Subgroups — C₁₃×#3371 #13112429

Order 1664.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#559	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#490	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#489	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#488	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#479	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#478	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#464	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#464	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#460	C₂	1	1	yes	maximal, characteristic
12	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
13	C₁₃×#441	C₂	1	1	yes	maximal, characteristic, complemented
14	C₁₃×#440	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
19	C₁₃×C₄⋊Q₈	—	2	1	no	—
20	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₁₃×#127	C₂²	1	1	yes	characteristic
22	C₁₃×#125	—	2	1	no	—
23	C₁₃×#125	—	2	1	no	—
24	C₁₃×#125	C₂²	1	1	yes	characteristic
25	C₁₃×#125	C₂²	1	1	yes	characteristic
26	C₁₃×#124	—	2	1	no	—
27	C₁₃×#124	C₂²	1	1	yes	characteristic
28	C₁₃×#124	C₂²	1	1	yes	characteristic
29	C₁₃×#123	—	2	1	no	—
30	C₁₃×#123	—	2	1	no	—
31	C₄×Q₈×C₁₃	—	2	1	no	—
32	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
33	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
37	C₄×D₄×C₁₃	—	2	1	no	—
38	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
40	C₁₃×#107	C₂²	1	1	yes	characteristic
41	C₁₃×#106	C₂²	1	1	yes	characteristic
42	C₁₃×#106	C₂²	1	1	yes	characteristic
43	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
45	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₁₃×#103	C₂²	1	1	yes	characteristic
47	C₁₃×#103	C₂²	1	1	yes	characteristic
48	C₁₃×#103	C₂²	1	1	yes	characteristic
49	C₁₃×#103	C₂²	1	1	yes	characteristic
50	C₁₃×#102	C₂²	1	1	yes	characteristic
51	C₁₃×#102	C₂²	1	1	yes	characteristic
52	C₁₃×#102	C₂²	1	1	yes	characteristic
53	C₁₃×#102	C₂²	1	1	yes	characteristic
54	C₁₃×#102	C₂²	1	1	yes	characteristic
55	C₁₃×#102	C₂²	1	1	yes	characteristic
56	C₁₃×#97	C₂²	1	1	yes	characteristic
57	C₁₃×#97	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	—	2	1	no	—
60	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
61	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₁₃	—	2	1	no	—
64	C₂×D₄×C₁₃	—	2	1	no	—
65	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	—	2	1	no	—
68	C₂²×C₄×C₁₃	—	2	1	no	—
69	C₁₃×Q₁₆	—	2	2	no	—
70	C₁₃×SD₁₆	—	2	2	no	—
71	C₁₃×SD₁₆	—	2	2	no	—
72	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	C₁₃×C₄⋊C₄	—	2	1	no	—
77	C₁₃×C₄⋊C₄	—	2	1	no	—
78	C₁₃×C₄⋊C₄	—	2	1	no	—
79	C₁₃×C₄⋊C₄	—	2	1	no	—
80	C₁₃×C₄⋊C₄	—	2	1	no	—
81	C₁₃×C₄⋊C₄	—	2	1	no	—
82	C₁₃×C₄⋊C₄	—	2	1	no	—
83	C₁₃×C₄⋊C₄	—	2	1	no	—
84	C₁₃×C₄⋊C₄	—	2	1	no	—
85	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
86	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
87	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
88	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
89	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
91	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
92	C₁₃×C₂²⋊C₄	—	2	1	no	—
93	C₁₃×C₂²⋊C₄	—	2	1	no	—
94	C₁₃×C₂²⋊C₄	—	2	1	no	—
95	C₁₃×C₂²⋊C₄	—	2	1	no	—
96	C₁₃×C₂²⋊C₄	—	2	1	no	—
97	C₁₃×C₂²⋊C₄	—	2	1	no	—
98	C₄²×C₁₃	—	2	1	no	—
99	C₄²×C₁₃	—	2	1	no	—
100	C₄²×C₁₃	—	2	1	no	—
101	C₄²×C₁₃	—	2	1	no	—
102	C₄²×C₁₃	C₂³	1	1	yes	characteristic
103	C₄²×C₁₃	C₂³	1	1	yes	characteristic
104	C₄²×C₁₃	C₂³	1	1	yes	characteristic
105	#3371	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₁₃	—	2	1	no	—
107	C₂³×C₁₃	—	2	1	no	—
108	Q₈×C₁₃	—	2	1	no	—
109	Q₈×C₁₃	—	2	1	no	—
110	Q₈×C₁₃	—	2	1	no	—
111	Q₈×C₁₃	—	4	1	no	—
112	Q₈×C₁₃	—	4	2	no	—
113	Q₈×C₁₃	#41	1	2	yes	—
114	D₄×C₁₃	—	2	1	no	—
115	D₄×C₁₃	—	2	1	no	—
116	D₄×C₁₃	—	2	1	no	—
117	D₄×C₁₃	—	2	1	no	—
118	D₄×C₁₃	—	4	2	no	—
119	D₄×C₁₃	—	8	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	8	1	no	—
127	C₂×C₄×C₁₃	—	8	1	no	—
128	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
135	C₈×C₁₃	—	2	1	no	—
136	C₈×C₁₃	—	2	1	no	—
137	C₈×C₁₃	—	2	1	no	—
138	C₈×C₁₃	—	2	1	no	—
139	#559	C₂×C₁₃	1	1	yes	characteristic
140	#557	C₂×C₁₃	1	1	yes	characteristic, complemented
141	#490	C₂×C₁₃	1	1	yes	characteristic
142	#489	C₂×C₁₃	1	1	yes	characteristic, complemented
143	#488	C₂×C₁₃	1	1	yes	characteristic, complemented
144	#479	C₂×C₁₃	1	1	yes	characteristic, complemented
145	#478	C₂×C₁₃	1	1	yes	characteristic, complemented
146	#476	C₂×C₁₃	1	1	yes	characteristic, complemented
147	#464	C₂×C₁₃	1	1	yes	characteristic, complemented
148	#464	C₂×C₁₃	1	1	yes	characteristic, complemented
149	#460	C₂×C₁₃	1	1	yes	characteristic
150	#446	C₂×C₁₃	1	1	yes	characteristic, complemented
151	#441	C₂×C₁₃	1	1	yes	characteristic, complemented
152	#440	C₂×C₁₃	1	1	yes	characteristic, complemented
153	C₄×SD₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
154	C₂²×C₁₃	—	4	1	no	—
155	C₂²×C₁₃	—	4	1	no	—
156	C₂²×C₁₃	—	4	1	no	—
157	C₂²×C₁₃	—	4	1	no	—
158	C₂²×C₁₃	—	4	1	no	—
159	C₂²×C₁₃	—	4	1	no	—
160	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
161	C₄×C₁₃	—	2	2	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	—	4	1	no	—
169	C₄×C₁₃	—	4	1	no	—
170	C₄×C₁₃	—	4	1	no	—
171	C₄×C₁₃	—	4	1	no	—
172	C₄×C₁₃	—	4	1	no	—
173	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
174	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
175	C₂×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
176	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
177	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
178	C₄⋊Q₈	—	2	1	no	—
179	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
180	#127	C₂²×C₁₃	1	1	yes	characteristic
181	#125	—	2	1	no	—
182	#125	—	2	1	no	—
183	#125	C₂²×C₁₃	1	1	yes	characteristic
184	#125	C₂²×C₁₃	1	1	yes	characteristic
185	#124	—	2	1	no	—
186	#124	C₂²×C₁₃	1	1	yes	characteristic
187	#124	C₂²×C₁₃	1	1	yes	characteristic
188	#123	—	2	1	no	—
189	#123	—	2	1	no	—
190	C₄×Q₈	—	2	1	no	—
191	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
192	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
193	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
194	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
195	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
196	C₄×D₄	—	2	1	no	—
197	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
198	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
199	#107	C₂²×C₁₃	1	1	yes	characteristic
200	#106	C₂²×C₁₃	1	1	yes	characteristic
201	#106	C₂²×C₁₃	1	1	yes	characteristic
202	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
203	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
204	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
205	#103	C₂²×C₁₃	1	1	yes	characteristic
206	#103	C₂²×C₁₃	1	1	yes	characteristic
207	#103	C₂²×C₁₃	1	1	yes	characteristic
208	#103	C₂²×C₁₃	1	1	yes	characteristic
209	#102	C₂²×C₁₃	1	1	yes	characteristic
210	#102	C₂²×C₁₃	1	1	yes	characteristic
211	#102	C₂²×C₁₃	1	1	yes	characteristic
212	#102	C₂²×C₁₃	1	1	yes	characteristic
213	#102	C₂²×C₁₃	1	1	yes	characteristic
214	#102	C₂²×C₁₃	1	1	yes	characteristic
215	#97	C₂²×C₁₃	1	1	yes	characteristic
216	#97	C₂²×C₁₃	1	1	yes	characteristic
217	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
218	C₂×C₁₃	—	8	1	no	—
219	C₂×C₁₃	—	8	1	no	—
220	C₂×C₁₃	#548	1	1	yes	characteristic
221	C₂×C₁₃	#575	1	1	yes	characteristic
222	C₂×C₁₃	#577	1	1	yes	characteristic
223	C₂×Q₈	—	2	1	no	—
224	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
225	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
226	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
230	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	Q₁₆	—	2	2	no	—
234	SD₁₆	—	2	2	no	—
235	SD₁₆	—	2	2	no	—
236	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
238	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
239	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
250	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
251	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
252	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
253	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
254	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
255	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
256	C₂²⋊C₄	—	2	1	no	—
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	C₂³×C₁₃	1	1	yes	characteristic
267	C₄²	C₂³×C₁₃	1	1	yes	characteristic
268	C₄²	C₂³×C₁₃	1	1	yes	characteristic
269	C₁₃	#3371	1	1	yes	characteristic, direct product factor, Sylow
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	4	1	no	—
276	Q₈	—	4	2	no	—
277	Q₈	C₁₃×#41	1	2	yes	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	2	1	no	—
281	D₄	—	2	1	no	—
282	D₄	—	4	2	no	—
283	D₄	—	8	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
294	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
298	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
323	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
324	C₂	—	8	1	no	—
325	C₂	—	8	1	no	—
326	C₂	C₁₃×#548	1	1	yes	characteristic
327	C₂	C₁₃×#575	1	1	yes	characteristic
328	C₂	C₁₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#557	C₂	semidirect
4. C₁₃×#489	C₂	semidirect
5. C₁₃×#488	C₂	semidirect
6. C₁₃×#479	C₂	semidirect
7. C₁₃×#478	C₂	semidirect
8. C₁₃×#476	C₂	semidirect
9. C₁₃×#464	C₂	semidirect
10. C₁₃×#464	C₂	semidirect
12. C₁₃×#446	C₂	semidirect
13. C₁₃×#441	C₂	semidirect
14. C₁₃×#440	C₂	semidirect
15. C₄×C₁₃×SD₁₆	C₂	semidirect
105. #3371	C₁₃	direct (N × H)
140. #557	C₂×C₁₃	semidirect
142. #489	C₂×C₁₃	semidirect
143. #488	C₂×C₁₃	semidirect
144. #479	C₂×C₁₃	semidirect
145. #478	C₂×C₁₃	semidirect
146. #476	C₂×C₁₃	semidirect
147. #464	C₂×C₁₃	semidirect
148. #464	C₂×C₁₃	semidirect
150. #446	C₂×C₁₃	semidirect
151. #441	C₂×C₁₃	semidirect
152. #440	C₂×C₁₃	semidirect
153. C₄×SD₁₆	C₂×C₁₃	semidirect
269. C₁₃	#3371	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×#559	2
C₁₃×#557	2
C₁₃×#490	2
C₁₃×#489	2
C₁₃×#488	2
C₁₃×#479	2
C₁₃×#478	2
C₁₃×#476	2
C₁₃×#464	2
C₁₃×#460	2
C₁₃×#446	2
C₁₃×#441	2
C₁₃×#440	2
C₄×C₁₃×SD₁₆	2
#3371	13

Subgroups — C13×#3371 #13112429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3371 #13112429