Subgroups — C₁₃×#18701 #15705040

Order 3328.

924 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3548	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#3254	C₂	1	2	yes	maximal, complemented
3	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1654	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1647	C₂	1	4	yes	maximal, complemented
6	C₁₃×#1561	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1492	C₂	1	2	yes	maximal, complemented
8	C₁₃×#562	—	2	2	no	—
9	C₁₃×#554	—	2	2	no	—
10	C₁₃×#550	—	2	2	no	—
11	C₁₃×#547	C₂²	1	4	yes	—
12	C₁₃×#540	—	2	2	no	—
13	C₁₃×#537	—	2	4	no	—
14	C₁₃×#533	C₂²	1	1	yes	characteristic
15	C₁₃×#533	C₂²	1	1	yes	characteristic
16	C₁₃×#532	C₂²	1	2	yes	—
17	C₁₃×#500	C₂²	1	2	yes	—
18	C₁₃×#499	—	2	2	no	—
19	C₁₃×#498	C₂²	1	2	yes	—
20	C₁₃×#477	—	2	2	no	—
21	C₁₃×#474	C₂²	1	4	yes	—
22	C₁₃×#468	—	2	4	no	—
23	C₁₃×#461	C₂²	1	4	yes	—
24	C₁₃×#460	—	2	2	no	—
25	C₁₃×#427	—	2	2	no	—
26	C₁₃×#423	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#332	C₂²	1	4	yes	—
29	C₁₃×D₄⋊C₈	C₂²	1	4	yes	—
30	C₁₃×D₄⋊C₈	C₂²	1	4	yes	—
31	C₂×C₁₃×#41	—	2	4	no	—
32	C₂×C₁₃×#41	—	2	2	no	—
33	C₂×C₁₃×SD₁₆	—	2	4	no	—
34	C₂×C₁₃×M₁₆	—	2	2	no	—
35	C₂²×C₈×C₁₃	—	2	1	no	—
36	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
37	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
38	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
39	C₁₃×#127	—	2	2	no	—
40	C₁₃×#127	D₄	1	2	yes	—
41	C₁₃×#125	—	2	2	no	—
42	C₁₃×#125	—	2	1	no	—
43	C₁₃×#125	D₄	1	2	yes	—
44	C₁₃×#124	—	2	2	no	—
45	C₁₃×#123	—	2	8	no	—
46	C₁₃×C₂²⋊Q₈	—	2	8	no	—
47	C₁₃×C₂²⋊Q₈	—	2	2	no	—
48	C₁₃×C₂²⋊Q₈	—	2	4	no	—
49	C₁₃×C₂²⋊Q₈	—	2	2	no	—
50	C₁₃×#121	—	2	8	no	—
51	C₁₃×#121	—	2	8	no	—
52	C₁₃×#121	—	2	4	no	—
53	C₁₃×#121	—	2	2	no	—
54	C₁₃×#121	D₄	1	4	yes	—
55	C₄×Q₈×C₁₃	—	2	2	no	—
56	C₄×Q₈×C₁₃	—	2	2	no	—
57	C₄×D₄×C₁₃	—	2	4	no	—
58	C₄×D₄×C₁₃	—	2	2	no	—
59	C₄×D₄×C₁₃	C₂³	1	4	yes	—
60	C₁₃×#117	—	2	1	no	—
61	C₁₃×#117	D₄	1	2	yes	—
62	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
63	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
64	C₁₃×#107	—	2	4	no	—
65	C₁₃×#107	—	2	4	no	—
66	C₁₃×#106	—	2	4	no	—
67	C₁₃×C₄⋊C₈	—	2	2	no	—
68	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
69	C₁₃×#103	—	2	4	no	—
70	C₁₃×#102	—	2	4	no	—
71	C₄×C₈×C₁₃	—	2	1	no	—
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	#18701	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₁₃×#41	—	8	2	no	—
76	C₁₃×#41	—	8	2	no	—
77	C₁₃×#41	—	8	4	no	—
78	C₁₃×#41	—	8	4	no	—
79	C₂×Q₈×C₁₃	—	2	2	no	—
80	C₂×Q₈×C₁₃	—	2	2	no	—
81	C₂×Q₈×C₁₃	—	2	2	no	—
82	C₂×D₄×C₁₃	—	2	8	no	—
83	C₂×D₄×C₁₃	—	2	4	no	—
84	C₂×D₄×C₁₃	—	2	2	no	—
85	C₂×D₄×C₁₃	—	2	2	no	—
86	C₂×D₄×C₁₃	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₁₃	—	2	8	no	—
88	C₂²×C₄×C₁₃	—	2	4	no	—
89	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
91	C₁₃×SD₁₆	—	8	4	no	—
92	C₁₃×M₁₆	—	8	2	no	—
93	C₂×C₈×C₁₃	—	2	4	no	—
94	C₂×C₈×C₁₃	—	2	1	no	—
95	C₂×C₈×C₁₃	—	2	1	no	—
96	C₂×C₈×C₁₃	—	2	4	no	—
97	C₁₃×C₄⋊C₄	—	2	8	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	4	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	2	2	no	—
103	C₁₃×C₄⋊C₄	—	4	2	no	—
104	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
105	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
106	C₁₃×C₂²⋊C₄	—	2	8	no	—
107	C₁₃×C₂²⋊C₄	—	2	2	no	—
108	C₁₃×C₂²⋊C₄	—	2	4	no	—
109	C₁₃×C₂²⋊C₄	—	2	4	no	—
110	C₁₃×C₂²⋊C₄	—	4	1	no	—
111	C₄²×C₁₃	—	2	2	no	—
112	C₄²×C₁₃	—	4	1	no	—
113	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₃	C₂×D₄	1	2	yes	—
115	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
116	#3548	C₂×C₁₃	1	1	yes	characteristic
117	#3254	C₂×C₁₃	1	2	yes	complemented
118	#3160	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#1654	C₂×C₁₃	1	4	yes	complemented
120	#1647	C₂×C₁₃	1	4	yes	complemented
121	#1561	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#1492	C₂×C₁₃	1	2	yes	complemented
123	C₂³×C₁₃	—	2	4	no	—
124	C₂³×C₁₃	#120	1	1	yes	characteristic
125	Q₈×C₁₃	—	4	2	no	—
126	Q₈×C₁₃	—	4	2	no	—
127	Q₈×C₁₃	—	4	2	no	—
128	Q₈×C₁₃	—	4	2	no	—
129	Q₈×C₁₃	—	8	2	no	—
130	D₄×C₁₃	—	2	4	no	—
131	D₄×C₁₃	—	2	4	no	—
132	D₄×C₁₃	—	4	4	no	—
133	D₄×C₁₃	—	8	2	no	—
134	D₄×C₁₃	—	8	4	no	—
135	D₄×C₁₃	—	8	8	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	4	no	—
140	C₂×C₄×C₁₃	—	2	4	no	—
141	C₂×C₄×C₁₃	—	4	4	no	—
142	C₂×C₄×C₁₃	—	4	4	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	8	8	no	—
145	C₂×C₄×C₁₃	#120	1	2	yes	—
146	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
149	C₈×C₁₃	—	2	2	no	—
150	C₈×C₁₃	—	2	2	no	—
151	C₈×C₁₃	—	4	4	no	—
152	#562	—	2	2	no	—
153	#554	—	2	2	no	—
154	#550	—	2	2	no	—
155	#547	C₂²×C₁₃	1	4	yes	—
156	#540	—	2	2	no	—
157	#537	—	2	4	no	—
158	#533	C₂²×C₁₃	1	1	yes	characteristic
159	#533	C₂²×C₁₃	1	1	yes	characteristic
160	#532	C₂²×C₁₃	1	2	yes	—
161	#500	C₂²×C₁₃	1	2	yes	—
162	#499	—	2	2	no	—
163	#498	C₂²×C₁₃	1	2	yes	—
164	#477	—	2	2	no	—
165	#474	C₂²×C₁₃	1	4	yes	—
166	#468	—	2	4	no	—
167	#461	C₂²×C₁₃	1	4	yes	—
168	#460	—	2	2	no	—
169	#427	—	2	2	no	—
170	#423	C₂²×C₁₃	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
172	#332	C₂²×C₁₃	1	4	yes	—
173	D₄⋊C₈	C₂²×C₁₃	1	4	yes	—
174	D₄⋊C₈	C₂²×C₁₃	1	4	yes	—
175	C₂²×C₁₃	—	2	2	no	—
176	C₂²×C₁₃	—	2	1	no	—
177	C₂²×C₁₃	—	4	4	no	—
178	C₂²×C₁₃	—	4	4	no	—
179	C₂²×C₁₃	—	4	4	no	—
180	C₂²×C₁₃	C₂×#120	1	1	yes	characteristic
181	C₄×C₁₃	—	2	2	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	4	4	no	—
184	C₄×C₁₃	—	4	4	no	—
185	C₄×C₁₃	—	8	2	no	—
186	C₄×C₁₃	#456	1	2	yes	—
187	C₄×C₁₃	#576	1	2	yes	—
188	C₄×C₁₃	#577	1	2	yes	—
189	C₂×#41	—	2	4	no	—
190	C₂×#41	—	2	2	no	—
191	C₂×SD₁₆	—	2	4	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂²×C₈	—	2	1	no	—
194	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
195	C₄⋊Q₈	C₂³×C₁₃	1	2	yes	—
196	C₄⋊Q₈	C₂³×C₁₃	1	2	yes	—
197	#127	—	2	2	no	—
198	#127	D₄×C₁₃	1	2	yes	—
199	#125	—	2	2	no	—
200	#125	—	2	1	no	—
201	#125	D₄×C₁₃	1	2	yes	—
202	#124	—	2	2	no	—
203	#123	—	2	8	no	—
204	C₂²⋊Q₈	—	2	8	no	—
205	C₂²⋊Q₈	—	2	2	no	—
206	C₂²⋊Q₈	—	2	4	no	—
207	C₂²⋊Q₈	—	2	2	no	—
208	#121	—	2	8	no	—
209	#121	—	2	8	no	—
210	#121	—	2	4	no	—
211	#121	—	2	2	no	—
212	#121	D₄×C₁₃	1	4	yes	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×D₄	—	2	4	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	C₂³×C₁₃	1	4	yes	—
218	#117	—	2	1	no	—
219	#117	D₄×C₁₃	1	2	yes	—
220	C₂×C₄⋊C₄	—	2	4	no	—
221	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
222	#107	—	2	4	no	—
223	#107	—	2	4	no	—
224	#106	—	2	4	no	—
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	C₂³×C₁₃	1	4	yes	—
227	#103	—	2	4	no	—
228	#102	—	2	4	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
232	C₂×C₁₃	—	4	1	no	—
233	C₂×C₁₃	—	8	4	no	—
234	C₂×C₁₃	#3012	1	2	yes	—
235	C₂×C₁₃	C₂×#456	1	1	yes	characteristic
236	#41	—	8	2	no	—
237	#41	—	8	2	no	—
238	#41	—	8	4	no	—
239	#41	—	8	4	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×D₄	—	2	8	no	—
244	C₂×D₄	—	2	4	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	C₂×D₄×C₁₃	1	4	yes	—
248	C₂²×C₄	—	2	8	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
252	SD₁₆	—	8	4	no	—
253	M₁₆	—	8	2	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	4	no	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
266	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
267	C₂²⋊C₄	—	2	8	no	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	4	no	—
270	C₂²⋊C₄	—	2	4	no	—
271	C₂²⋊C₄	—	4	1	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	4	1	no	—
274	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
275	C₄²	C₂×D₄×C₁₃	1	2	yes	—
276	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
277	C₁₃	#18701	1	1	yes	characteristic, direct product factor, Sylow
278	C₂³	—	2	4	no	—
279	C₂³	C₁₃×#120	1	1	yes	characteristic
280	Q₈	—	4	2	no	—
281	Q₈	—	4	2	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	8	2	no	—
285	D₄	—	2	4	no	—
286	D₄	—	2	4	no	—
287	D₄	—	4	4	no	—
288	D₄	—	8	2	no	—
289	D₄	—	8	4	no	—
290	D₄	—	8	8	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	8	no	—
300	C₂×C₄	C₁₃×#120	1	2	yes	—
301	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
304	C₈	—	2	2	no	—
305	C₈	—	2	2	no	—
306	C₈	—	4	4	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	C₂×C₁₃×#120	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	4	no	—
316	C₄	—	4	4	no	—
317	C₄	—	8	2	no	—
318	C₄	C₁₃×#456	1	2	yes	—
319	C₄	C₁₃×#576	1	2	yes	—
320	C₄	C₁₃×#577	1	2	yes	—
321	C₂	—	4	1	no	—
322	C₂	—	8	4	no	—
323	C₂	C₁₃×#3012	1	2	yes	—
324	C₂	C₂×C₁₃×#456	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃
prime radical	C₁₃×#3548	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#3254	C₂	semidirect
3. C₁₃×#3160	C₂	semidirect
4. C₁₃×#1654	C₂	semidirect
5. C₁₃×#1647	C₂	semidirect
6. C₁₃×#1561	C₂	semidirect
7. C₁₃×#1492	C₂	semidirect
74. #18701	C₁₃	direct (N × H)
117. #3254	C₂×C₁₃	semidirect
118. #3160	C₂×C₁₃	semidirect
119. #1654	C₂×C₁₃	semidirect
120. #1647	C₂×C₁₃	semidirect
121. #1561	C₂×C₁₃	semidirect
122. #1492	C₂×C₁₃	semidirect
277. C₁₃	#18701	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×#3548	2
C₁₃×#3254	2
C₁₃×#3160	2
C₁₃×#1654	2
C₁₃×#1647	2
C₁₃×#1561	2
C₁₃×#1492	2
#18701	13

Subgroups — C13×#18701 #15705040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#18701 #15705040