Subgroups — C₁₃×#33940 #15720279

Order 3328.

520 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3396	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2243	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2226	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2225	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2210	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2206	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#2177	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#559	—	2	1	no	—
13	D₄×Q₈×C₁₃	—	2	1	no	—
14	C₂×C₁₃×SD₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
16	C₁₃×#500	C₂²	1	1	yes	characteristic
17	C₁₃×#500	C₂²	1	2	yes	—
18	C₁₃×#495	C₂²	1	2	yes	—
19	C₁₃×#493	C₂²	1	1	yes	characteristic
20	C₁₃×#474	—	2	1	no	—
21	C₁₃×#474	—	2	2	no	—
22	C₁₃×#463	—	2	2	no	—
23	C₁₃×#460	—	2	1	no	—
24	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
26	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
27	C₄×C₁₃×D₈	C₂²	1	2	yes	—
28	C₄×C₁₃×D₈	C₂²	1	2	yes	—
29	C₁₃×#367	C₂²	1	1	yes	characteristic
30	C₁₃×#367	C₂²	1	2	yes	—
31	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₁₃×#358	C₂²	1	1	yes	characteristic
34	C₁₃×#358	C₂²	1	2	yes	—
35	C₁₃×#357	C₂²	1	2	yes	—
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	C₂²×Q₈×C₁₃	—	4	1	no	—
42	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
43	C₂×C₁₃×SD₁₆	—	2	2	no	—
44	C₂×C₁₃×D₈	—	2	1	no	—
45	C₂×C₁₃×D₈	C₂³	1	2	yes	—
46	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
47	C₁₃×C₄⋊Q₈	—	2	2	no	—
48	C₁₃×C₄⋊Q₈	—	2	1	no	—
49	C₁₃×#127	—	2	1	no	—
50	C₁₃×#124	—	2	2	no	—
51	C₁₃×#124	—	4	1	no	—
52	C₁₃×#123	—	4	2	no	—
53	C₁₃×C₂²⋊Q₈	—	4	1	no	—
54	C₁₃×C₂²⋊Q₈	—	4	2	no	—
55	C₄×Q₈×C₁₃	—	2	1	no	—
56	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
57	C₄×D₄×C₁₃	—	2	2	no	—
58	C₄×D₄×C₁₃	—	2	2	no	—
59	C₄×D₄×C₁₃	—	2	1	no	—
60	C₄×D₄×C₁₃	—	4	1	no	—
61	C₁₃×SD₃₂	—	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	2	no	—
67	C₁₃×#107	—	2	1	no	—
68	C₁₃×#107	C₂³	1	2	yes	—
69	C₁₃×#107	C₂³	1	1	yes	characteristic
70	C₁₃×#107	C₂³	1	1	yes	characteristic
71	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
72	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
73	C₁₃×#103	—	2	1	no	—
74	C₁₃×#102	—	2	2	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	#33940	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×Q₈×C₁₃	—	2	1	no	—
81	C₂×Q₈×C₁₃	—	4	2	no	—
82	C₂×Q₈×C₁₃	—	4	1	no	—
83	C₂×Q₈×C₁₃	—	8	4	no	—
84	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
85	C₂×D₄×C₁₃	—	2	1	no	—
86	C₂×D₄×C₁₃	—	2	2	no	—
87	C₂×D₄×C₁₃	—	4	1	no	—
88	C₂×D₄×C₁₃	—	4	1	no	—
89	C₂²×C₄×C₁₃	—	4	2	no	—
90	C₂²×C₄×C₁₃	—	4	1	no	—
91	C₂²×C₄×C₁₃	—	4	2	no	—
92	C₁₃×Q₁₆	—	2	1	no	—
93	C₁₃×Q₁₆	—	2	1	no	—
94	C₁₃×SD₁₆	—	8	2	no	—
95	C₁₃×D₈	—	2	2	no	—
96	C₁₃×D₈	—	2	1	no	—
97	C₁₃×D₈	—	2	2	no	—
98	C₁₃×D₈	—	8	1	no	—
99	C₁₃×D₈	#41	1	2	yes	—
100	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
103	C₁₃×C₄⋊C₄	—	2	1	no	—
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	2	1	no	—
106	C₁₃×C₄⋊C₄	—	4	1	no	—
107	C₁₃×C₄⋊C₄	—	4	2	no	—
108	C₁₃×C₄⋊C₄	—	4	1	no	—
109	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
110	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
111	C₁₃×C₂²⋊C₄	—	4	2	no	—
112	C₁₃×C₂²⋊C₄	—	4	1	no	—
113	C₁₃×C₂²⋊C₄	—	4	2	no	—
114	C₁₃×C₂²⋊C₄	—	4	2	no	—
115	C₁₃×C₂²⋊C₄	—	4	2	no	—
116	C₄²×C₁₃	—	4	1	no	—
117	C₄²×C₁₃	C₂×D₄	1	2	yes	—
118	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₁₃×C₁₆	—	2	2	no	—
120	C₁₃×C₁₆	—	2	1	no	—
121	C₁₃×C₁₆	—	2	1	no	—
122	#3396	C₂×C₁₃	1	1	yes	characteristic, complemented
123	Q₈×D₈	C₂×C₁₃	1	1	yes	characteristic, complemented
124	#2243	C₂×C₁₃	1	2	yes	complemented
125	#2240	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#2226	C₂×C₁₃	1	2	yes	complemented
127	#2225	C₂×C₁₃	1	1	yes	characteristic, complemented
128	#2210	C₂×C₁₃	1	2	yes	complemented
129	#2206	C₂×C₁₃	1	1	yes	characteristic
130	#2183	C₂×C₁₃	1	1	yes	characteristic, complemented
131	#2177	C₂×C₁₃	1	2	yes	complemented
132	C₄×SD₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
133	C₂³×C₁₃	—	4	1	no	—
134	C₂³×C₁₃	—	4	2	no	—
135	Q₈×C₁₃	—	2	2	no	—
136	Q₈×C₁₃	—	4	1	no	—
137	Q₈×C₁₃	—	4	1	no	—
138	Q₈×C₁₃	—	8	4	no	—
139	Q₈×C₁₃	—	8	2	no	—
140	D₄×C₁₃	—	2	2	no	—
141	D₄×C₁₃	—	4	2	no	—
142	D₄×C₁₃	—	4	1	no	—
143	D₄×C₁₃	—	4	2	no	—
144	D₄×C₁₃	—	8	2	no	—
145	D₄×C₁₃	—	16	1	no	—
146	C₂×C₄×C₁₃	—	4	1	no	—
147	C₂×C₄×C₁₃	—	4	1	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	4	1	no	—
150	C₂×C₄×C₁₃	—	8	4	no	—
151	C₂×C₄×C₁₃	—	8	2	no	—
152	C₂×C₄×C₁₃	—	16	2	no	—
153	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
154	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
155	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
158	C₈×C₁₃	—	2	1	no	—
159	C₈×C₁₃	—	2	2	no	—
160	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
161	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
162	#559	—	2	1	no	—
163	D₄×Q₈	—	2	1	no	—
164	C₂×SD₃₂	C₂²×C₁₃	1	1	yes	characteristic
165	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
166	#500	C₂²×C₁₃	1	1	yes	characteristic
167	#500	C₂²×C₁₃	1	2	yes	—
168	#495	C₂²×C₁₃	1	2	yes	—
169	#493	C₂²×C₁₃	1	1	yes	characteristic
170	#474	—	2	1	no	—
171	#474	—	2	2	no	—
172	#463	—	2	2	no	—
173	#460	—	2	1	no	—
174	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
175	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
176	C₄×D₈	C₂²×C₁₃	1	1	yes	characteristic
177	C₄×D₈	C₂²×C₁₃	1	2	yes	—
178	C₄×D₈	C₂²×C₁₃	1	2	yes	—
179	#367	C₂²×C₁₃	1	1	yes	characteristic
180	#367	C₂²×C₁₃	1	2	yes	—
181	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
182	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
183	#358	C₂²×C₁₃	1	1	yes	characteristic
184	#358	C₂²×C₁₃	1	2	yes	—
185	#357	C₂²×C₁₃	1	1	yes	characteristic
186	#357	C₂²×C₁₃	1	2	yes	—
187	#357	C₂²×C₁₃	1	2	yes	—
188	#357	C₂²×C₁₃	1	2	yes	—
189	#346	C₂²×C₁₃	1	2	yes	—
190	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
191	C₂²×C₁₃	—	4	2	no	—
192	C₂²×C₁₃	—	8	1	no	—
193	C₂²×C₁₃	—	8	1	no	—
194	C₂²×C₁₃	—	8	2	no	—
195	C₂²×C₁₃	—	8	2	no	—
196	C₂²×C₁₃	—	8	2	no	—
197	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
198	C₄×C₁₃	—	2	2	no	—
199	C₄×C₁₃	—	2	2	no	—
200	C₄×C₁₃	—	2	1	no	—
201	C₄×C₁₃	—	8	1	no	—
202	C₄×C₁₃	—	8	2	no	—
203	C₄×C₁₃	—	8	1	no	—
204	C₄×C₁₃	—	8	1	no	—
205	C₄×C₁₃	#509	1	2	yes	—
206	C₄×C₁₃	#548	1	1	yes	characteristic
207	C₄×C₁₃	#577	1	1	yes	characteristic
208	C₂²×Q₈	—	4	1	no	—
209	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
210	C₂×SD₁₆	—	2	2	no	—
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	C₂³×C₁₃	1	2	yes	—
213	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
214	C₄⋊Q₈	—	2	2	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	#127	—	2	1	no	—
217	#124	—	2	2	no	—
218	#124	—	4	1	no	—
219	#123	—	4	2	no	—
220	C₂²⋊Q₈	—	4	1	no	—
221	C₂²⋊Q₈	—	4	2	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
224	C₄×D₄	—	2	2	no	—
225	C₄×D₄	—	2	1	no	—
226	C₄×D₄	—	2	2	no	—
227	C₄×D₄	—	4	1	no	—
228	SD₃₂	—	2	2	no	—
229	D₁₆	—	2	4	no	—
230	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
231	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
232	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
233	#107	—	2	2	no	—
234	#107	—	2	1	no	—
235	#107	C₂³×C₁₃	1	2	yes	—
236	#107	C₂³×C₁₃	1	1	yes	characteristic
237	#107	C₂³×C₁₃	1	1	yes	characteristic
238	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
239	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
240	#103	—	2	1	no	—
241	#102	—	2	2	no	—
242	#102	—	2	2	no	—
243	#102	—	2	1	no	—
244	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
245	C₄×C₈	C₂³×C₁₃	1	2	yes	—
246	C₂×C₁₃	—	8	2	no	—
247	C₂×C₁₃	—	16	2	no	—
248	C₂×C₁₃	#3358	1	1	yes	characteristic
249	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
250	C₂×C₁₃	#3416	1	1	yes	characteristic
251	C₂×Q₈	—	2	1	no	—
252	C₂×Q₈	—	4	2	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×Q₈	—	8	4	no	—
255	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	4	2	no	—
263	Q₁₆	—	2	1	no	—
264	Q₁₆	—	2	1	no	—
265	SD₁₆	—	8	2	no	—
266	D₈	—	2	2	no	—
267	D₈	—	2	1	no	—
268	D₈	—	2	2	no	—
269	D₈	—	8	1	no	—
270	D₈	C₁₃×#41	1	2	yes	—
271	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
272	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
273	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
281	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₄²	—	4	1	no	—
288	C₄²	C₂×D₄×C₁₃	1	2	yes	—
289	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
290	C₁₆	—	2	2	no	—
291	C₁₆	—	2	1	no	—
292	C₁₆	—	2	1	no	—
293	C₁₃	#33940	1	1	yes	characteristic, direct product factor, Sylow
294	C₂³	—	4	1	no	—
295	C₂³	—	4	2	no	—
296	Q₈	—	2	2	no	—
297	Q₈	—	4	1	no	—
298	Q₈	—	4	1	no	—
299	Q₈	—	8	4	no	—
300	Q₈	—	8	2	no	—
301	D₄	—	2	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	1	no	—
304	D₄	—	4	2	no	—
305	D₄	—	8	2	no	—
306	D₄	—	16	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	16	2	no	—
314	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
315	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
316	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
317	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	2	2	no	—
321	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
322	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
323	C₂²	—	4	2	no	—
324	C₂²	—	8	1	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	1	no	—
337	C₄	C₁₃×#509	1	2	yes	—
338	C₄	C₁₃×#548	1	1	yes	characteristic
339	C₄	C₁₃×#577	1	1	yes	characteristic
340	C₂	—	8	2	no	—
341	C₂	—	16	2	no	—
342	C₂	C₁₃×#3358	1	1	yes	characteristic
343	C₂	C₂×C₁₃×#509	1	1	yes	characteristic
344	C₂	C₁₃×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#2206	C₂
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×#3396	C₂	semidirect
2. Q₈×C₁₃×D₈	C₂	semidirect
3. C₁₃×#2243	C₂	semidirect
4. C₁₃×#2240	C₂	semidirect
5. C₁₃×#2226	C₂	semidirect
6. C₁₃×#2225	C₂	semidirect
7. C₁₃×#2210	C₂	semidirect
9. C₁₃×#2183	C₂	semidirect
10. C₁₃×#2177	C₂	semidirect
11. C₄×C₁₃×SD₃₂	C₂	semidirect
79. #33940	C₁₃	direct (N × H)
122. #3396	C₂×C₁₃	semidirect
123. Q₈×D₈	C₂×C₁₃	semidirect
124. #2243	C₂×C₁₃	semidirect
125. #2240	C₂×C₁₃	semidirect
126. #2226	C₂×C₁₃	semidirect
127. #2225	C₂×C₁₃	semidirect
128. #2210	C₂×C₁₃	semidirect
130. #2183	C₂×C₁₃	semidirect
131. #2177	C₂×C₁₃	semidirect
132. C₄×SD₃₂	C₂×C₁₃	semidirect
293. C₁₃	#33940	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×#3396	2
Q₈×C₁₃×D₈	2
C₁₃×#2243	2
C₁₃×#2240	2
C₁₃×#2226	2
C₁₃×#2225	2
C₁₃×#2210	2
C₁₃×#2206	2
C₁₃×#2183	2
C₁₃×#2177	2
C₄×C₁₃×SD₃₂	2
#33940	13

Subgroups — C13×#33940 #15720279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33940 #15720279