Subgroups — C₁₃×#3062 #13112120

Order 1664.

504 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#537	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#535	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#469	C₂	1	2	yes	maximal, complemented
6	C₁₃×#468	C₂	1	2	yes	maximal, complemented
7	C₁₃×#466	C₂	1	2	yes	maximal, complemented
8	C₁₃×#465	C₂	1	2	yes	maximal, complemented
9	C₁₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#416	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₁₃×#102	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×AGL(1,ℤ/8)	—	2	4	no	—
16	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
17	C₂×C₁₃×D₈	C₂²	1	2	yes	—
18	C₁₃×#131	—	2	4	no	—
19	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×M₁₆	C₂²	1	2	yes	complemented
21	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂²	1	2	yes	complemented
23	C₁₃×#127	—	2	1	no	—
24	C₁₃×#124	—	2	1	no	—
25	C₁₃×C₂²⋊Q₈	—	2	1	no	—
26	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
27	C₁₃×#121	—	2	1	no	—
28	C₁₃×#121	—	2	2	no	—
29	C₁₃×#121	C₂²	1	2	yes	—
30	C₁₃×#121	C₂²	1	2	yes	—
31	C₁₃×#121	C₂²	1	2	yes	complemented
32	C₁₃×#120	—	2	2	no	—
33	C₁₃×#117	C₂²	1	1	yes	characteristic
34	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
35	C₁₃×#107	C₂²	1	2	yes	complemented
36	C₁₃×#106	C₂²	1	2	yes	complemented
37	C₁₃×#103	C₂²	1	1	yes	characteristic, complemented
38	C₁₃×#103	C₂²	1	1	yes	characteristic, complemented
39	C₁₃×#102	C₂²	1	2	yes	—
40	C₁₃×#102	C₂²	1	2	yes	—
41	C₁₃×#102	C₂²	1	1	yes	characteristic, complemented
42	C₁₃×#102	C₂²	1	1	yes	characteristic, complemented
43	C₂⁴×C₁₃	—	2	1	no	—
44	C₁₃×#41	—	2	2	no	—
45	C₁₃×#41	—	2	2	no	—
46	C₁₃×#41	—	2	2	no	—
47	C₁₃×#41	—	2	2	no	—
48	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
49	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₁₃	—	2	1	no	—
51	C₂×D₄×C₁₃	—	2	2	no	—
52	C₂×D₄×C₁₃	—	2	2	no	—
53	C₂×D₄×C₁₃	—	2	2	no	—
54	C₂×D₄×C₁₃	—	2	2	no	—
55	C₂×D₄×C₁₃	—	2	1	no	—
56	C₂×D₄×C₁₃	—	2	2	no	—
57	C₂×D₄×C₁₃	D₄	1	2	yes	—
58	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
59	C₂×D₄×C₁₃	C₂³	1	2	yes	—
60	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	—	2	1	no	—
63	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
65	C₁₃×SD₁₆	—	2	4	no	—
66	C₁₃×D₈	—	2	4	no	—
67	C₁₃×M₁₆	—	2	4	no	—
68	C₁₃×M₁₆	D₄	1	4	yes	complemented
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	4	no	—
71	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂³	1	2	yes	—
74	C₁₃×C₄⋊C₄	—	2	1	no	—
75	C₁₃×C₄⋊C₄	—	2	2	no	—
76	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
77	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
78	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
79	C₁₃×C₂²⋊C₄	—	2	1	no	—
80	C₁₃×C₂²⋊C₄	—	2	2	no	—
81	C₁₃×C₂²⋊C₄	—	2	2	no	—
82	C₁₃×C₂²⋊C₄	—	2	2	no	—
83	C₁₃×C₂²⋊C₄	—	2	2	no	—
84	C₄²×C₁₃	—	2	1	no	—
85	#3062	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₁₃	—	2	1	no	—
87	C₂³×C₁₃	—	2	2	no	—
88	C₂³×C₁₃	—	4	2	no	—
89	C₂³×C₁₃	—	8	1	no	—
90	C₂³×C₁₃	—	8	1	no	—
91	C₂³×C₁₃	C₂×D₄	1	2	yes	—
92	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
93	Q₈×C₁₃	—	2	2	no	—
94	Q₈×C₁₃	—	2	1	no	—
95	Q₈×C₁₃	—	2	1	no	—
96	D₄×C₁₃	—	2	4	no	—
97	D₄×C₁₃	—	2	2	no	—
98	D₄×C₁₃	—	2	1	no	—
99	D₄×C₁₃	—	2	2	no	—
100	D₄×C₁₃	—	2	2	no	—
101	D₄×C₁₃	—	2	1	no	—
102	D₄×C₁₃	—	4	2	no	—
103	D₄×C₁₃	—	4	2	no	—
104	D₄×C₁₃	—	8	2	no	—
105	D₄×C₁₃	—	8	2	no	—
106	D₄×C₁₃	—	8	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	—	8	1	no	—
115	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₁₃	#41	1	2	yes	—
119	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
120	C₈×C₁₃	—	2	1	no	—
121	C₈×C₁₃	—	2	1	no	—
122	C₈×C₁₃	C₂×D₄	1	4	yes	—
123	C₂×AGL(1,ℤ/8)	C₂×C₁₃	1	1	yes	characteristic, complemented
124	C₂×#131	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#537	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#535	C₂×C₁₃	1	1	yes	characteristic, complemented
127	#469	C₂×C₁₃	1	2	yes	complemented
128	#468	C₂×C₁₃	1	2	yes	complemented
129	#466	C₂×C₁₃	1	2	yes	complemented
130	#465	C₂×C₁₃	1	2	yes	complemented
131	#428	C₂×C₁₃	1	1	yes	characteristic, complemented
132	#416	C₂×C₁₃	1	1	yes	characteristic, complemented
133	C₂×#102	C₂×C₁₃	1	1	yes	characteristic, complemented
134	C₂²×C₁₃	—	2	1	no	—
135	C₂²×C₁₃	—	2	1	no	—
136	C₂²×C₁₃	—	2	2	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	—	2	2	no	—
139	C₂²×C₁₃	—	4	1	no	—
140	C₂²×C₁₃	—	4	2	no	—
141	C₂²×C₁₃	—	4	1	no	—
142	C₂²×C₁₃	—	4	2	no	—
143	C₂²×C₁₃	—	4	2	no	—
144	C₂²×C₁₃	—	4	1	no	—
145	C₂²×C₁₃	—	8	4	no	—
146	C₂²×C₁₃	#121	1	2	yes	—
147	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	—	4	1	no	—
150	C₄×C₁₃	—	4	2	no	—
151	C₄×C₁₃	—	4	1	no	—
152	C₄×C₁₃	—	4	1	no	—
153	C₄×C₁₃	#121	1	2	yes	—
154	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
155	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
156	C₂×#41	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×#41	C₂²×C₁₃	1	1	yes	characteristic
158	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
159	AGL(1,ℤ/8)	—	2	4	no	—
160	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
161	C₂×D₈	C₂²×C₁₃	1	2	yes	—
162	#131	—	2	4	no	—
163	C₂×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
164	C₂×M₁₆	C₂²×C₁₃	1	2	yes	complemented
165	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
166	C₂²×C₈	C₂²×C₁₃	1	2	yes	complemented
167	#127	—	2	1	no	—
168	#124	—	2	1	no	—
169	C₂²⋊Q₈	—	2	1	no	—
170	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	complemented
171	#121	—	2	1	no	—
172	#121	—	2	2	no	—
173	#121	C₂²×C₁₃	1	2	yes	—
174	#121	C₂²×C₁₃	1	2	yes	—
175	#121	C₂²×C₁₃	1	2	yes	complemented
176	#120	—	2	2	no	—
177	#117	C₂²×C₁₃	1	1	yes	characteristic
178	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
179	#107	C₂²×C₁₃	1	2	yes	complemented
180	#106	C₂²×C₁₃	1	2	yes	complemented
181	#103	C₂²×C₁₃	1	1	yes	characteristic, complemented
182	#103	C₂²×C₁₃	1	1	yes	characteristic, complemented
183	#102	C₂²×C₁₃	1	2	yes	—
184	#102	C₂²×C₁₃	1	2	yes	—
185	#102	C₂²×C₁₃	1	1	yes	characteristic, complemented
186	#102	C₂²×C₁₃	1	1	yes	characteristic, complemented
187	C₂×C₁₃	—	2	2	no	—
188	C₂×C₁₃	—	4	2	no	—
189	C₂×C₁₃	—	8	1	no	—
190	C₂×C₁₃	—	8	2	no	—
191	C₂×C₁₃	C₂×#121	1	1	yes	characteristic
192	C₂×C₁₃	#576	1	1	yes	characteristic
193	C₂×C₁₃	#577	1	1	yes	characteristic
194	C₂⁴	—	2	1	no	—
195	#41	—	2	2	no	—
196	#41	—	2	2	no	—
197	#41	—	2	2	no	—
198	#41	—	2	2	no	—
199	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
200	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	1	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	D₄×C₁₃	1	2	yes	—
209	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
210	C₂×D₄	C₂³×C₁₃	1	2	yes	—
211	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
215	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
216	SD₁₆	—	2	4	no	—
217	D₈	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	D₄×C₁₃	1	4	yes	complemented
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
223	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
224	C₂×C₈	C₂³×C₁₃	1	2	yes	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
228	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
229	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
230	C₂²⋊C₄	—	2	1	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	—	2	2	no	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₄²	—	2	1	no	—
236	C₁₃	#3062	1	1	yes	characteristic, direct product factor, Sylow
237	C₂³	—	2	1	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	4	2	no	—
240	C₂³	—	8	1	no	—
241	C₂³	—	8	1	no	—
242	C₂³	C₂×D₄×C₁₃	1	2	yes	—
243	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
244	Q₈	—	2	2	no	—
245	Q₈	—	2	1	no	—
246	Q₈	—	2	1	no	—
247	D₄	—	2	4	no	—
248	D₄	—	2	2	no	—
249	D₄	—	2	1	no	—
250	D₄	—	2	2	no	—
251	D₄	—	2	2	no	—
252	D₄	—	2	1	no	—
253	D₄	—	4	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	8	2	no	—
256	D₄	—	8	2	no	—
257	D₄	—	8	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	8	1	no	—
266	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
267	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
269	C₂×C₄	C₁₃×#41	1	2	yes	—
270	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	C₂×D₄×C₁₃	1	4	yes	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	8	4	no	—
286	C₂²	C₁₃×#121	1	2	yes	—
287	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	2	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	C₁₃×#121	1	2	yes	—
294	C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
295	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
296	C₂	—	2	2	no	—
297	C₂	—	4	2	no	—
298	C₂	—	8	2	no	—
299	C₂	—	8	1	no	—
300	C₂	C₂×C₁₃×#121	1	1	yes	characteristic
301	C₂	C₁₃×#576	1	1	yes	characteristic
302	C₂	C₁₃×#577	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 ◁ C₂×C₁₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×AGL(1,ℤ/8)	C₂	semidirect
2. C₂×C₁₃×#131	C₂	semidirect
3. C₁₃×#537	C₂	semidirect
4. C₁₃×#535	C₂	semidirect
5. C₁₃×#469	C₂	semidirect
6. C₁₃×#468	C₂	semidirect
7. C₁₃×#466	C₂	semidirect
8. C₁₃×#465	C₂	semidirect
9. C₁₃×#428	C₂	semidirect
10. C₁₃×#416	C₂	semidirect
11. C₂×C₁₃×#102	C₂	semidirect
20. C₂×C₁₃×M₁₆	C₂²	semidirect
22. C₂²×C₈×C₁₃	C₂²	semidirect
26. C₁₃×C₂²⋊Q₈	C₂²	semidirect
31. C₁₃×#121	C₂²	semidirect
35. C₁₃×#107	C₂²	semidirect
36. C₁₃×#106	C₂²	semidirect
37. C₁₃×#103	C₂²	semidirect
38. C₁₃×#103	C₂²	semidirect
41. C₁₃×#102	C₂²	semidirect
42. C₁₃×#102	C₂²	semidirect
68. C₁₃×M₁₆	D₄	semidirect
85. #3062	C₁₃	direct (N × H)
123. C₂×AGL(1,ℤ/8)	C₂×C₁₃	semidirect
124. C₂×#131	C₂×C₁₃	semidirect
125. #537	C₂×C₁₃	semidirect
126. #535	C₂×C₁₃	semidirect
127. #469	C₂×C₁₃	semidirect
128. #468	C₂×C₁₃	semidirect
129. #466	C₂×C₁₃	semidirect
130. #465	C₂×C₁₃	semidirect
131. #428	C₂×C₁₃	semidirect
132. #416	C₂×C₁₃	semidirect
133. C₂×#102	C₂×C₁₃	semidirect
164. C₂×M₁₆	C₂²×C₁₃	semidirect
166. C₂²×C₈	C₂²×C₁₃	semidirect
170. C₂²⋊Q₈	C₂²×C₁₃	semidirect
175. #121	C₂²×C₁₃	semidirect
179. #107	C₂²×C₁₃	semidirect
180. #106	C₂²×C₁₃	semidirect
181. #103	C₂²×C₁₃	semidirect
182. #103	C₂²×C₁₃	semidirect
185. #102	C₂²×C₁₃	semidirect
186. #102	C₂²×C₁₃	semidirect
219. M₁₆	D₄×C₁₃	semidirect
236. C₁₃	#3062	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×AGL(1,ℤ/8)	2
C₂×C₁₃×#131	2
C₁₃×#537	2
C₁₃×#535	2
C₁₃×#469	2
C₁₃×#468	2
C₁₃×#466	2
C₁₃×#465	2
C₁₃×#428	2
C₁₃×#416	2
C₂×C₁₃×#102	2
#3062	13

Subgroups — C13×#3062 #13112120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3062 #13112120