Subgroups — C₁₃×#13393 #15699732

Order 3328.

412 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#472	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#357	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2113	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1816	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1387	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×AGL(1,ℤ/8)	—	2	1	no	—
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₁₃×#472	—	2	2	no	—
14	C₁₃×#472	—	2	2	no	—
15	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
18	C₁₃×#417	—	2	1	no	—
19	C₂×C₁₃×#100	—	2	1	no	—
20	C₁₃×#368	—	2	2	no	—
21	C₁₃×#361	—	2	2	no	—
22	C₁₃×#357	—	2	2	no	—
23	C₁₃×C₂²⋊C₁₆	—	2	2	no	—
24	C₁₃×#343	—	2	1	no	—
25	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
26	C₂²×D₄×C₁₃	—	2	1	no	—
27	C₁₃×AGL(1,ℤ/8)	—	2	4	no	—
28	C₁₃×AGL(1,ℤ/8)	—	4	2	no	—
29	C₂×C₁₃×SD₁₆	—	4	1	no	—
30	C₂×C₁₃×D₈	—	2	1	no	—
31	C₂×C₁₃×D₈	—	2	1	no	—
32	C₂×C₁₃×D₈	—	2	1	no	—
33	C₂×C₁₃×D₈	—	2	1	no	—
34	C₂×C₁₃×D₈	—	2	1	no	—
35	C₂×C₁₃×D₈	—	4	1	no	—
36	C₂×C₁₃×D₈	D₄	1	2	yes	—
37	C₁₃×#131	—	2	1	no	—
38	C₁₃×#131	—	2	1	no	—
39	C₁₃×#131	—	2	1	no	—
40	C₁₃×#131	—	2	1	no	—
41	C₂×C₁₃×M₁₆	—	2	1	no	—
42	C₂×C₁₃×M₁₆	—	2	1	no	—
43	C₂×C₁₃×M₁₆	—	2	1	no	—
44	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	—	2	1	no	—
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
48	C₁₃×M₃₂	—	2	1	no	—
49	C₁₃×M₃₂	—	2	1	no	—
50	C₂×C₁₃×C₁₆	—	2	1	no	—
51	C₂×C₁₃×C₁₆	—	2	1	no	—
52	C₂×C₁₃×C₁₆	—	2	1	no	—
53	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
54	C₁₃×#108	—	2	1	no	—
55	C₁₃×#108	—	2	1	no	—
56	C₁₃×#108	—	2	1	no	—
57	C₁₃×#108	D₄	1	2	yes	—
58	C₁₃×#107	—	2	1	no	—
59	C₁₃×#107	—	2	1	no	—
60	C₁₃×#103	—	4	1	no	—
61	C₁₃×#102	—	4	1	no	—
62	C₁₃×#100	—	2	4	no	—
63	#13393	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂⁴×C₁₃	—	4	1	no	—
65	C₁₃×#41	—	2	1	no	—
66	C₁₃×#41	—	2	1	no	—
67	C₁₃×#41	—	2	1	no	—
68	C₁₃×#41	—	2	1	no	—
69	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
70	C₂×D₄×C₁₃	—	2	1	no	—
71	C₂×D₄×C₁₃	—	2	2	no	—
72	C₂×D₄×C₁₃	—	4	1	no	—
73	C₂×D₄×C₁₃	—	4	1	no	—
74	C₂×D₄×C₁₃	—	4	1	no	—
75	C₂×D₄×C₁₃	—	4	1	no	—
76	C₂×D₄×C₁₃	—	4	1	no	—
77	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	—	2	1	no	—
79	C₂²×C₄×C₁₃	—	4	1	no	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₁₃×SD₁₆	—	4	4	no	—
82	C₁₃×D₈	—	2	2	no	—
83	C₁₃×D₈	—	2	2	no	—
84	C₁₃×D₈	—	4	4	no	—
85	C₁₃×D₈	—	4	1	no	—
86	C₁₃×D₈	—	4	1	no	—
87	C₁₃×M₁₆	—	2	1	no	—
88	C₁₃×M₁₆	—	2	2	no	—
89	C₁₃×M₁₆	—	2	1	no	—
90	C₁₃×M₁₆	—	4	1	no	—
91	C₁₃×M₁₆	—	4	1	no	—
92	C₁₃×M₁₆	—	4	1	no	—
93	C₁₃×M₁₆	—	4	1	no	—
94	C₁₃×M₁₆	—	4	1	no	—
95	C₂×C₈×C₁₃	—	2	1	no	—
96	C₂×C₈×C₁₃	—	4	1	no	—
97	C₂×C₈×C₁₃	—	4	1	no	—
98	C₂×C₈×C₁₃	—	4	1	no	—
99	C₂×C₈×C₁₃	—	4	1	no	—
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
104	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
105	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
106	C₁₃×C₄⋊C₄	—	4	1	no	—
107	C₁₃×C₄⋊C₄	—	4	1	no	—
108	C₁₃×C₁₆	—	2	2	no	—
109	C₁₃×C₁₆	—	4	1	no	—
110	C₂×#472	C₂×C₁₃	1	1	yes	characteristic
111	C₂×#368	C₂×C₁₃	1	1	yes	characteristic, complemented
112	C₂×#361	C₂×C₁₃	1	1	yes	characteristic, complemented
113	C₂×#357	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#2113	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#1816	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#1387	C₂×C₁₃	1	1	yes	characteristic, complemented
117	C₂³×C₁₃	—	2	1	no	—
118	C₂³×C₁₃	—	4	2	no	—
119	C₂³×C₁₃	—	8	1	no	—
120	C₂³×C₁₃	—	8	1	no	—
121	C₂³×C₁₃	—	16	1	no	—
122	C₂³×C₁₃	—	16	1	no	—
123	C₂³×C₁₃	#135	1	1	yes	characteristic
124	Q₈×C₁₃	—	2	1	no	—
125	Q₈×C₁₃	—	2	1	no	—
126	D₄×C₁₃	—	2	1	no	—
127	D₄×C₁₃	—	2	1	no	—
128	D₄×C₁₃	—	4	1	no	—
129	D₄×C₁₃	—	4	1	no	—
130	D₄×C₁₃	—	4	2	no	—
131	D₄×C₁₃	—	4	2	no	—
132	D₄×C₁₃	—	8	1	no	—
133	D₄×C₁₃	—	8	1	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	—	4	1	no	—
137	C₂×C₄×C₁₃	—	8	1	no	—
138	C₂×C₄×C₁₃	—	16	1	no	—
139	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
145	C₈×C₁₃	—	4	1	no	—
146	C₈×C₁₃	—	4	2	no	—
147	C₈×C₁₃	—	8	1	no	—
148	C₈×C₁₃	#121	1	1	yes	characteristic
149	C₈×C₁₃	#121	1	1	yes	characteristic
150	C₈×C₁₃	#123	1	1	yes	characteristic
151	C₈×C₁₃	#124	1	1	yes	characteristic
152	C₂×AGL(1,ℤ/8)	—	2	1	no	—
153	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
156	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
157	#472	—	2	2	no	—
158	#472	—	2	2	no	—
159	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
160	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
162	#417	—	2	1	no	—
163	C₂×#100	—	2	1	no	—
164	#368	—	2	2	no	—
165	#361	—	2	2	no	—
166	#357	—	2	2	no	—
167	C₂²⋊C₁₆	—	2	2	no	—
168	#343	—	2	1	no	—
169	C₂²×C₁₃	—	2	1	no	—
170	C₂²×C₁₃	—	2	1	no	—
171	C₂²×C₁₃	—	4	1	no	—
172	C₂²×C₁₃	—	4	1	no	—
173	C₂²×C₁₃	—	4	1	no	—
174	C₂²×C₁₃	—	8	2	no	—
175	C₂²×C₁₃	—	16	2	no	—
176	C₂²×C₁₃	—	16	2	no	—
177	C₂²×C₁₃	—	16	1	no	—
178	C₂²×C₁₃	—	16	1	no	—
179	C₂²×C₁₃	#452	1	1	yes	characteristic
180	C₂²×C₁₃	#466	1	1	yes	characteristic
181	C₂²×C₁₃	#482	1	1	yes	characteristic
182	C₄×C₁₃	—	4	1	no	—
183	C₄×C₁₃	—	16	1	no	—
184	C₄×C₁₃	#394	1	1	yes	characteristic
185	C₄×C₁₃	#447	1	1	yes	characteristic
186	C₄×C₁₃	#468	1	1	yes	characteristic
187	C₄×C₁₃	#480	1	1	yes	characteristic
188	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
189	C₂²×D₄	—	2	1	no	—
190	AGL(1,ℤ/8)	—	2	4	no	—
191	AGL(1,ℤ/8)	—	4	2	no	—
192	C₂×SD₁₆	—	4	1	no	—
193	C₂×D₈	—	2	1	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	—	2	1	no	—
197	C₂×D₈	—	2	1	no	—
198	C₂×D₈	—	4	1	no	—
199	C₂×D₈	D₄×C₁₃	1	2	yes	—
200	#131	—	2	1	no	—
201	#131	—	2	1	no	—
202	#131	—	2	1	no	—
203	#131	—	2	1	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
210	C₂×C₄⋊C₄	—	2	1	no	—
211	M₃₂	—	2	1	no	—
212	M₃₂	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
217	#108	—	2	1	no	—
218	#108	—	2	1	no	—
219	#108	—	2	1	no	—
220	#108	D₄×C₁₃	1	2	yes	—
221	#107	—	2	1	no	—
222	#107	—	2	1	no	—
223	#103	—	4	1	no	—
224	#102	—	4	1	no	—
225	#100	—	2	4	no	—
226	C₂×C₁₃	—	2	1	no	—
227	C₂×C₁₃	—	2	1	no	—
228	C₂×C₁₃	—	8	1	no	—
229	C₂×C₁₃	—	16	2	no	—
230	C₂×C₁₃	#2033	1	1	yes	characteristic
231	C₂×C₁₃	#2221	1	1	yes	characteristic
232	C₂×C₁₃	#2223	1	1	yes	characteristic
233	C₂⁴	—	4	1	no	—
234	#41	—	2	1	no	—
235	#41	—	2	1	no	—
236	#41	—	2	1	no	—
237	#41	—	2	1	no	—
238	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
250	SD₁₆	—	4	4	no	—
251	D₈	—	2	2	no	—
252	D₈	—	2	2	no	—
253	D₈	—	4	4	no	—
254	D₈	—	4	1	no	—
255	D₈	—	4	1	no	—
256	M₁₆	—	2	1	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	2	1	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	4	1	no	—
261	M₁₆	—	4	1	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	—	4	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
270	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
272	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
273	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
274	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₁₆	—	2	2	no	—
278	C₁₆	—	4	1	no	—
279	C₁₃	#13393	1	1	yes	characteristic, direct product factor, Sylow
280	C₂³	—	2	1	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	8	1	no	—
283	C₂³	—	8	1	no	—
284	C₂³	—	16	1	no	—
285	C₂³	—	16	1	no	—
286	C₂³	C₁₃×#135	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	4	1	no	—
292	D₄	—	4	1	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	8	1	no	—
296	D₄	—	8	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	16	1	no	—
302	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
303	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
304	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
305	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
306	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
307	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
308	C₈	—	4	1	no	—
309	C₈	—	4	2	no	—
310	C₈	—	8	1	no	—
311	C₈	C₁₃×#121	1	1	yes	characteristic
312	C₈	C₁₃×#121	1	1	yes	characteristic
313	C₈	C₁₃×#123	1	1	yes	characteristic
314	C₈	C₁₃×#124	1	1	yes	characteristic
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	16	2	no	—
322	C₂²	—	16	2	no	—
323	C₂²	—	16	1	no	—
324	C₂²	—	16	1	no	—
325	C₂²	C₁₃×#452	1	1	yes	characteristic
326	C₂²	C₁₃×#466	1	1	yes	characteristic
327	C₂²	C₁₃×#482	1	1	yes	characteristic
328	C₄	—	4	1	no	—
329	C₄	—	16	1	no	—
330	C₄	C₁₃×#394	1	1	yes	characteristic
331	C₄	C₁₃×#447	1	1	yes	characteristic
332	C₄	C₁₃×#468	1	1	yes	characteristic
333	C₄	C₁₃×#480	1	1	yes	characteristic
334	C₂	—	2	1	no	—
335	C₂	—	2	1	no	—
336	C₂	—	8	1	no	—
337	C₂	—	16	2	no	—
338	C₂	C₁₃×#2033	1	1	yes	characteristic
339	C₂	C₁₃×#2221	1	1	yes	characteristic
340	C₂	C₁₃×#2223	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#472	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₁₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#368	C₂	semidirect
3. C₂×C₁₃×#361	C₂	semidirect
4. C₂×C₁₃×#357	C₂	semidirect
5. C₁₃×#2113	C₂	semidirect
6. C₁₃×#1816	C₂	semidirect
7. C₁₃×#1387	C₂	semidirect
63. #13393	C₁₃	direct (N × H)
111. C₂×#368	C₂×C₁₃	semidirect
112. C₂×#361	C₂×C₁₃	semidirect
113. C₂×#357	C₂×C₁₃	semidirect
114. #2113	C₂×C₁₃	semidirect
115. #1816	C₂×C₁₃	semidirect
116. #1387	C₂×C₁₃	semidirect
279. C₁₃	#13393	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×#472	2
C₂×C₁₃×#368	2
C₂×C₁₃×#361	2
C₂×C₁₃×#357	2
C₁₃×#2113	2
C₁₃×#1816	2
C₁₃×#1387	2
#13393	13

Subgroups — C13×#13393 #15699732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13393 #15699732