Subgroups — C₁₃×#33951 #15720290

Order 3328.

498 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×#3380	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2255	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2254	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2228	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2226	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2226	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2224	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
11	C₄×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic
12	C₁₃×#554	—	2	1	no	—
13	D₄×Q₈×C₁₃	—	2	1	no	—
14	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
15	C₁₃×#500	C₂²	1	1	yes	characteristic
16	C₁₃×#500	C₂²	1	1	yes	characteristic
17	C₁₃×#500	C₂²	1	2	yes	—
18	C₁₃×#499	C₂²	1	2	yes	—
19	C₁₃×#478	—	2	2	no	—
20	C₁₃×#476	—	2	1	no	—
21	C₁₃×#474	—	2	1	no	—
22	C₁₃×#474	—	2	2	no	—
23	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
25	C₄×C₁₃×D₈	C₂²	1	2	yes	—
26	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
27	C₄×C₁₃×D₈	C₂²	1	2	yes	—
28	C₁₃×#367	C₂²	1	2	yes	—
29	C₁₃×#367	C₂²	1	2	yes	—
30	C₁₃×#367	C₂²	1	2	yes	—
31	C₁₃×#366	C₂²	1	1	yes	characteristic
32	C₁₃×#366	C₂²	1	2	yes	—
33	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
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×#346	C₂²	1	2	yes	—
40	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₁₃	—	4	1	no	—
42	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
43	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
44	C₁₃×C₄⋊Q₈	—	2	2	no	—
45	C₁₃×C₄⋊Q₈	—	2	1	no	—
46	C₁₃×C₄⋊Q₈	—	2	1	no	—
47	C₁₃×#125	—	2	2	no	—
48	C₁₃×C₂²⋊Q₈	—	4	1	no	—
49	C₁₃×C₂²⋊Q₈	—	4	1	no	—
50	C₁₃×C₂²⋊Q₈	—	4	2	no	—
51	C₁₃×C₂²⋊Q₈	—	4	2	no	—
52	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
53	C₄×D₄×C₁₃	—	2	2	no	—
54	C₄×D₄×C₁₃	—	2	1	no	—
55	C₄×D₄×C₁₃	—	2	1	no	—
56	C₄×D₄×C₁₃	—	2	2	no	—
57	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
58	C₁₃×D₁₆	Q₈	1	4	yes	complemented
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
61	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
62	C₁₃×#107	—	2	2	no	—
63	C₁₃×#107	—	2	1	no	—
64	C₁₃×#107	—	2	1	no	—
65	C₁₃×#107	C₂³	1	1	yes	characteristic
66	C₁₃×#107	C₂³	1	1	yes	characteristic
67	C₁₃×#107	C₂³	1	2	yes	—
68	C₁₃×#107	C₂³	1	2	yes	—
69	C₁₃×#106	—	2	2	no	—
70	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
71	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
72	C₁₃×#102	—	2	2	no	—
73	C₁₃×#102	—	2	1	no	—
74	C₁₃×#102	—	2	2	no	—
75	C₁₃×#102	—	2	1	no	—
76	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₁₃	C₂³	1	2	yes	—
78	#33951	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂×Q₈×C₁₃	—	4	2	no	—
80	C₂×Q₈×C₁₃	—	4	1	no	—
81	C₂×Q₈×C₁₃	—	4	1	no	—
82	C₂×Q₈×C₁₃	—	8	4	no	—
83	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
84	C₂×D₄×C₁₃	—	2	1	no	—
85	C₂×D₄×C₁₃	—	2	1	no	—
86	C₂²×C₄×C₁₃	—	4	1	no	—
87	C₂²×C₄×C₁₃	—	4	2	no	—
88	C₂²×C₄×C₁₃	—	4	1	no	—
89	C₂²×C₄×C₁₃	—	4	2	no	—
90	C₁₃×D₈	—	2	1	no	—
91	C₁₃×D₈	—	2	1	no	—
92	C₁₃×D₈	C₂×Q₈	1	2	yes	—
93	C₁₃×D₈	C₂×Q₈	1	2	yes	complemented
94	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	2	1	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	2	1	no	—
101	C₁₃×C₄⋊C₄	—	4	2	no	—
102	C₁₃×C₄⋊C₄	—	4	1	no	—
103	C₁₃×C₄⋊C₄	—	4	2	no	—
104	C₁₃×C₄⋊C₄	—	4	1	no	—
105	C₁₃×C₄⋊C₄	—	4	2	no	—
106	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
107	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
108	C₁₃×C₂²⋊C₄	—	4	1	no	—
109	C₁₃×C₂²⋊C₄	—	4	2	no	—
110	C₁₃×C₂²⋊C₄	—	4	1	no	—
111	C₁₃×C₂²⋊C₄	—	4	2	no	—
112	C₄²×C₁₃	C₂×D₄	1	2	yes	—
113	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₁₃×C₁₆	—	2	1	no	—
115	C₁₃×C₁₆	—	2	2	no	—
116	C₁₃×C₁₆	C₂×Q₈	1	2	yes	complemented
117	#3380	C₂×C₁₃	1	1	yes	characteristic, complemented
118	Q₈×D₈	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#2255	C₂×C₁₃	1	2	yes	complemented
120	#2254	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#2228	C₂×C₁₃	1	2	yes	complemented
122	#2226	C₂×C₁₃	1	2	yes	complemented
123	#2226	C₂×C₁₃	1	1	yes	characteristic, complemented
124	#2224	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#2183	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#2177	C₂×C₁₃	1	2	yes	—
127	C₄×D₁₆	C₂×C₁₃	1	1	yes	characteristic
128	C₂³×C₁₃	—	4	1	no	—
129	C₂³×C₁₃	—	4	1	no	—
130	Q₈×C₁₃	—	2	2	no	—
131	Q₈×C₁₃	—	8	4	no	—
132	Q₈×C₁₃	—	8	2	no	—
133	Q₈×C₁₃	—	16	1	no	—
134	D₄×C₁₃	—	2	2	no	—
135	D₄×C₁₃	—	2	2	no	—
136	D₄×C₁₃	—	4	1	no	—
137	D₄×C₁₃	—	4	1	no	—
138	C₂×C₄×C₁₃	—	4	1	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	4	1	no	—
142	C₂×C₄×C₁₃	—	8	4	no	—
143	C₂×C₄×C₁₃	—	8	2	no	—
144	C₂×C₄×C₁₃	—	8	2	no	—
145	C₂×C₄×C₁₃	—	16	2	no	—
146	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
149	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₁₃	—	2	2	no	—
152	C₈×C₁₃	—	2	1	no	—
153	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
154	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
155	#554	—	2	1	no	—
156	D₄×Q₈	—	2	1	no	—
157	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
158	#500	C₂²×C₁₃	1	1	yes	characteristic
159	#500	C₂²×C₁₃	1	2	yes	—
160	#500	C₂²×C₁₃	1	1	yes	characteristic
161	#499	C₂²×C₁₃	1	2	yes	—
162	#478	—	2	2	no	—
163	#476	—	2	1	no	—
164	#474	—	2	1	no	—
165	#474	—	2	2	no	—
166	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
167	C₄×D₈	C₂²×C₁₃	1	1	yes	characteristic
168	C₄×D₈	C₂²×C₁₃	1	2	yes	—
169	C₄×D₈	C₂²×C₁₃	1	1	yes	characteristic
170	C₄×D₈	C₂²×C₁₃	1	2	yes	—
171	#367	C₂²×C₁₃	1	2	yes	—
172	#367	C₂²×C₁₃	1	2	yes	—
173	#367	C₂²×C₁₃	1	2	yes	—
174	#366	C₂²×C₁₃	1	1	yes	characteristic
175	#366	C₂²×C₁₃	1	2	yes	—
176	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
177	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
178	#357	C₂²×C₁₃	1	1	yes	characteristic
179	#357	C₂²×C₁₃	1	1	yes	characteristic
180	#357	C₂²×C₁₃	1	2	yes	—
181	#357	C₂²×C₁₃	1	2	yes	—
182	#346	C₂²×C₁₃	1	2	yes	—
183	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
184	C₂²×C₁₃	—	4	2	no	—
185	C₂²×C₁₃	—	4	2	no	—
186	C₂²×C₁₃	—	8	1	no	—
187	C₂²×C₁₃	—	8	1	no	—
188	C₂²×C₁₃	—	8	1	no	—
189	C₂²×C₁₃	—	8	1	no	—
190	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	2	1	no	—
194	C₄×C₁₃	—	8	1	no	—
195	C₄×C₁₃	—	8	2	no	—
196	C₄×C₁₃	—	8	1	no	—
197	C₄×C₁₃	—	8	2	no	—
198	C₄×C₁₃	#509	1	2	yes	—
199	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
200	C₄×C₁₃	#578	1	1	yes	characteristic
201	C₂²×Q₈	—	4	1	no	—
202	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
203	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	—	2	1	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	#125	—	2	2	no	—
208	C₂²⋊Q₈	—	4	1	no	—
209	C₂²⋊Q₈	—	4	1	no	—
210	C₂²⋊Q₈	—	4	2	no	—
211	C₂²⋊Q₈	—	4	2	no	—
212	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
213	C₄×D₄	—	2	2	no	—
214	C₄×D₄	—	2	1	no	—
215	C₄×D₄	—	2	1	no	—
216	C₄×D₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	4	1	no	—
218	D₁₆	Q₈×C₁₃	1	4	yes	complemented
219	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
220	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
221	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
222	#107	—	2	2	no	—
223	#107	—	2	1	no	—
224	#107	—	2	1	no	—
225	#107	C₂³×C₁₃	1	1	yes	characteristic
226	#107	C₂³×C₁₃	1	1	yes	characteristic
227	#107	C₂³×C₁₃	1	2	yes	—
228	#107	C₂³×C₁₃	1	2	yes	—
229	#106	—	2	2	no	—
230	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
231	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
232	#102	—	2	2	no	—
233	#102	—	2	1	no	—
234	#102	—	2	2	no	—
235	#102	—	2	1	no	—
236	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
237	C₄×C₈	C₂³×C₁₃	1	2	yes	—
238	C₂×C₁₃	—	8	2	no	—
239	C₂×C₁₃	—	8	2	no	—
240	C₂×C₁₃	Q₈×D₈	1	1	yes	characteristic
241	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
242	C₂×C₁₃	#3416	1	1	yes	characteristic
243	C₂×Q₈	—	4	2	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×Q₈	—	8	4	no	—
247	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	—	4	2	no	—
254	D₈	—	2	1	no	—
255	D₈	—	2	1	no	—
256	D₈	C₂×Q₈×C₁₃	1	2	yes	—
257	D₈	C₂×Q₈×C₁₃	1	2	yes	complemented
258	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
259	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
271	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	1	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₄²	C₂×D₄×C₁₃	1	2	yes	—
277	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
278	C₁₆	—	2	1	no	—
279	C₁₆	—	2	2	no	—
280	C₁₆	C₂×Q₈×C₁₃	1	2	yes	complemented
281	C₁₃	#33951	1	1	yes	characteristic, direct product factor, Sylow
282	C₂³	—	4	1	no	—
283	C₂³	—	4	1	no	—
284	Q₈	—	2	2	no	—
285	Q₈	—	8	4	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	16	1	no	—
288	D₄	—	2	2	no	—
289	D₄	—	2	2	no	—
290	D₄	—	4	1	no	—
291	D₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	8	4	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	16	2	no	—
300	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
301	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
302	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
303	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	2	1	no	—
307	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
308	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
309	C₂²	—	4	2	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	8	1	no	—
312	C₂²	—	8	1	no	—
313	C₂²	—	8	1	no	—
314	C₂²	—	8	1	no	—
315	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	8	1	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	1	no	—
322	C₄	—	8	2	no	—
323	C₄	C₁₃×#509	1	2	yes	—
324	C₄	D₄×Q₈×C₁₃	1	1	yes	characteristic
325	C₄	C₁₃×#578	1	1	yes	characteristic
326	C₂	—	8	2	no	—
327	C₂	—	8	2	no	—
328	C₂	Q₈×C₁₃×D₈	1	1	yes	characteristic
329	C₂	C₂×C₁₃×#509	1	1	yes	characteristic
330	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	C₂×C₁₃×D₁₆	C₂²
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×#3380	C₂	semidirect
2. Q₈×C₁₃×D₈	C₂	semidirect
3. C₁₃×#2255	C₂	semidirect
4. C₁₃×#2254	C₂	semidirect
5. C₁₃×#2228	C₂	semidirect
6. C₁₃×#2226	C₂	semidirect
7. C₁₃×#2226	C₂	semidirect
8. C₁₃×#2224	C₂	semidirect
9. C₁₃×#2183	C₂	semidirect
58. C₁₃×D₁₆	Q₈	semidirect
78. #33951	C₁₃	direct (N × H)
93. C₁₃×D₈	C₂×Q₈	semidirect
116. C₁₃×C₁₆	C₂×Q₈	semidirect
117. #3380	C₂×C₁₃	semidirect
118. Q₈×D₈	C₂×C₁₃	semidirect
119. #2255	C₂×C₁₃	semidirect
120. #2254	C₂×C₁₃	semidirect
121. #2228	C₂×C₁₃	semidirect
122. #2226	C₂×C₁₃	semidirect
123. #2226	C₂×C₁₃	semidirect
124. #2224	C₂×C₁₃	semidirect
125. #2183	C₂×C₁₃	semidirect
218. D₁₆	Q₈×C₁₃	semidirect
257. D₈	C₂×Q₈×C₁₃	semidirect
280. C₁₆	C₂×Q₈×C₁₃	semidirect
281. C₁₃	#33951	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×#3380	2
Q₈×C₁₃×D₈	2
C₁₃×#2255	2
C₁₃×#2254	2
C₁₃×#2228	2
C₁₃×#2226	2
C₁₃×#2224	2
C₁₃×#2183	2
C₁₃×#2177	2
C₄×C₁₃×D₁₆	2
#33951	13

Subgroups — C13×#33951 #15720290

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33951 #15720290