Subgroups — C₁₃×#33743 #15720082

Order 3328.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 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×#3176	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2245	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2244	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2220	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2217	C₂	1	2	yes	maximal, complemented
9	C₁₃×#2175	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₃×SD₃₂	C₂	1	2	yes	maximal, complemented
11	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#541	—	2	1	no	—
13	C₁₃×#540	—	2	1	no	—
14	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
15	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
16	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
18	C₁₃×#499	C₂²	1	1	yes	characteristic
19	C₁₃×#499	C₂²	1	1	yes	characteristic
20	C₁₃×#494	C₂²	1	1	yes	characteristic
21	C₁₃×#493	C₂²	1	1	yes	characteristic
22	C₁₃×#491	—	2	1	no	—
23	C₁₃×#490	—	2	1	no	—
24	C₁₃×#470	—	2	1	no	—
25	C₁₃×#469	—	2	1	no	—
26	C₁₃×#467	C₂²	1	2	yes	—
27	C₁₃×#466	C₂²	1	2	yes	—
28	C₁₃×#442	—	2	1	no	—
29	C₁₃×#441	—	2	1	no	—
30	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
31	C₄×C₁₃×D₈	C₂²	1	2	yes	—
32	C₁₃×#405	C₂²	1	1	yes	characteristic
33	C₁₃×#367	C₂²	1	2	yes	—
34	C₁₃×#366	C₂²	1	2	yes	—
35	C₁₃×#358	C₂²	1	2	yes	—
36	C₁₃×#358	C₂²	1	2	yes	—
37	C₁₃×#357	C₂²	1	2	yes	—
38	C₁₃×#357	C₂²	1	2	yes	—
39	C₁₃×#346	C₂²	1	1	yes	characteristic
40	C₁₃×#346	C₂²	1	1	yes	characteristic
41	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
42	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
43	C₂×C₁₃×Q₁₆	—	2	1	no	—
44	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
45	C₂×C₁₃×D₈	—	2	1	no	—
46	C₂×C₁₃×D₈	C₂³	1	2	yes	—
47	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×C₄⋊Q₈	—	2	1	no	—
50	C₁₃×#127	—	2	1	no	—
51	C₁₃×#126	—	4	2	no	—
52	C₁₃×#125	—	2	1	no	—
53	C₁₃×#125	—	2	1	no	—
54	C₁₃×#125	—	4	2	no	—
55	C₁₃×#123	—	4	2	no	—
56	C₁₃×C₂²⋊Q₈	—	2	2	no	—
57	C₁₃×#121	—	2	2	no	—
58	C₁₃×#121	—	4	2	no	—
59	C₄×Q₈×C₁₃	—	2	2	no	—
60	C₄×D₄×C₁₃	—	2	2	no	—
61	C₁₃×#117	D₄	1	1	yes	characteristic
62	C₁₃×SD₃₂	—	2	4	no	—
63	C₁₃×SD₃₂	—	2	4	no	—
64	C₁₃×M₃₂	—	2	2	no	—
65	C₂×C₁₃×C₁₆	—	2	2	no	—
66	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
68	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
70	C₁₃×#107	C₂³	1	2	yes	—
71	C₁₃×#107	C₂³	1	2	yes	—
72	C₁₃×#106	—	2	1	no	—
73	C₁₃×#106	—	2	1	no	—
74	C₁₃×#103	—	2	2	no	—
75	C₁₃×#103	—	2	2	no	—
76	C₁₃×#102	—	2	2	no	—
77	C₁₃×#102	—	2	2	no	—
78	C₁₃×#97	D₄	1	1	yes	characteristic
79	C₁₃×#97	D₄	1	1	yes	characteristic
80	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
82	#33743	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂×Q₈×C₁₃	—	2	2	no	—
84	C₂×D₄×C₁₃	—	2	2	no	—
85	C₂×D₄×C₁₃	—	4	2	no	—
86	C₂×D₄×C₁₃	—	4	2	no	—
87	C₂²×C₄×C₁₃	—	4	2	no	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₁₃×Q₁₆	—	2	2	no	—
90	C₁₃×Q₁₆	—	2	2	no	—
91	C₁₃×Q₁₆	—	4	2	no	—
92	C₁₃×D₈	—	2	2	no	—
93	C₁₃×D₈	—	2	2	no	—
94	C₁₃×D₈	—	4	2	no	—
95	C₁₃×M₁₆	—	2	2	no	—
96	C₂×C₈×C₁₃	—	2	1	no	—
97	C₂×C₈×C₁₃	—	2	1	no	—
98	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
102	C₁₃×C₄⋊C₄	—	2	2	no	—
103	C₁₃×C₄⋊C₄	—	2	2	no	—
104	C₁₃×C₄⋊C₄	—	4	2	no	—
105	C₁₃×C₄⋊C₄	—	4	2	no	—
106	C₁₃×C₄⋊C₄	—	4	2	no	—
107	C₁₃×C₄⋊C₄	—	4	2	no	—
108	C₁₃×C₄⋊C₄	—	4	2	no	—
109	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
110	C₁₃×C₂²⋊C₄	—	4	2	no	—
111	C₁₃×C₂²⋊C₄	—	4	2	no	—
112	C₁₃×C₂²⋊C₄	—	4	2	no	—
113	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
114	C₄²×C₁₃	—	4	2	no	—
115	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₁₃×C₁₆	—	2	1	no	—
118	C₁₃×C₁₆	—	2	1	no	—
119	C₁₃×C₁₆	#41	1	2	yes	—
120	C₁₃×C₁₆	#41	1	2	yes	—
121	#3177	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#3176	C₂×C₁₃	1	1	yes	characteristic
123	#2253	C₂×C₁₃	1	1	yes	characteristic, complemented
124	#2248	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#2245	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#2244	C₂×C₁₃	1	1	yes	characteristic, complemented
127	#2220	C₂×C₁₃	1	2	yes	complemented
128	#2217	C₂×C₁₃	1	2	yes	complemented
129	#2175	C₂×C₁₃	1	2	yes	complemented
130	C₄×SD₃₂	C₂×C₁₃	1	2	yes	complemented
131	#2108	C₂×C₁₃	1	1	yes	characteristic, complemented
132	C₂³×C₁₃	—	4	2	no	—
133	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
134	Q₈×C₁₃	—	4	2	no	—
135	Q₈×C₁₃	—	4	2	no	—
136	D₄×C₁₃	—	4	2	no	—
137	D₄×C₁₃	—	4	2	no	—
138	D₄×C₁₃	—	16	2	no	—
139	D₄×C₁₃	—	16	2	no	—
140	C₂×C₄×C₁₃	—	2	1	no	—
141	C₂×C₄×C₁₃	—	2	1	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	—	16	2	no	—
146	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
147	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	—	2	1	no	—
152	C₈×C₁₃	#135	1	2	yes	—
153	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
154	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
155	#541	—	2	1	no	—
156	#540	—	2	1	no	—
157	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
158	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
159	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
160	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
161	#499	C₂²×C₁₃	1	1	yes	characteristic
162	#499	C₂²×C₁₃	1	1	yes	characteristic
163	#494	C₂²×C₁₃	1	1	yes	characteristic
164	#493	C₂²×C₁₃	1	1	yes	characteristic
165	#491	—	2	1	no	—
166	#490	—	2	1	no	—
167	#470	—	2	1	no	—
168	#469	—	2	1	no	—
169	#467	C₂²×C₁₃	1	2	yes	—
170	#466	C₂²×C₁₃	1	2	yes	—
171	#442	—	2	1	no	—
172	#441	—	2	1	no	—
173	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
174	C₄×D₈	C₂²×C₁₃	1	2	yes	—
175	#405	C₂²×C₁₃	1	1	yes	characteristic
176	#367	C₂²×C₁₃	1	2	yes	—
177	#366	C₂²×C₁₃	1	2	yes	—
178	#358	C₂²×C₁₃	1	2	yes	—
179	#358	C₂²×C₁₃	1	2	yes	—
180	#357	C₂²×C₁₃	1	2	yes	—
181	#357	C₂²×C₁₃	1	2	yes	—
182	#346	C₂²×C₁₃	1	1	yes	characteristic
183	#346	C₂²×C₁₃	1	1	yes	characteristic
184	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
185	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
186	C₂²×C₁₃	—	2	2	no	—
187	C₂²×C₁₃	—	2	1	no	—
188	C₂²×C₁₃	—	8	2	no	—
189	C₂²×C₁₃	—	8	2	no	—
190	C₂²×C₁₃	—	8	2	no	—
191	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	2	2	no	—
194	C₄×C₁₃	—	2	1	no	—
195	C₄×C₁₃	—	8	2	no	—
196	C₄×C₁₃	—	8	2	no	—
197	C₄×C₁₃	—	8	2	no	—
198	C₄×C₁₃	#532	1	1	yes	characteristic
199	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
200	C₂×Q₁₆	—	2	1	no	—
201	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	C₂³×C₁₃	1	2	yes	—
204	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
206	C₄⋊Q₈	—	2	1	no	—
207	#127	—	2	1	no	—
208	#126	—	4	2	no	—
209	#125	—	2	1	no	—
210	#125	—	2	1	no	—
211	#125	—	4	2	no	—
212	#123	—	4	2	no	—
213	C₂²⋊Q₈	—	2	2	no	—
214	#121	—	2	2	no	—
215	#121	—	4	2	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×D₄	—	2	2	no	—
218	#117	D₄×C₁₃	1	1	yes	characteristic
219	SD₃₂	—	2	4	no	—
220	SD₃₂	—	2	4	no	—
221	M₃₂	—	2	2	no	—
222	C₂×C₁₆	—	2	2	no	—
223	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
224	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
225	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
226	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
227	#107	C₂³×C₁₃	1	2	yes	—
228	#107	C₂³×C₁₃	1	2	yes	—
229	#106	—	2	1	no	—
230	#106	—	2	1	no	—
231	#103	—	2	2	no	—
232	#103	—	2	2	no	—
233	#102	—	2	2	no	—
234	#102	—	2	2	no	—
235	#97	D₄×C₁₃	1	1	yes	characteristic
236	#97	D₄×C₁₃	1	1	yes	characteristic
237	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
239	C₂×C₁₃	—	4	1	no	—
240	C₂×C₁₃	—	16	2	no	—
241	C₂×C₁₃	#3169	1	1	yes	characteristic
242	C₂×C₁₃	#3416	1	2	yes	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
249	Q₁₆	—	2	2	no	—
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	4	2	no	—
252	D₈	—	2	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	4	2	no	—
255	M₁₆	—	2	2	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
274	C₄²	—	4	2	no	—
275	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
276	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
277	C₁₆	—	2	1	no	—
278	C₁₆	—	2	1	no	—
279	C₁₆	C₁₃×#41	1	2	yes	—
280	C₁₆	C₁₃×#41	1	2	yes	—
281	C₁₃	#33743	1	1	yes	characteristic, direct product factor, Sylow
282	C₂³	—	4	2	no	—
283	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	16	2	no	—
289	D₄	—	16	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	16	2	no	—
296	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
297	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
298	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	C₁₃×#135	1	2	yes	—
303	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
304	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
305	C₂²	—	2	2	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	2	no	—
310	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	2	no	—
317	C₄	C₁₃×#532	1	1	yes	characteristic
318	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
319	C₂	—	4	1	no	—
320	C₂	—	16	2	no	—
321	C₂	C₁₃×#3169	1	1	yes	characteristic
322	C₂	C₁₃×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#3176	C₂
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×#2253	C₂	semidirect
4. C₁₃×#2248	C₂	semidirect
5. C₁₃×#2245	C₂	semidirect
6. C₁₃×#2244	C₂	semidirect
7. C₁₃×#2220	C₂	semidirect
8. C₁₃×#2217	C₂	semidirect
9. C₁₃×#2175	C₂	semidirect
10. C₄×C₁₃×SD₃₂	C₂	semidirect
11. C₁₃×#2108	C₂	semidirect
82. #33743	C₁₃	direct (N × H)
121. #3177	C₂×C₁₃	semidirect
123. #2253	C₂×C₁₃	semidirect
124. #2248	C₂×C₁₃	semidirect
125. #2245	C₂×C₁₃	semidirect
126. #2244	C₂×C₁₃	semidirect
127. #2220	C₂×C₁₃	semidirect
128. #2217	C₂×C₁₃	semidirect
129. #2175	C₂×C₁₃	semidirect
130. C₄×SD₃₂	C₂×C₁₃	semidirect
131. #2108	C₂×C₁₃	semidirect
281. C₁₃	#33743	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×#3176	2
C₁₃×#2253	2
C₁₃×#2248	2
C₁₃×#2245	2
C₁₃×#2244	2
C₁₃×#2220	2
C₁₃×#2217	2
C₁₃×#2175	2
C₄×C₁₃×SD₃₂	2
C₁₃×#2108	2
#33743	13

Subgroups — C13×#33743 #15720082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33743 #15720082