Subgroups — C₁₃×#10148 #15696487

Order 3328.

614 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3227	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3151	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1997	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1545	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1492	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×AGL(1,ℤ/8)	—	2	2	no	—
9	C₁₃×#532	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
12	C₁₃×#498	—	2	1	no	—
13	C₁₃×#496	—	2	2	no	—
14	C₁₃×#493	—	2	1	no	—
15	C₁₃×#492	—	2	1	no	—
16	C₁₃×#486	—	2	2	no	—
17	C₁₃×#486	—	2	1	no	—
18	C₁₃×#486	—	2	2	no	—
19	C₁₃×#481	—	2	2	no	—
20	C₁₃×#476	—	2	2	no	—
21	C₁₃×#431	C₂²	1	1	yes	characteristic
22	C₁₃×#423	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×#102	—	2	2	no	—
24	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×#343	—	2	2	no	—
27	C₁₃×#331	—	2	2	no	—
28	C₁₃×D₄⋊C₈	—	2	2	no	—
29	C₂×C₁₃×#41	—	2	1	no	—
30	C₂²×Q₈×C₁₃	—	2	1	no	—
31	C₂²×D₄×C₁₃	—	2	1	no	—
32	C₂²×D₄×C₁₃	—	2	1	no	—
33	C₂²×D₄×C₁₃	—	2	1	no	—
34	C₁₃×AGL(1,ℤ/8)	—	4	4	no	—
35	C₁₃×AGL(1,ℤ/8)	—	4	4	no	—
36	C₂×C₁₃×SD₁₆	—	2	4	no	—
37	C₂×C₁₃×D₈	—	2	4	no	—
38	C₂×C₁₃×M₁₆	—	2	1	no	—
39	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
40	C₂²×C₈×C₁₃	—	2	1	no	—
41	C₁₃×C₄⋊Q₈	—	2	1	no	—
42	C₁₃×C₄⋊Q₈	—	2	1	no	—
43	C₁₃×C₄⋊Q₈	—	2	1	no	—
44	C₁₃×C₄⋊Q₈	—	2	1	no	—
45	C₁₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
46	C₁₃×#127	—	2	1	no	—
47	C₁₃×#127	—	2	1	no	—
48	C₁₃×#127	—	2	1	no	—
49	C₁₃×#127	D₄	1	1	yes	characteristic
50	C₁₃×#127	D₄	1	2	yes	—
51	C₁₃×#124	—	2	1	no	—
52	C₁₃×#121	—	2	1	no	—
53	C₁₃×#121	—	2	1	no	—
54	C₄×D₄×C₁₃	—	2	1	no	—
55	C₄×D₄×C₁₃	—	2	1	no	—
56	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
57	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
58	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	C₁₃×#106	—	2	4	no	—
60	C₁₃×C₄⋊C₈	—	2	1	no	—
61	C₁₃×C₄⋊C₈	—	2	1	no	—
62	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
63	C₁₃×#102	—	2	8	no	—
64	C₁₃×#102	—	2	4	no	—
65	C₁₃×C₂²⋊C₈	—	2	1	no	—
66	C₁₃×#97	—	2	1	no	—
67	C₄×C₈×C₁₃	—	2	1	no	—
68	C₄×C₈×C₁₃	—	2	1	no	—
69	C₄×C₈×C₁₃	—	2	1	no	—
70	C₄×C₈×C₁₃	D₄	1	2	yes	—
71	#10148	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂⁴×C₁₃	—	4	1	no	—
73	C₁₃×#41	—	4	2	no	—
74	C₁₃×#41	—	4	2	no	—
75	C₂×Q₈×C₁₃	—	2	1	no	—
76	C₂×Q₈×C₁₃	—	4	1	no	—
77	C₂×Q₈×C₁₃	—	4	1	no	—
78	C₂×Q₈×C₁₃	—	8	2	no	—
79	C₂×D₄×C₁₃	—	2	1	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	—	2	1	no	—
84	C₂×D₄×C₁₃	—	2	2	no	—
85	C₂×D₄×C₁₃	—	4	4	no	—
86	C₂×D₄×C₁₃	—	4	1	no	—
87	C₂×D₄×C₁₃	—	4	1	no	—
88	C₂×D₄×C₁₃	—	4	1	no	—
89	C₂×D₄×C₁₃	—	4	1	no	—
90	C₂×D₄×C₁₃	—	4	2	no	—
91	C₂×D₄×C₁₃	—	4	2	no	—
92	C₂×D₄×C₁₃	—	8	2	no	—
93	C₂²×C₄×C₁₃	—	2	2	no	—
94	C₂²×C₄×C₁₃	—	4	1	no	—
95	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
98	C₁₃×SD₁₆	—	4	8	no	—
99	C₁₃×D₈	—	4	8	no	—
100	C₁₃×M₁₆	—	2	2	no	—
101	C₁₃×M₁₆	—	4	2	no	—
102	C₁₃×M₁₆	SD₁₆	1	4	yes	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	—	4	2	no	—
107	C₂×C₈×C₁₃	—	4	1	no	—
108	C₁₃×C₄⋊C₄	—	2	2	no	—
109	C₁₃×C₄⋊C₄	—	2	2	no	—
110	C₁₃×C₄⋊C₄	—	2	1	no	—
111	C₁₃×C₄⋊C₄	—	2	1	no	—
112	C₁₃×C₄⋊C₄	—	4	1	no	—
113	C₁₃×C₄⋊C₄	—	4	1	no	—
114	C₁₃×C₂²⋊C₄	—	4	1	no	—
115	C₁₃×C₂²⋊C₄	—	4	1	no	—
116	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
120	#3227	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#3151	C₂×C₁₃	1	1	yes	characteristic
122	C₂×#486	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#1997	C₂×C₁₃	1	1	yes	characteristic, complemented
124	#1545	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#1492	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#1278	C₂×C₁₃	1	1	yes	characteristic, complemented
127	C₂³×C₁₃	—	4	1	no	—
128	C₂³×C₁₃	—	4	2	no	—
129	C₂³×C₁₃	—	8	2	no	—
130	C₂³×C₁₃	—	8	2	no	—
131	C₂³×C₁₃	—	8	2	no	—
132	C₂³×C₁₃	#121	1	1	yes	characteristic
133	Q₈×C₁₃	—	4	2	no	—
134	Q₈×C₁₃	—	8	2	no	—
135	Q₈×C₁₃	—	8	2	no	—
136	D₄×C₁₃	—	4	4	no	—
137	D₄×C₁₃	—	4	2	no	—
138	D₄×C₁₃	—	4	4	no	—
139	D₄×C₁₃	—	4	2	no	—
140	D₄×C₁₃	—	4	2	no	—
141	D₄×C₁₃	—	8	2	no	—
142	D₄×C₁₃	—	8	2	no	—
143	D₄×C₁₃	—	8	2	no	—
144	D₄×C₁₃	—	8	2	no	—
145	D₄×C₁₃	—	8	1	no	—
146	D₄×C₁₃	—	8	1	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	4	1	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	—	8	1	no	—
152	C₂×C₄×C₁₃	—	8	1	no	—
153	C₂×C₄×C₁₃	—	8	2	no	—
154	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
155	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
156	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
157	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
158	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
159	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
161	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
162	C₈×C₁₃	—	2	4	no	—
163	C₈×C₁₃	—	4	2	no	—
164	C₈×C₁₃	—	4	2	no	—
165	C₈×C₁₃	—	8	1	no	—
166	C₂×AGL(1,ℤ/8)	—	2	2	no	—
167	#532	C₂²×C₁₃	1	1	yes	characteristic
168	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
169	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
170	#498	—	2	1	no	—
171	#496	—	2	2	no	—
172	#493	—	2	1	no	—
173	#492	—	2	1	no	—
174	#486	—	2	2	no	—
175	#486	—	2	1	no	—
176	#486	—	2	2	no	—
177	#481	—	2	2	no	—
178	#476	—	2	2	no	—
179	#431	C₂²×C₁₃	1	1	yes	characteristic
180	#423	C₂²×C₁₃	1	1	yes	characteristic
181	C₂×#102	—	2	2	no	—
182	C₄×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
183	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
184	#343	—	2	2	no	—
185	#331	—	2	2	no	—
186	D₄⋊C₈	—	2	2	no	—
187	C₂²×C₁₃	—	2	1	no	—
188	C₂²×C₁₃	—	2	1	no	—
189	C₂²×C₁₃	—	8	2	no	—
190	C₂²×C₁₃	—	8	1	no	—
191	C₂²×C₁₃	—	8	1	no	—
192	C₂²×C₁₃	—	8	2	no	—
193	C₂²×C₁₃	—	8	1	no	—
194	C₂²×C₁₃	—	8	2	no	—
195	C₂²×C₁₃	—	16	4	no	—
196	C₂²×C₁₃	#392	1	1	yes	characteristic
197	C₂²×C₁₃	#460	1	2	yes	—
198	C₄×C₁₃	—	2	1	no	—
199	C₄×C₁₃	—	2	1	no	—
200	C₄×C₁₃	—	8	1	no	—
201	C₄×C₁₃	—	8	2	no	—
202	C₄×C₁₃	#450	1	2	yes	—
203	C₄×C₁₃	#454	1	2	yes	—
204	C₄×C₁₃	#472	1	2	yes	—
205	C₄×C₁₃	#492	1	1	yes	characteristic
206	C₄×C₁₃	#496	1	1	yes	characteristic
207	C₂×#41	—	2	1	no	—
208	C₂²×Q₈	—	2	1	no	—
209	C₂²×D₄	—	2	1	no	—
210	C₂²×D₄	—	2	1	no	—
211	C₂²×D₄	—	2	1	no	—
212	AGL(1,ℤ/8)	—	4	4	no	—
213	AGL(1,ℤ/8)	—	4	4	no	—
214	C₂×SD₁₆	—	2	4	no	—
215	C₂×D₈	—	2	4	no	—
216	C₂×M₁₆	—	2	1	no	—
217	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
218	C₂²×C₈	—	2	1	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	2	1	no	—
223	C₄⋊Q₈	D₄×C₁₃	1	1	yes	characteristic
224	#127	—	2	1	no	—
225	#127	—	2	1	no	—
226	#127	—	2	1	no	—
227	#127	D₄×C₁₃	1	2	yes	—
228	#127	D₄×C₁₃	1	1	yes	characteristic
229	#124	—	2	1	no	—
230	#121	—	2	1	no	—
231	#121	—	2	1	no	—
232	C₄×D₄	—	2	1	no	—
233	C₄×D₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
236	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
237	#106	—	2	4	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
241	#102	—	2	8	no	—
242	#102	—	2	4	no	—
243	C₂²⋊C₈	—	2	1	no	—
244	#97	—	2	1	no	—
245	C₄×C₈	—	2	1	no	—
246	C₄×C₈	—	2	1	no	—
247	C₄×C₈	—	2	1	no	—
248	C₄×C₈	D₄×C₁₃	1	2	yes	—
249	C₂×C₁₃	—	2	2	no	—
250	C₂×C₁₃	—	16	1	no	—
251	C₂×C₁₃	—	16	2	no	—
252	C₂×C₁₃	#2003	1	1	yes	characteristic
253	C₂×C₁₃	#2014	1	1	yes	characteristic
254	C₂×C₁₃	#2018	1	1	yes	characteristic
255	C₂⁴	—	4	1	no	—
256	#41	—	4	2	no	—
257	#41	—	4	2	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	—	4	1	no	—
260	C₂×Q₈	—	4	1	no	—
261	C₂×Q₈	—	8	2	no	—
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	1	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	1	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	4	4	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂×D₄	—	4	1	no	—
272	C₂×D₄	—	4	1	no	—
273	C₂×D₄	—	4	2	no	—
274	C₂×D₄	—	4	2	no	—
275	C₂×D₄	—	8	2	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	4	1	no	—
278	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
280	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
281	SD₁₆	—	4	8	no	—
282	D₈	—	4	8	no	—
283	M₁₆	—	2	2	no	—
284	M₁₆	—	4	2	no	—
285	M₁₆	C₁₃×SD₁₆	1	4	yes	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	4	2	no	—
290	C₂×C₈	—	4	1	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₂²⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	—	4	1	no	—
299	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
300	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
301	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
302	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
303	C₁₃	#10148	1	1	yes	characteristic, direct product factor, Sylow
304	C₂³	—	4	1	no	—
305	C₂³	—	4	2	no	—
306	C₂³	—	8	2	no	—
307	C₂³	—	8	2	no	—
308	C₂³	—	8	2	no	—
309	C₂³	C₁₃×#121	1	1	yes	characteristic
310	Q₈	—	4	2	no	—
311	Q₈	—	8	2	no	—
312	Q₈	—	8	2	no	—
313	D₄	—	4	4	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	4	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	8	2	no	—
319	D₄	—	8	2	no	—
320	D₄	—	8	2	no	—
321	D₄	—	8	2	no	—
322	D₄	—	8	1	no	—
323	D₄	—	8	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	8	1	no	—
329	C₂×C₄	—	8	1	no	—
330	C₂×C₄	—	8	2	no	—
331	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
332	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
333	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
334	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
335	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
336	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
337	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
338	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
339	C₈	—	2	4	no	—
340	C₈	—	4	2	no	—
341	C₈	—	4	2	no	—
342	C₈	—	8	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	1	no	—
347	C₂²	—	8	1	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	8	1	no	—
350	C₂²	—	8	2	no	—
351	C₂²	—	16	4	no	—
352	C₂²	C₁₃×#392	1	1	yes	characteristic
353	C₂²	C₁₃×#460	1	2	yes	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	8	1	no	—
357	C₄	—	8	2	no	—
358	C₄	C₁₃×#450	1	2	yes	—
359	C₄	C₁₃×#454	1	2	yes	—
360	C₄	C₁₃×#472	1	2	yes	—
361	C₄	C₁₃×#492	1	1	yes	characteristic
362	C₄	C₁₃×#496	1	1	yes	characteristic
363	C₂	—	2	2	no	—
364	C₂	—	16	2	no	—
365	C₂	—	16	1	no	—
366	C₂	C₁₃×#2003	1	1	yes	characteristic
367	C₂	C₁₃×#2014	1	1	yes	characteristic
368	C₂	C₁₃×#2018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₁₃×#3151	C₂
LCS	G ▷ C₂×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. C₁₃×#3227	C₂	semidirect
3. C₂×C₁₃×#486	C₂	semidirect
4. C₁₃×#1997	C₂	semidirect
5. C₁₃×#1545	C₂	semidirect
6. C₁₃×#1492	C₂	semidirect
7. C₁₃×#1278	C₂	semidirect
71. #10148	C₁₃	direct (N × H)
120. #3227	C₂×C₁₃	semidirect
122. C₂×#486	C₂×C₁₃	semidirect
123. #1997	C₂×C₁₃	semidirect
124. #1545	C₂×C₁₃	semidirect
125. #1492	C₂×C₁₃	semidirect
126. #1278	C₂×C₁₃	semidirect
303. C₁₃	#10148	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×#3227	2
C₁₃×#3151	2
C₂×C₁₃×#486	2
C₁₃×#1997	2
C₁₃×#1545	2
C₁₃×#1492	2
C₁₃×#1278	2
#10148	13

Subgroups — C13×#10148 #15696487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10148 #15696487