Subgroups — C₁₃×#13037 #15699376

Order 3328.

538 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#509	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#442	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#360	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1975	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1383	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1375	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃	—	2	1	no	—
11	C₁₃×#509	—	2	2	no	—
12	C₁₃×#509	—	2	2	no	—
13	C₂×C₁₃×SD₃₂	C₄	1	2	yes	complemented
14	C₂×C₁₃×D₁₆	C₄	1	2	yes	complemented
15	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
17	C₁₃×#442	—	2	2	no	—
18	C₁₃×#442	—	2	2	no	—
19	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×#104	—	2	1	no	—
22	C₂×C₁₃×#102	—	2	1	no	—
23	C₁₃×#408	—	2	1	no	—
24	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
25	C₁₃×#361	—	2	2	no	—
26	C₁₃×#360	—	2	2	no	—
27	C₂×C₁₃×#41	—	2	1	no	—
28	C₂²×D₄×C₁₃	—	2	1	no	—
29	C₂³×C₄×C₁₃	—	4	1	no	—
30	C₁₃×#135	—	2	1	no	—
31	C₁₃×#135	—	2	1	no	—
32	C₁₃×#135	—	2	1	no	—
33	C₁₃×#135	—	2	1	no	—
34	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
35	C₂×C₁₃×SD₁₆	—	2	1	no	—
36	C₂×C₁₃×D₈	—	2	1	no	—
37	C₂×C₁₃×D₈	—	2	1	no	—
38	C₂×C₁₃×D₈	—	2	1	no	—
39	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
40	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic, complemented
41	C₂×C₁₃×D₈	D₄	1	2	yes	—
42	C₂×C₁₃×D₈	D₄	1	2	yes	—
43	C₂×C₁₃×M₁₆	—	2	1	no	—
44	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₄×D₄×C₁₃	—	2	4	no	—
47	C₄×D₄×C₁₃	—	4	2	no	—
48	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
49	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
50	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
51	C₁₃×SD₃₂	—	2	4	no	—
52	C₁₃×D₁₆	—	2	4	no	—
53	C₁₃×M₃₂	—	2	1	no	—
54	C₁₃×M₃₂	—	2	1	no	—
55	C₁₃×M₃₂	—	2	1	no	—
56	C₁₃×M₃₂	—	2	1	no	—
57	C₂×C₁₃×C₁₆	—	2	1	no	—
58	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	complemented
59	C₁₃×#108	—	2	1	no	—
60	C₁₃×#108	—	2	1	no	—
61	C₁₃×#106	—	2	1	no	—
62	C₁₃×#106	D₄	1	2	yes	—
63	C₁₃×#104	—	4	2	no	—
64	C₁₃×#102	—	2	4	no	—
65	C₁₃×C₂²⋊C₈	—	4	2	no	—
66	C₁₃×#97	—	2	1	no	—
67	C₁₃×#97	—	2	1	no	—
68	#13037	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂⁴×C₁₃	—	4	1	no	—
70	C₁₃×#41	—	4	1	no	—
71	C₁₃×#41	—	4	1	no	—
72	C₁₃×#41	—	4	1	no	—
73	C₁₃×#41	—	4	1	no	—
74	C₂×Q₈×C₁₃	—	2	1	no	—
75	C₂×D₄×C₁₃	—	2	2	no	—
76	C₂×D₄×C₁₃	—	2	1	no	—
77	C₂×D₄×C₁₃	—	2	2	no	—
78	C₂×D₄×C₁₃	—	2	2	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	—	2	1	no	—
84	C₂²×C₄×C₁₃	—	4	4	no	—
85	C₂²×C₄×C₁₃	—	4	1	no	—
86	C₂²×C₄×C₁₃	—	4	1	no	—
87	C₂²×C₄×C₁₃	—	8	4	no	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₁₃×Q₁₆	—	2	1	no	—
90	C₁₃×Q₁₆	—	2	1	no	—
91	C₁₃×SD₁₆	—	4	1	no	—
92	C₁₃×SD₁₆	—	4	1	no	—
93	C₁₃×D₈	—	2	1	no	—
94	C₁₃×D₈	—	2	2	no	—
95	C₁₃×D₈	—	2	1	no	—
96	C₁₃×D₈	—	2	2	no	—
97	C₁₃×D₈	—	2	2	no	—
98	C₁₃×D₈	C₂²⋊C₄	1	4	yes	—
99	C₁₃×M₁₆	—	4	1	no	—
100	C₁₃×M₁₆	—	4	1	no	—
101	C₂×C₈×C₁₃	—	2	1	no	—
102	C₂×C₈×C₁₃	—	2	1	no	—
103	C₂×C₈×C₁₃	—	2	1	no	—
104	C₂×C₈×C₁₃	—	4	1	no	—
105	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
110	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
111	C₁₃×C₄⋊C₄	—	2	2	no	—
112	C₁₃×C₄⋊C₄	—	4	1	no	—
113	C₁₃×C₂²⋊C₄	—	4	4	no	—
114	C₄²×C₁₃	—	2	1	no	—
115	C₄²×C₁₃	D₈	1	2	yes	—
116	C₁₃×C₁₆	—	2	2	no	—
117	C₁₃×C₁₆	—	4	1	no	—
118	C₂×#509	C₂×C₁₃	1	1	yes	characteristic
119	C₂×#442	C₂×C₁₃	1	1	yes	characteristic, complemented
120	C₂×#361	C₂×C₁₃	1	1	yes	characteristic, complemented
121	C₂×#360	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#1975	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#1383	C₂×C₁₃	1	1	yes	characteristic, complemented
124	#1375	C₂×C₁₃	1	1	yes	characteristic, complemented
125	C₂³×C₁₃	—	4	2	no	—
126	C₂³×C₁₃	—	4	2	no	—
127	C₂³×C₁₃	—	4	2	no	—
128	C₂³×C₁₃	—	4	1	no	—
129	C₂³×C₁₃	—	8	2	no	—
130	C₂³×C₁₃	—	8	2	no	—
131	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
132	Q₈×C₁₃	—	4	1	no	—
133	Q₈×C₁₃	—	4	1	no	—
134	D₄×C₁₃	—	2	4	no	—
135	D₄×C₁₃	—	4	1	no	—
136	D₄×C₁₃	—	4	2	no	—
137	D₄×C₁₃	—	4	1	no	—
138	D₄×C₁₃	—	4	2	no	—
139	D₄×C₁₃	—	4	2	no	—
140	D₄×C₁₃	—	8	1	no	—
141	D₄×C₁₃	—	8	1	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	4	1	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	—	8	8	no	—
147	C₂×C₄×C₁₃	—	8	2	no	—
148	C₂×C₄×C₁₃	—	8	1	no	—
149	C₂×C₄×C₁₃	—	8	1	no	—
150	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
152	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
155	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₈×C₁₃	—	4	1	no	—
157	C₈×C₁₃	—	8	1	no	—
158	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
160	C₈×C₁₃	#121	1	1	yes	characteristic
161	C₈×C₁₃	#123	1	1	yes	characteristic
162	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
163	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
164	C₂×C₄×D₄	—	2	1	no	—
165	#509	—	2	2	no	—
166	#509	—	2	2	no	—
167	C₂×SD₃₂	C₄×C₁₃	1	2	yes	complemented
168	C₂×D₁₆	C₄×C₁₃	1	2	yes	complemented
169	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
170	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
171	#442	—	2	2	no	—
172	#442	—	2	2	no	—
173	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
174	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
175	C₂×#104	—	2	1	no	—
176	C₂×#102	—	2	1	no	—
177	#408	—	2	1	no	—
178	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
179	#361	—	2	2	no	—
180	#360	—	2	2	no	—
181	C₂²×C₁₃	—	2	2	no	—
182	C₂²×C₁₃	—	4	4	no	—
183	C₂²×C₁₃	—	8	2	no	—
184	C₂²×C₁₃	—	8	2	no	—
185	C₂²×C₁₃	—	8	4	no	—
186	C₂²×C₁₃	—	8	4	no	—
187	C₂²×C₁₃	—	8	1	no	—
188	C₂²×C₁₃	—	8	2	no	—
189	C₂²×C₁₃	#442	1	1	yes	characteristic
190	C₂²×C₁₃	#447	1	1	yes	characteristic
191	C₂²×C₁₃	#463	1	1	yes	characteristic
192	C₄×C₁₃	—	4	2	no	—
193	C₄×C₁₃	—	8	2	no	—
194	C₄×C₁₃	—	8	1	no	—
195	C₄×C₁₃	#386	1	1	yes	characteristic
196	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
197	C₄×C₁₃	#449	1	1	yes	characteristic
198	C₄×C₁₃	#459	1	1	yes	characteristic
199	C₂×#41	—	2	1	no	—
200	C₂²×D₄	—	2	1	no	—
201	C₂³×C₄	—	4	1	no	—
202	#135	—	2	1	no	—
203	#135	—	2	1	no	—
204	#135	—	2	1	no	—
205	#135	—	2	1	no	—
206	C₂×Q₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic, complemented
207	C₂×SD₁₆	—	2	1	no	—
208	C₂×D₈	—	2	1	no	—
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	—	2	1	no	—
211	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
212	C₂×D₈	C₂×C₄×C₁₃	1	1	yes	characteristic, complemented
213	C₂×D₈	D₄×C₁₃	1	2	yes	—
214	C₂×D₈	D₄×C₁₃	1	2	yes	—
215	C₂×M₁₆	—	2	1	no	—
216	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
217	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
218	C₄×D₄	—	2	4	no	—
219	C₄×D₄	—	4	2	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₂²⋊C₄	—	4	1	no	—
222	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
223	SD₃₂	—	2	4	no	—
224	D₁₆	—	2	4	no	—
225	M₃₂	—	2	1	no	—
226	M₃₂	—	2	1	no	—
227	M₃₂	—	2	1	no	—
228	M₃₂	—	2	1	no	—
229	C₂×C₁₆	—	2	1	no	—
230	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	complemented
231	#108	—	2	1	no	—
232	#108	—	2	1	no	—
233	#106	—	2	1	no	—
234	#106	D₄×C₁₃	1	2	yes	—
235	#104	—	4	2	no	—
236	#102	—	2	4	no	—
237	C₂²⋊C₈	—	4	2	no	—
238	#97	—	2	1	no	—
239	#97	—	2	1	no	—
240	C₂×C₁₃	—	2	1	no	—
241	C₂×C₁₃	—	2	1	no	—
242	C₂×C₁₃	—	8	4	no	—
243	C₂×C₁₃	—	16	1	no	—
244	C₂×C₁₃	#1879	1	1	yes	characteristic
245	C₂×C₁₃	AGL(1,ℤ/16)	1	2	yes	—
246	C₂⁴	—	4	1	no	—
247	#41	—	4	1	no	—
248	#41	—	4	1	no	—
249	#41	—	4	1	no	—
250	#41	—	4	1	no	—
251	C₂×Q₈	—	2	1	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	4	4	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	8	4	no	—
265	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
266	Q₁₆	—	2	1	no	—
267	Q₁₆	—	2	1	no	—
268	SD₁₆	—	4	1	no	—
269	SD₁₆	—	4	1	no	—
270	D₈	—	2	1	no	—
271	D₈	—	2	2	no	—
272	D₈	—	2	1	no	—
273	D₈	—	2	2	no	—
274	D₈	—	2	2	no	—
275	D₈	C₁₃×C₂²⋊C₄	1	4	yes	—
276	M₁₆	—	4	1	no	—
277	M₁₆	—	4	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
283	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
284	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
285	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
286	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
287	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	4	4	no	—
291	C₄²	—	2	1	no	—
292	C₄²	C₁₃×D₈	1	2	yes	—
293	C₁₆	—	2	2	no	—
294	C₁₆	—	4	1	no	—
295	C₁₃	#13037	1	1	yes	characteristic, direct product factor, Sylow
296	C₂³	—	4	2	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	1	no	—
300	C₂³	—	8	2	no	—
301	C₂³	—	8	2	no	—
302	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
303	Q₈	—	4	1	no	—
304	Q₈	—	4	1	no	—
305	D₄	—	2	4	no	—
306	D₄	—	4	1	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	1	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	8	1	no	—
312	D₄	—	8	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	8	8	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	—	8	1	no	—
320	C₂×C₄	—	8	1	no	—
321	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
322	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
323	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
324	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
325	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
326	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
327	C₈	—	4	1	no	—
328	C₈	—	8	1	no	—
329	C₈	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
330	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
331	C₈	C₁₃×#121	1	1	yes	characteristic
332	C₈	C₁₃×#123	1	1	yes	characteristic
333	C₂²	—	2	2	no	—
334	C₂²	—	4	4	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	4	no	—
338	C₂²	—	8	4	no	—
339	C₂²	—	8	1	no	—
340	C₂²	—	8	2	no	—
341	C₂²	C₁₃×#442	1	1	yes	characteristic
342	C₂²	C₁₃×#447	1	1	yes	characteristic
343	C₂²	C₁₃×#463	1	1	yes	characteristic
344	C₄	—	4	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	1	no	—
347	C₄	C₁₃×#386	1	1	yes	characteristic
348	C₄	C₄×C₁₃×D₈	1	1	yes	characteristic
349	C₄	C₁₃×#449	1	1	yes	characteristic
350	C₄	C₁₃×#459	1	1	yes	characteristic
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	8	4	no	—
354	C₂	—	16	1	no	—
355	C₂	C₁₃×#1879	1	1	yes	characteristic
356	C₂	C₁₃×AGL(1,ℤ/16)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	#442
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₁₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#442	C₂	semidirect
3. C₂×C₁₃×#361	C₂	semidirect
4. C₂×C₁₃×#360	C₂	semidirect
5. C₁₃×#1975	C₂	semidirect
6. C₁₃×#1383	C₂	semidirect
7. C₁₃×#1375	C₂	semidirect
13. C₂×C₁₃×SD₃₂	C₄	semidirect
14. C₂×C₁₃×D₁₆	C₄	semidirect
34. C₂×C₁₃×Q₁₆	C₂×C₄	semidirect
40. C₂×C₁₃×D₈	C₂×C₄	semidirect
58. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
68. #13037	C₁₃	direct (N × H)
119. C₂×#442	C₂×C₁₃	semidirect
120. C₂×#361	C₂×C₁₃	semidirect
121. C₂×#360	C₂×C₁₃	semidirect
122. #1975	C₂×C₁₃	semidirect
123. #1383	C₂×C₁₃	semidirect
124. #1375	C₂×C₁₃	semidirect
167. C₂×SD₃₂	C₄×C₁₃	semidirect
168. C₂×D₁₆	C₄×C₁₃	semidirect
206. C₂×Q₁₆	C₂×C₄×C₁₃	semidirect
212. C₂×D₈	C₂×C₄×C₁₃	semidirect
230. C₂×C₁₆	C₂×C₄×C₁₃	semidirect
295. C₁₃	#13037	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×#509	2
C₂×C₁₃×#442	2
C₂×C₁₃×#361	2
C₂×C₁₃×#360	2
C₁₃×#1975	2
C₁₃×#1383	2
C₁₃×#1375	2
#13037	13

Subgroups — C13×#13037 #15699376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13037 #15699376