Subgroups — C₁₃×#13017 #15699356

Order 3328.

516 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2948	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2140	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1978	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₁₃×#518	—	2	1	no	—
11	C₂×C₁₃×D₁₆	—	2	2	no	—
12	C₂×C₁₃×D₁₆	—	2	2	no	—
13	C₂×C₁₃×D₁₆	C₄	1	4	yes	complemented
14	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₁₃×#442	—	2	2	no	—
17	C₄×C₁₃×D₈	—	2	2	no	—
18	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
19	C₁₃×#427	C₂²	1	1	yes	characteristic
20	C₁₃×#418	—	2	1	no	—
21	C₁₃×#418	—	2	1	no	—
22	C₂×C₁₃×#100	—	2	1	no	—
23	C₁₃×#405	C₂²	1	1	yes	characteristic
24	C₁₃×#361	—	2	2	no	—
25	C₁₃×#357	—	2	2	no	—
26	C₂²×D₄×C₁₃	—	2	1	no	—
27	C₂²×D₄×C₁₃	—	2	1	no	—
28	C₂×C₁₃×D₈	—	2	1	no	—
29	C₂×C₁₃×D₈	—	2	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	—	2	1	no	—
36	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
37	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	complemented
38	C₂×C₁₃×D₈	D₄	1	2	yes	—
39	C₂×C₁₃×D₈	D₄	1	2	yes	—
40	C₂×C₁₃×M₁₆	—	2	1	no	—
41	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₄×D₄×C₁₃	—	2	4	no	—
44	C₄×D₄×C₁₃	—	4	2	no	—
45	C₁₃×#117	—	2	1	no	—
46	C₁₃×#117	D₄	1	1	yes	characteristic
47	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
48	C₁₃×D₁₆	—	2	8	no	—
49	C₁₃×M₃₂	—	2	1	no	—
50	C₁₃×M₃₂	—	2	1	no	—
51	C₂×C₁₃×C₁₆	—	2	1	no	—
52	C₂×C₁₃×C₁₆	—	2	1	no	—
53	C₂×C₁₃×C₁₆	—	2	1	no	—
54	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	complemented
55	C₁₃×#108	—	2	1	no	—
56	C₁₃×#108	—	2	1	no	—
57	C₁₃×#107	D₄	1	2	yes	—
58	C₁₃×#106	—	2	1	no	—
59	C₁₃×#102	—	2	4	no	—
60	C₁₃×#102	—	4	2	no	—
61	C₁₃×#100	—	4	2	no	—
62	C₁₃×#97	—	2	1	no	—
63	C₄×C₈×C₁₃	—	2	1	no	—
64	#13017	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂⁴×C₁₃	—	4	1	no	—
66	C₂⁴×C₁₃	—	4	1	no	—
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	—	2	2	no	—
69	C₂×D₄×C₁₃	—	2	2	no	—
70	C₂×D₄×C₁₃	—	2	2	no	—
71	C₂×D₄×C₁₃	—	4	1	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	—	4	1	no	—
78	C₂×D₄×C₁₃	—	4	1	no	—
79	C₂²×C₄×C₁₃	—	4	4	no	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₁₃×D₈	—	2	2	no	—
82	C₁₃×D₈	—	2	2	no	—
83	C₁₃×D₈	—	2	2	no	—
84	C₁₃×D₈	—	2	2	no	—
85	C₁₃×D₈	—	2	2	no	—
86	C₁₃×D₈	—	4	1	no	—
87	C₁₃×D₈	—	4	1	no	—
88	C₁₃×D₈	C₂²⋊C₄	1	4	yes	—
89	C₁₃×M₁₆	—	2	1	no	—
90	C₁₃×M₁₆	—	2	1	no	—
91	C₁₃×M₁₆	—	4	1	no	—
92	C₁₃×M₁₆	—	4	1	no	—
93	C₂×C₈×C₁₃	—	2	1	no	—
94	C₂×C₈×C₁₃	—	4	1	no	—
95	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
100	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
103	C₁₃×C₂²⋊C₄	—	2	1	no	—
104	C₁₃×C₂²⋊C₄	—	4	4	no	—
105	C₁₃×C₂²⋊C₄	—	4	1	no	—
106	C₄²×C₁₃	—	2	1	no	—
107	C₄²×C₁₃	—	4	1	no	—
108	C₁₃×C₁₆	—	2	2	no	—
109	C₁₃×C₁₆	—	4	1	no	—
110	C₂²×D₁₆	C₂×C₁₃	1	1	yes	characteristic
111	#2948	C₂×C₁₃	1	1	yes	characteristic, complemented
112	C₂×#361	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#2140	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#1978	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#1382	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#1379	C₂×C₁₃	1	1	yes	characteristic, complemented
117	C₂³×C₁₃	—	4	2	no	—
118	C₂³×C₁₃	—	4	2	no	—
119	C₂³×C₁₃	—	4	2	no	—
120	C₂³×C₁₃	—	4	2	no	—
121	C₂³×C₁₃	—	8	1	no	—
122	C₂³×C₁₃	—	8	1	no	—
123	C₂³×C₁₃	—	16	2	no	—
124	C₂³×C₁₃	—	16	2	no	—
125	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
126	D₄×C₁₃	—	2	4	no	—
127	D₄×C₁₃	—	4	2	no	—
128	D₄×C₁₃	—	4	2	no	—
129	D₄×C₁₃	—	4	2	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	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	16	4	no	—
137	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
142	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₈×C₁₃	—	4	1	no	—
144	C₈×C₁₃	—	8	1	no	—
145	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
147	C₈×C₁₃	#121	1	1	yes	characteristic
148	C₈×C₁₃	#123	1	1	yes	characteristic
149	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
150	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
151	#518	—	2	1	no	—
152	C₂×D₁₆	—	2	2	no	—
153	C₂×D₁₆	—	2	2	no	—
154	C₂×D₁₆	C₄×C₁₃	1	4	yes	complemented
155	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
156	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
157	#442	—	2	2	no	—
158	C₄×D₈	—	2	2	no	—
159	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
160	#427	C₂²×C₁₃	1	1	yes	characteristic
161	#418	—	2	1	no	—
162	#418	—	2	1	no	—
163	C₂×#100	—	2	1	no	—
164	#405	C₂²×C₁₃	1	1	yes	characteristic
165	#361	—	2	2	no	—
166	#357	—	2	2	no	—
167	C₂²×C₁₃	—	2	2	no	—
168	C₂²×C₁₃	—	4	4	no	—
169	C₂²×C₁₃	—	8	4	no	—
170	C₂²×C₁₃	—	8	2	no	—
171	C₂²×C₁₃	—	16	2	no	—
172	C₂²×C₁₃	—	16	2	no	—
173	C₂²×C₁₃	—	16	2	no	—
174	C₂²×C₁₃	—	16	2	no	—
175	C₂²×C₁₃	—	16	2	no	—
176	C₂²×C₁₃	#442	1	1	yes	characteristic
177	C₂²×C₁₃	#447	1	1	yes	characteristic
178	C₂²×C₁₃	#463	1	1	yes	characteristic
179	C₄×C₁₃	—	4	2	no	—
180	C₄×C₁₃	—	8	2	no	—
181	C₄×C₁₃	#386	1	1	yes	characteristic
182	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
183	C₄×C₁₃	#449	1	1	yes	characteristic
184	C₄×C₁₃	#459	1	1	yes	characteristic
185	C₂²×D₄	—	2	1	no	—
186	C₂²×D₄	—	2	1	no	—
187	C₂×D₈	—	2	1	no	—
188	C₂×D₈	—	2	1	no	—
189	C₂×D₈	—	2	1	no	—
190	C₂×D₈	—	2	1	no	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	—	2	1	no	—
193	C₂×D₈	—	2	1	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
196	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	complemented
197	C₂×D₈	D₄×C₁₃	1	2	yes	—
198	C₂×D₈	D₄×C₁₃	1	2	yes	—
199	C₂×M₁₆	—	2	1	no	—
200	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
202	C₄×D₄	—	2	4	no	—
203	C₄×D₄	—	4	2	no	—
204	#117	—	2	1	no	—
205	#117	D₄×C₁₃	1	1	yes	characteristic
206	C₂×C₂²⋊C₄	—	4	2	no	—
207	D₁₆	—	2	8	no	—
208	M₃₂	—	2	1	no	—
209	M₃₂	—	2	1	no	—
210	C₂×C₁₆	—	2	1	no	—
211	C₂×C₁₆	—	2	1	no	—
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	complemented
214	#108	—	2	1	no	—
215	#108	—	2	1	no	—
216	#107	D₄×C₁₃	1	2	yes	—
217	#106	—	2	1	no	—
218	#102	—	2	4	no	—
219	#102	—	4	2	no	—
220	#100	—	4	2	no	—
221	#97	—	2	1	no	—
222	C₄×C₈	—	2	1	no	—
223	C₂×C₁₃	—	2	1	no	—
224	C₂×C₁₃	—	2	1	no	—
225	C₂×C₁₃	—	8	4	no	—
226	C₂×C₁₃	—	16	2	no	—
227	C₂×C₁₃	#1879	1	1	yes	characteristic
228	C₂×C₁₃	#2192	1	2	yes	—
229	C₂⁴	—	4	1	no	—
230	C₂⁴	—	4	1	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
245	D₈	—	2	2	no	—
246	D₈	—	2	2	no	—
247	D₈	—	2	2	no	—
248	D₈	—	2	2	no	—
249	D₈	—	2	2	no	—
250	D₈	—	4	1	no	—
251	D₈	—	4	1	no	—
252	D₈	C₁₃×C₂²⋊C₄	1	4	yes	—
253	M₁₆	—	2	1	no	—
254	M₁₆	—	2	1	no	—
255	M₁₆	—	4	1	no	—
256	M₁₆	—	4	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
263	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
264	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	4	4	no	—
269	C₂²⋊C₄	—	4	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	4	1	no	—
272	C₁₆	—	2	2	no	—
273	C₁₆	—	4	1	no	—
274	C₁₃	#13017	1	1	yes	characteristic, direct product factor, Sylow
275	C₂³	—	4	2	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	8	1	no	—
280	C₂³	—	8	1	no	—
281	C₂³	—	16	2	no	—
282	C₂³	—	16	2	no	—
283	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
284	D₄	—	2	4	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	8	1	no	—
291	D₄	—	8	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	16	4	no	—
295	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
296	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
297	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
298	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
299	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
300	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
301	C₈	—	4	1	no	—
302	C₈	—	8	1	no	—
303	C₈	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
304	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
305	C₈	C₁₃×#121	1	1	yes	characteristic
306	C₈	C₁₃×#123	1	1	yes	characteristic
307	C₂²	—	2	2	no	—
308	C₂²	—	4	4	no	—
309	C₂²	—	8	4	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	16	2	no	—
312	C₂²	—	16	2	no	—
313	C₂²	—	16	2	no	—
314	C₂²	—	16	2	no	—
315	C₂²	—	16	2	no	—
316	C₂²	C₁₃×#442	1	1	yes	characteristic
317	C₂²	C₁₃×#447	1	1	yes	characteristic
318	C₂²	C₁₃×#463	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	8	2	no	—
321	C₄	C₁₃×#386	1	1	yes	characteristic
322	C₄	C₄×C₁₃×D₈	1	1	yes	characteristic
323	C₄	C₁₃×#449	1	1	yes	characteristic
324	C₄	C₁₃×#459	1	1	yes	characteristic
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	8	4	no	—
328	C₂	—	16	2	no	—
329	C₂	C₁₃×#1879	1	1	yes	characteristic
330	C₂	C₁₃×#2192	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	#447
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₁₆	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2948	C₂	semidirect
3. C₂×C₁₃×#361	C₂	semidirect
4. C₁₃×#2140	C₂	semidirect
5. C₁₃×#1978	C₂	semidirect
6. C₁₃×#1382	C₂	semidirect
7. C₁₃×#1379	C₂	semidirect
13. C₂×C₁₃×D₁₆	C₄	semidirect
37. C₂×C₁₃×D₈	C₂×C₄	semidirect
54. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
64. #13017	C₁₃	direct (N × H)
111. #2948	C₂×C₁₃	semidirect
112. C₂×#361	C₂×C₁₃	semidirect
113. #2140	C₂×C₁₃	semidirect
114. #1978	C₂×C₁₃	semidirect
115. #1382	C₂×C₁₃	semidirect
116. #1379	C₂×C₁₃	semidirect
154. C₂×D₁₆	C₄×C₁₃	semidirect
196. C₂×D₈	C₂×C₄×C₁₃	semidirect
213. C₂×C₁₆	C₂×C₄×C₁₃	semidirect
274. C₁₃	#13017	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×D₁₆	2
C₁₃×#2948	2
C₂×C₁₃×#361	2
C₁₃×#2140	2
C₁₃×#1978	2
C₁₃×#1382	2
C₁₃×#1379	2
#13017	13

Subgroups — C13×#13017 #15699356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13017 #15699356