Subgroups — C₁₃×#18714 #15705053

Order 3328.

1132 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3549	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3205	C₂	1	2	yes	maximal, complemented
3	C₁₃×#3151	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1624	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1619	C₂	1	4	yes	maximal, complemented
6	C₁₃×#1543	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1484	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₃×AGL(1,ℤ/8)	—	2	2	no	—
9	C₁₃×H(3,𝔽₄)	—	2	2	no	—
10	C₁₃×#555	—	2	2	no	—
11	C₁₃×#555	C₂²	1	2	yes	complemented
12	D₄²×C₁₃	—	2	2	no	—
13	D₄²×C₁₃	C₂²	1	2	yes	complemented
14	C₁₃×#535	—	2	2	no	—
15	C₁₃×#535	—	2	4	no	—
16	C₁₃×#532	C₂²	1	1	yes	characteristic
17	C₁₃×#532	C₂²	1	2	yes	—
18	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic, complemented
19	C₁₃×#496	—	2	2	no	—
20	C₁₃×#493	—	2	2	no	—
21	C₁₃×#493	C₂²	1	2	yes	—
22	C₁₃×#492	C₂²	1	2	yes	complemented
23	C₁₃×#486	—	2	2	no	—
24	C₁₃×#459	C₂²	1	8	yes	complemented
25	C₁₃×#447	—	2	4	no	—
26	C₁₃×#431	C₂²	1	2	yes	complemented
27	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic, complemented
28	C₁₃×#333	C₂²	1	2	yes	complemented
29	C₁₃×#331	C₂²	1	2	yes	complemented
30	C₁₃×D₄⋊C₈	C₂²	1	8	yes	complemented
31	C₂×C₁₃×#41	—	2	2	no	—
32	C₂×C₁₃×#41	D₄	1	2	yes	—
33	C₂²×D₄×C₁₃	—	2	2	no	—
34	C₂²×D₄×C₁₃	—	2	4	no	—
35	C₂²×D₄×C₁₃	—	2	1	no	—
36	C₂²×D₄×C₁₃	—	2	2	no	—
37	C₁₃×AGL(1,ℤ/8)	—	4	8	no	—
38	C₂×C₁₃×SD₁₆	—	2	4	no	—
39	C₂×C₁₃×D₈	—	2	4	no	—
40	C₂×C₁₃×D₈	—	2	8	no	—
41	C₂×C₁₃×M₁₆	—	2	1	no	—
42	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
43	C₁₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
44	C₁₃×#127	—	2	4	no	—
45	C₁₃×#127	—	2	2	no	—
46	C₁₃×#127	—	2	1	no	—
47	C₁₃×#127	D₄	1	2	yes	—
48	C₁₃×#127	C₂³	1	2	yes	—
49	C₁₃×#127	C₂³	1	1	yes	characteristic
50	C₁₃×#124	—	2	8	no	—
51	C₁₃×#124	—	2	4	no	—
52	C₁₃×#124	—	2	2	no	—
53	C₁₃×#124	D₄	1	2	yes	—
54	C₁₃×#121	—	2	4	no	—
55	C₁₃×#121	—	2	4	no	—
56	C₁₃×#121	—	2	2	no	—
57	C₁₃×#121	—	2	2	no	—
58	C₁₃×#121	—	2	4	no	—
59	C₁₃×#121	D₄	1	4	yes	—
60	C₁₃×#120	—	2	4	no	—
61	C₁₃×#120	—	2	8	no	—
62	C₁₃×#120	—	2	4	no	—
63	C₄×D₄×C₁₃	—	2	2	no	—
64	C₄×D₄×C₁₃	—	2	2	no	—
65	C₄×D₄×C₁₃	C₂³	1	4	yes	—
66	C₁₃×#117	—	2	2	no	—
67	C₁₃×#117	—	2	4	no	—
68	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
69	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
70	C₁₃×#102	—	2	8	no	—
71	C₁₃×C₂²⋊C₈	—	2	2	no	—
72	C₁₃×#97	—	2	1	no	—
73	C₄×C₈×C₁₃	—	2	2	no	—
74	C₄×C₈×C₁₃	C₂³	1	2	yes	—
75	#18714	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂⁴×C₁₃	—	2	4	no	—
77	C₂⁴×C₁₃	—	4	1	no	—
78	C₁₃×#41	—	4	4	no	—
79	C₁₃×#41	—	8	4	no	—
80	C₂×Q₈×C₁₃	—	2	2	no	—
81	C₂×Q₈×C₁₃	C₂×D₄	1	2	yes	—
82	C₂×D₄×C₁₃	—	2	4	no	—
83	C₂×D₄×C₁₃	—	2	8	no	—
84	C₂×D₄×C₁₃	—	2	2	no	—
85	C₂×D₄×C₁₃	—	2	4	no	—
86	C₂×D₄×C₁₃	—	2	2	no	—
87	C₂×D₄×C₁₃	—	2	2	no	—
88	C₂×D₄×C₁₃	—	2	2	no	—
89	C₂×D₄×C₁₃	—	2	4	no	—
90	C₂×D₄×C₁₃	—	2	2	no	—
91	C₂×D₄×C₁₃	—	4	4	no	—
92	C₂×D₄×C₁₃	—	4	2	no	—
93	C₂×D₄×C₁₃	—	4	1	no	—
94	C₂×D₄×C₁₃	—	8	4	no	—
95	C₂×D₄×C₁₃	—	8	8	no	—
96	C₂×D₄×C₁₃	—	8	4	no	—
97	C₂×D₄×C₁₃	C₂×D₄	1	4	yes	—
98	C₂×D₄×C₁₃	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₁₃	—	2	2	no	—
100	C₂²×C₄×C₁₃	—	2	4	no	—
101	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₁₃×SD₁₆	—	4	8	no	—
104	C₁₃×D₈	—	4	8	no	—
105	C₁₃×D₈	—	8	8	no	—
106	C₁₃×M₁₆	—	2	4	no	—
107	C₂×C₈×C₁₃	—	2	2	no	—
108	C₂×C₈×C₁₃	—	2	4	no	—
109	C₁₃×C₄⋊C₄	—	2	2	no	—
110	C₁₃×C₄⋊C₄	—	2	2	no	—
111	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
112	C₁₃×C₂²⋊C₄	—	2	8	no	—
113	C₁₃×C₂²⋊C₄	—	2	4	no	—
114	C₁₃×C₂²⋊C₄	—	2	4	no	—
115	C₁₃×C₂²⋊C₄	—	2	4	no	—
116	C₁₃×C₂²⋊C₄	—	2	4	no	—
117	C₄²×C₁₃	—	2	4	no	—
118	C₄²×C₁₃	C₂×D₄	1	2	yes	—
119	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
121	#3549	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#3205	C₂×C₁₃	1	2	yes	complemented
123	#3151	C₂×C₁₃	1	1	yes	characteristic, complemented
124	#1624	C₂×C₁₃	1	4	yes	complemented
125	#1619	C₂×C₁₃	1	4	yes	complemented
126	#1543	C₂×C₁₃	1	1	yes	characteristic, complemented
127	#1484	C₂×C₁₃	1	2	yes	complemented
128	C₂³×C₁₃	—	2	4	no	—
129	C₂³×C₁₃	—	2	2	no	—
130	C₂³×C₁₃	—	4	2	no	—
131	C₂³×C₁₃	—	8	2	no	—
132	C₂³×C₁₃	—	8	4	no	—
133	C₂³×C₁₃	—	8	8	no	—
134	C₂³×C₁₃	—	8	4	no	—
135	C₂³×C₁₃	#120	1	1	yes	characteristic
136	Q₈×C₁₃	—	4	2	no	—
137	Q₈×C₁₃	—	8	2	no	—
138	D₄×C₁₃	—	2	8	no	—
139	D₄×C₁₃	—	4	2	no	—
140	D₄×C₁₃	—	4	4	no	—
141	D₄×C₁₃	—	4	4	no	—
142	D₄×C₁₃	—	4	8	no	—
143	D₄×C₁₃	—	4	4	no	—
144	D₄×C₁₃	—	8	2	no	—
145	D₄×C₁₃	—	8	4	no	—
146	D₄×C₁₃	—	8	2	no	—
147	D₄×C₁₃	—	8	4	no	—
148	D₄×C₁₃	—	8	8	no	—
149	D₄×C₁₃	—	8	4	no	—
150	D₄×C₁₃	—	8	2	no	—
151	C₂×C₄×C₁₃	—	2	4	no	—
152	C₂×C₄×C₁₃	—	2	2	no	—
153	C₂×C₄×C₁₃	—	2	2	no	—
154	C₂×C₄×C₁₃	—	2	4	no	—
155	C₂×C₄×C₁₃	—	4	8	no	—
156	C₂×C₄×C₁₃	—	4	4	no	—
157	C₂×C₄×C₁₃	#120	1	2	yes	—
158	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
161	C₈×C₁₃	—	2	4	no	—
162	C₈×C₁₃	—	4	4	no	—
163	C₂×AGL(1,ℤ/8)	—	2	2	no	—
164	H(3,𝔽₄)	—	2	2	no	—
165	#555	—	2	2	no	—
166	#555	C₂²×C₁₃	1	2	yes	complemented
167	D₄²	—	2	2	no	—
168	D₄²	C₂²×C₁₃	1	2	yes	complemented
169	#535	—	2	2	no	—
170	#535	—	2	4	no	—
171	#532	C₂²×C₁₃	1	1	yes	characteristic
172	#532	C₂²×C₁₃	1	2	yes	—
173	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic, complemented
174	#496	—	2	2	no	—
175	#493	—	2	2	no	—
176	#493	C₂²×C₁₃	1	2	yes	—
177	#492	C₂²×C₁₃	1	2	yes	complemented
178	#486	—	2	2	no	—
179	#459	C₂²×C₁₃	1	8	yes	complemented
180	#447	—	2	4	no	—
181	#431	C₂²×C₁₃	1	2	yes	complemented
182	C₄×M₁₆	C₂²×C₁₃	1	1	yes	characteristic, complemented
183	#333	C₂²×C₁₃	1	2	yes	complemented
184	#331	C₂²×C₁₃	1	2	yes	complemented
185	D₄⋊C₈	C₂²×C₁₃	1	8	yes	complemented
186	C₂²×C₁₃	—	2	2	no	—
187	C₂²×C₁₃	—	2	1	no	—
188	C₂²×C₁₃	—	4	4	no	—
189	C₂²×C₁₃	—	4	4	no	—
190	C₂²×C₁₃	—	4	4	no	—
191	C₂²×C₁₃	—	4	2	no	—
192	C₂²×C₁₃	—	4	4	no	—
193	C₂²×C₁₃	—	8	4	no	—
194	C₂²×C₁₃	—	8	2	no	—
195	C₂²×C₁₃	—	16	4	no	—
196	C₂²×C₁₃	—	16	8	no	—
197	C₂²×C₁₃	C₂×#120	1	1	yes	characteristic
198	C₄×C₁₃	—	2	2	no	—
199	C₄×C₁₃	—	2	1	no	—
200	C₄×C₁₃	—	4	4	no	—
201	C₄×C₁₃	—	4	2	no	—
202	C₄×C₁₃	#453	1	2	yes	—
203	C₄×C₁₃	#576	1	4	yes	—
204	C₂×#41	—	2	2	no	—
205	C₂×#41	D₄×C₁₃	1	2	yes	—
206	C₂²×D₄	—	2	2	no	—
207	C₂²×D₄	—	2	4	no	—
208	C₂²×D₄	—	2	1	no	—
209	C₂²×D₄	—	2	2	no	—
210	AGL(1,ℤ/8)	—	4	8	no	—
211	C₂×SD₁₆	—	2	4	no	—
212	C₂×D₈	—	2	4	no	—
213	C₂×D₈	—	2	8	no	—
214	C₂×M₁₆	—	2	1	no	—
215	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
216	C₄⋊Q₈	C₂³×C₁₃	1	1	yes	characteristic
217	#127	—	2	4	no	—
218	#127	—	2	2	no	—
219	#127	—	2	1	no	—
220	#127	D₄×C₁₃	1	2	yes	—
221	#127	C₂³×C₁₃	1	2	yes	—
222	#127	C₂³×C₁₃	1	1	yes	characteristic
223	#124	—	2	8	no	—
224	#124	—	2	4	no	—
225	#124	—	2	2	no	—
226	#124	D₄×C₁₃	1	2	yes	—
227	#121	—	2	4	no	—
228	#121	—	2	4	no	—
229	#121	—	2	2	no	—
230	#121	—	2	2	no	—
231	#121	—	2	4	no	—
232	#121	D₄×C₁₃	1	4	yes	—
233	#120	—	2	4	no	—
234	#120	—	2	8	no	—
235	#120	—	2	4	no	—
236	C₄×D₄	—	2	2	no	—
237	C₄×D₄	—	2	2	no	—
238	C₄×D₄	C₂³×C₁₃	1	4	yes	—
239	#117	—	2	2	no	—
240	#117	—	2	4	no	—
241	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
242	C₄⋊C₈	C₂³×C₁₃	1	4	yes	—
243	#102	—	2	8	no	—
244	C₂²⋊C₈	—	2	2	no	—
245	#97	—	2	1	no	—
246	C₄×C₈	—	2	2	no	—
247	C₄×C₈	C₂³×C₁₃	1	2	yes	—
248	C₂×C₁₃	—	4	1	no	—
249	C₂×C₁₃	—	8	4	no	—
250	C₂×C₁₃	—	8	2	no	—
251	C₂×C₁₃	—	16	2	no	—
252	C₂×C₁₃	#3008	1	2	yes	—
253	C₂×C₁₃	C₂×#453	1	1	yes	characteristic
254	C₂⁴	—	2	4	no	—
255	C₂⁴	—	4	1	no	—
256	#41	—	4	4	no	—
257	#41	—	8	4	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	C₂×D₄×C₁₃	1	2	yes	—
260	C₂×D₄	—	2	4	no	—
261	C₂×D₄	—	2	8	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	4	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	4	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	4	4	no	—
270	C₂×D₄	—	4	2	no	—
271	C₂×D₄	—	4	1	no	—
272	C₂×D₄	—	8	4	no	—
273	C₂×D₄	—	8	8	no	—
274	C₂×D₄	—	8	4	no	—
275	C₂×D₄	C₂×D₄×C₁₃	1	4	yes	—
276	C₂×D₄	C₂×D₄×C₁₃	1	2	yes	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	—	2	4	no	—
279	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
280	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
281	SD₁₆	—	4	8	no	—
282	D₈	—	4	8	no	—
283	D₈	—	8	8	no	—
284	M₁₆	—	2	4	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	4	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
290	C₂²⋊C₄	—	2	8	no	—
291	C₂²⋊C₄	—	2	4	no	—
292	C₂²⋊C₄	—	2	4	no	—
293	C₂²⋊C₄	—	2	4	no	—
294	C₂²⋊C₄	—	2	4	no	—
295	C₄²	—	2	4	no	—
296	C₄²	C₂×D₄×C₁₃	1	2	yes	—
297	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
298	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
299	C₁₃	#18714	1	1	yes	characteristic, direct product factor, Sylow
300	C₂³	—	2	4	no	—
301	C₂³	—	2	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	8	2	no	—
304	C₂³	—	8	4	no	—
305	C₂³	—	8	8	no	—
306	C₂³	—	8	4	no	—
307	C₂³	C₁₃×#120	1	1	yes	characteristic
308	Q₈	—	4	2	no	—
309	Q₈	—	8	2	no	—
310	D₄	—	2	8	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	4	no	—
313	D₄	—	4	4	no	—
314	D₄	—	4	4	no	—
315	D₄	—	4	8	no	—
316	D₄	—	8	2	no	—
317	D₄	—	8	4	no	—
318	D₄	—	8	2	no	—
319	D₄	—	8	4	no	—
320	D₄	—	8	8	no	—
321	D₄	—	8	4	no	—
322	D₄	—	8	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	4	no	—
327	C₂×C₄	—	4	4	no	—
328	C₂×C₄	—	4	8	no	—
329	C₂×C₄	C₁₃×#120	1	2	yes	—
330	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
331	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
332	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
333	C₈	—	2	4	no	—
334	C₈	—	4	4	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	4	4	no	—
338	C₂²	—	4	4	no	—
339	C₂²	—	4	4	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	4	4	no	—
342	C₂²	—	8	4	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	16	4	no	—
345	C₂²	—	16	8	no	—
346	C₂²	C₂×C₁₃×#120	1	1	yes	characteristic
347	C₄	—	2	2	no	—
348	C₄	—	2	1	no	—
349	C₄	—	4	4	no	—
350	C₄	—	4	2	no	—
351	C₄	C₁₃×#453	1	2	yes	—
352	C₄	C₁₃×#576	1	4	yes	—
353	C₂	—	4	1	no	—
354	C₂	—	8	4	no	—
355	C₂	—	8	2	no	—
356	C₂	—	16	2	no	—
357	C₂	C₁₃×#3008	1	2	yes	—
358	C₂	C₂×C₁₃×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3549	C₂	semidirect
2. C₁₃×#3205	C₂	semidirect
3. C₁₃×#3151	C₂	semidirect
4. C₁₃×#1624	C₂	semidirect
5. C₁₃×#1619	C₂	semidirect
6. C₁₃×#1543	C₂	semidirect
7. C₁₃×#1484	C₂	semidirect
11. C₁₃×#555	C₂²	semidirect
13. D₄²×C₁₃	C₂²	semidirect
18. C₂×C₁₃×#127	C₂²	semidirect
22. C₁₃×#492	C₂²	semidirect
24. C₁₃×#459	C₂²	semidirect
26. C₁₃×#431	C₂²	semidirect
27. C₄×C₁₃×M₁₆	C₂²	semidirect
28. C₁₃×#333	C₂²	semidirect
29. C₁₃×#331	C₂²	semidirect
30. C₁₃×D₄⋊C₈	C₂²	semidirect
75. #18714	C₁₃	direct (N × H)
121. #3549	C₂×C₁₃	semidirect
122. #3205	C₂×C₁₃	semidirect
123. #3151	C₂×C₁₃	semidirect
124. #1624	C₂×C₁₃	semidirect
125. #1619	C₂×C₁₃	semidirect
126. #1543	C₂×C₁₃	semidirect
127. #1484	C₂×C₁₃	semidirect
166. #555	C₂²×C₁₃	semidirect
168. D₄²	C₂²×C₁₃	semidirect
173. C₂×#127	C₂²×C₁₃	semidirect
177. #492	C₂²×C₁₃	semidirect
179. #459	C₂²×C₁₃	semidirect
181. #431	C₂²×C₁₃	semidirect
182. C₄×M₁₆	C₂²×C₁₃	semidirect
183. #333	C₂²×C₁₃	semidirect
184. #331	C₂²×C₁₃	semidirect
185. D₄⋊C₈	C₂²×C₁₃	semidirect
299. C₁₃	#18714	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×#3549	2
C₁₃×#3205	2
C₁₃×#3151	2
C₁₃×#1624	2
C₁₃×#1619	2
C₁₃×#1543	2
C₁₃×#1484	2
#18714	13

Subgroups — C13×#18714 #15705053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#18714 #15705053