Subgroups — C₁₃×#25913 #15712252

Order 3328.

1218 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#490	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2856	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2025	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1992	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1874	C₂	1	4	yes	maximal, complemented
7	C₁₃×#1749	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#125	—	2	8	no	—
10	C₂×C₁₃×#125	—	2	2	no	—
11	C₂×C₁₃×#125	C₂²	1	2	yes	—
12	C₂×C₁₃×#125	C₂²	1	2	yes	—
13	C₂×C₁₃×#121	—	2	2	no	—
14	C₂×C₁₃×#121	C₂²	1	2	yes	—
15	C₂²×C₁₃×C₄⋊C₄	—	2	2	no	—
16	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#490	—	2	8	no	—
19	C₂×C₁₃×#102	C₂²	1	8	yes	—
20	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₂×C₁₃×#97	C₂²	1	2	yes	—
22	C₁₃×#390	—	2	2	no	—
23	C₁₃×#390	C₂²	1	2	yes	—
24	C₁₃×#384	C₂²	1	1	yes	characteristic
25	C₁₃×#340	C₂²	1	8	yes	—
26	C₁₃×#336	C₂²	1	2	yes	—
27	C₂²×D₄×C₁₃	—	2	2	no	—
28	C₂²×D₄×C₁₃	—	2	2	no	—
29	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
30	C₂³×C₄×C₁₃	—	2	2	no	—
31	C₂³×C₄×C₁₃	D₄	1	2	yes	—
32	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
34	C₁₃×#127	—	2	4	no	—
35	C₁₃×#125	—	2	16	no	—
36	C₁₃×#125	—	2	8	no	—
37	C₁₃×#125	—	2	4	no	—
38	C₁₃×#125	D₄	1	8	yes	complemented
39	C₁₃×#121	—	2	8	no	—
40	C₁₃×#121	—	4	8	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	16	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
49	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
50	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
51	C₂×C₄²×C₁₃	—	2	4	no	—
52	C₂×C₄²×C₁₃	D₄	1	2	yes	—
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	C₁₃×#102	—	2	16	no	—
56	C₁₃×C₂²⋊C₈	—	2	4	no	—
57	C₁₃×#97	—	2	4	no	—
58	#25913	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂⁴×C₁₃	—	2	2	no	—
60	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×D₄×C₁₃	—	2	4	no	—
62	C₂×D₄×C₁₃	—	2	4	no	—
63	C₂×D₄×C₁₃	—	2	4	no	—
64	C₂×D₄×C₁₃	—	4	4	no	—
65	C₂×D₄×C₁₃	—	4	4	no	—
66	C₂×D₄×C₁₃	—	4	4	no	—
67	C₂×D₄×C₁₃	—	4	8	no	—
68	C₂×D₄×C₁₃	—	4	4	no	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	—	2	2	no	—
71	C₂²×C₄×C₁₃	—	2	8	no	—
72	C₂²×C₄×C₁₃	—	2	4	no	—
73	C₂²×C₄×C₁₃	—	2	4	no	—
74	C₂²×C₄×C₁₃	—	2	4	no	—
75	C₂²×C₄×C₁₃	—	2	2	no	—
76	C₂²×C₄×C₁₃	—	4	2	no	—
77	C₂²×C₄×C₁₃	—	4	4	no	—
78	C₂²×C₄×C₁₃	—	4	4	no	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	#41	1	4	yes	—
83	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₁₃	—	2	4	no	—
85	C₂×C₈×C₁₃	—	2	4	no	—
86	C₂×C₈×C₁₃	—	2	4	no	—
87	C₁₃×C₄⋊C₄	—	2	8	no	—
88	C₁₃×C₄⋊C₄	—	2	16	no	—
89	C₁₃×C₄⋊C₄	—	2	16	no	—
90	C₁₃×C₄⋊C₄	—	2	4	no	—
91	C₁₃×C₄⋊C₄	—	2	4	no	—
92	C₁₃×C₄⋊C₄	—	4	4	no	—
93	C₁₃×C₄⋊C₄	C₂×D₄	1	8	yes	—
94	C₁₃×C₂²⋊C₄	—	4	8	no	—
95	C₁₃×C₂²⋊C₄	—	4	4	no	—
96	C₄²×C₁₃	—	2	4	no	—
97	C₄²×C₁₃	—	2	2	no	—
98	C₄²×C₁₃	C₂×D₄	1	4	yes	—
99	C₂²×#125	C₂×C₁₃	1	1	yes	characteristic, complemented
100	C₂×#490	C₂×C₁₃	1	2	yes	complemented
101	#2856	C₂×C₁₃	1	1	yes	characteristic
102	#2025	C₂×C₁₃	1	4	yes	complemented
103	#1992	C₂×C₁₃	1	2	yes	complemented
104	#1874	C₂×C₁₃	1	4	yes	complemented
105	#1749	C₂×C₁₃	1	1	yes	characteristic, complemented
106	C₂³×C₁₃	—	2	1	no	—
107	C₂³×C₁₃	—	2	1	no	—
108	C₂³×C₁₃	—	2	2	no	—
109	C₂³×C₁₃	—	4	2	no	—
110	C₂³×C₁₃	—	4	2	no	—
111	C₂³×C₁₃	—	4	2	no	—
112	C₂³×C₁₃	—	4	4	no	—
113	C₂³×C₁₃	—	4	4	no	—
114	C₂³×C₁₃	#124	1	2	yes	—
115	C₂³×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
116	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	D₄×C₁₃	—	4	8	no	—
118	D₄×C₁₃	—	8	8	no	—
119	D₄×C₁₃	—	8	8	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	8	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	8	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	4	8	no	—
130	C₂×C₄×C₁₃	—	4	16	no	—
131	C₂×C₄×C₁₃	—	4	4	no	—
132	C₂×C₄×C₁₃	—	4	4	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	4	no	—
135	C₂×C₄×C₁₃	#120	1	4	yes	—
136	C₂×C₄×C₁₃	#124	1	2	yes	—
137	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
138	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
139	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
140	C₈×C₁₃	—	4	4	no	—
141	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
142	C₂×#125	—	2	8	no	—
143	C₂×#125	—	2	2	no	—
144	C₂×#125	C₂²×C₁₃	1	2	yes	—
145	C₂×#125	C₂²×C₁₃	1	2	yes	—
146	C₂×#121	—	2	2	no	—
147	C₂×#121	C₂²×C₁₃	1	2	yes	—
148	C₂²×C₄⋊C₄	—	2	2	no	—
149	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
150	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
151	#490	—	2	8	no	—
152	C₂×#102	C₂²×C₁₃	1	8	yes	—
153	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
154	C₂×#97	C₂²×C₁₃	1	2	yes	—
155	#390	—	2	2	no	—
156	#390	C₂²×C₁₃	1	2	yes	—
157	#384	C₂²×C₁₃	1	1	yes	characteristic
158	#340	C₂²×C₁₃	1	8	yes	—
159	#336	C₂²×C₁₃	1	2	yes	—
160	C₂²×C₁₃	—	2	2	no	—
161	C₂²×C₁₃	—	2	2	no	—
162	C₂²×C₁₃	—	2	4	no	—
163	C₂²×C₁₃	—	2	4	no	—
164	C₂²×C₁₃	—	8	2	no	—
165	C₂²×C₁₃	—	8	2	no	—
166	C₂²×C₁₃	—	8	2	no	—
167	C₂²×C₁₃	—	8	4	no	—
168	C₂²×C₁₃	—	8	4	no	—
169	C₂²×C₁₃	#490	1	4	yes	complemented
170	C₂²×C₁₃	C₂×#120	1	1	yes	characteristic
171	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
172	C₂²×C₁₃	#536	1	1	yes	characteristic
173	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
174	C₂²×C₁₃	#575	1	2	yes	—
175	C₄×C₁₃	—	2	2	no	—
176	C₄×C₁₃	—	2	4	no	—
177	C₄×C₁₃	—	4	2	no	—
178	C₄×C₁₃	—	4	8	no	—
179	C₄×C₁₃	—	8	2	no	—
180	C₄×C₁₃	#548	1	4	yes	—
181	C₂²×D₄	—	2	2	no	—
182	C₂²×D₄	—	2	2	no	—
183	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
184	C₂³×C₄	—	2	2	no	—
185	C₂³×C₄	D₄×C₁₃	1	2	yes	—
186	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
188	#127	—	2	4	no	—
189	#125	—	2	16	no	—
190	#125	—	2	8	no	—
191	#125	—	2	4	no	—
192	#125	D₄×C₁₃	1	8	yes	complemented
193	#121	—	2	8	no	—
194	#121	—	4	8	no	—
195	C₂×C₄⋊C₄	—	2	16	no	—
196	C₂×C₄⋊C₄	—	2	4	no	—
197	C₂×C₄⋊C₄	—	2	4	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	8	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
203	C₂×C₂²⋊C₄	—	2	4	no	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₄²	—	2	4	no	—
206	C₂×C₄²	D₄×C₁₃	1	2	yes	—
207	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
209	#102	—	2	16	no	—
210	C₂²⋊C₈	—	2	4	no	—
211	#97	—	2	4	no	—
212	C₂×C₁₃	—	2	4	no	—
213	C₂×C₁₃	—	16	2	no	—
214	C₂×C₁₃	#2436	1	1	yes	characteristic
215	C₂×C₁₃	#3006	1	2	yes	—
216	C₂×C₁₃	C₂×#490	1	1	yes	characteristic
217	C₂×C₁₃	#3142	1	1	yes	characteristic
218	C₂×C₁₃	#3234	1	2	yes	—
219	C₂⁴	—	2	2	no	—
220	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
221	C₂×D₄	—	2	4	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	4	4	no	—
225	C₂×D₄	—	4	4	no	—
226	C₂×D₄	—	4	4	no	—
227	C₂×D₄	—	4	8	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	8	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	—	4	4	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
241	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
242	C₂²×C₄	C₁₃×#41	1	4	yes	—
243	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₄⋊C₄	—	2	8	no	—
248	C₄⋊C₄	—	2	16	no	—
249	C₄⋊C₄	—	2	16	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	C₂×D₄×C₁₃	1	8	yes	—
254	C₂²⋊C₄	—	4	8	no	—
255	C₂²⋊C₄	—	4	4	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	2	2	no	—
258	C₄²	C₂×D₄×C₁₃	1	4	yes	—
259	C₁₃	#25913	1	1	yes	characteristic, direct product factor, Sylow
260	C₂³	—	2	1	no	—
261	C₂³	—	2	1	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	2	no	—
266	C₂³	—	4	4	no	—
267	C₂³	—	4	4	no	—
268	C₂³	C₁₃×#124	1	2	yes	—
269	C₂³	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
270	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
271	D₄	—	4	8	no	—
272	D₄	—	8	8	no	—
273	D₄	—	8	8	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	8	no	—
284	C₂×C₄	—	4	16	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	C₁₃×#120	1	4	yes	—
290	C₂×C₄	C₁₃×#124	1	2	yes	—
291	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
292	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
293	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
294	C₈	—	4	4	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	8	4	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	4	no	—
304	C₂²	C₁₃×#490	1	4	yes	complemented
305	C₂²	C₂×C₁₃×#120	1	1	yes	characteristic
306	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
307	C₂²	C₁₃×#536	1	1	yes	characteristic
308	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	2	yes	—
309	C₂²	C₁₃×#575	1	2	yes	—
310	C₄	—	2	4	no	—
311	C₄	—	2	2	no	—
312	C₄	—	4	2	no	—
313	C₄	—	4	8	no	—
314	C₄	—	8	2	no	—
315	C₄	C₁₃×#548	1	4	yes	—
316	C₂	—	2	4	no	—
317	C₂	—	16	2	no	—
318	C₂	C₁₃×#2436	1	1	yes	characteristic
319	C₂	C₁₃×#3006	1	2	yes	—
320	C₂	C₂×C₁₃×#490	1	1	yes	characteristic
321	C₂	C₁₃×#3142	1	1	yes	characteristic
322	C₂	C₁₃×#3234	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#2856	C₂
LCS	G ▷ C₂²×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
1. C₂²×C₁₃×#125	C₂	semidirect
2. C₂×C₁₃×#490	C₂	semidirect
4. C₁₃×#2025	C₂	semidirect
5. C₁₃×#1992	C₂	semidirect
6. C₁₃×#1874	C₂	semidirect
7. C₁₃×#1749	C₂	semidirect
38. C₁₃×#125	D₄	semidirect
58. #25913	C₁₃	direct (N × H)
99. C₂²×#125	C₂×C₁₃	semidirect
100. C₂×#490	C₂×C₁₃	semidirect
102. #2025	C₂×C₁₃	semidirect
103. #1992	C₂×C₁₃	semidirect
104. #1874	C₂×C₁₃	semidirect
105. #1749	C₂×C₁₃	semidirect
169. C₂²×C₁₃	#490	semidirect
192. #125	D₄×C₁₃	semidirect
259. C₁₃	#25913	direct (N × H)
304. C₂²	C₁₃×#490	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×#125	2
C₂×C₁₃×#490	2
C₁₃×#2856	2
C₁₃×#2025	2
C₁₃×#1992	2
C₁₃×#1874	2
C₁₃×#1749	2
#25913	13

Subgroups — C13×#25913 #15712252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#25913 #15712252