Subgroups — C₁₃×#13398 #15699737

Order 3328.

458 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#469	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×#349	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1811	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#121	—	2	1	no	—
11	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₁₃×#469	—	2	2	no	—
14	C₁₃×#469	—	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×#102	—	2	1	no	—
19	C₁₃×#407	—	2	1	no	—
20	C₁₃×#365	—	2	2	no	—
21	C₁₃×#361	—	2	2	no	—
22	C₁₃×#361	—	2	2	no	—
23	C₁₃×#349	—	2	2	no	—
24	C₁₃×#341	—	2	1	no	—
25	C₂²×D₄×C₁₃	—	2	1	no	—
26	C₂²×D₄×C₁₃	—	4	1	no	—
27	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
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	D₄	1	2	yes	—
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	C₂×C₁₃×M₁₆	—	2	1	no	—
42	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×#121	—	2	4	no	—
45	C₁₃×#121	—	4	2	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	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	complemented
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×#102	—	2	4	no	—
61	C₁₃×C₂²⋊C₈	—	4	2	no	—
62	#13398	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂⁴×C₁₃	—	4	1	no	—
64	C₂⁴×C₁₃	SD₁₆	1	1	yes	characteristic
65	C₂×D₄×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	—	4	4	no	—
67	C₂×D₄×C₁₃	—	4	1	no	—
68	C₂×D₄×C₁₃	—	4	1	no	—
69	C₂×D₄×C₁₃	—	4	1	no	—
70	C₂×D₄×C₁₃	—	4	1	no	—
71	C₂×D₄×C₁₃	—	4	1	no	—
72	C₂×D₄×C₁₃	—	8	4	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	—	2	1	no	—
77	C₂²×C₄×C₁₃	—	2	1	no	—
78	C₂²×C₄×C₁₃	—	2	1	no	—
79	C₂²×C₄×C₁₃	—	4	1	no	—
80	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₁₃×D₈	—	2	2	no	—
83	C₁₃×D₈	—	2	2	no	—
84	C₁₃×D₈	—	4	1	no	—
85	C₁₃×D₈	—	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	C₁₃×M₁₆	—	4	1	no	—
93	C₁₃×M₁₆	—	4	1	no	—
94	C₂×C₈×C₁₃	—	2	1	no	—
95	C₂×C₈×C₁₃	—	2	1	no	—
96	C₂×C₈×C₁₃	—	2	1	no	—
97	C₂×C₈×C₁₃	—	4	1	no	—
98	C₂×C₈×C₁₃	—	4	1	no	—
99	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
103	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
104	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
105	C₁₃×C₄⋊C₄	—	2	2	no	—
106	C₁₃×C₄⋊C₄	—	4	1	no	—
107	C₁₃×C₂²⋊C₄	—	4	4	no	—
108	C₁₃×C₁₆	—	2	2	no	—
109	C₁₃×C₁₆	—	4	1	no	—
110	C₂×#469	C₂×C₁₃	1	1	yes	characteristic
111	C₂×#365	C₂×C₁₃	1	1	yes	characteristic, complemented
112	C₂×#361	C₂×C₁₃	1	1	yes	characteristic, complemented
113	C₂×#361	C₂×C₁₃	1	1	yes	characteristic, complemented
114	C₂×#349	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#1811	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#1383	C₂×C₁₃	1	1	yes	characteristic, complemented
117	C₂³×C₁₃	—	2	1	no	—
118	C₂³×C₁₃	—	2	1	no	—
119	C₂³×C₁₃	—	2	1	no	—
120	C₂³×C₁₃	—	4	2	no	—
121	C₂³×C₁₃	—	4	2	no	—
122	C₂³×C₁₃	—	8	2	no	—
123	C₂³×C₁₃	—	8	2	no	—
124	C₂³×C₁₃	—	8	1	no	—
125	C₂³×C₁₃	—	8	1	no	—
126	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
127	D₄×C₁₃	—	4	2	no	—
128	D₄×C₁₃	—	4	2	no	—
129	D₄×C₁₃	—	8	8	no	—
130	D₄×C₁₃	—	8	1	no	—
131	D₄×C₁₃	—	8	1	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	2	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	—	4	1	no	—
138	C₂×C₄×C₁₃	—	4	1	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	8	2	no	—
141	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
142	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
144	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
145	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₈×C₁₃	—	4	1	no	—
148	C₈×C₁₃	—	8	1	no	—
149	C₈×C₁₃	—	8	1	no	—
150	C₈×C₁₃	#121	1	1	yes	characteristic
151	C₈×C₁₃	#121	1	1	yes	characteristic
152	C₈×C₁₃	#123	1	1	yes	characteristic
153	C₈×C₁₃	#124	1	1	yes	characteristic
154	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
155	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
156	C₂×#121	—	2	1	no	—
157	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
159	#469	—	2	2	no	—
160	#469	—	2	2	no	—
161	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
162	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
164	C₂×#102	—	2	1	no	—
165	#407	—	2	1	no	—
166	#365	—	2	2	no	—
167	#361	—	2	2	no	—
168	#361	—	2	2	no	—
169	#349	—	2	2	no	—
170	#341	—	2	1	no	—
171	C₂²×C₁₃	—	2	2	no	—
172	C₂²×C₁₃	—	4	2	no	—
173	C₂²×C₁₃	—	4	2	no	—
174	C₂²×C₁₃	—	4	1	no	—
175	C₂²×C₁₃	—	8	1	no	—
176	C₂²×C₁₃	—	8	2	no	—
177	C₂²×C₁₃	—	8	2	no	—
178	C₂²×C₁₃	—	8	2	no	—
179	C₂²×C₁₃	—	16	2	no	—
180	C₂²×C₁₃	—	16	2	no	—
181	C₂²×C₁₃	#450	1	1	yes	characteristic
182	C₂²×C₁₃	#465	1	1	yes	characteristic
183	C₂²×C₁₃	#481	1	1	yes	characteristic
184	C₄×C₁₃	—	4	1	no	—
185	C₄×C₁₃	—	8	2	no	—
186	C₄×C₁₃	#394	1	1	yes	characteristic
187	C₄×C₁₃	#449	1	1	yes	characteristic
188	C₄×C₁₃	#469	1	1	yes	characteristic
189	C₄×C₁₃	#482	1	1	yes	characteristic
190	C₂²×D₄	—	2	1	no	—
191	C₂²×D₄	—	4	1	no	—
192	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
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	D₄×C₁₃	1	2	yes	—
199	C₂×M₁₆	—	2	1	no	—
200	C₂×M₁₆	—	2	1	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	—	2	1	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
209	#121	—	2	4	no	—
210	#121	—	4	2	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₂²⋊C₄	—	4	1	no	—
213	M₃₂	—	2	1	no	—
214	M₃₂	—	2	1	no	—
215	M₃₂	—	2	1	no	—
216	M₃₂	D₄×C₁₃	1	2	yes	complemented
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	2	1	no	—
219	#108	—	2	1	no	—
220	#108	—	2	1	no	—
221	#108	—	2	1	no	—
222	#108	—	2	1	no	—
223	#106	—	2	1	no	—
224	#106	D₄×C₁₃	1	2	yes	—
225	#102	—	2	4	no	—
226	C₂²⋊C₈	—	4	2	no	—
227	C₂×C₁₃	—	2	1	no	—
228	C₂×C₁₃	—	2	1	no	—
229	C₂×C₁₃	—	8	1	no	—
230	C₂×C₁₃	—	16	2	no	—
231	C₂×C₁₃	#2034	1	1	yes	characteristic
232	C₂×C₁₃	#2238	1	2	yes	—
233	C₂⁴	—	4	1	no	—
234	C₂⁴	C₁₃×SD₁₆	1	1	yes	characteristic
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	4	4	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	—	8	4	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
252	D₈	—	2	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	4	1	no	—
255	D₈	—	4	1	no	—
256	M₁₆	—	4	1	no	—
257	M₁₆	—	4	1	no	—
258	M₁₆	—	4	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	—	2	1	no	—
266	C₂×C₈	—	2	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	—	2	2	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₂²⋊C₄	—	4	4	no	—
278	C₁₆	—	2	2	no	—
279	C₁₆	—	4	1	no	—
280	C₁₃	#13398	1	1	yes	characteristic, direct product factor, Sylow
281	C₂³	—	2	1	no	—
282	C₂³	—	2	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	4	2	no	—
285	C₂³	—	4	2	no	—
286	C₂³	—	8	2	no	—
287	C₂³	—	8	2	no	—
288	C₂³	—	8	1	no	—
289	C₂³	—	8	1	no	—
290	C₂³	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	8	8	no	—
294	D₄	—	8	1	no	—
295	D₄	—	8	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	8	2	no	—
305	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
306	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
307	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
308	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
309	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
310	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
311	C₈	—	4	1	no	—
312	C₈	—	8	1	no	—
313	C₈	—	8	1	no	—
314	C₈	C₁₃×#121	1	1	yes	characteristic
315	C₈	C₁₃×#121	1	1	yes	characteristic
316	C₈	C₁₃×#123	1	1	yes	characteristic
317	C₈	C₁₃×#124	1	1	yes	characteristic
318	C₂²	—	2	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	8	1	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	16	2	no	—
327	C₂²	—	16	2	no	—
328	C₂²	C₁₃×#450	1	1	yes	characteristic
329	C₂²	C₁₃×#465	1	1	yes	characteristic
330	C₂²	C₁₃×#481	1	1	yes	characteristic
331	C₄	—	4	1	no	—
332	C₄	—	8	2	no	—
333	C₄	C₁₃×#394	1	1	yes	characteristic
334	C₄	C₁₃×#449	1	1	yes	characteristic
335	C₄	C₁₃×#469	1	1	yes	characteristic
336	C₄	C₁₃×#482	1	1	yes	characteristic
337	C₂	—	2	1	no	—
338	C₂	—	2	1	no	—
339	C₂	—	8	1	no	—
340	C₂	—	16	2	no	—
341	C₂	C₁₃×#2034	1	1	yes	characteristic
342	C₂	C₁₃×#2238	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	#481
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#469	C₂
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×#361	C₂	semidirect
4. C₂×C₁₃×#361	C₂	semidirect
5. C₂×C₁₃×#349	C₂	semidirect
6. C₁₃×#1811	C₂	semidirect
7. C₁₃×#1383	C₂	semidirect
51. C₁₃×M₃₂	D₄	semidirect
62. #13398	C₁₃	direct (N × H)
111. C₂×#365	C₂×C₁₃	semidirect
112. C₂×#361	C₂×C₁₃	semidirect
113. C₂×#361	C₂×C₁₃	semidirect
114. C₂×#349	C₂×C₁₃	semidirect
115. #1811	C₂×C₁₃	semidirect
116. #1383	C₂×C₁₃	semidirect
216. M₃₂	D₄×C₁₃	semidirect
280. C₁₃	#13398	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×#469	2
C₂×C₁₃×#365	2
C₂×C₁₃×#361	2
C₂×C₁₃×#349	2
C₁₃×#1811	2
C₁₃×#1383	2
#13398	13

Subgroups — C13×#13398 #15699737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13398 #15699737