Subgroups — C₁₃×#3344 #13112402

Order 1664.

344 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#554	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#552	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#501	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#485	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#484	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#483	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#483	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#479	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#470	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#441	C₂	1	1	yes	maximal, characteristic, complemented
13	C₁₃×#435	C₂	1	1	yes	maximal, characteristic
14	C₁₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₁₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₁₃×#126	—	2	1	no	—
21	C₁₃×#125	—	2	1	no	—
22	C₁₃×#125	—	2	1	no	—
23	C₁₃×#125	C₂²	1	1	yes	characteristic
24	C₁₃×#123	—	2	1	no	—
25	C₁₃×C₂²⋊Q₈	—	2	1	no	—
26	C₁₃×C₂²⋊Q₈	—	2	1	no	—
27	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
28	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
29	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
30	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
31	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
32	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
33	C₄×D₄×C₁₃	—	2	1	no	—
34	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
35	C₁₃×#117	—	2	1	no	—
36	C₁₃×#117	C₂²	1	1	yes	characteristic
37	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
38	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
39	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
40	C₁₃×#107	C₂²	1	1	yes	characteristic
41	C₁₃×#107	C₂²	1	1	yes	characteristic
42	C₁₃×#107	C₂²	1	1	yes	characteristic
43	C₁₃×#106	C₂²	1	1	yes	characteristic
44	C₁₃×#106	C₂²	1	1	yes	characteristic
45	C₁₃×#106	C₂²	1	1	yes	characteristic
46	C₁₃×#106	C₂²	1	1	yes	characteristic
47	C₁₃×#106	C₂²	1	1	yes	characteristic
48	C₁₃×#106	C₂²	1	1	yes	characteristic
49	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
50	C₁₃×#103	C₂²	1	1	yes	characteristic
51	C₁₃×#103	C₂²	1	1	yes	characteristic
52	C₁₃×#103	C₂²	1	1	yes	characteristic
53	C₁₃×#103	C₂²	1	1	yes	characteristic
54	C₁₃×#103	C₂²	1	1	yes	characteristic
55	C₁₃×#103	C₂²	1	1	yes	characteristic
56	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
57	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
58	C₁₃×#97	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	—	2	1	no	—
60	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	—	2	1	no	—
64	C₂²×C₄×C₁₃	—	2	1	no	—
65	C₂²×C₄×C₁₃	—	2	1	no	—
66	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
68	C₁₃×Q₁₆	—	2	2	no	—
69	C₁₃×M₁₆	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
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	C₁₃×C₄⋊C₄	—	2	1	no	—
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	—	2	1	no	—
86	C₁₃×C₄⋊C₄	—	2	1	no	—
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×C₄⋊C₄	C₂³	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
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	C₁₃×C₂²⋊C₄	—	2	1	no	—
99	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
100	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
101	C₄²×C₁₃	—	2	1	no	—
102	C₄²×C₁₃	—	2	1	no	—
103	C₄²×C₁₃	—	2	1	no	—
104	C₄²×C₁₃	C₂³	1	1	yes	characteristic
105	#3344	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
108	Q₈×C₁₃	—	2	1	no	—
109	Q₈×C₁₃	—	2	1	no	—
110	Q₈×C₁₃	—	2	1	no	—
111	Q₈×C₁₃	—	2	1	no	—
112	Q₈×C₁₃	—	8	1	no	—
113	D₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	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	—	4	2	no	—
127	C₂×C₄×C₁₃	—	8	1	no	—
128	C₂×C₄×C₁₃	—	8	1	no	—
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	C₂⁴	1	1	yes	characteristic
134	C₈×C₁₃	—	2	1	no	—
135	C₈×C₁₃	—	2	1	no	—
136	C₈×C₁₃	—	2	1	no	—
137	C₈×C₁₃	#41	1	2	yes	—
138	#554	C₂×C₁₃	1	1	yes	characteristic
139	#552	C₂×C₁₃	1	1	yes	characteristic
140	#501	C₂×C₁₃	1	1	yes	characteristic, complemented
141	#485	C₂×C₁₃	1	1	yes	characteristic, complemented
142	#484	C₂×C₁₃	1	1	yes	characteristic, complemented
143	#483	C₂×C₁₃	1	1	yes	characteristic, complemented
144	#483	C₂×C₁₃	1	1	yes	characteristic, complemented
145	#479	C₂×C₁₃	1	1	yes	characteristic, complemented
146	#475	C₂×C₁₃	1	1	yes	characteristic, complemented
147	#470	C₂×C₁₃	1	1	yes	characteristic, complemented
148	#467	C₂×C₁₃	1	1	yes	characteristic, complemented
149	#441	C₂×C₁₃	1	1	yes	characteristic, complemented
150	#435	C₂×C₁₃	1	1	yes	characteristic
151	#428	C₂×C₁₃	1	1	yes	characteristic, complemented
152	C₂×#106	C₂×C₁₃	1	1	yes	characteristic, complemented
153	C₂²×C₁₃	—	2	1	no	—
154	C₂²×C₁₃	—	2	1	no	—
155	C₂²×C₁₃	—	2	1	no	—
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	#137	1	2	yes	—
159	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	4	1	no	—
165	C₄×C₁₃	—	4	1	no	—
166	C₄×C₁₃	—	4	1	no	—
167	C₄×C₁₃	—	4	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	C₂×#41	1	1	yes	characteristic
173	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
174	C₂×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
175	C₂×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
176	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
177	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
178	#126	—	2	1	no	—
179	#125	—	2	1	no	—
180	#125	—	2	1	no	—
181	#125	C₂²×C₁₃	1	1	yes	characteristic
182	#123	—	2	1	no	—
183	C₂²⋊Q₈	—	2	1	no	—
184	C₂²⋊Q₈	—	2	1	no	—
185	C₂²⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
186	C₂²⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
187	C₂²⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
188	C₂²⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
189	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
190	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
191	C₄×D₄	—	2	1	no	—
192	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
193	#117	—	2	1	no	—
194	#117	C₂²×C₁₃	1	1	yes	characteristic
195	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
196	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
197	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
198	#107	C₂²×C₁₃	1	1	yes	characteristic
199	#107	C₂²×C₁₃	1	1	yes	characteristic
200	#107	C₂²×C₁₃	1	1	yes	characteristic
201	#106	C₂²×C₁₃	1	1	yes	characteristic
202	#106	C₂²×C₁₃	1	1	yes	characteristic
203	#106	C₂²×C₁₃	1	1	yes	characteristic
204	#106	C₂²×C₁₃	1	1	yes	characteristic
205	#106	C₂²×C₁₃	1	1	yes	characteristic
206	#106	C₂²×C₁₃	1	1	yes	characteristic
207	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
208	#103	C₂²×C₁₃	1	1	yes	characteristic
209	#103	C₂²×C₁₃	1	1	yes	characteristic
210	#103	C₂²×C₁₃	1	1	yes	characteristic
211	#103	C₂²×C₁₃	1	1	yes	characteristic
212	#103	C₂²×C₁₃	1	1	yes	characteristic
213	#103	C₂²×C₁₃	1	1	yes	characteristic
214	C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
215	C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
216	#97	C₂²×C₁₃	1	1	yes	characteristic
217	C₂×C₁₃	—	2	2	no	—
218	C₂×C₁₃	—	4	1	no	—
219	C₂×C₁₃	#547	1	1	yes	characteristic
220	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
221	C₂×C₁₃	#577	1	1	yes	characteristic
222	C₂×Q₈	—	2	1	no	—
223	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
224	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
225	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
230	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
231	Q₁₆	—	2	2	no	—
232	M₁₆	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
235	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
236	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
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	—	2	1	no	—
250	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
251	C₄⋊C₄	C₂³×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	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
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	C₂²⋊C₄	D₄×C₁₃	1	1	yes	characteristic
263	C₂²⋊C₄	D₄×C₁₃	1	1	yes	characteristic
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	C₂³×C₁₃	1	1	yes	characteristic
268	C₁₃	#3344	1	1	yes	characteristic, direct product factor, Sylow
269	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
270	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
271	Q₈	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	8	1	no	—
276	D₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	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	—	4	2	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	C₂⁴×C₁₃	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	C₁₃×#41	1	2	yes	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	C₁₃×#137	1	2	yes	—
307	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
321	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	1	no	—
324	C₂	C₁₃×#547	1	1	yes	characteristic
325	C₂	C₂×C₁₃×#137	1	1	yes	characteristic
326	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	C₂×D₄×C₁₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#501	C₂	semidirect
4. C₁₃×#485	C₂	semidirect
5. C₁₃×#484	C₂	semidirect
6. C₁₃×#483	C₂	semidirect
7. C₁₃×#483	C₂	semidirect
8. C₁₃×#479	C₂	semidirect
9. C₁₃×#475	C₂	semidirect
10. C₁₃×#470	C₂	semidirect
11. C₁₃×#467	C₂	semidirect
12. C₁₃×#441	C₂	semidirect
14. C₁₃×#428	C₂	semidirect
15. C₂×C₁₃×#106	C₂	semidirect
105. #3344	C₁₃	direct (N × H)
140. #501	C₂×C₁₃	semidirect
141. #485	C₂×C₁₃	semidirect
142. #484	C₂×C₁₃	semidirect
143. #483	C₂×C₁₃	semidirect
144. #483	C₂×C₁₃	semidirect
145. #479	C₂×C₁₃	semidirect
146. #475	C₂×C₁₃	semidirect
147. #470	C₂×C₁₃	semidirect
148. #467	C₂×C₁₃	semidirect
149. #441	C₂×C₁₃	semidirect
151. #428	C₂×C₁₃	semidirect
152. C₂×#106	C₂×C₁₃	semidirect
268. C₁₃	#3344	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×#554	2
C₁₃×#552	2
C₁₃×#501	2
C₁₃×#485	2
C₁₃×#484	2
C₁₃×#483	2
C₁₃×#479	2
C₁₃×#475	2
C₁₃×#470	2
C₁₃×#467	2
C₁₃×#441	2
C₁₃×#435	2
C₁₃×#428	2
C₂×C₁₃×#106	2
#3344	13

Subgroups — C13×#3344 #13112402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3344 #13112402