Subgroups — C₁₃×#13035 #15699374

Order 3328.

456 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#510	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2957	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2143	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1979	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1385	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₁₃×#568	—	2	1	no	—
11	C₁₃×#510	—	2	2	no	—
12	C₁₃×#510	—	2	2	no	—
13	C₂×C₁₃×Q₃₂	C₄	1	2	yes	complemented
14	C₂×C₁₃×SD₃₂	C₄	1	2	yes	complemented
15	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
17	C₁₃×#444	—	2	2	no	—
18	C₁₃×#443	—	2	2	no	—
19	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
21	C₁₃×#421	—	2	1	no	—
22	C₁₃×#419	—	2	1	no	—
23	C₂×C₁₃×#101	—	2	1	no	—
24	C₁₃×#405	C₂²	1	1	yes	characteristic
25	C₁₃×#362	—	2	2	no	—
26	C₁₃×#359	—	2	2	no	—
27	C₂×C₁₃×#41	—	2	1	no	—
28	C₂²×Q₈×C₁₃	—	2	1	no	—
29	C₁₃×#135	—	2	1	no	—
30	C₁₃×#135	—	2	1	no	—
31	C₁₃×#135	D₄	1	2	yes	—
32	C₁₃×#135	D₄	1	2	yes	—
33	C₂×C₁₃×Q₁₆	—	2	1	no	—
34	C₂×C₁₃×Q₁₆	—	2	1	no	—
35	C₂×C₁₃×Q₁₆	—	2	1	no	—
36	C₂×C₁₃×Q₁₆	—	2	1	no	—
37	C₂×C₁₃×Q₁₆	—	2	1	no	—
38	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₃×SD₁₆	—	2	1	no	—
41	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
42	C₁₃×#131	—	2	4	no	—
43	C₁₃×#131	—	4	2	no	—
44	C₂×C₁₃×M₁₆	—	2	1	no	—
45	C₂×C₁₃×M₁₆	—	2	1	no	—
46	C₂×C₁₃×M₁₆	—	4	2	no	—
47	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×#117	D₄	1	1	yes	characteristic
50	C₁₃×Q₃₂	—	2	4	no	—
51	C₁₃×SD₃₂	—	2	4	no	—
52	C₁₃×M₃₂	—	2	1	no	—
53	C₁₃×M₃₂	—	2	1	no	—
54	C₁₃×M₃₂	—	2	1	no	—
55	C₁₃×M₃₂	—	2	1	no	—
56	C₂×C₁₃×C₁₆	—	2	1	no	—
57	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
58	C₁₃×#108	—	2	1	no	—
59	C₁₃×#108	—	2	1	no	—
60	C₁₃×#108	—	2	1	no	—
61	C₁₃×#108	D₄	1	2	yes	—
62	C₁₃×#104	—	2	4	no	—
63	C₁₃×#103	—	4	2	no	—
64	C₁₃×#101	—	4	2	no	—
65	C₁₃×#97	—	2	1	no	—
66	C₄×C₈×C₁₃	—	2	1	no	—
67	#13035	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₁₃×#41	—	2	2	no	—
69	C₁₃×#41	—	2	2	no	—
70	C₁₃×#41	—	4	1	no	—
71	C₁₃×#41	—	4	1	no	—
72	C₂×Q₈×C₁₃	—	2	1	no	—
73	C₂×Q₈×C₁₃	—	2	2	no	—
74	C₂×Q₈×C₁₃	—	4	1	no	—
75	C₂×Q₈×C₁₃	—	4	1	no	—
76	C₂×Q₈×C₁₃	—	4	1	no	—
77	C₂×Q₈×C₁₃	—	4	1	no	—
78	C₂×Q₈×C₁₃	—	4	1	no	—
79	C₂×D₄×C₁₃	—	2	1	no	—
80	C₂×D₄×C₁₃	—	4	1	no	—
81	C₂²×C₄×C₁₃	—	4	1	no	—
82	C₂²×C₄×C₁₃	—	4	1	no	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₁₃×Q₁₆	—	2	2	no	—
85	C₁₃×Q₁₆	—	2	1	no	—
86	C₁₃×Q₁₆	—	2	2	no	—
87	C₁₃×Q₁₆	—	4	1	no	—
88	C₁₃×Q₁₆	—	4	1	no	—
89	C₁₃×Q₁₆	C₂²⋊C₄	1	2	yes	—
90	C₁₃×SD₁₆	—	2	2	no	—
91	C₁₃×SD₁₆	—	2	2	no	—
92	C₁₃×D₈	—	2	1	no	—
93	C₁₃×D₈	C₂²⋊C₄	1	2	yes	—
94	C₁₃×M₁₆	—	2	1	no	—
95	C₁₃×M₁₆	—	2	2	no	—
96	C₁₃×M₁₆	—	4	4	no	—
97	C₁₃×M₁₆	—	4	1	no	—
98	C₁₃×M₁₆	—	4	1	no	—
99	C₁₃×M₁₆	—	4	1	no	—
100	C₁₃×M₁₆	D₈	1	2	yes	—
101	C₂×C₈×C₁₃	—	2	1	no	—
102	C₂×C₈×C₁₃	—	4	4	no	—
103	C₂×C₈×C₁₃	—	4	1	no	—
104	C₂×C₈×C₁₃	—	4	1	no	—
105	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
110	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
111	C₁₃×C₄⋊C₄	—	2	1	no	—
112	C₁₃×C₂²⋊C₄	—	2	1	no	—
113	C₄²×C₁₃	—	2	1	no	—
114	C₁₃×C₁₆	—	2	2	no	—
115	C₁₃×C₁₆	—	4	1	no	—
116	C₂×#510	C₂×C₁₃	1	1	yes	characteristic
117	#2957	C₂×C₁₃	1	1	yes	characteristic
118	C₂×#362	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#2143	C₂×C₁₃	1	1	yes	characteristic
120	#1979	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#1385	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#1378	C₂×C₁₃	1	1	yes	characteristic, complemented
123	C₂³×C₁₃	—	4	1	no	—
124	C₂³×C₁₃	#135	1	1	yes	characteristic
125	Q₈×C₁₃	—	2	2	no	—
126	Q₈×C₁₃	—	4	1	no	—
127	Q₈×C₁₃	—	4	2	no	—
128	Q₈×C₁₃	—	4	2	no	—
129	Q₈×C₁₃	—	8	1	no	—
130	Q₈×C₁₃	—	8	1	no	—
131	D₄×C₁₃	—	2	2	no	—
132	D₄×C₁₃	—	4	2	no	—
133	D₄×C₁₃	—	4	2	no	—
134	D₄×C₁₃	—	4	1	no	—
135	C₂×C₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	1	no	—
140	C₂×C₄×C₁₃	—	8	1	no	—
141	C₂×C₄×C₁₃	—	8	1	no	—
142	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
144	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₈×C₁₃	—	2	2	no	—
149	C₈×C₁₃	—	4	2	no	—
150	C₈×C₁₃	—	8	1	no	—
151	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
153	C₈×C₁₃	#121	1	1	yes	characteristic
154	C₈×C₁₃	#123	1	1	yes	characteristic
155	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
156	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
157	#568	—	2	1	no	—
158	#510	—	2	2	no	—
159	#510	—	2	2	no	—
160	C₂×Q₃₂	C₄×C₁₃	1	2	yes	complemented
161	C₂×SD₃₂	C₄×C₁₃	1	2	yes	complemented
162	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
164	#444	—	2	2	no	—
165	#443	—	2	2	no	—
166	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
167	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
168	#421	—	2	1	no	—
169	#419	—	2	1	no	—
170	C₂×#101	—	2	1	no	—
171	#405	C₂²×C₁₃	1	1	yes	characteristic
172	#362	—	2	2	no	—
173	#359	—	2	2	no	—
174	C₂²×C₁₃	—	4	1	no	—
175	C₂²×C₁₃	—	4	2	no	—
176	C₂²×C₁₃	—	16	1	no	—
177	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
178	C₂²×C₁₃	#449	1	1	yes	characteristic
179	C₂²×C₁₃	#463	1	1	yes	characteristic
180	C₄×C₁₃	—	4	2	no	—
181	C₄×C₁₃	—	8	1	no	—
182	C₄×C₁₃	—	8	2	no	—
183	C₄×C₁₃	#386	1	1	yes	characteristic
184	C₄×C₁₃	#442	1	1	yes	characteristic
185	C₄×C₁₃	#447	1	1	yes	characteristic
186	C₄×C₁₃	#459	1	1	yes	characteristic
187	C₂×#41	—	2	1	no	—
188	C₂²×Q₈	—	2	1	no	—
189	#135	—	2	1	no	—
190	#135	—	2	1	no	—
191	#135	D₄×C₁₃	1	2	yes	—
192	#135	D₄×C₁₃	1	2	yes	—
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	—	2	1	no	—
195	C₂×Q₁₆	—	2	1	no	—
196	C₂×Q₁₆	—	2	1	no	—
197	C₂×Q₁₆	—	2	1	no	—
198	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
199	C₂×Q₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
200	C₂×SD₁₆	—	2	1	no	—
201	C₂×D₈	C₂×C₄×C₁₃	1	1	yes	characteristic
202	#131	—	2	4	no	—
203	#131	—	4	2	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	4	2	no	—
207	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
209	#117	D₄×C₁₃	1	1	yes	characteristic
210	Q₃₂	—	2	4	no	—
211	SD₃₂	—	2	4	no	—
212	M₃₂	—	2	1	no	—
213	M₃₂	—	2	1	no	—
214	M₃₂	—	2	1	no	—
215	M₃₂	—	2	1	no	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
218	#108	—	2	1	no	—
219	#108	—	2	1	no	—
220	#108	—	2	1	no	—
221	#108	D₄×C₁₃	1	2	yes	—
222	#104	—	2	4	no	—
223	#103	—	4	2	no	—
224	#101	—	4	2	no	—
225	#97	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₂×C₁₃	—	2	1	no	—
228	C₂×C₁₃	—	2	1	no	—
229	C₂×C₁₃	—	2	1	no	—
230	C₂×C₁₃	—	8	2	no	—
231	C₂×C₁₃	#1879	1	1	yes	characteristic
232	#41	—	2	2	no	—
233	#41	—	2	2	no	—
234	#41	—	4	1	no	—
235	#41	—	4	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
248	Q₁₆	—	2	2	no	—
249	Q₁₆	—	2	1	no	—
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	4	1	no	—
252	Q₁₆	—	4	1	no	—
253	Q₁₆	C₁₃×C₂²⋊C₄	1	2	yes	—
254	SD₁₆	—	2	2	no	—
255	SD₁₆	—	2	2	no	—
256	D₈	—	2	1	no	—
257	D₈	C₁₃×C₂²⋊C₄	1	2	yes	—
258	M₁₆	—	2	1	no	—
259	M₁₆	—	2	2	no	—
260	M₁₆	—	4	4	no	—
261	M₁₆	—	4	1	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	—	4	1	no	—
264	M₁₆	C₁₃×D₈	1	2	yes	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	4	4	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
270	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
273	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
274	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₄²	—	2	1	no	—
278	C₁₆	—	2	2	no	—
279	C₁₆	—	4	1	no	—
280	C₁₃	#13035	1	1	yes	characteristic, direct product factor, Sylow
281	C₂³	—	4	1	no	—
282	C₂³	C₁₃×#135	1	1	yes	characteristic
283	Q₈	—	2	2	no	—
284	Q₈	—	4	1	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	8	1	no	—
288	Q₈	—	8	1	no	—
289	D₄	—	2	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
301	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
302	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
303	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
304	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
305	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	4	2	no	—
308	C₈	—	8	1	no	—
309	C₈	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
310	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
311	C₈	C₁₃×#121	1	1	yes	characteristic
312	C₈	C₁₃×#123	1	1	yes	characteristic
313	C₂²	—	4	1	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	16	1	no	—
316	C₂²	C₄×C₁₃×D₈	1	1	yes	characteristic
317	C₂²	C₁₃×#449	1	1	yes	characteristic
318	C₂²	C₁₃×#463	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	2	no	—
322	C₄	C₁₃×#386	1	1	yes	characteristic
323	C₄	C₁₃×#442	1	1	yes	characteristic
324	C₄	C₁₃×#447	1	1	yes	characteristic
325	C₄	C₁₃×#459	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	8	2	no	—
330	C₂	C₁₃×#1879	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	#1879
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₃×#362	C₂	semidirect
5. C₁₃×#1979	C₂	semidirect
6. C₁₃×#1385	C₂	semidirect
7. C₁₃×#1378	C₂	semidirect
13. C₂×C₁₃×Q₃₂	C₄	semidirect
14. C₂×C₁₃×SD₃₂	C₄	semidirect
67. #13035	C₁₃	direct (N × H)
118. C₂×#362	C₂×C₁₃	semidirect
120. #1979	C₂×C₁₃	semidirect
121. #1385	C₂×C₁₃	semidirect
122. #1378	C₂×C₁₃	semidirect
160. C₂×Q₃₂	C₄×C₁₃	semidirect
161. C₂×SD₃₂	C₄×C₁₃	semidirect
280. C₁₃	#13035	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×#510	2
C₁₃×#2957	2
C₂×C₁₃×#362	2
C₁₃×#2143	2
C₁₃×#1979	2
C₁₃×#1385	2
C₁₃×#1378	2
#13035	13

Subgroups — C13×#13035 #15699374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13035 #15699374