Subgroups — C₁₃×#33801 #15720140

Order 3328.

500 conjugacy classes of subgroups.

The subgroup lattice has 314 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×#2255	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2244	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2234	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2226	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2224	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#533	—	2	1	no	—
10	C₁₃×#532	—	2	1	no	—
11	C₁₃×#500	C₂²	1	1	yes	characteristic
12	C₁₃×#500	C₂²	1	1	yes	characteristic
13	C₁₃×#500	C₂²	1	1	yes	characteristic
14	C₁₃×#499	—	2	1	no	—
15	C₁₃×#499	C₂²	1	2	yes	—
16	C₁₃×#498	—	2	1	no	—
17	C₁₃×#493	C₂²	1	1	yes	characteristic
18	C₁₃×#486	—	2	1	no	—
19	C₁₃×#466	—	2	1	no	—
20	C₁₃×#466	C₂²	1	2	yes	—
21	C₄×C₁₃×D₈	—	2	1	no	—
22	C₄×C₁₃×D₈	C₂²	1	2	yes	—
23	C₁₃×#427	—	2	1	no	—
24	C₁₃×#427	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×#367	C₂²	1	4	yes	—
27	C₁₃×#366	C₂²	1	4	yes	—
28	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₁₃×#357	C₂²	1	4	yes	—
30	C₁₃×#357	C₂²	1	4	yes	—
31	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₁₃×#346	C₂²	1	2	yes	—
33	C₂×C₁₃×#41	—	4	1	no	—
34	C₂×C₁₃×D₈	—	2	1	no	—
35	C₂×C₁₃×D₈	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×C₄⋊Q₈	—	2	1	no	—
39	C₁₃×C₄⋊Q₈	—	2	1	no	—
40	C₁₃×C₄⋊Q₈	—	2	1	no	—
41	C₁₃×#127	—	2	1	no	—
42	C₁₃×#125	—	2	2	no	—
43	C₁₃×#124	—	4	1	no	—
44	C₁₃×C₂²⋊Q₈	—	4	1	no	—
45	C₁₃×C₂²⋊Q₈	—	4	1	no	—
46	C₁₃×#121	—	2	2	no	—
47	C₁₃×#121	—	4	1	no	—
48	C₄×Q₈×C₁₃	—	4	1	no	—
49	C₄×Q₈×C₁₃	—	4	1	no	—
50	C₄×D₄×C₁₃	—	2	2	no	—
51	C₄×D₄×C₁₃	—	4	1	no	—
52	C₁₃×#117	—	2	2	no	—
53	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
54	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
55	C₁₃×#107	—	2	1	no	—
56	C₁₃×#107	—	2	1	no	—
57	C₁₃×#107	—	2	1	no	—
58	C₁₃×#107	C₂³	1	2	yes	—
59	C₁₃×#107	C₂³	1	2	yes	—
60	C₁₃×#107	C₂³	1	2	yes	—
61	C₁₃×#106	—	2	2	no	—
62	C₁₃×#106	—	2	2	no	—
63	C₁₃×#102	—	2	2	no	—
64	C₁₃×#102	—	2	2	no	—
65	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
66	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	#33801	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₁₃×#41	—	8	4	no	—
71	C₂×Q₈×C₁₃	—	4	1	no	—
72	C₂×Q₈×C₁₃	—	4	1	no	—
73	C₂×Q₈×C₁₃	—	4	1	no	—
74	C₂×D₄×C₁₃	—	2	2	no	—
75	C₂×D₄×C₁₃	—	4	2	no	—
76	C₂×D₄×C₁₃	—	4	1	no	—
77	C₂×D₄×C₁₃	—	4	1	no	—
78	C₂²×C₄×C₁₃	—	4	2	no	—
79	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₁₃×D₈	—	2	4	no	—
83	C₁₃×D₈	—	4	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	4	1	no	—
94	C₁₃×C₄⋊C₄	—	4	2	no	—
95	C₁₃×C₄⋊C₄	—	4	2	no	—
96	C₁₃×C₄⋊C₄	—	4	1	no	—
97	C₁₃×C₄⋊C₄	—	4	1	no	—
98	C₁₃×C₂²⋊C₄	—	4	2	no	—
99	C₁₃×C₂²⋊C₄	—	4	2	no	—
100	C₁₃×C₂²⋊C₄	—	4	2	no	—
101	C₄²×C₁₃	—	4	2	no	—
102	C₄²×C₁₃	D₈	1	1	yes	characteristic
103	C₄²×C₁₃	D₈	1	1	yes	characteristic
104	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₁₃×C₁₆	—	2	4	no	—
107	#3169	C₂×C₁₃	1	1	yes	characteristic
108	#3160	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#2255	C₂×C₁₃	1	2	yes	complemented
110	#2244	C₂×C₁₃	1	2	yes	complemented
111	#2234	C₂×C₁₃	1	4	yes	complemented
112	#2226	C₂×C₁₃	1	2	yes	complemented
113	#2224	C₂×C₁₃	1	2	yes	complemented
114	#2163	C₂×C₁₃	1	1	yes	characteristic, complemented
115	C₂³×C₁₃	—	4	2	no	—
116	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
117	Q₈×C₁₃	—	8	2	no	—
118	Q₈×C₁₃	—	8	2	no	—
119	Q₈×C₁₃	—	8	2	no	—
120	D₄×C₁₃	—	4	2	no	—
121	D₄×C₁₃	—	4	2	no	—
122	D₄×C₁₃	—	8	2	no	—
123	D₄×C₁₃	—	8	2	no	—
124	D₄×C₁₃	—	8	4	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	4	1	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	8	4	no	—
133	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₁₃	—	2	1	no	—
140	C₈×C₁₃	—	2	1	no	—
141	C₈×C₁₃	#137	1	2	yes	—
142	C₈×C₁₃	2₋¹⁺⁴	1	2	yes	—
143	#533	—	2	1	no	—
144	#532	—	2	1	no	—
145	#500	C₂²×C₁₃	1	1	yes	characteristic
146	#500	C₂²×C₁₃	1	1	yes	characteristic
147	#500	C₂²×C₁₃	1	1	yes	characteristic
148	#499	—	2	1	no	—
149	#499	C₂²×C₁₃	1	2	yes	—
150	#498	—	2	1	no	—
151	#493	C₂²×C₁₃	1	1	yes	characteristic
152	#486	—	2	1	no	—
153	#466	—	2	1	no	—
154	#466	C₂²×C₁₃	1	2	yes	—
155	C₄×D₈	—	2	1	no	—
156	C₄×D₈	C₂²×C₁₃	1	2	yes	—
157	#427	—	2	1	no	—
158	#427	C₂²×C₁₃	1	2	yes	—
159	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
160	#367	C₂²×C₁₃	1	4	yes	—
161	#366	C₂²×C₁₃	1	4	yes	—
162	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
163	#357	C₂²×C₁₃	1	4	yes	—
164	#357	C₂²×C₁₃	1	4	yes	—
165	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
166	#346	C₂²×C₁₃	1	2	yes	—
167	C₂²×C₁₃	—	2	1	no	—
168	C₂²×C₁₃	—	2	1	no	—
169	C₂²×C₁₃	—	2	1	no	—
170	C₂²×C₁₃	—	8	2	no	—
171	C₂²×C₁₃	—	8	2	no	—
172	C₂²×C₁₃	—	8	2	no	—
173	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
174	C₄×C₁₃	—	2	1	no	—
175	C₄×C₁₃	—	2	1	no	—
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	—	2	1	no	—
178	C₄×C₁₃	—	8	2	no	—
179	C₄×C₁₃	—	8	2	no	—
180	C₄×C₁₃	—	8	2	no	—
181	C₄×C₁₃	#509	1	2	yes	—
182	C₄×C₁₃	#536	1	1	yes	characteristic
183	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
184	C₂×#41	—	4	1	no	—
185	C₂×D₈	—	2	1	no	—
186	C₂×D₈	C₂³×C₁₃	1	2	yes	—
187	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
189	C₄⋊Q₈	—	2	1	no	—
190	C₄⋊Q₈	—	2	1	no	—
191	C₄⋊Q₈	—	2	1	no	—
192	#127	—	2	1	no	—
193	#125	—	2	2	no	—
194	#124	—	4	1	no	—
195	C₂²⋊Q₈	—	4	1	no	—
196	C₂²⋊Q₈	—	4	1	no	—
197	#121	—	2	2	no	—
198	#121	—	4	1	no	—
199	C₄×Q₈	—	4	1	no	—
200	C₄×Q₈	—	4	1	no	—
201	C₄×D₄	—	2	2	no	—
202	C₄×D₄	—	4	1	no	—
203	#117	—	2	2	no	—
204	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
205	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
206	#107	—	2	1	no	—
207	#107	—	2	1	no	—
208	#107	—	2	1	no	—
209	#107	C₂³×C₁₃	1	2	yes	—
210	#107	C₂³×C₁₃	1	2	yes	—
211	#107	C₂³×C₁₃	1	2	yes	—
212	#106	—	2	2	no	—
213	#106	—	2	2	no	—
214	#102	—	2	2	no	—
215	#102	—	2	2	no	—
216	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
217	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
218	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
219	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
220	C₂×C₁₃	—	4	1	no	—
221	C₂×C₁₃	—	16	2	no	—
222	C₂×C₁₃	#3226	1	1	yes	characteristic
223	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
224	C₂×C₁₃	#3414	1	1	yes	characteristic
225	#41	—	8	4	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂×D₄	—	4	1	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
235	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
237	D₈	—	2	4	no	—
238	D₈	—	4	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
242	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
243	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
244	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₄²	—	4	2	no	—
257	C₄²	C₁₃×D₈	1	1	yes	characteristic
258	C₄²	C₁₃×D₈	1	1	yes	characteristic
259	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₁₆	—	2	4	no	—
262	C₁₃	#33801	1	1	yes	characteristic, direct product factor, Sylow
263	C₂³	—	4	2	no	—
264	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
265	Q₈	—	8	2	no	—
266	Q₈	—	8	2	no	—
267	Q₈	—	8	2	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	2	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	D₄	—	8	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	8	4	no	—
281	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
282	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
283	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
284	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
285	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
286	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	C₁₃×#137	1	2	yes	—
290	C₈	C₁₃×2₋¹⁺⁴	1	2	yes	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	8	2	no	—
296	C₂²	—	8	2	no	—
297	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	8	2	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	C₁₃×#509	1	2	yes	—
306	C₄	C₁₃×#536	1	1	yes	characteristic
307	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
308	C₂	—	4	1	no	—
309	C₂	—	16	2	no	—
310	C₂	C₁₃×#3226	1	1	yes	characteristic
311	C₂	C₂×C₁₃×#509	1	1	yes	characteristic
312	C₂	C₁₃×#3414	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×#2255	C₂	semidirect
4. C₁₃×#2244	C₂	semidirect
5. C₁₃×#2234	C₂	semidirect
6. C₁₃×#2226	C₂	semidirect
7. C₁₃×#2224	C₂	semidirect
8. C₁₃×#2163	C₂	semidirect
69. #33801	C₁₃	direct (N × H)
108. #3160	C₂×C₁₃	semidirect
109. #2255	C₂×C₁₃	semidirect
110. #2244	C₂×C₁₃	semidirect
111. #2234	C₂×C₁₃	semidirect
112. #2226	C₂×C₁₃	semidirect
113. #2224	C₂×C₁₃	semidirect
114. #2163	C₂×C₁₃	semidirect
262. C₁₃	#33801	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×#2255	2
C₁₃×#2244	2
C₁₃×#2234	2
C₁₃×#2226	2
C₁₃×#2224	2
C₁₃×#2163	2
#33801	13

Subgroups — C13×#33801 #15720140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33801 #15720140