Subgroups — C₁₃×#33781 #15720120

Order 3328.

598 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×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#495	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×#2211	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2188	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2187	C₂	1	2	yes	maximal
8	C₁₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×SD₁₆	—	2	1	no	—
11	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
12	C₁₃×#533	—	2	1	no	—
13	C₂×C₁₃×#124	—	2	1	no	—
14	C₂×C₁₃×Q₃₂	C₂²	1	4	yes	complemented
15	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	—
16	C₁₃×#500	C₂²	1	1	yes	characteristic, complemented
17	C₁₃×#495	—	2	1	no	—
18	C₁₃×#495	—	2	1	no	—
19	C₁₃×#495	C₂²	1	2	yes	—
20	C₁₃×#495	C₂²	1	2	yes	—
21	C₁₃×#494	C₂²	1	1	yes	characteristic, complemented
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	complemented
27	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
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	C₁₃×#346	C₂²	1	2	yes	complemented
33	C₂²×Q₈×C₁₃	—	2	1	no	—
34	C₂²×D₄×C₁₃	—	2	1	no	—
35	C₂×C₁₃×Q₁₆	—	2	1	no	—
36	C₂×C₁₃×Q₁₆	—	2	1	no	—
37	C₂×C₁₃×Q₁₆	—	4	2	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×SD₁₆	—	4	4	no	—
43	C₂×C₁₃×D₈	—	2	1	no	—
44	C₂×C₁₃×D₈	—	4	2	no	—
45	C₂×C₁₃×D₈	C₂³	1	2	yes	—
46	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×C₄⋊Q₈	—	2	1	no	—
49	C₁₃×C₄⋊Q₈	—	2	1	no	—
50	C₁₃×#125	—	4	1	no	—
51	C₁₃×#124	—	2	2	no	—
52	C₁₃×#124	—	2	2	no	—
53	C₁₃×#124	—	4	1	no	—
54	C₁₃×#124	—	4	1	no	—
55	C₁₃×C₂²⋊Q₈	—	2	2	no	—
56	C₁₃×C₂²⋊Q₈	—	4	1	no	—
57	C₄×Q₈×C₁₃	—	2	2	no	—
58	C₄×Q₈×C₁₃	—	4	1	no	—
59	C₁₃×#117	—	4	1	no	—
60	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
61	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
62	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
63	C₁₃×Q₃₂	—	4	4	no	—
64	C₁₃×SD₃₂	—	4	4	no	—
65	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
66	C₁₃×#107	—	2	1	no	—
67	C₁₃×#107	C₂³	1	2	yes	—
68	C₁₃×#103	—	2	2	no	—
69	C₁₃×#103	—	2	2	no	—
70	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
71	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	#33781	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂⁴×C₁₃	—	4	1	no	—
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	—	4	2	no	—
79	C₂×Q₈×C₁₃	—	4	2	no	—
80	C₂×Q₈×C₁₃	—	4	1	no	—
81	C₂×Q₈×C₁₃	—	4	1	no	—
82	C₂×D₄×C₁₃	—	2	2	no	—
83	C₂×D₄×C₁₃	—	4	2	no	—
84	C₂×D₄×C₁₃	—	4	2	no	—
85	C₂×D₄×C₁₃	—	4	1	no	—
86	C₂²×C₄×C₁₃	—	4	1	no	—
87	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₁₃×Q₁₆	—	2	4	no	—
91	C₁₃×Q₁₆	—	2	4	no	—
92	C₁₃×Q₁₆	—	4	2	no	—
93	C₁₃×Q₁₆	—	4	2	no	—
94	C₁₃×SD₁₆	—	4	4	no	—
95	C₁₃×SD₁₆	—	4	4	no	—
96	C₁₃×D₈	—	2	4	no	—
97	C₁₃×D₈	—	4	2	no	—
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	—	2	2	no	—
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
104	C₁₃×C₄⋊C₄	—	2	2	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	C₁₃×C₂²⋊C₄	—	4	2	no	—
114	C₁₃×C₂²⋊C₄	—	4	2	no	—
115	C₄²×C₁₃	—	4	2	no	—
116	C₄²×C₁₃	D₈	1	1	yes	characteristic
117	C₄²×C₁₃	D₈	1	1	yes	characteristic
118	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
120	C₁₃×C₁₆	—	2	4	no	—
121	#3170	C₂×C₁₃	1	1	yes	characteristic, complemented
122	C₂×#495	C₂×C₁₃	1	1	yes	characteristic
123	#2251	C₂×C₁₃	1	2	yes	complemented
124	#2243	C₂×C₁₃	1	2	yes	complemented
125	#2211	C₂×C₁₃	1	4	yes	complemented
126	#2188	C₂×C₁₃	1	2	yes	complemented
127	#2187	C₂×C₁₃	1	2	yes	—
128	#2163	C₂×C₁₃	1	1	yes	characteristic, complemented
129	C₂³×C₁₃	—	4	2	no	—
130	C₂³×C₁₃	—	8	2	no	—
131	C₂³×C₁₃	—	8	2	no	—
132	C₂³×C₁₃	—	8	2	no	—
133	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
134	Q₈×C₁₃	—	4	2	no	—
135	Q₈×C₁₃	—	4	2	no	—
136	Q₈×C₁₃	—	4	2	no	—
137	Q₈×C₁₃	—	4	2	no	—
138	Q₈×C₁₃	—	8	2	no	—
139	Q₈×C₁₃	—	8	2	no	—
140	D₄×C₁₃	—	4	2	no	—
141	D₄×C₁₃	—	4	2	no	—
142	D₄×C₁₃	—	8	2	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	2	1	no	—
145	C₂×C₄×C₁₃	—	2	1	no	—
146	C₂×C₄×C₁₃	—	4	1	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	—	8	2	no	—
151	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
157	C₈×C₁₃	—	2	1	no	—
158	C₈×C₁₃	—	2	1	no	—
159	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
160	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
161	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
162	C₂²×SD₁₆	—	2	1	no	—
163	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
164	#533	—	2	1	no	—
165	C₂×#124	—	2	1	no	—
166	C₂×Q₃₂	C₂²×C₁₃	1	4	yes	complemented
167	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	—
168	#500	C₂²×C₁₃	1	1	yes	characteristic, complemented
169	#495	—	2	1	no	—
170	#495	—	2	1	no	—
171	#495	C₂²×C₁₃	1	2	yes	—
172	#495	C₂²×C₁₃	1	2	yes	—
173	#494	C₂²×C₁₃	1	1	yes	characteristic, complemented
174	#487	—	2	1	no	—
175	#467	—	2	1	no	—
176	#467	C₂²×C₁₃	1	2	yes	—
177	C₄×Q₁₆	—	2	1	no	—
178	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	complemented
179	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
180	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	complemented
181	#358	C₂²×C₁₃	1	4	yes	complemented
182	#357	C₂²×C₁₃	1	4	yes	—
183	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
184	#346	C₂²×C₁₃	1	2	yes	complemented
185	C₂²×C₁₃	—	2	1	no	—
186	C₂²×C₁₃	—	2	1	no	—
187	C₂²×C₁₃	—	8	2	no	—
188	C₂²×C₁₃	—	8	2	no	—
189	C₂²×C₁₃	—	8	2	no	—
190	C₂²×C₁₃	—	16	4	no	—
191	C₂²×C₁₃	#510	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	—	2	1	no	—
198	C₄×C₁₃	—	8	2	no	—
199	C₄×C₁₃	—	8	2	no	—
200	C₄×C₁₃	—	8	2	no	—
201	C₄×C₁₃	#535	1	1	yes	characteristic
202	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
203	C₂²×Q₈	—	2	1	no	—
204	C₂²×D₄	—	2	1	no	—
205	C₂×Q₁₆	—	2	1	no	—
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	—	4	2	no	—
208	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
209	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
210	C₂×SD₁₆	—	2	2	no	—
211	C₂×SD₁₆	—	2	2	no	—
212	C₂×SD₁₆	—	4	4	no	—
213	C₂×D₈	—	2	1	no	—
214	C₂×D₈	—	4	2	no	—
215	C₂×D₈	C₂³×C₁₃	1	2	yes	—
216	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
217	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	#125	—	4	1	no	—
221	#124	—	2	2	no	—
222	#124	—	2	2	no	—
223	#124	—	4	1	no	—
224	#124	—	4	1	no	—
225	C₂²⋊Q₈	—	2	2	no	—
226	C₂²⋊Q₈	—	4	1	no	—
227	C₄×Q₈	—	2	2	no	—
228	C₄×Q₈	—	4	1	no	—
229	#117	—	4	1	no	—
230	C₂×C₂²⋊C₄	—	4	1	no	—
231	C₂×C₂²⋊C₄	—	4	1	no	—
232	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
233	Q₃₂	—	4	4	no	—
234	SD₃₂	—	4	4	no	—
235	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
236	#107	—	2	1	no	—
237	#107	C₂³×C₁₃	1	2	yes	—
238	#103	—	2	2	no	—
239	#103	—	2	2	no	—
240	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
241	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
244	C₂×C₁₃	—	2	2	no	—
245	C₂×C₁₃	—	16	2	no	—
246	C₂×C₁₃	#3205	1	1	yes	characteristic
247	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
248	C₂×C₁₃	#3414	1	1	yes	characteristic
249	C₂⁴	—	4	1	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	4	2	no	—
253	C₂×Q₈	—	4	2	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
262	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
264	Q₁₆	—	2	4	no	—
265	Q₁₆	—	2	4	no	—
266	Q₁₆	—	4	2	no	—
267	Q₁₆	—	4	2	no	—
268	SD₁₆	—	4	4	no	—
269	SD₁₆	—	4	4	no	—
270	D₈	—	2	4	no	—
271	D₈	—	4	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
275	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
277	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₄²	—	4	2	no	—
290	C₄²	C₁₃×D₈	1	1	yes	characteristic
291	C₄²	C₁₃×D₈	1	1	yes	characteristic
292	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
293	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
294	C₁₆	—	2	4	no	—
295	C₁₃	#33781	1	1	yes	characteristic, direct product factor, Sylow
296	C₂³	—	4	2	no	—
297	C₂³	—	8	2	no	—
298	C₂³	—	8	2	no	—
299	C₂³	—	8	2	no	—
300	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	8	2	no	—
306	Q₈	—	8	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	8	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	8	2	no	—
318	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
319	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
320	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
321	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
322	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
327	C₈	C₁₃×H(5,𝔽₂)	1	2	yes	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	16	4	no	—
334	C₂²	C₁₃×#510	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	—	2	1	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	—	8	2	no	—
344	C₄	C₁₃×#535	1	1	yes	characteristic
345	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
346	C₂	—	2	2	no	—
347	C₂	—	16	2	no	—
348	C₂	C₁₃×#3205	1	1	yes	characteristic
349	C₂	C₂×C₁₃×#510	1	1	yes	characteristic
350	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	C₂²×C₁₃×D₈	C₂²
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×#2211	C₂	semidirect
6. C₁₃×#2188	C₂	semidirect
8. C₁₃×#2163	C₂	semidirect
14. C₂×C₁₃×Q₃₂	C₂²	semidirect
16. C₁₃×#500	C₂²	semidirect
21. C₁₃×#494	C₂²	semidirect
26. C₄×C₁₃×Q₁₆	C₂²	semidirect
28. C₁₃×C₄⋊C₁₆	C₂²	semidirect
29. C₁₃×#358	C₂²	semidirect
32. C₁₃×#346	C₂²	semidirect
74. #33781	C₁₃	direct (N × H)
121. #3170	C₂×C₁₃	semidirect
123. #2251	C₂×C₁₃	semidirect
124. #2243	C₂×C₁₃	semidirect
125. #2211	C₂×C₁₃	semidirect
126. #2188	C₂×C₁₃	semidirect
128. #2163	C₂×C₁₃	semidirect
166. C₂×Q₃₂	C₂²×C₁₃	semidirect
168. #500	C₂²×C₁₃	semidirect
173. #494	C₂²×C₁₃	semidirect
178. C₄×Q₁₆	C₂²×C₁₃	semidirect
180. C₄⋊C₁₆	C₂²×C₁₃	semidirect
181. #358	C₂²×C₁₃	semidirect
184. #346	C₂²×C₁₃	semidirect
295. C₁₃	#33781	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
C₂×C₁₃×#495	2
C₁₃×#2251	2
C₁₃×#2243	2
C₁₃×#2211	2
C₁₃×#2188	2
C₁₃×#2187	2
C₁₃×#2163	2
#33781	13

Subgroups — C13×#33781 #15720120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33781 #15720120