Subgroups — C₁₃×#18718 #15705057

Order 3328.

1004 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3550	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3208	C₂	1	2	yes	maximal, complemented
3	C₁₃×#3151	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1623	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1620	C₂	1	4	yes	maximal, complemented
6	C₁₃×#1543	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1485	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₃×AGL(1,ℤ/8)	—	2	2	no	—
9	C₁₃×#563	—	2	2	no	—
10	C₁₃×#559	C₂²	1	2	yes	complemented
11	Q₈²×C₁₃	C₂²	1	2	yes	complemented
12	C₁₃×#556	—	2	2	no	—
13	D₄×Q₈×C₁₃	—	2	2	no	—
14	C₁₃×#542	—	2	4	no	—
15	C₁₃×#536	—	2	2	no	—
16	C₁₃×#533	C₂²	1	2	yes	—
17	C₁₃×#532	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
19	C₁₃×#496	—	2	2	no	—
20	C₁₃×#493	C₂²	1	2	yes	—
21	C₁₃×#492	—	2	2	no	—
22	C₁₃×#492	C₂²	1	2	yes	—
23	C₁₃×#486	—	2	2	no	—
24	C₁₃×#460	C₂²	1	8	yes	—
25	C₁₃×#450	—	2	4	no	—
26	C₁₃×#431	C₂²	1	2	yes	—
27	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
28	C₁₃×#333	C₂²	1	2	yes	complemented
29	C₁₃×#331	C₂²	1	2	yes	complemented
30	C₁₃×Q₈⋊C₈	C₂²	1	8	yes	complemented
31	C₂×C₁₃×#41	D₄	1	2	yes	—
32	C₂²×Q₈×C₁₃	—	2	2	no	—
33	C₂²×D₄×C₁₃	—	2	1	no	—
34	C₂²×D₄×C₁₃	—	2	2	no	—
35	C₁₃×AGL(1,ℤ/8)	—	4	8	no	—
36	C₂×C₁₃×SD₁₆	—	2	4	no	—
37	C₂×C₁₃×SD₁₆	—	2	8	no	—
38	C₂×C₁₃×D₈	—	2	4	no	—
39	C₂×C₁₃×M₁₆	—	2	1	no	—
40	C₁₃×C₄⋊Q₈	—	2	4	no	—
41	C₁₃×C₄⋊Q₈	—	2	4	no	—
42	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
43	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
44	C₁₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
45	C₁₃×#127	—	2	2	no	—
46	C₁₃×#127	—	2	1	no	—
47	C₁₃×#127	C₂³	1	2	yes	—
48	C₁₃×#127	C₂³	1	1	yes	characteristic
49	C₁₃×#126	—	2	8	no	—
50	C₁₃×#125	—	2	2	no	—
51	C₁₃×#124	—	2	4	no	—
52	C₁₃×#124	—	2	4	no	—
53	C₁₃×#124	D₄	1	2	yes	—
54	C₁₃×#123	—	2	4	no	—
55	C₁₃×#123	—	2	4	no	—
56	C₁₃×C₂²⋊Q₈	—	2	4	no	—
57	C₁₃×C₂²⋊Q₈	—	2	8	no	—
58	C₁₃×C₂²⋊Q₈	—	2	2	no	—
59	C₁₃×C₂²⋊Q₈	D₄	1	4	yes	—
60	C₁₃×#121	—	2	2	no	—
61	C₄×Q₈×C₁₃	—	2	4	no	—
62	C₄×Q₈×C₁₃	—	2	2	no	—
63	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
64	C₄×D₄×C₁₃	—	2	4	no	—
65	C₄×D₄×C₁₃	—	2	2	no	—
66	C₁₃×#117	—	2	2	no	—
67	C₁₃×#117	—	2	2	no	—
68	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
69	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
70	C₁₃×#102	—	2	8	no	—
71	C₁₃×C₂²⋊C₈	—	2	2	no	—
72	C₁₃×#97	—	2	1	no	—
73	C₄×C₈×C₁₃	—	2	2	no	—
74	C₄×C₈×C₁₃	C₂³	1	2	yes	—
75	#18718	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂⁴×C₁₃	—	4	1	no	—
77	C₁₃×#41	—	4	4	no	—
78	C₂×Q₈×C₁₃	—	2	4	no	—
79	C₂×Q₈×C₁₃	—	2	2	no	—
80	C₂×Q₈×C₁₃	—	8	4	no	—
81	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
82	C₂×Q₈×C₁₃	C₂×D₄	1	2	yes	—
83	C₂×D₄×C₁₃	—	2	2	no	—
84	C₂×D₄×C₁₃	—	2	2	no	—
85	C₂×D₄×C₁₃	—	2	4	no	—
86	C₂×D₄×C₁₃	—	2	2	no	—
87	C₂×D₄×C₁₃	—	4	4	no	—
88	C₂×D₄×C₁₃	—	4	2	no	—
89	C₂×D₄×C₁₃	—	4	1	no	—
90	C₂×D₄×C₁₃	—	8	4	no	—
91	C₂×D₄×C₁₃	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₁₃	—	2	4	no	—
93	C₂²×C₄×C₁₃	—	2	2	no	—
94	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₁₃×SD₁₆	—	4	8	no	—
97	C₁₃×SD₁₆	—	8	8	no	—
98	C₁₃×D₈	—	4	8	no	—
99	C₁₃×M₁₆	—	2	4	no	—
100	C₂×C₈×C₁₃	—	2	2	no	—
101	C₂×C₈×C₁₃	—	2	4	no	—
102	C₁₃×C₄⋊C₄	—	2	8	no	—
103	C₁₃×C₄⋊C₄	—	2	2	no	—
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	2	2	no	—
106	C₁₃×C₄⋊C₄	—	2	4	no	—
107	C₁₃×C₄⋊C₄	—	2	2	no	—
108	C₁₃×C₄⋊C₄	—	2	4	no	—
109	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
110	C₁₃×C₂²⋊C₄	—	2	4	no	—
111	C₁₃×C₂²⋊C₄	—	2	8	no	—
112	C₁₃×C₂²⋊C₄	—	2	4	no	—
113	C₁₃×C₂²⋊C₄	—	2	4	no	—
114	C₄²×C₁₃	—	2	4	no	—
115	C₄²×C₁₃	—	2	4	no	—
116	C₄²×C₁₃	C₂×D₄	1	2	yes	—
117	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
119	#3550	C₂×C₁₃	1	1	yes	characteristic, complemented
120	#3208	C₂×C₁₃	1	2	yes	complemented
121	#3151	C₂×C₁₃	1	1	yes	characteristic
122	#1623	C₂×C₁₃	1	4	yes	complemented
123	#1620	C₂×C₁₃	1	4	yes	complemented
124	#1543	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#1485	C₂×C₁₃	1	2	yes	complemented
126	C₂³×C₁₃	—	2	2	no	—
127	C₂³×C₁₃	—	4	2	no	—
128	C₂³×C₁₃	—	8	2	no	—
129	C₂³×C₁₃	—	8	4	no	—
130	C₂³×C₁₃	#120	1	1	yes	characteristic
131	Q₈×C₁₃	—	2	8	no	—
132	Q₈×C₁₃	—	4	2	no	—
133	Q₈×C₁₃	—	4	4	no	—
134	Q₈×C₁₃	—	8	4	no	—
135	D₄×C₁₃	—	4	2	no	—
136	D₄×C₁₃	—	4	4	no	—
137	D₄×C₁₃	—	4	4	no	—
138	D₄×C₁₃	—	4	8	no	—
139	D₄×C₁₃	—	8	4	no	—
140	D₄×C₁₃	—	8	2	no	—
141	D₄×C₁₃	—	8	2	no	—
142	C₂×C₄×C₁₃	—	2	4	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	2	4	no	—
145	C₂×C₄×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	—	2	4	no	—
147	C₂×C₄×C₁₃	—	4	4	no	—
148	C₂×C₄×C₁₃	—	8	4	no	—
149	C₂×C₄×C₁₃	#120	1	2	yes	—
150	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
153	C₈×C₁₃	—	2	4	no	—
154	C₈×C₁₃	—	4	4	no	—
155	C₂×AGL(1,ℤ/8)	—	2	2	no	—
156	#563	—	2	2	no	—
157	#559	C₂²×C₁₃	1	2	yes	complemented
158	Q₈²	C₂²×C₁₃	1	2	yes	complemented
159	#556	—	2	2	no	—
160	D₄×Q₈	—	2	2	no	—
161	#542	—	2	4	no	—
162	#536	—	2	2	no	—
163	#533	C₂²×C₁₃	1	2	yes	—
164	#532	C₂²×C₁₃	1	1	yes	characteristic
165	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
166	#496	—	2	2	no	—
167	#493	C₂²×C₁₃	1	2	yes	—
168	#492	—	2	2	no	—
169	#492	C₂²×C₁₃	1	2	yes	—
170	#486	—	2	2	no	—
171	#460	C₂²×C₁₃	1	8	yes	—
172	#450	—	2	4	no	—
173	#431	C₂²×C₁₃	1	2	yes	—
174	C₄×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
175	#333	C₂²×C₁₃	1	2	yes	complemented
176	#331	C₂²×C₁₃	1	2	yes	complemented
177	Q₈⋊C₈	C₂²×C₁₃	1	8	yes	complemented
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	4	4	no	—
181	C₂²×C₁₃	—	4	2	no	—
182	C₂²×C₁₃	—	8	4	no	—
183	C₂²×C₁₃	—	8	2	no	—
184	C₂²×C₁₃	—	16	4	no	—
185	C₂²×C₁₃	C₂×#120	1	1	yes	characteristic
186	C₄×C₁₃	—	2	2	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	4	4	no	—
189	C₄×C₁₃	—	4	2	no	—
190	C₄×C₁₃	—	4	4	no	—
191	C₄×C₁₃	#453	1	2	yes	—
192	C₄×C₁₃	#577	1	4	yes	—
193	C₂×#41	D₄×C₁₃	1	2	yes	—
194	C₂²×Q₈	—	2	2	no	—
195	C₂²×D₄	—	2	1	no	—
196	C₂²×D₄	—	2	2	no	—
197	AGL(1,ℤ/8)	—	4	8	no	—
198	C₂×SD₁₆	—	2	4	no	—
199	C₂×SD₁₆	—	2	8	no	—
200	C₂×D₈	—	2	4	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₄⋊Q₈	—	2	4	no	—
203	C₄⋊Q₈	—	2	4	no	—
204	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
205	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
206	C₄⋊Q₈	C₂³×C₁₃	1	1	yes	characteristic
207	#127	—	2	2	no	—
208	#127	—	2	1	no	—
209	#127	C₂³×C₁₃	1	2	yes	—
210	#127	C₂³×C₁₃	1	1	yes	characteristic
211	#126	—	2	8	no	—
212	#125	—	2	2	no	—
213	#124	—	2	4	no	—
214	#124	—	2	4	no	—
215	#124	D₄×C₁₃	1	2	yes	—
216	#123	—	2	4	no	—
217	#123	—	2	4	no	—
218	C₂²⋊Q₈	—	2	4	no	—
219	C₂²⋊Q₈	—	2	8	no	—
220	C₂²⋊Q₈	—	2	2	no	—
221	C₂²⋊Q₈	D₄×C₁₃	1	4	yes	—
222	#121	—	2	2	no	—
223	C₄×Q₈	—	2	4	no	—
224	C₄×Q₈	—	2	2	no	—
225	C₄×Q₈	C₂³×C₁₃	1	4	yes	—
226	C₄×D₄	—	2	4	no	—
227	C₄×D₄	—	2	2	no	—
228	#117	—	2	2	no	—
229	#117	—	2	2	no	—
230	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
231	C₄⋊C₈	C₂³×C₁₃	1	4	yes	—
232	#102	—	2	8	no	—
233	C₂²⋊C₈	—	2	2	no	—
234	#97	—	2	1	no	—
235	C₄×C₈	—	2	2	no	—
236	C₄×C₈	C₂³×C₁₃	1	2	yes	—
237	C₂×C₁₃	—	4	1	no	—
238	C₂×C₁₃	—	8	2	no	—
239	C₂×C₁₃	—	16	2	no	—
240	C₂×C₁₃	#3010	1	2	yes	—
241	C₂×C₁₃	C₂×#453	1	1	yes	characteristic
242	C₂⁴	—	4	1	no	—
243	#41	—	4	4	no	—
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	8	4	no	—
247	C₂×Q₈	C₂×D₄×C₁₃	1	4	yes	—
248	C₂×Q₈	C₂×D₄×C₁₃	1	2	yes	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	4	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	4	4	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	8	4	no	—
257	C₂×D₄	C₂×D₄×C₁₃	1	2	yes	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
261	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
262	SD₁₆	—	4	8	no	—
263	SD₁₆	—	8	8	no	—
264	D₈	—	4	8	no	—
265	M₁₆	—	2	4	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	4	no	—
268	C₄⋊C₄	—	2	8	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	4	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
276	C₂²⋊C₄	—	2	4	no	—
277	C₂²⋊C₄	—	2	8	no	—
278	C₂²⋊C₄	—	2	4	no	—
279	C₂²⋊C₄	—	2	4	no	—
280	C₄²	—	2	4	no	—
281	C₄²	—	2	4	no	—
282	C₄²	C₂×D₄×C₁₃	1	2	yes	—
283	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
284	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
285	C₁₃	#18718	1	1	yes	characteristic, direct product factor, Sylow
286	C₂³	—	2	2	no	—
287	C₂³	—	4	2	no	—
288	C₂³	—	8	2	no	—
289	C₂³	—	8	4	no	—
290	C₂³	C₁₃×#120	1	1	yes	characteristic
291	Q₈	—	2	8	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	4	no	—
294	Q₈	—	8	4	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	4	no	—
297	D₄	—	4	4	no	—
298	D₄	—	4	8	no	—
299	D₄	—	8	4	no	—
300	D₄	—	8	2	no	—
301	D₄	—	8	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	C₁₃×#120	1	2	yes	—
310	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
311	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
313	C₈	—	2	4	no	—
314	C₈	—	4	4	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	8	4	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	16	4	no	—
322	C₂²	C₂×C₁₃×#120	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	4	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	4	no	—
328	C₄	C₁₃×#453	1	2	yes	—
329	C₄	C₁₃×#577	1	4	yes	—
330	C₂	—	4	1	no	—
331	C₂	—	8	2	no	—
332	C₂	—	16	2	no	—
333	C₂	C₁₃×#3010	1	2	yes	—
334	C₂	C₂×C₁₃×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃
prime radical	C₁₃×#3151	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
1. C₁₃×#3550	C₂	semidirect
2. C₁₃×#3208	C₂	semidirect
4. C₁₃×#1623	C₂	semidirect
5. C₁₃×#1620	C₂	semidirect
6. C₁₃×#1543	C₂	semidirect
7. C₁₃×#1485	C₂	semidirect
10. C₁₃×#559	C₂²	semidirect
11. Q₈²×C₁₃	C₂²	semidirect
28. C₁₃×#333	C₂²	semidirect
29. C₁₃×#331	C₂²	semidirect
30. C₁₃×Q₈⋊C₈	C₂²	semidirect
75. #18718	C₁₃	direct (N × H)
119. #3550	C₂×C₁₃	semidirect
120. #3208	C₂×C₁₃	semidirect
122. #1623	C₂×C₁₃	semidirect
123. #1620	C₂×C₁₃	semidirect
124. #1543	C₂×C₁₃	semidirect
125. #1485	C₂×C₁₃	semidirect
157. #559	C₂²×C₁₃	semidirect
158. Q₈²	C₂²×C₁₃	semidirect
175. #333	C₂²×C₁₃	semidirect
176. #331	C₂²×C₁₃	semidirect
177. Q₈⋊C₈	C₂²×C₁₃	semidirect
285. C₁₃	#18718	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×#3550	2
C₁₃×#3208	2
C₁₃×#3151	2
C₁₃×#1623	2
C₁₃×#1620	2
C₁₃×#1543	2
C₁₃×#1485	2
#18718	13

Subgroups — C13×#18718 #15705057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#18718 #15705057