Subgroups — C₁₃×#13399 #15699738

Order 3328.

406 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	C₂×C₁₃×#470	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×#349	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1811	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₂²⋊Q₈	—	2	1	no	—
11	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₁₃×#470	—	2	2	no	—
14	C₁₃×#470	—	2	2	no	—
15	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#103	—	2	1	no	—
19	C₁₃×#407	—	2	1	no	—
20	C₁₃×#365	—	2	2	no	—
21	C₁₃×#362	—	2	2	no	—
22	C₁₃×#362	—	2	2	no	—
23	C₁₃×#349	—	2	2	no	—
24	C₁₃×#341	—	2	1	no	—
25	C₂²×Q₈×C₁₃	—	2	1	no	—
26	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
27	C₂×C₁₃×Q₁₆	—	2	1	no	—
28	C₂×C₁₃×Q₁₆	—	2	1	no	—
29	C₂×C₁₃×Q₁₆	—	2	1	no	—
30	C₂×C₁₃×Q₁₆	—	2	1	no	—
31	C₂×C₁₃×Q₁₆	—	2	1	no	—
32	C₂×C₁₃×Q₁₆	D₄	1	2	yes	—
33	C₂×C₁₃×M₁₆	—	2	1	no	—
34	C₂×C₁₃×M₁₆	—	2	1	no	—
35	C₂×C₁₃×M₁₆	—	2	1	no	—
36	C₂×C₁₃×M₁₆	—	2	1	no	—
37	C₂×C₁₃×M₁₆	—	2	1	no	—
38	C₂×C₁₃×M₁₆	—	2	1	no	—
39	C₂×C₁₃×M₁₆	—	2	1	no	—
40	C₂×C₁₃×M₁₆	—	2	1	no	—
41	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×C₂²⋊Q₈	—	2	4	no	—
44	C₁₃×C₂²⋊Q₈	—	4	2	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
46	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
47	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
48	C₁₃×M₃₂	—	2	1	no	—
49	C₁₃×M₃₂	—	2	1	no	—
50	C₁₃×M₃₂	—	2	1	no	—
51	C₁₃×M₃₂	D₄	1	2	yes	—
52	C₂×C₁₃×C₁₆	—	2	1	no	—
53	C₂×C₁₃×C₁₆	—	2	1	no	—
54	C₁₃×#108	—	2	1	no	—
55	C₁₃×#108	—	2	1	no	—
56	C₁₃×#108	—	2	1	no	—
57	C₁₃×#108	—	2	1	no	—
58	C₁₃×#106	—	2	1	no	—
59	C₁₃×#106	D₄	1	2	yes	—
60	C₁₃×#103	—	2	4	no	—
61	C₁₃×C₂²⋊C₈	—	4	2	no	—
62	#13399	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂⁴×C₁₃	SD₁₆	1	1	yes	characteristic
64	C₂×Q₈×C₁₃	—	2	2	no	—
65	C₂×Q₈×C₁₃	—	4	1	no	—
66	C₂×Q₈×C₁₃	—	4	1	no	—
67	C₂×Q₈×C₁₃	—	4	1	no	—
68	C₂×Q₈×C₁₃	—	4	1	no	—
69	C₂×Q₈×C₁₃	—	4	1	no	—
70	C₂²×C₄×C₁₃	—	2	1	no	—
71	C₂²×C₄×C₁₃	—	2	1	no	—
72	C₂²×C₄×C₁₃	—	2	1	no	—
73	C₂²×C₄×C₁₃	—	2	1	no	—
74	C₂²×C₄×C₁₃	—	2	1	no	—
75	C₂²×C₄×C₁₃	—	2	1	no	—
76	C₂²×C₄×C₁₃	—	4	1	no	—
77	C₂²×C₄×C₁₃	—	4	1	no	—
78	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₁₃×Q₁₆	—	2	2	no	—
81	C₁₃×Q₁₆	—	2	2	no	—
82	C₁₃×Q₁₆	—	4	1	no	—
83	C₁₃×Q₁₆	—	4	1	no	—
84	C₁₃×M₁₆	—	4	1	no	—
85	C₁₃×M₁₆	—	4	1	no	—
86	C₁₃×M₁₆	—	4	1	no	—
87	C₁₃×M₁₆	—	4	1	no	—
88	C₁₃×M₁₆	—	4	1	no	—
89	C₁₃×M₁₆	—	4	1	no	—
90	C₁₃×M₁₆	—	4	1	no	—
91	C₁₃×M₁₆	—	4	1	no	—
92	C₂×C₈×C₁₃	—	2	1	no	—
93	C₂×C₈×C₁₃	—	2	1	no	—
94	C₂×C₈×C₁₃	—	2	1	no	—
95	C₂×C₈×C₁₃	—	4	1	no	—
96	C₂×C₈×C₁₃	—	4	1	no	—
97	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
101	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
102	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
103	C₁₃×C₄⋊C₄	—	2	2	no	—
104	C₁₃×C₄⋊C₄	—	4	4	no	—
105	C₁₃×C₄⋊C₄	—	4	1	no	—
106	C₁₃×C₂²⋊C₄	—	4	4	no	—
107	C₁₃×C₁₆	—	2	2	no	—
108	C₁₃×C₁₆	—	4	1	no	—
109	C₂×#470	C₂×C₁₃	1	1	yes	characteristic
110	C₂×#365	C₂×C₁₃	1	1	yes	characteristic, complemented
111	C₂×#362	C₂×C₁₃	1	1	yes	characteristic, complemented
112	C₂×#362	C₂×C₁₃	1	1	yes	characteristic, complemented
113	C₂×#349	C₂×C₁₃	1	1	yes	characteristic
114	#1811	C₂×C₁₃	1	1	yes	characteristic
115	#1383	C₂×C₁₃	1	1	yes	characteristic, complemented
116	C₂³×C₁₃	—	2	1	no	—
117	C₂³×C₁₃	—	2	1	no	—
118	C₂³×C₁₃	—	2	1	no	—
119	C₂³×C₁₃	—	4	2	no	—
120	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
121	Q₈×C₁₃	—	4	2	no	—
122	Q₈×C₁₃	—	4	2	no	—
123	Q₈×C₁₃	—	8	1	no	—
124	Q₈×C₁₃	—	8	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	4	1	no	—
129	C₂×C₄×C₁₃	—	4	1	no	—
130	C₂×C₄×C₁₃	—	4	1	no	—
131	C₂×C₄×C₁₃	—	4	1	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	8	2	no	—
135	C₂×C₄×C₁₃	—	8	1	no	—
136	C₂×C₄×C₁₃	—	8	1	no	—
137	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
141	C₂×C₄×C₁₃	AGL(1,ℤ/8)	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	—	8	1	no	—
146	C₈×C₁₃	#121	1	1	yes	characteristic
147	C₈×C₁₃	#121	1	1	yes	characteristic
148	C₈×C₁₃	#123	1	1	yes	characteristic
149	C₈×C₁₃	#124	1	1	yes	characteristic
150	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
151	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
152	C₂×C₂²⋊Q₈	—	2	1	no	—
153	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
155	#470	—	2	2	no	—
156	#470	—	2	2	no	—
157	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
159	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
160	C₂×#103	—	2	1	no	—
161	#407	—	2	1	no	—
162	#365	—	2	2	no	—
163	#362	—	2	2	no	—
164	#362	—	2	2	no	—
165	#349	—	2	2	no	—
166	#341	—	2	1	no	—
167	C₂²×C₁₃	—	2	2	no	—
168	C₂²×C₁₃	—	4	2	no	—
169	C₂²×C₁₃	—	4	2	no	—
170	C₂²×C₁₃	—	4	1	no	—
171	C₂²×C₁₃	—	8	1	no	—
172	C₂²×C₁₃	#450	1	1	yes	characteristic
173	C₂²×C₁₃	#465	1	1	yes	characteristic
174	C₂²×C₁₃	#481	1	1	yes	characteristic
175	C₄×C₁₃	—	4	1	no	—
176	C₄×C₁₃	—	8	2	no	—
177	C₄×C₁₃	—	8	2	no	—
178	C₄×C₁₃	#394	1	1	yes	characteristic
179	C₄×C₁₃	#449	1	1	yes	characteristic
180	C₄×C₁₃	#469	1	1	yes	characteristic
181	C₄×C₁₃	#482	1	1	yes	characteristic
182	C₂²×Q₈	—	2	1	no	—
183	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
184	C₂×Q₁₆	—	2	1	no	—
185	C₂×Q₁₆	—	2	1	no	—
186	C₂×Q₁₆	—	2	1	no	—
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	—	2	1	no	—
189	C₂×Q₁₆	D₄×C₁₃	1	2	yes	—
190	C₂×M₁₆	—	2	1	no	—
191	C₂×M₁₆	—	2	1	no	—
192	C₂×M₁₆	—	2	1	no	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₂×M₁₆	—	2	1	no	—
196	C₂×M₁₆	—	2	1	no	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
200	C₂²⋊Q₈	—	2	4	no	—
201	C₂²⋊Q₈	—	4	2	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	4	1	no	—
204	C₂×C₂²⋊C₄	—	4	1	no	—
205	M₃₂	—	2	1	no	—
206	M₃₂	—	2	1	no	—
207	M₃₂	—	2	1	no	—
208	M₃₂	D₄×C₁₃	1	2	yes	—
209	C₂×C₁₆	—	2	1	no	—
210	C₂×C₁₆	—	2	1	no	—
211	#108	—	2	1	no	—
212	#108	—	2	1	no	—
213	#108	—	2	1	no	—
214	#108	—	2	1	no	—
215	#106	—	2	1	no	—
216	#106	D₄×C₁₃	1	2	yes	—
217	#103	—	2	4	no	—
218	C₂²⋊C₈	—	4	2	no	—
219	C₂×C₁₃	—	2	1	no	—
220	C₂×C₁₃	—	2	1	no	—
221	C₂×C₁₃	—	8	1	no	—
222	C₂×C₁₃	#2034	1	1	yes	characteristic
223	C₂×C₁₃	#2239	1	2	yes	—
224	C₂⁴	C₁₃×SD₁₆	1	1	yes	characteristic
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×Q₈	—	4	1	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
241	Q₁₆	—	2	2	no	—
242	Q₁₆	—	2	2	no	—
243	Q₁₆	—	4	1	no	—
244	Q₁₆	—	4	1	no	—
245	M₁₆	—	4	1	no	—
246	M₁₆	—	4	1	no	—
247	M₁₆	—	4	1	no	—
248	M₁₆	—	4	1	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
262	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
263	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	4	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₂²⋊C₄	—	4	4	no	—
268	C₁₆	—	2	2	no	—
269	C₁₆	—	4	1	no	—
270	C₁₃	#13399	1	1	yes	characteristic, direct product factor, Sylow
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	4	2	no	—
275	C₂³	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	8	1	no	—
279	Q₈	—	8	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	8	2	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
293	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
294	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
295	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
296	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
297	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
298	C₈	—	4	1	no	—
299	C₈	—	8	1	no	—
300	C₈	—	8	1	no	—
301	C₈	C₁₃×#121	1	1	yes	characteristic
302	C₈	C₁₃×#121	1	1	yes	characteristic
303	C₈	C₁₃×#123	1	1	yes	characteristic
304	C₈	C₁₃×#124	1	1	yes	characteristic
305	C₂²	—	2	2	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	8	1	no	—
310	C₂²	C₁₃×#450	1	1	yes	characteristic
311	C₂²	C₁₃×#465	1	1	yes	characteristic
312	C₂²	C₁₃×#481	1	1	yes	characteristic
313	C₄	—	4	1	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	C₁₃×#394	1	1	yes	characteristic
317	C₄	C₁₃×#449	1	1	yes	characteristic
318	C₄	C₁₃×#469	1	1	yes	characteristic
319	C₄	C₁₃×#482	1	1	yes	characteristic
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	8	1	no	—
323	C₂	C₁₃×#2034	1	1	yes	characteristic
324	C₂	C₁₃×#2239	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	#481
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂⁴×C₁₃	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₁₃×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#365	C₂	semidirect
3. C₂×C₁₃×#362	C₂	semidirect
4. C₂×C₁₃×#362	C₂	semidirect
7. C₁₃×#1383	C₂	semidirect
62. #13399	C₁₃	direct (N × H)
110. C₂×#365	C₂×C₁₃	semidirect
111. C₂×#362	C₂×C₁₃	semidirect
112. C₂×#362	C₂×C₁₃	semidirect
115. #1383	C₂×C₁₃	semidirect
270. C₁₃	#13399	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×#470	2
C₂×C₁₃×#365	2
C₂×C₁₃×#362	2
C₂×C₁₃×#349	2
C₁₃×#1811	2
C₁₃×#1383	2
#13399	13

Subgroups — C13×#13399 #15699738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13399 #15699738