Subgroups — C₁₃×#18715 #15705054

Order 3328.

1056 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	C₁₃×#3549	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#3227	C₂	1	2	yes	maximal, complemented
3	C₁₃×#3152	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1634	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1629	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×#137	—	2	2	no	—
9	C₁₃×H(3,𝔽₄)	—	2	2	no	—
10	C₁₃×#555	—	2	2	no	—
11	C₁₃×#555	C₂²	1	2	yes	—
12	D₄²×C₁₃	—	2	2	no	—
13	D₄²×C₁₃	C₂²	1	2	yes	—
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×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	C₁₃×#498	—	2	2	no	—
20	C₁₃×#497	—	2	2	no	—
21	C₁₃×#494	C₂²	1	2	yes	complemented
22	C₁₃×#492	C₂²	1	2	yes	—
23	C₁₃×#486	—	2	2	no	—
24	C₁₃×#481	—	2	4	no	—
25	C₁₃×#476	C₂²	1	8	yes	complemented
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	—
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×Q₈×C₁₃	—	2	1	no	—
34	C₂²×D₄×C₁₃	—	2	2	no	—
35	C₂²×D₄×C₁₃	—	2	4	no	—
36	C₁₃×#137	—	4	8	no	—
37	C₂×C₁₃×Q₁₆	—	2	4	no	—
38	C₂×C₁₃×SD₁₆	—	2	4	no	—
39	C₂×C₁₃×M₁₆	—	2	1	no	—
40	C₁₃×C₄⋊Q₈	—	2	2	no	—
41	C₁₃×C₄⋊Q₈	—	2	1	no	—
42	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
43	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
44	C₁₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
45	C₁₃×#127	—	2	4	no	—
46	C₁₃×#127	D₄	1	2	yes	—
47	C₁₃×#127	C₂³	1	1	yes	characteristic
48	C₁₃×#124	—	2	8	no	—
49	C₁₃×#124	—	2	4	no	—
50	C₁₃×#124	—	2	2	no	—
51	C₁₃×#124	D₄	1	2	yes	—
52	C₁₃×#121	—	2	4	no	—
53	C₁₃×#121	—	2	4	no	—
54	C₁₃×#121	—	2	2	no	—
55	C₁₃×#121	—	2	2	no	—
56	C₁₃×#121	—	2	4	no	—
57	C₁₃×#121	D₄	1	4	yes	—
58	C₁₃×#120	—	2	4	no	—
59	C₁₃×#120	—	2	8	no	—
60	C₁₃×#120	—	2	4	no	—
61	C₄×D₄×C₁₃	—	2	2	no	—
62	C₄×D₄×C₁₃	—	2	2	no	—
63	C₄×D₄×C₁₃	C₂³	1	4	yes	—
64	C₁₃×#117	—	2	2	no	—
65	C₁₃×#117	—	2	4	no	—
66	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
68	C₁₃×#106	—	2	8	no	—
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	#18715	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂⁴×C₁₃	—	2	4	no	—
77	C₁₃×#41	—	4	4	no	—
78	C₁₃×#41	—	8	4	no	—
79	C₂×Q₈×C₁₃	—	2	2	no	—
80	C₂×Q₈×C₁₃	—	2	2	no	—
81	C₂×Q₈×C₁₃	—	4	4	no	—
82	C₂×Q₈×C₁₃	—	4	1	no	—
83	C₂×Q₈×C₁₃	C₂×D₄	1	2	yes	—
84	C₂×D₄×C₁₃	—	2	4	no	—
85	C₂×D₄×C₁₃	—	2	8	no	—
86	C₂×D₄×C₁₃	—	2	2	no	—
87	C₂×D₄×C₁₃	—	2	4	no	—
88	C₂×D₄×C₁₃	—	2	2	no	—
89	C₂×D₄×C₁₃	—	2	2	no	—
90	C₂×D₄×C₁₃	—	2	2	no	—
91	C₂×D₄×C₁₃	—	8	8	no	—
92	C₂×D₄×C₁₃	—	8	4	no	—
93	C₂×D₄×C₁₃	C₂×D₄	1	4	yes	—
94	C₂×D₄×C₁₃	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₁₃	—	2	2	no	—
96	C₂²×C₄×C₁₃	—	2	4	no	—
97	C₂²×C₄×C₁₃	—	4	1	no	—
98	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₁₃×Q₁₆	—	4	8	no	—
101	C₁₃×SD₁₆	—	4	8	no	—
102	C₁₃×M₁₆	—	2	4	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	2	4	no	—
105	C₁₃×C₄⋊C₄	—	2	2	no	—
106	C₁₃×C₄⋊C₄	—	2	2	no	—
107	C₁₃×C₄⋊C₄	—	2	4	no	—
108	C₁₃×C₄⋊C₄	—	4	2	no	—
109	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
110	C₁₃×C₂²⋊C₄	—	2	8	no	—
111	C₁₃×C₂²⋊C₄	—	2	4	no	—
112	C₁₃×C₂²⋊C₄	—	2	4	no	—
113	C₁₃×C₂²⋊C₄	—	2	4	no	—
114	C₁₃×C₂²⋊C₄	—	2	4	no	—
115	C₄²×C₁₃	—	2	4	no	—
116	C₄²×C₁₃	C₂×D₄	1	2	yes	—
117	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
119	#3549	C₂×C₁₃	1	1	yes	characteristic
120	#3227	C₂×C₁₃	1	2	yes	complemented
121	#3152	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#1634	C₂×C₁₃	1	4	yes	complemented
123	#1629	C₂×C₁₃	1	4	yes	complemented
124	#1543	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#1484	C₂×C₁₃	1	2	yes	complemented
126	C₂³×C₁₃	—	2	4	no	—
127	C₂³×C₁₃	—	2	2	no	—
128	C₂³×C₁₃	—	8	8	no	—
129	C₂³×C₁₃	—	8	4	no	—
130	C₂³×C₁₃	#120	1	1	yes	characteristic
131	Q₈×C₁₃	—	4	2	no	—
132	Q₈×C₁₃	—	4	4	no	—
133	Q₈×C₁₃	—	8	2	no	—
134	Q₈×C₁₃	—	8	2	no	—
135	D₄×C₁₃	—	2	8	no	—
136	D₄×C₁₃	—	4	2	no	—
137	D₄×C₁₃	—	4	4	no	—
138	D₄×C₁₃	—	4	4	no	—
139	D₄×C₁₃	—	8	2	no	—
140	D₄×C₁₃	—	8	2	no	—
141	D₄×C₁₃	—	8	4	no	—
142	D₄×C₁₃	—	8	8	no	—
143	D₄×C₁₃	—	8	4	no	—
144	C₂×C₄×C₁₃	—	2	4	no	—
145	C₂×C₄×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	2	4	no	—
148	C₂×C₄×C₁₃	—	4	8	no	—
149	C₂×C₄×C₁₃	—	4	4	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	—	8	2	no	—
152	C₂×C₄×C₁₃	#120	1	2	yes	—
153	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
156	C₈×C₁₃	—	2	4	no	—
157	C₈×C₁₃	—	4	4	no	—
158	C₂×#137	—	2	2	no	—
159	H(3,𝔽₄)	—	2	2	no	—
160	#555	—	2	2	no	—
161	#555	C₂²×C₁₃	1	2	yes	—
162	D₄²	—	2	2	no	—
163	D₄²	C₂²×C₁₃	1	2	yes	—
164	#535	—	2	2	no	—
165	#535	—	2	4	no	—
166	#532	C₂²×C₁₃	1	1	yes	characteristic
167	#532	C₂²×C₁₃	1	2	yes	—
168	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
169	#498	—	2	2	no	—
170	#497	—	2	2	no	—
171	#494	C₂²×C₁₃	1	2	yes	complemented
172	#492	C₂²×C₁₃	1	2	yes	—
173	#486	—	2	2	no	—
174	#481	—	2	4	no	—
175	#476	C₂²×C₁₃	1	8	yes	complemented
176	#431	C₂²×C₁₃	1	2	yes	complemented
177	C₄×M₁₆	C₂²×C₁₃	1	1	yes	characteristic, complemented
178	#333	C₂²×C₁₃	1	2	yes	complemented
179	#331	C₂²×C₁₃	1	2	yes	—
180	D₄⋊C₈	C₂²×C₁₃	1	8	yes	complemented
181	C₂²×C₁₃	—	2	2	no	—
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	—	4	4	no	—
184	C₂²×C₁₃	—	4	4	no	—
185	C₂²×C₁₃	—	4	4	no	—
186	C₂²×C₁₃	—	4	2	no	—
187	C₂²×C₁₃	—	4	4	no	—
188	C₂²×C₁₃	—	16	8	no	—
189	C₂²×C₁₃	C₂×#120	1	1	yes	characteristic
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	—	2	1	no	—
192	C₄×C₁₃	—	4	4	no	—
193	C₄×C₁₃	—	4	2	no	—
194	C₄×C₁₃	—	8	2	no	—
195	C₄×C₁₃	#453	1	2	yes	—
196	C₄×C₁₃	#577	1	4	yes	—
197	C₂×#41	—	2	2	no	—
198	C₂×#41	D₄×C₁₃	1	2	yes	—
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×D₄	—	2	2	no	—
201	C₂²×D₄	—	2	4	no	—
202	#137	—	4	8	no	—
203	C₂×Q₁₆	—	2	4	no	—
204	C₂×SD₁₆	—	2	4	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₄⋊Q₈	—	2	2	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
209	C₄⋊Q₈	C₂³×C₁₃	1	2	yes	—
210	C₄⋊Q₈	C₂³×C₁₃	1	1	yes	characteristic
211	#127	—	2	4	no	—
212	#127	D₄×C₁₃	1	2	yes	—
213	#127	C₂³×C₁₃	1	1	yes	characteristic
214	#124	—	2	8	no	—
215	#124	—	2	4	no	—
216	#124	—	2	2	no	—
217	#124	D₄×C₁₃	1	2	yes	—
218	#121	—	2	4	no	—
219	#121	—	2	4	no	—
220	#121	—	2	2	no	—
221	#121	—	2	2	no	—
222	#121	—	2	4	no	—
223	#121	D₄×C₁₃	1	4	yes	—
224	#120	—	2	4	no	—
225	#120	—	2	8	no	—
226	#120	—	2	4	no	—
227	C₄×D₄	—	2	2	no	—
228	C₄×D₄	—	2	2	no	—
229	C₄×D₄	C₂³×C₁₃	1	4	yes	—
230	#117	—	2	2	no	—
231	#117	—	2	4	no	—
232	C₂×C₄⋊C₄	—	2	2	no	—
233	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
234	#106	—	2	8	no	—
235	C₄⋊C₈	C₂³×C₁₃	1	4	yes	—
236	#102	—	2	8	no	—
237	C₂²⋊C₈	—	2	2	no	—
238	#97	—	2	1	no	—
239	C₄×C₈	—	2	2	no	—
240	C₄×C₈	C₂³×C₁₃	1	2	yes	—
241	C₂×C₁₃	—	4	1	no	—
242	C₂×C₁₃	—	8	4	no	—
243	C₂×C₁₃	—	8	2	no	—
244	C₂×C₁₃	#3009	1	2	yes	—
245	C₂×C₁₃	C₂×#453	1	1	yes	characteristic
246	C₂⁴	—	2	4	no	—
247	#41	—	4	4	no	—
248	#41	—	8	4	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	4	4	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×Q₈	C₂×D₄×C₁₃	1	2	yes	—
254	C₂×D₄	—	2	4	no	—
255	C₂×D₄	—	2	8	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	4	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	8	8	no	—
262	C₂×D₄	—	8	4	no	—
263	C₂×D₄	C₂×D₄×C₁₃	1	4	yes	—
264	C₂×D₄	C₂×D₄×C₁₃	1	2	yes	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	4	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
269	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
270	Q₁₆	—	4	8	no	—
271	SD₁₆	—	4	8	no	—
272	M₁₆	—	2	4	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	4	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
280	C₂²⋊C₄	—	2	8	no	—
281	C₂²⋊C₄	—	2	4	no	—
282	C₂²⋊C₄	—	2	4	no	—
283	C₂²⋊C₄	—	2	4	no	—
284	C₂²⋊C₄	—	2	4	no	—
285	C₄²	—	2	4	no	—
286	C₄²	C₂×D₄×C₁₃	1	2	yes	—
287	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
288	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
289	C₁₃	#18715	1	1	yes	characteristic, direct product factor, Sylow
290	C₂³	—	2	4	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	8	8	no	—
293	C₂³	—	8	4	no	—
294	C₂³	C₁₃×#120	1	1	yes	characteristic
295	Q₈	—	4	2	no	—
296	Q₈	—	4	4	no	—
297	Q₈	—	8	2	no	—
298	Q₈	—	8	2	no	—
299	D₄	—	2	8	no	—
300	D₄	—	4	2	no	—
301	D₄	—	4	4	no	—
302	D₄	—	4	4	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	2	no	—
305	D₄	—	8	4	no	—
306	D₄	—	8	8	no	—
307	D₄	—	8	4	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	4	8	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	C₁₃×#120	1	2	yes	—
317	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
319	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
320	C₈	—	2	4	no	—
321	C₈	—	4	4	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	4	4	no	—
326	C₂²	—	4	4	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	4	4	no	—
329	C₂²	—	16	8	no	—
330	C₂²	C₂×C₁₃×#120	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	4	4	no	—
334	C₄	—	4	2	no	—
335	C₄	—	8	2	no	—
336	C₄	C₁₃×#453	1	2	yes	—
337	C₄	C₁₃×#577	1	4	yes	—
338	C₂	—	4	1	no	—
339	C₂	—	8	4	no	—
340	C₂	—	8	2	no	—
341	C₂	C₁₃×#3009	1	2	yes	—
342	C₂	C₂×C₁₃×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃
prime radical	C₁₃×#3549	C₂
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
2. C₁₃×#3227	C₂	semidirect
3. C₁₃×#3152	C₂	semidirect
4. C₁₃×#1634	C₂	semidirect
5. C₁₃×#1629	C₂	semidirect
6. C₁₃×#1543	C₂	semidirect
7. C₁₃×#1484	C₂	semidirect
18. C₂×C₁₃×C₄⋊Q₈	C₂²	semidirect
21. C₁₃×#494	C₂²	semidirect
25. C₁₃×#476	C₂²	semidirect
26. C₁₃×#431	C₂²	semidirect
27. C₄×C₁₃×M₁₆	C₂²	semidirect
28. C₁₃×#333	C₂²	semidirect
30. C₁₃×D₄⋊C₈	C₂²	semidirect
75. #18715	C₁₃	direct (N × H)
120. #3227	C₂×C₁₃	semidirect
121. #3152	C₂×C₁₃	semidirect
122. #1634	C₂×C₁₃	semidirect
123. #1629	C₂×C₁₃	semidirect
124. #1543	C₂×C₁₃	semidirect
125. #1484	C₂×C₁₃	semidirect
168. C₂×C₄⋊Q₈	C₂²×C₁₃	semidirect
171. #494	C₂²×C₁₃	semidirect
175. #476	C₂²×C₁₃	semidirect
176. #431	C₂²×C₁₃	semidirect
177. C₄×M₁₆	C₂²×C₁₃	semidirect
178. #333	C₂²×C₁₃	semidirect
180. D₄⋊C₈	C₂²×C₁₃	semidirect
289. C₁₃	#18715	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×#3227	2
C₁₃×#3152	2
C₁₃×#1634	2
C₁₃×#1629	2
C₁₃×#1543	2
C₁₃×#1484	2
#18715	13

Subgroups — C13×#18715 #15705054

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#18715 #15705054