Subgroups — C₁₃×#20823 #15707162

Order 3328.

898 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2285	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2283	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1847	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1836	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1808	C₂	1	2	yes	maximal
7	C₁₃×#1807	C₂	1	2	yes	maximal
8	C₁₃×#1776	C₂	1	4	yes	maximal, complemented
9	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
10	C₂×C₁₃×#117	—	2	1	no	—
11	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	C₂²	1	4	yes	—
13	C₂×C₁₃×#106	C₂²	1	4	yes	—
14	C₂×C₁₃×#97	C₄	1	8	yes	complemented
15	C₂×C₁₃×#97	C₂²	1	2	yes	—
16	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
18	C₁₃×#384	C₂²	1	2	yes	—
19	C₁₃×#383	—	2	1	no	—
20	C₁₃×#382	—	2	1	no	—
21	C₁₃×#382	C₂²	1	2	yes	—
22	C₁₃×#381	—	2	1	no	—
23	C₁₃×#380	—	2	1	no	—
24	C₁₃×#380	C₂²	1	1	yes	characteristic
25	C₁₃×#380	C₂²	1	1	yes	characteristic
26	C₁₃×#380	C₂²	1	2	yes	—
27	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
28	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
29	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
30	C₄×C₁₃×C₂²⋊C₄	—	2	1	no	—
31	C₁₃×#376	—	2	1	no	—
32	C₁₃×#340	C₂²	1	4	yes	—
33	C₁₃×#340	C₂²	1	4	yes	—
34	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	—	2	8	no	—
38	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
41	C₁₃×#117	—	4	4	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
47	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
48	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
49	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
50	C₂×C₄²×C₁₃	—	2	2	no	—
51	C₂×C₄²×C₁₃	—	2	2	no	—
52	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
56	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₁₃×#107	—	2	8	no	—
59	C₁₃×#106	—	2	8	no	—
60	C₁₃×#97	C₂×C₄	1	4	yes	—
61	C₁₃×#97	C₂×C₄	1	8	yes	—
62	C₁₃×#97	C₂×C₄	1	4	yes	—
63	C₁₃×#95	—	2	2	no	—
64	C₁₃×#95	—	2	2	no	—
65	C₁₃×#95	—	2	2	no	—
66	C₁₃×#95	—	2	2	no	—
67	C₁₃×#95	—	2	1	no	—
68	C₁₃×#95	—	2	1	no	—
69	C₁₃×#95	—	2	2	no	—
70	#20823	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	—	2	2	no	—
73	C₂²×C₄×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	—	2	4	no	—
75	C₂²×C₄×C₁₃	—	2	2	no	—
76	C₂²×C₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	—	2	4	no	—
78	C₂²×C₄×C₁₃	—	4	2	no	—
79	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
80	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
81	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
88	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₁₃	—	2	4	no	—
90	C₂×C₈×C₁₃	—	2	16	no	—
91	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
93	C₂×C₈×C₁₃	#41	1	8	yes	—
94	C₁₃×C₄⋊C₄	—	2	4	no	—
95	C₁₃×C₄⋊C₄	—	2	4	no	—
96	C₁₃×C₄⋊C₄	—	2	4	no	—
97	C₁₃×C₄⋊C₄	—	2	4	no	—
98	C₁₃×C₄⋊C₄	—	4	4	no	—
99	C₁₃×C₄⋊C₄	—	4	4	no	—
100	C₁₃×C₂²⋊C₄	—	4	2	no	—
101	C₁₃×C₂²⋊C₄	—	4	4	no	—
102	C₁₃×C₂²⋊C₄	—	4	2	no	—
103	C₄²×C₁₃	—	4	4	no	—
104	C₄²×C₁₃	—	4	4	no	—
105	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
106	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
107	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
108	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
109	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
110	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
111	C₂²×#97	C₂×C₁₃	1	1	yes	characteristic
112	#2285	C₂×C₁₃	1	1	yes	characteristic
113	#2283	C₂×C₁₃	1	1	yes	characteristic
114	#1847	C₂×C₁₃	1	2	yes	complemented
115	#1836	C₂×C₁₃	1	2	yes	complemented
116	#1808	C₂×C₁₃	1	2	yes	—
117	#1807	C₂×C₁₃	1	2	yes	—
118	#1776	C₂×C₁₃	1	4	yes	complemented
119	C₂³×C₁₃	—	2	2	no	—
120	C₂³×C₁₃	—	2	2	no	—
121	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
122	C₂³×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
123	C₂³×C₁₃	#137	1	2	yes	—
124	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	—	2	2	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	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	8	no	—
133	C₂×C₄×C₁₃	—	2	4	no	—
134	C₂×C₄×C₁₃	—	2	4	no	—
135	C₂×C₄×C₁₃	—	4	4	no	—
136	C₂×C₄×C₁₃	—	4	4	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	4	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	4	no	—
145	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
146	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
147	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
148	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
149	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
150	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
153	C₈×C₁₃	—	2	4	no	—
154	C₈×C₁₃	#117	1	8	yes	complemented
155	C₂³×C₈	C₂²×C₁₃	1	2	yes	—
156	C₂×#117	—	2	1	no	—
157	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×#107	C₂²×C₁₃	1	4	yes	—
159	C₂×#106	C₂²×C₁₃	1	4	yes	—
160	C₂×#97	C₄×C₁₃	1	8	yes	complemented
161	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
162	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×#97	C₂²×C₁₃	1	2	yes	—
164	#384	C₂²×C₁₃	1	2	yes	—
165	#383	—	2	1	no	—
166	#382	—	2	1	no	—
167	#382	C₂²×C₁₃	1	2	yes	—
168	#381	—	2	1	no	—
169	#380	—	2	1	no	—
170	#380	C₂²×C₁₃	1	1	yes	characteristic
171	#380	C₂²×C₁₃	1	2	yes	—
172	#380	C₂²×C₁₃	1	1	yes	characteristic
173	C₄×C₄⋊C₄	—	2	1	no	—
174	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
175	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
176	C₄×C₂²⋊C₄	—	2	1	no	—
177	#376	—	2	1	no	—
178	#340	C₂²×C₁₃	1	4	yes	—
179	#340	C₂²×C₁₃	1	4	yes	—
180	C₂²×C₁₃	—	2	2	no	—
181	C₂²×C₁₃	—	2	4	no	—
182	C₂²×C₁₃	—	2	4	no	—
183	C₂²×C₁₃	—	2	2	no	—
184	C₂²×C₁₃	#428	1	4	yes	—
185	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
186	C₂²×C₁₃	#532	1	2	yes	—
187	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
188	C₂²×C₁₃	C₂×#137	1	1	yes	characteristic
189	C₂²×C₁₃	#575	1	2	yes	—
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	—	2	2	no	—
195	C₄×C₁₃	—	8	4	no	—
196	C₄×C₁₃	—	8	2	no	—
197	C₄×C₁₃	—	8	2	no	—
198	C₄×C₁₃	C₂×#117	1	2	yes	—
199	C₄×C₁₃	#533	1	2	yes	—
200	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
201	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
202	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
203	C₂²×C₈	—	2	8	no	—
204	C₂²×C₈	C₂×C₄×C₁₃	1	8	yes	—
205	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
206	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
207	#117	—	4	4	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
211	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
213	C₂×C₂²⋊C₄	—	2	2	no	—
214	C₂×C₂²⋊C₄	—	2	1	no	—
215	C₂×C₂²⋊C₄	—	2	1	no	—
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
219	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
220	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
221	C₂×C₄²	Q₈×C₁₃	1	4	yes	—
222	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
223	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
224	#107	—	2	8	no	—
225	#106	—	2	8	no	—
226	#97	C₂×C₄×C₁₃	1	4	yes	—
227	#97	C₂×C₄×C₁₃	1	8	yes	—
228	#97	C₂×C₄×C₁₃	1	4	yes	—
229	#95	—	2	1	no	—
230	#95	—	2	1	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	C₂×C₁₃	—	2	4	no	—
237	C₂×C₁₃	#2285	1	1	yes	characteristic
238	C₂×C₁₃	C₂×#428	1	1	yes	characteristic
239	C₂×C₁₃	#2911	1	1	yes	characteristic
240	C₂×C₁₃	#3180	1	2	yes	—
241	C₂×C₁₃	#3181	1	2	yes	—
242	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	C₁₃×C₄⋊C₄	1	4	yes	—
251	C₂²×C₄	C₁₃×C₄⋊C₄	1	4	yes	—
252	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
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×Q₈×C₁₃	1	2	yes	—
259	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	2	16	no	—
262	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
263	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
264	C₂×C₈	C₁₃×#41	1	8	yes	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	4	4	no	—
270	C₄⋊C₄	—	4	4	no	—
271	C₂²⋊C₄	—	4	4	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₄²	—	4	4	no	—
275	C₄²	—	4	4	no	—
276	C₄²	C₁₃×C₄⋊C₄	1	4	yes	—
277	C₄²	C₁₃×C₄⋊C₄	1	4	yes	—
278	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
279	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
280	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
281	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
282	C₁₃	#20823	1	1	yes	characteristic, direct product factor, Sylow
283	C₂³	—	2	2	no	—
284	C₂³	—	2	2	no	—
285	C₂³	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
286	C₂³	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
287	C₂³	C₁₃×#137	1	2	yes	—
288	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
289	C₂×C₄	—	2	8	no	—
290	C₂×C₄	—	2	2	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	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	4	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	4	no	—
309	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
310	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
311	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
312	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
313	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
314	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
315	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
316	C₂×C₄	C₂×C₁₃×#41	1	4	yes	—
317	C₈	—	2	4	no	—
318	C₈	C₁₃×#117	1	8	yes	complemented
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	4	no	—
322	C₂²	—	2	4	no	—
323	C₂²	C₁₃×#428	1	4	yes	—
324	C₂²	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
325	C₂²	C₁₃×#532	1	2	yes	—
326	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
327	C₂²	C₂×C₁₃×#137	1	1	yes	characteristic
328	C₂²	C₁₃×#575	1	2	yes	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	8	4	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	C₂×C₁₃×#117	1	2	yes	—
338	C₄	C₁₃×#533	1	2	yes	—
339	C₂	—	2	4	no	—
340	C₂	C₁₃×#2285	1	1	yes	characteristic
341	C₂	C₂×C₁₃×#428	1	1	yes	characteristic
342	C₂	C₁₃×#2911	1	1	yes	characteristic
343	C₂	C₁₃×#3180	1	2	yes	—
344	C₂	C₁₃×#3181	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₃×#1847	C₂	semidirect
5. C₁₃×#1836	C₂	semidirect
8. C₁₃×#1776	C₂	semidirect
14. C₂×C₁₃×#97	C₄	semidirect
70. #20823	C₁₃	direct (N × H)
114. #1847	C₂×C₁₃	semidirect
115. #1836	C₂×C₁₃	semidirect
118. #1776	C₂×C₁₃	semidirect
154. C₈×C₁₃	#117	semidirect
160. C₂×#97	C₄×C₁₃	semidirect
282. C₁₃	#20823	direct (N × H)
318. C₈	C₁₃×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×#97	2
C₁₃×#2285	2
C₁₃×#2283	2
C₁₃×#1847	2
C₁₃×#1836	2
C₁₃×#1808	2
C₁₃×#1807	2
C₁₃×#1776	2
#20823	13

Subgroups — C13×#20823 #15707162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20823 #15707162