Subgroups — C₁₃×#32410 #15718749

Order 3328.

624 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2497	C₂	1	2	yes	maximal
2	C₁₃×#2087	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2083	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2076	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#2076	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1947	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1921	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1850	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#1808	C₂	1	2	yes	maximal
10	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#107	C₂²	1	2	yes	—
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
17	C₁₃×#400	—	2	2	no	—
18	C₁₃×#396	—	2	2	no	—
19	C₁₃×#389	C₂²	1	2	yes	—
20	C₁₃×#389	C₂²	1	2	yes	—
21	C₁₃×#388	—	2	2	no	—
22	C₁₃×#387	—	2	2	no	—
23	C₁₃×#384	C₂²	1	2	yes	—
24	C₁₃×#380	C₂²	1	2	yes	—
25	C₁₃×#380	C₂²	1	2	yes	—
26	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
27	C₄×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₁₃×#340	C₂²	1	2	yes	—
29	C₁₃×#340	C₂²	1	2	yes	—
30	C₁₃×#340	C₂²	1	2	yes	—
31	C₁₃×#340	C₂²	1	1	yes	characteristic
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#340	C₂²	1	1	yes	characteristic
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	—	2	4	no	—
37	C₂²×C₈×C₁₃	—	4	1	no	—
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
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	C₂³	1	2	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
50	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
51	C₂×C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
52	C₂×C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₁₃	—	2	2	no	—
54	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
55	C₁₃×#107	—	2	4	no	—
56	C₁₃×#107	—	4	2	no	—
57	C₁₃×#107	—	4	1	no	—
58	C₁₃×#107	—	4	1	no	—
59	C₁₃×C₄⋊C₈	—	2	4	no	—
60	C₁₃×C₂²⋊C₈	—	4	1	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	#32410	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	—	2	1	no	—
71	C₂²×C₄×C₁₃	—	2	1	no	—
72	C₂²×C₄×C₁₃	—	2	2	no	—
73	C₂²×C₄×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	—	2	2	no	—
75	C₂²×C₄×C₁₃	—	2	2	no	—
76	C₂²×C₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
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	—	2	1	no	—
85	C₂×C₈×C₁₃	—	2	1	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	4	4	no	—
88	C₂×C₈×C₁₃	#41	1	4	yes	—
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	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	4	4	no	—
98	C₁₃×C₄⋊C₄	—	8	2	no	—
99	C₁₃×C₄⋊C₄	—	8	2	no	—
100	C₁₃×C₄⋊C₄	—	8	2	no	—
101	C₁₃×C₂²⋊C₄	—	4	1	no	—
102	C₁₃×C₂²⋊C₄	—	8	2	no	—
103	C₁₃×C₂²⋊C₄	—	8	2	no	—
104	C₁₃×C₂²⋊C₄	Q₁₆	1	4	yes	—
105	C₄²×C₁₃	—	2	2	no	—
106	C₄²×C₁₃	—	2	2	no	—
107	C₄²×C₁₃	—	8	2	no	—
108	#2497	C₂×C₁₃	1	2	yes	—
109	#2087	C₂×C₁₃	1	2	yes	complemented
110	#2083	C₂×C₁₃	1	2	yes	complemented
111	#2076	C₂×C₁₃	1	1	yes	characteristic
112	#2076	C₂×C₁₃	1	1	yes	characteristic
113	#1947	C₂×C₁₃	1	2	yes	complemented
114	#1921	C₂×C₁₃	1	2	yes	complemented
115	#1850	C₂×C₁₃	1	1	yes	characteristic
116	#1808	C₂×C₁₃	1	2	yes	—
117	C₂³×C₁₃	—	2	1	no	—
118	C₂³×C₁₃	—	2	2	no	—
119	C₂³×C₁₃	—	2	1	no	—
120	C₂³×C₁₃	—	2	2	no	—
121	C₂³×C₁₃	C₂×Q₁₆	1	2	yes	—
122	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	1	no	—
132	C₂×C₄×C₁₃	—	4	1	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	—	4	2	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
146	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
147	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
149	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
150	C₈×C₁₃	—	2	4	no	—
151	C₈×C₁₃	—	4	1	no	—
152	C₈×C₁₃	—	4	1	no	—
153	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×#107	C₂²×C₁₃	1	2	yes	—
155	C₂×#107	C₂²×C₁₃	1	2	yes	—
156	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
159	C₂×C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
160	#400	—	2	2	no	—
161	#396	—	2	2	no	—
162	#389	C₂²×C₁₃	1	2	yes	—
163	#389	C₂²×C₁₃	1	2	yes	—
164	#388	—	2	2	no	—
165	#387	—	2	2	no	—
166	#384	C₂²×C₁₃	1	2	yes	—
167	#380	C₂²×C₁₃	1	2	yes	—
168	#380	C₂²×C₁₃	1	2	yes	—
169	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
170	C₄×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
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	1	yes	characteristic
175	#340	C₂²×C₁₃	1	2	yes	—
176	#340	C₂²×C₁₃	1	1	yes	characteristic
177	#336	C₂²×C₁₃	1	2	yes	—
178	C₂²×C₁₃	—	2	4	no	—
179	C₂²×C₁₃	—	2	2	no	—
180	C₂²×C₁₃	—	4	2	no	—
181	C₂²×C₁₃	—	4	1	no	—
182	C₂²×C₁₃	—	4	1	no	—
183	C₂²×C₁₃	#537	1	1	yes	characteristic
184	C₂²×C₁₃	#547	1	2	yes	—
185	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
186	C₂²×C₁₃	#575	1	1	yes	characteristic
187	C₂²×C₁₃	#576	1	2	yes	—
188	C₄×C₁₃	—	2	4	no	—
189	C₄×C₁₃	—	4	2	no	—
190	C₄×C₁₃	—	4	1	no	—
191	C₄×C₁₃	—	8	2	no	—
192	C₄×C₁₃	—	8	2	no	—
193	C₄×C₁₃	—	8	2	no	—
194	C₄×C₁₃	—	8	2	no	—
195	C₄×C₁₃	#541	1	2	yes	—
196	C₄×C₁₃	#551	1	1	yes	characteristic
197	C₄×C₁₃	#563	1	1	yes	characteristic
198	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	—	4	1	no	—
201	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
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×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₂²⋊C₄	—	2	2	no	—
214	C₂×C₂²⋊C₄	D₄×C₁₃	1	1	yes	characteristic
215	C₂×C₂²⋊C₄	D₄×C₁₃	1	1	yes	characteristic
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
218	#107	—	2	4	no	—
219	#107	—	4	2	no	—
220	#107	—	4	1	no	—
221	#107	—	4	1	no	—
222	C₄⋊C₈	—	2	4	no	—
223	C₂²⋊C₈	—	4	1	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	D₄×C₁₃	1	2	yes	—
230	C₂×C₁₃	—	4	2	no	—
231	C₂×C₁₃	#2722	1	1	yes	characteristic
232	C₂×C₁₃	#3249	1	1	yes	characteristic
233	C₂×C₁₃	#3259	1	1	yes	characteristic
234	C₂×C₁₃	#3338	1	2	yes	—
235	C₂×C₁₃	#3347	1	2	yes	—
236	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
248	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	4	no	—
256	C₂×C₈	C₁₃×#41	1	4	yes	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	4	no	—
266	C₄⋊C₄	—	8	2	no	—
267	C₄⋊C₄	—	8	2	no	—
268	C₄⋊C₄	—	8	2	no	—
269	C₂²⋊C₄	—	4	1	no	—
270	C₂²⋊C₄	—	8	2	no	—
271	C₂²⋊C₄	—	8	2	no	—
272	C₂²⋊C₄	C₁₃×Q₁₆	1	4	yes	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	8	2	no	—
276	C₁₃	#32410	1	1	yes	characteristic, direct product factor, Sylow
277	C₂³	—	2	1	no	—
278	C₂³	—	2	2	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	2	2	no	—
281	C₂³	C₂×C₁₃×Q₁₆	1	2	yes	—
282	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	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	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	C₂×C₁₃×Q₁₆	1	4	yes	—
306	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
307	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
308	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
309	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
310	C₈	—	2	4	no	—
311	C₈	—	4	1	no	—
312	C₈	—	4	1	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	C₁₃×#537	1	1	yes	characteristic
319	C₂²	C₁₃×#547	1	2	yes	—
320	C₂²	C₂²×C₁₃×Q₁₆	1	1	yes	characteristic
321	C₂²	C₁₃×#575	1	1	yes	characteristic
322	C₂²	C₁₃×#576	1	2	yes	—
323	C₄	—	2	4	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	1	no	—
326	C₄	—	8	2	no	—
327	C₄	—	8	2	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	2	no	—
330	C₄	C₁₃×#541	1	2	yes	—
331	C₄	C₁₃×#551	1	1	yes	characteristic
332	C₄	C₁₃×#563	1	1	yes	characteristic
333	C₂	—	4	2	no	—
334	C₂	C₁₃×#2722	1	1	yes	characteristic
335	C₂	C₁₃×#3249	1	1	yes	characteristic
336	C₂	C₁₃×#3259	1	1	yes	characteristic
337	C₂	C₁₃×#3338	1	2	yes	—
338	C₂	C₁₃×#3347	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2087	C₂	semidirect
3. C₁₃×#2083	C₂	semidirect
6. C₁₃×#1947	C₂	semidirect
7. C₁₃×#1921	C₂	semidirect
67. #32410	C₁₃	direct (N × H)
109. #2087	C₂×C₁₃	semidirect
110. #2083	C₂×C₁₃	semidirect
113. #1947	C₂×C₁₃	semidirect
114. #1921	C₂×C₁₃	semidirect
276. C₁₃	#32410	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×#2497	2
C₁₃×#2087	2
C₁₃×#2083	2
C₁₃×#2076	2
C₁₃×#1947	2
C₁₃×#1921	2
C₁₃×#1850	2
C₁₃×#1808	2
#32410	13

Subgroups — C13×#32410 #15718749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32410 #15718749