Subgroups — C₁₃×#29223 #15715562

Order 3328.

678 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×#2859	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×#2099	C₂	1	2	yes	maximal
5	C₁₃×#1993	C₂	1	2	yes	maximal
6	C₁₃×#1989	C₂	1	4	yes	maximal
7	C₁₃×#1836	C₂	1	2	yes	maximal
8	C₁₃×#1767	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#125	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₃×#97	C₂²	1	2	yes	—
14	C₁₃×#391	—	2	1	no	—
15	C₁₃×#389	—	2	1	no	—
16	C₁₃×#389	—	2	1	no	—
17	C₁₃×#389	—	2	1	no	—
18	C₁₃×#389	—	2	1	no	—
19	C₁₃×#389	—	2	1	no	—
20	C₁₃×#389	C₂²	1	2	yes	—
21	C₁₃×#384	C₂²	1	1	yes	characteristic
22	C₁₃×#384	C₂²	1	1	yes	characteristic
23	C₁₃×#384	C₂²	1	1	yes	characteristic
24	C₁₃×#382	C₂²	1	2	yes	—
25	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
26	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
27	C₄³×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#340	C₂²	1	4	yes	—
29	C₁₃×#340	C₂²	1	4	yes	—
30	C₁₃×#340	C₂²	1	4	yes	—
31	C₁₃×#340	C₂²	1	4	yes	—
32	C₁₃×#336	C₂²	1	2	yes	—
33	C₂²×Q₈×C₁₃	—	2	1	no	—
34	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
35	C₁₃×C₄⋊Q₈	—	2	2	no	—
36	C₁₃×C₄⋊Q₈	Q₈	1	4	yes	—
37	C₁₃×#125	—	2	8	no	—
38	C₄×Q₈×C₁₃	—	4	4	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	1	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	2	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×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
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×C₄⋊C₈	—	2	4	no	—
62	C₁₃×#97	—	2	4	no	—
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	#29223	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×Q₈×C₁₃	—	2	4	no	—
69	C₂×Q₈×C₁₃	—	4	2	no	—
70	C₂×Q₈×C₁₃	—	4	2	no	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	—	2	2	no	—
73	C₂²×C₄×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	—	2	2	no	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₁₃	—	2	4	no	—
83	C₂×C₈×C₁₃	—	2	4	no	—
84	C₂×C₈×C₁₃	—	2	4	no	—
85	C₁₃×C₄⋊C₄	—	2	4	no	—
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₁₃×C₄⋊C₄	—	2	4	no	—
88	C₁₃×C₄⋊C₄	—	2	4	no	—
89	C₁₃×C₄⋊C₄	—	2	4	no	—
90	C₁₃×C₄⋊C₄	—	4	2	no	—
91	C₁₃×C₄⋊C₄	—	4	2	no	—
92	C₁₃×C₄⋊C₄	—	4	2	no	—
93	C₁₃×C₄⋊C₄	—	4	2	no	—
94	C₁₃×C₄⋊C₄	—	4	4	no	—
95	C₁₃×C₄⋊C₄	—	4	4	no	—
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₁₃×C₄⋊C₄	—	4	4	no	—
98	C₁₃×C₄⋊C₄	—	4	2	no	—
99	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
100	C₄²×C₁₃	—	2	2	no	—
101	C₄²×C₁₃	—	2	2	no	—
102	C₄²×C₁₃	—	2	2	no	—
103	C₄²×C₁₃	—	4	1	no	—
104	C₄²×C₁₃	—	4	1	no	—
105	C₄²×C₁₃	—	4	4	no	—
106	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
107	C₄²×C₁₃	#41	1	2	yes	—
108	C₄²×C₁₃	#41	1	4	yes	—
109	#2859	C₂×C₁₃	1	1	yes	characteristic
110	C₄×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic
111	#2100	C₂×C₁₃	1	2	yes	—
112	#2099	C₂×C₁₃	1	2	yes	—
113	#1993	C₂×C₁₃	1	2	yes	—
114	#1989	C₂×C₁₃	1	4	yes	—
115	#1836	C₂×C₁₃	1	2	yes	—
116	#1767	C₂×C₁₃	1	1	yes	characteristic
117	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	Q₈×C₁₃	—	4	8	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	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	4	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	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	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	C₂²⋊Q₈	1	2	yes	—
144	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
145	C₂×C₄×C₁₃	#124	1	2	yes	—
146	C₂×C₄×C₁₃	#124	1	2	yes	—
147	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₁₃	#137	1	2	yes	—
149	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
152	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
153	C₈×C₁₃	—	4	4	no	—
154	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×#125	C₂²×C₁₃	1	2	yes	—
156	C₂×C₄×Q₈	—	2	1	no	—
157	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
158	C₂×#97	C₂²×C₁₃	1	2	yes	—
159	#391	—	2	1	no	—
160	#389	—	2	1	no	—
161	#389	—	2	1	no	—
162	#389	—	2	1	no	—
163	#389	—	2	1	no	—
164	#389	—	2	1	no	—
165	#389	C₂²×C₁₃	1	2	yes	—
166	#384	C₂²×C₁₃	1	1	yes	characteristic
167	#384	C₂²×C₁₃	1	1	yes	characteristic
168	#384	C₂²×C₁₃	1	1	yes	characteristic
169	#382	C₂²×C₁₃	1	2	yes	—
170	C₄×C₄⋊C₄	—	2	1	no	—
171	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
172	C₄³	C₂²×C₁₃	1	1	yes	characteristic
173	#340	C₂²×C₁₃	1	4	yes	—
174	#340	C₂²×C₁₃	1	4	yes	—
175	#340	C₂²×C₁₃	1	4	yes	—
176	#340	C₂²×C₁₃	1	4	yes	—
177	#336	C₂²×C₁₃	1	2	yes	—
178	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
179	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
180	C₂²×C₁₃	#536	1	1	yes	characteristic
181	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂²×C₁₃	C₂×#137	1	1	yes	characteristic
183	C₂²×C₁₃	#575	1	1	yes	characteristic
184	C₂²×C₁₃	#575	1	1	yes	characteristic
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	—	2	2	no	—
187	C₄×C₁₃	—	2	2	no	—
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	2	2	no	—
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	—	4	4	no	—
192	C₄×C₁₃	—	8	2	no	—
193	C₄×C₁₃	—	8	2	no	—
194	C₄×C₁₃	—	8	2	no	—
195	C₄×C₁₃	#556	1	2	yes	—
196	C₄×C₁₃	#557	1	2	yes	—
197	C₂²×Q₈	—	2	1	no	—
198	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	Q₈×C₁₃	1	4	yes	—
201	#125	—	2	8	no	—
202	C₄×Q₈	—	4	4	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	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	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
214	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
215	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
216	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
219	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
220	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
221	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
223	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
224	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
225	C₄⋊C₈	—	2	4	no	—
226	#97	—	2	4	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	C₂×C₁₃	#2568	1	1	yes	characteristic
232	C₂×C₁₃	#3077	1	1	yes	characteristic
233	C₂×C₁₃	#3078	1	1	yes	characteristic
234	C₂×C₁₃	#3135	1	1	yes	characteristic
235	C₂×C₁₃	#3141	1	1	yes	characteristic
236	C₂×C₁₃	#3234	1	1	yes	characteristic
237	C₂×C₁₃	#3236	1	1	yes	characteristic
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
251	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	4	no	—
265	C₄⋊C₄	—	4	4	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	C₂×Q₈×C₁₃	1	4	yes	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	4	1	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	4	4	no	—
276	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
277	C₄²	C₁₃×#41	1	2	yes	—
278	C₄²	C₁₃×#41	1	4	yes	—
279	C₁₃	#29223	1	1	yes	characteristic, direct product factor, Sylow
280	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
281	Q₈	—	4	8	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	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	4	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	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	C₁₃×C₂²⋊Q₈	1	2	yes	—
307	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
308	C₂×C₄	C₁₃×#124	1	2	yes	—
309	C₂×C₄	C₁₃×#124	1	2	yes	—
310	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
311	C₂×C₄	C₁₃×#137	1	2	yes	—
312	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
313	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
314	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
315	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
316	C₈	—	4	4	no	—
317	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
318	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
319	C₂²	C₁₃×#536	1	1	yes	characteristic
320	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
321	C₂²	C₂×C₁₃×#137	1	1	yes	characteristic
322	C₂²	C₁₃×#575	1	1	yes	characteristic
323	C₂²	C₁₃×#575	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	4	4	no	—
331	C₄	—	8	2	no	—
332	C₄	—	8	2	no	—
333	C₄	—	8	2	no	—
334	C₄	C₁₃×#556	1	2	yes	—
335	C₄	C₁₃×#557	1	2	yes	—
336	C₂	C₁₃×#2568	1	1	yes	characteristic
337	C₂	C₁₃×#3077	1	1	yes	characteristic
338	C₂	C₁₃×#3078	1	1	yes	characteristic
339	C₂	C₁₃×#3135	1	1	yes	characteristic
340	C₂	C₁₃×#3141	1	1	yes	characteristic
341	C₂	C₁₃×#3234	1	1	yes	characteristic
342	C₂	C₁₃×#3236	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
67. #29223	C₁₃	direct (N × H)
279. C₁₃	#29223	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×#2859	2
C₄×C₁₃×C₄⋊Q₈	2
C₁₃×#2100	2
C₁₃×#2099	2
C₁₃×#1993	2
C₁₃×#1989	2
C₁₃×#1836	2
C₁₃×#1767	2
#29223	13

Subgroups — C13×#29223 #15715562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29223 #15715562