Subgroups — C₁₃×#33735 #15720074

Order 3328.

526 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3168	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2242	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2217	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2216	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2215	C₂	1	1	yes	maximal, characteristic
8	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₁₃×SD₃₂	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#529	—	2	2	no	—
13	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
15	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
18	C₁₃×#499	C₂²	1	2	yes	—
19	C₁₃×#495	C₂²	1	2	yes	—
20	C₁₃×#488	—	2	2	no	—
21	C₁₃×#467	C₂²	1	2	yes	—
22	C₁₃×#466	C₂²	1	2	yes	—
23	C₁₃×#465	—	2	2	no	—
24	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
25	C₄×C₁₃×SD₁₆	—	2	2	no	—
26	C₄×C₁₃×D₈	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#367	C₂²	1	2	yes	—
29	C₁₃×#366	C₂²	1	1	yes	characteristic
30	C₁₃×#366	C₂²	1	1	yes	characteristic
31	C₁₃×#358	C₂²	1	2	yes	—
32	C₁₃×#358	C₂²	1	2	yes	—
33	C₁₃×#357	C₂²	1	2	yes	—
34	C₁₃×#357	C₂²	1	2	yes	—
35	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
36	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
37	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
38	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
39	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
40	C₂×C₁₃×SD₁₆	—	2	2	no	—
41	C₂×C₁₃×D₈	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×#126	—	4	2	no	—
45	C₁₃×#125	—	2	2	no	—
46	C₁₃×#124	—	2	2	no	—
47	C₁₃×#123	—	4	2	no	—
48	C₁₃×C₂²⋊Q₈	—	2	2	no	—
49	C₁₃×#121	—	2	2	no	—
50	C₄×Q₈×C₁₃	—	2	2	no	—
51	C₄×D₄×C₁₃	—	2	2	no	—
52	C₄×D₄×C₁₃	—	4	2	no	—
53	C₁₃×#117	—	4	2	no	—
54	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
55	C₁₃×Q₃₂	—	2	2	no	—
56	C₁₃×SD₃₂	—	2	4	no	—
57	C₁₃×D₁₆	—	2	2	no	—
58	C₂×C₁₃×C₁₆	—	2	2	no	—
59	C₂×C₁₃×C₁₆	—	2	2	no	—
60	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
64	C₁₃×#107	C₂³	1	2	yes	—
65	C₁₃×#107	C₂³	1	2	yes	—
66	C₁₃×#106	—	2	2	no	—
67	C₁₃×#103	—	2	2	no	—
68	C₁₃×#103	—	2	2	no	—
69	C₁₃×#102	—	2	2	no	—
70	C₁₃×#102	—	2	2	no	—
71	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
72	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
73	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	#33735	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×D₄×C₁₃	—	2	2	no	—
78	C₂×D₄×C₁₃	—	4	2	no	—
79	C₂²×C₄×C₁₃	—	4	2	no	—
80	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₁₃×Q₁₆	—	2	2	no	—
84	C₁₃×Q₁₆	—	2	2	no	—
85	C₁₃×SD₁₆	—	4	4	no	—
86	C₁₃×D₈	—	2	2	no	—
87	C₁₃×D₈	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	1	no	—
90	C₂×C₈×C₁₃	—	2	1	no	—
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	4	2	no	—
98	C₁₃×C₄⋊C₄	—	4	2	no	—
99	C₁₃×C₄⋊C₄	—	4	2	no	—
100	C₁₃×C₄⋊C₄	—	4	2	no	—
101	C₁₃×C₂²⋊C₄	—	4	2	no	—
102	C₁₃×C₂²⋊C₄	—	4	2	no	—
103	C₁₃×C₂²⋊C₄	—	4	2	no	—
104	C₁₃×C₂²⋊C₄	—	4	2	no	—
105	C₁₃×C₂²⋊C₄	—	4	2	no	—
106	C₄²×C₁₃	—	4	2	no	—
107	C₄²×C₁₃	D₈	1	1	yes	characteristic
108	C₄²×C₁₃	D₈	1	1	yes	characteristic
109	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₁₃×C₁₆	—	2	1	no	—
112	C₁₃×C₁₆	—	2	1	no	—
113	C₁₃×C₁₆	#41	1	2	yes	—
114	C₁₃×C₁₆	#41	1	2	yes	—
115	#3168	C₂×C₁₃	1	2	yes	complemented
116	#2253	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#2249	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#2242	C₂×C₁₃	1	2	yes	complemented
119	#2217	C₂×C₁₃	1	2	yes	complemented
120	#2216	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#2215	C₂×C₁₃	1	1	yes	characteristic
122	C₄×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
123	C₄×SD₃₂	C₂×C₁₃	1	2	yes	complemented
124	C₄×D₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
125	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
126	C₂³×C₁₃	—	4	2	no	—
127	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
128	Q₈×C₁₃	—	4	2	no	—
129	Q₈×C₁₃	—	4	2	no	—
130	D₄×C₁₃	—	4	2	no	—
131	D₄×C₁₃	—	4	2	no	—
132	D₄×C₁₃	—	8	4	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	2	1	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	8	4	no	—
141	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
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	C₂²×D₄	1	1	yes	characteristic
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	#135	1	2	yes	—
150	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
151	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
152	#529	—	2	2	no	—
153	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
155	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
156	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
157	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
158	#499	C₂²×C₁₃	1	2	yes	—
159	#495	C₂²×C₁₃	1	2	yes	—
160	#488	—	2	2	no	—
161	#467	C₂²×C₁₃	1	2	yes	—
162	#466	C₂²×C₁₃	1	2	yes	—
163	#465	—	2	2	no	—
164	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
165	C₄×SD₁₆	—	2	2	no	—
166	C₄×D₈	C₂²×C₁₃	1	2	yes	—
167	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
168	#367	C₂²×C₁₃	1	2	yes	—
169	#366	C₂²×C₁₃	1	1	yes	characteristic
170	#366	C₂²×C₁₃	1	1	yes	characteristic
171	#358	C₂²×C₁₃	1	2	yes	—
172	#358	C₂²×C₁₃	1	2	yes	—
173	#357	C₂²×C₁₃	1	2	yes	—
174	#357	C₂²×C₁₃	1	2	yes	—
175	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
176	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
177	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
178	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	—	8	2	no	—
182	C₂²×C₁₃	—	8	2	no	—
183	C₂²×C₁₃	—	8	2	no	—
184	C₂²×C₁₃	#508	1	2	yes	—
185	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	8	2	no	—
191	C₄×C₁₃	—	8	2	no	—
192	C₄×C₁₃	—	8	2	no	—
193	C₄×C₁₃	#508	1	2	yes	—
194	C₄×C₁₃	#532	1	1	yes	characteristic
195	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
196	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
197	C₂×SD₁₆	—	2	2	no	—
198	C₂×D₈	C₂³×C₁₃	1	2	yes	—
199	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
200	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
201	#126	—	4	2	no	—
202	#125	—	2	2	no	—
203	#124	—	2	2	no	—
204	#123	—	4	2	no	—
205	C₂²⋊Q₈	—	2	2	no	—
206	#121	—	2	2	no	—
207	C₄×Q₈	—	2	2	no	—
208	C₄×D₄	—	2	2	no	—
209	C₄×D₄	—	4	2	no	—
210	#117	—	4	2	no	—
211	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
212	Q₃₂	—	2	2	no	—
213	SD₃₂	—	2	4	no	—
214	D₁₆	—	2	2	no	—
215	C₂×C₁₆	—	2	2	no	—
216	C₂×C₁₆	—	2	2	no	—
217	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
218	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
219	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
220	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
221	#107	C₂³×C₁₃	1	2	yes	—
222	#107	C₂³×C₁₃	1	2	yes	—
223	#106	—	2	2	no	—
224	#103	—	2	2	no	—
225	#103	—	2	2	no	—
226	#102	—	2	2	no	—
227	#102	—	2	2	no	—
228	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
229	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
230	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
232	C₂×C₁₃	—	2	2	no	—
233	C₂×C₁₃	—	16	2	no	—
234	C₂×C₁₃	#3169	1	1	yes	characteristic
235	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
236	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
237	C₂×Q₈	—	2	2	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
242	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
244	Q₁₆	—	2	2	no	—
245	Q₁₆	—	2	2	no	—
246	SD₁₆	—	4	4	no	—
247	D₈	—	2	2	no	—
248	D₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
253	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
254	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
255	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	C₁₃×D₈	1	1	yes	characteristic
269	C₄²	C₁₃×D₈	1	1	yes	characteristic
270	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
271	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
272	C₁₆	—	2	1	no	—
273	C₁₆	—	2	1	no	—
274	C₁₆	C₁₃×#41	1	2	yes	—
275	C₁₆	C₁₃×#41	1	2	yes	—
276	C₁₃	#33735	1	1	yes	characteristic, direct product factor, Sylow
277	C₂³	—	4	2	no	—
278	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
279	Q₈	—	4	2	no	—
280	Q₈	—	4	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	8	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	8	4	no	—
292	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
293	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
294	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
295	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
296	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	C₁₃×#135	1	2	yes	—
301	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
302	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	C₁₃×#508	1	2	yes	—
309	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	2	no	—
317	C₄	C₁₃×#508	1	2	yes	—
318	C₄	C₁₃×#532	1	1	yes	characteristic
319	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	16	2	no	—
322	C₂	C₁₃×#3169	1	1	yes	characteristic
323	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
324	C₂	C₂×C₁₃×#508	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	C₁₃×#2215	C₂
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×#3168	C₂	semidirect
2. C₁₃×#2253	C₂	semidirect
3. C₁₃×#2249	C₂	semidirect
4. C₁₃×#2242	C₂	semidirect
5. C₁₃×#2217	C₂	semidirect
6. C₁₃×#2216	C₂	semidirect
8. C₄×C₁₃×Q₃₂	C₂	semidirect
9. C₄×C₁₃×SD₃₂	C₂	semidirect
10. C₄×C₁₃×D₁₆	C₂	semidirect
11. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
75. #33735	C₁₃	direct (N × H)
115. #3168	C₂×C₁₃	semidirect
116. #2253	C₂×C₁₃	semidirect
117. #2249	C₂×C₁₃	semidirect
118. #2242	C₂×C₁₃	semidirect
119. #2217	C₂×C₁₃	semidirect
120. #2216	C₂×C₁₃	semidirect
122. C₄×Q₃₂	C₂×C₁₃	semidirect
123. C₄×SD₃₂	C₂×C₁₃	semidirect
124. C₄×D₁₆	C₂×C₁₃	semidirect
125. C₂×C₄×C₁₆	C₂×C₁₃	semidirect
276. C₁₃	#33735	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×#3168	2
C₁₃×#2253	2
C₁₃×#2249	2
C₁₃×#2242	2
C₁₃×#2217	2
C₁₃×#2216	2
C₁₃×#2215	2
C₄×C₁₃×Q₃₂	2
C₄×C₁₃×SD₃₂	2
C₄×C₁₃×D₁₆	2
C₂×C₄×C₁₃×C₁₆	2
#33735	13

Subgroups — C13×#33735 #15720074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33735 #15720074