Subgroups — C₁₃×#33797 #15720136

Order 3328.

460 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2237	C₂	1	4	yes	maximal
8	C₁₃×#2229	C₂	1	4	yes	maximal, complemented
9	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic
10	C₁₃×#533	—	2	1	no	—
11	C₁₃×#533	—	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×#494	C₂²	1	1	yes	characteristic
19	C₁₃×#487	—	2	1	no	—
20	C₁₃×#467	—	2	1	no	—
21	C₁₃×#467	C₂²	1	2	yes	—
22	C₄×C₁₃×Q₁₆	—	2	1	no	—
23	C₄×C₁₃×Q₁₆	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×#358	C₂²	1	4	yes	—
31	C₁₃×#358	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×Q₁₆	—	2	1	no	—
36	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₁₃×C₄⋊Q₈	—	2	1	no	—
40	C₁₃×C₄⋊Q₈	—	2	1	no	—
41	C₁₃×C₄⋊Q₈	—	2	1	no	—
42	C₁₃×C₄⋊Q₈	—	2	1	no	—
43	C₁₃×#125	—	2	2	no	—
44	C₁₃×#125	—	4	1	no	—
45	C₁₃×C₂²⋊Q₈	—	2	2	no	—
46	C₁₃×C₂²⋊Q₈	—	4	1	no	—
47	C₁₃×C₂²⋊Q₈	—	4	1	no	—
48	C₁₃×C₂²⋊Q₈	—	4	1	no	—
49	C₄×Q₈×C₁₃	—	2	2	no	—
50	C₄×Q₈×C₁₃	—	4	1	no	—
51	C₄×Q₈×C₁₃	—	4	1	no	—
52	C₄×Q₈×C₁₃	—	4	1	no	—
53	C₁₃×#117	—	2	2	no	—
54	C₁₃×#117	—	4	1	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×#103	—	2	2	no	—
67	C₁₃×#103	—	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	#33797	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂×Q₈×C₁₃	—	2	2	no	—
74	C₂×Q₈×C₁₃	—	4	1	no	—
75	C₂×Q₈×C₁₃	—	4	1	no	—
76	C₂×Q₈×C₁₃	—	4	1	no	—
77	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₁₃×Q₁₆	—	2	4	no	—
81	C₁₃×Q₁₆	—	4	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	—	4	1	no	—
92	C₁₃×C₄⋊C₄	—	4	1	no	—
93	C₁₃×C₄⋊C₄	—	4	2	no	—
94	C₁₃×C₄⋊C₄	—	4	2	no	—
95	C₁₃×C₄⋊C₄	—	4	2	no	—
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₁₃×C₄⋊C₄	—	4	1	no	—
98	C₁₃×C₄⋊C₄	—	4	1	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₂²⋊C₄	—	4	2	no	—
101	C₁₃×C₂²⋊C₄	—	4	2	no	—
102	C₄²×C₁₃	—	4	2	no	—
103	C₄²×C₁₃	—	4	2	no	—
104	C₄²×C₁₃	D₈	1	1	yes	characteristic
105	C₄²×C₁₃	D₈	1	1	yes	characteristic
106	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₁₃×C₁₆	—	2	2	no	—
109	C₁₃×C₁₆	—	2	2	no	—
110	#3170	C₂×C₁₃	1	1	yes	characteristic
111	#3160	C₂×C₁₃	1	1	yes	characteristic
112	#2254	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#2252	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#2241	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#2241	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#2237	C₂×C₁₃	1	4	yes	—
117	#2229	C₂×C₁₃	1	4	yes	complemented
118	#2162	C₂×C₁₃	1	1	yes	characteristic
119	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₁₃	—	4	2	no	—
121	Q₈×C₁₃	—	4	2	no	—
122	Q₈×C₁₃	—	8	2	no	—
123	Q₈×C₁₃	—	8	2	no	—
124	Q₈×C₁₃	—	8	2	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	—	2	2	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	—	4	2	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	#533	—	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	#494	C₂²×C₁₃	1	1	yes	characteristic
152	#487	—	2	1	no	—
153	#467	—	2	1	no	—
154	#467	C₂²×C₁₃	1	2	yes	—
155	C₄×Q₁₆	—	2	1	no	—
156	C₄×Q₁₆	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	#358	C₂²×C₁₃	1	4	yes	—
164	#358	C₂²×C₁₃	1	4	yes	—
165	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
166	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
167	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
168	C₂²×C₁₃	—	2	1	no	—
169	C₂²×C₁₃	—	2	1	no	—
170	C₂²×C₁₃	—	2	1	no	—
171	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
172	C₄×C₁₃	—	2	1	no	—
173	C₄×C₁₃	—	2	1	no	—
174	C₄×C₁₃	—	2	1	no	—
175	C₄×C₁₃	—	8	2	no	—
176	C₄×C₁₃	—	8	2	no	—
177	C₄×C₁₃	—	8	2	no	—
178	C₄×C₁₃	—	8	2	no	—
179	C₄×C₁₃	#508	1	2	yes	—
180	C₄×C₁₃	#510	1	2	yes	—
181	C₄×C₁₃	#536	1	1	yes	characteristic
182	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
183	C₂×Q₁₆	—	2	1	no	—
184	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
185	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
187	C₄⋊Q₈	—	2	1	no	—
188	C₄⋊Q₈	—	2	1	no	—
189	C₄⋊Q₈	—	2	1	no	—
190	C₄⋊Q₈	—	2	1	no	—
191	#125	—	2	2	no	—
192	#125	—	4	1	no	—
193	C₂²⋊Q₈	—	2	2	no	—
194	C₂²⋊Q₈	—	4	1	no	—
195	C₂²⋊Q₈	—	4	1	no	—
196	C₂²⋊Q₈	—	4	1	no	—
197	C₄×Q₈	—	2	2	no	—
198	C₄×Q₈	—	4	1	no	—
199	C₄×Q₈	—	4	1	no	—
200	C₄×Q₈	—	4	1	no	—
201	#117	—	2	2	no	—
202	#117	—	4	1	no	—
203	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
204	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
205	C₂×C₁₆	C₂³×C₁₃	1	2	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	#103	—	2	2	no	—
215	#103	—	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	#3226	1	1	yes	characteristic
222	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
223	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	4	1	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
229	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
231	Q₁₆	—	2	4	no	—
232	Q₁₆	—	4	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
236	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
238	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₂²⋊C₄	—	4	2	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₄²	—	4	2	no	—
254	C₄²	—	4	2	no	—
255	C₄²	C₁₃×D₈	1	1	yes	characteristic
256	C₄²	C₁₃×D₈	1	1	yes	characteristic
257	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₁₆	—	2	2	no	—
260	C₁₆	—	2	2	no	—
261	C₁₃	#33797	1	1	yes	characteristic, direct product factor, Sylow
262	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
263	Q₈	—	4	2	no	—
264	Q₈	—	4	2	no	—
265	Q₈	—	8	2	no	—
266	Q₈	—	8	2	no	—
267	Q₈	—	8	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
277	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
278	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
279	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
280	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	C₁₃×#137	1	2	yes	—
285	C₈	C₁₃×2₋¹⁺⁴	1	2	yes	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	8	2	no	—
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	—	8	2	no	—
297	C₄	C₁₃×#508	1	2	yes	—
298	C₄	C₁₃×#510	1	2	yes	—
299	C₄	C₁₃×#536	1	1	yes	characteristic
300	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
301	C₂	—	4	1	no	—
302	C₂	C₁₃×#3226	1	1	yes	characteristic
303	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
304	C₂	C₂×C₁₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂³×C₁₃	C₂×D₈
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
3. C₁₃×#2254	C₂	semidirect
4. C₁₃×#2252	C₂	semidirect
5. C₁₃×#2241	C₂	semidirect
6. C₁₃×#2241	C₂	semidirect
8. C₁₃×#2229	C₂	semidirect
72. #33797	C₁₃	direct (N × H)
112. #2254	C₂×C₁₃	semidirect
113. #2252	C₂×C₁₃	semidirect
114. #2241	C₂×C₁₃	semidirect
115. #2241	C₂×C₁₃	semidirect
117. #2229	C₂×C₁₃	semidirect
261. C₁₃	#33797	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×#3170	2
C₁₃×#3160	2
C₁₃×#2254	2
C₁₃×#2252	2
C₁₃×#2241	2
C₁₃×#2237	2
C₁₃×#2229	2
C₁₃×#2162	2
#33797	13

Subgroups — C13×#33797 #15720136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33797 #15720136