Subgroups — C₁₃×#32667 #15719006

Order 3328.

666 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2582	C₂	1	2	yes	maximal
2	C₁₃×#2096	C₂	1	2	yes	maximal
3	C₁₃×#2084	C₂	1	2	yes	maximal
4	C₁₃×#2066	C₂	1	2	yes	maximal
5	C₁₃×#1985	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1971	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1944	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#1878	C₂	1	2	yes	maximal
9	C₁₃×#1863	C₂	1	2	yes	maximal
10	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#106	C₂²	1	2	yes	—
16	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#103	C₂²	1	2	yes	—
19	C₂×C₁₃×#103	C₂²	1	2	yes	—
20	C₂×C₁₃×#103	C₂²	1	2	yes	—
21	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
22	C₁₃×#398	—	2	2	no	—
23	C₁₃×#398	—	2	2	no	—
24	C₁₃×#391	—	2	2	no	—
25	C₁₃×#391	C₂²	1	2	yes	—
26	C₁₃×#391	C₂²	1	2	yes	—
27	C₁₃×#389	—	2	2	no	—
28	C₁₃×#389	C₂²	1	2	yes	—
29	C₁₃×#389	C₂²	1	2	yes	—
30	C₁₃×#384	C₂²	1	2	yes	—
31	C₁₃×#340	C₂²	1	2	yes	—
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#336	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
37	C₂×C₁₃×Q₁₆	—	4	1	no	—
38	C₂×C₁₃×Q₁₆	—	4	1	no	—
39	C₂×C₁₃×Q₁₆	Q₈	1	4	yes	complemented
40	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₁₃	—	4	1	no	—
44	C₄×Q₈×C₁₃	—	4	1	no	—
45	C₄×Q₈×C₁₃	—	4	2	no	—
46	C₄×Q₈×C₁₃	—	4	2	no	—
47	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
49	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
50	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
52	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
53	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
54	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
55	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
56	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
57	C₂×C₄²×C₁₃	—	2	2	no	—
58	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
59	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
60	C₁₃×#107	—	4	1	no	—
61	C₁₃×#106	—	2	2	no	—
62	C₁₃×#106	—	4	2	no	—
63	C₁₃×#106	—	4	1	no	—
64	C₁₃×C₄⋊C₈	—	4	1	no	—
65	C₁₃×#103	—	2	4	no	—
66	C₁₃×#103	—	4	2	no	—
67	C₁₃×#103	—	4	2	no	—
68	C₁₃×#97	—	4	1	no	—
69	C₁₃×#95	—	2	2	no	—
70	C₁₃×#95	—	2	2	no	—
71	C₁₃×#95	—	2	2	no	—
72	C₁₃×#95	—	2	2	no	—
73	C₁₃×#95	—	2	2	no	—
74	C₁₃×#95	—	2	2	no	—
75	#32667	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂×Q₈×C₁₃	—	2	4	no	—
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	—	2	2	no	—
79	C₂×Q₈×C₁₃	—	2	2	no	—
80	C₂×Q₈×C₁₃	—	2	2	no	—
81	C₂×Q₈×C₁₃	—	2	2	no	—
82	C₂×Q₈×C₁₃	—	4	1	no	—
83	C₂×Q₈×C₁₃	C₂×Q₈	1	4	yes	—
84	C₂²×C₄×C₁₃	—	2	2	no	—
85	C₂²×C₄×C₁₃	—	2	2	no	—
86	C₂²×C₄×C₁₃	—	2	2	no	—
87	C₂²×C₄×C₁₃	—	2	2	no	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₁₃×Q₁₆	—	4	4	no	—
94	C₂×C₈×C₁₃	—	2	1	no	—
95	C₂×C₈×C₁₃	—	2	1	no	—
96	C₂×C₈×C₁₃	—	2	1	no	—
97	C₂×C₈×C₁₃	—	2	1	no	—
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	—	2	2	no	—
100	C₂×C₈×C₁₃	—	2	2	no	—
101	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
102	C₂×C₈×C₁₃	#41	1	2	yes	—
103	C₁₃×C₄⋊C₄	—	2	2	no	—
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	2	2	no	—
106	C₁₃×C₄⋊C₄	—	2	2	no	—
107	C₁₃×C₄⋊C₄	—	2	2	no	—
108	C₁₃×C₄⋊C₄	—	2	2	no	—
109	C₁₃×C₄⋊C₄	—	4	2	no	—
110	C₁₃×C₄⋊C₄	—	4	1	no	—
111	C₁₃×C₄⋊C₄	—	4	4	no	—
112	C₁₃×C₄⋊C₄	—	4	4	no	—
113	C₁₃×C₄⋊C₄	—	8	2	no	—
114	C₁₃×C₄⋊C₄	—	8	2	no	—
115	C₁₃×C₄⋊C₄	—	8	2	no	—
116	C₄²×C₁₃	—	2	1	no	—
117	C₄²×C₁₃	—	2	1	no	—
118	C₄²×C₁₃	—	2	2	no	—
119	C₄²×C₁₃	—	2	2	no	—
120	C₄²×C₁₃	—	4	4	no	—
121	#2582	C₂×C₁₃	1	2	yes	—
122	#2096	C₂×C₁₃	1	2	yes	—
123	#2084	C₂×C₁₃	1	2	yes	—
124	#2066	C₂×C₁₃	1	2	yes	—
125	#1985	C₂×C₁₃	1	1	yes	characteristic
126	#1971	C₂×C₁₃	1	1	yes	characteristic
127	#1944	C₂×C₁₃	1	1	yes	characteristic
128	#1878	C₂×C₁₃	1	2	yes	—
129	#1863	C₂×C₁₃	1	2	yes	—
130	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
131	Q₈×C₁₃	—	2	4	no	—
132	Q₈×C₁₃	—	2	4	no	—
133	Q₈×C₁₃	—	4	4	no	—
134	Q₈×C₁₃	—	4	2	no	—
135	Q₈×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	1	no	—
140	C₂×C₄×C₁₃	—	2	1	no	—
141	C₂×C₄×C₁₃	—	2	4	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	2	2	no	—
145	C₂×C₄×C₁₃	—	2	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	#137	1	4	yes	—
158	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
160	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
164	C₈×C₁₃	—	2	2	no	—
165	C₈×C₁₃	—	2	2	no	—
166	C₈×C₁₃	—	4	2	no	—
167	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
168	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
169	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
170	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
171	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
172	C₂×#106	C₂²×C₁₃	1	2	yes	—
173	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
174	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
175	C₂×#103	C₂²×C₁₃	1	2	yes	—
176	C₂×#103	C₂²×C₁₃	1	2	yes	—
177	C₂×#103	C₂²×C₁₃	1	2	yes	—
178	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
179	#398	—	2	2	no	—
180	#398	—	2	2	no	—
181	#391	—	2	2	no	—
182	#391	C₂²×C₁₃	1	2	yes	—
183	#391	C₂²×C₁₃	1	2	yes	—
184	#389	—	2	2	no	—
185	#389	C₂²×C₁₃	1	2	yes	—
186	#389	C₂²×C₁₃	1	2	yes	—
187	#384	C₂²×C₁₃	1	2	yes	—
188	#340	C₂²×C₁₃	1	2	yes	—
189	#340	C₂²×C₁₃	1	2	yes	—
190	#336	C₂²×C₁₃	1	2	yes	—
191	#336	C₂²×C₁₃	1	2	yes	—
192	C₂²×C₁₃	#534	1	1	yes	characteristic
193	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
194	C₂²×C₁₃	#550	1	1	yes	characteristic
195	C₂²×C₁₃	C₂×#137	1	2	yes	—
196	C₂²×C₁₃	#577	1	2	yes	—
197	C₄×C₁₃	—	2	4	no	—
198	C₄×C₁₃	—	4	1	no	—
199	C₄×C₁₃	—	4	2	no	—
200	C₄×C₁₃	—	4	1	no	—
201	C₄×C₁₃	—	4	4	no	—
202	C₄×C₁₃	—	8	2	no	—
203	C₄×C₁₃	—	8	2	no	—
204	C₄×C₁₃	—	8	2	no	—
205	C₄×C₁₃	#539	1	1	yes	characteristic
206	C₄×C₁₃	#543	1	1	yes	characteristic
207	C₄×C₁₃	#554	1	1	yes	characteristic
208	C₄×C₁₃	#562	1	1	yes	characteristic
209	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
210	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
211	C₂×Q₁₆	—	4	1	no	—
212	C₂×Q₁₆	—	4	1	no	—
213	C₂×Q₁₆	Q₈×C₁₃	1	4	yes	complemented
214	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
215	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
217	C₄×Q₈	—	4	1	no	—
218	C₄×Q₈	—	4	1	no	—
219	C₄×Q₈	—	4	2	no	—
220	C₄×Q₈	—	4	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
227	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
228	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
229	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
230	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
231	C₂×C₄²	—	2	2	no	—
232	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
233	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
234	#107	—	4	1	no	—
235	#106	—	2	2	no	—
236	#106	—	4	2	no	—
237	#106	—	4	1	no	—
238	C₄⋊C₈	—	4	1	no	—
239	#103	—	2	4	no	—
240	#103	—	4	2	no	—
241	#103	—	4	2	no	—
242	#97	—	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	#95	—	2	2	no	—
249	C₂×C₁₃	#2732	1	1	yes	characteristic
250	C₂×C₁₃	#3195	1	1	yes	characteristic
251	C₂×C₁₃	#3204	1	1	yes	characteristic
252	C₂×C₁₃	#3387	1	2	yes	—
253	C₂×C₁₃	#3400	1	2	yes	—
254	C₂×Q₈	—	2	4	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	4	1	no	—
261	C₂×Q₈	C₂×Q₈×C₁₃	1	4	yes	—
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	C₂×D₄×C₁₃	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
269	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
270	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
271	Q₁₆	—	4	4	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
280	C₂×C₈	C₁₃×#41	1	2	yes	—
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	2	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	4	4	no	—
290	C₄⋊C₄	—	4	4	no	—
291	C₄⋊C₄	—	8	2	no	—
292	C₄⋊C₄	—	8	2	no	—
293	C₄⋊C₄	—	8	2	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	4	4	no	—
299	C₁₃	#32667	1	1	yes	characteristic, direct product factor, Sylow
300	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
301	Q₈	—	2	4	no	—
302	Q₈	—	2	4	no	—
303	Q₈	—	4	4	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	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	C₁₃×#137	1	4	yes	—
328	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
329	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
330	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
331	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
332	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
333	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	2	2	no	—
336	C₈	—	4	2	no	—
337	C₂²	C₁₃×#534	1	1	yes	characteristic
338	C₂²	D₄×Q₈×C₁₃	1	1	yes	characteristic
339	C₂²	C₁₃×#550	1	1	yes	characteristic
340	C₂²	C₂×C₁₃×#137	1	2	yes	—
341	C₂²	C₁₃×#577	1	2	yes	—
342	C₄	—	2	4	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	4	no	—
347	C₄	—	8	2	no	—
348	C₄	—	8	2	no	—
349	C₄	—	8	2	no	—
350	C₄	C₁₃×#539	1	1	yes	characteristic
351	C₄	C₁₃×#543	1	1	yes	characteristic
352	C₄	C₁₃×#554	1	1	yes	characteristic
353	C₄	C₁₃×#562	1	1	yes	characteristic
354	C₂	C₁₃×#2732	1	1	yes	characteristic
355	C₂	C₁₃×#3195	1	1	yes	characteristic
356	C₂	C₁₃×#3204	1	1	yes	characteristic
357	C₂	C₁₃×#3387	1	2	yes	—
358	C₂	C₁₃×#3400	1	2	yes	—

	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 ◁ C₂×C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
39. C₂×C₁₃×Q₁₆	Q₈	semidirect
75. #32667	C₁₃	direct (N × H)
213. C₂×Q₁₆	Q₈×C₁₃	semidirect
299. C₁₃	#32667	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×#2582	2
C₁₃×#2096	2
C₁₃×#2084	2
C₁₃×#2066	2
C₁₃×#1985	2
C₁₃×#1971	2
C₁₃×#1944	2
C₁₃×#1878	2
C₁₃×#1863	2
#32667	13

Subgroups — C13×#32667 #15719006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32667 #15719006