Subgroups — #13129604

Order 1664.

1424 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	#11719650	C₂	1	2	yes	maximal, complemented
2	C₁₃×#576	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719541	C₂	1	2	yes	maximal, complemented
4	#11719480	C₂	1	6	yes	maximal, complemented
5	#11719478	C₂	1	6	yes	maximal, complemented
6	#11719476	C₂	1	3	yes	maximal, complemented
7	D₁₃×#135	C₂	1	3	yes	maximal, complemented
8	#11719462	C₂	1	3	yes	maximal, complemented
9	C₂×#88166	C₂	1	3	yes	maximal, complemented
10	#11719458	C₂	1	1	yes	maximal, characteristic, complemented
11	D₁₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×H(5,𝔽₂)	C₂²	1	2	yes	complemented
13	#88257	—	2	2	no	—
14	#88257	C₂²	1	6	yes	complemented
15	#88257	C₂²	1	2	yes	complemented
16	#41×D₁₃	—	2	2	no	—
17	#41×D₁₃	C₂²	1	1	yes	characteristic
18	#41×D₁₃	C₂²	1	6	yes	complemented
19	#88251	—	2	6	no	—
20	#88251	C₂²	1	6	yes	complemented
21	C₂×#6129	—	2	6	no	—
22	C₂×#6129	C₂²	1	6	yes	complemented
23	C₁₃×AGL(1,ℤ/8)	C₂²	1	6	yes	complemented
24	C₁₃×#135	C₂²	1	3	yes	complemented
25	C₂×C₁₃×D₈	C₂²	1	3	yes	complemented
26	C₁₃×#131	C₂²	1	1	yes	characteristic, complemented
27	#88205	C₂²	1	6	yes	complemented
28	#88204	C₂²	1	6	yes	complemented
29	#88202	C₂²	1	1	yes	characteristic, complemented
30	C₂×#6105	C₂²	1	6	yes	complemented
31	#88186	C₂²	1	6	yes	complemented
32	#88173	C₂²	1	1	yes	characteristic, complemented
33	Q₁₆×D₁₃	C₂²	1	3	yes	complemented
34	#88170	C₂²	1	6	yes	complemented
35	#88169	C₂²	1	12	yes	complemented
36	SD₁₆×D₁₃	C₂²	1	6	yes	complemented
37	#88166	C₂²	1	3	yes	complemented
38	#88166	C₂²	1	6	yes	complemented
39	#88166	C₂²	1	6	yes	complemented
40	#88166	C₂²	1	6	yes	complemented
41	#88165	C₂²	1	12	yes	complemented
42	D₈×D₁₃	C₂²	1	3	yes	complemented
43	#88163	C₂²	1	6	yes	complemented
44	#88161	C₂²	1	3	yes	complemented
45	M₁₆×D₁₃	C₂²	1	3	yes	complemented
46	C₂×Dic₅₂	C₂²	1	3	yes	complemented
47	#88158	C₂²	1	3	yes	complemented
48	#88155	C₂²	1	3	yes	complemented
49	C₂×C₈×D₁₃	C₂²	1	3	yes	complemented
50	C₁₃×#41	—	2	2	no	—
51	C₁₃×#41	C₂³	1	6	yes	—
52	C₁₃×#41	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₁₃	—	2	6	no	—
54	C₂×D₄×C₁₃	C₂³	1	6	yes	—
55	C₂×C₂²⋊D₁₃	—	2	6	no	—
56	C₂×C₂²⋊D₁₃	—	2	12	no	—
57	C₂²×Dic₁₃	—	2	6	no	—
58	#6131	D₄	1	1	yes	characteristic
59	#6131	C₂³	1	2	yes	—
60	Q₈×D₁₃	—	2	2	no	—
61	Q₈×D₁₃	D₄	1	1	yes	characteristic
62	Q₈×D₁₃	C₂³	1	6	yes	—
63	#6129	—	2	6	no	—
64	#6129	—	2	12	no	—
65	#6129	—	2	6	no	—
66	#6129	—	2	6	no	—
67	#6129	—	2	6	no	—
68	#6129	—	2	6	no	—
69	#6129	D₄	1	3	yes	—
70	#6129	C₂³	1	12	yes	—
71	#6129	C₂³	1	6	yes	—
72	D₄×D₁₃	—	2	6	no	—
73	D₄×D₁₃	—	2	6	no	—
74	D₄×D₁₃	D₄	1	3	yes	—
75	D₄×D₁₃	C₂³	1	6	yes	—
76	#6127	—	2	6	no	—
77	#6127	—	2	6	no	—
78	#6127	C₂³	1	3	yes	—
79	#6127	C₂³	1	6	yes	—
80	C₂×C₄×D₁₃	—	2	6	no	—
81	C₂×C₄×D₁₃	C₂³	1	3	yes	—
82	C₂×Dic₂₆	—	2	6	no	—
83	C₂×Dic₂₆	C₂³	1	6	yes	—
84	C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
85	C₁₃×SD₁₆	C₂³	1	6	yes	—
86	C₁₃×D₈	C₂³	1	3	yes	—
87	C₁₃×D₈	C₂³	1	6	yes	—
88	C₁₃×M₁₆	C₂³	1	3	yes	—
89	C₂×C₈×C₁₃	C₂³	1	3	yes	—
90	#6107	C₂³	1	6	yes	—
91	#6106	C₂³	1	2	yes	—
92	#6105	C₂³	1	12	yes	—
93	#6105	C₂³	1	6	yes	—
94	#6104	C₂³	1	6	yes	—
95	#6099	C₂³	1	3	yes	—
96	C₂×#1055	C₂³	1	3	yes	—
97	Dic₅₂	C₂³	1	3	yes	—
98	Dic₅₂	C₂³	1	6	yes	—
99	D₁₀₄	C₂³	1	1	yes	characteristic
100	#6095	C₂³	1	6	yes	—
101	#6094	C₂³	1	6	yes	—
102	C₈×D₁₃	C₂³	1	1	yes	characteristic
103	C₈×D₁₃	C₂³	1	3	yes	—
104	C₈×D₁₃	C₂³	1	3	yes	—
105	C₈×D₁₃	C₂³	1	3	yes	—
106	#3584	—	13	1	no	maximal, Sylow
107	C₂³×C₁₃	—	2	6	no	—
108	C₂²×D₁₃	—	2	6	no	—
109	C₂²×D₁₃	C₂×D₄	1	3	yes	—
110	Q₈×C₁₃	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₁₃	C₂⁴	1	2	yes	—
112	D₄×C₁₃	—	2	6	no	—
113	D₄×C₁₃	—	2	6	no	—
114	D₄×C₁₃	C₂×D₄	1	3	yes	—
115	D₄×C₁₃	C₂⁴	1	6	yes	—
116	C₂×C₄×C₁₃	—	2	6	no	—
117	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
118	C₂²⋊D₁₃	—	2	6	no	—
119	C₂²⋊D₁₃	—	2	12	no	—
120	C₂²⋊D₁₃	—	2	6	no	—
121	C₂²⋊D₁₃	—	2	6	no	—
122	C₂²⋊D₁₃	C₂×D₄	1	6	yes	—
123	C₂×Dic₁₃	—	2	6	no	—
124	C₂×Dic₁₃	—	2	12	no	—
125	C₂×Dic₁₃	—	2	6	no	—
126	C₂×Dic₁₃	C₂×D₄	1	3	yes	—
127	D₅₂	—	2	2	no	—
128	D₅₂	C₂×D₄	1	3	yes	—
129	D₅₂	C₂⁴	1	2	yes	—
130	C₄×D₁₃	—	2	6	no	—
131	C₄×D₁₃	—	2	6	no	—
132	C₄×D₁₃	—	2	2	no	—
133	C₄×D₁₃	C₂×D₄	1	3	yes	—
134	C₄×D₁₃	C₂×D₄	1	3	yes	—
135	C₄×D₁₃	C₂⁴	1	1	yes	characteristic
136	C₄×D₁₃	C₂⁴	1	3	yes	—
137	Dic₂₆	—	2	6	no	—
138	Dic₂₆	—	2	6	no	—
139	Dic₂₆	C₂×D₄	1	3	yes	—
140	Dic₂₆	C₂⁴	1	6	yes	—
141	C₈×C₁₃	C₂⁴	1	1	yes	characteristic
142	C₈×C₁₃	C₂⁴	1	3	yes	—
143	#1055	C₂⁴	1	1	yes	characteristic
144	#1055	C₂⁴	1	3	yes	—
145	#585	—	13	2	no	—
146	#578	—	13	3	no	—
147	#578	—	13	1	no	—
148	#577	—	13	2	no	—
149	#577	—	13	6	no	—
150	#576	—	13	3	no	—
151	#576	D₁₃	1	1	yes	characteristic, complemented
152	#575	—	13	6	no	—
153	C₂×#135	—	13	3	no	—
154	C₂×#135	—	13	3	no	—
155	C₂×#131	—	13	1	no	—
156	C₂²×C₁₃	—	2	6	no	—
157	C₂²×C₁₃	—	8	6	no	—
158	C₂²×C₁₃	C₂²×D₄	1	3	yes	—
159	C₂×D₁₃	—	2	2	no	—
160	C₂×D₁₃	—	4	3	no	—
161	C₂×D₁₃	—	8	6	no	—
162	C₂×D₁₃	C₂²×D₄	1	3	yes	—
163	C₂×D₁₃	C₂²×D₄	1	1	yes	characteristic
164	C₄×C₁₃	—	2	2	no	—
165	C₄×C₁₃	C₂²×D₄	1	3	yes	—
166	C₄×C₁₃	C₂⁵	1	1	yes	characteristic
167	Dic₁₃	—	2	6	no	—
168	Dic₁₃	C₂²×D₄	1	3	yes	—
169	Dic₁₃	C₂²×D₄	1	1	yes	characteristic
170	2₋¹⁺⁴	—	13	6	no	—
171	2₋¹⁺⁴	—	13	2	no	—
172	2₋¹⁺⁴	—	26	2	no	—
173	H(5,𝔽₂)	—	13	6	no	—
174	H(5,𝔽₂)	—	26	6	no	—
175	H(5,𝔽₂)	C₂×D₁₃	1	2	yes	complemented
176	C₂×#41	—	13	1	no	—
177	C₂×#41	—	13	6	no	—
178	C₂×#41	—	13	6	no	—
179	C₂×#41	—	26	6	no	—
180	C₂×#41	—	26	2	no	—
181	#137	—	13	6	no	—
182	#137	—	13	12	no	—
183	#137	—	13	6	no	—
184	AGL(1,ℤ/8)	—	13	6	no	—
185	AGL(1,ℤ/8)	—	13	12	no	—
186	AGL(1,ℤ/8)	C₂×D₁₃	1	6	yes	complemented
187	#135	—	13	6	no	—
188	#135	—	13	1	no	—
189	#135	—	13	6	no	—
190	#135	—	13	3	no	—
191	#135	—	13	6	no	—
192	#135	—	13	6	no	—
193	#135	—	13	6	no	—
194	#135	—	13	3	no	—
195	#135	C₂×D₁₃	1	3	yes	complemented
196	C₂×Q₁₆	—	13	3	no	—
197	C₂×Q₁₆	—	13	3	no	—
198	C₂×SD₁₆	—	13	6	no	—
199	C₂×SD₁₆	—	13	6	no	—
200	C₂×D₈	—	13	3	no	—
201	C₂×D₈	C₂×D₁₃	1	3	yes	complemented
202	#131	—	13	1	no	—
203	#131	—	13	3	no	—
204	#131	—	13	3	no	—
205	#131	C₂×D₁₃	1	1	yes	characteristic, complemented
206	C₂×M₁₆	—	13	3	no	—
207	C₂²×C₈	—	13	3	no	—
208	C₂×C₁₃	—	2	3	no	—
209	C₂×C₁₃	—	4	6	no	—
210	C₂×C₁₃	C₂³×D₄	1	1	yes	characteristic
211	D₁₃	—	2	1	no	—
212	D₁₃	—	2	3	no	—
213	D₁₃	—	4	2	no	—
214	#41	—	2	2	no	—
215	#41	—	13	1	no	—
216	#41	—	13	3	no	—
217	#41	—	13	3	no	—
218	#41	—	13	2	no	—
219	#41	—	13	6	no	—
220	#41	—	13	12	no	—
221	#41	—	13	6	no	—
222	#41	—	26	6	no	—
223	#41	—	26	6	no	—
224	#41	—	26	6	no	—
225	#41	—	26	6	no	—
226	#41	—	26	6	no	—
227	#41	—	26	12	no	—
228	#41	—	26	6	no	—
229	#41	—	26	6	no	—
230	#41	C₂²×D₁₃	1	6	yes	—
231	#41	C₂²×D₁₃	1	1	yes	characteristic
232	C₂×Q₈	—	13	1	no	—
233	C₂×Q₈	—	13	6	no	—
234	C₂×Q₈	—	13	6	no	—
235	C₂×Q₈	—	26	6	no	—
236	C₂×Q₈	—	26	2	no	—
237	C₂×D₄	—	2	6	no	—
238	C₂×D₄	—	13	3	no	—
239	C₂×D₄	—	13	6	no	—
240	C₂×D₄	—	26	6	no	—
241	C₂×D₄	—	26	6	no	—
242	C₂×D₄	—	26	12	no	—
243	C₂×D₄	—	26	6	no	—
244	C₂×D₄	C₂²×D₁₃	1	6	yes	—
245	C₂²×C₄	—	13	3	no	—
246	C₂²×C₄	—	26	6	no	—
247	C₂²×C₄	—	26	6	no	—
248	Q₁₆	—	13	6	no	—
249	Q₁₆	—	13	3	no	—
250	Q₁₆	—	13	6	no	—
251	Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
252	SD₁₆	—	13	2	no	—
253	SD₁₆	—	13	6	no	—
254	SD₁₆	—	13	12	no	—
255	SD₁₆	—	13	6	no	—
256	SD₁₆	C₂²×D₁₃	1	6	yes	—
257	D₈	—	13	6	no	—
258	D₈	—	13	1	no	—
259	D₈	C₂²×D₁₃	1	3	yes	—
260	D₈	C₂²×D₁₃	1	6	yes	—
261	M₁₆	—	13	3	no	—
262	M₁₆	—	13	6	no	—
263	M₁₆	C₂²×D₁₃	1	3	yes	—
264	C₂×C₈	—	13	3	no	—
265	C₂×C₈	—	13	1	no	—
266	C₂×C₈	—	13	3	no	—
267	C₂×C₈	—	13	3	no	—
268	C₂×C₈	—	13	3	no	—
269	C₂×C₈	C₂²×D₁₃	1	3	yes	—
270	C₁₃	#3584	1	1	yes	characteristic, complemented, Sylow
271	C₂³	—	2	6	no	—
272	C₂³	—	13	3	no	—
273	C₂³	—	26	6	no	—
274	Q₈	—	13	3	no	—
275	Q₈	—	13	6	no	—
276	Q₈	—	26	6	no	—
277	Q₈	—	26	6	no	—
278	Q₈	D₄×D₁₃	1	1	yes	characteristic
279	Q₈	C₂³×D₁₃	1	2	yes	—
280	D₄	—	2	6	no	—
281	D₄	—	2	6	no	—
282	D₄	—	13	6	no	—
283	D₄	—	13	3	no	—
284	D₄	—	13	2	no	—
285	D₄	—	26	2	no	—
286	D₄	—	26	6	no	—
287	D₄	—	26	6	no	—
288	D₄	—	26	12	no	—
289	D₄	—	26	6	no	—
290	D₄	D₄×D₁₃	1	3	yes	—
291	D₄	C₂³×D₁₃	1	6	yes	—
292	C₂×C₄	—	2	6	no	—
293	C₂×C₄	—	13	3	no	—
294	C₂×C₄	—	13	3	no	—
295	C₂×C₄	—	13	3	no	—
296	C₂×C₄	—	13	1	no	—
297	C₂×C₄	—	13	3	no	—
298	C₂×C₄	—	26	2	no	—
299	C₂×C₄	—	26	6	no	—
300	C₂×C₄	—	26	6	no	—
301	C₂×C₄	—	26	12	no	—
302	C₂×C₄	—	26	6	no	—
303	C₂×C₄	—	26	6	no	—
304	C₂×C₄	C₂³×D₁₃	1	3	yes	—
305	C₈	—	13	1	no	—
306	C₈	—	13	3	no	—
307	C₈	C₂³×D₁₃	1	1	yes	characteristic
308	C₈	C₂³×D₁₃	1	3	yes	—
309	C₂²	—	2	6	no	—
310	C₂²	—	8	6	no	—
311	C₂²	—	13	1	no	—
312	C₂²	—	13	3	no	—
313	C₂²	—	26	2	no	—
314	C₂²	—	52	3	no	—
315	C₂²	—	104	6	no	—
316	C₂²	C₂×D₄×D₁₃	1	3	yes	—
317	C₄	—	2	2	no	—
318	C₄	—	13	1	no	—
319	C₄	—	13	3	no	—
320	C₄	—	26	6	no	—
321	C₄	C₂×D₄×D₁₃	1	3	yes	—
322	C₄	C₂⁴×D₁₃	1	1	yes	characteristic
323	C₂	—	2	3	no	—
324	C₂	—	4	6	no	—
325	C₂	—	26	1	no	—
326	C₂	—	26	3	no	—
327	C₂	—	52	2	no	—
328	C₂	C₂²×D₄×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄×D₁₃
commutator, cosocle	C₄×C₁₃	C₂⁵
Frattini	C₄	C₂⁴×D₁₃
Fitting	C₁₃×#576	C₂
socle	C₂×C₁₃	C₂³×D₄
hypercenter	#576	D₁₃
hypocenter	C₁₃	#3584
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #576

Complements

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

N	H ≅ G/N	type
1. #11719650	C₂	semidirect
2. C₁₃×#576	C₂	semidirect
3. #11719541	C₂	semidirect
4. #11719480	C₂	semidirect
5. #11719478	C₂	semidirect
6. #11719476	C₂	semidirect
7. D₁₃×#135	C₂	semidirect
8. #11719462	C₂	semidirect
9. C₂×#88166	C₂	semidirect
10. #11719458	C₂	semidirect
11. D₁₃×#131	C₂	semidirect
12. C₁₃×H(5,𝔽₂)	C₂²	semidirect
14. #88257	C₂²	semidirect
15. #88257	C₂²	semidirect
18. #41×D₁₃	C₂²	semidirect
20. #88251	C₂²	semidirect
22. C₂×#6129	C₂²	semidirect
23. C₁₃×AGL(1,ℤ/8)	C₂²	semidirect
24. C₁₃×#135	C₂²	semidirect
25. C₂×C₁₃×D₈	C₂²	semidirect
26. C₁₃×#131	C₂²	semidirect
27. #88205	C₂²	semidirect
28. #88204	C₂²	semidirect
29. #88202	C₂²	semidirect
30. C₂×#6105	C₂²	semidirect
31. #88186	C₂²	semidirect
32. #88173	C₂²	semidirect
33. Q₁₆×D₁₃	C₂²	semidirect
34. #88170	C₂²	semidirect
35. #88169	C₂²	semidirect
36. SD₁₆×D₁₃	C₂²	semidirect
37. #88166	C₂²	semidirect
38. #88166	C₂²	semidirect
39. #88166	C₂²	semidirect
40. #88166	C₂²	semidirect
41. #88165	C₂²	semidirect
42. D₈×D₁₃	C₂²	semidirect
43. #88163	C₂²	semidirect
44. #88161	C₂²	semidirect
45. M₁₆×D₁₃	C₂²	semidirect
46. C₂×Dic₅₂	C₂²	semidirect
47. #88158	C₂²	semidirect
48. #88155	C₂²	semidirect
49. C₂×C₈×D₁₃	C₂²	semidirect
151. #576	D₁₃	semidirect
175. H(5,𝔽₂)	C₂×D₁₃	semidirect
186. AGL(1,ℤ/8)	C₂×D₁₃	semidirect
195. #135	C₂×D₁₃	semidirect
201. C₂×D₈	C₂×D₁₃	semidirect
205. #131	C₂×D₁₃	semidirect
270. C₁₃	#3584	semidirect

Maximal subgroups

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

subgroup	index
#11719650	2
C₁₃×#576	2
#11719541	2
#11719480	2
#11719478	2
#11719476	2
D₁₃×#135	2
#11719462	2
C₂×#88166	2
#11719458	2
D₁₃×#131	2
#3584	13