Subgroups — C₁₃×#9870 #15696209

Order 3328.

566 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1842	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1279	C₂	1	4	yes	maximal
5	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
7	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
8	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	—	2	1	no	—
11	C₂×C₄×C₈×C₁₃	—	2	1	no	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
13	C₁₃×#391	—	2	1	no	—
14	C₁₃×#389	—	2	1	no	—
15	C₁₃×#389	C₄	1	2	yes	—
16	C₁₃×#384	C₄	1	1	yes	characteristic
17	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
18	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₄³×C₁₃	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂²×C₈×C₁₃	—	2	2	no	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₁₃×C₄⋊Q₈	—	2	2	no	—
24	C₁₃×C₄⋊Q₈	C₈	1	4	yes	complemented
25	C₄×Q₈×C₁₃	—	4	2	no	—
26	C₄×Q₈×C₁₃	—	4	2	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₃	—	2	2	no	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×C₄⋊C₈	—	2	4	no	—
42	C₁₃×C₄⋊C₈	—	4	2	no	—
43	C₁₃×C₄⋊C₈	—	4	2	no	—
44	C₄×C₈×C₁₃	—	2	4	no	—
45	C₄×C₈×C₁₃	—	2	4	no	—
46	C₄×C₈×C₁₃	—	2	2	no	—
47	C₄×C₈×C₁₃	Q₈	1	4	yes	complemented
48	C₁₃×#95	—	2	2	no	—
49	C₁₃×#95	—	2	2	no	—
50	#9870	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂×Q₈×C₁₃	—	2	4	no	—
52	C₂×Q₈×C₁₃	—	4	2	no	—
53	C₂×Q₈×C₁₃	—	4	2	no	—
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	—	2	4	no	—
64	C₂×C₈×C₁₃	—	2	4	no	—
65	C₂×C₈×C₁₃	—	2	4	no	—
66	C₂×C₈×C₁₃	—	4	2	no	—
67	C₂×C₈×C₁₃	—	4	2	no	—
68	C₂×C₈×C₁₃	—	4	2	no	—
69	C₁₃×C₄⋊C₄	—	2	4	no	—
70	C₁₃×C₄⋊C₄	—	2	2	no	—
71	C₁₃×C₄⋊C₄	—	2	4	no	—
72	C₁₃×C₄⋊C₄	—	4	4	no	—
73	C₁₃×C₄⋊C₄	—	4	2	no	—
74	C₁₃×C₄⋊C₄	—	4	2	no	—
75	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
76	C₄²×C₁₃	—	2	2	no	—
77	C₄²×C₁₃	—	2	2	no	—
78	C₄²×C₁₃	—	2	2	no	—
79	C₄²×C₁₃	—	4	2	no	—
80	C₄²×C₁₃	—	4	2	no	—
81	C₄²×C₁₃	C₂×C₈	1	2	yes	—
82	C₄²×C₁₃	M₁₆	1	2	yes	—
83	C₄²×C₁₃	D₈	1	2	yes	—
84	C₄²×C₁₃	SD₁₆	1	2	yes	—
85	C₄²×C₁₃	SD₁₆	1	2	yes	—
86	C₄²×C₁₃	Q₁₆	1	2	yes	—
87	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
88	C₄²×C₁₃	#41	1	2	yes	—
89	C₄×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic
90	#1842	C₂×C₁₃	1	1	yes	characteristic
91	C₄²×C₈	C₂×C₁₃	1	1	yes	characteristic
92	#1279	C₂×C₁₃	1	4	yes	—
93	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	Q₈×C₁₃	—	4	8	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
108	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
109	C₂×C₄×C₁₃	#102	1	2	yes	—
110	C₂×C₄×C₁₃	#102	1	2	yes	—
111	C₂×C₄×C₁₃	#103	1	2	yes	—
112	C₂×C₄×C₁₃	#103	1	2	yes	—
113	C₂×C₄×C₁₃	#104	1	2	yes	—
114	C₂×C₄×C₁₃	#104	1	2	yes	—
115	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
116	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#131	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
126	C₈×C₁₃	—	2	8	no	—
127	C₈×C₁₃	—	8	2	no	—
128	C₂×C₄⋊Q₈	C₄×C₁₃	1	1	yes	characteristic
129	C₂×C₄×Q₈	—	2	1	no	—
130	C₂×C₄⋊C₈	—	2	1	no	—
131	C₂×C₄⋊C₈	—	2	1	no	—
132	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
133	C₂×C₄×C₈	—	2	1	no	—
134	C₂×C₄×C₈	—	2	1	no	—
135	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
136	#391	—	2	1	no	—
137	#389	—	2	1	no	—
138	#389	C₄×C₁₃	1	2	yes	—
139	#384	C₄×C₁₃	1	1	yes	characteristic
140	C₄×C₄⋊C₄	—	2	1	no	—
141	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
142	C₄³	C₂²×C₁₃	1	1	yes	characteristic
143	C₂²×C₁₃	#391	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₂²×C₁₃	#408	1	1	yes	characteristic
146	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
147	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
148	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
149	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	4	4	no	—
154	C₄×C₁₃	—	8	2	no	—
155	C₄×C₁₃	D₄⋊C₈	1	4	yes	—
156	C₄×C₁₃	Q₈⋊C₈	1	4	yes	—
157	C₄×C₁₃	C₈×Q₈	1	2	yes	—
158	C₄×C₁₃	#446	1	2	yes	—
159	C₄×C₁₃	#486	1	1	yes	characteristic
160	C₄×C₁₃	#487	1	1	yes	characteristic
161	C₄×C₁₃	#498	1	1	yes	characteristic
162	C₄×C₁₃	#500	1	1	yes	characteristic
163	C₂²×Q₈	—	2	1	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₄⋊Q₈	—	2	2	no	—
167	C₄⋊Q₈	C₈×C₁₃	1	4	yes	complemented
168	C₄×Q₈	—	4	2	no	—
169	C₄×Q₈	—	4	2	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
175	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
176	C₂×C₄²	—	2	2	no	—
177	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
178	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
179	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
180	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
181	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
182	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
184	C₄⋊C₈	—	2	4	no	—
185	C₄⋊C₈	—	4	2	no	—
186	C₄⋊C₈	—	4	2	no	—
187	C₄×C₈	—	2	4	no	—
188	C₄×C₈	—	2	4	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	Q₈×C₁₃	1	4	yes	complemented
191	#95	—	2	2	no	—
192	#95	—	2	2	no	—
193	C₂×C₁₃	C₂×D₄⋊C₈	1	1	yes	characteristic
194	C₂×C₁₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
195	C₂×C₁₃	#1492	1	1	yes	characteristic
196	C₂×C₁₃	#1493	1	1	yes	characteristic
197	C₂×C₁₃	#1984	1	1	yes	characteristic
198	C₂×C₁₃	#1990	1	1	yes	characteristic
199	C₂×C₁₃	#1994	1	1	yes	characteristic
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	4	2	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
206	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
208	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	C₂×C₈×C₁₃	1	4	yes	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	4	2	no	—
229	C₄²	—	4	2	no	—
230	C₄²	C₂×C₈×C₁₃	1	2	yes	—
231	C₄²	C₁₃×M₁₆	1	2	yes	—
232	C₄²	C₁₃×D₈	1	2	yes	—
233	C₄²	C₁₃×SD₁₆	1	2	yes	—
234	C₄²	C₁₃×SD₁₆	1	2	yes	—
235	C₄²	C₁₃×Q₁₆	1	2	yes	—
236	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
237	C₄²	C₁₃×#41	1	2	yes	—
238	C₁₃	#9870	1	1	yes	characteristic, direct product factor, Sylow
239	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
240	Q₈	—	4	8	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₁₃×C₂²⋊C₈	1	2	yes	—
254	C₂×C₄	C₁₃×C₂²⋊C₈	1	2	yes	—
255	C₂×C₄	C₁₃×#102	1	2	yes	—
256	C₂×C₄	C₁₃×#102	1	2	yes	—
257	C₂×C₄	C₁₃×#103	1	2	yes	—
258	C₂×C₄	C₁₃×#103	1	2	yes	—
259	C₂×C₄	C₁₃×#104	1	2	yes	—
260	C₂×C₄	C₁₃×#104	1	2	yes	—
261	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
262	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
263	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
264	C₂×C₄	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
266	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
267	C₂×C₄	C₁₃×#131	1	2	yes	—
268	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
269	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
270	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
271	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
272	C₈	—	2	8	no	—
273	C₈	—	8	2	no	—
274	C₂²	C₁₃×#391	1	1	yes	characteristic
275	C₂²	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
276	C₂²	C₁₃×#408	1	1	yes	characteristic
277	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
278	C₂²	C₂×C₁₃×#103	1	1	yes	characteristic
279	C₂²	C₂×C₁₃×#104	1	1	yes	characteristic
280	C₂²	C₂×C₁₃×#104	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	4	4	no	—
285	C₄	—	8	2	no	—
286	C₄	C₁₃×D₄⋊C₈	1	4	yes	—
287	C₄	C₁₃×Q₈⋊C₈	1	4	yes	—
288	C₄	C₈×Q₈×C₁₃	1	2	yes	—
289	C₄	C₁₃×#446	1	2	yes	—
290	C₄	C₁₃×#486	1	1	yes	characteristic
291	C₄	C₁₃×#487	1	1	yes	characteristic
292	C₄	C₁₃×#498	1	1	yes	characteristic
293	C₄	C₁₃×#500	1	1	yes	characteristic
294	C₂	C₂×C₁₃×D₄⋊C₈	1	1	yes	characteristic
295	C₂	C₂×C₁₃×Q₈⋊C₈	1	1	yes	characteristic
296	C₂	C₁₃×#1492	1	1	yes	characteristic
297	C₂	C₁₃×#1493	1	1	yes	characteristic
298	C₂	C₁₃×#1984	1	1	yes	characteristic
299	C₂	C₁₃×#1990	1	1	yes	characteristic
300	C₂	C₁₃×#1994	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
24. C₁₃×C₄⋊Q₈	C₈	semidirect
47. C₄×C₈×C₁₃	Q₈	semidirect
50. #9870	C₁₃	direct (N × H)
167. C₄⋊Q₈	C₈×C₁₃	semidirect
190. C₄×C₈	Q₈×C₁₃	semidirect
238. C₁₃	#9870	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₁₃×C₄⋊Q₈	2
C₁₃×#1842	2
C₄²×C₈×C₁₃	2
C₁₃×#1279	2
#9870	13

Subgroups — C13×#9870 #15696209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9870 #15696209