Subgroups — C₁₃×#33782 #15720121

Order 3328.

588 conjugacy classes of subgroups.

The subgroup lattice has 342 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	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2251	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2243	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2209	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2207	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2189	C₂	1	4	yes	maximal, complemented
8	C₁₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	—	2	1	no	—
10	C₂×C₁₃×#135	C₂²	1	2	yes	—
11	C₁₃×#533	—	2	1	no	—
12	C₁₃×#532	—	2	1	no	—
13	C₂×C₁₃×Q₃₂	C₂²	1	4	yes	—
14	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	—
15	C₁₃×#500	C₂²	1	1	yes	characteristic
16	C₁₃×#495	—	2	1	no	—
17	C₁₃×#495	C₂²	1	2	yes	—
18	C₁₃×#494	C₂²	1	1	yes	characteristic
19	C₁₃×#494	C₂²	1	1	yes	characteristic
20	C₁₃×#493	C₂²	1	1	yes	characteristic
21	C₁₃×#492	—	2	1	no	—
22	C₁₃×#487	—	2	1	no	—
23	C₁₃×#467	—	2	1	no	—
24	C₁₃×#467	C₂²	1	2	yes	—
25	C₄×C₁₃×Q₁₆	—	2	1	no	—
26	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₁₃×#358	C₂²	1	4	yes	—
30	C₁₃×#357	C₂²	1	4	yes	—
31	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₁₃×#346	C₂²	1	2	yes	—
33	C₂×C₁₃×#41	—	2	2	no	—
34	C₁₃×#135	—	4	4	no	—
35	C₁₃×#135	—	4	4	no	—
36	C₂×C₁₃×Q₁₆	—	2	1	no	—
37	C₂×C₁₃×Q₁₆	—	2	1	no	—
38	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
39	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
40	C₂×C₁₃×SD₁₆	—	2	2	no	—
41	C₂×C₁₃×SD₁₆	—	2	2	no	—
42	C₂×C₁₃×D₈	—	2	1	no	—
43	C₂×C₁₃×D₈	C₂³	1	2	yes	—
44	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₁₃×C₄⋊Q₈	—	2	1	no	—
47	C₁₃×C₄⋊Q₈	—	2	1	no	—
48	C₁₃×C₄⋊Q₈	—	2	1	no	—
49	C₁₃×#127	—	2	1	no	—
50	C₁₃×#125	—	4	1	no	—
51	C₁₃×#124	—	2	2	no	—
52	C₁₃×C₂²⋊Q₈	—	2	2	no	—
53	C₁₃×C₂²⋊Q₈	—	4	1	no	—
54	C₁₃×#121	—	4	1	no	—
55	C₁₃×#121	—	4	1	no	—
56	C₄×Q₈×C₁₃	—	2	2	no	—
57	C₄×Q₈×C₁₃	—	4	1	no	—
58	C₄×D₄×C₁₃	—	4	1	no	—
59	C₄×D₄×C₁₃	—	4	1	no	—
60	C₁₃×#117	—	4	1	no	—
61	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
62	C₁₃×Q₃₂	—	4	4	no	—
63	C₁₃×SD₃₂	—	4	4	no	—
64	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
65	C₁₃×#107	—	2	1	no	—
66	C₁₃×#107	C₂³	1	2	yes	—
67	C₁₃×#103	—	2	2	no	—
68	C₁₃×#103	—	2	2	no	—
69	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	#33782	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₁₃×#41	—	4	4	no	—
75	C₁₃×#41	—	4	4	no	—
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	—	4	1	no	—
79	C₂×D₄×C₁₃	—	2	2	no	—
80	C₂×D₄×C₁₃	—	4	2	no	—
81	C₂×D₄×C₁₃	—	4	1	no	—
82	C₂×D₄×C₁₃	—	4	1	no	—
83	C₂²×C₄×C₁₃	—	4	2	no	—
84	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₁₃×Q₁₆	—	2	2	no	—
88	C₁₃×Q₁₆	—	2	2	no	—
89	C₁₃×Q₁₆	—	2	4	no	—
90	C₁₃×Q₁₆	—	4	2	no	—
91	C₁₃×Q₁₆	—	4	2	no	—
92	C₁₃×SD₁₆	—	4	4	no	—
93	C₁₃×SD₁₆	—	4	4	no	—
94	C₁₃×D₈	—	2	4	no	—
95	C₁₃×D₈	—	4	2	no	—
96	C₂×C₈×C₁₃	—	2	2	no	—
97	C₂×C₈×C₁₃	—	2	2	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	—	4	1	no	—
104	C₁₃×C₄⋊C₄	—	4	1	no	—
105	C₁₃×C₄⋊C₄	—	4	1	no	—
106	C₁₃×C₄⋊C₄	—	4	1	no	—
107	C₁₃×C₄⋊C₄	—	4	2	no	—
108	C₁₃×C₄⋊C₄	—	4	2	no	—
109	C₁₃×C₄⋊C₄	—	4	1	no	—
110	C₁₃×C₂²⋊C₄	—	4	2	no	—
111	C₁₃×C₂²⋊C₄	—	4	2	no	—
112	C₁₃×C₂²⋊C₄	—	4	2	no	—
113	C₄²×C₁₃	—	4	2	no	—
114	C₄²×C₁₃	D₈	1	1	yes	characteristic
115	C₄²×C₁₃	D₈	1	1	yes	characteristic
116	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₁₃×C₁₆	—	2	4	no	—
119	#3170	C₂×C₁₃	1	1	yes	characteristic, complemented
120	#3147	C₂×C₁₃	1	1	yes	characteristic
121	#2251	C₂×C₁₃	1	2	yes	complemented
122	#2243	C₂×C₁₃	1	2	yes	complemented
123	#2209	C₂×C₁₃	1	2	yes	complemented
124	#2207	C₂×C₁₃	1	2	yes	complemented
125	#2189	C₂×C₁₃	1	4	yes	complemented
126	#2163	C₂×C₁₃	1	1	yes	characteristic, complemented
127	C₂³×C₁₃	—	4	2	no	—
128	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
129	Q₈×C₁₃	—	4	2	no	—
130	Q₈×C₁₃	—	4	2	no	—
131	Q₈×C₁₃	—	4	2	no	—
132	Q₈×C₁₃	—	4	2	no	—
133	Q₈×C₁₃	—	8	2	no	—
134	D₄×C₁₃	—	4	2	no	—
135	D₄×C₁₃	—	4	2	no	—
136	D₄×C₁₃	—	8	2	no	—
137	D₄×C₁₃	—	8	4	no	—
138	D₄×C₁₃	—	8	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	1	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	—	8	4	no	—
147	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
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	C₂²×D₄	1	1	yes	characteristic
153	C₈×C₁₃	—	2	1	no	—
154	C₈×C₁₃	—	2	1	no	—
155	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
156	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
157	C₂×#135	—	2	1	no	—
158	C₂×#135	C₂²×C₁₃	1	2	yes	—
159	#533	—	2	1	no	—
160	#532	—	2	1	no	—
161	C₂×Q₃₂	C₂²×C₁₃	1	4	yes	—
162	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	—
163	#500	C₂²×C₁₃	1	1	yes	characteristic
164	#495	—	2	1	no	—
165	#495	C₂²×C₁₃	1	2	yes	—
166	#494	C₂²×C₁₃	1	1	yes	characteristic
167	#494	C₂²×C₁₃	1	1	yes	characteristic
168	#493	C₂²×C₁₃	1	1	yes	characteristic
169	#492	—	2	1	no	—
170	#487	—	2	1	no	—
171	#467	—	2	1	no	—
172	#467	C₂²×C₁₃	1	2	yes	—
173	C₄×Q₁₆	—	2	1	no	—
174	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
175	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
176	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
177	#358	C₂²×C₁₃	1	4	yes	—
178	#357	C₂²×C₁₃	1	4	yes	—
179	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
180	#346	C₂²×C₁₃	1	2	yes	—
181	C₂²×C₁₃	—	2	1	no	—
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	—	2	1	no	—
184	C₂²×C₁₃	—	8	2	no	—
185	C₂²×C₁₃	—	8	2	no	—
186	C₂²×C₁₃	—	8	2	no	—
187	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	2	1	no	—
191	C₄×C₁₃	—	2	1	no	—
192	C₄×C₁₃	—	8	2	no	—
193	C₄×C₁₃	—	8	2	no	—
194	C₄×C₁₃	—	8	2	no	—
195	C₄×C₁₃	#510	1	2	yes	—
196	C₄×C₁₃	#535	1	1	yes	characteristic
197	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
198	C₂×#41	—	2	2	no	—
199	#135	—	4	4	no	—
200	#135	—	4	4	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
204	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
205	C₂×SD₁₆	—	2	2	no	—
206	C₂×SD₁₆	—	2	2	no	—
207	C₂×D₈	—	2	1	no	—
208	C₂×D₈	C₂³×C₁₃	1	2	yes	—
209	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	#127	—	2	1	no	—
215	#125	—	4	1	no	—
216	#124	—	2	2	no	—
217	C₂²⋊Q₈	—	2	2	no	—
218	C₂²⋊Q₈	—	4	1	no	—
219	#121	—	4	1	no	—
220	#121	—	4	1	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	—	4	1	no	—
223	C₄×D₄	—	4	1	no	—
224	C₄×D₄	—	4	1	no	—
225	#117	—	4	1	no	—
226	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
227	Q₃₂	—	4	4	no	—
228	SD₃₂	—	4	4	no	—
229	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
230	#107	—	2	1	no	—
231	#107	C₂³×C₁₃	1	2	yes	—
232	#103	—	2	2	no	—
233	#103	—	2	2	no	—
234	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
235	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
237	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
238	C₂×C₁₃	—	4	1	no	—
239	C₂×C₁₃	—	16	2	no	—
240	C₂×C₁₃	#3205	1	1	yes	characteristic
241	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
242	C₂×C₁₃	#3414	1	1	yes	characteristic
243	#41	—	4	4	no	—
244	#41	—	4	4	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
254	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	2	2	no	—
258	Q₁₆	—	2	4	no	—
259	Q₁₆	—	4	2	no	—
260	Q₁₆	—	4	2	no	—
261	SD₁₆	—	4	4	no	—
262	SD₁₆	—	4	4	no	—
263	D₈	—	2	4	no	—
264	D₈	—	4	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
269	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
270	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₄²	—	4	2	no	—
283	C₄²	C₁₃×D₈	1	1	yes	characteristic
284	C₄²	C₁₃×D₈	1	1	yes	characteristic
285	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
286	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
287	C₁₆	—	2	4	no	—
288	C₁₃	#33782	1	1	yes	characteristic, direct product factor, Sylow
289	C₂³	—	4	2	no	—
290	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
291	Q₈	—	4	2	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	8	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	8	2	no	—
299	D₄	—	8	4	no	—
300	D₄	—	8	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
310	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
311	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
312	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
313	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
318	C₈	C₁₃×H(5,𝔽₂)	1	2	yes	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	8	2	no	—
331	C₄	—	8	2	no	—
332	C₄	—	8	2	no	—
333	C₄	C₁₃×#510	1	2	yes	—
334	C₄	C₁₃×#535	1	1	yes	characteristic
335	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
336	C₂	—	4	1	no	—
337	C₂	—	16	2	no	—
338	C₂	C₁₃×#3205	1	1	yes	characteristic
339	C₂	C₂×C₁₃×#510	1	1	yes	characteristic
340	C₂	C₁₃×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#3147	C₂
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×#3170	C₂	semidirect
3. C₁₃×#2251	C₂	semidirect
4. C₁₃×#2243	C₂	semidirect
5. C₁₃×#2209	C₂	semidirect
6. C₁₃×#2207	C₂	semidirect
7. C₁₃×#2189	C₂	semidirect
8. C₁₃×#2163	C₂	semidirect
73. #33782	C₁₃	direct (N × H)
119. #3170	C₂×C₁₃	semidirect
121. #2251	C₂×C₁₃	semidirect
122. #2243	C₂×C₁₃	semidirect
123. #2209	C₂×C₁₃	semidirect
124. #2207	C₂×C₁₃	semidirect
125. #2189	C₂×C₁₃	semidirect
126. #2163	C₂×C₁₃	semidirect
288. C₁₃	#33782	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×#3170	2
C₁₃×#3147	2
C₁₃×#2251	2
C₁₃×#2243	2
C₁₃×#2209	2
C₁₃×#2207	2
C₁₃×#2189	2
C₁₃×#2163	2
#33782	13

Subgroups — C13×#33782 #15720121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33782 #15720121