Subgroups — C₁₃×#33933 #15720272

Order 3328.

532 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3391	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3380	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2244	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2228	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2226	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2206	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2206	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#2177	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#554	—	2	1	no	—
13	C₁₃×#550	—	2	1	no	—
14	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
16	C₁₃×#500	C₂²	1	1	yes	characteristic
17	C₁₃×#499	C₂²	1	2	yes	—
18	C₁₃×#493	C₂²	1	1	yes	characteristic
19	C₁₃×#493	C₂²	1	2	yes	—
20	C₁₃×#478	—	2	2	no	—
21	C₁₃×#476	—	2	1	no	—
22	C₁₃×#459	—	2	1	no	—
23	C₁₃×#459	—	2	2	no	—
24	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
26	C₄×C₁₃×D₈	C₂²	1	2	yes	—
27	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
28	C₄×C₁₃×D₈	C₂²	1	2	yes	—
29	C₁₃×#367	C₂²	1	2	yes	—
30	C₁₃×#366	C₂²	1	1	yes	characteristic
31	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₁₃×#357	C₂²	1	2	yes	—
34	C₁₃×#357	C₂²	1	2	yes	—
35	C₁₃×#357	C₂²	1	1	yes	characteristic
36	C₁₃×#357	C₂²	1	2	yes	—
37	C₁₃×#357	C₂²	1	2	yes	—
38	C₁₃×#357	C₂²	1	1	yes	characteristic
39	C₁₃×#346	C₂²	1	2	yes	—
40	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
41	C₂×C₁₃×#41	—	4	1	no	—
42	C₂×C₁₃×D₈	—	2	2	no	—
43	C₂×C₁₃×D₈	—	2	1	no	—
44	C₂×C₁₃×D₈	C₂³	1	2	yes	—
45	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
46	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
47	C₁₃×C₄⋊Q₈	—	2	1	no	—
48	C₁₃×#127	—	2	2	no	—
49	C₁₃×#127	—	2	1	no	—
50	C₁₃×#125	—	2	2	no	—
51	C₁₃×C₂²⋊Q₈	—	4	1	no	—
52	C₁₃×C₂²⋊Q₈	—	4	2	no	—
53	C₁₃×#121	—	4	1	no	—
54	C₁₃×#121	—	4	2	no	—
55	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
56	C₄×D₄×C₁₃	—	2	2	no	—
57	C₄×D₄×C₁₃	—	2	1	no	—
58	C₄×D₄×C₁₃	—	2	1	no	—
59	C₄×D₄×C₁₃	—	2	2	no	—
60	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
61	C₁₃×D₁₆	—	2	2	no	—
62	C₁₃×D₁₆	—	2	4	no	—
63	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
65	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
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×#106	—	2	2	no	—
71	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
72	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
73	C₁₃×#102	—	2	2	no	—
74	C₁₃×#102	—	2	1	no	—
75	C₁₃×#102	—	2	2	no	—
76	C₁₃×#102	—	2	1	no	—
77	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₃	C₂³	1	2	yes	—
79	#33933	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	D₈	1	1	yes	characteristic
83	C₂×D₄×C₁₃	—	2	1	no	—
84	C₂×D₄×C₁₃	—	2	2	no	—
85	C₂×D₄×C₁₃	—	2	1	no	—
86	C₂×D₄×C₁₃	—	4	2	no	—
87	C₂×D₄×C₁₃	—	4	1	no	—
88	C₂×D₄×C₁₃	—	4	1	no	—
89	C₂×D₄×C₁₃	—	4	2	no	—
90	C₂²×C₄×C₁₃	—	4	1	no	—
91	C₂²×C₄×C₁₃	—	4	2	no	—
92	C₂²×C₄×C₁₃	—	4	1	no	—
93	C₂²×C₄×C₁₃	—	4	2	no	—
94	C₁₃×D₈	—	2	1	no	—
95	C₁₃×D₈	—	2	2	no	—
96	C₁₃×D₈	—	2	1	no	—
97	C₁₃×D₈	—	2	2	no	—
98	C₁₃×D₈	—	2	1	no	—
99	C₁₃×D₈	—	8	2	no	—
100	C₁₃×D₈	—	8	1	no	—
101	C₁₃×D₈	#41	1	2	yes	complemented
102	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
105	C₁₃×C₄⋊C₄	—	2	1	no	—
106	C₁₃×C₄⋊C₄	—	2	1	no	—
107	C₁₃×C₄⋊C₄	—	2	2	no	—
108	C₁₃×C₄⋊C₄	—	4	2	no	—
109	C₁₃×C₄⋊C₄	—	4	1	no	—
110	C₁₃×C₄⋊C₄	—	4	2	no	—
111	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
112	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
113	C₁₃×C₂²⋊C₄	—	4	1	no	—
114	C₁₃×C₂²⋊C₄	—	4	2	no	—
115	C₁₃×C₂²⋊C₄	—	4	1	no	—
116	C₁₃×C₂²⋊C₄	—	4	2	no	—
117	C₄²×C₁₃	C₂×D₄	1	2	yes	—
118	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₁₃×C₁₆	—	2	1	no	—
120	C₁₃×C₁₆	—	2	2	no	—
121	C₁₃×C₁₆	—	2	1	no	—
122	#3391	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#3380	C₂×C₁₃	1	1	yes	characteristic, complemented
124	#2244	C₂×C₁₃	1	2	yes	complemented
125	#2240	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#2228	C₂×C₁₃	1	2	yes	complemented
127	#2226	C₂×C₁₃	1	1	yes	characteristic, complemented
128	#2206	C₂×C₁₃	1	2	yes	complemented
129	#2206	C₂×C₁₃	1	1	yes	characteristic, complemented
130	#2183	C₂×C₁₃	1	1	yes	characteristic, complemented
131	#2177	C₂×C₁₃	1	2	yes	complemented
132	C₄×D₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
133	C₂³×C₁₃	—	4	1	no	—
134	C₂³×C₁₃	—	4	2	no	—
135	C₂³×C₁₃	—	4	1	no	—
136	Q₈×C₁₃	—	2	2	no	—
137	Q₈×C₁₃	—	16	1	no	—
138	D₄×C₁₃	—	2	2	no	—
139	D₄×C₁₃	—	4	1	no	—
140	D₄×C₁₃	—	4	2	no	—
141	D₄×C₁₃	—	4	1	no	—
142	D₄×C₁₃	—	4	1	no	—
143	D₄×C₁₃	—	4	2	no	—
144	D₄×C₁₃	—	8	4	no	—
145	D₄×C₁₃	—	8	2	no	—
146	D₄×C₁₃	—	16	2	no	—
147	D₄×C₁₃	—	16	1	no	—
148	C₂×C₄×C₁₃	—	4	1	no	—
149	C₂×C₄×C₁₃	—	4	1	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	—	8	4	no	—
152	C₂×C₄×C₁₃	—	8	2	no	—
153	C₂×C₄×C₁₃	—	8	2	no	—
154	C₂×C₄×C₁₃	—	16	2	no	—
155	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
157	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
158	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
160	C₈×C₁₃	—	2	2	no	—
161	C₈×C₁₃	—	2	1	no	—
162	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
163	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
164	#554	—	2	1	no	—
165	#550	—	2	1	no	—
166	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
167	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
168	#500	C₂²×C₁₃	1	1	yes	characteristic
169	#499	C₂²×C₁₃	1	2	yes	—
170	#493	C₂²×C₁₃	1	1	yes	characteristic
171	#493	C₂²×C₁₃	1	2	yes	—
172	#478	—	2	2	no	—
173	#476	—	2	1	no	—
174	#459	—	2	1	no	—
175	#459	—	2	2	no	—
176	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
177	C₄×D₈	C₂²×C₁₃	1	1	yes	characteristic
178	C₄×D₈	C₂²×C₁₃	1	2	yes	—
179	C₄×D₈	C₂²×C₁₃	1	1	yes	characteristic
180	C₄×D₈	C₂²×C₁₃	1	2	yes	—
181	#367	C₂²×C₁₃	1	2	yes	—
182	#366	C₂²×C₁₃	1	1	yes	characteristic
183	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
184	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
185	#357	C₂²×C₁₃	1	1	yes	characteristic
186	#357	C₂²×C₁₃	1	1	yes	characteristic
187	#357	C₂²×C₁₃	1	2	yes	—
188	#357	C₂²×C₁₃	1	2	yes	—
189	#357	C₂²×C₁₃	1	2	yes	—
190	#357	C₂²×C₁₃	1	2	yes	—
191	#346	C₂²×C₁₃	1	2	yes	—
192	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
193	C₂²×C₁₃	—	4	2	no	—
194	C₂²×C₁₃	—	8	1	no	—
195	C₂²×C₁₃	—	8	1	no	—
196	C₂²×C₁₃	—	8	2	no	—
197	C₂²×C₁₃	—	8	1	no	—
198	C₂²×C₁₃	—	8	2	no	—
199	C₂²×C₁₃	—	8	1	no	—
200	C₂²×C₁₃	—	8	1	no	—
201	C₂²×C₁₃	—	8	2	no	—
202	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
203	C₄×C₁₃	—	2	2	no	—
204	C₄×C₁₃	—	2	2	no	—
205	C₄×C₁₃	—	2	1	no	—
206	C₄×C₁₃	—	8	1	no	—
207	C₄×C₁₃	—	8	1	no	—
208	C₄×C₁₃	—	8	2	no	—
209	C₄×C₁₃	#509	1	2	yes	—
210	C₄×C₁₃	#548	1	1	yes	characteristic
211	C₄×C₁₃	#577	1	1	yes	characteristic
212	C₂×#41	—	4	1	no	—
213	C₂×D₈	—	2	2	no	—
214	C₂×D₈	—	2	1	no	—
215	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
216	C₂×D₈	C₂³×C₁₃	1	2	yes	—
217	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
218	C₄⋊Q₈	—	2	1	no	—
219	#127	—	2	2	no	—
220	#127	—	2	1	no	—
221	#125	—	2	2	no	—
222	C₂²⋊Q₈	—	4	1	no	—
223	C₂²⋊Q₈	—	4	2	no	—
224	#121	—	4	1	no	—
225	#121	—	4	2	no	—
226	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
227	C₄×D₄	—	2	2	no	—
228	C₄×D₄	—	2	1	no	—
229	C₄×D₄	—	2	1	no	—
230	C₄×D₄	—	2	2	no	—
231	C₂×C₄⋊C₄	—	4	1	no	—
232	D₁₆	—	2	2	no	—
233	D₁₆	—	2	4	no	—
234	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
235	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
236	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
237	#107	—	2	1	no	—
238	#107	C₂³×C₁₃	1	1	yes	characteristic
239	#107	C₂³×C₁₃	1	2	yes	—
240	#107	C₂³×C₁₃	1	1	yes	characteristic
241	#106	—	2	2	no	—
242	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
243	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
244	#102	—	2	2	no	—
245	#102	—	2	1	no	—
246	#102	—	2	2	no	—
247	#102	—	2	1	no	—
248	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
249	C₄×C₈	C₂³×C₁₃	1	2	yes	—
250	C₂×C₁₃	—	8	2	no	—
251	C₂×C₁₃	—	16	2	no	—
252	C₂×C₁₃	—	16	1	no	—
253	C₂×C₁₃	#3358	1	1	yes	characteristic
254	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
255	C₂×C₁₃	#3415	1	1	yes	characteristic
256	#41	—	8	4	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂²×C₄	—	4	1	no	—
267	C₂²×C₄	—	4	2	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	4	2	no	—
270	D₈	—	2	1	no	—
271	D₈	—	2	2	no	—
272	D₈	—	2	1	no	—
273	D₈	—	2	1	no	—
274	D₈	—	2	2	no	—
275	D₈	—	8	2	no	—
276	D₈	—	8	1	no	—
277	D₈	C₁₃×#41	1	2	yes	complemented
278	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
279	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
280	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
288	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	1	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₄²	C₂×D₄×C₁₃	1	2	yes	—
294	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
295	C₁₆	—	2	1	no	—
296	C₁₆	—	2	2	no	—
297	C₁₆	—	2	1	no	—
298	C₁₃	#33933	1	1	yes	characteristic, direct product factor, Sylow
299	C₂³	—	4	1	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	1	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	16	1	no	—
304	D₄	—	2	2	no	—
305	D₄	—	4	1	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	1	no	—
308	D₄	—	4	1	no	—
309	D₄	—	4	2	no	—
310	D₄	—	8	4	no	—
311	D₄	—	8	2	no	—
312	D₄	—	16	2	no	—
313	D₄	—	16	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	8	4	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	—	8	2	no	—
320	C₂×C₄	—	16	2	no	—
321	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
322	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
323	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
324	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
325	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	—	2	1	no	—
328	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
329	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
330	C₂²	—	4	2	no	—
331	C₂²	—	8	1	no	—
332	C₂²	—	8	1	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	1	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	8	2	no	—
339	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	8	1	no	—
344	C₄	—	8	1	no	—
345	C₄	—	8	2	no	—
346	C₄	C₁₃×#509	1	2	yes	—
347	C₄	C₁₃×#548	1	1	yes	characteristic
348	C₄	C₁₃×#577	1	1	yes	characteristic
349	C₂	—	8	2	no	—
350	C₂	—	16	2	no	—
351	C₂	—	16	1	no	—
352	C₂	C₁₃×#3358	1	1	yes	characteristic
353	C₂	C₂×C₁₃×#509	1	1	yes	characteristic
354	C₂	C₁₃×#3415	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×Q₈×C₁₃ ◁ C₈×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3391	C₂	semidirect
2. C₁₃×#3380	C₂	semidirect
3. C₁₃×#2244	C₂	semidirect
4. C₁₃×#2240	C₂	semidirect
5. C₁₃×#2228	C₂	semidirect
6. C₁₃×#2226	C₂	semidirect
7. C₁₃×#2206	C₂	semidirect
8. C₁₃×#2206	C₂	semidirect
9. C₁₃×#2183	C₂	semidirect
10. C₁₃×#2177	C₂	semidirect
11. C₄×C₁₃×D₁₆	C₂	semidirect
79. #33933	C₁₃	direct (N × H)
101. C₁₃×D₈	#41	semidirect
122. #3391	C₂×C₁₃	semidirect
123. #3380	C₂×C₁₃	semidirect
124. #2244	C₂×C₁₃	semidirect
125. #2240	C₂×C₁₃	semidirect
126. #2228	C₂×C₁₃	semidirect
127. #2226	C₂×C₁₃	semidirect
128. #2206	C₂×C₁₃	semidirect
129. #2206	C₂×C₁₃	semidirect
130. #2183	C₂×C₁₃	semidirect
131. #2177	C₂×C₁₃	semidirect
132. C₄×D₁₆	C₂×C₁₃	semidirect
277. D₈	C₁₃×#41	semidirect
298. C₁₃	#33933	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×#3391	2
C₁₃×#3380	2
C₁₃×#2244	2
C₁₃×#2240	2
C₁₃×#2228	2
C₁₃×#2226	2
C₁₃×#2206	2
C₁₃×#2183	2
C₁₃×#2177	2
C₄×C₁₃×D₁₆	2
#33933	13

Subgroups — C13×#33933 #15720272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33933 #15720272