Subgroups — C₁₃×#32509 #15718848

Order 3328.

564 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2508	C₂	1	2	yes	maximal
2	C₁₃×#2088	C₂	1	2	yes	maximal
3	C₁₃×#2086	C₂	1	2	yes	maximal
4	C₁₃×#2085	C₂	1	2	yes	maximal
5	C₁₃×#1949	C₂	1	2	yes	maximal
6	C₁₃×#1942	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1942	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#1939	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#1921	C₂	1	2	yes	maximal
10	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#106	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#400	—	2	2	no	—
19	C₁₃×#398	—	2	2	no	—
20	C₁₃×#389	C₂²	1	2	yes	—
21	C₁₃×#389	C₂²	1	2	yes	—
22	C₁₃×#389	C₂²	1	2	yes	—
23	C₁₃×#389	C₂²	1	2	yes	—
24	C₁₃×#387	—	2	2	no	—
25	C₁₃×#382	—	2	2	no	—
26	C₁₃×#382	C₂²	1	2	yes	—
27	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
28	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₁₃×#340	C₂²	1	2	yes	—
30	C₁₃×#340	C₂²	1	2	yes	—
31	C₁₃×#340	C₂²	1	2	yes	—
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#340	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
43	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
44	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₄²×C₁₃	—	2	2	no	—
49	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₁₃×#107	—	2	2	no	—
52	C₁₃×#107	—	4	1	no	—
53	C₁₃×#107	—	4	1	no	—
54	C₁₃×#107	Q₈	1	4	yes	—
55	C₁₃×#106	—	4	2	no	—
56	C₁₃×C₄⋊C₈	—	2	4	no	—
57	C₁₃×C₄⋊C₈	—	4	1	no	—
58	C₄×C₈×C₁₃	—	4	1	no	—
59	C₁₃×#95	—	2	2	no	—
60	C₁₃×#95	—	2	2	no	—
61	C₁₃×#95	—	2	2	no	—
62	C₁₃×#95	—	2	2	no	—
63	C₁₃×#95	—	2	2	no	—
64	C₁₃×#95	—	2	2	no	—
65	C₁₃×#95	—	2	2	no	—
66	C₁₃×#95	D₄	1	2	yes	—
67	#32509	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	—	2	2	no	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	—	2	1	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₁₃	#41	1	2	yes	—
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₁₃×C₄⋊C₄	—	2	2	no	—
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	4	1	no	—
95	C₁₃×C₄⋊C₄	—	4	4	no	—
96	C₁₃×C₄⋊C₄	—	8	2	no	—
97	C₁₃×C₄⋊C₄	—	8	2	no	—
98	C₁₃×C₄⋊C₄	—	8	2	no	—
99	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
100	C₄²×C₁₃	—	2	2	no	—
101	C₄²×C₁₃	—	2	1	no	—
102	C₄²×C₁₃	—	2	1	no	—
103	C₄²×C₁₃	—	2	2	no	—
104	C₄²×C₁₃	—	4	4	no	—
105	#2508	C₂×C₁₃	1	2	yes	—
106	#2088	C₂×C₁₃	1	2	yes	—
107	#2086	C₂×C₁₃	1	2	yes	—
108	#2085	C₂×C₁₃	1	2	yes	—
109	#1949	C₂×C₁₃	1	2	yes	—
110	#1942	C₂×C₁₃	1	1	yes	characteristic
111	#1942	C₂×C₁₃	1	1	yes	characteristic
112	#1939	C₂×C₁₃	1	1	yes	characteristic
113	#1921	C₂×C₁₃	1	2	yes	—
114	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	4	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	#135	1	2	yes	—
139	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
144	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
145	C₈×C₁₃	—	2	2	no	—
146	C₈×C₁₃	—	2	2	no	—
147	C₈×C₁₃	—	4	2	no	—
148	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
150	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
151	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
152	C₂×#106	C₂²×C₁₃	1	2	yes	—
153	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
154	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
156	#400	—	2	2	no	—
157	#398	—	2	2	no	—
158	#389	C₂²×C₁₃	1	2	yes	—
159	#389	C₂²×C₁₃	1	2	yes	—
160	#389	C₂²×C₁₃	1	2	yes	—
161	#389	C₂²×C₁₃	1	2	yes	—
162	#387	—	2	2	no	—
163	#382	—	2	2	no	—
164	#382	C₂²×C₁₃	1	2	yes	—
165	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
166	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
167	#340	C₂²×C₁₃	1	2	yes	—
168	#340	C₂²×C₁₃	1	2	yes	—
169	#340	C₂²×C₁₃	1	2	yes	—
170	#340	C₂²×C₁₃	1	2	yes	—
171	#340	C₂²×C₁₃	1	2	yes	—
172	#336	C₂²×C₁₃	1	2	yes	—
173	C₂²×C₁₃	#537	1	1	yes	characteristic
174	C₂²×C₁₃	#547	1	1	yes	characteristic
175	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
176	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
177	C₂²×C₁₃	#575	1	1	yes	characteristic
178	C₂²×C₁₃	#576	1	1	yes	characteristic
179	C₂²×C₁₃	#578	1	1	yes	characteristic
180	C₄×C₁₃	—	2	2	no	—
181	C₄×C₁₃	—	4	2	no	—
182	C₄×C₁₃	—	4	2	no	—
183	C₄×C₁₃	—	4	4	no	—
184	C₄×C₁₃	—	4	1	no	—
185	C₄×C₁₃	—	8	2	no	—
186	C₄×C₁₃	—	8	2	no	—
187	C₄×C₁₃	—	8	2	no	—
188	C₄×C₁₃	#539	1	1	yes	characteristic
189	C₄×C₁₃	#544	1	1	yes	characteristic
190	C₄×C₁₃	#557	1	1	yes	characteristic
191	C₄×C₁₃	#564	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
193	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
200	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
201	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
207	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
208	#107	—	2	2	no	—
209	#107	—	4	1	no	—
210	#107	—	4	1	no	—
211	#107	Q₈×C₁₃	1	4	yes	—
212	#106	—	4	2	no	—
213	C₄⋊C₈	—	2	4	no	—
214	C₄⋊C₈	—	4	1	no	—
215	C₄×C₈	—	4	1	no	—
216	#95	—	2	2	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	D₄×C₁₃	1	2	yes	—
224	C₂×C₁₃	#2726	1	1	yes	characteristic
225	C₂×C₁₃	#3250	1	1	yes	characteristic
226	C₂×C₁₃	#3262	1	1	yes	characteristic
227	C₂×C₁₃	#3341	1	1	yes	characteristic
228	C₂×C₁₃	#3352	1	1	yes	characteristic
229	C₂×C₁₃	#3383	1	1	yes	characteristic
230	C₂×C₁₃	#3389	1	1	yes	characteristic
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
239	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
248	C₂×C₈	C₁₃×#41	1	2	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	4	no	—
259	C₄⋊C₄	—	8	2	no	—
260	C₄⋊C₄	—	8	2	no	—
261	C₄⋊C₄	—	8	2	no	—
262	C₄⋊C₄	C₂×Q₈×C₁₃	1	4	yes	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	4	4	no	—
268	C₁₃	#32509	1	1	yes	characteristic, direct product factor, Sylow
269	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	C₁₃×#135	1	2	yes	—
294	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
295	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
296	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
297	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
298	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
299	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
300	C₈	—	2	2	no	—
301	C₈	—	2	2	no	—
302	C₈	—	4	2	no	—
303	C₂²	C₁₃×#537	1	1	yes	characteristic
304	C₂²	C₁₃×#547	1	1	yes	characteristic
305	C₂²	D₄×Q₈×C₁₃	1	1	yes	characteristic
306	C₂²	C₂×C₁₃×#135	1	1	yes	characteristic
307	C₂²	C₁₃×#575	1	1	yes	characteristic
308	C₂²	C₁₃×#576	1	1	yes	characteristic
309	C₂²	C₁₃×#578	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	2	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	4	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	2	no	—
317	C₄	—	8	2	no	—
318	C₄	C₁₃×#539	1	1	yes	characteristic
319	C₄	C₁₃×#544	1	1	yes	characteristic
320	C₄	C₁₃×#557	1	1	yes	characteristic
321	C₄	C₁₃×#564	1	1	yes	characteristic
322	C₂	C₁₃×#2726	1	1	yes	characteristic
323	C₂	C₁₃×#3250	1	1	yes	characteristic
324	C₂	C₁₃×#3262	1	1	yes	characteristic
325	C₂	C₁₃×#3341	1	1	yes	characteristic
326	C₂	C₁₃×#3352	1	1	yes	characteristic
327	C₂	C₁₃×#3383	1	1	yes	characteristic
328	C₂	C₁₃×#3389	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₄×C₁₃×C₄⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
67. #32509	C₁₃	direct (N × H)
268. C₁₃	#32509	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×#2508	2
C₁₃×#2088	2
C₁₃×#2086	2
C₁₃×#2085	2
C₁₃×#1949	2
C₁₃×#1942	2
C₁₃×#1939	2
C₁₃×#1921	2
#32509	13

Subgroups — C13×#32509 #15718848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32509 #15718848