Subgroups — C₁₃×#3211 #13112269

Order 1664.

410 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#495	C₂	1	2	yes	maximal, complemented
4	C₁₃×#488	C₂	1	2	yes	maximal, complemented
5	C₁₃×#464	C₂	1	2	yes	maximal, complemented
6	C₁₃×#463	C₂	1	2	yes	maximal, complemented
7	C₁₃×#451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#448	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#447	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	complemented
15	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
16	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	complemented
17	C₂×C₁₃×D₈	C₂²	1	2	yes	—
18	C₁₃×#126	—	2	1	no	—
19	C₁₃×#125	C₂²	1	1	yes	characteristic, complemented
20	C₁₃×#124	—	2	1	no	—
21	C₁₃×#124	—	2	1	no	—
22	C₁₃×#124	C₂²	1	2	yes	—
23	C₁₃×#124	C₂²	1	2	yes	—
24	C₁₃×#124	C₂²	1	1	yes	characteristic, complemented
25	C₁₃×#123	—	2	1	no	—
26	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
27	C₁₃×#121	C₂²	1	1	yes	characteristic
28	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
29	C₄×D₄×C₁₃	—	2	1	no	—
30	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic, complemented
31	C₁₃×#117	—	2	1	no	—
32	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
33	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
34	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
35	C₁₃×C₄⋊C₈	C₂²	1	2	yes	complemented
36	C₁₃×#103	C₂²	1	2	yes	complemented
37	C₁₃×#103	C₂²	1	2	yes	complemented
38	C₁₃×#102	C₂²	1	2	yes	—
39	C₁₃×#102	C₂²	1	2	yes	—
40	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₁₃	C₂²	1	2	yes	complemented
43	C₂⁴×C₁₃	—	2	1	no	—
44	C₂×Q₈×C₁₃	—	2	1	no	—
45	C₂×Q₈×C₁₃	—	4	2	no	—
46	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
47	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₁₃	—	2	1	no	—
49	C₂×D₄×C₁₃	—	2	1	no	—
50	C₂×D₄×C₁₃	—	4	2	no	—
51	C₂×D₄×C₁₃	C₂³	1	2	yes	—
52	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
58	C₁₃×Q₁₆	—	4	2	no	—
59	C₁₃×SD₁₆	—	4	2	no	—
60	C₁₃×SD₁₆	—	4	2	no	—
61	C₁₃×D₈	—	4	2	no	—
62	C₂×C₈×C₁₃	C₂³	1	2	yes	—
63	C₂×C₈×C₁₃	C₂³	1	2	yes	—
64	C₁₃×C₄⋊C₄	—	2	1	no	—
65	C₁₃×C₄⋊C₄	—	2	1	no	—
66	C₁₃×C₄⋊C₄	—	2	1	no	—
67	C₁₃×C₄⋊C₄	—	2	1	no	—
68	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
70	C₁₃×C₂²⋊C₄	—	2	1	no	—
71	C₁₃×C₂²⋊C₄	—	2	2	no	—
72	C₁₃×C₂²⋊C₄	—	2	2	no	—
73	C₁₃×C₂²⋊C₄	—	2	2	no	—
74	C₁₃×C₂²⋊C₄	—	2	2	no	—
75	C₁₃×C₂²⋊C₄	—	2	1	no	—
76	C₁₃×C₂²⋊C₄	—	2	1	no	—
77	C₁₃×C₂²⋊C₄	—	2	1	no	—
78	C₁₃×C₂²⋊C₄	—	2	1	no	—
79	C₄²×C₁₃	—	2	1	no	—
80	C₄²×C₁₃	D₄	1	1	yes	characteristic
81	C₄²×C₁₃	D₄	1	1	yes	characteristic
82	C₄²×C₁₃	C₂³	1	1	yes	characteristic
83	C₄²×C₁₃	C₂³	1	1	yes	characteristic
84	#3211	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₁₃	—	2	1	no	—
86	C₂³×C₁₃	—	2	2	no	—
87	C₂³×C₁₃	—	4	2	no	—
88	C₂³×C₁₃	—	4	2	no	—
89	C₂³×C₁₃	—	4	2	no	—
90	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
91	Q₈×C₁₃	—	2	2	no	—
92	Q₈×C₁₃	—	2	2	no	—
93	Q₈×C₁₃	—	2	2	no	—
94	Q₈×C₁₃	—	4	2	no	—
95	D₄×C₁₃	—	2	2	no	—
96	D₄×C₁₃	—	2	2	no	—
97	D₄×C₁₃	—	2	2	no	—
98	D₄×C₁₃	—	4	2	no	—
99	D₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	4	1	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
115	C₈×C₁₃	—	2	2	no	—
116	C₈×C₁₃	—	2	2	no	—
117	#529	C₂×C₁₃	1	1	yes	characteristic, complemented
118	C₂×#124	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#495	C₂×C₁₃	1	2	yes	complemented
120	#488	C₂×C₁₃	1	2	yes	complemented
121	#464	C₂×C₁₃	1	2	yes	complemented
122	#463	C₂×C₁₃	1	2	yes	complemented
123	#451	C₂×C₁₃	1	1	yes	characteristic, complemented
124	#450	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#448	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#447	C₂×C₁₃	1	1	yes	characteristic
127	#431	C₂×C₁₃	1	1	yes	characteristic, complemented
128	C₂²×C₁₃	—	2	1	no	—
129	C₂²×C₁₃	—	2	1	no	—
130	C₂²×C₁₃	—	4	2	no	—
131	C₂²×C₁₃	—	4	1	no	—
132	C₂²×C₁₃	—	4	1	no	—
133	C₂²×C₁₃	—	4	2	no	—
134	C₂²×C₁₃	—	4	1	no	—
135	C₂²×C₁₃	—	4	2	no	—
136	C₂²×C₁₃	—	8	4	no	—
137	C₂²×C₁₃	#135	1	2	yes	—
138	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₁₃	—	2	1	no	—
140	C₄×C₁₃	—	2	1	no	—
141	C₄×C₁₃	—	2	1	no	—
142	C₄×C₁₃	—	2	1	no	—
143	C₄×C₁₃	—	2	1	no	—
144	C₄×C₁₃	—	4	2	no	—
145	C₄×C₁₃	—	4	1	no	—
146	C₄×C₁₃	—	4	1	no	—
147	C₄×C₁₃	—	4	1	no	—
148	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
149	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
150	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
151	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	complemented
152	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
153	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	complemented
154	C₂×D₈	C₂²×C₁₃	1	2	yes	—
155	#126	—	2	1	no	—
156	#125	C₂²×C₁₃	1	1	yes	characteristic, complemented
157	#124	—	2	1	no	—
158	#124	—	2	1	no	—
159	#124	C₂²×C₁₃	1	2	yes	—
160	#124	C₂²×C₁₃	1	2	yes	—
161	#124	C₂²×C₁₃	1	1	yes	characteristic, complemented
162	#123	—	2	1	no	—
163	C₂²⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
164	#121	C₂²×C₁₃	1	1	yes	characteristic
165	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
166	C₄×D₄	—	2	1	no	—
167	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic, complemented
168	#117	—	2	1	no	—
169	C₂×C₂²⋊C₄	—	2	1	no	—
170	C₂×C₂²⋊C₄	—	2	1	no	—
171	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
172	C₄⋊C₈	C₂²×C₁₃	1	2	yes	complemented
173	#103	C₂²×C₁₃	1	2	yes	complemented
174	#103	C₂²×C₁₃	1	2	yes	complemented
175	#102	C₂²×C₁₃	1	2	yes	—
176	#102	C₂²×C₁₃	1	2	yes	—
177	C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
178	C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
179	C₄×C₈	C₂²×C₁₃	1	2	yes	complemented
180	C₂×C₁₃	—	2	2	no	—
181	C₂×C₁₃	—	8	1	no	—
182	C₂×C₁₃	—	8	2	no	—
183	C₂×C₁₃	#535	1	1	yes	characteristic
184	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
185	C₂×C₁₃	#575	1	1	yes	characteristic
186	C₂⁴	—	2	1	no	—
187	C₂×Q₈	—	2	1	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
190	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂×D₄	C₂³×C₁₃	1	2	yes	—
195	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
199	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
201	Q₁₆	—	4	2	no	—
202	SD₁₆	—	4	2	no	—
203	SD₁₆	—	4	2	no	—
204	D₈	—	4	2	no	—
205	C₂×C₈	C₂³×C₁₃	1	2	yes	—
206	C₂×C₈	C₂³×C₁₃	1	2	yes	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
212	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	2	1	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	D₄×C₁₃	1	1	yes	characteristic
224	C₄²	D₄×C₁₃	1	1	yes	characteristic
225	C₄²	C₂³×C₁₃	1	1	yes	characteristic
226	C₄²	C₂³×C₁₃	1	1	yes	characteristic
227	C₁₃	#3211	1	1	yes	characteristic, direct product factor, Sylow
228	C₂³	—	2	1	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	4	2	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
234	Q₈	—	2	2	no	—
235	Q₈	—	2	2	no	—
236	Q₈	—	2	2	no	—
237	Q₈	—	4	2	no	—
238	D₄	—	2	2	no	—
239	D₄	—	2	2	no	—
240	D₄	—	2	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
253	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
254	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
255	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
257	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	2	2	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	8	4	no	—
269	C₂²	C₁₃×#135	1	2	yes	—
270	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	2	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	8	2	no	—
283	C₂	—	8	1	no	—
284	C₂	C₁₃×#535	1	1	yes	characteristic
285	C₂	C₂×C₁₃×#135	1	1	yes	characteristic
286	C₂	C₁₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#447	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#529	C₂	semidirect
2. C₂×C₁₃×#124	C₂	semidirect
3. C₁₃×#495	C₂	semidirect
4. C₁₃×#488	C₂	semidirect
5. C₁₃×#464	C₂	semidirect
6. C₁₃×#463	C₂	semidirect
7. C₁₃×#451	C₂	semidirect
8. C₁₃×#450	C₂	semidirect
9. C₁₃×#448	C₂	semidirect
11. C₁₃×#431	C₂	semidirect
14. C₂×C₁₃×Q₁₆	C₂²	semidirect
16. C₂×C₁₃×SD₁₆	C₂²	semidirect
19. C₁₃×#125	C₂²	semidirect
24. C₁₃×#124	C₂²	semidirect
28. C₄×Q₈×C₁₃	C₂²	semidirect
30. C₄×D₄×C₁₃	C₂²	semidirect
35. C₁₃×C₄⋊C₈	C₂²	semidirect
36. C₁₃×#103	C₂²	semidirect
37. C₁₃×#103	C₂²	semidirect
42. C₄×C₈×C₁₃	C₂²	semidirect
84. #3211	C₁₃	direct (N × H)
117. #529	C₂×C₁₃	semidirect
118. C₂×#124	C₂×C₁₃	semidirect
119. #495	C₂×C₁₃	semidirect
120. #488	C₂×C₁₃	semidirect
121. #464	C₂×C₁₃	semidirect
122. #463	C₂×C₁₃	semidirect
123. #451	C₂×C₁₃	semidirect
124. #450	C₂×C₁₃	semidirect
125. #448	C₂×C₁₃	semidirect
127. #431	C₂×C₁₃	semidirect
151. C₂×Q₁₆	C₂²×C₁₃	semidirect
153. C₂×SD₁₆	C₂²×C₁₃	semidirect
156. #125	C₂²×C₁₃	semidirect
161. #124	C₂²×C₁₃	semidirect
165. C₄×Q₈	C₂²×C₁₃	semidirect
167. C₄×D₄	C₂²×C₁₃	semidirect
172. C₄⋊C₈	C₂²×C₁₃	semidirect
173. #103	C₂²×C₁₃	semidirect
174. #103	C₂²×C₁₃	semidirect
179. C₄×C₈	C₂²×C₁₃	semidirect
227. C₁₃	#3211	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×#529	2
C₂×C₁₃×#124	2
C₁₃×#495	2
C₁₃×#488	2
C₁₃×#464	2
C₁₃×#463	2
C₁₃×#451	2
C₁₃×#450	2
C₁₃×#448	2
C₁₃×#447	2
C₁₃×#431	2
#3211	13

Subgroups — C13×#3211 #13112269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3211 #13112269