Subgroups — C₁₃×#33683 #15720022

Order 3328.

556 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2948	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2237	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2228	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2208	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2187	C₂	1	2	yes	maximal
7	C₁₃×#2175	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×SD₃₂	C₂	1	2	yes	maximal, complemented
9	C₁₃×#2164	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
11	C₁₃×#541	—	2	1	no	—
12	C₁₃×#518	—	2	1	no	—
13	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
14	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
15	C₁₃×#499	C₂²	1	1	yes	characteristic, complemented
16	C₁₃×#494	C₂²	1	1	yes	characteristic, complemented
17	C₁₃×#491	—	2	1	no	—
18	C₁₃×#470	—	2	1	no	—
19	C₁₃×#467	C₂²	1	2	yes	—
20	C₁₃×#442	—	2	1	no	—
21	C₁₃×#442	—	2	1	no	—
22	C₁₃×#441	—	2	1	no	—
23	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	complemented
24	C₄×C₁₃×D₈	C₂²	1	2	yes	—
25	C₄×C₁₃×D₈	C₂²	1	2	yes	—
26	C₁₃×#427	C₂²	1	1	yes	characteristic
27	C₁₃×#418	—	2	1	no	—
28	C₁₃×#405	C₂²	1	1	yes	characteristic
29	C₁₃×#367	C₂²	1	2	yes	complemented
30	C₁₃×#366	C₂²	1	2	yes	complemented
31	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
32	C₁₃×#358	C₂²	1	2	yes	complemented
33	C₁₃×#358	C₂²	1	2	yes	complemented
34	C₁₃×#357	C₂²	1	2	yes	—
35	C₁₃×#357	C₂²	1	2	yes	—
36	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
37	C₁₃×#346	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₄×C₁₃	—	2	1	no	—
40	C₂×C₁₃×Q₁₆	—	2	1	no	—
41	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
42	C₂×C₁₃×D₈	—	2	4	no	—
43	C₂×C₁₃×D₈	—	2	1	no	—
44	C₂×C₁₃×D₈	C₂³	1	2	yes	—
45	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×C₄⋊Q₈	—	2	1	no	—
48	C₁₃×#126	—	4	2	no	—
49	C₁₃×#125	—	2	1	no	—
50	C₁₃×#125	—	4	2	no	—
51	C₁₃×C₂²⋊Q₈	—	2	2	no	—
52	C₄×Q₈×C₁₃	—	2	2	no	—
53	C₄×D₄×C₁₃	—	2	2	no	—
54	C₄×D₄×C₁₃	—	2	2	no	—
55	C₄×D₄×C₁₃	—	4	2	no	—
56	C₁₃×#117	—	2	1	no	—
57	C₁₃×#117	D₄	1	1	yes	characteristic
58	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
59	C₁₃×SD₃₂	—	2	4	no	—
60	C₁₃×SD₃₂	—	2	4	no	—
61	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
62	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
63	C₁₃×#107	C₂³	1	2	yes	—
64	C₁₃×#107	C₂³	1	2	yes	—
65	C₁₃×#106	—	2	1	no	—
66	C₁₃×#106	—	2	1	no	—
67	C₁₃×#103	—	2	2	no	—
68	C₁₃×#103	—	2	2	no	—
69	C₁₃×#102	—	2	2	no	—
70	C₁₃×#102	—	2	2	no	—
71	C₁₃×#97	D₄	1	1	yes	characteristic
72	C₁₃×#97	D₄	1	1	yes	characteristic
73	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	#33683	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂⁴×C₁₃	—	4	1	no	—
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×D₄×C₁₃	—	2	4	no	—
79	C₂×D₄×C₁₃	—	2	2	no	—
80	C₂×D₄×C₁₃	—	4	2	no	—
81	C₂×D₄×C₁₃	—	4	1	no	—
82	C₂²×C₄×C₁₃	—	4	2	no	—
83	C₂²×C₄×C₁₃	—	4	2	no	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₁₃×Q₁₆	—	2	2	no	—
86	C₁₃×Q₁₆	—	2	2	no	—
87	C₁₃×Q₁₆	—	4	2	no	—
88	C₁₃×D₈	—	2	4	no	—
89	C₁₃×D₈	—	2	2	no	—
90	C₁₃×D₈	#41	1	4	yes	—
91	C₁₃×M₁₆	—	4	1	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	4	2	no	—
100	C₁₃×C₄⋊C₄	—	4	2	no	—
101	C₁₃×C₄⋊C₄	—	4	2	no	—
102	C₁₃×C₄⋊C₄	—	4	2	no	—
103	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
104	C₁₃×C₂²⋊C₄	—	4	1	no	—
105	C₁₃×C₂²⋊C₄	—	4	2	no	—
106	C₁₃×C₂²⋊C₄	—	4	2	no	—
107	C₁₃×C₂²⋊C₄	—	4	2	no	—
108	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
109	C₄²×C₁₃	—	4	1	no	—
110	C₄²×C₁₃	—	4	2	no	—
111	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₁₃×C₁₆	—	2	2	no	—
114	C₁₃×C₁₆	—	2	2	no	—
115	#3177	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#2948	C₂×C₁₃	1	1	yes	characteristic
117	#2237	C₂×C₁₃	1	2	yes	complemented
118	#2228	C₂×C₁₃	1	2	yes	complemented
119	#2208	C₂×C₁₃	1	2	yes	complemented
120	#2187	C₂×C₁₃	1	2	yes	—
121	#2175	C₂×C₁₃	1	2	yes	complemented
122	C₄×SD₃₂	C₂×C₁₃	1	2	yes	complemented
123	#2164	C₂×C₁₃	1	1	yes	characteristic, complemented
124	C₂³×C₁₃	—	4	4	no	—
125	C₂³×C₁₃	—	4	2	no	—
126	C₂³×C₁₃	—	16	2	no	—
127	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
128	Q₈×C₁₃	—	4	2	no	—
129	Q₈×C₁₃	—	4	2	no	—
130	D₄×C₁₃	—	2	4	no	—
131	D₄×C₁₃	—	4	4	no	—
132	D₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	16	2	no	—
139	C₂×C₄×C₁₃	—	16	2	no	—
140	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
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	—	2	1	no	—
147	C₈×C₁₃	C₂×#41	1	2	yes	—
148	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
149	#541	—	2	1	no	—
150	#518	—	2	1	no	—
151	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
152	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
153	#499	C₂²×C₁₃	1	1	yes	characteristic, complemented
154	#494	C₂²×C₁₃	1	1	yes	characteristic, complemented
155	#491	—	2	1	no	—
156	#470	—	2	1	no	—
157	#467	C₂²×C₁₃	1	2	yes	—
158	#442	—	2	1	no	—
159	#442	—	2	1	no	—
160	#441	—	2	1	no	—
161	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	complemented
162	C₄×D₈	C₂²×C₁₃	1	2	yes	—
163	C₄×D₈	C₂²×C₁₃	1	2	yes	—
164	#427	C₂²×C₁₃	1	1	yes	characteristic
165	#418	—	2	1	no	—
166	#405	C₂²×C₁₃	1	1	yes	characteristic
167	#367	C₂²×C₁₃	1	2	yes	complemented
168	#366	C₂²×C₁₃	1	2	yes	complemented
169	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	complemented
170	#358	C₂²×C₁₃	1	2	yes	complemented
171	#358	C₂²×C₁₃	1	2	yes	complemented
172	#357	C₂²×C₁₃	1	2	yes	—
173	#357	C₂²×C₁₃	1	2	yes	—
174	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
175	#346	C₂²×C₁₃	1	1	yes	characteristic, complemented
176	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic, complemented
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	—	4	4	no	—
180	C₂²×C₁₃	—	8	2	no	—
181	C₂²×C₁₃	—	8	2	no	—
182	C₂²×C₁₃	—	16	4	no	—
183	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	—	2	2	no	—
187	C₄×C₁₃	—	8	2	no	—
188	C₄×C₁₃	—	8	2	no	—
189	C₄×C₁₃	—	8	2	no	—
190	C₄×C₁₃	#525	1	1	yes	characteristic
191	C₄×C₁₃	#575	1	1	yes	characteristic
192	C₂²×D₄	—	2	1	no	—
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
195	C₂×D₈	—	2	4	no	—
196	C₂×D₈	—	2	1	no	—
197	C₂×D₈	C₂³×C₁₃	1	2	yes	—
198	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
200	C₄⋊Q₈	—	2	1	no	—
201	#126	—	4	2	no	—
202	#125	—	2	1	no	—
203	#125	—	4	2	no	—
204	C₂²⋊Q₈	—	2	2	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×D₄	—	2	2	no	—
207	C₄×D₄	—	2	2	no	—
208	C₄×D₄	—	4	2	no	—
209	#117	—	2	1	no	—
210	#117	D₄×C₁₃	1	1	yes	characteristic
211	C₂×C₂²⋊C₄	—	4	2	no	—
212	SD₃₂	—	2	4	no	—
213	SD₃₂	—	2	4	no	—
214	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
215	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
216	#107	C₂³×C₁₃	1	2	yes	—
217	#107	C₂³×C₁₃	1	2	yes	—
218	#106	—	2	1	no	—
219	#106	—	2	1	no	—
220	#103	—	2	2	no	—
221	#103	—	2	2	no	—
222	#102	—	2	2	no	—
223	#102	—	2	2	no	—
224	#97	D₄×C₁₃	1	1	yes	characteristic
225	#97	D₄×C₁₃	1	1	yes	characteristic
226	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
227	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
228	C₂×C₁₃	—	4	1	no	—
229	C₂×C₁₃	—	8	4	no	—
230	C₂×C₁₃	#3100	1	1	yes	characteristic
231	C₂×C₁₃	#3416	1	2	yes	—
232	C₂⁴	—	4	1	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×D₄	—	2	4	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂²×C₄	—	4	2	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
241	Q₁₆	—	2	2	no	—
242	Q₁₆	—	2	2	no	—
243	Q₁₆	—	4	2	no	—
244	D₈	—	2	4	no	—
245	D₈	—	2	2	no	—
246	D₈	C₁₃×#41	1	4	yes	—
247	M₁₆	—	4	1	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
252	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
260	C₂²⋊C₄	—	4	1	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
265	C₄²	—	4	1	no	—
266	C₄²	—	4	2	no	—
267	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
268	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
269	C₁₆	—	2	2	no	—
270	C₁₆	—	2	2	no	—
271	C₁₃	#33683	1	1	yes	characteristic, direct product factor, Sylow
272	C₂³	—	4	4	no	—
273	C₂³	—	4	2	no	—
274	C₂³	—	16	2	no	—
275	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	D₄	—	2	4	no	—
279	D₄	—	4	4	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	16	2	no	—
287	C₂×C₄	—	16	2	no	—
288	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
289	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
290	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
291	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	C₂×C₁₃×#41	1	2	yes	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	16	4	no	—
302	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
303	C₄	—	2	1	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	C₁₃×#525	1	1	yes	characteristic
310	C₄	C₁₃×#575	1	1	yes	characteristic
311	C₂	—	4	1	no	—
312	C₂	—	8	4	no	—
313	C₂	C₁₃×#3100	1	1	yes	characteristic
314	C₂	C₁₃×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3177	C₂	semidirect
3. C₁₃×#2237	C₂	semidirect
4. C₁₃×#2228	C₂	semidirect
5. C₁₃×#2208	C₂	semidirect
7. C₁₃×#2175	C₂	semidirect
8. C₄×C₁₃×SD₃₂	C₂	semidirect
9. C₁₃×#2164	C₂	semidirect
15. C₁₃×#499	C₂²	semidirect
16. C₁₃×#494	C₂²	semidirect
23. C₄×C₁₃×Q₁₆	C₂²	semidirect
29. C₁₃×#367	C₂²	semidirect
30. C₁₃×#366	C₂²	semidirect
31. C₁₃×C₄⋊C₁₆	C₂²	semidirect
32. C₁₃×#358	C₂²	semidirect
33. C₁₃×#358	C₂²	semidirect
37. C₁₃×#346	C₂²	semidirect
38. C₄×C₁₃×C₁₆	C₂²	semidirect
75. #33683	C₁₃	direct (N × H)
115. #3177	C₂×C₁₃	semidirect
117. #2237	C₂×C₁₃	semidirect
118. #2228	C₂×C₁₃	semidirect
119. #2208	C₂×C₁₃	semidirect
121. #2175	C₂×C₁₃	semidirect
122. C₄×SD₃₂	C₂×C₁₃	semidirect
123. #2164	C₂×C₁₃	semidirect
153. #499	C₂²×C₁₃	semidirect
154. #494	C₂²×C₁₃	semidirect
161. C₄×Q₁₆	C₂²×C₁₃	semidirect
167. #367	C₂²×C₁₃	semidirect
168. #366	C₂²×C₁₃	semidirect
169. C₄⋊C₁₆	C₂²×C₁₃	semidirect
170. #358	C₂²×C₁₃	semidirect
171. #358	C₂²×C₁₃	semidirect
175. #346	C₂²×C₁₃	semidirect
176. C₄×C₁₆	C₂²×C₁₃	semidirect
271. C₁₃	#33683	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×#3177	2
C₁₃×#2948	2
C₁₃×#2237	2
C₁₃×#2228	2
C₁₃×#2208	2
C₁₃×#2187	2
C₁₃×#2175	2
C₄×C₁₃×SD₃₂	2
C₁₃×#2164	2
#33683	13

Subgroups — C13×#33683 #15720022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33683 #15720022