Subgroups — C₁₃×#22813 #15709152

Order 3328.

922 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2354	C₂	1	2	yes	maximal, complemented
3	C₁₃×#1972	C₂	1	2	yes	maximal
4	C₁₃×#1926	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1844	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1802	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1776	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
11	C₂×C₁₃×#117	—	2	2	no	—
12	C₄×C₁₃×D₈	C₄	1	8	yes	complemented
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#106	C₂²	1	2	yes	—
15	C₂×C₁₃×#106	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#102	C₂²	1	4	yes	—
18	C₂×C₁₃×#102	C₂²	1	2	yes	—
19	C₂×C₁₃×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	C₁₃×#390	C₂²	1	4	yes	—
22	C₁₃×#388	—	2	4	no	—
23	C₁₃×#384	C₂²	1	2	yes	—
24	C₁₃×#380	—	2	2	no	—
25	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
26	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₁₃×#340	C₂²	1	4	yes	—
28	C₁₃×#336	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
30	C₂³×C₄×C₁₃	—	2	2	no	—
31	C₂×C₁₃×D₈	—	2	2	no	—
32	C₂×C₁₃×D₈	—	2	2	no	—
33	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₄×D₄×C₁₃	—	2	4	no	—
38	C₄×D₄×C₁₃	C₂×C₄	1	8	yes	—
39	C₁₃×#117	—	4	8	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
43	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
44	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
45	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
46	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
47	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
48	C₂×C₄²×C₁₃	—	2	4	no	—
49	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₁₃×#107	C₂×C₄	1	4	yes	—
52	C₁₃×#106	—	2	4	no	—
53	C₁₃×#106	—	4	2	no	—
54	C₁₃×C₄⋊C₈	—	2	2	no	—
55	C₁₃×#102	—	4	4	no	—
56	C₁₃×#102	C₂×C₄	1	8	yes	—
57	C₁₃×#97	—	2	4	no	—
58	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
59	C₁₃×#95	—	2	4	no	—
60	C₁₃×#95	D₄	1	2	yes	—
61	#22813	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂⁴×C₁₃	—	2	2	no	—
63	C₂×D₄×C₁₃	—	2	2	no	—
64	C₂×D₄×C₁₃	—	2	2	no	—
65	C₂×D₄×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	—	2	2	no	—
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	—	2	8	no	—
71	C₂²×C₄×C₁₃	—	2	4	no	—
72	C₂²×C₄×C₁₃	—	4	4	no	—
73	C₂²×C₄×C₁₃	—	4	4	no	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₁₃×D₈	—	2	8	no	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	—	2	1	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃	#41	1	2	yes	—
87	C₂×C₈×C₁₃	#41	1	2	yes	—
88	C₁₃×C₄⋊C₄	—	2	4	no	—
89	C₁₃×C₄⋊C₄	—	2	2	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	—	8	2	no	—
94	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
95	C₁₃×C₂²⋊C₄	—	2	8	no	—
96	C₁₃×C₂²⋊C₄	—	4	8	no	—
97	C₁₃×C₂²⋊C₄	—	8	4	no	—
98	C₄²×C₁₃	—	2	2	no	—
99	C₄²×C₁₃	—	2	2	no	—
100	C₄²×C₁₃	—	4	8	no	—
101	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
102	C₄²×C₁₃	#41	1	2	yes	—
103	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
104	#2354	C₂×C₁₃	1	2	yes	complemented
105	#1972	C₂×C₁₃	1	2	yes	—
106	#1926	C₂×C₁₃	1	4	yes	complemented
107	#1844	C₂×C₁₃	1	1	yes	characteristic, complemented
108	#1802	C₂×C₁₃	1	2	yes	complemented
109	#1776	C₂×C₁₃	1	2	yes	complemented
110	#1770	C₂×C₁₃	1	1	yes	characteristic, complemented
111	C₂³×C₁₃	—	2	4	no	—
112	C₂³×C₁₃	—	4	2	no	—
113	C₂³×C₁₃	—	4	2	no	—
114	C₂³×C₁₃	—	4	2	no	—
115	C₂³×C₁₃	—	4	2	no	—
116	C₂³×C₁₃	—	4	2	no	—
117	C₂³×C₁₃	—	4	2	no	—
118	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	D₄×C₁₃	—	2	4	no	—
120	D₄×C₁₃	—	2	4	no	—
121	D₄×C₁₃	—	4	2	no	—
122	D₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	—	2	4	no	—
126	C₂×C₄×C₁₃	—	2	4	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	2	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	4	no	—
134	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
137	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
143	C₈×C₁₃	—	2	2	no	—
144	C₈×C₁₃	—	4	2	no	—
145	C₈×C₁₃	#117	1	4	yes	complemented
146	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
147	C₂×C₄×D₄	C₂²×C₁₃	1	2	yes	—
148	C₂×#117	—	2	2	no	—
149	C₄×D₈	C₄×C₁₃	1	8	yes	complemented
150	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
151	C₂×#106	C₂²×C₁₃	1	2	yes	—
152	C₂×#106	C₂²×C₁₃	1	2	yes	—
153	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×#102	C₂²×C₁₃	1	4	yes	—
155	C₂×#102	C₂²×C₁₃	1	2	yes	—
156	C₂×#97	C₂²×C₁₃	1	2	yes	—
157	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
158	#390	C₂²×C₁₃	1	4	yes	—
159	#388	—	2	4	no	—
160	#384	C₂²×C₁₃	1	2	yes	—
161	#380	—	2	2	no	—
162	C₄×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
163	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
164	#340	C₂²×C₁₃	1	4	yes	—
165	#336	C₂²×C₁₃	1	2	yes	—
166	C₂²×C₁₃	—	4	4	no	—
167	C₂²×C₁₃	—	4	4	no	—
168	C₂²×C₁₃	—	4	4	no	—
169	C₂²×C₁₃	—	8	2	no	—
170	C₂²×C₁₃	—	8	2	no	—
171	C₂²×C₁₃	—	8	2	no	—
172	C₂²×C₁₃	—	8	2	no	—
173	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂²×C₁₃	#547	1	1	yes	characteristic
175	C₂²×C₁₃	#550	1	1	yes	characteristic
176	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
177	C₂²×C₁₃	#577	1	2	yes	—
178	C₄×C₁₃	—	2	2	no	—
179	C₄×C₁₃	—	4	4	no	—
180	C₄×C₁₃	—	4	4	no	—
181	C₄×C₁₃	—	8	4	no	—
182	C₄×C₁₃	#442	1	4	yes	—
183	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
184	C₄×C₁₃	#520	1	1	yes	characteristic
185	C₄×C₁₃	#555	1	1	yes	characteristic
186	C₄×C₁₃	#556	1	1	yes	characteristic
187	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
188	C₂³×C₄	—	2	2	no	—
189	C₂×D₈	—	2	2	no	—
190	C₂×D₈	—	2	2	no	—
191	C₂×D₈	C₂×C₄×C₁₃	1	4	yes	—
192	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
193	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
195	C₄×D₄	—	2	4	no	—
196	C₄×D₄	C₂×C₄×C₁₃	1	8	yes	—
197	#117	—	4	8	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
202	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
203	C₂×C₂²⋊C₄	—	2	2	no	—
204	C₂×C₂²⋊C₄	—	2	4	no	—
205	C₂×C₂²⋊C₄	—	2	4	no	—
206	C₂×C₄²	—	2	4	no	—
207	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
208	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
209	#107	C₂×C₄×C₁₃	1	4	yes	—
210	#106	—	2	4	no	—
211	#106	—	4	2	no	—
212	C₄⋊C₈	—	2	2	no	—
213	#102	—	4	4	no	—
214	#102	C₂×C₄×C₁₃	1	8	yes	—
215	#97	—	2	4	no	—
216	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
217	#95	—	2	4	no	—
218	#95	D₄×C₁₃	1	2	yes	—
219	C₂×C₁₃	—	8	4	no	—
220	C₂×C₁₃	#2354	1	1	yes	characteristic
221	C₂×C₁₃	C₂×#442	1	1	yes	characteristic
222	C₂×C₁₃	#2952	1	1	yes	characteristic
223	C₂×C₁₃	#3354	1	2	yes	—
224	C₂×C₁₃	#3397	1	2	yes	—
225	C₂⁴	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	C₂²×C₄×C₁₃	1	4	yes	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	8	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	4	4	no	—
236	C₂²×C₄	—	4	4	no	—
237	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
238	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
240	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
241	D₈	—	2	8	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
248	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
249	C₂×C₈	C₁₃×#41	1	2	yes	—
250	C₂×C₈	C₁₃×#41	1	2	yes	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	—	8	2	no	—
257	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
258	C₂²⋊C₄	—	2	8	no	—
259	C₂²⋊C₄	—	4	8	no	—
260	C₂²⋊C₄	—	8	4	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	4	8	no	—
264	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
265	C₄²	C₁₃×#41	1	2	yes	—
266	C₁₃	#22813	1	1	yes	characteristic, direct product factor, Sylow
267	C₂³	—	2	4	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	4	2	no	—
274	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
275	D₄	—	2	4	no	—
276	D₄	—	2	4	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	16	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
291	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
292	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
293	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
294	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
295	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
296	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
297	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
298	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
299	C₈	—	2	2	no	—
300	C₈	—	4	2	no	—
301	C₈	C₁₃×#117	1	4	yes	complemented
302	C₂²	—	4	4	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
310	C₂²	C₁₃×#547	1	1	yes	characteristic
311	C₂²	C₁₃×#550	1	1	yes	characteristic
312	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	2	yes	—
313	C₂²	C₁₃×#577	1	2	yes	—
314	C₄	—	2	2	no	—
315	C₄	—	4	4	no	—
316	C₄	—	4	4	no	—
317	C₄	—	8	4	no	—
318	C₄	C₁₃×#442	1	4	yes	—
319	C₄	C₂×C₁₃×#117	1	1	yes	characteristic
320	C₄	C₁₃×#520	1	1	yes	characteristic
321	C₄	C₁₃×#555	1	1	yes	characteristic
322	C₄	C₁₃×#556	1	1	yes	characteristic
323	C₂	—	8	4	no	—
324	C₂	C₁₃×#2354	1	1	yes	characteristic
325	C₂	C₂×C₁₃×#442	1	1	yes	characteristic
326	C₂	C₁₃×#2952	1	1	yes	characteristic
327	C₂	C₁₃×#3354	1	2	yes	—
328	C₂	C₁₃×#3397	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	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₄×C₁₃×C₄⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2354	C₂	semidirect
4. C₁₃×#1926	C₂	semidirect
5. C₁₃×#1844	C₂	semidirect
6. C₁₃×#1802	C₂	semidirect
7. C₁₃×#1776	C₂	semidirect
8. C₁₃×#1770	C₂	semidirect
12. C₄×C₁₃×D₈	C₄	semidirect
61. #22813	C₁₃	direct (N × H)
104. #2354	C₂×C₁₃	semidirect
106. #1926	C₂×C₁₃	semidirect
107. #1844	C₂×C₁₃	semidirect
108. #1802	C₂×C₁₃	semidirect
109. #1776	C₂×C₁₃	semidirect
110. #1770	C₂×C₁₃	semidirect
145. C₈×C₁₃	#117	semidirect
149. C₄×D₈	C₄×C₁₃	semidirect
266. C₁₃	#22813	direct (N × H)
301. 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
C₂×C₄×C₁₃×D₈	2
C₁₃×#2354	2
C₁₃×#1972	2
C₁₃×#1926	2
C₁₃×#1844	2
C₁₃×#1802	2
C₁₃×#1776	2
C₁₃×#1770	2
#22813	13

Subgroups — C13×#22813 #15709152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#22813 #15709152