2	no	—
61	C₂×Q₈×C₁₃	—	4	1	no	—
62	C₂×Q₈×C₁₃	—	4	2	no	—
63	C₂×Q₈×C₁₃	—	4	2	no	—
64	C₂×D₄×C₁₃	—	2	2	no	—
65	C₂×D₄×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	—	2	2	no	—
67	C₂×D₄×C₁₃	—	4	1	no	—
68	C₂×D₄×C₁₃	—	4	1	no	—
69	C₂×D₄×C₁₃	—	4	1	no	—
70	C₂×D₄×C₁₃	—	4	2	no	—
71	C₂×D₄×C₁₃	—	4	2	no	—
72	C₂×D₄×C₁₃	—	4	2	no	—
73	C₂×D₄×C₁₃	—	4	2	no	—
74	C₂×D₄×C₁₃	—	4	2	no	—
75	C₂×D₄×C₁₃	—	4	2	no	—
76	C₂²×C₄×C₁₃	—	4	1	no	—
77	C₂²×C₄×C₁₃	—	4	1	no	—
78	C₂²×C₄×C₁₃	—	4	1	no	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
89	C₂×C₈×C₁₃	D₈	1	4	yes	—
90	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	4	1	no	—
94	C₁₃×C₄⋊C₄	—	4	1	no	—
95	C₁₃×C₄⋊C₄	—	4	2	no	—
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₁₃×C₄⋊C₄	—	4	2	no	—
98	C₁₃×C₄⋊C₄	—	4	2	no	—
99	C₁₃×C₄⋊C₄	—	4	2	no	—
100	C₁₃×C₄⋊C₄	—	4	2	no	—
101	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₃	#41	1	1	yes	characteristic
104	C₄²×C₁₃	#41	1	1	yes	characteristic
105	C₂×#486	C₂×C₁₃	1	1	yes	characteristic
106	#1990	C₂×C₁₃	1	1	yes	characteristic, complemented
107	#1957	C₂×C₁₃	1	1	yes	characteristic
108	#1957	C₂×C₁₃	1	1	yes	characteristic
109	#1956	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#1834	C₂×C₁₃	1	1	yes	characteristic, complemented
111	C₂×C₈²	C₂×C₁₃	1	1	yes	characteristic, complemented
112	C₂³×C₁₃	—	4	2	no	—
113	C₂³×C₁₃	—	8	1	no	—
114	C₂³×C₁₃	—	8	1	no	—
115	C₂³×C₁₃	—	8	1	no	—
116	C₂³×C₁₃	—	8	1	no	—
117	C₂³×C₁₃	—	8	1	no	—
118	C₂³×C₁₃	—	8	1	no	—
119	C₂³×C₁₃	#127	1	1	yes	characteristic
120	Q₈×C₁₃	—	4	4	no	—
121	Q₈×C₁₃	—	8	2	no	—
122	D₄×C₁₃	—	4	4	no	—
123	D₄×C₁₃	—	4	4	no	—
124	D₄×C₁₃	—	4	4	no	—
125	D₄×C₁₃	—	8	2	no	—
126	D₄×C₁₃	—	8	2	no	—
127	D₄×C₁₃	—	8	2	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	8	1	no	—
130	C₂×C₄×C₁₃	—	8	1	no	—
131	C₂×C₄×C₁₃	—	8	1	no	—
132	C₂×C₄×C₁₃	—	8	1	no	—
133	C₂×C₄×C₁₃	—	8	1	no	—
134	C₂×C₄×C₁₃	—	8	1	no	—
135	C₂×C₄×C₁₃	—	8	1	no	—
136	C₂×C₄×C₁₃	—	8	1	no	—
137	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
142	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
145	C₂×C₄×C₁₃	#135	1	2	yes	—
146	C₂×C₄×C₁₃	#135	1	2	yes	—
147	C₂×C₄×C₁₃	#135	1	2	yes	—
148	C₂×C₄×C₁₃	#135	1	2	yes	—
149	C₈×C₁₃	—	2	4	no	—
150	C₈×C₁₃	—	2	4	no	—
151	C₈×C₁₃	—	2	4	no	—
152	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
153	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
154	#486	—	2	2	no	—
155	#486	C₄×C₁₃	1	4	yes	complemented
156	C₂×#103	—	2	2	no	—
157	C₂×#102	—	2	2	no	—
158	C₂×#102	—	2	2	no	—
159	C₂×#102	—	2	2	no	—
160	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
163	#384	C₂²×C₁₃	1	1	yes	characteristic
164	#384	C₂²×C₁₃	1	1	yes	characteristic
165	#340	—	2	2	no	—
166	#340	—	2	2	no	—
167	C₈²	—	2	2	no	—
168	C₂²×C₁₃	—	8	2	no	—
169	C₂²×C₁₃	—	8	2	no	—
170	C₂²×C₁₃	—	8	2	no	—
171	C₂²×C₁₃	—	16	1	no	—
172	C₂²×C₁₃	—	16	1	no	—
173	C₂²×C₁₃	—	16	1	no	—
174	C₂²×C₁₃	—	16	1	no	—
175	C₂²×C₁₃	#390	1	1	yes	characteristic
176	C₂²×C₁₃	#492	1	1	yes	characteristic
177	C₂²×C₁₃	#493	1	1	yes	characteristic
178	C₂²×C₁₃	#495	1	1	yes	characteristic
179	C₂²×C₁₃	#495	1	1	yes	characteristic
180	C₂²×C₁₃	#495	1	1	yes	characteristic
181	C₂²×C₁₃	#495	1	1	yes	characteristic
182	C₄×C₁₃	—	8	2	no	—
183	C₄×C₁₃	—	16	1	no	—
184	C₄×C₁₃	—	16	1	no	—
185	C₄×C₁₃	#416	1	2	yes	—
186	C₄×C₁₃	C₄×D₈	1	2	yes	—
187	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
188	C₄×C₁₃	#465	1	2	yes	—
189	C₄×C₁₃	#466	1	2	yes	—
190	C₄×C₁₃	#488	1	2	yes	—
191	C₂²×Q₈	—	2	1	no	—
192	C₂²×D₄	—	2	1	no	—
193	C₂²×D₄	—	2	1	no	—
194	C₂²×D₄	—	2	1	no	—
195	C₂²×C₈	D₄×C₁₃	1	2	yes	—
196	C₂²×C₈	D₄×C₁₃	1	2	yes	—
197	C₂²×C₈	D₄×C₁₃	1	2	yes	—
198	C₄⋊Q₈	—	2	1	no	—
199	C₄⋊Q₈	—	2	1	no	—
200	C₄⋊Q₈	—	2	1	no	—
201	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
202	#127	—	2	1	no	—
203	#127	—	2	1	no	—
204	#127	—	2	1	no	—
205	#127	C₂×C₄×C₁₃	1	2	yes	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
215	#103	—	4	4	no	—
216	#102	—	2	8	no	—
217	#102	—	4	4	no	—
218	#102	—	4	4	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
224	C₄×C₈	D₄×C₁₃	1	2	yes	—
225	C₂×C₁₃	—	16	2	no	—
226	C₂×C₁₃	#1706	1	1	yes	characteristic
227	C₂×C₁₃	#1708	1	1	yes	characteristic
228	C₂×C₁₃	#1709	1	1	yes	characteristic
229	C₂×C₁₃	#1711	1	1	yes	characteristic
230	C₂×C₁₃	#1958	1	1	yes	characteristic
231	C₂×C₁₃	#1968	1	1	yes	characteristic
232	C₂×C₁₃	#1970	1	1	yes	characteristic
233	C₂⁴	—	4	1	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	C₁₃×C₂²⋊C₄	1	4	yes	—
263	C₂×C₈	C₁₃×D₈	1	4	yes	—
264	C₂×C₈	C₁₃×SD₁₆	1	4	yes	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
276	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
277	C₄²	C₁₃×#41	1	1	yes	characteristic
278	C₄²	C₁₃×#41	1	1	yes	characteristic
279	C₁₃	#9822	1	1	yes	characteristic, direct product factor, Sylow
280	C₂³	—	4	2	no	—
281	C₂³	—	8	1	no	—
282	C₂³	—	8	1	no	—
283	C₂³	—	8	1	no	—
284	C₂³	—	8	1	no	—
285	C₂³	—	8	1	no	—
286	C₂³	—	8	1	no	—
287	C₂³	C₁₃×#127	1	1	yes	characteristic
288	Q₈	—	4	4	no	—
289	Q₈	—	8	2	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	4	4	no	—
293	D₄	—	8	2	no	—
294	D₄	—	8	2	no	—
295	D₄	—	8	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
306	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
307	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
308	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
309	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
310	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
311	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
312	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
313	C₂×C₄	C₁₃×#135	1	2	yes	—
314	C₂×C₄	C₁₃×#135	1	2	yes	—
315	C₂×C₄	C₁₃×#135	1	2	yes	—
316	C₂×C₄	C₁₃×#135	1	2	yes	—
317	C₈	—	2	4	no	—
318	C₈	—	2	4	no	—
319	C₈	—	2	4	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	16	1	no	—
324	C₂²	—	16	1	no	—
325	C₂²	—	16	1	no	—
326	C₂²	—	16	1	no	—
327	C₂²	C₁₃×#390	1	1	yes	characteristic
328	C₂²	C₁₃×#492	1	1	yes	characteristic
329	C₂²	C₁₃×#493	1	1	yes	characteristic
330	C₂²	C₁₃×#495	1	1	yes	characteristic
331	C₂²	C₁₃×#495	1	1	yes	characteristic
332	C₂²	C₁₃×#495	1	1	yes	characteristic
333	C₂²	C₁₃×#495	1	1	yes	characteristic
334	C₄	—	8	2	no	—
335	C₄	—	16	1	no	—
336	C₄	—	16	1	no	—
337	C₄	C₁₃×#416	1	2	yes	—
338	C₄	C₄×C₁₃×D₈	1	2	yes	—
339	C₄	C₄×C₁₃×SD₁₆	1	2	yes	—
340	C₄	C₁₃×#465	1	2	yes	—
341	C₄	C₁₃×#466	1	2	yes	—
342	C₄	C₁₃×#488	1	2	yes	—
343	C₂	—	16	2	no	—
344	C₂	C₁₃×#1706	1	1	yes	characteristic
345	C₂	C₁₃×#1708	1	1	yes	characteristic
346	C₂	C₁₃×#1709	1	1	yes	characteristic
347	C₂	C₁₃×#1711	1	1	yes	characteristic
348	C₂	C₁₃×#1958	1	1	yes	characteristic
349	C₂	C₁₃×#1968	1	1	yes	characteristic
350	C₂	C₁₃×#1970	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₁₃×#1990	C₂	semidirect
5. C₁₃×#1956	C₂	semidirect
6. C₁₃×#1834	C₂	semidirect
7. C₂×C₈²×C₁₃	C₂	semidirect
11. C₁₃×#486	C₄	semidirect
58. #9822	C₁₃	direct (N × H)
106. #1990	C₂×C₁₃	semidirect
109. #1956	C₂×C₁₃	semidirect
110. #1834	C₂×C₁₃	semidirect
111. C₂×C₈²	C₂×C₁₃	semidirect
155. #486	C₄×C₁₃	semidirect
279. C₁₃	#9822	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×#486	2
C₁₃×#1990	2
C₁₃×#1957	2
C₁₃×#1956	2
C₁₃×#1834	2
C₂×C₈²×C₁₃	2
#9822	13

Subgroups — C13×#9822 #15696161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9822 #15696161