Subgroups — C₁₃×#33921 #15720260

Order 3328.

514 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×#3341	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3334	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2234	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2233	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2226	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2224	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2218	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#2169	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2159	C₂	1	2	yes	maximal, complemented
12	C₁₃×#555	—	2	1	no	—
13	C₁₃×#547	—	2	1	no	—
14	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
16	C₁₃×#500	C₂²	1	1	yes	characteristic
17	C₁₃×#500	C₂²	1	1	yes	characteristic
18	C₁₃×#482	—	2	2	no	—
19	C₁₃×#480	—	2	2	no	—
20	C₁₃×#476	—	2	1	no	—
21	C₁₃×#474	—	2	1	no	—
22	C₁₃×#466	C₂²	1	2	yes	—
23	C₁₃×#466	C₂²	1	2	yes	—
24	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
25	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#427	C₂²	1	2	yes	—
28	C₂×C₁₃×#107	C₂²	1	2	yes	—
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	1	yes	characteristic
33	C₁₃×#366	C₂²	1	2	yes	—
34	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₁₃×#357	C₂²	1	1	yes	characteristic
36	C₁₃×#357	C₂²	1	1	yes	characteristic
37	C₁₃×#357	C₂²	1	2	yes	—
38	C₁₃×#357	C₂²	1	2	yes	—
39	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
41	C₂×C₁₃×#41	—	4	1	no	—
42	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
43	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
45	C₁₃×C₄⋊Q₈	—	2	1	no	—
46	C₁₃×C₄⋊Q₈	—	2	1	no	—
47	C₁₃×#124	—	4	2	no	—
48	C₁₃×#123	—	4	2	no	—
49	C₁₃×C₂²⋊Q₈	—	4	1	no	—
50	C₁₃×#121	—	2	2	no	—
51	C₁₃×#121	—	2	2	no	—
52	C₁₃×#121	—	4	1	no	—
53	C₄×D₄×C₁₃	—	2	1	no	—
54	C₄×D₄×C₁₃	—	2	1	no	—
55	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
56	C₁₃×#117	—	2	2	no	—
57	C₁₃×#117	—	4	1	no	—
58	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×D₁₆	—	2	2	no	—
60	C₁₃×M₃₂	—	2	4	no	—
61	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
63	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
64	C₁₃×#107	—	2	2	no	—
65	C₁₃×#107	—	2	1	no	—
66	C₁₃×#107	—	2	1	no	—
67	C₁₃×#107	C₂³	1	1	yes	characteristic
68	C₁₃×#107	C₂³	1	2	yes	—
69	C₁₃×#107	C₂³	1	1	yes	characteristic
70	C₁₃×#107	C₂³	1	2	yes	—
71	C₁₃×#106	—	2	2	no	—
72	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
73	C₁₃×#102	—	2	1	no	—
74	C₁₃×#102	—	2	1	no	—
75	C₁₃×#102	—	2	2	no	—
76	C₁₃×#102	—	2	2	no	—
77	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
78	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
79	#33921	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₁₃×#41	—	8	4	no	—
81	C₂×Q₈×C₁₃	—	4	1	no	—
82	C₂×Q₈×C₁₃	—	4	1	no	—
83	C₂×D₄×C₁₃	—	2	1	no	—
84	C₂×D₄×C₁₃	—	2	1	no	—
85	C₂×D₄×C₁₃	—	4	2	no	—
86	C₂×D₄×C₁₃	—	4	2	no	—
87	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	—	4	2	no	—
89	C₂²×C₄×C₁₃	—	4	1	no	—
90	C₂²×C₄×C₁₃	—	4	1	no	—
91	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
92	C₁₃×D₈	—	2	1	no	—
93	C₁₃×D₈	—	2	1	no	—
94	C₁₃×D₈	—	2	1	no	—
95	C₁₃×D₈	—	2	1	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	2	yes	—
100	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
101	C₁₃×C₄⋊C₄	—	2	1	no	—
102	C₁₃×C₄⋊C₄	—	2	2	no	—
103	C₁₃×C₄⋊C₄	—	2	1	no	—
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	4	2	no	—
106	C₁₃×C₄⋊C₄	—	4	1	no	—
107	C₁₃×C₄⋊C₄	—	4	1	no	—
108	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
109	C₁₃×C₂²⋊C₄	—	4	2	no	—
110	C₁₃×C₂²⋊C₄	—	4	2	no	—
111	C₁₃×C₂²⋊C₄	—	4	1	no	—
112	C₁₃×C₂²⋊C₄	—	4	1	no	—
113	C₁₃×C₂²⋊C₄	—	4	2	no	—
114	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
115	C₄²×C₁₃	—	4	2	no	—
116	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₁₃×C₁₆	—	2	2	no	—
118	C₁₃×C₁₆	—	2	1	no	—
119	C₁₃×C₁₆	#41	1	2	yes	complemented
120	#3341	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#3334	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#2254	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#2234	C₂×C₁₃	1	2	yes	complemented
124	#2233	C₂×C₁₃	1	2	yes	complemented
125	#2226	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#2224	C₂×C₁₃	1	1	yes	characteristic, complemented
127	#2218	C₂×C₁₃	1	2	yes	complemented
128	C₄×D₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
129	#2169	C₂×C₁₃	1	1	yes	characteristic, complemented
130	#2159	C₂×C₁₃	1	2	yes	complemented
131	C₂³×C₁₃	—	4	1	no	—
132	C₂³×C₁₃	—	4	1	no	—
133	C₂³×C₁₃	C₂×D₈	1	2	yes	—
134	Q₈×C₁₃	—	8	2	no	—
135	Q₈×C₁₃	—	16	1	no	—
136	D₄×C₁₃	—	2	2	no	—
137	D₄×C₁₃	—	4	1	no	—
138	D₄×C₁₃	—	4	1	no	—
139	D₄×C₁₃	—	4	1	no	—
140	D₄×C₁₃	—	4	1	no	—
141	D₄×C₁₃	—	8	4	no	—
142	D₄×C₁₃	—	16	2	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	2	2	no	—
145	C₂×C₄×C₁₃	—	4	1	no	—
146	C₂×C₄×C₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	—	4	1	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	8	4	no	—
150	C₂×C₄×C₁₃	—	8	2	no	—
151	C₂×C₄×C₁₃	—	8	2	no	—
152	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
153	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
156	C₈×C₁₃	—	2	2	no	—
157	C₈×C₁₃	—	2	1	no	—
158	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
159	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
160	#555	—	2	1	no	—
161	#547	—	2	1	no	—
162	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
164	#500	C₂²×C₁₃	1	1	yes	characteristic
165	#500	C₂²×C₁₃	1	1	yes	characteristic
166	#482	—	2	2	no	—
167	#480	—	2	2	no	—
168	#476	—	2	1	no	—
169	#474	—	2	1	no	—
170	#466	C₂²×C₁₃	1	2	yes	—
171	#466	C₂²×C₁₃	1	2	yes	—
172	C₄×D₈	C₂²×C₁₃	1	1	yes	characteristic
173	C₄×D₈	C₂²×C₁₃	1	1	yes	characteristic
174	C₈×D₄	C₂²×C₁₃	1	1	yes	characteristic
175	#427	C₂²×C₁₃	1	2	yes	—
176	C₂×#107	C₂²×C₁₃	1	2	yes	—
177	#367	C₂²×C₁₃	1	2	yes	—
178	#367	C₂²×C₁₃	1	2	yes	—
179	#367	C₂²×C₁₃	1	2	yes	—
180	#366	C₂²×C₁₃	1	1	yes	characteristic
181	#366	C₂²×C₁₃	1	2	yes	—
182	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
183	#357	C₂²×C₁₃	1	1	yes	characteristic
184	#357	C₂²×C₁₃	1	1	yes	characteristic
185	#357	C₂²×C₁₃	1	2	yes	—
186	#357	C₂²×C₁₃	1	2	yes	—
187	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
188	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
189	C₂²×C₁₃	—	2	2	no	—
190	C₂²×C₁₃	—	2	2	no	—
191	C₂²×C₁₃	—	2	2	no	—
192	C₂²×C₁₃	—	8	1	no	—
193	C₂²×C₁₃	—	8	1	no	—
194	C₂²×C₁₃	—	8	1	no	—
195	C₂²×C₁₃	—	8	1	no	—
196	C₂²×C₁₃	—	8	1	no	—
197	C₂²×C₁₃	—	8	1	no	—
198	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
199	C₄×C₁₃	—	2	2	no	—
200	C₄×C₁₃	—	2	1	no	—
201	C₄×C₁₃	—	8	1	no	—
202	C₄×C₁₃	—	8	1	no	—
203	C₄×C₁₃	—	8	2	no	—
204	C₄×C₁₃	—	8	2	no	—
205	C₄×C₁₃	#509	1	2	yes	—
206	C₄×C₁₃	#547	1	1	yes	characteristic
207	C₄×C₁₃	#578	1	1	yes	characteristic
208	C₂×#41	—	4	1	no	—
209	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
210	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	#124	—	4	2	no	—
215	#123	—	4	2	no	—
216	C₂²⋊Q₈	—	4	1	no	—
217	#121	—	2	2	no	—
218	#121	—	2	2	no	—
219	#121	—	4	1	no	—
220	C₄×D₄	—	2	1	no	—
221	C₄×D₄	—	2	1	no	—
222	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
223	#117	—	2	2	no	—
224	#117	—	4	1	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	D₁₆	—	2	2	no	—
227	M₃₂	—	2	4	no	—
228	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
229	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
230	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
231	#107	—	2	2	no	—
232	#107	—	2	1	no	—
233	#107	—	2	1	no	—
234	#107	C₂³×C₁₃	1	1	yes	characteristic
235	#107	C₂³×C₁₃	1	1	yes	characteristic
236	#107	C₂³×C₁₃	1	2	yes	—
237	#107	C₂³×C₁₃	1	2	yes	—
238	#106	—	2	2	no	—
239	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
240	#102	—	2	1	no	—
241	#102	—	2	1	no	—
242	#102	—	2	2	no	—
243	#102	—	2	2	no	—
244	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
245	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
246	C₂×C₁₃	—	4	2	no	—
247	C₂×C₁₃	—	16	1	no	—
248	C₂×C₁₃	—	16	1	no	—
249	C₂×C₁₃	#3334	1	1	yes	characteristic
250	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
251	C₂×C₁₃	#3416	1	1	yes	characteristic
252	#41	—	8	4	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
264	D₈	—	2	1	no	—
265	D₈	—	2	1	no	—
266	D₈	—	2	1	no	—
267	D₈	—	2	1	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
272	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
287	C₄²	—	4	2	no	—
288	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
289	C₁₆	—	2	2	no	—
290	C₁₆	—	2	1	no	—
291	C₁₆	C₁₃×#41	1	2	yes	complemented
292	C₁₃	#33921	1	1	yes	characteristic, direct product factor, Sylow
293	C₂³	—	4	1	no	—
294	C₂³	—	4	1	no	—
295	C₂³	C₂×C₁₃×D₈	1	2	yes	—
296	Q₈	—	8	2	no	—
297	Q₈	—	16	1	no	—
298	D₄	—	2	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	4	1	no	—
302	D₄	—	4	1	no	—
303	D₄	—	8	4	no	—
304	D₄	—	16	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
315	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
316	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
317	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	—	2	1	no	—
320	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
321	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	—	8	1	no	—
331	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	C₁₃×#509	1	2	yes	—
339	C₄	C₁₃×#547	1	1	yes	characteristic
340	C₄	C₁₃×#578	1	1	yes	characteristic
341	C₂	—	4	2	no	—
342	C₂	—	16	1	no	—
343	C₂	—	16	1	no	—
344	C₂	C₁₃×#3334	1	1	yes	characteristic
345	C₂	C₂×C₁₃×#509	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
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×D₄×C₁₃ ◁ C₈×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3341	C₂	semidirect
2. C₁₃×#3334	C₂	semidirect
3. C₁₃×#2254	C₂	semidirect
4. C₁₃×#2234	C₂	semidirect
5. C₁₃×#2233	C₂	semidirect
6. C₁₃×#2226	C₂	semidirect
7. C₁₃×#2224	C₂	semidirect
8. C₁₃×#2218	C₂	semidirect
9. C₄×C₁₃×D₁₆	C₂	semidirect
10. C₁₃×#2169	C₂	semidirect
11. C₁₃×#2159	C₂	semidirect
79. #33921	C₁₃	direct (N × H)
119. C₁₃×C₁₆	#41	semidirect
120. #3341	C₂×C₁₃	semidirect
121. #3334	C₂×C₁₃	semidirect
122. #2254	C₂×C₁₃	semidirect
123. #2234	C₂×C₁₃	semidirect
124. #2233	C₂×C₁₃	semidirect
125. #2226	C₂×C₁₃	semidirect
126. #2224	C₂×C₁₃	semidirect
127. #2218	C₂×C₁₃	semidirect
128. C₄×D₁₆	C₂×C₁₃	semidirect
129. #2169	C₂×C₁₃	semidirect
130. #2159	C₂×C₁₃	semidirect
291. C₁₆	C₁₃×#41	semidirect
292. C₁₃	#33921	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×#3341	2
C₁₃×#3334	2
C₁₃×#2254	2
C₁₃×#2234	2
C₁₃×#2233	2
C₁₃×#2226	2
C₁₃×#2224	2
C₁₃×#2218	2
C₄×C₁₃×D₁₆	2
C₁₃×#2169	2
C₁₃×#2159	2
#33921	13

Subgroups — C13×#33921 #15720260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33921 #15720260