Subgroups — C₁₃×#33661 #15720000

Order 3328.

610 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×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2235	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2226	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2208	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2187	C₂	1	2	yes	maximal
7	C₄×C₁₃×SD₃₂	C₂	1	4	yes	maximal, complemented
8	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₁₃×#533	—	2	1	no	—
11	C₂×C₄×D₄×C₁₃	—	2	1	no	—
12	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	—
13	C₁₃×#500	C₂²	1	1	yes	characteristic, complemented
14	C₁₃×#494	C₂²	1	1	yes	characteristic, complemented
15	C₁₃×#487	—	2	1	no	—
16	C₁₃×#467	—	2	1	no	—
17	C₁₃×#467	C₂²	1	2	yes	—
18	C₄×C₁₃×Q₁₆	—	2	1	no	—
19	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	complemented
20	C₄×C₁₃×D₈	—	2	1	no	—
21	C₄×C₁₃×D₈	—	2	1	no	—
22	C₄×C₁₃×D₈	C₂²	1	2	yes	—
23	C₄×C₁₃×D₈	C₂²	1	2	yes	—
24	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
25	C₂×C₁₃×#102	—	2	1	no	—
26	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#367	C₂²	1	4	yes	complemented
28	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
29	C₁₃×#358	C₂²	1	4	yes	complemented
30	C₁₃×#357	C₂²	1	4	yes	—
31	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
33	C₂²×D₄×C₁₃	—	2	1	no	—
34	C₂³×C₄×C₁₃	—	4	1	no	—
35	C₂×C₁₃×Q₁₆	—	2	1	no	—
36	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
37	C₂×C₁₃×D₈	—	2	4	no	—
38	C₂×C₁₃×D₈	—	2	1	no	—
39	C₂×C₁₃×D₈	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×C₄⋊Q₈	—	2	1	no	—
43	C₁₃×C₄⋊Q₈	—	2	1	no	—
44	C₁₃×#125	—	4	1	no	—
45	C₁₃×C₂²⋊Q₈	—	2	2	no	—
46	C₁₃×C₂²⋊Q₈	—	4	1	no	—
47	C₄×Q₈×C₁₃	—	2	2	no	—
48	C₄×Q₈×C₁₃	—	4	1	no	—
49	C₄×D₄×C₁₃	—	2	2	no	—
50	C₄×D₄×C₁₃	—	2	2	no	—
51	C₄×D₄×C₁₃	—	4	1	no	—
52	C₄×D₄×C₁₃	—	4	1	no	—
53	C₁₃×#117	—	4	1	no	—
54	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
55	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
56	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
57	C₁₃×SD₃₂	—	2	8	no	—
58	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
59	C₁₃×#107	—	2	1	no	—
60	C₁₃×#107	—	2	1	no	—
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×#107	C₂³	1	2	yes	—
63	C₁₃×#103	—	2	2	no	—
64	C₁₃×#103	—	2	2	no	—
65	C₁₃×#102	—	2	2	no	—
66	C₁₃×#102	—	2	2	no	—
67	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
68	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	#33661	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂⁴×C₁₃	—	4	1	no	—
73	C₂×Q₈×C₁₃	—	2	2	no	—
74	C₂×Q₈×C₁₃	—	4	1	no	—
75	C₂×D₄×C₁₃	—	2	4	no	—
76	C₂×D₄×C₁₃	—	2	2	no	—
77	C₂×D₄×C₁₃	—	4	2	no	—
78	C₂×D₄×C₁₃	—	4	1	no	—
79	C₂²×C₄×C₁₃	—	4	1	no	—
80	C₂²×C₄×C₁₃	—	4	2	no	—
81	C₂²×C₄×C₁₃	—	4	2	no	—
82	C₂²×C₄×C₁₃	—	8	4	no	—
83	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₁₃×Q₁₆	—	2	4	no	—
87	C₁₃×Q₁₆	—	4	2	no	—
88	C₁₃×D₈	—	2	4	no	—
89	C₁₃×D₈	—	2	2	no	—
90	C₁₃×D₈	#41	1	4	yes	—
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	4	1	no	—
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₄⋊C₄	—	4	1	no	—
101	C₁₃×C₄⋊C₄	—	4	1	no	—
102	C₁₃×C₄⋊C₄	—	4	2	no	—
103	C₁₃×C₄⋊C₄	—	4	2	no	—
104	C₁₃×C₄⋊C₄	—	4	1	no	—
105	C₁₃×C₂²⋊C₄	—	4	2	no	—
106	C₁₃×C₂²⋊C₄	—	4	2	no	—
107	C₁₃×C₂²⋊C₄	—	4	2	no	—
108	C₄²×C₁₃	—	4	2	no	—
109	C₄²×C₁₃	D₈	1	1	yes	characteristic
110	C₄²×C₁₃	D₈	1	1	yes	characteristic
111	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₁₃×C₁₆	—	2	4	no	—
114	#3170	C₂×C₁₃	1	1	yes	characteristic, complemented
115	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
116	#2235	C₂×C₁₃	1	2	yes	complemented
117	#2226	C₂×C₁₃	1	2	yes	complemented
118	#2208	C₂×C₁₃	1	2	yes	complemented
119	#2187	C₂×C₁₃	1	2	yes	—
120	C₄×SD₃₂	C₂×C₁₃	1	4	yes	complemented
121	#2162	C₂×C₁₃	1	1	yes	characteristic, complemented
122	C₂³×C₁₃	—	4	4	no	—
123	C₂³×C₁₃	—	4	2	no	—
124	C₂³×C₁₃	—	8	2	no	—
125	C₂³×C₁₃	—	8	2	no	—
126	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
127	Q₈×C₁₃	—	4	2	no	—
128	Q₈×C₁₃	—	4	2	no	—
129	Q₈×C₁₃	—	8	2	no	—
130	D₄×C₁₃	—	2	4	no	—
131	D₄×C₁₃	—	4	4	no	—
132	D₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	4	1	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	—	8	2	no	—
140	C₂×C₄×C₁₃	—	8	4	no	—
141	C₂×C₄×C₁₃	—	8	4	no	—
142	C₂×C₄×C₁₃	SD₃₂	1	4	yes	complemented
143	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	—	2	1	no	—
152	C₈×C₁₃	C₂×#41	1	2	yes	—
153	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
154	#533	—	2	1	no	—
155	C₂×C₄×D₄	—	2	1	no	—
156	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	—
157	#500	C₂²×C₁₃	1	1	yes	characteristic, complemented
158	#494	C₂²×C₁₃	1	1	yes	characteristic, complemented
159	#487	—	2	1	no	—
160	#467	—	2	1	no	—
161	#467	C₂²×C₁₃	1	2	yes	—
162	C₄×Q₁₆	—	2	1	no	—
163	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	complemented
164	C₄×D₈	—	2	1	no	—
165	C₄×D₈	—	2	1	no	—
166	C₄×D₈	C₂²×C₁₃	1	2	yes	—
167	C₄×D₈	C₂²×C₁₃	1	2	yes	—
168	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
169	C₂×#102	—	2	1	no	—
170	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
171	#367	C₂²×C₁₃	1	4	yes	complemented
172	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	complemented
173	#358	C₂²×C₁₃	1	4	yes	complemented
174	#357	C₂²×C₁₃	1	4	yes	—
175	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
176	C₄×C₁₆	C₂²×C₁₃	1	2	yes	complemented
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	—	2	1	no	—
179	C₂²×C₁₃	—	4	4	no	—
180	C₂²×C₁₃	—	8	8	no	—
181	C₂²×C₁₃	—	8	2	no	—
182	C₂²×C₁₃	—	8	2	no	—
183	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
184	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	8	2	no	—
189	C₄×C₁₃	—	8	2	no	—
190	C₄×C₁₃	—	8	2	no	—
191	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
192	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
193	C₄×C₁₃	#525	1	1	yes	characteristic
194	C₄×C₁₃	#575	1	1	yes	characteristic
195	C₂²×D₄	—	2	1	no	—
196	C₂³×C₄	—	4	1	no	—
197	C₂×Q₁₆	—	2	1	no	—
198	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
199	C₂×D₈	—	2	4	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	C₂³×C₁₃	1	2	yes	—
202	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
204	C₄⋊Q₈	—	2	1	no	—
205	C₄⋊Q₈	—	2	1	no	—
206	#125	—	4	1	no	—
207	C₂²⋊Q₈	—	2	2	no	—
208	C₂²⋊Q₈	—	4	1	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	4	1	no	—
211	C₄×D₄	—	2	2	no	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	—	4	1	no	—
214	C₄×D₄	—	4	1	no	—
215	#117	—	4	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₂²⋊C₄	—	4	1	no	—
218	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
219	SD₃₂	—	2	8	no	—
220	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
221	#107	—	2	1	no	—
222	#107	—	2	1	no	—
223	#107	C₂³×C₁₃	1	2	yes	—
224	#107	C₂³×C₁₃	1	2	yes	—
225	#103	—	2	2	no	—
226	#103	—	2	2	no	—
227	#102	—	2	2	no	—
228	#102	—	2	2	no	—
229	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
230	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
232	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
233	C₂×C₁₃	—	2	2	no	—
234	C₂×C₁₃	—	8	4	no	—
235	C₂×C₁₃	#3100	1	1	yes	characteristic
236	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
237	C₂×C₁₃	#3414	1	1	yes	characteristic
238	C₂⁴	—	4	1	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×D₄	—	2	4	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	8	4	no	—
249	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
250	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
252	Q₁₆	—	2	4	no	—
253	Q₁₆	—	4	2	no	—
254	D₈	—	2	4	no	—
255	D₈	—	2	2	no	—
256	D₈	C₁₃×#41	1	4	yes	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₄²	—	4	2	no	—
275	C₄²	C₁₃×D₈	1	1	yes	characteristic
276	C₄²	C₁₃×D₈	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
278	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
279	C₁₆	—	2	4	no	—
280	C₁₃	#33661	1	1	yes	characteristic, direct product factor, Sylow
281	C₂³	—	4	4	no	—
282	C₂³	—	4	2	no	—
283	C₂³	—	8	2	no	—
284	C₂³	—	8	2	no	—
285	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
286	Q₈	—	4	2	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	8	2	no	—
289	D₄	—	2	4	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	—	8	4	no	—
301	C₂×C₄	C₁₃×SD₃₂	1	4	yes	complemented
302	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
303	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
304	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
305	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
306	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	C₂×C₁₃×#41	1	2	yes	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	8	8	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	C₂×C₁₃×SD₃₂	1	2	yes	—
319	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	C₂×C₁₃×SD₃₂	1	2	yes	—
327	C₄	C₂×C₁₃×SD₃₂	1	2	yes	—
328	C₄	C₁₃×#525	1	1	yes	characteristic
329	C₄	C₁₃×#575	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	8	4	no	—
332	C₂	C₁₃×#3100	1	1	yes	characteristic
333	C₂	C₂²×C₁₃×SD₃₂	1	1	yes	characteristic
334	C₂	C₁₃×#3414	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3170	C₂	semidirect
3. C₁₃×#2235	C₂	semidirect
4. C₁₃×#2226	C₂	semidirect
5. C₁₃×#2208	C₂	semidirect
7. C₄×C₁₃×SD₃₂	C₂	semidirect
8. C₁₃×#2162	C₂	semidirect
13. C₁₃×#500	C₂²	semidirect
14. C₁₃×#494	C₂²	semidirect
19. C₄×C₁₃×Q₁₆	C₂²	semidirect
27. C₁₃×#367	C₂²	semidirect
28. C₁₃×C₄⋊C₁₆	C₂²	semidirect
29. C₁₃×#358	C₂²	semidirect
32. C₄×C₁₃×C₁₆	C₂²	semidirect
71. #33661	C₁₃	direct (N × H)
114. #3170	C₂×C₁₃	semidirect
116. #2235	C₂×C₁₃	semidirect
117. #2226	C₂×C₁₃	semidirect
118. #2208	C₂×C₁₃	semidirect
120. C₄×SD₃₂	C₂×C₁₃	semidirect
121. #2162	C₂×C₁₃	semidirect
142. C₂×C₄×C₁₃	SD₃₂	semidirect
157. #500	C₂²×C₁₃	semidirect
158. #494	C₂²×C₁₃	semidirect
163. C₄×Q₁₆	C₂²×C₁₃	semidirect
171. #367	C₂²×C₁₃	semidirect
172. C₄⋊C₁₆	C₂²×C₁₃	semidirect
173. #358	C₂²×C₁₃	semidirect
176. C₄×C₁₆	C₂²×C₁₃	semidirect
280. C₁₃	#33661	direct (N × H)
301. C₂×C₄	C₁₃×SD₃₂	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₃×#3170	2
C₂×C₄×C₁₃×D₈	2
C₁₃×#2235	2
C₁₃×#2226	2
C₁₃×#2208	2
C₁₃×#2187	2
C₄×C₁₃×SD₃₂	2
C₁₃×#2162	2
#33661	13

Subgroups — C13×#33661 #15720000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33661 #15720000