Subgroups — C₁₃×#33669 #15720008

Order 3328.

650 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2233	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2226	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2206	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2184	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2177	C₂	1	4	yes	maximal, complemented
8	C₁₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₁₃×#532	—	2	1	no	—
11	C₂×C₄×D₄×C₁₃	—	2	1	no	—
12	C₂×C₁₃×D₁₆	C₂²	1	4	yes	—
13	C₁₃×#500	C₂²	1	1	yes	characteristic, complemented
14	C₁₃×#493	C₂²	1	1	yes	characteristic, complemented
15	C₁₃×#486	—	2	1	no	—
16	C₁₃×#466	—	2	1	no	—
17	C₁₃×#466	C₂²	1	2	yes	—
18	C₄×C₁₃×D₈	—	2	1	no	—
19	C₄×C₁₃×D₈	—	2	1	no	—
20	C₄×C₁₃×D₈	—	2	1	no	—
21	C₄×C₁₃×D₈	C₂²	1	2	yes	—
22	C₄×C₁₃×D₈	C₂²	1	2	yes	—
23	C₄×C₁₃×D₈	C₂²	1	2	yes	complemented
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×#357	C₂²	1	4	yes	—
30	C₁₃×#357	C₂²	1	4	yes	complemented
31	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₁₃×#346	C₂²	1	2	yes	complemented
33	C₂×C₁₃×#41	—	4	1	no	—
34	C₂²×D₄×C₁₃	—	2	1	no	—
35	C₂³×C₄×C₁₃	—	4	1	no	—
36	C₂×C₁₃×D₈	—	2	4	no	—
37	C₂×C₁₃×D₈	—	2	1	no	—
38	C₂×C₁₃×D₈	—	2	1	no	—
39	C₂×C₁₃×D₈	C₂³	1	2	yes	—
40	C₂×C₁₃×D₈	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×C₄⋊Q₈	—	2	1	no	—
44	C₁₃×#127	—	2	1	no	—
45	C₁₃×#124	—	4	1	no	—
46	C₁₃×#121	—	2	2	no	—
47	C₁₃×#121	—	4	1	no	—
48	C₄×D₄×C₁₃	—	2	2	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	C₄×D₄×C₁₃	—	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×D₁₆	—	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×#102	—	2	2	no	—
64	C₁₃×#102	—	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	#33669	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂⁴×C₁₃	—	4	1	no	—
73	C₁₃×#41	—	8	4	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	—	2	2	no	—
78	C₂×D₄×C₁₃	—	4	2	no	—
79	C₂×D₄×C₁₃	—	4	2	no	—
80	C₂×D₄×C₁₃	—	4	1	no	—
81	C₂×D₄×C₁₃	—	4	1	no	—
82	C₂×D₄×C₁₃	—	4	1	no	—
83	C₂²×C₄×C₁₃	—	4	1	no	—
84	C₂²×C₄×C₁₃	—	4	2	no	—
85	C₂²×C₄×C₁₃	—	4	2	no	—
86	C₂²×C₄×C₁₃	—	4	2	no	—
87	C₂²×C₄×C₁₃	—	8	4	no	—
88	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₁₃×D₈	—	2	4	no	—
92	C₁₃×D₈	—	2	2	no	—
93	C₁₃×D₈	—	2	4	no	—
94	C₁₃×D₈	—	4	2	no	—
95	C₁₃×D₈	#41	1	4	yes	complemented
96	C₂×C₈×C₁₃	—	2	2	no	—
97	C₂×C₈×C₁₃	—	4	1	no	—
98	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
102	C₁₃×C₄⋊C₄	—	2	2	no	—
103	C₁₃×C₄⋊C₄	—	2	2	no	—
104	C₁₃×C₄⋊C₄	—	4	1	no	—
105	C₁₃×C₄⋊C₄	—	4	1	no	—
106	C₁₃×C₂²⋊C₄	—	4	2	no	—
107	C₁₃×C₂²⋊C₄	—	4	2	no	—
108	C₁₃×C₂²⋊C₄	—	4	2	no	—
109	C₁₃×C₂²⋊C₄	—	4	2	no	—
110	C₄²×C₁₃	D₈	1	1	yes	characteristic
111	C₄²×C₁₃	D₈	1	1	yes	characteristic
112	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₁₃×C₁₆	—	2	4	no	—
115	#3169	C₂×C₁₃	1	1	yes	characteristic, complemented
116	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#2233	C₂×C₁₃	1	2	yes	complemented
118	#2226	C₂×C₁₃	1	2	yes	complemented
119	#2206	C₂×C₁₃	1	2	yes	complemented
120	#2184	C₂×C₁₃	1	2	yes	complemented
121	#2177	C₂×C₁₃	1	4	yes	complemented
122	#2163	C₂×C₁₃	1	1	yes	characteristic, complemented
123	C₂³×C₁₃	—	4	4	no	—
124	C₂³×C₁₃	—	4	2	no	—
125	C₂³×C₁₃	—	4	2	no	—
126	C₂³×C₁₃	—	8	2	no	—
127	C₂³×C₁₃	—	8	2	no	—
128	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
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	D₄×C₁₃	—	4	2	no	—
134	D₄×C₁₃	—	4	2	no	—
135	D₄×C₁₃	—	8	2	no	—
136	D₄×C₁₃	—	8	4	no	—
137	D₄×C₁₃	—	8	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	2	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	8	4	no	—
145	C₂×C₄×C₁₃	—	8	4	no	—
146	C₂×C₄×C₁₃	—	8	2	no	—
147	C₂×C₄×C₁₃	—	8	4	no	—
148	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
154	C₈×C₁₃	—	2	1	no	—
155	C₈×C₁₃	—	2	1	no	—
156	C₈×C₁₃	—	2	1	no	—
157	C₈×C₁₃	C₂×#41	1	2	yes	—
158	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
159	#532	—	2	1	no	—
160	C₂×C₄×D₄	—	2	1	no	—
161	C₂×D₁₆	C₂²×C₁₃	1	4	yes	—
162	#500	C₂²×C₁₃	1	1	yes	characteristic, complemented
163	#493	C₂²×C₁₃	1	1	yes	characteristic, complemented
164	#486	—	2	1	no	—
165	#466	—	2	1	no	—
166	#466	C₂²×C₁₃	1	2	yes	—
167	C₄×D₈	—	2	1	no	—
168	C₄×D₈	—	2	1	no	—
169	C₄×D₈	—	2	1	no	—
170	C₄×D₈	C₂²×C₁₃	1	2	yes	—
171	C₄×D₈	C₂²×C₁₃	1	2	yes	—
172	C₄×D₈	C₂²×C₁₃	1	2	yes	complemented
173	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
174	C₂×#102	—	2	1	no	—
175	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
176	#367	C₂²×C₁₃	1	4	yes	complemented
177	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	complemented
178	#357	C₂²×C₁₃	1	4	yes	—
179	#357	C₂²×C₁₃	1	4	yes	complemented
180	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
181	#346	C₂²×C₁₃	1	2	yes	complemented
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	—	2	1	no	—
184	C₂²×C₁₃	—	4	4	no	—
185	C₂²×C₁₃	—	8	8	no	—
186	C₂²×C₁₃	—	8	2	no	—
187	C₂²×C₁₃	—	8	2	no	—
188	C₂²×C₁₃	—	8	2	no	—
189	C₂²×C₁₃	—	8	2	no	—
190	C₂²×C₁₃	—	8	2	no	—
191	C₂²×C₁₃	#509	1	2	yes	—
192	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
193	C₄×C₁₃	—	2	1	no	—
194	C₄×C₁₃	—	2	1	no	—
195	C₄×C₁₃	—	2	1	no	—
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	8	2	no	—
198	C₄×C₁₃	—	8	2	no	—
199	C₄×C₁₃	#509	1	2	yes	—
200	C₄×C₁₃	#525	1	1	yes	characteristic
201	C₄×C₁₃	#575	1	1	yes	characteristic
202	C₂×#41	—	4	1	no	—
203	C₂²×D₄	—	2	1	no	—
204	C₂³×C₄	—	4	1	no	—
205	C₂×D₈	—	2	4	no	—
206	C₂×D₈	—	2	1	no	—
207	C₂×D₈	—	2	1	no	—
208	C₂×D₈	C₂³×C₁₃	1	2	yes	—
209	C₂×D₈	C₂³×C₁₃	1	2	yes	—
210	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
212	C₄⋊Q₈	—	2	1	no	—
213	#127	—	2	1	no	—
214	#124	—	4	1	no	—
215	#121	—	2	2	no	—
216	#121	—	4	1	no	—
217	C₄×D₄	—	2	2	no	—
218	C₄×D₄	—	2	2	no	—
219	C₄×D₄	—	2	2	no	—
220	C₄×D₄	—	4	1	no	—
221	C₄×D₄	—	4	1	no	—
222	C₄×D₄	—	4	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₂²⋊C₄	—	4	1	no	—
225	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
226	D₁₆	—	2	8	no	—
227	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
228	#107	—	2	1	no	—
229	#107	—	2	1	no	—
230	#107	C₂³×C₁₃	1	2	yes	—
231	#107	C₂³×C₁₃	1	2	yes	—
232	#102	—	2	2	no	—
233	#102	—	2	2	no	—
234	#102	—	2	2	no	—
235	#102	—	2	2	no	—
236	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
237	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
240	C₂×C₁₃	—	2	2	no	—
241	C₂×C₁₃	—	8	4	no	—
242	C₂×C₁₃	—	16	2	no	—
243	C₂×C₁₃	#3100	1	1	yes	characteristic
244	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
245	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
246	C₂⁴	—	4	1	no	—
247	#41	—	8	4	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×D₄	—	2	4	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	8	4	no	—
262	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
263	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
265	D₈	—	2	4	no	—
266	D₈	—	2	2	no	—
267	D₈	—	2	4	no	—
268	D₈	—	4	2	no	—
269	D₈	C₁₃×#41	1	4	yes	complemented
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
273	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
274	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
275	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₄²	C₁₃×D₈	1	1	yes	characteristic
285	C₄²	C₁₃×D₈	1	1	yes	characteristic
286	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
287	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
288	C₁₆	—	2	4	no	—
289	C₁₃	#33669	1	1	yes	characteristic, direct product factor, Sylow
290	C₂³	—	4	4	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	8	2	no	—
294	C₂³	—	8	2	no	—
295	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
296	Q₈	—	8	2	no	—
297	D₄	—	2	4	no	—
298	D₄	—	4	4	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	8	2	no	—
303	D₄	—	8	4	no	—
304	D₄	—	8	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	—	8	4	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	8	4	no	—
315	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
316	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
317	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
318	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
319	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
320	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	C₂×C₁₃×#41	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	4	4	no	—
328	C₂²	—	8	8	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	C₁₃×#509	1	2	yes	—
335	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	C₁₃×#509	1	2	yes	—
343	C₄	C₁₃×#525	1	1	yes	characteristic
344	C₄	C₁₃×#575	1	1	yes	characteristic
345	C₂	—	2	2	no	—
346	C₂	—	8	4	no	—
347	C₂	—	16	2	no	—
348	C₂	C₁₃×#3100	1	1	yes	characteristic
349	C₂	C₂×C₁₃×#509	1	1	yes	characteristic
350	C₂	C₂×C₁₃×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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×#3169	C₂	semidirect
2. C₂×C₄×C₁₃×D₈	C₂	semidirect
3. C₁₃×#2233	C₂	semidirect
4. C₁₃×#2226	C₂	semidirect
5. C₁₃×#2206	C₂	semidirect
6. C₁₃×#2184	C₂	semidirect
7. C₁₃×#2177	C₂	semidirect
8. C₁₃×#2163	C₂	semidirect
13. C₁₃×#500	C₂²	semidirect
14. C₁₃×#493	C₂²	semidirect
23. C₄×C₁₃×D₈	C₂²	semidirect
27. C₁₃×#367	C₂²	semidirect
28. C₁₃×C₄⋊C₁₆	C₂²	semidirect
30. C₁₃×#357	C₂²	semidirect
32. C₁₃×#346	C₂²	semidirect
71. #33669	C₁₃	direct (N × H)
95. C₁₃×D₈	#41	semidirect
115. #3169	C₂×C₁₃	semidirect
116. C₂×C₄×D₈	C₂×C₁₃	semidirect
117. #2233	C₂×C₁₃	semidirect
118. #2226	C₂×C₁₃	semidirect
119. #2206	C₂×C₁₃	semidirect
120. #2184	C₂×C₁₃	semidirect
121. #2177	C₂×C₁₃	semidirect
122. #2163	C₂×C₁₃	semidirect
162. #500	C₂²×C₁₃	semidirect
163. #493	C₂²×C₁₃	semidirect
172. C₄×D₈	C₂²×C₁₃	semidirect
176. #367	C₂²×C₁₃	semidirect
177. C₄⋊C₁₆	C₂²×C₁₃	semidirect
179. #357	C₂²×C₁₃	semidirect
181. #346	C₂²×C₁₃	semidirect
269. D₈	C₁₃×#41	semidirect
289. C₁₃	#33669	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×#3169	2
C₂×C₄×C₁₃×D₈	2
C₁₃×#2233	2
C₁₃×#2226	2
C₁₃×#2206	2
C₁₃×#2184	2
C₁₃×#2177	2
C₁₃×#2163	2
#33669	13

Subgroups — C13×#33669 #15720008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33669 #15720008