Subgroups — C₁₃×#32504 #15718843

Order 3328.

620 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2506	C₂	1	2	yes	maximal
2	C₁₃×#2086	C₂	1	2	yes	maximal
3	C₁₃×#2086	C₂	1	2	yes	maximal
4	C₁₃×#2083	C₂	1	2	yes	maximal
5	C₁₃×#1947	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×#1918	C₂	1	2	yes	maximal
10	C₂×C₁₃×#125	—	2	2	no	—
11	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#107	C₂²	1	2	yes	—
14	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#400	—	2	2	no	—
20	C₁₃×#398	—	2	2	no	—
21	C₁₃×#389	C₂²	1	2	yes	—
22	C₁₃×#389	C₂²	1	2	yes	—
23	C₁₃×#389	C₂²	1	2	yes	—
24	C₁₃×#389	C₂²	1	2	yes	—
25	C₁₃×#387	—	2	2	no	—
26	C₁₃×#384	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	C₁₃×#125	—	4	8	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	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
46	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₄²×C₁₃	—	2	2	no	—
52	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×#107	—	2	2	no	—
55	C₁₃×#107	—	4	1	no	—
56	C₁₃×#107	—	4	1	no	—
57	C₁₃×#107	—	4	2	no	—
58	C₁₃×#107	Q₈	1	4	yes	—
59	C₁₃×C₄⋊C₈	—	2	4	no	—
60	C₁₃×C₄⋊C₈	—	4	1	no	—
61	C₄×C₈×C₁₃	—	4	1	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	—	2	2	no	—
67	C₁₃×#95	D₄	1	2	yes	—
68	#32504	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
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	—	2	2	no	—
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	2	yes	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
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	1	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₁₃	#41	1	2	yes	—
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	—	4	4	no	—
97	C₁₃×C₄⋊C₄	—	4	4	no	—
98	C₁₃×C₄⋊C₄	—	4	4	no	—
99	C₁₃×C₄⋊C₄	—	4	4	no	—
100	C₁₃×C₄⋊C₄	—	4	4	no	—
101	C₁₃×C₄⋊C₄	D₈	1	4	yes	—
102	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
103	C₄²×C₁₃	—	2	2	no	—
104	C₄²×C₁₃	—	2	1	no	—
105	C₄²×C₁₃	—	2	1	no	—
106	C₄²×C₁₃	—	2	2	no	—
107	C₄²×C₁₃	—	4	4	no	—
108	#2506	C₂×C₁₃	1	2	yes	—
109	#2086	C₂×C₁₃	1	2	yes	—
110	#2086	C₂×C₁₃	1	2	yes	—
111	#2083	C₂×C₁₃	1	2	yes	—
112	#1947	C₂×C₁₃	1	2	yes	—
113	#1942	C₂×C₁₃	1	1	yes	characteristic
114	#1942	C₂×C₁₃	1	1	yes	characteristic
115	#1939	C₂×C₁₃	1	1	yes	characteristic
116	#1918	C₂×C₁₃	1	2	yes	—
117	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	1	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	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
141	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
147	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
148	C₈×C₁₃	—	2	2	no	—
149	C₈×C₁₃	—	2	2	no	—
150	C₈×C₁₃	—	4	2	no	—
151	C₂×#125	—	2	2	no	—
152	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
153	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×#107	C₂²×C₁₃	1	2	yes	—
155	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
156	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
158	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
160	#400	—	2	2	no	—
161	#398	—	2	2	no	—
162	#389	C₂²×C₁₃	1	2	yes	—
163	#389	C₂²×C₁₃	1	2	yes	—
164	#389	C₂²×C₁₃	1	2	yes	—
165	#389	C₂²×C₁₃	1	2	yes	—
166	#387	—	2	2	no	—
167	#384	C₂²×C₁₃	1	2	yes	—
168	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
169	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
170	#340	C₂²×C₁₃	1	2	yes	—
171	#340	C₂²×C₁₃	1	2	yes	—
172	#340	C₂²×C₁₃	1	2	yes	—
173	#340	C₂²×C₁₃	1	2	yes	—
174	#340	C₂²×C₁₃	1	2	yes	—
175	#336	C₂²×C₁₃	1	2	yes	—
176	C₂²×C₁₃	#537	1	1	yes	characteristic
177	C₂²×C₁₃	#547	1	1	yes	characteristic
178	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
179	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
180	C₂²×C₁₃	#575	1	1	yes	characteristic
181	C₂²×C₁₃	#578	1	1	yes	characteristic
182	C₂²×C₁₃	#578	1	1	yes	characteristic
183	C₄×C₁₃	—	2	4	no	—
184	C₄×C₁₃	—	2	2	no	—
185	C₄×C₁₃	—	4	2	no	—
186	C₄×C₁₃	—	4	4	no	—
187	C₄×C₁₃	—	4	1	no	—
188	C₄×C₁₃	—	8	2	no	—
189	C₄×C₁₃	—	8	2	no	—
190	C₄×C₁₃	—	8	2	no	—
191	C₄×C₁₃	#539	1	1	yes	characteristic
192	C₄×C₁₃	#544	1	1	yes	characteristic
193	C₄×C₁₃	#557	1	1	yes	characteristic
194	C₄×C₁₃	#564	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
196	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
198	#125	—	4	8	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
206	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
207	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
210	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
213	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
214	#107	—	2	2	no	—
215	#107	—	4	1	no	—
216	#107	—	4	1	no	—
217	#107	—	4	2	no	—
218	#107	Q₈×C₁₃	1	4	yes	—
219	C₄⋊C₈	—	2	4	no	—
220	C₄⋊C₈	—	4	1	no	—
221	C₄×C₈	—	4	1	no	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	#95	—	2	2	no	—
227	#95	D₄×C₁₃	1	2	yes	—
228	C₂×C₁₃	#2726	1	1	yes	characteristic
229	C₂×C₁₃	#3245	1	1	yes	characteristic
230	C₂×C₁₃	#3261	1	1	yes	characteristic
231	C₂×C₁₃	#3334	1	1	yes	characteristic
232	C₂×C₁₃	#3352	1	1	yes	characteristic
233	C₂×C₁₃	Q₈×D₈	1	1	yes	characteristic
234	C₂×C₁₃	#3390	1	1	yes	characteristic
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
241	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
242	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
243	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
252	C₂×C₈	C₁₃×#41	1	2	yes	—
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	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	—	4	4	no	—
263	C₄⋊C₄	—	4	4	no	—
264	C₄⋊C₄	—	4	4	no	—
265	C₄⋊C₄	—	4	4	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₄⋊C₄	C₁₃×D₈	1	4	yes	—
268	C₄⋊C₄	C₂×Q₈×C₁₃	1	4	yes	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	4	4	no	—
274	C₁₃	#32504	1	1	yes	characteristic, direct product factor, Sylow
275	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
276	C₂×C₄	—	2	1	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	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	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	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	C₂×C₁₃×D₈	1	4	yes	—
299	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
300	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
301	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
302	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
303	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
304	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
305	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	2	no	—
308	C₈	—	4	2	no	—
309	C₂²	C₁₃×#537	1	1	yes	characteristic
310	C₂²	C₁₃×#547	1	1	yes	characteristic
311	C₂²	D₄×Q₈×C₁₃	1	1	yes	characteristic
312	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
313	C₂²	C₁₃×#575	1	1	yes	characteristic
314	C₂²	C₁₃×#578	1	1	yes	characteristic
315	C₂²	C₁₃×#578	1	1	yes	characteristic
316	C₄	—	2	4	no	—
317	C₄	—	2	2	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	4	no	—
320	C₄	—	4	1	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	C₁₃×#539	1	1	yes	characteristic
325	C₄	C₁₃×#544	1	1	yes	characteristic
326	C₄	C₁₃×#557	1	1	yes	characteristic
327	C₄	C₁₃×#564	1	1	yes	characteristic
328	C₂	C₁₃×#2726	1	1	yes	characteristic
329	C₂	C₁₃×#3245	1	1	yes	characteristic
330	C₂	C₁₃×#3261	1	1	yes	characteristic
331	C₂	C₁₃×#3334	1	1	yes	characteristic
332	C₂	C₁₃×#3352	1	1	yes	characteristic
333	C₂	Q₈×C₁₃×D₈	1	1	yes	characteristic
334	C₂	C₁₃×#3390	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
68. #32504	C₁₃	direct (N × H)
274. C₁₃	#32504	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×#2506	2
C₁₃×#2086	2
C₁₃×#2083	2
C₁₃×#1947	2
C₁₃×#1942	2
C₁₃×#1939	2
C₁₃×#1918	2
#32504	13

Subgroups — C13×#32504 #15718843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32504 #15718843