Subgroups — C₁₃×#33672 #15720011

Order 3328.

586 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2232	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2227	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2207	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2185	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2175	C₂	1	4	yes	maximal, complemented
8	C₁₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₁₃×#532	—	2	1	no	—
11	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
12	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	—
13	C₁₃×#500	C₂²	1	1	yes	characteristic
14	C₁₃×#493	C₂²	1	1	yes	characteristic
15	C₁₃×#486	—	2	1	no	—
16	C₁₃×#466	—	2	1	no	—
17	C₁₃×#466	C₂²	1	2	yes	—
18	C₄×C₁₃×Q₁₆	—	2	1	no	—
19	C₄×C₁₃×Q₁₆	—	2	1	no	—
20	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
21	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
22	C₄×C₁₃×D₈	—	2	1	no	—
23	C₄×C₁₃×D₈	C₂²	1	2	yes	—
24	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
25	C₂×C₁₃×#103	—	2	1	no	—
26	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#366	C₂²	1	4	yes	—
28	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₁₃×#358	C₂²	1	4	yes	—
30	C₁₃×#357	C₂²	1	4	yes	—
31	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₁₃×#346	C₂²	1	2	yes	—
33	C₂×C₁₃×#41	—	4	1	no	—
34	C₂²×Q₈×C₁₃	—	2	1	no	—
35	C₂×C₁₃×Q₁₆	—	2	4	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×#127	—	2	1	no	—
44	C₁₃×#124	—	4	1	no	—
45	C₁₃×#121	—	2	2	no	—
46	C₁₃×#121	—	4	1	no	—
47	C₄×Q₈×C₁₃	—	2	2	no	—
48	C₄×Q₈×C₁₃	—	2	2	no	—
49	C₄×Q₈×C₁₃	—	4	1	no	—
50	C₄×Q₈×C₁₃	—	4	1	no	—
51	C₄×D₄×C₁₃	—	2	2	no	—
52	C₄×D₄×C₁₃	—	4	1	no	—
53	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
54	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
55	C₂×C₄²×C₁₃	—	4	1	no	—
56	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
57	C₁₃×SD₃₂	—	2	8	no	—
58	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
59	C₁₃×#107	—	2	1	no	—
60	C₁₃×#107	—	2	1	no	—
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×#107	C₂³	1	2	yes	—
63	C₁₃×#103	—	2	2	no	—
64	C₁₃×#103	—	2	2	no	—
65	C₁₃×#102	—	2	2	no	—
66	C₁₃×#102	—	2	2	no	—
67	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
68	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	#33672	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₁₃×#41	—	8	4	no	—
73	C₂×Q₈×C₁₃	—	2	4	no	—
74	C₂×Q₈×C₁₃	—	2	2	no	—
75	C₂×Q₈×C₁₃	—	4	2	no	—
76	C₂×Q₈×C₁₃	—	4	1	no	—
77	C₂×Q₈×C₁₃	—	4	1	no	—
78	C₂×D₄×C₁₃	—	2	2	no	—
79	C₂×D₄×C₁₃	—	4	2	no	—
80	C₂×D₄×C₁₃	—	4	1	no	—
81	C₂×D₄×C₁₃	—	4	1	no	—
82	C₂²×C₄×C₁₃	—	4	1	no	—
83	C₂²×C₄×C₁₃	—	4	1	no	—
84	C₂²×C₄×C₁₃	—	4	2	no	—
85	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₁₃×Q₁₆	—	2	4	no	—
89	C₁₃×Q₁₆	—	2	2	no	—
90	C₁₃×Q₁₆	#41	1	4	yes	complemented
91	C₁₃×D₈	—	2	4	no	—
92	C₁₃×D₈	—	4	2	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	4	1	no	—
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	2	2	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	C₄²×C₁₃	—	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	4	no	—
114	#3169	C₂×C₁₃	1	1	yes	characteristic
115	C₂×C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#2232	C₂×C₁₃	1	2	yes	complemented
117	#2227	C₂×C₁₃	1	2	yes	complemented
118	#2207	C₂×C₁₃	1	2	yes	complemented
119	#2185	C₂×C₁₃	1	2	yes	complemented
120	#2175	C₂×C₁₃	1	4	yes	complemented
121	#2163	C₂×C₁₃	1	1	yes	characteristic, complemented
122	C₂³×C₁₃	—	4	2	no	—
123	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
124	Q₈×C₁₃	—	2	4	no	—
125	Q₈×C₁₃	—	4	4	no	—
126	Q₈×C₁₃	—	4	2	no	—
127	Q₈×C₁₃	—	8	2	no	—
128	D₄×C₁₃	—	4	2	no	—
129	D₄×C₁₃	—	4	2	no	—
130	D₄×C₁₃	—	8	2	no	—
131	D₄×C₁₃	—	8	4	no	—
132	D₄×C₁₃	—	8	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	2	1	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	4	4	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	—	8	2	no	—
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	—	2	1	no	—
152	C₈×C₁₃	C₂×#41	1	2	yes	—
153	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
154	#532	—	2	1	no	—
155	C₂×C₄×Q₈	—	2	1	no	—
156	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	—
157	#500	C₂²×C₁₃	1	1	yes	characteristic
158	#493	C₂²×C₁₃	1	1	yes	characteristic
159	#486	—	2	1	no	—
160	#466	—	2	1	no	—
161	#466	C₂²×C₁₃	1	2	yes	—
162	C₄×Q₁₆	—	2	1	no	—
163	C₄×Q₁₆	—	2	1	no	—
164	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
165	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
166	C₄×D₈	—	2	1	no	—
167	C₄×D₈	C₂²×C₁₃	1	2	yes	—
168	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
169	C₂×#103	—	2	1	no	—
170	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
171	#366	C₂²×C₁₃	1	4	yes	—
172	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
173	#358	C₂²×C₁₃	1	4	yes	—
174	#357	C₂²×C₁₃	1	4	yes	—
175	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
176	#346	C₂²×C₁₃	1	2	yes	—
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	—	2	1	no	—
179	C₂²×C₁₃	—	8	2	no	—
180	C₂²×C₁₃	—	8	2	no	—
181	C₂²×C₁₃	—	8	2	no	—
182	C₂²×C₁₃	#509	1	2	yes	—
183	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	4	4	no	—
189	C₄×C₁₃	—	8	2	no	—
190	C₄×C₁₃	—	8	2	no	—
191	C₄×C₁₃	#510	1	2	yes	—
192	C₄×C₁₃	#525	1	1	yes	characteristic
193	C₄×C₁₃	#575	1	1	yes	characteristic
194	C₂×#41	—	4	1	no	—
195	C₂²×Q₈	—	2	1	no	—
196	C₂×Q₁₆	—	2	4	no	—
197	C₂×Q₁₆	—	2	1	no	—
198	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	C₂³×C₁₃	1	2	yes	—
201	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
203	C₄⋊Q₈	—	2	1	no	—
204	#127	—	2	1	no	—
205	#124	—	4	1	no	—
206	#121	—	2	2	no	—
207	#121	—	4	1	no	—
208	C₄×Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	4	1	no	—
211	C₄×Q₈	—	4	1	no	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	—	4	1	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	4	1	no	—
216	C₂×C₄²	—	4	1	no	—
217	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
218	SD₃₂	—	2	8	no	—
219	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
220	#107	—	2	1	no	—
221	#107	—	2	1	no	—
222	#107	C₂³×C₁₃	1	2	yes	—
223	#107	C₂³×C₁₃	1	2	yes	—
224	#103	—	2	2	no	—
225	#103	—	2	2	no	—
226	#102	—	2	2	no	—
227	#102	—	2	2	no	—
228	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
229	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
230	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
232	C₂×C₁₃	—	2	2	no	—
233	C₂×C₁₃	—	16	2	no	—
234	C₂×C₁₃	#3100	1	1	yes	characteristic
235	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
236	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
237	#41	—	8	4	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	4	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	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
251	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
253	Q₁₆	—	2	4	no	—
254	Q₁₆	—	2	2	no	—
255	Q₁₆	C₁₃×#41	1	4	yes	complemented
256	D₈	—	2	4	no	—
257	D₈	—	4	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
263	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₄²	—	4	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	C₁₃×D₈	1	1	yes	characteristic
275	C₄²	C₁₃×D₈	1	1	yes	characteristic
276	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
278	C₁₆	—	2	4	no	—
279	C₁₃	#33672	1	1	yes	characteristic, direct product factor, Sylow
280	C₂³	—	4	2	no	—
281	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
282	Q₈	—	2	4	no	—
283	Q₈	—	4	4	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	8	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	8	2	no	—
289	D₄	—	8	4	no	—
290	D₄	—	8	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	—	8	2	no	—
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	C₂×C₁₃×D₈	1	1	yes	characteristic
305	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	C₂×C₁₃×#41	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	C₁₃×#509	1	2	yes	—
317	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	4	4	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	C₁₃×#510	1	2	yes	—
326	C₄	C₁₃×#525	1	1	yes	characteristic
327	C₄	C₁₃×#575	1	1	yes	characteristic
328	C₂	—	2	2	no	—
329	C₂	—	16	2	no	—
330	C₂	C₁₃×#3100	1	1	yes	characteristic
331	C₂	C₂×C₁₃×#509	1	1	yes	characteristic
332	C₂	C₂×C₁₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
2. C₂×C₄×C₁₃×Q₁₆	C₂	semidirect
3. C₁₃×#2232	C₂	semidirect
4. C₁₃×#2227	C₂	semidirect
5. C₁₃×#2207	C₂	semidirect
6. C₁₃×#2185	C₂	semidirect
7. C₁₃×#2175	C₂	semidirect
8. C₁₃×#2163	C₂	semidirect
71. #33672	C₁₃	direct (N × H)
90. C₁₃×Q₁₆	#41	semidirect
115. C₂×C₄×Q₁₆	C₂×C₁₃	semidirect
116. #2232	C₂×C₁₃	semidirect
117. #2227	C₂×C₁₃	semidirect
118. #2207	C₂×C₁₃	semidirect
119. #2185	C₂×C₁₃	semidirect
120. #2175	C₂×C₁₃	semidirect
121. #2163	C₂×C₁₃	semidirect
255. Q₁₆	C₁₃×#41	semidirect
279. C₁₃	#33672	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×#3169	2
C₂×C₄×C₁₃×Q₁₆	2
C₁₃×#2232	2
C₁₃×#2227	2
C₁₃×#2207	2
C₁₃×#2185	2
C₁₃×#2175	2
C₁₃×#2163	2
#33672	13

Subgroups — C13×#33672 #15720011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33672 #15720011