Subgroups — C₁₃×#33736 #15720075

Order 3328.

558 conjugacy classes of subgroups.

The subgroup lattice has 332 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×#3169	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2217	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₃×SD₃₂	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#533	—	2	1	no	—
11	C₁₃×#532	—	2	1	no	—
12	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	—
13	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
14	C₁₃×#500	C₂²	1	1	yes	characteristic
15	C₁₃×#500	C₂²	1	1	yes	characteristic
16	C₁₃×#494	C₂²	1	1	yes	characteristic
17	C₁₃×#493	C₂²	1	1	yes	characteristic
18	C₁₃×#487	—	2	1	no	—
19	C₁₃×#486	—	2	1	no	—
20	C₁₃×#467	—	2	1	no	—
21	C₁₃×#467	C₂²	1	2	yes	—
22	C₁₃×#466	—	2	1	no	—
23	C₁₃×#466	C₂²	1	2	yes	—
24	C₄×C₁₃×Q₁₆	—	2	1	no	—
25	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
26	C₄×C₁₃×D₈	—	2	1	no	—
27	C₄×C₁₃×D₈	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
29	C₁₃×#367	C₂²	1	4	yes	—
30	C₁₃×#358	C₂²	1	4	yes	—
31	C₁₃×#357	C₂²	1	4	yes	—
32	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
33	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
34	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
35	C₂×C₁₃×#41	—	4	1	no	—
36	C₂×C₁₃×Q₁₆	—	2	1	no	—
37	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
38	C₂×C₁₃×D₈	—	2	1	no	—
39	C₂×C₁₃×D₈	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×C₄⋊Q₈	—	2	1	no	—
43	C₁₃×C₄⋊Q₈	—	2	1	no	—
44	C₁₃×C₄⋊Q₈	—	2	1	no	—
45	C₁₃×#127	—	2	1	no	—
46	C₁₃×#125	—	4	1	no	—
47	C₁₃×#124	—	4	1	no	—
48	C₁₃×C₂²⋊Q₈	—	2	2	no	—
49	C₁₃×C₂²⋊Q₈	—	4	1	no	—
50	C₁₃×#121	—	2	2	no	—
51	C₁₃×#121	—	4	1	no	—
52	C₄×Q₈×C₁₃	—	2	2	no	—
53	C₄×Q₈×C₁₃	—	4	1	no	—
54	C₄×D₄×C₁₃	—	2	2	no	—
55	C₄×D₄×C₁₃	—	4	1	no	—
56	C₁₃×#117	—	4	1	no	—
57	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
58	C₁₃×SD₃₂	—	2	8	no	—
59	C₂×C₁₃×C₁₆	—	2	4	no	—
60	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
61	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
62	C₁₃×#107	—	2	1	no	—
63	C₁₃×#107	—	2	1	no	—
64	C₁₃×#107	C₂³	1	2	yes	—
65	C₁₃×#107	C₂³	1	2	yes	—
66	C₁₃×#103	—	2	2	no	—
67	C₁₃×#103	—	2	2	no	—
68	C₁₃×#102	—	2	2	no	—
69	C₁₃×#102	—	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	#33736	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₁₃×#41	—	8	4	no	—
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	4	1	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	4	no	—
88	C₁₃×Q₁₆	—	4	2	no	—
89	C₁₃×D₈	—	2	4	no	—
90	C₁₃×D₈	—	4	2	no	—
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₄⋊C₄	—	4	1	no	—
101	C₁₃×C₄⋊C₄	—	4	1	no	—
102	C₁₃×C₄⋊C₄	—	4	2	no	—
103	C₁₃×C₄⋊C₄	—	4	2	no	—
104	C₁₃×C₄⋊C₄	—	4	1	no	—
105	C₁₃×C₂²⋊C₄	—	4	2	no	—
106	C₁₃×C₂²⋊C₄	—	4	2	no	—
107	C₁₃×C₂²⋊C₄	—	4	2	no	—
108	C₄²×C₁₃	—	4	2	no	—
109	C₄²×C₁₃	D₈	1	1	yes	characteristic
110	C₄²×C₁₃	D₈	1	1	yes	characteristic
111	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₁₃×C₁₆	—	2	2	no	—
114	C₁₃×C₁₆	#41	1	4	yes	complemented
115	#3170	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#3169	C₂×C₁₃	1	1	yes	characteristic
117	#2254	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#2248	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#2241	C₂×C₁₃	1	1	yes	characteristic, complemented
120	#2240	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#2217	C₂×C₁₃	1	4	yes	complemented
122	C₄×SD₃₂	C₂×C₁₃	1	4	yes	complemented
123	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
124	C₂³×C₁₃	—	4	2	no	—
125	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
126	Q₈×C₁₃	—	4	2	no	—
127	Q₈×C₁₃	—	4	2	no	—
128	Q₈×C₁₃	—	8	2	no	—
129	Q₈×C₁₃	—	8	2	no	—
130	D₄×C₁₃	—	4	2	no	—
131	D₄×C₁₃	—	4	2	no	—
132	D₄×C₁₃	—	8	4	no	—
133	D₄×C₁₃	—	8	2	no	—
134	D₄×C₁₃	—	8	2	no	—
135	C₂×C₄×C₁₃	—	2	1	no	—
136	C₂×C₄×C₁₃	—	2	1	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	8	4	no	—
142	C₂×C₄×C₁₃	SD₃₂	1	4	yes	complemented
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	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	#135	1	2	yes	—
152	C₈×C₁₃	C₂×#41	1	2	yes	—
153	#533	—	2	1	no	—
154	#532	—	2	1	no	—
155	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	—
156	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
157	#500	C₂²×C₁₃	1	1	yes	characteristic
158	#500	C₂²×C₁₃	1	1	yes	characteristic
159	#494	C₂²×C₁₃	1	1	yes	characteristic
160	#493	C₂²×C₁₃	1	1	yes	characteristic
161	#487	—	2	1	no	—
162	#486	—	2	1	no	—
163	#467	—	2	1	no	—
164	#467	C₂²×C₁₃	1	2	yes	—
165	#466	—	2	1	no	—
166	#466	C₂²×C₁₃	1	2	yes	—
167	C₄×Q₁₆	—	2	1	no	—
168	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
169	C₄×D₈	—	2	1	no	—
170	C₄×D₈	C₂²×C₁₃	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
172	#367	C₂²×C₁₃	1	4	yes	—
173	#358	C₂²×C₁₃	1	4	yes	—
174	#357	C₂²×C₁₃	1	4	yes	—
175	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
176	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
177	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
178	C₂²×C₁₃	—	2	1	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	8	2	no	—
181	C₂²×C₁₃	—	8	2	no	—
182	C₂²×C₁₃	—	8	2	no	—
183	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
184	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	8	2	no	—
188	C₄×C₁₃	—	8	2	no	—
189	C₄×C₁₃	—	8	2	no	—
190	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
191	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
192	C₄×C₁₃	#508	1	2	yes	—
193	C₄×C₁₃	#532	1	1	yes	characteristic
194	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
195	C₂×#41	—	4	1	no	—
196	C₂×Q₁₆	—	2	1	no	—
197	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
198	C₂×D₈	—	2	1	no	—
199	C₂×D₈	C₂³×C₁₃	1	2	yes	—
200	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
202	C₄⋊Q₈	—	2	1	no	—
203	C₄⋊Q₈	—	2	1	no	—
204	C₄⋊Q₈	—	2	1	no	—
205	#127	—	2	1	no	—
206	#125	—	4	1	no	—
207	#124	—	4	1	no	—
208	C₂²⋊Q₈	—	2	2	no	—
209	C₂²⋊Q₈	—	4	1	no	—
210	#121	—	2	2	no	—
211	#121	—	4	1	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×Q₈	—	4	1	no	—
214	C₄×D₄	—	2	2	no	—
215	C₄×D₄	—	4	1	no	—
216	#117	—	4	1	no	—
217	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
218	SD₃₂	—	2	8	no	—
219	C₂×C₁₆	—	2	4	no	—
220	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
221	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
222	#107	—	2	1	no	—
223	#107	—	2	1	no	—
224	#107	C₂³×C₁₃	1	2	yes	—
225	#107	C₂³×C₁₃	1	2	yes	—
226	#103	—	2	2	no	—
227	#103	—	2	2	no	—
228	#102	—	2	2	no	—
229	#102	—	2	2	no	—
230	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
231	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
232	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
233	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
234	C₂×C₁₃	—	2	2	no	—
235	C₂×C₁₃	—	16	2	no	—
236	C₂×C₁₃	#3169	1	1	yes	characteristic
237	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
238	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
239	#41	—	8	4	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
249	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
251	Q₁₆	—	2	4	no	—
252	Q₁₆	—	4	2	no	—
253	D₈	—	2	4	no	—
254	D₈	—	4	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₄²	—	4	2	no	—
273	C₄²	C₁₃×D₈	1	1	yes	characteristic
274	C₄²	C₁₃×D₈	1	1	yes	characteristic
275	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
276	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
277	C₁₆	—	2	2	no	—
278	C₁₆	C₁₃×#41	1	4	yes	complemented
279	C₁₃	#33736	1	1	yes	characteristic, direct product factor, Sylow
280	C₂³	—	4	2	no	—
281	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
282	Q₈	—	4	2	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	8	2	no	—
285	Q₈	—	8	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	8	4	no	—
289	D₄	—	8	2	no	—
290	D₄	—	8	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	C₁₃×SD₃₂	1	4	yes	complemented
299	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
300	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
301	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
302	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
303	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	C₁₃×#135	1	2	yes	—
308	C₈	C₂×C₁₃×#41	1	2	yes	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	C₂×C₁₃×SD₃₂	1	2	yes	—
315	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	C₂×C₁₃×SD₃₂	1	2	yes	—
322	C₄	C₂×C₁₃×SD₃₂	1	2	yes	—
323	C₄	C₁₃×#508	1	2	yes	—
324	C₄	C₁₃×#532	1	1	yes	characteristic
325	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	16	2	no	—
328	C₂	C₁₃×#3169	1	1	yes	characteristic
329	C₂	C₂²×C₁₃×SD₃₂	1	1	yes	characteristic
330	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×#3169	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×#3170	C₂	semidirect
3. C₁₃×#2254	C₂	semidirect
4. C₁₃×#2248	C₂	semidirect
5. C₁₃×#2241	C₂	semidirect
6. C₁₃×#2240	C₂	semidirect
7. C₁₃×#2217	C₂	semidirect
8. C₄×C₁₃×SD₃₂	C₂	semidirect
9. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
74. #33736	C₁₃	direct (N × H)
114. C₁₃×C₁₆	#41	semidirect
115. #3170	C₂×C₁₃	semidirect
117. #2254	C₂×C₁₃	semidirect
118. #2248	C₂×C₁₃	semidirect
119. #2241	C₂×C₁₃	semidirect
120. #2240	C₂×C₁₃	semidirect
121. #2217	C₂×C₁₃	semidirect
122. C₄×SD₃₂	C₂×C₁₃	semidirect
123. C₂×C₄×C₁₆	C₂×C₁₃	semidirect
142. C₂×C₄×C₁₃	SD₃₂	semidirect
278. C₁₆	C₁₃×#41	semidirect
279. C₁₃	#33736	direct (N × H)
298. C₂×C₄	C₁₃×SD₃₂	semidirect

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×#3169	2
C₁₃×#2254	2
C₁₃×#2248	2
C₁₃×#2241	2
C₁₃×#2240	2
C₁₃×#2217	2
C₄×C₁₃×SD₃₂	2
C₂×C₄×C₁₃×C₁₆	2
#33736	13

Subgroups — C13×#33736 #15720075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33736 #15720075