Subgroups — C₁₃×#12937 #15699276

Order 3328.

490 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#357	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#357	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1942	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1380	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₁₃	—	2	1	no	—
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₁₃×D₈	—	2	2	no	—
13	C₄×C₁₃×D₈	—	2	2	no	—
14	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#102	—	2	1	no	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#389	—	2	1	no	—
20	C₁₃×#367	—	2	2	no	—
21	C₁₃×#357	—	2	2	no	—
22	C₁₃×#357	C₄	1	4	yes	complemented
23	C₁₃×#340	—	2	1	no	—
24	C₂²×D₄×C₁₃	—	2	1	no	—
25	C₂³×C₄×C₁₃	—	4	1	no	—
26	C₂×C₁₃×D₈	—	2	1	no	—
27	C₂×C₁₃×D₈	—	2	1	no	—
28	C₂×C₁₃×D₈	—	2	1	no	—
29	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
30	C₂×C₁₃×D₈	D₄	1	2	yes	—
31	C₂×C₁₃×D₈	Q₈	1	2	yes	—
32	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₄×D₄×C₁₃	—	2	4	no	—
35	C₄×D₄×C₁₃	—	4	2	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
39	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
40	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
41	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
42	C₂×C₁₃×C₁₆	—	2	1	no	—
43	C₂×C₁₃×C₁₆	—	2	1	no	—
44	C₂×C₁₃×C₁₆	—	2	1	no	—
45	C₂×C₁₃×C₁₆	—	2	1	no	—
46	C₂×C₁₃×C₁₆	—	2	1	no	—
47	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	complemented
48	C₁₃×#107	—	2	1	no	—
49	C₁₃×#107	—	2	1	no	—
50	C₁₃×#107	—	2	1	no	—
51	C₁₃×#107	—	2	1	no	—
52	C₁₃×#107	C₂×C₄	1	2	yes	complemented
53	C₁₃×#107	D₄	1	2	yes	—
54	C₁₃×#102	—	2	4	no	—
55	C₄×C₈×C₁₃	—	2	1	no	—
56	C₄×C₈×C₁₃	—	2	1	no	—
57	C₁₃×#95	—	4	1	no	—
58	#12937	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂⁴×C₁₃	—	4	1	no	—
60	C₂×D₄×C₁₃	—	2	2	no	—
61	C₂×D₄×C₁₃	—	2	2	no	—
62	C₂×D₄×C₁₃	—	2	2	no	—
63	C₂×D₄×C₁₃	—	4	1	no	—
64	C₂×D₄×C₁₃	—	4	1	no	—
65	C₂×D₄×C₁₃	—	4	1	no	—
66	C₂²×C₄×C₁₃	—	4	4	no	—
67	C₂²×C₄×C₁₃	—	4	1	no	—
68	C₂²×C₄×C₁₃	—	4	1	no	—
69	C₂²×C₄×C₁₃	—	4	1	no	—
70	C₂²×C₄×C₁₃	—	8	2	no	—
71	C₂²×C₄×C₁₃	—	8	2	no	—
72	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₁₃×D₈	—	2	2	no	—
76	C₁₃×D₈	—	2	2	no	—
77	C₁₃×D₈	—	2	2	no	—
78	C₁₃×D₈	C₄⋊C₄	1	4	yes	complemented
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	—	2	1	no	—
81	C₂×C₈×C₁₃	—	2	1	no	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
87	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	4	1	no	—
91	C₁₃×C₄⋊C₄	—	4	1	no	—
92	C₁₃×C₄⋊C₄	—	4	1	no	—
93	C₁₃×C₄⋊C₄	—	4	1	no	—
94	C₁₃×C₄⋊C₄	—	8	2	no	—
95	C₁₃×C₂²⋊C₄	—	4	4	no	—
96	C₄²×C₁₃	—	2	1	no	—
97	C₄²×C₁₃	—	2	1	no	—
98	C₁₃×C₁₆	—	2	2	no	—
99	C₁₃×C₁₆	—	4	1	no	—
100	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
101	C₂×#367	C₂×C₁₃	1	1	yes	characteristic, complemented
102	C₂×#357	C₂×C₁₃	1	1	yes	characteristic
103	C₂×#357	C₂×C₁₃	1	1	yes	characteristic
104	#1942	C₂×C₁₃	1	1	yes	characteristic, complemented
105	#1380	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#1374	C₂×C₁₃	1	1	yes	characteristic, complemented
107	C₂³×C₁₃	—	4	2	no	—
108	C₂³×C₁₃	—	4	2	no	—
109	C₂³×C₁₃	—	4	2	no	—
110	C₂³×C₁₃	—	8	1	no	—
111	C₂³×C₁₃	—	8	1	no	—
112	C₂³×C₁₃	—	8	1	no	—
113	C₂³×C₁₃	—	8	1	no	—
114	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
115	D₄×C₁₃	—	2	4	no	—
116	D₄×C₁₃	—	4	2	no	—
117	D₄×C₁₃	—	4	2	no	—
118	D₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	8	8	no	—
127	C₂×C₄×C₁₃	—	8	1	no	—
128	C₂×C₄×C₁₃	—	8	1	no	—
129	C₂×C₄×C₁₃	—	8	1	no	—
130	C₂×C₄×C₁₃	—	8	1	no	—
131	C₂×C₄×C₁₃	—	8	1	no	—
132	C₂×C₄×C₁₃	—	8	1	no	—
133	C₂×C₄×C₁₃	—	8	1	no	—
134	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
135	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
139	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
141	C₈×C₁₃	—	4	1	no	—
142	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
144	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
145	C₈×C₁₃	#123	1	1	yes	characteristic
146	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
147	C₂×C₄×D₄	—	2	1	no	—
148	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
149	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
150	C₄×D₈	—	2	2	no	—
151	C₄×D₈	—	2	2	no	—
152	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
153	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×#102	—	2	1	no	—
156	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
157	#389	—	2	1	no	—
158	#367	—	2	2	no	—
159	#357	—	2	2	no	—
160	#357	C₄×C₁₃	1	4	yes	complemented
161	#340	—	2	1	no	—
162	C₂²×C₁₃	—	4	4	no	—
163	C₂²×C₁₃	—	8	2	no	—
164	C₂²×C₁₃	—	8	2	no	—
165	C₂²×C₁₃	—	8	2	no	—
166	C₂²×C₁₃	—	8	2	no	—
167	C₂²×C₁₃	—	8	2	no	—
168	C₂²×C₁₃	—	8	2	no	—
169	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
170	C₂²×C₁₃	#447	1	1	yes	characteristic
171	C₂²×C₁₃	#474	1	1	yes	characteristic
172	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
173	C₂²×C₁₃	#508	1	1	yes	characteristic
174	C₂²×C₁₃	#509	1	1	yes	characteristic
175	C₂²×C₁₃	#509	1	1	yes	characteristic
176	C₄×C₁₃	—	2	2	no	—
177	C₄×C₁₃	—	4	1	no	—
178	C₄×C₁₃	—	8	2	no	—
179	C₄×C₁₃	—	8	2	no	—
180	C₄×C₁₃	—	16	1	no	—
181	C₄×C₁₃	#385	1	1	yes	characteristic
182	C₄×C₁₃	#440	1	1	yes	characteristic
183	C₄×C₁₃	#452	1	1	yes	characteristic
184	C₄×C₁₃	#478	1	1	yes	characteristic
185	C₂²×D₄	—	2	1	no	—
186	C₂³×C₄	—	4	1	no	—
187	C₂×D₈	—	2	1	no	—
188	C₂×D₈	—	2	1	no	—
189	C₂×D₈	—	2	1	no	—
190	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
191	C₂×D₈	D₄×C₁₃	1	2	yes	—
192	C₂×D₈	Q₈×C₁₃	1	2	yes	—
193	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
195	C₄×D₄	—	2	4	no	—
196	C₄×D₄	—	4	2	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	4	1	no	—
201	C₂×C₂²⋊C₄	—	4	1	no	—
202	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	complemented
209	#107	—	2	1	no	—
210	#107	—	2	1	no	—
211	#107	—	2	1	no	—
212	#107	—	2	1	no	—
213	#107	C₂×C₄×C₁₃	1	2	yes	complemented
214	#107	D₄×C₁₃	1	2	yes	—
215	#102	—	2	4	no	—
216	C₄×C₈	—	2	1	no	—
217	C₄×C₈	—	2	1	no	—
218	#95	—	4	1	no	—
219	C₂×C₁₃	—	8	4	no	—
220	C₂×C₁₃	#1862	1	1	yes	characteristic
221	C₂×C₁₃	C₄×SD₃₂	1	1	yes	characteristic
222	C₂×C₁₃	#2177	1	1	yes	characteristic
223	C₂×C₁₃	#2186	1	1	yes	characteristic
224	C₂×C₁₃	#2187	1	1	yes	characteristic
225	C₂×C₁₃	#2226	1	1	yes	characteristic
226	C₂×C₁₃	#2228	1	1	yes	characteristic
227	C₂⁴	—	4	1	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂×D₄	—	4	1	no	—
233	C₂×D₄	—	4	1	no	—
234	C₂²×C₄	—	4	4	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	8	2	no	—
239	C₂²×C₄	—	8	2	no	—
240	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
241	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
243	D₈	—	2	2	no	—
244	D₈	—	2	2	no	—
245	D₈	—	2	2	no	—
246	D₈	C₁₃×C₄⋊C₄	1	4	yes	complemented
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
252	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
253	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
254	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
255	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	8	2	no	—
263	C₂²⋊C₄	—	4	4	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₁₆	—	2	2	no	—
267	C₁₆	—	4	1	no	—
268	C₁₃	#12937	1	1	yes	characteristic, direct product factor, Sylow
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	8	1	no	—
273	C₂³	—	8	1	no	—
274	C₂³	—	8	1	no	—
275	C₂³	—	8	1	no	—
276	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
277	D₄	—	2	4	no	—
278	D₄	—	4	2	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	8	8	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	8	1	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	C₁₃×SD₃₂	1	2	yes	—
297	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
298	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
299	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
300	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
301	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
302	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
303	C₈	—	4	1	no	—
304	C₈	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
305	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
306	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
307	C₈	C₁₃×#123	1	1	yes	characteristic
308	C₂²	—	4	4	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₄×C₁₃×D₈	1	1	yes	characteristic
316	C₂²	C₁₃×#447	1	1	yes	characteristic
317	C₂²	C₁₃×#474	1	1	yes	characteristic
318	C₂²	C₂×C₁₃×SD₃₂	1	1	yes	characteristic
319	C₂²	C₁₃×#508	1	1	yes	characteristic
320	C₂²	C₁₃×#509	1	1	yes	characteristic
321	C₂²	C₁₃×#509	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	4	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	16	1	no	—
327	C₄	C₁₃×#385	1	1	yes	characteristic
328	C₄	C₁₃×#440	1	1	yes	characteristic
329	C₄	C₁₃×#452	1	1	yes	characteristic
330	C₄	C₁₃×#478	1	1	yes	characteristic
331	C₂	—	8	4	no	—
332	C₂	C₁₃×#1862	1	1	yes	characteristic
333	C₂	C₄×C₁₃×SD₃₂	1	1	yes	characteristic
334	C₂	C₁₃×#2177	1	1	yes	characteristic
335	C₂	C₁₃×#2186	1	1	yes	characteristic
336	C₂	C₁₃×#2187	1	1	yes	characteristic
337	C₂	C₁₃×#2226	1	1	yes	characteristic
338	C₂	C₁₃×#2228	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#367	C₂	semidirect
5. C₁₃×#1942	C₂	semidirect
6. C₁₃×#1380	C₂	semidirect
7. C₁₃×#1374	C₂	semidirect
22. C₁₃×#357	C₄	semidirect
47. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
52. C₁₃×#107	C₂×C₄	semidirect
58. #12937	C₁₃	direct (N × H)
78. C₁₃×D₈	C₄⋊C₄	semidirect
101. C₂×#367	C₂×C₁₃	semidirect
104. #1942	C₂×C₁₃	semidirect
105. #1380	C₂×C₁₃	semidirect
106. #1374	C₂×C₁₃	semidirect
160. #357	C₄×C₁₃	semidirect
208. C₂×C₁₆	C₂×C₄×C₁₃	semidirect
213. #107	C₂×C₄×C₁₃	semidirect
246. D₈	C₁₃×C₄⋊C₄	semidirect
268. C₁₃	#12937	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×C₄×C₁₃×D₈	2
C₂×C₁₃×#367	2
C₂×C₁₃×#357	2
C₁₃×#1942	2
C₁₃×#1380	2
C₁₃×#1374	2
#12937	13

Subgroups — C13×#12937 #15699276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12937 #15699276