Subgroups — C₁₃×#13215 #15699554

Order 3328.

466 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3155	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2145	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2143	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2140	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2110	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1833	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×AGL(1,ℤ/8)	—	2	1	no	—
9	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
11	C₁₃×#535	—	2	1	no	—
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₁₃×#496	—	2	2	no	—
15	C₁₃×#495	C₄	1	2	yes	complemented
16	C₁₃×#493	C₄	1	2	yes	complemented
17	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
18	C₁₃×#427	C₂²	1	1	yes	characteristic
19	C₁₃×#405	C₂²	1	1	yes	characteristic
20	C₁₃×#361	—	2	2	no	—
21	C₁₃×#360	—	2	2	no	—
22	C₁₃×#359	—	2	2	no	—
23	C₁₃×#357	—	2	2	no	—
24	C₁₃×#347	—	2	2	no	—
25	C₁₃×#344	—	2	1	no	—
26	C₁₃×H(3,ℤ/4)	—	2	1	no	—
27	C₂×C₁₃×#41	—	2	1	no	—
28	C₂²×D₄×C₁₃	—	2	1	no	—
29	C₁₃×AGL(1,ℤ/8)	—	4	2	no	—
30	C₁₃×AGL(1,ℤ/8)	—	4	2	no	—
31	C₁₃×#135	—	2	1	no	—
32	C₁₃×#135	—	2	1	no	—
33	C₁₃×#135	—	2	1	no	—
34	C₁₃×#135	—	2	1	no	—
35	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₃×SD₁₆	—	2	2	no	—
37	C₂×C₁₃×SD₁₆	—	2	1	no	—
38	C₂×C₁₃×D₈	—	2	2	no	—
39	C₂×C₁₃×D₈	—	2	1	no	—
40	C₂×C₁₃×D₈	—	2	1	no	—
41	C₂×C₁₃×D₈	—	2	1	no	—
42	C₂×C₁₃×D₈	—	2	1	no	—
43	C₂×C₁₃×D₈	—	2	1	no	—
44	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
46	C₂×C₁₃×M₁₆	—	2	1	no	—
47	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×#127	—	2	2	no	—
50	C₁₃×#124	—	2	2	no	—
51	C₁₃×#121	—	4	2	no	—
52	C₁₃×#120	—	4	2	no	—
53	C₁₃×#117	—	2	1	no	—
54	C₁₃×#117	D₄	1	1	yes	characteristic
55	C₁₃×M₃₂	—	2	1	no	—
56	C₁₃×M₃₂	—	2	1	no	—
57	C₁₃×M₃₂	—	2	1	no	—
58	C₁₃×M₃₂	D₄	1	2	yes	complemented
59	C₂×C₁₃×C₁₆	—	2	1	no	—
60	C₂×C₁₃×C₁₆	D₄	1	2	yes	complemented
61	C₁₃×#108	—	2	1	no	—
62	C₁₃×#108	—	2	1	no	—
63	C₁₃×#107	—	2	1	no	—
64	C₁₃×#106	—	2	1	no	—
65	C₁₃×#97	D₄	1	2	yes	—
66	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
67	#13215	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂⁴×C₁₃	—	4	1	no	—
69	C₁₃×#41	—	4	1	no	—
70	C₁₃×#41	—	4	1	no	—
71	C₁₃×#41	—	4	1	no	—
72	C₁₃×#41	—	4	1	no	—
73	C₂×Q₈×C₁₃	—	2	1	no	—
74	C₂×D₄×C₁₃	—	2	2	no	—
75	C₂×D₄×C₁₃	—	2	1	no	—
76	C₂×D₄×C₁₃	—	4	4	no	—
77	C₂×D₄×C₁₃	—	4	1	no	—
78	C₂×D₄×C₁₃	—	4	1	no	—
79	C₂×D₄×C₁₃	—	4	1	no	—
80	C₂×D₄×C₁₃	—	4	1	no	—
81	C₂×D₄×C₁₃	—	4	1	no	—
82	C₂×D₄×C₁₃	—	4	1	no	—
83	C₂²×C₄×C₁₃	—	4	1	no	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₁₃×Q₁₆	—	2	1	no	—
86	C₁₃×Q₁₆	—	2	1	no	—
87	C₁₃×SD₁₆	—	4	4	no	—
88	C₁₃×SD₁₆	—	4	1	no	—
89	C₁₃×SD₁₆	—	4	1	no	—
90	C₁₃×D₈	—	2	2	no	—
91	C₁₃×D₈	—	2	2	no	—
92	C₁₃×D₈	—	2	1	no	—
93	C₁₃×D₈	—	2	1	no	—
94	C₁₃×D₈	—	4	4	no	—
95	C₁₃×D₈	—	4	1	no	—
96	C₁₃×D₈	—	4	1	no	—
97	C₁₃×M₁₆	—	2	1	no	—
98	C₁₃×M₁₆	—	2	1	no	—
99	C₁₃×M₁₆	—	4	1	no	—
100	C₁₃×M₁₆	—	4	1	no	—
101	C₂×C₈×C₁₃	—	4	1	no	—
102	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
103	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
108	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
109	C₁₃×C₄⋊C₄	—	2	1	no	—
110	C₁₃×C₄⋊C₄	—	4	1	no	—
111	C₁₃×C₂²⋊C₄	—	2	1	no	—
112	C₁₃×C₂²⋊C₄	—	4	4	no	—
113	C₁₃×C₂²⋊C₄	—	4	1	no	—
114	C₄²×C₁₃	—	4	1	no	—
115	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
116	C₁₃×C₁₆	—	2	2	no	—
117	C₁₃×C₁₆	—	2	2	no	—
118	#3155	C₂×C₁₃	1	1	yes	characteristic
119	C₂×#361	C₂×C₁₃	1	1	yes	characteristic, complemented
120	#2145	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#2143	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#2140	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#2110	C₂×C₁₃	1	1	yes	characteristic, complemented
124	#1833	C₂×C₁₃	1	1	yes	characteristic, complemented
125	C₂³×C₁₃	—	4	2	no	—
126	C₂³×C₁₃	—	4	1	no	—
127	C₂³×C₁₃	—	8	1	no	—
128	C₂³×C₁₃	—	8	1	no	—
129	C₂³×C₁₃	—	16	2	no	—
130	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
131	Q₈×C₁₃	—	4	1	no	—
132	Q₈×C₁₃	—	4	1	no	—
133	D₄×C₁₃	—	4	2	no	—
134	D₄×C₁₃	—	4	1	no	—
135	D₄×C₁₃	—	4	1	no	—
136	D₄×C₁₃	—	4	2	no	—
137	D₄×C₁₃	—	8	1	no	—
138	D₄×C₁₃	—	8	1	no	—
139	D₄×C₁₃	—	8	1	no	—
140	D₄×C₁₃	—	8	1	no	—
141	D₄×C₁₃	—	16	4	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	4	1	no	—
144	C₂×C₄×C₁₃	—	8	1	no	—
145	C₂×C₄×C₁₃	—	8	1	no	—
146	C₂×C₄×C₁₃	—	8	1	no	—
147	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
149	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
150	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₈×C₁₃	—	2	2	no	—
154	C₈×C₁₃	—	8	1	no	—
155	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
156	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
157	C₈×C₁₃	#121	1	1	yes	characteristic
158	C₈×C₁₃	#121	1	1	yes	characteristic
159	C₂×AGL(1,ℤ/8)	—	2	1	no	—
160	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
161	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
162	#535	—	2	1	no	—
163	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
164	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
165	#496	—	2	2	no	—
166	#495	C₄×C₁₃	1	2	yes	complemented
167	#493	C₄×C₁₃	1	2	yes	complemented
168	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
169	#427	C₂²×C₁₃	1	1	yes	characteristic
170	#405	C₂²×C₁₃	1	1	yes	characteristic
171	#361	—	2	2	no	—
172	#360	—	2	2	no	—
173	#359	—	2	2	no	—
174	#357	—	2	2	no	—
175	#347	—	2	2	no	—
176	#344	—	2	1	no	—
177	H(3,ℤ/4)	—	2	1	no	—
178	C₂²×C₁₃	—	4	1	no	—
179	C₂²×C₁₃	—	8	2	no	—
180	C₂²×C₁₃	—	8	4	no	—
181	C₂²×C₁₃	—	8	1	no	—
182	C₂²×C₁₃	—	8	2	no	—
183	C₂²×C₁₃	—	16	2	no	—
184	C₂²×C₁₃	—	16	2	no	—
185	C₂²×C₁₃	#418	1	1	yes	characteristic
186	C₂²×C₁₃	#490	1	1	yes	characteristic
187	C₂²×C₁₃	#496	1	1	yes	characteristic
188	C₄×C₁₃	—	4	2	no	—
189	C₄×C₁₃	—	8	1	no	—
190	C₄×C₁₃	—	16	1	no	—
191	C₄×C₁₃	#390	1	1	yes	characteristic
192	C₄×C₁₃	#418	1	1	yes	characteristic
193	C₄×C₁₃	#490	1	1	yes	characteristic
194	C₄×C₁₃	#496	1	1	yes	characteristic
195	C₂×#41	—	2	1	no	—
196	C₂²×D₄	—	2	1	no	—
197	AGL(1,ℤ/8)	—	4	2	no	—
198	AGL(1,ℤ/8)	—	4	2	no	—
199	#135	—	2	1	no	—
200	#135	—	2	1	no	—
201	#135	—	2	1	no	—
202	#135	—	2	1	no	—
203	C₂×Q₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
204	C₂×SD₁₆	—	2	2	no	—
205	C₂×SD₁₆	—	2	1	no	—
206	C₂×D₈	—	2	2	no	—
207	C₂×D₈	—	2	1	no	—
208	C₂×D₈	—	2	1	no	—
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	—	2	1	no	—
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
213	C₂×D₈	C₂×C₄×C₁₃	1	1	yes	characteristic
214	C₂×M₁₆	—	2	1	no	—
215	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
217	#127	—	2	2	no	—
218	#124	—	2	2	no	—
219	#121	—	4	2	no	—
220	#120	—	4	2	no	—
221	#117	—	2	1	no	—
222	#117	D₄×C₁₃	1	1	yes	characteristic
223	M₃₂	—	2	1	no	—
224	M₃₂	—	2	1	no	—
225	M₃₂	—	2	1	no	—
226	M₃₂	D₄×C₁₃	1	2	yes	complemented
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	D₄×C₁₃	1	2	yes	complemented
229	#108	—	2	1	no	—
230	#108	—	2	1	no	—
231	#107	—	2	1	no	—
232	#106	—	2	1	no	—
233	#97	D₄×C₁₃	1	2	yes	—
234	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
235	C₂×C₁₃	—	2	1	no	—
236	C₂×C₁₃	—	2	1	no	—
237	C₂×C₁₃	—	2	1	no	—
238	C₂×C₁₃	—	16	1	no	—
239	C₂×C₁₃	—	16	2	no	—
240	C₂×C₁₃	#1961	1	1	yes	characteristic
241	C₂⁴	—	4	1	no	—
242	#41	—	4	1	no	—
243	#41	—	4	1	no	—
244	#41	—	4	1	no	—
245	#41	—	4	1	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	4	4	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
258	Q₁₆	—	2	1	no	—
259	Q₁₆	—	2	1	no	—
260	SD₁₆	—	4	4	no	—
261	SD₁₆	—	4	1	no	—
262	SD₁₆	—	4	1	no	—
263	D₈	—	2	2	no	—
264	D₈	—	2	2	no	—
265	D₈	—	2	1	no	—
266	D₈	—	2	1	no	—
267	D₈	—	4	4	no	—
268	D₈	—	4	1	no	—
269	D₈	—	4	1	no	—
270	M₁₆	—	2	1	no	—
271	M₁₆	—	2	1	no	—
272	M₁₆	—	4	1	no	—
273	M₁₆	—	4	1	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
276	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
277	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
278	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
279	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
280	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
281	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	—	4	4	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₄²	—	4	1	no	—
288	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
289	C₁₆	—	2	2	no	—
290	C₁₆	—	2	2	no	—
291	C₁₃	#13215	1	1	yes	characteristic, direct product factor, Sylow
292	C₂³	—	4	2	no	—
293	C₂³	—	4	1	no	—
294	C₂³	—	8	1	no	—
295	C₂³	—	8	1	no	—
296	C₂³	—	16	2	no	—
297	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
298	Q₈	—	4	1	no	—
299	Q₈	—	4	1	no	—
300	D₄	—	4	2	no	—
301	D₄	—	4	1	no	—
302	D₄	—	4	1	no	—
303	D₄	—	4	2	no	—
304	D₄	—	8	1	no	—
305	D₄	—	8	1	no	—
306	D₄	—	8	1	no	—
307	D₄	—	8	1	no	—
308	D₄	—	16	4	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	8	1	no	—
312	C₂×C₄	—	8	1	no	—
313	C₂×C₄	—	8	1	no	—
314	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
315	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
316	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
317	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
318	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
319	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	8	1	no	—
322	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
323	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
324	C₈	C₁₃×#121	1	1	yes	characteristic
325	C₈	C₁₃×#121	1	1	yes	characteristic
326	C₂²	—	4	1	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	4	no	—
329	C₂²	—	8	1	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	16	2	no	—
332	C₂²	—	16	2	no	—
333	C₂²	C₁₃×#418	1	1	yes	characteristic
334	C₂²	C₁₃×#490	1	1	yes	characteristic
335	C₂²	C₁₃×#496	1	1	yes	characteristic
336	C₄	—	4	2	no	—
337	C₄	—	8	1	no	—
338	C₄	—	16	1	no	—
339	C₄	C₁₃×#390	1	1	yes	characteristic
340	C₄	C₁₃×#418	1	1	yes	characteristic
341	C₄	C₁₃×#490	1	1	yes	characteristic
342	C₄	C₁₃×#496	1	1	yes	characteristic
343	C₂	—	2	1	no	—
344	C₂	—	2	1	no	—
345	C₂	—	2	1	no	—
346	C₂	—	16	2	no	—
347	C₂	—	16	1	no	—
348	C₂	C₁₃×#1961	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	#1961
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₁₃×#3155	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#361	C₂	semidirect
3. C₁₃×#2145	C₂	semidirect
4. C₁₃×#2143	C₂	semidirect
5. C₁₃×#2140	C₂	semidirect
6. C₁₃×#2110	C₂	semidirect
7. C₁₃×#1833	C₂	semidirect
15. C₁₃×#495	C₄	semidirect
16. C₁₃×#493	C₄	semidirect
58. C₁₃×M₃₂	D₄	semidirect
60. C₂×C₁₃×C₁₆	D₄	semidirect
67. #13215	C₁₃	direct (N × H)
119. C₂×#361	C₂×C₁₃	semidirect
120. #2145	C₂×C₁₃	semidirect
121. #2143	C₂×C₁₃	semidirect
122. #2140	C₂×C₁₃	semidirect
123. #2110	C₂×C₁₃	semidirect
124. #1833	C₂×C₁₃	semidirect
166. #495	C₄×C₁₃	semidirect
167. #493	C₄×C₁₃	semidirect
226. M₃₂	D₄×C₁₃	semidirect
228. C₂×C₁₆	D₄×C₁₃	semidirect
291. C₁₃	#13215	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×#3155	2
C₂×C₁₃×#361	2
C₁₃×#2145	2
C₁₃×#2143	2
C₁₃×#2140	2
C₁₃×#2110	2
C₁₃×#1833	2
#13215	13

Subgroups — C13×#13215 #15699554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13215 #15699554