Subgroups — C₁₃×#32461 #15718800

Order 3328.

676 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2485	C₂	1	2	yes	maximal
2	C₁₃×#2087	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2086	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2077	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2077	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1954	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1941	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1854	C₂	1	2	yes	maximal, complemented
9	C₁₃×#1854	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#117	C₂²	1	2	yes	—
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×#106	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#397	—	2	2	no	—
21	C₁₃×#394	—	2	2	no	—
22	C₁₃×#389	C₂²	1	2	yes	—
23	C₁₃×#389	C₂²	1	2	yes	—
24	C₁₃×#388	—	2	2	no	—
25	C₁₃×#387	—	2	2	no	—
26	C₁₃×#380	C₂²	1	2	yes	—
27	C₁₃×#380	C₂²	1	2	yes	—
28	C₁₃×#380	C₂²	1	2	yes	—
29	C₁₃×#340	C₂²	1	1	yes	characteristic
30	C₁₃×#340	C₂²	1	1	yes	characteristic
31	C₁₃×#340	C₂²	1	2	yes	—
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#340	C₂²	1	2	yes	—
34	C₁₃×#340	C₂²	1	2	yes	—
35	C₂²×D₄×C₁₃	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
40	C₄×D₄×C₁₃	—	4	1	no	—
41	C₄×D₄×C₁₃	—	4	1	no	—
42	C₁₃×#117	—	4	2	no	—
43	C₁₃×#117	—	4	2	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
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×C₁₃×C₂²⋊C₄	—	2	2	no	—
52	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
53	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
54	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
55	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
56	C₂×C₄²×C₁₃	—	2	2	no	—
57	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₁₃×#107	—	4	1	no	—
59	C₁₃×#107	—	4	1	no	—
60	C₁₃×#106	—	2	4	no	—
61	C₁₃×#106	—	4	2	no	—
62	C₁₃×C₄⋊C₈	—	4	1	no	—
63	C₁₃×C₂²⋊C₈	—	2	4	no	—
64	C₁₃×C₂²⋊C₈	—	2	4	no	—
65	C₄×C₈×C₁₃	—	4	1	no	—
66	C₁₃×#95	—	2	2	no	—
67	C₁₃×#95	—	2	2	no	—
68	C₁₃×#95	—	2	2	no	—
69	C₁₃×#95	—	2	2	no	—
70	C₁₃×#95	—	2	2	no	—
71	#32461	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂⁴×C₁₃	C₂×D₄	1	2	yes	—
73	C₂×D₄×C₁₃	—	2	2	no	—
74	C₂×D₄×C₁₃	—	4	1	no	—
75	C₂×D₄×C₁₃	—	4	1	no	—
76	C₂²×C₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	—	2	2	no	—
78	C₂²×C₄×C₁₃	—	2	2	no	—
79	C₂²×C₄×C₁₃	—	2	2	no	—
80	C₂²×C₄×C₁₃	—	2	2	no	—
81	C₂²×C₄×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	—	2	2	no	—
84	C₂²×C₄×C₁₃	—	2	2	no	—
85	C₂²×C₄×C₁₃	—	2	2	no	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	1	no	—
95	C₂×C₈×C₁₃	—	2	1	no	—
96	C₂×C₈×C₁₃	#41	1	4	yes	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	2	2	no	—
103	C₁₃×C₄⋊C₄	—	2	2	no	—
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	4	1	no	—
106	C₁₃×C₄⋊C₄	—	4	4	no	—
107	C₁₃×C₄⋊C₄	—	8	2	no	—
108	C₁₃×C₂²⋊C₄	—	4	2	no	—
109	C₁₃×C₂²⋊C₄	—	4	4	no	—
110	C₁₃×C₂²⋊C₄	—	4	4	no	—
111	C₁₃×C₂²⋊C₄	—	8	2	no	—
112	C₁₃×C₂²⋊C₄	—	8	2	no	—
113	C₄²×C₁₃	—	2	1	no	—
114	C₄²×C₁₃	—	2	1	no	—
115	C₄²×C₁₃	—	4	4	no	—
116	#2485	C₂×C₁₃	1	2	yes	—
117	#2087	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#2086	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#2077	C₂×C₁₃	1	2	yes	complemented
120	#2077	C₂×C₁₃	1	2	yes	complemented
121	#1954	C₂×C₁₃	1	1	yes	characteristic
122	#1941	C₂×C₁₃	1	2	yes	complemented
123	#1854	C₂×C₁₃	1	2	yes	complemented
124	#1854	C₂×C₁₃	1	2	yes	complemented
125	C₂³×C₁₃	—	2	2	no	—
126	C₂³×C₁₃	—	2	2	no	—
127	C₂³×C₁₃	—	2	2	no	—
128	C₂³×C₁₃	—	2	2	no	—
129	C₂³×C₁₃	—	2	2	no	—
130	C₂³×C₁₃	—	2	2	no	—
131	C₂³×C₁₃	—	2	2	no	—
132	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	D₄×C₁₃	—	4	4	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	2	1	no	—
136	C₂×C₄×C₁₃	—	2	1	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	2	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	—	4	2	no	—
146	C₂×C₄×C₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	—	4	2	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	4	2	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	—	4	2	no	—
153	C₂×C₄×C₁₃	—	4	2	no	—
154	C₂×C₄×C₁₃	—	4	2	no	—
155	C₂×C₄×C₁₃	—	4	2	no	—
156	C₂×C₄×C₁₃	—	4	2	no	—
157	C₂×C₄×C₁₃	#135	1	2	yes	—
158	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
159	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
162	C₈×C₁₃	—	2	4	no	—
163	C₈×C₁₃	—	4	1	no	—
164	C₈×C₁₃	—	4	1	no	—
165	C₂×C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
166	C₂×#117	C₂²×C₁₃	1	2	yes	—
167	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
168	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
169	C₂×#106	C₂²×C₁₃	1	2	yes	—
170	C₂×#106	C₂²×C₁₃	1	2	yes	—
171	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
172	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
173	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
174	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
175	#397	—	2	2	no	—
176	#394	—	2	2	no	—
177	#389	C₂²×C₁₃	1	2	yes	—
178	#389	C₂²×C₁₃	1	2	yes	—
179	#388	—	2	2	no	—
180	#387	—	2	2	no	—
181	#380	C₂²×C₁₃	1	2	yes	—
182	#380	C₂²×C₁₃	1	2	yes	—
183	#380	C₂²×C₁₃	1	2	yes	—
184	#340	C₂²×C₁₃	1	1	yes	characteristic
185	#340	C₂²×C₁₃	1	1	yes	characteristic
186	#340	C₂²×C₁₃	1	2	yes	—
187	#340	C₂²×C₁₃	1	2	yes	—
188	#340	C₂²×C₁₃	1	2	yes	—
189	#340	C₂²×C₁₃	1	2	yes	—
190	C₂²×C₁₃	—	4	2	no	—
191	C₂²×C₁₃	—	4	2	no	—
192	C₂²×C₁₃	—	4	2	no	—
193	C₂²×C₁₃	—	4	2	no	—
194	C₂²×C₁₃	—	4	2	no	—
195	C₂²×C₁₃	—	4	2	no	—
196	C₂²×C₁₃	—	4	2	no	—
197	C₂²×C₁₃	#534	1	1	yes	characteristic
198	C₂²×C₁₃	#547	1	2	yes	—
199	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
200	C₂²×C₁₃	#575	1	1	yes	characteristic
201	C₂²×C₁₃	#577	1	2	yes	—
202	C₄×C₁₃	—	2	2	no	—
203	C₄×C₁₃	—	4	1	no	—
204	C₄×C₁₃	—	4	2	no	—
205	C₄×C₁₃	—	8	2	no	—
206	C₄×C₁₃	—	8	2	no	—
207	C₄×C₁₃	—	8	2	no	—
208	C₄×C₁₃	—	8	2	no	—
209	C₄×C₁₃	#539	1	2	yes	—
210	C₄×C₁₃	#559	1	1	yes	characteristic
211	C₄×C₁₃	#563	1	1	yes	characteristic
212	C₂²×D₄	D₄×C₁₃	1	1	yes	characteristic
213	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
214	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
215	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
217	C₄×D₄	—	4	1	no	—
218	C₄×D₄	—	4	1	no	—
219	#117	—	4	2	no	—
220	#117	—	4	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
224	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
225	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
226	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
227	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
228	C₂×C₂²⋊C₄	—	2	2	no	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₂²⋊C₄	—	2	2	no	—
232	C₂×C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
233	C₂×C₄²	—	2	2	no	—
234	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
235	#107	—	4	1	no	—
236	#107	—	4	1	no	—
237	#106	—	2	4	no	—
238	#106	—	4	2	no	—
239	C₄⋊C₈	—	4	1	no	—
240	C₂²⋊C₈	—	2	4	no	—
241	C₂²⋊C₈	—	2	4	no	—
242	C₄×C₈	—	4	1	no	—
243	#95	—	2	2	no	—
244	#95	—	2	2	no	—
245	#95	—	2	2	no	—
246	#95	—	2	2	no	—
247	#95	—	2	2	no	—
248	C₂×C₁₃	—	8	2	no	—
249	C₂×C₁₃	#2724	1	1	yes	characteristic
250	C₂×C₁₃	#3191	1	1	yes	characteristic
251	C₂×C₁₃	#3204	1	1	yes	characteristic
252	C₂×C₁₃	#3340	1	2	yes	—
253	C₂×C₁₃	#3351	1	2	yes	—
254	C₂⁴	C₂×D₄×C₁₃	1	2	yes	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
269	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
271	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	C₁₃×#41	1	4	yes	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	4	no	—
289	C₄⋊C₄	—	8	2	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	4	no	—
292	C₂²⋊C₄	—	4	4	no	—
293	C₂²⋊C₄	—	8	2	no	—
294	C₂²⋊C₄	—	8	2	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	4	4	no	—
298	C₁₃	#32461	1	1	yes	characteristic, direct product factor, Sylow
299	C₂³	—	2	2	no	—
300	C₂³	—	2	2	no	—
301	C₂³	—	2	2	no	—
302	C₂³	—	2	2	no	—
303	C₂³	—	2	2	no	—
304	C₂³	—	2	2	no	—
305	C₂³	—	2	2	no	—
306	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
307	D₄	—	4	4	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	C₁₃×#135	1	2	yes	—
332	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
333	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
334	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
335	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
336	C₈	—	2	4	no	—
337	C₈	—	4	1	no	—
338	C₈	—	4	1	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	C₁₃×#534	1	1	yes	characteristic
347	C₂²	C₁₃×#547	1	2	yes	—
348	C₂²	C₂×C₁₃×#135	1	1	yes	characteristic
349	C₂²	C₁₃×#575	1	1	yes	characteristic
350	C₂²	C₁₃×#577	1	2	yes	—
351	C₄	—	2	2	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	2	no	—
354	C₄	—	8	2	no	—
355	C₄	—	8	2	no	—
356	C₄	—	8	2	no	—
357	C₄	—	8	2	no	—
358	C₄	C₁₃×#539	1	2	yes	—
359	C₄	C₁₃×#559	1	1	yes	characteristic
360	C₄	C₁₃×#563	1	1	yes	characteristic
361	C₂	—	8	2	no	—
362	C₂	C₁₃×#2724	1	1	yes	characteristic
363	C₂	C₁₃×#3191	1	1	yes	characteristic
364	C₂	C₁₃×#3204	1	1	yes	characteristic
365	C₂	C₁₃×#3340	1	2	yes	—
366	C₂	C₁₃×#3351	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2087	C₂	semidirect
3. C₁₃×#2086	C₂	semidirect
4. C₁₃×#2077	C₂	semidirect
5. C₁₃×#2077	C₂	semidirect
7. C₁₃×#1941	C₂	semidirect
8. C₁₃×#1854	C₂	semidirect
9. C₁₃×#1854	C₂	semidirect
71. #32461	C₁₃	direct (N × H)
117. #2087	C₂×C₁₃	semidirect
118. #2086	C₂×C₁₃	semidirect
119. #2077	C₂×C₁₃	semidirect
120. #2077	C₂×C₁₃	semidirect
122. #1941	C₂×C₁₃	semidirect
123. #1854	C₂×C₁₃	semidirect
124. #1854	C₂×C₁₃	semidirect
298. C₁₃	#32461	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×#2485	2
C₁₃×#2087	2
C₁₃×#2086	2
C₁₃×#2077	2
C₁₃×#1954	2
C₁₃×#1941	2
C₁₃×#1854	2
#32461	13

Subgroups — C13×#32461 #15718800

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32461 #15718800