Subgroups — C₁₃×#15197 #15701536

Order 3328.

1254 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#389	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#384	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1442	C₂	1	4	yes	maximal
4	C₁₃×#1441	C₂	1	4	yes	maximal
5	C₁₃×C₄⋊#95	C₂	1	2	yes	maximal
6	C₁₃×C₄⋊#95	C₂	1	2	yes	maximal
7	C₄×C₁₃×#95	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂²×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂²×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#389	—	2	4	no	—
17	C₁₃×#384	—	2	4	no	—
18	C₂×C₁₃×#95	C₂²	1	2	yes	—
19	C₂×C₁₃×#95	C₂²	1	2	yes	—
20	C₂×C₁₃×#95	C₂²	1	8	yes	—
21	C₂×C₁₃×#95	C₂²	1	8	yes	—
22	C₂×C₁₃×#95	C₂²	1	2	yes	—
23	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
40	C₂×C₁₃×C₄⋊C₄	—	4	4	no	—
41	C₂×C₁₃×C₄⋊C₄	—	4	4	no	—
42	C₂×C₁₃×C₄⋊C₄	Q₈	1	4	yes	—
43	C₂×C₄²×C₁₃	—	2	4	no	—
44	C₂×C₄²×C₁₃	—	2	4	no	—
45	C₂×C₄²×C₁₃	—	2	4	no	—
46	C₂×C₄²×C₁₃	—	2	4	no	—
47	C₂×C₄²×C₁₃	D₄	1	4	yes	—
48	C₁₃×#95	—	2	4	no	—
49	C₁₃×#95	—	2	4	no	—
50	C₁₃×#95	—	4	8	no	—
51	C₁₃×#95	—	4	8	no	—
52	C₁₃×#95	Q₈	1	8	yes	—
53	#15197	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	—	2	2	no	—
60	C₂²×C₄×C₁₃	—	2	4	no	—
61	C₂²×C₄×C₁₃	—	2	4	no	—
62	C₂²×C₄×C₁₃	—	2	4	no	—
63	C₂²×C₄×C₁₃	—	2	4	no	—
64	C₂²×C₄×C₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₃	—	2	4	no	—
66	C₂²×C₄×C₁₃	—	2	4	no	—
67	C₂²×C₄×C₁₃	—	2	4	no	—
68	C₂²×C₄×C₁₃	—	2	4	no	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	—	2	4	no	—
71	C₂²×C₄×C₁₃	—	2	4	no	—
72	C₂²×C₄×C₁₃	—	2	8	no	—
73	C₂²×C₄×C₁₃	—	2	4	no	—
74	C₂²×C₄×C₁₃	—	2	4	no	—
75	C₂²×C₄×C₁₃	—	2	8	no	—
76	C₂²×C₄×C₁₃	—	2	8	no	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
81	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
82	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
83	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₁₃	#41	1	4	yes	—
85	C₂²×C₄×C₁₃	#41	1	2	yes	—
86	C₂²×C₄×C₁₃	#41	1	4	yes	—
87	C₂²×C₄×C₁₃	#41	1	4	yes	—
88	C₁₃×C₄⋊C₄	—	2	8	no	—
89	C₁₃×C₄⋊C₄	—	4	8	no	—
90	C₁₃×C₄⋊C₄	—	4	8	no	—
91	C₁₃×C₄⋊C₄	—	4	4	no	—
92	C₁₃×C₄⋊C₄	—	4	8	no	—
93	C₁₃×C₄⋊C₄	—	4	8	no	—
94	C₄²×C₁₃	—	2	8	no	—
95	C₄²×C₁₃	—	4	8	no	—
96	C₂×#389	C₂×C₁₃	1	1	yes	characteristic
97	C₂×#384	C₂×C₁₃	1	1	yes	characteristic
98	#1442	C₂×C₁₃	1	4	yes	—
99	#1441	C₂×C₁₃	1	4	yes	—
100	C₄⋊#95	C₂×C₁₃	1	2	yes	—
101	C₄⋊#95	C₂×C₁₃	1	2	yes	—
102	C₄×#95	C₂×C₁₃	1	1	yes	characteristic
103	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
104	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
105	C₂³×C₁₃	C₂²×Q₈	1	2	yes	—
106	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
107	C₂³×C₁₃	C₂×#41	1	2	yes	—
108	C₂³×C₁₃	C₂×#41	1	2	yes	—
109	C₂³×C₁₃	C₂×#41	1	2	yes	—
110	C₂³×C₁₃	H(5,𝔽₂)	1	2	yes	—
111	C₂³×C₁₃	H(5,𝔽₂)	1	2	yes	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	8	no	—
117	C₂×C₄×C₁₃	—	2	8	no	—
118	C₂×C₄×C₁₃	—	2	4	no	—
119	C₂×C₄×C₁₃	—	2	8	no	—
120	C₂×C₄×C₁₃	—	2	8	no	—
121	C₂×C₄×C₁₃	—	2	8	no	—
122	C₂×C₄×C₁₃	—	2	8	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	4	4	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	4	4	no	—
127	C₂×C₄×C₁₃	—	4	4	no	—
128	C₂×C₄×C₁₃	—	4	4	no	—
129	C₂×C₄×C₁₃	—	4	4	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	4	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	4	no	—
134	C₂×C₄×C₁₃	—	4	4	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	16	no	—
137	C₂×C₄×C₁₃	—	4	8	no	—
138	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
139	C₂×C₄×C₁₃	#123	1	4	yes	—
140	C₂×C₄×C₁₃	#124	1	4	yes	—
141	C₂×C₄×C₁₃	#124	1	4	yes	—
142	C₂×C₄×C₁₃	#125	1	8	yes	—
143	C₂×C₄×C₁₃	#126	1	8	yes	—
144	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
145	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
146	C₂²×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
147	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
148	C₂²×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
149	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
150	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
151	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
152	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
153	#389	—	2	4	no	—
154	#384	—	2	4	no	—
155	C₂×#95	C₂²×C₁₃	1	2	yes	—
156	C₂×#95	C₂²×C₁₃	1	2	yes	—
157	C₂×#95	C₂²×C₁₃	1	8	yes	—
158	C₂×#95	C₂²×C₁₃	1	8	yes	—
159	C₂×#95	C₂²×C₁₃	1	2	yes	—
160	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#123	1	1	yes	characteristic
162	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
163	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
164	C₂²×C₁₃	C₂×#125	1	2	yes	—
165	C₂²×C₁₃	C₂×#126	1	2	yes	—
166	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
167	C₂²×C₁₃	#533	1	4	yes	—
168	C₂²×C₁₃	#535	1	1	yes	characteristic
169	C₂²×C₁₃	#535	1	1	yes	characteristic
170	C₂²×C₁₃	#540	1	4	yes	—
171	C₂²×C₁₃	#553	1	4	yes	—
172	C₂²×C₁₃	#554	1	2	yes	—
173	C₂²×C₁₃	#554	1	2	yes	—
174	C₂²×C₁₃	#554	1	4	yes	—
175	C₂²×C₁₃	#555	1	2	yes	—
176	C₂²×C₁₃	Q₈²	1	2	yes	—
177	C₄×C₁₃	—	2	4	no	—
178	C₄×C₁₃	—	2	4	no	—
179	C₄×C₁₃	—	4	4	no	—
180	C₄×C₁₃	—	4	8	no	—
181	C₄×C₁₃	—	8	4	no	—
182	C₄×C₁₃	—	8	2	no	—
183	C₄×C₁₃	#400	1	8	yes	—
184	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
185	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
186	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
187	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
188	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
189	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
190	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	4	no	—
198	C₂×C₄⋊C₄	—	2	4	no	—
199	C₂×C₄⋊C₄	—	2	8	no	—
200	C₂×C₄⋊C₄	—	2	8	no	—
201	C₂×C₄⋊C₄	—	4	4	no	—
202	C₂×C₄⋊C₄	—	4	4	no	—
203	C₂×C₄⋊C₄	Q₈×C₁₃	1	4	yes	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	—	2	4	no	—
206	C₂×C₄²	—	2	4	no	—
207	C₂×C₄²	—	2	4	no	—
208	C₂×C₄²	D₄×C₁₃	1	4	yes	—
209	#95	—	2	4	no	—
210	#95	—	2	4	no	—
211	#95	—	4	8	no	—
212	#95	—	4	8	no	—
213	#95	Q₈×C₁₃	1	8	yes	—
214	C₂×C₁₃	C₂×#400	1	1	yes	characteristic
215	C₂×C₁₃	#2497	1	4	yes	—
216	C₂×C₁₃	#2502	1	1	yes	characteristic
217	C₂×C₁₃	#2504	1	1	yes	characteristic
218	C₂×C₁₃	#2507	1	4	yes	—
219	C₂×C₁₃	#2563	1	1	yes	characteristic
220	C₂×C₁₃	#2567	1	1	yes	characteristic
221	C₂×C₁₃	#2641	1	1	yes	characteristic
222	C₂×C₁₃	#2862	1	1	yes	characteristic
223	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	8	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	8	no	—
245	C₂²×C₄	—	2	8	no	—
246	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
249	C₂²×C₄	C₂×Q₈×C₁₃	1	8	yes	—
250	C₂²×C₄	C₂×Q₈×C₁₃	1	4	yes	—
251	C₂²×C₄	C₂×Q₈×C₁₃	1	4	yes	—
252	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
253	C₂²×C₄	C₁₃×#41	1	4	yes	—
254	C₂²×C₄	C₁₃×#41	1	2	yes	—
255	C₂²×C₄	C₁₃×#41	1	4	yes	—
256	C₂²×C₄	C₁₃×#41	1	4	yes	—
257	C₄⋊C₄	—	2	8	no	—
258	C₄⋊C₄	—	4	8	no	—
259	C₄⋊C₄	—	4	8	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₄⋊C₄	—	4	8	no	—
262	C₄⋊C₄	—	4	8	no	—
263	C₄²	—	2	8	no	—
264	C₄²	—	4	8	no	—
265	C₁₃	#15197	1	1	yes	characteristic, direct product factor, Sylow
266	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
267	C₂³	C₂²×Q₈×C₁₃	1	1	yes	characteristic
268	C₂³	C₂²×Q₈×C₁₃	1	2	yes	—
269	C₂³	C₂×C₁₃×#41	1	1	yes	characteristic
270	C₂³	C₂×C₁₃×#41	1	2	yes	—
271	C₂³	C₂×C₁₃×#41	1	2	yes	—
272	C₂³	C₂×C₁₃×#41	1	2	yes	—
273	C₂³	C₁₃×H(5,𝔽₂)	1	2	yes	—
274	C₂³	C₁₃×H(5,𝔽₂)	1	2	yes	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	2	8	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	16	no	—
300	C₂×C₄	—	4	8	no	—
301	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
302	C₂×C₄	C₁₃×#123	1	4	yes	—
303	C₂×C₄	C₁₃×#124	1	4	yes	—
304	C₂×C₄	C₁₃×#124	1	4	yes	—
305	C₂×C₄	C₁₃×#125	1	8	yes	—
306	C₂×C₄	C₁₃×#126	1	8	yes	—
307	C₂×C₄	C₁₃×C₄⋊Q₈	1	4	yes	—
308	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
309	C₂²	C₂×C₁₃×#123	1	1	yes	characteristic
310	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
311	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
312	C₂²	C₂×C₁₃×#125	1	2	yes	—
313	C₂²	C₂×C₁₃×#126	1	2	yes	—
314	C₂²	C₂×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
315	C₂²	C₁₃×#533	1	4	yes	—
316	C₂²	C₁₃×#535	1	1	yes	characteristic
317	C₂²	C₁₃×#535	1	1	yes	characteristic
318	C₂²	C₁₃×#540	1	4	yes	—
319	C₂²	C₁₃×#553	1	4	yes	—
320	C₂²	C₁₃×#554	1	2	yes	—
321	C₂²	C₁₃×#554	1	4	yes	—
322	C₂²	C₁₃×#554	1	2	yes	—
323	C₂²	C₁₃×#555	1	2	yes	—
324	C₂²	Q₈²×C₁₃	1	2	yes	—
325	C₄	—	2	4	no	—
326	C₄	—	2	4	no	—
327	C₄	—	4	8	no	—
328	C₄	—	4	4	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	4	no	—
331	C₄	C₁₃×#400	1	8	yes	—
332	C₂	C₂×C₁₃×#400	1	1	yes	characteristic
333	C₂	C₁₃×#2497	1	4	yes	—
334	C₂	C₁₃×#2502	1	1	yes	characteristic
335	C₂	C₁₃×#2504	1	1	yes	characteristic
336	C₂	C₁₃×#2507	1	4	yes	—
337	C₂	C₁₃×#2563	1	1	yes	characteristic
338	C₂	C₁₃×#2567	1	1	yes	characteristic
339	C₂	C₁₃×#2641	1	1	yes	characteristic
340	C₂	C₁₃×#2862	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₁₃
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
53. #15197	C₁₃	direct (N × H)
265. C₁₃	#15197	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#389	2
C₂×C₁₃×#384	2
C₁₃×#1442	2
C₁₃×#1441	2
C₁₃×C₄⋊#95	2
C₄×C₁₃×#95	2
#15197	13

Subgroups — C13×#15197 #15701536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#15197 #15701536