Subgroups — C₁₃×#33634 #15719973

Order 3328.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3164	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2951	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2229	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2228	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2226	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2225	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2159	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#2139	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#2138	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
13	C₁₃×#544	—	2	1	no	—
14	C₁₃×#520	—	2	1	no	—
15	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
17	C₁₃×#501	—	2	2	no	—
18	C₁₃×#500	C₂²	1	2	yes	—
19	C₁₃×#499	C₂²	1	2	yes	—
20	C₁₃×#440	—	2	2	no	—
21	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
22	C₄×C₁₃×D₈	C₂²	1	2	yes	—
23	C₁₃×#428	—	2	1	no	—
24	C₁₃×#427	C₂²	1	1	yes	characteristic
25	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
26	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
27	C₁₃×#417	—	2	1	no	—
28	C₁₃×#405	C₂²	1	1	yes	characteristic
29	C₁₃×#367	C₂²	1	2	yes	—
30	C₁₃×#367	C₂²	1	2	yes	—
31	C₁₃×#367	C₂²	1	2	yes	—
32	C₁₃×#366	C₂²	1	2	yes	—
33	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₁₃×#358	C₂²	1	1	yes	characteristic
36	C₁₃×#358	C₂²	1	1	yes	characteristic
37	C₁₃×#358	C₂²	1	2	yes	—
38	C₁₃×#357	C₂²	1	1	yes	characteristic
39	C₁₃×#357	C₂²	1	1	yes	characteristic
40	C₁₃×#357	C₂²	1	2	yes	—
41	C₂×C₁₃×#41	—	2	1	no	—
42	C₁₃×#135	—	2	2	no	—
43	C₁₃×#135	Q₈	1	4	yes	complemented
44	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
45	C₂×C₁₃×SD₁₆	—	2	1	no	—
46	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
47	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×C₄⋊Q₈	—	2	2	no	—
50	C₁₃×C₄⋊Q₈	—	4	1	no	—
51	C₁₃×#125	—	2	2	no	—
52	C₁₃×#125	—	4	1	no	—
53	C₁₃×C₂²⋊Q₈	—	4	2	no	—
54	C₄×Q₈×C₁₃	—	2	2	no	—
55	C₄×D₄×C₁₃	—	2	2	no	—
56	C₄×D₄×C₁₃	—	4	2	no	—
57	C₁₃×#117	—	2	1	no	—
58	C₁₃×#117	—	4	1	no	—
59	C₁₃×#117	D₄	1	1	yes	characteristic
60	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
61	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
62	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
63	C₁₃×M₃₂	—	2	2	no	—
64	C₂×C₁₃×C₁₆	—	2	2	no	—
65	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
66	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
68	C₁₃×#107	—	2	1	no	—
69	C₁₃×#107	—	2	1	no	—
70	C₁₃×#107	—	2	2	no	—
71	C₁₃×#107	C₂³	1	2	yes	—
72	C₁₃×#107	C₂³	1	2	yes	—
73	C₁₃×#107	C₂³	1	1	yes	characteristic
74	C₁₃×#107	C₂³	1	1	yes	characteristic
75	C₁₃×#106	—	2	1	no	—
76	C₁₃×#106	—	2	2	no	—
77	C₁₃×#103	—	2	2	no	—
78	C₁₃×#102	—	2	2	no	—
79	C₁₃×#97	D₄	1	2	yes	—
80	C₄×C₈×C₁₃	C₂³	1	2	yes	—
81	#33634	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₁₃×#41	—	2	4	no	—
83	C₁₃×#41	—	4	2	no	—
84	C₂×Q₈×C₁₃	—	2	1	no	—
85	C₂×Q₈×C₁₃	—	4	2	no	—
86	C₂×D₄×C₁₃	—	2	1	no	—
87	C₂×D₄×C₁₃	—	4	1	no	—
88	C₂²×C₄×C₁₃	—	4	1	no	—
89	C₂²×C₄×C₁₃	—	4	1	no	—
90	C₂²×C₄×C₁₃	—	4	1	no	—
91	C₂²×C₄×C₁₃	—	4	2	no	—
92	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₁₃×Q₁₆	—	2	1	no	—
94	C₁₃×Q₁₆	C₂×Q₈	1	2	yes	—
95	C₁₃×SD₁₆	—	2	4	no	—
96	C₁₃×D₈	—	2	1	no	—
97	C₁₃×D₈	C₂×Q₈	1	2	yes	—
98	C₁₃×M₁₆	—	2	2	no	—
99	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
102	C₂×C₈×C₁₃	#41	1	2	yes	—
103	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	2	1	no	—
106	C₁₃×C₄⋊C₄	—	2	2	no	—
107	C₁₃×C₄⋊C₄	—	2	1	no	—
108	C₁₃×C₄⋊C₄	—	4	1	no	—
109	C₁₃×C₄⋊C₄	—	4	1	no	—
110	C₁₃×C₄⋊C₄	—	4	2	no	—
111	C₁₃×C₄⋊C₄	—	4	2	no	—
112	C₁₃×C₄⋊C₄	—	4	2	no	—
113	C₁₃×C₄⋊C₄	—	4	2	no	—
114	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
115	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
116	C₁₃×C₂²⋊C₄	—	4	1	no	—
117	C₁₃×C₂²⋊C₄	—	4	2	no	—
118	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
119	C₄²×C₁₃	—	4	1	no	—
120	C₄²×C₁₃	—	4	2	no	—
121	C₄²×C₁₃	C₂×D₄	1	2	yes	—
122	C₁₃×C₁₆	—	2	1	no	—
123	C₁₃×C₁₆	—	2	1	no	—
124	C₁₃×C₁₆	—	2	2	no	—
125	#3164	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#2951	C₂×C₁₃	1	1	yes	characteristic
127	#2229	C₂×C₁₃	1	2	yes	complemented
128	#2228	C₂×C₁₃	1	2	yes	complemented
129	#2226	C₂×C₁₃	1	2	yes	complemented
130	#2225	C₂×C₁₃	1	2	yes	complemented
131	#2159	C₂×C₁₃	1	1	yes	characteristic, complemented
132	C₂×#367	C₂×C₁₃	1	1	yes	characteristic, complemented
133	#2151	C₂×C₁₃	1	1	yes	characteristic, complemented
134	#2139	C₂×C₁₃	1	1	yes	characteristic
135	#2138	C₂×C₁₃	1	1	yes	characteristic
136	C₂³×C₁₃	—	4	1	no	—
137	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
138	Q₈×C₁₃	—	2	2	no	—
139	Q₈×C₁₃	—	4	1	no	—
140	Q₈×C₁₃	—	16	2	no	—
141	D₄×C₁₃	—	2	2	no	—
142	D₄×C₁₃	—	4	4	no	—
143	D₄×C₁₃	—	4	1	no	—
144	C₂×C₄×C₁₃	—	4	1	no	—
145	C₂×C₄×C₁₃	—	4	4	no	—
146	C₂×C₄×C₁₃	—	4	1	no	—
147	C₂×C₄×C₁₃	—	4	2	no	—
148	C₂×C₄×C₁₃	—	4	1	no	—
149	C₂×C₄×C₁₃	—	4	2	no	—
150	C₂×C₄×C₁₃	—	16	2	no	—
151	C₂×C₄×C₁₃	—	16	1	no	—
152	C₂×C₄×C₁₃	—	16	1	no	—
153	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
154	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
155	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
156	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₁₃	#137	1	2	yes	—
158	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
159	C₈×C₁₃	—	2	2	no	—
160	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
161	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
162	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
163	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
164	#544	—	2	1	no	—
165	#520	—	2	1	no	—
166	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
167	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
168	#501	—	2	2	no	—
169	#500	C₂²×C₁₃	1	2	yes	—
170	#499	C₂²×C₁₃	1	2	yes	—
171	#440	—	2	2	no	—
172	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
173	C₄×D₈	C₂²×C₁₃	1	2	yes	—
174	#428	—	2	1	no	—
175	#427	C₂²×C₁₃	1	1	yes	characteristic
176	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
177	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
178	#417	—	2	1	no	—
179	#405	C₂²×C₁₃	1	1	yes	characteristic
180	#367	C₂²×C₁₃	1	2	yes	—
181	#367	C₂²×C₁₃	1	2	yes	—
182	#367	C₂²×C₁₃	1	2	yes	—
183	#366	C₂²×C₁₃	1	2	yes	—
184	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
185	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
186	#358	C₂²×C₁₃	1	1	yes	characteristic
187	#358	C₂²×C₁₃	1	1	yes	characteristic
188	#358	C₂²×C₁₃	1	2	yes	—
189	#357	C₂²×C₁₃	1	1	yes	characteristic
190	#357	C₂²×C₁₃	1	1	yes	characteristic
191	#357	C₂²×C₁₃	1	2	yes	—
192	C₂²×C₁₃	—	2	1	no	—
193	C₂²×C₁₃	—	2	1	no	—
194	C₂²×C₁₃	—	4	2	no	—
195	C₂²×C₁₃	—	8	1	no	—
196	C₂²×C₁₃	—	8	1	no	—
197	C₂²×C₁₃	#474	1	2	yes	—
198	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
199	C₄×C₁₃	—	2	2	no	—
200	C₄×C₁₃	—	2	2	no	—
201	C₄×C₁₃	—	4	2	no	—
202	C₄×C₁₃	—	8	1	no	—
203	C₄×C₁₃	—	8	2	no	—
204	C₄×C₁₃	—	8	2	no	—
205	C₄×C₁₃	—	8	1	no	—
206	C₄×C₁₃	#474	1	2	yes	—
207	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
208	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
209	C₂×#41	—	2	1	no	—
210	#135	—	2	2	no	—
211	#135	Q₈×C₁₃	1	4	yes	complemented
212	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
213	C₂×SD₁₆	—	2	1	no	—
214	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
215	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
217	C₄⋊Q₈	—	2	2	no	—
218	C₄⋊Q₈	—	4	1	no	—
219	#125	—	2	2	no	—
220	#125	—	4	1	no	—
221	C₂²⋊Q₈	—	4	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×D₄	—	2	2	no	—
224	C₄×D₄	—	4	2	no	—
225	#117	—	2	1	no	—
226	#117	—	4	1	no	—
227	#117	D₄×C₁₃	1	1	yes	characteristic
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	—	4	1	no	—
231	M₃₂	—	2	2	no	—
232	C₂×C₁₆	—	2	2	no	—
233	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
234	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
235	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
236	#107	—	2	1	no	—
237	#107	—	2	1	no	—
238	#107	—	2	2	no	—
239	#107	C₂³×C₁₃	1	1	yes	characteristic
240	#107	C₂³×C₁₃	1	2	yes	—
241	#107	C₂³×C₁₃	1	1	yes	characteristic
242	#107	C₂³×C₁₃	1	2	yes	—
243	#106	—	2	1	no	—
244	#106	—	2	2	no	—
245	#103	—	2	2	no	—
246	#102	—	2	2	no	—
247	#97	D₄×C₁₃	1	2	yes	—
248	C₄×C₈	C₂³×C₁₃	1	2	yes	—
249	C₂×C₁₃	—	2	2	no	—
250	C₂×C₁₃	—	8	2	no	—
251	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
252	C₂×C₁₃	#3416	1	1	yes	characteristic
253	C₂×C₁₃	#3416	1	1	yes	characteristic
254	#41	—	2	4	no	—
255	#41	—	4	2	no	—
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	4	2	no	—
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	2	no	—
264	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
265	Q₁₆	—	2	1	no	—
266	Q₁₆	C₂×Q₈×C₁₃	1	2	yes	—
267	SD₁₆	—	2	4	no	—
268	D₈	—	2	1	no	—
269	D₈	C₂×Q₈×C₁₃	1	2	yes	—
270	M₁₆	—	2	2	no	—
271	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
272	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
273	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
274	C₂×C₈	C₁₃×#41	1	2	yes	—
275	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
287	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
288	C₂²⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
291	C₄²	—	4	1	no	—
292	C₄²	—	4	2	no	—
293	C₄²	C₂×D₄×C₁₃	1	2	yes	—
294	C₁₆	—	2	1	no	—
295	C₁₆	—	2	1	no	—
296	C₁₆	—	2	2	no	—
297	C₁₃	#33634	1	1	yes	characteristic, direct product factor, Sylow
298	C₂³	—	4	1	no	—
299	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	Q₈	—	4	1	no	—
302	Q₈	—	16	2	no	—
303	D₄	—	2	2	no	—
304	D₄	—	4	4	no	—
305	D₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	16	2	no	—
313	C₂×C₄	—	16	1	no	—
314	C₂×C₄	—	16	1	no	—
315	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
316	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
317	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
318	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
319	C₂×C₄	C₁₃×#137	1	2	yes	—
320	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	C₁₃×C₂²⋊Q₈	1	2	yes	—
323	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
324	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	C₁₃×#474	1	2	yes	—
331	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	4	2	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	—	8	1	no	—
339	C₄	C₁₃×#474	1	2	yes	—
340	C₄	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
341	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	8	2	no	—
344	C₂	C₂×C₁₃×#474	1	1	yes	characteristic
345	C₂	C₁₃×#3416	1	1	yes	characteristic
346	C₂	C₁₃×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3164	C₂	semidirect
3. C₁₃×#2229	C₂	semidirect
4. C₁₃×#2228	C₂	semidirect
5. C₁₃×#2226	C₂	semidirect
6. C₁₃×#2225	C₂	semidirect
7. C₁₃×#2159	C₂	semidirect
8. C₂×C₁₃×#367	C₂	semidirect
9. C₁₃×#2151	C₂	semidirect
43. C₁₃×#135	Q₈	semidirect
81. #33634	C₁₃	direct (N × H)
125. #3164	C₂×C₁₃	semidirect
127. #2229	C₂×C₁₃	semidirect
128. #2228	C₂×C₁₃	semidirect
129. #2226	C₂×C₁₃	semidirect
130. #2225	C₂×C₁₃	semidirect
131. #2159	C₂×C₁₃	semidirect
132. C₂×#367	C₂×C₁₃	semidirect
133. #2151	C₂×C₁₃	semidirect
211. #135	Q₈×C₁₃	semidirect
297. C₁₃	#33634	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×#3164	2
C₁₃×#2951	2
C₁₃×#2229	2
C₁₃×#2228	2
C₁₃×#2226	2
C₁₃×#2225	2
C₁₃×#2159	2
C₂×C₁₃×#367	2
C₁₃×#2151	2
C₁₃×#2139	2
C₁₃×#2138	2
#33634	13

Subgroups — C13×#33634 #15719973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33634 #15719973