Subgroups — C₁₃×#12811 #15699150

Order 3328.

510 conjugacy classes of subgroups.

The subgroup lattice has 306 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	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#357	C₂	1	2	yes	maximal
4	C₁₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1380	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃	—	2	1	no	—
8	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
9	C₄×C₁₃×D₈	—	2	2	no	—
10	C₄×C₁₃×D₈	C₄	1	4	yes	complemented
11	C₂×C₁₃×#107	—	2	1	no	—
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#102	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#384	—	2	1	no	—
17	C₁₃×C₄⋊C₁₆	—	2	2	no	—
18	C₁₃×#357	—	2	4	no	—
19	C₂²×D₄×C₁₃	—	2	1	no	—
20	C₂³×C₄×C₁₃	—	4	1	no	—
21	C₂×C₁₃×D₈	—	2	1	no	—
22	C₂×C₁₃×D₈	—	2	1	no	—
23	C₂×C₁₃×D₈	—	2	1	no	—
24	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
25	C₂×C₁₃×D₈	D₄	1	2	yes	—
26	C₂×C₁₃×D₈	Q₈	1	2	yes	—
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₄×D₄×C₁₃	—	2	4	no	—
30	C₄×D₄×C₁₃	—	4	2	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
34	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
35	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₂×C₁₃×C₁₆	—	2	2	no	—
38	C₂×C₁₃×C₁₆	—	2	2	no	—
39	C₂×C₁₃×C₁₆	—	2	2	no	—
40	C₁₃×#107	—	2	1	no	—
41	C₁₃×#107	—	2	2	no	—
42	C₁₃×#107	—	2	2	no	—
43	C₁₃×#107	—	4	2	no	—
44	C₁₃×#107	C₂×C₄	1	2	yes	—
45	C₁₃×#102	—	2	4	no	—
46	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₁₃	D₄	1	2	yes	—
48	#12811	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₃	—	4	1	no	—
50	C₂×D₄×C₁₃	—	2	2	no	—
51	C₂×D₄×C₁₃	—	2	2	no	—
52	C₂×D₄×C₁₃	—	2	2	no	—
53	C₂×D₄×C₁₃	—	4	1	no	—
54	C₂×D₄×C₁₃	—	4	1	no	—
55	C₂×D₄×C₁₃	—	4	1	no	—
56	C₂²×C₄×C₁₃	—	4	1	no	—
57	C₂²×C₄×C₁₃	—	4	4	no	—
58	C₂²×C₄×C₁₃	—	4	2	no	—
59	C₂²×C₄×C₁₃	—	8	2	no	—
60	C₂²×C₄×C₁₃	—	8	2	no	—
61	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₁₃×D₈	—	2	2	no	—
65	C₁₃×D₈	—	2	2	no	—
66	C₁₃×D₈	—	2	2	no	—
67	C₁₃×D₈	C₄⋊C₄	1	4	yes	complemented
68	C₂×C₈×C₁₃	—	2	1	no	—
69	C₂×C₈×C₁₃	—	2	1	no	—
70	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
76	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
77	C₁₃×C₄⋊C₄	—	2	2	no	—
78	C₁₃×C₄⋊C₄	—	4	1	no	—
79	C₁₃×C₄⋊C₄	—	4	2	no	—
80	C₁₃×C₄⋊C₄	—	4	2	no	—
81	C₁₃×C₄⋊C₄	—	8	2	no	—
82	C₁₃×C₄⋊C₄	—	8	2	no	—
83	C₁₃×C₂²⋊C₄	—	4	4	no	—
84	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₁₃	SD₁₆	1	2	yes	—
86	C₁₃×C₁₆	—	4	2	no	—
87	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
88	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
89	C₂×#357	C₂×C₁₃	1	2	yes	—
90	#1845	C₂×C₁₃	1	1	yes	characteristic, complemented
91	#1380	C₂×C₁₃	1	2	yes	complemented
92	C₂³×C₁₃	—	4	2	no	—
93	C₂³×C₁₃	—	4	2	no	—
94	C₂³×C₁₃	—	4	2	no	—
95	C₂³×C₁₃	—	8	1	no	—
96	C₂³×C₁₃	—	8	1	no	—
97	C₂³×C₁₃	—	8	1	no	—
98	C₂³×C₁₃	—	8	1	no	—
99	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
100	D₄×C₁₃	—	2	4	no	—
101	D₄×C₁₃	—	4	2	no	—
102	D₄×C₁₃	—	4	2	no	—
103	D₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	—	8	1	no	—
108	C₂×C₄×C₁₃	—	8	1	no	—
109	C₂×C₄×C₁₃	—	8	8	no	—
110	C₂×C₄×C₁₃	—	8	2	no	—
111	C₂×C₄×C₁₃	—	8	2	no	—
112	C₂×C₄×C₁₃	—	8	2	no	—
113	C₂×C₄×C₁₃	#102	1	2	yes	—
114	C₂×C₄×C₁₃	#102	1	2	yes	—
115	C₂×C₄×C₁₃	D₁₆	1	2	yes	—
116	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
123	C₈×C₁₃	—	2	2	no	—
124	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₈×C₁₃	#117	1	1	yes	characteristic
126	C₈×C₁₃	#121	1	1	yes	characteristic
127	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
129	C₂×C₄×D₄	—	2	1	no	—
130	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
131	C₄×D₈	—	2	2	no	—
132	C₄×D₈	C₄×C₁₃	1	4	yes	complemented
133	C₂×#107	—	2	1	no	—
134	C₂×#107	C₂²×C₁₃	1	2	yes	—
135	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×#102	—	2	1	no	—
137	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
138	#384	—	2	1	no	—
139	C₄⋊C₁₆	—	2	2	no	—
140	#357	—	2	4	no	—
141	C₂²×C₁₃	—	4	4	no	—
142	C₂²×C₁₃	—	8	2	no	—
143	C₂²×C₁₃	—	8	2	no	—
144	C₂²×C₁₃	—	8	2	no	—
145	C₂²×C₁₃	—	8	2	no	—
146	C₂²×C₁₃	—	8	2	no	—
147	C₂²×C₁₃	—	8	2	no	—
148	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
149	C₂²×C₁₃	#459	1	1	yes	characteristic
150	C₂²×C₁₃	#474	1	1	yes	characteristic
151	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
152	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
153	C₂²×C₁₃	#509	1	1	yes	characteristic
154	C₂²×C₁₃	#510	1	1	yes	characteristic
155	C₄×C₁₃	—	2	2	no	—
156	C₄×C₁₃	—	8	2	no	—
157	C₄×C₁₃	—	16	2	no	—
158	C₄×C₁₃	#357	1	4	yes	complemented
159	C₄×C₁₃	#380	1	1	yes	characteristic
160	C₄×C₁₃	#417	1	1	yes	characteristic
161	C₄×C₁₃	#461	1	1	yes	characteristic
162	C₄×C₁₃	#476	1	1	yes	characteristic
163	C₂²×D₄	—	2	1	no	—
164	C₂³×C₄	—	4	1	no	—
165	C₂×D₈	—	2	1	no	—
166	C₂×D₈	—	2	1	no	—
167	C₂×D₈	—	2	1	no	—
168	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
169	C₂×D₈	D₄×C₁₃	1	2	yes	—
170	C₂×D₈	Q₈×C₁₃	1	2	yes	—
171	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
173	C₄×D₄	—	2	4	no	—
174	C₄×D₄	—	4	2	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	4	1	no	—
178	C₂×C₄⋊C₄	—	4	1	no	—
179	C₂×C₂²⋊C₄	—	4	1	no	—
180	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
181	C₂×C₁₆	—	2	2	no	—
182	C₂×C₁₆	—	2	2	no	—
183	C₂×C₁₆	—	2	2	no	—
184	#107	—	2	1	no	—
185	#107	—	2	2	no	—
186	#107	—	2	2	no	—
187	#107	—	4	2	no	—
188	#107	C₂×C₄×C₁₃	1	2	yes	—
189	#102	—	2	4	no	—
190	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
191	C₄×C₈	D₄×C₁₃	1	2	yes	—
192	C₂×C₁₃	—	8	4	no	—
193	C₂×C₁₃	#1802	1	1	yes	characteristic
194	C₂×C₁₃	C₂×#357	1	1	yes	characteristic
195	C₂×C₁₃	#2139	1	1	yes	characteristic
196	C₂×C₁₃	#2206	1	1	yes	characteristic
197	C₂×C₁₃	#2208	1	1	yes	characteristic
198	C₂×C₁₃	#2224	1	1	yes	characteristic
199	C₂×C₁₃	#2226	1	1	yes	characteristic
200	C₂⁴	—	4	1	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	8	2	no	—
211	C₂²×C₄	—	8	2	no	—
212	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
213	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
215	D₈	—	2	2	no	—
216	D₈	—	2	2	no	—
217	D₈	—	2	2	no	—
218	D₈	C₁₃×C₄⋊C₄	1	4	yes	complemented
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
222	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
223	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
224	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
225	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
226	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
227	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	8	2	no	—
233	C₄⋊C₄	—	8	2	no	—
234	C₂²⋊C₄	—	4	4	no	—
235	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
236	C₄²	C₁₃×SD₁₆	1	2	yes	—
237	C₁₆	—	4	2	no	—
238	C₁₃	#12811	1	1	yes	characteristic, direct product factor, Sylow
239	C₂³	—	4	2	no	—
240	C₂³	—	4	2	no	—
241	C₂³	—	4	2	no	—
242	C₂³	—	8	1	no	—
243	C₂³	—	8	1	no	—
244	C₂³	—	8	1	no	—
245	C₂³	—	8	1	no	—
246	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
247	D₄	—	2	4	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	8	1	no	—
256	C₂×C₄	—	8	8	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	8	2	no	—
260	C₂×C₄	C₁₃×#102	1	2	yes	—
261	C₂×C₄	C₁₃×#102	1	2	yes	—
262	C₂×C₄	C₁₃×D₁₆	1	2	yes	—
263	C₂×C₄	C₁₃×SD₃₂	1	2	yes	—
264	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
266	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
267	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
268	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
269	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
272	C₈	C₁₃×#117	1	1	yes	characteristic
273	C₈	C₁₃×#121	1	1	yes	characteristic
274	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
275	C₂²	—	4	4	no	—
276	C₂²	—	8	2	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	8	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
283	C₂²	C₁₃×#459	1	1	yes	characteristic
284	C₂²	C₁₃×#474	1	1	yes	characteristic
285	C₂²	C₂×C₁₃×D₁₆	1	1	yes	characteristic
286	C₂²	C₂×C₁₃×SD₃₂	1	1	yes	characteristic
287	C₂²	C₁₃×#509	1	1	yes	characteristic
288	C₂²	C₁₃×#510	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	8	2	no	—
291	C₄	—	16	2	no	—
292	C₄	C₁₃×#357	1	4	yes	complemented
293	C₄	C₁₃×#380	1	1	yes	characteristic
294	C₄	C₁₃×#417	1	1	yes	characteristic
295	C₄	C₁₃×#461	1	1	yes	characteristic
296	C₄	C₁₃×#476	1	1	yes	characteristic
297	C₂	—	8	4	no	—
298	C₂	C₁₃×#1802	1	1	yes	characteristic
299	C₂	C₂×C₁₃×#357	1	1	yes	characteristic
300	C₂	C₁₃×#2139	1	1	yes	characteristic
301	C₂	C₁₃×#2206	1	1	yes	characteristic
302	C₂	C₁₃×#2208	1	1	yes	characteristic
303	C₂	C₁₃×#2224	1	1	yes	characteristic
304	C₂	C₁₃×#2226	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
4. C₁₃×#1845	C₂	semidirect
5. C₁₃×#1380	C₂	semidirect
10. C₄×C₁₃×D₈	C₄	semidirect
48. #12811	C₁₃	direct (N × H)
67. C₁₃×D₈	C₄⋊C₄	semidirect
88. C₂×C₄⋊C₁₆	C₂×C₁₃	semidirect
90. #1845	C₂×C₁₃	semidirect
91. #1380	C₂×C₁₃	semidirect
132. C₄×D₈	C₄×C₁₃	semidirect
158. C₄×C₁₃	#357	semidirect
218. D₈	C₁₃×C₄⋊C₄	semidirect
238. C₁₃	#12811	direct (N × H)
292. C₄	C₁₃×#357	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
C₂×C₁₃×C₄⋊C₁₆	2
C₂×C₁₃×#357	2
C₁₃×#1845	2
C₁₃×#1380	2
#12811	13

Subgroups — C13×#12811 #15699150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12811 #15699150