Subgroups — C₁₃×#23816 #15710155

Order 3328.

778 conjugacy classes of subgroups.

The subgroup lattice has 350 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	C₄×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2100	C₂	1	2	yes	maximal
4	C₁₃×#2072	C₂	1	2	yes	maximal
5	C₁₃×#2043	C₂	1	2	yes	maximal
6	C₁₃×#2024	C₂	1	2	yes	maximal
7	C₄×C₁₃×#103	C₂	1	4	yes	maximal
8	C₁₃×#1744	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
12	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
13	C₂×C₁₃×#103	C₂²	1	4	yes	—
14	C₂×C₁₃×#103	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
16	C₁₃×#391	—	2	1	no	—
17	C₁₃×#391	C₂²	1	2	yes	—
18	C₁₃×#389	—	2	1	no	—
19	C₁₃×#389	—	2	1	no	—
20	C₁₃×#389	—	2	1	no	—
21	C₁₃×#389	C₂²	1	2	yes	—
22	C₁₃×#384	C₂²	1	1	yes	characteristic
23	C₁₃×#384	C₂²	1	1	yes	characteristic
24	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
25	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
26	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
27	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
28	C₄³×C₁₃	C₂²	1	1	yes	characteristic
29	C₁₃×#340	C₂²	1	4	yes	—
30	C₁₃×#340	C₂²	1	4	yes	—
31	C₁₃×#336	C₂²	1	2	yes	—
32	C₁₃×#336	C₂²	1	2	yes	—
33	C₂²×Q₈×C₁₃	—	2	1	no	—
34	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
36	C₁₃×C₄⋊Q₈	—	2	4	no	—
37	C₁₃×C₄⋊Q₈	—	2	4	no	—
38	C₄×Q₈×C₁₃	—	2	8	no	—
39	C₄×Q₈×C₁₃	—	4	4	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
47	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
49	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
52	C₂×C₄²×C₁₃	—	2	2	no	—
53	C₂×C₄²×C₁₃	—	2	2	no	—
54	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
61	C₁₃×#103	—	2	8	no	—
62	C₁₃×#103	—	2	8	no	—
63	C₄×C₈×C₁₃	—	2	4	no	—
64	C₁₃×#95	—	2	2	no	—
65	C₁₃×#95	—	2	2	no	—
66	C₁₃×#95	—	2	2	no	—
67	C₁₃×#95	—	2	2	no	—
68	#23816	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×Q₈×C₁₃	—	2	4	no	—
70	C₂×Q₈×C₁₃	—	2	2	no	—
71	C₂×Q₈×C₁₃	—	2	4	no	—
72	C₂×Q₈×C₁₃	—	2	4	no	—
73	C₂×Q₈×C₁₃	—	4	2	no	—
74	C₂×Q₈×C₁₃	—	4	2	no	—
75	C₂×Q₈×C₁₃	#41	1	4	yes	—
76	C₂²×C₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	—	2	2	no	—
78	C₂²×C₄×C₁₃	—	2	2	no	—
79	C₂²×C₄×C₁₃	—	2	2	no	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₂×C₈×C₁₃	—	2	4	no	—
88	C₂×C₈×C₁₃	—	2	4	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	2	no	—
94	C₁₃×C₄⋊C₄	—	2	4	no	—
95	C₁₃×C₄⋊C₄	—	2	4	no	—
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₁₃×C₄⋊C₄	—	4	2	no	—
98	C₁₃×C₄⋊C₄	—	4	2	no	—
99	C₁₃×C₄⋊C₄	—	4	4	no	—
100	C₁₃×C₄⋊C₄	—	4	4	no	—
101	C₁₃×C₄⋊C₄	—	4	2	no	—
102	C₁₃×C₄⋊C₄	#41	1	4	yes	—
103	C₄²×C₁₃	—	2	2	no	—
104	C₄²×C₁₃	—	2	2	no	—
105	C₄²×C₁₃	—	4	1	no	—
106	C₄²×C₁₃	—	4	1	no	—
107	C₄²×C₁₃	—	4	4	no	—
108	C₄²×C₁₃	—	4	4	no	—
109	C₄²×C₁₃	SD₁₆	1	4	yes	—
110	C₄²×C₁₃	Q₁₆	1	4	yes	—
111	C₄²×C₁₃	#41	1	4	yes	—
112	C₄×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic
113	C₄×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic
114	#2100	C₂×C₁₃	1	2	yes	—
115	#2072	C₂×C₁₃	1	2	yes	—
116	#2043	C₂×C₁₃	1	2	yes	—
117	#2024	C₂×C₁₃	1	2	yes	—
118	C₄×#103	C₂×C₁₃	1	4	yes	—
119	#1744	C₂×C₁₃	1	1	yes	characteristic
120	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	Q₈×C₁₃	—	2	8	no	—
122	Q₈×C₁₃	—	4	8	no	—
123	Q₈×C₁₃	—	4	4	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	4	no	—
135	C₂×C₄×C₁₃	—	2	4	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	#124	1	2	yes	—
147	C₂×C₄×C₁₃	#124	1	2	yes	—
148	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
149	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
150	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
151	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
152	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
153	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
154	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
155	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
156	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
157	C₈×C₁₃	—	4	4	no	—
158	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
159	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
160	C₂×C₄×Q₈	—	2	1	no	—
161	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
162	C₂×#103	C₂²×C₁₃	1	4	yes	—
163	C₂×#103	C₂²×C₁₃	1	4	yes	—
164	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
165	#391	—	2	1	no	—
166	#391	C₂²×C₁₃	1	2	yes	—
167	#389	—	2	1	no	—
168	#389	—	2	1	no	—
169	#389	—	2	1	no	—
170	#389	C₂²×C₁₃	1	2	yes	—
171	#384	C₂²×C₁₃	1	1	yes	characteristic
172	#384	C₂²×C₁₃	1	1	yes	characteristic
173	C₄×C₄⋊C₄	—	2	1	no	—
174	C₄×C₄⋊C₄	—	2	1	no	—
175	C₄×C₄⋊C₄	—	2	1	no	—
176	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
177	C₄³	C₂²×C₁₃	1	1	yes	characteristic
178	#340	C₂²×C₁₃	1	4	yes	—
179	#340	C₂²×C₁₃	1	4	yes	—
180	#336	C₂²×C₁₃	1	2	yes	—
181	#336	C₂²×C₁₃	1	2	yes	—
182	C₂²×C₁₃	#525	1	1	yes	characteristic
183	C₂²×C₁₃	#525	1	1	yes	characteristic
184	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
185	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
186	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
187	C₂²×C₁₃	#575	1	1	yes	characteristic
188	C₂²×C₁₃	#575	1	1	yes	characteristic
189	C₄×C₁₃	—	2	2	no	—
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	2	2	no	—
194	C₄×C₁₃	—	4	4	no	—
195	C₄×C₁₃	—	4	4	no	—
196	C₄×C₁₃	—	8	2	no	—
197	C₄×C₁₃	—	8	2	no	—
198	C₄×C₁₃	#487	1	4	yes	—
199	C₄×C₁₃	#529	1	4	yes	—
200	C₂²×Q₈	—	2	1	no	—
201	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
202	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
203	C₄⋊Q₈	—	2	4	no	—
204	C₄⋊Q₈	—	2	4	no	—
205	C₄×Q₈	—	2	8	no	—
206	C₄×Q₈	—	4	4	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	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	C₂³×C₁₃	1	2	yes	—
217	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
218	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
222	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
223	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
224	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
225	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
226	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
227	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
228	#103	—	2	8	no	—
229	#103	—	2	8	no	—
230	C₄×C₈	—	2	4	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	C₂×C₁₃	#2398	1	1	yes	characteristic
236	C₂×C₁₃	#3101	1	1	yes	characteristic
237	C₂×C₁₃	#3103	1	1	yes	characteristic
238	C₂×C₁₃	#3104	1	1	yes	characteristic
239	C₂×C₁₃	#3104	1	1	yes	characteristic
240	C₂×C₁₃	C₂×#487	1	1	yes	characteristic
241	C₂×C₁₃	#3135	1	1	yes	characteristic
242	C₂×Q₈	—	2	4	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂×Q₈	C₁₃×#41	1	4	yes	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	4	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	4	no	—
273	C₄⋊C₄	—	4	4	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	C₁₃×#41	1	4	yes	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	4	1	no	—
280	C₄²	—	4	4	no	—
281	C₄²	—	4	4	no	—
282	C₄²	C₁₃×SD₁₆	1	4	yes	—
283	C₄²	C₁₃×Q₁₆	1	4	yes	—
284	C₄²	C₁₃×#41	1	4	yes	—
285	C₁₃	#23816	1	1	yes	characteristic, direct product factor, Sylow
286	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
287	Q₈	—	2	8	no	—
288	Q₈	—	4	8	no	—
289	Q₈	—	4	4	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	C₁₃×#124	1	2	yes	—
313	C₂×C₄	C₁₃×#124	1	2	yes	—
314	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
315	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
316	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
317	C₂×C₄	C₂×C₁₃×Q₁₆	1	2	yes	—
318	C₂×C₄	C₂×C₁₃×Q₁₆	1	2	yes	—
319	C₂×C₄	C₂×C₁₃×Q₁₆	1	2	yes	—
320	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
321	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
322	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
323	C₈	—	4	4	no	—
324	C₂²	C₁₃×#525	1	1	yes	characteristic
325	C₂²	C₁₃×#525	1	1	yes	characteristic
326	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
327	C₂²	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
328	C₂²	C₂²×C₁₃×Q₁₆	1	1	yes	characteristic
329	C₂²	C₁₃×#575	1	1	yes	characteristic
330	C₂²	C₁₃×#575	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	4	4	no	—
337	C₄	—	4	4	no	—
338	C₄	—	8	2	no	—
339	C₄	—	8	2	no	—
340	C₄	C₁₃×#487	1	4	yes	—
341	C₄	C₁₃×#529	1	4	yes	—
342	C₂	C₁₃×#2398	1	1	yes	characteristic
343	C₂	C₁₃×#3101	1	1	yes	characteristic
344	C₂	C₁₃×#3103	1	1	yes	characteristic
345	C₂	C₁₃×#3104	1	1	yes	characteristic
346	C₂	C₁₃×#3104	1	1	yes	characteristic
347	C₂	C₂×C₁₃×#487	1	1	yes	characteristic
348	C₂	C₁₃×#3135	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle, prime radical	C₂³×C₁₃	C₂²×D₄
LCS	G ▷ C₂²×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
68. #23816	C₁₃	direct (N × H)
285. C₁₃	#23816	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×#2100	2
C₁₃×#2072	2
C₁₃×#2043	2
C₁₃×#2024	2
C₄×C₁₃×#103	2
C₁₃×#1744	2
#23816	13

Subgroups — C13×#23816 #15710155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#23816 #15710155