Subgroups — C₁₃×#18404 #15704743

Order 3328.

910 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3502	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#3209	C₂	1	2	yes	maximal
3	C₁₃×#2567	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1605	C₂	1	8	yes	maximal
5	C₁₃×#1503	C₂	1	2	yes	maximal
6	C₁₃×#1461	C₂	1	1	yes	maximal, characteristic
7	Q₈²×C₁₃	—	2	2	no	—
8	C₁₃×#557	—	2	2	no	—
9	C₁₃×#557	—	2	4	no	—
10	C₁₃×#544	—	2	1	no	—
11	C₁₃×#544	C₂²	1	4	yes	—
12	C₁₃×#533	—	2	2	no	—
13	C₁₃×#533	C₂²	1	2	yes	—
14	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₃×#519	—	2	1	no	—
16	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#494	—	2	2	no	—
18	C₁₃×#487	—	2	2	no	—
19	C₁₃×#462	—	2	4	no	—
20	C₁₃×#451	C₂²	1	8	yes	—
21	C₁₃×#431	C₂²	1	2	yes	—
22	C₁₃×#395	C₂²	1	4	yes	—
23	C₁₃×#391	—	2	1	no	—
24	C₁₃×#389	—	2	2	no	—
25	C₁₃×#384	—	2	1	no	—
26	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₁₃×#328	C₂²	1	8	yes	—
28	C₁₃×#324	C₂²	1	4	yes	—
29	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
31	C₂×C₁₃×Q₁₆	—	2	8	no	—
32	C₁₃×C₄⋊Q₈	—	2	2	no	—
33	C₁₃×C₄⋊Q₈	—	2	2	no	—
34	C₁₃×C₄⋊Q₈	—	2	4	no	—
35	C₁₃×C₄⋊Q₈	—	2	8	no	—
36	C₁₃×C₄⋊Q₈	—	2	2	no	—
37	C₁₃×C₄⋊Q₈	—	2	2	no	—
38	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
39	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
40	C₁₃×#125	—	2	4	no	—
41	C₁₃×#125	—	2	4	no	—
42	C₁₃×#125	—	2	8	no	—
43	C₁₃×#125	—	2	2	no	—
44	C₁₃×C₂²⋊Q₈	—	2	2	no	—
45	C₁₃×C₂²⋊Q₈	—	2	4	no	—
46	C₁₃×C₂²⋊Q₈	C₂³	1	4	yes	—
47	C₄×Q₈×C₁₃	—	2	4	no	—
48	C₄×Q₈×C₁₃	—	2	4	no	—
49	C₄×Q₈×C₁₃	—	2	4	no	—
50	C₄×Q₈×C₁₃	—	2	4	no	—
51	C₄×Q₈×C₁₃	—	2	2	no	—
52	C₄×Q₈×C₁₃	—	2	2	no	—
53	C₄×Q₈×C₁₃	—	2	2	no	—
54	C₄×Q₈×C₁₃	D₄	1	4	yes	—
55	C₁₃×#117	—	2	4	no	—
56	C₁₃×#117	—	2	2	no	—
57	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
58	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
59	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
60	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
61	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
62	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
63	C₂×C₄²×C₁₃	—	2	1	no	—
64	C₂×C₄²×C₁₃	D₄	1	2	yes	—
65	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
66	C₁₃×C₄⋊C₈	—	2	2	no	—
67	C₁₃×#103	—	2	8	no	—
68	C₁₃×C₂²⋊C₈	C₂³	1	4	yes	—
69	C₄×C₈×C₁₃	—	2	2	no	—
70	C₁₃×#95	—	2	4	no	—
71	C₁₃×#95	C₂³	1	2	yes	—
72	#18404	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂×Q₈×C₁₃	—	2	4	no	—
74	C₂×Q₈×C₁₃	—	2	2	no	—
75	C₂×Q₈×C₁₃	—	2	2	no	—
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	2	4	no	—
78	C₂×Q₈×C₁₃	—	8	2	no	—
79	C₂×Q₈×C₁₃	Q₁₆	1	4	yes	complemented
80	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₁₃×Q₁₆	—	8	8	no	—
89	C₂×C₈×C₁₃	—	2	4	no	—
90	C₁₃×C₄⋊C₄	—	2	4	no	—
91	C₁₃×C₄⋊C₄	—	2	4	no	—
92	C₁₃×C₄⋊C₄	—	2	4	no	—
93	C₁₃×C₄⋊C₄	—	2	4	no	—
94	C₁₃×C₄⋊C₄	—	2	8	no	—
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	4	no	—
100	C₁₃×C₄⋊C₄	—	2	4	no	—
101	C₁₃×C₄⋊C₄	—	2	4	no	—
102	C₁₃×C₄⋊C₄	—	4	1	no	—
103	C₁₃×C₄⋊C₄	—	4	2	no	—
104	C₁₃×C₄⋊C₄	—	4	1	no	—
105	C₁₃×C₄⋊C₄	—	8	4	no	—
106	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
107	C₁₃×C₂²⋊C₄	—	2	4	no	—
108	C₄²×C₁₃	—	2	8	no	—
109	C₄²×C₁₃	—	2	2	no	—
110	C₄²×C₁₃	—	2	2	no	—
111	C₄²×C₁₃	—	2	4	no	—
112	C₄²×C₁₃	—	4	2	no	—
113	C₄²×C₁₃	C₂×D₄	1	2	yes	—
114	C₄²×C₁₃	C₂×D₄	1	2	yes	—
115	#3502	C₂×C₁₃	1	1	yes	characteristic
116	#3209	C₂×C₁₃	1	2	yes	—
117	#2567	C₂×C₁₃	1	1	yes	characteristic
118	#1605	C₂×C₁₃	1	8	yes	—
119	#1503	C₂×C₁₃	1	2	yes	—
120	#1461	C₂×C₁₃	1	1	yes	characteristic
121	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	Q₈×C₁₃	—	2	8	no	—
123	Q₈×C₁₃	—	4	4	no	—
124	Q₈×C₁₃	—	4	8	no	—
125	Q₈×C₁₃	—	4	4	no	—
126	Q₈×C₁₃	—	8	4	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	4	no	—
131	C₂×C₄×C₁₃	—	2	4	no	—
132	C₂×C₄×C₁₃	—	2	4	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	4	1	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	4	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
141	C₂×C₄×C₁₃	#120	1	2	yes	—
142	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
144	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
145	C₂×C₄×C₁₃	#137	1	2	yes	—
146	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
147	C₈×C₁₃	—	4	4	no	—
148	Q₈²	—	2	2	no	—
149	#557	—	2	2	no	—
150	#557	—	2	4	no	—
151	#544	—	2	1	no	—
152	#544	C₂²×C₁₃	1	4	yes	—
153	#533	—	2	2	no	—
154	#533	C₂²×C₁₃	1	2	yes	—
155	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
156	#519	—	2	1	no	—
157	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
158	#494	—	2	2	no	—
159	#487	—	2	2	no	—
160	#462	—	2	4	no	—
161	#451	C₂²×C₁₃	1	8	yes	—
162	#431	C₂²×C₁₃	1	2	yes	—
163	#395	C₂²×C₁₃	1	4	yes	—
164	#391	—	2	1	no	—
165	#389	—	2	2	no	—
166	#384	—	2	1	no	—
167	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
168	#328	C₂²×C₁₃	1	8	yes	—
169	#324	C₂²×C₁₃	1	4	yes	—
170	C₂²×C₁₃	—	2	1	no	—
171	C₂²×C₁₃	—	2	1	no	—
172	C₂²×C₁₃	C₂×#120	1	1	yes	characteristic
173	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
174	C₂²×C₁₃	C₂×#137	1	1	yes	characteristic
175	C₄×C₁₃	—	2	4	no	—
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	—	2	2	no	—
178	C₄×C₁₃	—	4	4	no	—
179	C₄×C₁₃	—	4	4	no	—
180	C₄×C₁₃	—	4	2	no	—
181	C₄×C₁₃	—	4	4	no	—
182	C₄×C₁₃	—	8	2	no	—
183	C₄×C₁₃	#451	1	4	yes	—
184	C₄×C₁₃	#456	1	2	yes	—
185	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
186	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
187	C₂×Q₁₆	—	2	8	no	—
188	C₄⋊Q₈	—	2	2	no	—
189	C₄⋊Q₈	—	2	2	no	—
190	C₄⋊Q₈	—	2	4	no	—
191	C₄⋊Q₈	—	2	8	no	—
192	C₄⋊Q₈	—	2	2	no	—
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
195	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
196	#125	—	2	4	no	—
197	#125	—	2	4	no	—
198	#125	—	2	8	no	—
199	#125	—	2	2	no	—
200	C₂²⋊Q₈	—	2	2	no	—
201	C₂²⋊Q₈	—	2	4	no	—
202	C₂²⋊Q₈	C₂³×C₁₃	1	4	yes	—
203	C₄×Q₈	—	2	4	no	—
204	C₄×Q₈	—	2	4	no	—
205	C₄×Q₈	—	2	4	no	—
206	C₄×Q₈	—	2	4	no	—
207	C₄×Q₈	—	2	2	no	—
208	C₄×Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	D₄×C₁₃	1	4	yes	—
211	#117	—	2	4	no	—
212	#117	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
218	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
219	C₂×C₄²	—	2	1	no	—
220	C₂×C₄²	D₄×C₁₃	1	2	yes	—
221	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
222	C₄⋊C₈	—	2	2	no	—
223	#103	—	2	8	no	—
224	C₂²⋊C₈	C₂³×C₁₃	1	4	yes	—
225	C₄×C₈	—	2	2	no	—
226	#95	—	2	4	no	—
227	#95	C₂³×C₁₃	1	2	yes	—
228	C₂×C₁₃	—	2	1	no	—
229	C₂×C₁₃	—	2	1	no	—
230	C₂×C₁₃	C₂×#451	1	1	yes	characteristic
231	C₂×C₁₃	C₂×#456	1	1	yes	characteristic
232	C₂×C₁₃	#3027	1	1	yes	characteristic
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	—	8	2	no	—
239	C₂×Q₈	C₁₃×Q₁₆	1	4	yes	complemented
240	C₂×Q₈	C₂×D₄×C₁₃	1	4	yes	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
245	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
247	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
248	Q₁₆	—	8	8	no	—
249	C₂×C₈	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	8	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	8	4	no	—
266	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
267	C₂²⋊C₄	—	2	4	no	—
268	C₄²	—	2	8	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	4	no	—
272	C₄²	—	4	2	no	—
273	C₄²	C₂×D₄×C₁₃	1	2	yes	—
274	C₄²	C₂×D₄×C₁₃	1	2	yes	—
275	C₁₃	#18404	1	1	yes	characteristic, direct product factor, Sylow
276	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
277	Q₈	—	2	8	no	—
278	Q₈	—	4	4	no	—
279	Q₈	—	4	8	no	—
280	Q₈	—	4	4	no	—
281	Q₈	—	8	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
296	C₂×C₄	C₁₃×#120	1	2	yes	—
297	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
298	C₂×C₄	C₂×C₁₃×Q₁₆	1	4	yes	—
299	C₂×C₄	C₂×C₁₃×Q₁₆	1	2	yes	—
300	C₂×C₄	C₁₃×#137	1	2	yes	—
301	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
302	C₈	—	4	4	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	C₂×C₁₃×#120	1	1	yes	characteristic
306	C₂²	C₂²×C₁₃×Q₁₆	1	1	yes	characteristic
307	C₂²	C₂×C₁₃×#137	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	4	no	—
311	C₄	—	4	4	no	—
312	C₄	—	4	4	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	4	no	—
315	C₄	—	8	2	no	—
316	C₄	C₁₃×#451	1	4	yes	—
317	C₄	C₁₃×#456	1	2	yes	—
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	C₂×C₁₃×#451	1	1	yes	characteristic
321	C₂	C₂×C₁₃×#456	1	1	yes	characteristic
322	C₂	C₁₃×#3027	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂³×C₁₃	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ 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
72. #18404	C₁₃	direct (N × H)
79. C₂×Q₈×C₁₃	Q₁₆	semidirect
239. C₂×Q₈	C₁₃×Q₁₆	semidirect
275. C₁₃	#18404	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×#3502	2
C₁₃×#3209	2
C₁₃×#2567	2
C₁₃×#1605	2
C₁₃×#1503	2
C₁₃×#1461	2
#18404	13

Subgroups — C13×#18404 #15704743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#18404 #15704743