Subgroups — C₁₃×#17654 #15703993

Order 3328.

960 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3518	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2971	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2875	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1531	C₂	1	3	yes	maximal, complemented
5	C₁₃×#1529	C₂	1	3	yes	maximal, complemented
6	C₁₃×#1526	C₂	1	3	yes	maximal, complemented
7	C₁₃×#1524	C₂	1	3	yes	maximal, complemented
8	C₂×C₁₃×#131	—	2	1	no	—
9	Q₈²×C₁₃	C₄	1	1	yes	characteristic
10	C₁₃×#556	—	2	3	no	—
11	C₁₃×#556	C₄	1	3	yes	—
12	C₁₃×#555	C₄	1	3	yes	—
13	D₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
14	C₁₃×#548	—	2	1	no	—
15	C₁₃×#542	—	2	3	no	—
16	C₁₃×#533	C₂²	1	3	yes	—
17	C₁₃×#529	C₂²	1	3	yes	—
18	C₁₃×#519	—	2	1	no	—
19	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
20	C₈×Q₈×C₁₃	—	2	1	no	—
21	C₁₃×#436	—	2	3	no	—
22	C₁₃×#432	C₂²	1	3	yes	—
23	C₁₃×#405	—	2	3	no	—
24	C₄×C₁₃×M₁₆	C₂²	1	3	yes	—
25	C₄×C₁₃×M₁₆	C₂²	1	3	yes	—
26	C₂×C₁₃×#97	C₂²	1	3	yes	—
27	C₁₃×#330	C₂²	1	3	yes	—
28	C₁₃×#330	C₂²	1	3	yes	—
29	C₁₃×#330	C₂²	1	6	yes	—
30	C₁₃×#329	C₂²	1	1	yes	characteristic
31	C₁₃×#329	C₂²	1	3	yes	—
32	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
33	C₂²×Q₈×C₁₃	—	2	1	no	—
34	C₁₃×#131	—	2	8	no	—
35	C₂×C₁₃×M₁₆	—	2	3	no	—
36	C₂×C₁₃×M₁₆	—	2	3	no	—
37	C₂²×C₈×C₁₃	—	2	3	no	—
38	C₁₃×C₄⋊Q₈	—	2	3	no	—
39	C₁₃×C₄⋊Q₈	C₂×C₄	1	3	yes	—
40	C₁₃×C₄⋊Q₈	C₂×C₄	1	3	yes	—
41	C₁₃×#126	—	2	6	no	—
42	C₁₃×#126	—	2	3	no	—
43	C₁₃×#125	C₂³	1	3	yes	—
44	C₁₃×#124	—	2	3	no	—
45	C₁₃×#124	—	2	6	no	—
46	C₁₃×#124	C₂³	1	1	yes	characteristic
47	C₁₃×#123	—	2	3	no	—
48	C₁₃×C₂²⋊Q₈	—	2	3	no	—
49	C₁₃×C₂²⋊Q₈	—	2	3	no	—
50	C₁₃×C₂²⋊Q₈	—	2	3	no	—
51	C₁₃×C₂²⋊Q₈	C₂×C₄	1	3	yes	—
52	C₁₃×#121	—	2	3	no	—
53	C₁₃×#121	C₂×C₄	1	3	yes	—
54	C₄×Q₈×C₁₃	—	2	1	no	—
55	C₄×Q₈×C₁₃	—	2	3	no	—
56	C₄×Q₈×C₁₃	—	2	3	no	—
57	C₄×Q₈×C₁₃	—	2	3	no	—
58	C₄×Q₈×C₁₃	C₂×C₄	1	3	yes	—
59	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
60	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
61	C₄×D₄×C₁₃	—	2	3	no	—
62	C₄×D₄×C₁₃	C₂×C₄	1	3	yes	—
63	C₄×D₄×C₁₃	C₂×C₄	1	3	yes	—
64	C₄×D₄×C₁₃	D₄	1	3	yes	—
65	C₁₃×#117	—	2	3	no	—
66	C₁₃×#117	—	2	3	no	—
67	C₁₃×#117	C₂×C₄	1	6	yes	—
68	C₁₃×#117	D₄	1	3	yes	—
69	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
70	C₁₃×C₄⋊C₈	—	2	3	no	—
71	C₁₃×C₂²⋊C₈	—	2	3	no	—
72	C₁₃×#97	—	2	3	no	—
73	C₁₃×#97	C₂³	1	1	yes	characteristic
74	C₁₃×#97	C₂³	1	3	yes	—
75	C₁₃×#97	C₂³	1	1	yes	characteristic
76	C₁₃×#97	C₂³	1	3	yes	—
77	C₄×C₈×C₁₃	—	2	3	no	—
78	C₄×C₈×C₁₃	—	2	3	no	—
79	#17654	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₁₃×#41	—	2	4	no	—
81	C₂×Q₈×C₁₃	—	2	3	no	—
82	C₂×Q₈×C₁₃	—	2	3	no	—
83	C₂×Q₈×C₁₃	—	4	4	no	—
84	C₂×Q₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×Q₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×D₄×C₁₃	—	2	3	no	—
87	C₂×D₄×C₁₃	C₂²⋊C₄	1	3	yes	—
88	C₂×D₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	—	2	3	no	—
90	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
91	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
92	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
93	C₁₃×M₁₆	—	2	12	no	—
94	C₁₃×M₁₆	—	4	6	no	—
95	C₂×C₈×C₁₃	—	2	12	no	—
96	C₂×C₈×C₁₃	—	2	1	no	—
97	C₂×C₈×C₁₃	—	2	3	no	—
98	C₂×C₈×C₁₃	—	2	1	no	—
99	C₂×C₈×C₁₃	—	2	3	no	—
100	C₁₃×C₄⋊C₄	—	2	3	no	—
101	C₁₃×C₄⋊C₄	—	2	3	no	—
102	C₁₃×C₄⋊C₄	—	2	3	no	—
103	C₁₃×C₄⋊C₄	—	2	3	no	—
104	C₁₃×C₄⋊C₄	—	2	3	no	—
105	C₁₃×C₄⋊C₄	—	2	3	no	—
106	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	3	yes	—
107	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	3	yes	—
108	C₁₃×C₄⋊C₄	C₂²×C₄	1	3	yes	—
109	C₁₃×C₄⋊C₄	C₂²×C₄	1	3	yes	—
110	C₁₃×C₂²⋊C₄	—	2	6	no	—
111	C₁₃×C₂²⋊C₄	—	2	3	no	—
112	C₁₃×C₂²⋊C₄	—	2	3	no	—
113	C₁₃×C₂²⋊C₄	—	2	3	no	—
114	C₁₃×C₂²⋊C₄	—	2	1	no	—
115	C₁₃×C₂²⋊C₄	—	2	1	no	—
116	C₁₃×C₂²⋊C₄	C₂²⋊C₄	1	6	yes	—
117	C₄²×C₁₃	—	2	6	no	—
118	C₄²×C₁₃	—	2	3	no	—
119	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
120	C₄²×C₁₃	C₂×D₄	1	3	yes	—
121	C₄²×C₁₃	C₂×D₄	1	3	yes	—
122	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
123	#3518	C₂×C₁₃	1	1	yes	characteristic
124	#2971	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#2875	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#1531	C₂×C₁₃	1	3	yes	complemented
127	#1529	C₂×C₁₃	1	3	yes	complemented
128	#1526	C₂×C₁₃	1	3	yes	complemented
129	#1524	C₂×C₁₃	1	3	yes	complemented
130	C₂³×C₁₃	—	2	1	no	—
131	C₂³×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
132	Q₈×C₁₃	—	4	1	no	—
133	Q₈×C₁₃	—	4	6	no	—
134	Q₈×C₁₃	—	8	3	no	—
135	Q₈×C₁₃	#104	1	4	yes	—
136	D₄×C₁₃	—	2	6	no	—
137	D₄×C₁₃	—	4	1	no	—
138	D₄×C₁₃	—	8	3	no	—
139	C₂×C₄×C₁₃	—	2	6	no	—
140	C₂×C₄×C₁₃	—	2	1	no	—
141	C₂×C₄×C₁₃	—	2	3	no	—
142	C₂×C₄×C₁₃	—	2	3	no	—
143	C₂×C₄×C₁₃	—	4	3	no	—
144	C₂×C₄×C₁₃	—	4	3	no	—
145	C₂×C₄×C₁₃	—	4	6	no	—
146	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
147	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
148	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
149	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
152	C₈×C₁₃	—	2	2	no	—
153	C₈×C₁₃	—	2	6	no	—
154	C₈×C₁₃	—	2	2	no	—
155	C₈×C₁₃	—	4	3	no	—
156	C₂×#131	—	2	1	no	—
157	Q₈²	C₄×C₁₃	1	1	yes	characteristic
158	#556	—	2	3	no	—
159	#556	C₄×C₁₃	1	3	yes	—
160	#555	C₄×C₁₃	1	3	yes	—
161	D₄×Q₈	C₄×C₁₃	1	1	yes	characteristic
162	#548	—	2	1	no	—
163	#542	—	2	3	no	—
164	#533	C₂²×C₁₃	1	3	yes	—
165	#529	C₂²×C₁₃	1	3	yes	—
166	#519	—	2	1	no	—
167	C₄×#41	C₂²×C₁₃	1	1	yes	characteristic
168	C₈×Q₈	—	2	1	no	—
169	#436	—	2	3	no	—
170	#432	C₂²×C₁₃	1	3	yes	—
171	#405	—	2	3	no	—
172	C₄×M₁₆	C₂²×C₁₃	1	3	yes	—
173	C₄×M₁₆	C₂²×C₁₃	1	3	yes	—
174	C₂×#97	C₂²×C₁₃	1	3	yes	—
175	#330	C₂²×C₁₃	1	3	yes	—
176	#330	C₂²×C₁₃	1	3	yes	—
177	#330	C₂²×C₁₃	1	6	yes	—
178	#329	C₂²×C₁₃	1	1	yes	characteristic
179	#329	C₂²×C₁₃	1	3	yes	—
180	C₂²×C₁₃	—	2	3	no	—
181	C₂²×C₁₃	—	2	3	no	—
182	C₂²×C₁₃	—	2	3	no	—
183	C₂²×C₁₃	—	4	1	no	—
184	C₂²×C₁₃	—	4	1	no	—
185	C₂²×C₁₃	—	4	1	no	—
186	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	2	3	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	2	1	no	—
191	C₄×C₁₃	—	2	3	no	—
192	C₄×C₁₃	—	4	1	no	—
193	C₄×C₁₃	—	4	3	no	—
194	C₄×C₁₃	—	4	3	no	—
195	C₄×C₁₃	C₂×#104	1	6	yes	—
196	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
197	C₂²×Q₈	—	2	1	no	—
198	#131	—	2	8	no	—
199	C₂×M₁₆	—	2	3	no	—
200	C₂×M₁₆	—	2	3	no	—
201	C₂²×C₈	—	2	3	no	—
202	C₄⋊Q₈	—	2	3	no	—
203	C₄⋊Q₈	C₂×C₄×C₁₃	1	3	yes	—
204	C₄⋊Q₈	C₂×C₄×C₁₃	1	3	yes	—
205	#126	—	2	6	no	—
206	#126	—	2	3	no	—
207	#125	C₂³×C₁₃	1	3	yes	—
208	#124	—	2	3	no	—
209	#124	—	2	6	no	—
210	#124	C₂³×C₁₃	1	1	yes	characteristic
211	#123	—	2	3	no	—
212	C₂²⋊Q₈	—	2	3	no	—
213	C₂²⋊Q₈	—	2	3	no	—
214	C₂²⋊Q₈	—	2	3	no	—
215	C₂²⋊Q₈	C₂×C₄×C₁₃	1	3	yes	—
216	#121	—	2	3	no	—
217	#121	C₂×C₄×C₁₃	1	3	yes	—
218	C₄×Q₈	—	2	1	no	—
219	C₄×Q₈	—	2	3	no	—
220	C₄×Q₈	—	2	3	no	—
221	C₄×Q₈	—	2	3	no	—
222	C₄×Q₈	C₂×C₄×C₁₃	1	3	yes	—
223	C₄×Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
224	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
225	C₄×D₄	—	2	3	no	—
226	C₄×D₄	C₂×C₄×C₁₃	1	3	yes	—
227	C₄×D₄	C₂×C₄×C₁₃	1	3	yes	—
228	C₄×D₄	D₄×C₁₃	1	3	yes	—
229	#117	—	2	3	no	—
230	#117	—	2	3	no	—
231	#117	C₂×C₄×C₁₃	1	6	yes	—
232	#117	D₄×C₁₃	1	3	yes	—
233	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
234	C₄⋊C₈	—	2	3	no	—
235	C₂²⋊C₈	—	2	3	no	—
236	#97	—	2	3	no	—
237	#97	C₂³×C₁₃	1	1	yes	characteristic
238	#97	C₂³×C₁₃	1	3	yes	—
239	#97	C₂³×C₁₃	1	1	yes	characteristic
240	#97	C₂³×C₁₃	1	3	yes	—
241	C₄×C₈	—	2	3	no	—
242	C₄×C₈	—	2	3	no	—
243	C₂×C₁₃	—	4	3	no	—
244	C₂×C₁₃	—	8	1	no	—
245	C₂×C₁₃	#2889	1	1	yes	characteristic
246	C₂×C₁₃	C₂²×#104	1	1	yes	characteristic
247	C₂×C₁₃	#2901	1	1	yes	characteristic
248	#41	—	2	4	no	—
249	C₂×Q₈	—	2	3	no	—
250	C₂×Q₈	—	2	3	no	—
251	C₂×Q₈	—	4	4	no	—
252	C₂×Q₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
253	C₂×Q₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
254	C₂×D₄	—	2	3	no	—
255	C₂×D₄	C₁₃×C₂²⋊C₄	1	3	yes	—
256	C₂×D₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
257	C₂²×C₄	—	2	3	no	—
258	C₂²×C₄	C₂²×C₄×C₁₃	1	3	yes	—
259	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
260	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
261	M₁₆	—	2	12	no	—
262	M₁₆	—	4	6	no	—
263	C₂×C₈	—	2	12	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	3	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	3	no	—
268	C₄⋊C₄	—	2	3	no	—
269	C₄⋊C₄	—	2	3	no	—
270	C₄⋊C₄	—	2	3	no	—
271	C₄⋊C₄	—	2	3	no	—
272	C₄⋊C₄	—	2	3	no	—
273	C₄⋊C₄	—	2	3	no	—
274	C₄⋊C₄	C₁₃×C₂²⋊C₄	1	3	yes	—
275	C₄⋊C₄	C₁₃×C₂²⋊C₄	1	3	yes	—
276	C₄⋊C₄	C₂²×C₄×C₁₃	1	3	yes	—
277	C₄⋊C₄	C₂²×C₄×C₁₃	1	3	yes	—
278	C₂²⋊C₄	—	2	6	no	—
279	C₂²⋊C₄	—	2	3	no	—
280	C₂²⋊C₄	—	2	3	no	—
281	C₂²⋊C₄	—	2	3	no	—
282	C₂²⋊C₄	—	2	1	no	—
283	C₂²⋊C₄	—	2	1	no	—
284	C₂²⋊C₄	C₁₃×C₂²⋊C₄	1	6	yes	—
285	C₄²	—	2	6	no	—
286	C₄²	—	2	3	no	—
287	C₄²	C₂²×C₄×C₁₃	1	3	yes	—
288	C₄²	C₂×D₄×C₁₃	1	3	yes	—
289	C₄²	C₂×D₄×C₁₃	1	3	yes	—
290	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
291	C₁₃	#17654	1	1	yes	characteristic, direct product factor, Sylow
292	C₂³	—	2	1	no	—
293	C₂³	C₂×C₁₃×C₂²⋊C₄	1	3	yes	—
294	Q₈	—	4	1	no	—
295	Q₈	—	4	6	no	—
296	Q₈	—	8	3	no	—
297	Q₈	C₁₃×#104	1	4	yes	—
298	D₄	—	2	6	no	—
299	D₄	—	4	1	no	—
300	D₄	—	8	3	no	—
301	C₂×C₄	—	2	6	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	3	no	—
304	C₂×C₄	—	2	3	no	—
305	C₂×C₄	—	4	3	no	—
306	C₂×C₄	—	4	3	no	—
307	C₂×C₄	—	4	6	no	—
308	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	3	yes	—
309	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	3	yes	—
310	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	3	yes	—
311	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
313	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	—	2	6	no	—
316	C₈	—	2	2	no	—
317	C₈	—	4	3	no	—
318	C₂²	—	2	3	no	—
319	C₂²	—	2	3	no	—
320	C₂²	—	2	3	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	3	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	3	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	3	no	—
332	C₄	—	4	3	no	—
333	C₄	C₂×C₁₃×#104	1	6	yes	—
334	C₂	—	4	3	no	—
335	C₂	—	8	1	no	—
336	C₂	C₁₃×#2889	1	1	yes	characteristic
337	C₂	C₂²×C₁₃×#104	1	1	yes	characteristic
338	C₂	C₁₃×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₃
prime radical	C₁₃×#3518	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×C₁₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2971	C₂	semidirect
3. C₁₃×#2875	C₂	semidirect
4. C₁₃×#1531	C₂	semidirect
5. C₁₃×#1529	C₂	semidirect
6. C₁₃×#1526	C₂	semidirect
7. C₁₃×#1524	C₂	semidirect
79. #17654	C₁₃	direct (N × H)
124. #2971	C₂×C₁₃	semidirect
125. #2875	C₂×C₁₃	semidirect
126. #1531	C₂×C₁₃	semidirect
127. #1529	C₂×C₁₃	semidirect
128. #1526	C₂×C₁₃	semidirect
129. #1524	C₂×C₁₃	semidirect
291. C₁₃	#17654	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×#3518	2
C₁₃×#2971	2
C₁₃×#2875	2
C₁₃×#1531	2
C₁₃×#1529	2
C₁₃×#1526	2
C₁₃×#1524	2
#17654	13

Subgroups — C13×#17654 #15703993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#17654 #15703993