Subgroups — C₁₃×#3325 #13112383

Order 1664.

460 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#555	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#482	C₂	1	2	yes	maximal, complemented
5	C₁₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#469	C₂	1	2	yes	maximal, complemented
7	C₁₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#447	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#418	C₂	1	2	yes	maximal, complemented
12	C₂²×D₄×C₁₃	—	2	1	no	—
13	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
14	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×D₈	C₂²	1	2	yes	complemented
16	C₂×C₁₃×M₁₆	C₂²	1	2	yes	complemented
17	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
18	C₁₃×#127	C₂²	1	1	yes	characteristic
19	C₁₃×#124	—	2	2	no	—
20	C₁₃×#121	—	2	1	no	—
21	C₁₃×#121	—	2	1	no	—
22	C₁₃×#121	C₂²	1	2	yes	—
23	C₁₃×#121	C₂²	1	2	yes	complemented
24	C₁₃×#120	—	2	2	no	—
25	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic, complemented
28	C₁₃×#117	—	2	1	no	—
29	C₁₃×#117	C₂²	1	2	yes	complemented
30	C₁₃×#107	C₂²	1	1	yes	characteristic, complemented
31	C₁₃×#106	C₂²	1	2	yes	complemented
32	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
33	C₁₃×#102	C₂²	1	1	yes	characteristic
34	C₁₃×#102	C₂²	1	2	yes	—
35	C₁₃×#102	C₂²	1	2	yes	complemented
36	C₁₃×#102	C₂²	1	2	yes	complemented
37	C₁₃×#102	C₂²	1	1	yes	characteristic, complemented
38	C₁₃×#102	C₂²	1	2	yes	complemented
39	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
40	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
41	C₂⁴×C₁₃	—	2	2	no	—
42	C₂×Q₈×C₁₃	—	2	1	no	—
43	C₂×D₄×C₁₃	—	2	4	no	—
44	C₂×D₄×C₁₃	—	2	2	no	—
45	C₂×D₄×C₁₃	—	2	2	no	—
46	C₂×D₄×C₁₃	—	2	2	no	—
47	C₂×D₄×C₁₃	—	2	1	no	—
48	C₂×D₄×C₁₃	—	2	1	no	—
49	C₂×D₄×C₁₃	—	4	2	no	—
50	C₂×D₄×C₁₃	—	4	4	no	—
51	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
52	C₂×D₄×C₁₃	C₂³	1	2	yes	—
53	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	—	2	1	no	—
56	C₂²×C₄×C₁₃	—	2	1	no	—
57	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
58	C₁₃×D₈	—	2	2	no	—
59	C₁₃×D₈	—	4	2	no	—
60	C₁₃×M₁₆	—	4	2	no	—
61	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂³	1	2	yes	—
64	C₁₃×C₄⋊C₄	—	2	1	no	—
65	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
66	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
67	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₁₃×C₂²⋊C₄	—	2	2	no	—
70	C₁₃×C₂²⋊C₄	—	2	2	no	—
71	C₁₃×C₂²⋊C₄	—	2	1	no	—
72	C₁₃×C₂²⋊C₄	—	2	1	no	—
73	C₁₃×C₂²⋊C₄	—	2	2	no	—
74	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
75	C₄²×C₁₃	—	2	2	no	—
76	C₄²×C₁₃	C₂³	1	1	yes	characteristic
77	#3325	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₁₃	—	2	1	no	—
79	C₂³×C₁₃	—	2	1	no	—
80	C₂³×C₁₃	—	2	2	no	—
81	C₂³×C₁₃	—	4	4	no	—
82	C₂³×C₁₃	—	4	4	no	—
83	C₂³×C₁₃	—	8	2	no	—
84	C₂³×C₁₃	C₂×D₄	1	2	yes	—
85	Q₈×C₁₃	—	8	1	no	—
86	D₄×C₁₃	—	2	2	no	—
87	D₄×C₁₃	—	2	1	no	—
88	D₄×C₁₃	—	2	2	no	—
89	D₄×C₁₃	—	2	1	no	—
90	D₄×C₁₃	—	2	1	no	—
91	D₄×C₁₃	—	2	2	no	—
92	D₄×C₁₃	—	4	4	no	—
93	D₄×C₁₃	—	4	4	no	—
94	D₄×C₁₃	—	4	2	no	—
95	D₄×C₁₃	—	8	1	no	—
96	D₄×C₁₃	—	8	2	no	—
97	D₄×C₁₃	#41	1	2	yes	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
109	C₈×C₁₃	—	2	1	no	—
110	C₈×C₁₃	—	2	1	no	—
111	C₈×C₁₃	—	2	2	no	—
112	#555	C₂×C₁₃	1	1	yes	characteristic, complemented
113	D₄²	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#486	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#482	C₂×C₁₃	1	2	yes	complemented
116	#476	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#469	C₂×C₁₃	1	2	yes	complemented
118	#459	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#447	C₂×C₁₃	1	2	yes	complemented
120	C₄×D₈	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#436	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#418	C₂×C₁₃	1	2	yes	complemented
123	C₂²×C₁₃	—	2	2	no	—
124	C₂²×C₁₃	—	2	2	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	—	4	1	no	—
128	C₂²×C₁₃	—	4	1	no	—
129	C₂²×C₁₃	—	4	2	no	—
130	C₂²×C₁₃	—	4	1	no	—
131	C₂²×C₁₃	—	4	1	no	—
132	C₂²×C₁₃	—	4	2	no	—
133	C₂²×C₁₃	—	4	1	no	—
134	C₂²×C₁₃	—	4	2	no	—
135	C₂²×C₁₃	—	8	8	no	—
136	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
137	C₄×C₁₃	—	2	2	no	—
138	C₄×C₁₃	—	2	1	no	—
139	C₄×C₁₃	—	4	1	no	—
140	C₄×C₁₃	—	4	1	no	—
141	C₄×C₁₃	—	4	2	no	—
142	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
143	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
144	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂²×D₄	—	2	1	no	—
146	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
147	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
148	C₂×D₈	C₂²×C₁₃	1	2	yes	complemented
149	C₂×M₁₆	C₂²×C₁₃	1	2	yes	complemented
150	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
151	#127	C₂²×C₁₃	1	1	yes	characteristic
152	#124	—	2	2	no	—
153	#121	—	2	1	no	—
154	#121	—	2	1	no	—
155	#121	C₂²×C₁₃	1	2	yes	—
156	#121	C₂²×C₁₃	1	2	yes	complemented
157	#120	—	2	2	no	—
158	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
159	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
160	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic, complemented
161	#117	—	2	1	no	—
162	#117	C₂²×C₁₃	1	2	yes	complemented
163	#107	C₂²×C₁₃	1	1	yes	characteristic, complemented
164	#106	C₂²×C₁₃	1	2	yes	complemented
165	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
166	#102	C₂²×C₁₃	1	1	yes	characteristic
167	#102	C₂²×C₁₃	1	2	yes	—
168	#102	C₂²×C₁₃	1	2	yes	complemented
169	#102	C₂²×C₁₃	1	2	yes	complemented
170	#102	C₂²×C₁₃	1	1	yes	characteristic, complemented
171	#102	C₂²×C₁₃	1	2	yes	complemented
172	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	complemented
173	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
174	C₂×C₁₃	—	4	2	no	—
175	C₂×C₁₃	—	4	2	no	—
176	C₂×C₁₃	—	8	1	no	—
177	C₂×C₁₃	—	8	2	no	—
178	C₂×C₁₃	#546	1	1	yes	characteristic
179	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₂×C₁₃	#576	1	1	yes	characteristic
181	C₂⁴	—	2	2	no	—
182	C₂×Q₈	—	2	1	no	—
183	C₂×D₄	—	2	4	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	4	4	no	—
191	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
192	C₂×D₄	C₂³×C₁₃	1	2	yes	—
193	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
194	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
198	D₈	—	2	2	no	—
199	D₈	—	4	2	no	—
200	M₁₆	—	4	2	no	—
201	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
202	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
203	C₂×C₈	C₂³×C₁₃	1	2	yes	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
206	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
207	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
208	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
209	C₂²⋊C₄	—	2	2	no	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
215	C₄²	—	2	2	no	—
216	C₄²	C₂³×C₁₃	1	1	yes	characteristic
217	C₁₃	#3325	1	1	yes	characteristic, direct product factor, Sylow
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	4	4	no	—
222	C₂³	—	4	4	no	—
223	C₂³	—	8	2	no	—
224	C₂³	C₂×D₄×C₁₃	1	2	yes	—
225	Q₈	—	8	1	no	—
226	D₄	—	2	2	no	—
227	D₄	—	2	1	no	—
228	D₄	—	2	2	no	—
229	D₄	—	2	1	no	—
230	D₄	—	2	1	no	—
231	D₄	—	2	2	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	4	no	—
234	D₄	—	4	2	no	—
235	D₄	—	8	1	no	—
236	D₄	—	8	2	no	—
237	D₄	C₁₃×#41	1	2	yes	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
246	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
247	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
248	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	8	8	no	—
265	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	4	1	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	2	no	—
271	C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
272	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
273	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
274	C₂	—	4	2	no	—
275	C₂	—	4	2	no	—
276	C₂	—	8	2	no	—
277	C₂	—	8	1	no	—
278	C₂	C₁₃×#546	1	1	yes	characteristic
279	C₂	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
280	C₂	C₁₃×#576	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₁₃×#555	C₂	semidirect
2. D₄²×C₁₃	C₂	semidirect
3. C₁₃×#486	C₂	semidirect
4. C₁₃×#482	C₂	semidirect
5. C₁₃×#476	C₂	semidirect
6. C₁₃×#469	C₂	semidirect
7. C₁₃×#459	C₂	semidirect
8. C₁₃×#447	C₂	semidirect
9. C₄×C₁₃×D₈	C₂	semidirect
10. C₁₃×#436	C₂	semidirect
11. C₁₃×#418	C₂	semidirect
15. C₂×C₁₃×D₈	C₂²	semidirect
16. C₂×C₁₃×M₁₆	C₂²	semidirect
17. C₁₃×C₄⋊Q₈	C₂²	semidirect
23. C₁₃×#121	C₂²	semidirect
27. C₄×D₄×C₁₃	C₂²	semidirect
29. C₁₃×#117	C₂²	semidirect
30. C₁₃×#107	C₂²	semidirect
31. C₁₃×#106	C₂²	semidirect
32. C₁₃×C₄⋊C₈	C₂²	semidirect
35. C₁₃×#102	C₂²	semidirect
36. C₁₃×#102	C₂²	semidirect
37. C₁₃×#102	C₂²	semidirect
38. C₁₃×#102	C₂²	semidirect
39. C₁₃×C₂²⋊C₈	C₂²	semidirect
40. C₄×C₈×C₁₃	C₂²	semidirect
77. #3325	C₁₃	direct (N × H)
112. #555	C₂×C₁₃	semidirect
113. D₄²	C₂×C₁₃	semidirect
114. #486	C₂×C₁₃	semidirect
115. #482	C₂×C₁₃	semidirect
116. #476	C₂×C₁₃	semidirect
117. #469	C₂×C₁₃	semidirect
118. #459	C₂×C₁₃	semidirect
119. #447	C₂×C₁₃	semidirect
120. C₄×D₈	C₂×C₁₃	semidirect
121. #436	C₂×C₁₃	semidirect
122. #418	C₂×C₁₃	semidirect
148. C₂×D₈	C₂²×C₁₃	semidirect
149. C₂×M₁₆	C₂²×C₁₃	semidirect
150. C₄⋊Q₈	C₂²×C₁₃	semidirect
156. #121	C₂²×C₁₃	semidirect
160. C₄×D₄	C₂²×C₁₃	semidirect
162. #117	C₂²×C₁₃	semidirect
163. #107	C₂²×C₁₃	semidirect
164. #106	C₂²×C₁₃	semidirect
165. C₄⋊C₈	C₂²×C₁₃	semidirect
168. #102	C₂²×C₁₃	semidirect
169. #102	C₂²×C₁₃	semidirect
170. #102	C₂²×C₁₃	semidirect
171. #102	C₂²×C₁₃	semidirect
172. C₂²⋊C₈	C₂²×C₁₃	semidirect
173. C₄×C₈	C₂²×C₁₃	semidirect
217. C₁₃	#3325	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×#555	2
D₄²×C₁₃	2
C₁₃×#486	2
C₁₃×#482	2
C₁₃×#476	2
C₁₃×#469	2
C₁₃×#459	2
C₁₃×#447	2
C₄×C₁₃×D₈	2
C₁₃×#436	2
C₁₃×#418	2
#3325	13

Subgroups — C13×#3325 #13112383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3325 #13112383