Subgroups — C₁₃×#3178 #13112236

Order 1664.

350 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#542	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#541	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#501	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#497	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#488	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#470	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#468	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#465	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#441	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#440	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₁₃×#126	—	2	1	no	—
25	C₁₃×#126	—	2	1	no	—
26	C₁₃×#126	—	2	1	no	—
27	C₁₃×#125	—	2	1	no	—
28	C₁₃×#125	—	2	1	no	—
29	C₁₃×#125	C₂²	1	1	yes	characteristic
30	C₁₃×#124	—	2	1	no	—
31	C₁₃×#124	C₂²	1	1	yes	characteristic
32	C₁₃×#123	—	2	1	no	—
33	C₁₃×C₂²⋊Q₈	—	2	1	no	—
34	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
35	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
36	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
37	C₁₃×#121	C₂²	1	1	yes	characteristic
38	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
39	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
40	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
41	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
42	C₁₃×#117	C₂²	1	1	yes	characteristic
43	C₁₃×#107	C₂²	1	1	yes	characteristic
44	C₁₃×#107	C₂²	1	1	yes	characteristic
45	C₁₃×#106	C₂²	1	1	yes	characteristic
46	C₁₃×#106	C₂²	1	1	yes	characteristic
47	C₁₃×#103	C₂²	1	1	yes	characteristic
48	C₁₃×#103	C₂²	1	1	yes	characteristic
49	C₁₃×#103	C₂²	1	1	yes	characteristic
50	C₁₃×#103	C₂²	1	1	yes	characteristic
51	C₁₃×#103	C₂²	1	1	yes	characteristic
52	C₁₃×#103	C₂²	1	1	yes	characteristic
53	C₁₃×#102	C₂²	1	1	yes	characteristic
54	C₁₃×#102	C₂²	1	1	yes	characteristic
55	C₁₃×#97	C₂²	1	1	yes	characteristic
56	C₁₃×#97	C₂²	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	—	2	1	no	—
60	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₁₃	—	2	1	no	—
64	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	—	2	1	no	—
66	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
67	C₁₃×Q₁₆	—	2	2	no	—
68	C₁₃×Q₁₆	—	2	2	no	—
69	C₁₃×SD₁₆	—	2	2	no	—
70	C₁₃×SD₁₆	—	2	2	no	—
71	C₁₃×M₁₆	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
77	C₁₃×C₄⋊C₄	—	2	1	no	—
78	C₁₃×C₄⋊C₄	—	2	1	no	—
79	C₁₃×C₄⋊C₄	—	2	1	no	—
80	C₁₃×C₄⋊C₄	—	2	1	no	—
81	C₁₃×C₄⋊C₄	—	2	1	no	—
82	C₁₃×C₄⋊C₄	—	2	1	no	—
83	C₁₃×C₄⋊C₄	—	2	1	no	—
84	C₁₃×C₄⋊C₄	—	2	1	no	—
85	C₁₃×C₄⋊C₄	—	2	1	no	—
86	C₁₃×C₄⋊C₄	—	2	1	no	—
87	C₁₃×C₄⋊C₄	—	2	1	no	—
88	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
89	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
91	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
92	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
94	C₁₃×C₂²⋊C₄	—	2	1	no	—
95	C₁₃×C₂²⋊C₄	—	2	1	no	—
96	C₁₃×C₂²⋊C₄	—	2	1	no	—
97	C₁₃×C₂²⋊C₄	—	2	1	no	—
98	C₁₃×C₂²⋊C₄	—	2	1	no	—
99	C₁₃×C₂²⋊C₄	—	2	1	no	—
100	C₁₃×C₂²⋊C₄	—	2	1	no	—
101	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
102	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
103	C₄²×C₁₃	—	2	1	no	—
104	C₄²×C₁₃	—	2	1	no	—
105	C₄²×C₁₃	—	2	1	no	—
106	C₄²×C₁₃	C₂³	1	1	yes	characteristic
107	C₄²×C₁₃	C₂³	1	1	yes	characteristic
108	#3178	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂³×C₁₃	—	2	1	no	—
110	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₁₃	—	2	1	no	—
112	Q₈×C₁₃	—	2	1	no	—
113	Q₈×C₁₃	—	2	1	no	—
114	Q₈×C₁₃	—	2	1	no	—
115	Q₈×C₁₃	—	2	1	no	—
116	Q₈×C₁₃	—	2	1	no	—
117	Q₈×C₁₃	—	8	1	no	—
118	D₄×C₁₃	—	2	1	no	—
119	D₄×C₁₃	—	2	1	no	—
120	D₄×C₁₃	—	8	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	8	1	no	—
131	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
137	C₈×C₁₃	—	2	1	no	—
138	C₈×C₁₃	—	2	1	no	—
139	C₈×C₁₃	#41	1	2	yes	—
140	C₈×C₁₃	#41	1	2	yes	—
141	#542	C₂×C₁₃	1	1	yes	characteristic
142	#541	C₂×C₁₃	1	1	yes	characteristic, complemented
143	#501	C₂×C₁₃	1	1	yes	characteristic, complemented
144	#497	C₂×C₁₃	1	1	yes	characteristic, complemented
145	#488	C₂×C₁₃	1	1	yes	characteristic, complemented
146	#487	C₂×C₁₃	1	1	yes	characteristic, complemented
147	#470	C₂×C₁₃	1	1	yes	characteristic, complemented
148	#468	C₂×C₁₃	1	1	yes	characteristic
149	#467	C₂×C₁₃	1	1	yes	characteristic, complemented
150	#465	C₂×C₁₃	1	1	yes	characteristic
151	#441	C₂×C₁₃	1	1	yes	characteristic, complemented
152	#440	C₂×C₁₃	1	1	yes	characteristic, complemented
153	C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
154	C₄×SD₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
155	#405	C₂×C₁₃	1	1	yes	characteristic, complemented
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	—	2	1	no	—
159	C₂²×C₁₃	—	4	1	no	—
160	C₂²×C₁₃	—	4	1	no	—
161	C₂²×C₁₃	—	4	1	no	—
162	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	—	4	1	no	—
169	C₄×C₁₃	—	4	1	no	—
170	C₄×C₁₃	—	4	1	no	—
171	C₄×C₁₃	—	4	1	no	—
172	C₄×C₁₃	—	4	1	no	—
173	C₄×C₁₃	—	4	1	no	—
174	C₄×C₁₃	—	4	1	no	—
175	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
176	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
177	C₂×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
178	C₂×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
179	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
180	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
181	C₂×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
183	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
184	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
185	#126	—	2	1	no	—
186	#126	—	2	1	no	—
187	#126	—	2	1	no	—
188	#125	—	2	1	no	—
189	#125	—	2	1	no	—
190	#125	C₂²×C₁₃	1	1	yes	characteristic
191	#124	—	2	1	no	—
192	#124	C₂²×C₁₃	1	1	yes	characteristic
193	#123	—	2	1	no	—
194	C₂²⋊Q₈	—	2	1	no	—
195	C₂²⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
196	C₂²⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
197	C₂²⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
198	#121	C₂²×C₁₃	1	1	yes	characteristic
199	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
200	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
201	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
202	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
203	#117	C₂²×C₁₃	1	1	yes	characteristic
204	#107	C₂²×C₁₃	1	1	yes	characteristic
205	#107	C₂²×C₁₃	1	1	yes	characteristic
206	#106	C₂²×C₁₃	1	1	yes	characteristic
207	#106	C₂²×C₁₃	1	1	yes	characteristic
208	#103	C₂²×C₁₃	1	1	yes	characteristic
209	#103	C₂²×C₁₃	1	1	yes	characteristic
210	#103	C₂²×C₁₃	1	1	yes	characteristic
211	#103	C₂²×C₁₃	1	1	yes	characteristic
212	#103	C₂²×C₁₃	1	1	yes	characteristic
213	#103	C₂²×C₁₃	1	1	yes	characteristic
214	#102	C₂²×C₁₃	1	1	yes	characteristic
215	#102	C₂²×C₁₃	1	1	yes	characteristic
216	#97	C₂²×C₁₃	1	1	yes	characteristic
217	#97	C₂²×C₁₃	1	1	yes	characteristic
218	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
219	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
220	C₂×C₁₃	—	4	1	no	—
221	C₂×C₁₃	—	8	1	no	—
222	C₂×C₁₃	#532	1	1	yes	characteristic
223	C₂×C₁₃	#577	1	1	yes	characteristic
224	C₂×C₁₃	#578	1	1	yes	characteristic
225	C₂×Q₈	—	2	1	no	—
226	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
227	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
228	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
233	Q₁₆	—	2	2	no	—
234	Q₁₆	—	2	2	no	—
235	SD₁₆	—	2	2	no	—
236	SD₁₆	—	2	2	no	—
237	M₁₆	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
240	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
241	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
242	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
255	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
256	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
257	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
258	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
259	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	D₄×C₁₃	1	1	yes	characteristic
268	C₂²⋊C₄	D₄×C₁₃	1	1	yes	characteristic
269	C₄²	—	2	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	C₂³×C₁₃	1	1	yes	characteristic
273	C₄²	C₂³×C₁₃	1	1	yes	characteristic
274	C₁₃	#3178	1	1	yes	characteristic, direct product factor, Sylow
275	C₂³	—	2	1	no	—
276	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	8	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	8	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
299	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
301	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
302	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	C₁₃×#41	1	2	yes	—
306	C₈	C₁₃×#41	1	2	yes	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
327	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
328	C₂	—	4	1	no	—
329	C₂	—	8	1	no	—
330	C₂	C₁₃×#532	1	1	yes	characteristic
331	C₂	C₁₃×#577	1	1	yes	characteristic
332	C₂	C₁₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#541	C₂	semidirect
3. C₁₃×#501	C₂	semidirect
4. C₁₃×#497	C₂	semidirect
5. C₁₃×#488	C₂	semidirect
6. C₁₃×#487	C₂	semidirect
7. C₁₃×#470	C₂	semidirect
9. C₁₃×#467	C₂	semidirect
11. C₁₃×#441	C₂	semidirect
12. C₁₃×#440	C₂	semidirect
13. C₄×C₁₃×Q₁₆	C₂	semidirect
14. C₄×C₁₃×SD₁₆	C₂	semidirect
15. C₁₃×#405	C₂	semidirect
108. #3178	C₁₃	direct (N × H)
142. #541	C₂×C₁₃	semidirect
143. #501	C₂×C₁₃	semidirect
144. #497	C₂×C₁₃	semidirect
145. #488	C₂×C₁₃	semidirect
146. #487	C₂×C₁₃	semidirect
147. #470	C₂×C₁₃	semidirect
149. #467	C₂×C₁₃	semidirect
151. #441	C₂×C₁₃	semidirect
152. #440	C₂×C₁₃	semidirect
153. C₄×Q₁₆	C₂×C₁₃	semidirect
154. C₄×SD₁₆	C₂×C₁₃	semidirect
155. #405	C₂×C₁₃	semidirect
274. C₁₃	#3178	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×#542	2
C₁₃×#541	2
C₁₃×#501	2
C₁₃×#497	2
C₁₃×#488	2
C₁₃×#487	2
C₁₃×#470	2
C₁₃×#468	2
C₁₃×#467	2
C₁₃×#465	2
C₁₃×#441	2
C₁₃×#440	2
C₄×C₁₃×Q₁₆	2
C₄×C₁₃×SD₁₆	2
C₁₃×#405	2
#3178	13

Subgroups — C13×#3178 #13112236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3178 #13112236