Subgroups — C₁₃×#33794 #15720133

Order 3328.

504 conjugacy classes of subgroups.

The subgroup lattice has 322 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	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2234	C₂	1	4	yes	maximal, complemented
8	C₁₃×#2228	C₂	1	4	yes	maximal, complemented
9	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#533	—	2	1	no	—
11	C₁₃×#532	—	2	1	no	—
12	C₁₃×#500	C₂²	1	1	yes	characteristic
13	C₁₃×#500	C₂²	1	1	yes	characteristic
14	C₁₃×#500	C₂²	1	1	yes	characteristic
15	C₁₃×#499	—	2	1	no	—
16	C₁₃×#499	C₂²	1	2	yes	—
17	C₁₃×#498	—	2	1	no	—
18	C₁₃×#493	C₂²	1	1	yes	characteristic
19	C₁₃×#486	—	2	1	no	—
20	C₁₃×#466	—	2	1	no	—
21	C₁₃×#466	C₂²	1	2	yes	—
22	C₄×C₁₃×D₈	—	2	1	no	—
23	C₄×C₁₃×D₈	C₂²	1	2	yes	—
24	C₁₃×#427	—	2	1	no	—
25	C₁₃×#427	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#367	C₂²	1	4	yes	—
28	C₁₃×#366	C₂²	1	4	yes	—
29	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₁₃×#357	C₂²	1	4	yes	—
31	C₁₃×#357	C₂²	1	4	yes	—
32	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
34	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
35	C₂×C₁₃×#41	—	4	1	no	—
36	C₂×C₁₃×D₈	—	2	1	no	—
37	C₂×C₁₃×D₈	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×C₄⋊Q₈	—	2	1	no	—
41	C₁₃×C₄⋊Q₈	—	2	1	no	—
42	C₁₃×C₄⋊Q₈	—	2	1	no	—
43	C₁₃×#127	—	2	1	no	—
44	C₁₃×#125	—	2	2	no	—
45	C₁₃×#124	—	4	1	no	—
46	C₁₃×C₂²⋊Q₈	—	4	1	no	—
47	C₁₃×C₂²⋊Q₈	—	4	1	no	—
48	C₁₃×#121	—	2	2	no	—
49	C₁₃×#121	—	4	1	no	—
50	C₄×Q₈×C₁₃	—	4	1	no	—
51	C₄×Q₈×C₁₃	—	4	1	no	—
52	C₄×D₄×C₁₃	—	2	2	no	—
53	C₄×D₄×C₁₃	—	4	1	no	—
54	C₁₃×#117	—	2	2	no	—
55	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
56	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
57	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
58	C₁₃×#107	—	2	1	no	—
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×#107	C₂³	1	2	yes	—
64	C₁₃×#106	—	2	2	no	—
65	C₁₃×#106	—	2	2	no	—
66	C₁₃×#102	—	2	2	no	—
67	C₁₃×#102	—	2	2	no	—
68	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	#33794	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₁₃×#41	—	8	4	no	—
74	C₂×Q₈×C₁₃	—	4	1	no	—
75	C₂×Q₈×C₁₃	—	4	1	no	—
76	C₂×Q₈×C₁₃	—	4	1	no	—
77	C₂×D₄×C₁₃	—	2	2	no	—
78	C₂×D₄×C₁₃	—	4	2	no	—
79	C₂×D₄×C₁₃	—	4	1	no	—
80	C₂×D₄×C₁₃	—	4	1	no	—
81	C₂²×C₄×C₁₃	—	4	2	no	—
82	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₁₃×D₈	—	2	4	no	—
86	C₁₃×D₈	—	4	2	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	4	1	no	—
97	C₁₃×C₄⋊C₄	—	4	2	no	—
98	C₁₃×C₄⋊C₄	—	4	2	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₄⋊C₄	—	4	1	no	—
101	C₁₃×C₂²⋊C₄	—	4	2	no	—
102	C₁₃×C₂²⋊C₄	—	4	2	no	—
103	C₁₃×C₂²⋊C₄	—	4	2	no	—
104	C₄²×C₁₃	—	4	2	no	—
105	C₄²×C₁₃	D₈	1	1	yes	characteristic
106	C₄²×C₁₃	D₈	1	1	yes	characteristic
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₁₃×C₁₆	—	2	2	no	—
110	C₁₃×C₁₆	—	2	2	no	—
111	#3169	C₂×C₁₃	1	1	yes	characteristic
112	#3160	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#2254	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#2252	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#2240	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#2240	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#2234	C₂×C₁₃	1	4	yes	complemented
118	#2228	C₂×C₁₃	1	4	yes	complemented
119	#2162	C₂×C₁₃	1	1	yes	characteristic, complemented
120	C₂³×C₁₃	—	4	2	no	—
121	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
122	Q₈×C₁₃	—	8	2	no	—
123	Q₈×C₁₃	—	8	2	no	—
124	Q₈×C₁₃	—	8	2	no	—
125	D₄×C₁₃	—	4	2	no	—
126	D₄×C₁₃	—	4	2	no	—
127	D₄×C₁₃	—	8	4	no	—
128	D₄×C₁₃	—	8	2	no	—
129	D₄×C₁₃	—	8	2	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	8	4	no	—
138	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
144	C₈×C₁₃	—	2	1	no	—
145	C₈×C₁₃	—	2	1	no	—
146	C₈×C₁₃	#137	1	2	yes	—
147	C₈×C₁₃	2₋¹⁺⁴	1	2	yes	—
148	#533	—	2	1	no	—
149	#532	—	2	1	no	—
150	#500	C₂²×C₁₃	1	1	yes	characteristic
151	#500	C₂²×C₁₃	1	1	yes	characteristic
152	#500	C₂²×C₁₃	1	1	yes	characteristic
153	#499	—	2	1	no	—
154	#499	C₂²×C₁₃	1	2	yes	—
155	#498	—	2	1	no	—
156	#493	C₂²×C₁₃	1	1	yes	characteristic
157	#486	—	2	1	no	—
158	#466	—	2	1	no	—
159	#466	C₂²×C₁₃	1	2	yes	—
160	C₄×D₈	—	2	1	no	—
161	C₄×D₈	C₂²×C₁₃	1	2	yes	—
162	#427	—	2	1	no	—
163	#427	C₂²×C₁₃	1	2	yes	—
164	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
165	#367	C₂²×C₁₃	1	4	yes	—
166	#366	C₂²×C₁₃	1	4	yes	—
167	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
168	#357	C₂²×C₁₃	1	4	yes	—
169	#357	C₂²×C₁₃	1	4	yes	—
170	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
171	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
172	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	—	2	1	no	—
175	C₂²×C₁₃	—	2	1	no	—
176	C₂²×C₁₃	—	8	2	no	—
177	C₂²×C₁₃	—	8	2	no	—
178	C₂²×C₁₃	—	8	2	no	—
179	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
180	C₄×C₁₃	—	2	1	no	—
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	8	2	no	—
184	C₄×C₁₃	—	8	2	no	—
185	C₄×C₁₃	—	8	2	no	—
186	C₄×C₁₃	#508	1	2	yes	—
187	C₄×C₁₃	#509	1	2	yes	—
188	C₄×C₁₃	#536	1	1	yes	characteristic
189	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
190	C₂×#41	—	4	1	no	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	C₂³×C₁₃	1	2	yes	—
193	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
195	C₄⋊Q₈	—	2	1	no	—
196	C₄⋊Q₈	—	2	1	no	—
197	C₄⋊Q₈	—	2	1	no	—
198	#127	—	2	1	no	—
199	#125	—	2	2	no	—
200	#124	—	4	1	no	—
201	C₂²⋊Q₈	—	4	1	no	—
202	C₂²⋊Q₈	—	4	1	no	—
203	#121	—	2	2	no	—
204	#121	—	4	1	no	—
205	C₄×Q₈	—	4	1	no	—
206	C₄×Q₈	—	4	1	no	—
207	C₄×D₄	—	2	2	no	—
208	C₄×D₄	—	4	1	no	—
209	#117	—	2	2	no	—
210	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
211	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
212	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
213	#107	—	2	1	no	—
214	#107	—	2	1	no	—
215	#107	—	2	1	no	—
216	#107	C₂³×C₁₃	1	2	yes	—
217	#107	C₂³×C₁₃	1	2	yes	—
218	#107	C₂³×C₁₃	1	2	yes	—
219	#106	—	2	2	no	—
220	#106	—	2	2	no	—
221	#102	—	2	2	no	—
222	#102	—	2	2	no	—
223	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
224	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
225	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
226	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
227	C₂×C₁₃	—	4	1	no	—
228	C₂×C₁₃	—	16	2	no	—
229	C₂×C₁₃	#3226	1	1	yes	characteristic
230	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
231	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
232	#41	—	8	4	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
242	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
244	D₈	—	2	4	no	—
245	D₈	—	4	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
249	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
251	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₄²	—	4	2	no	—
264	C₄²	C₁₃×D₈	1	1	yes	characteristic
265	C₄²	C₁₃×D₈	1	1	yes	characteristic
266	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
267	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
268	C₁₆	—	2	2	no	—
269	C₁₆	—	2	2	no	—
270	C₁₃	#33794	1	1	yes	characteristic, direct product factor, Sylow
271	C₂³	—	4	2	no	—
272	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
273	Q₈	—	8	2	no	—
274	Q₈	—	8	2	no	—
275	Q₈	—	8	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	2	no	—
278	D₄	—	8	4	no	—
279	D₄	—	8	2	no	—
280	D₄	—	8	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	8	4	no	—
289	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
290	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
291	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
292	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
293	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	C₁₃×#137	1	2	yes	—
298	C₈	C₁₃×2₋¹⁺⁴	1	2	yes	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	C₁₃×#508	1	2	yes	—
313	C₄	C₁₃×#509	1	2	yes	—
314	C₄	C₁₃×#536	1	1	yes	characteristic
315	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
316	C₂	—	4	1	no	—
317	C₂	—	16	2	no	—
318	C₂	C₁₃×#3226	1	1	yes	characteristic
319	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
320	C₂	C₂×C₁₃×#509	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. C₁₃×#3160	C₂	semidirect
3. C₁₃×#2254	C₂	semidirect
4. C₁₃×#2252	C₂	semidirect
5. C₁₃×#2240	C₂	semidirect
6. C₁₃×#2240	C₂	semidirect
7. C₁₃×#2234	C₂	semidirect
8. C₁₃×#2228	C₂	semidirect
9. C₁₃×#2162	C₂	semidirect
72. #33794	C₁₃	direct (N × H)
112. #3160	C₂×C₁₃	semidirect
113. #2254	C₂×C₁₃	semidirect
114. #2252	C₂×C₁₃	semidirect
115. #2240	C₂×C₁₃	semidirect
116. #2240	C₂×C₁₃	semidirect
117. #2234	C₂×C₁₃	semidirect
118. #2228	C₂×C₁₃	semidirect
119. #2162	C₂×C₁₃	semidirect
270. C₁₃	#33794	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
C₁₃×#3160	2
C₁₃×#2254	2
C₁₃×#2252	2
C₁₃×#2240	2
C₁₃×#2234	2
C₁₃×#2228	2
C₁₃×#2162	2
#33794	13

Subgroups — C13×#33794 #15720133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33794 #15720133