Subgroups — #13128707

Order 1664.

1232 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719510	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88194	C₂	1	1	yes	maximal, characteristic, complemented
4	#11719267	C₂	1	4	yes	maximal, complemented
5	#11719264	C₂	1	4	yes	maximal, complemented
6	#11719258	C₂	1	4	yes	maximal, complemented
7	#11719253	C₂	1	4	yes	maximal, complemented
8	#11719230	C₂	1	1	yes	maximal, characteristic, complemented
9	#11719227	C₂	1	1	yes	maximal, characteristic, complemented
10	#11719226	C₂	1	1	yes	maximal, characteristic, complemented
11	#11719222	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719195	C₂	1	8	yes	maximal, complemented
13	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₁₃	C₂²	1	4	yes	—
15	C₂×#6127	C₂²	1	1	yes	characteristic
16	C₂×#6127	C₂²	1	8	yes	complemented
17	C₂²×D₅₂	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₃	C₂²	1	4	yes	complemented
19	C₄×D₄×C₁₃	C₂²	1	4	yes	complemented
20	C₁₃×#117	C₂²	1	1	yes	characteristic
21	C₁₃×#117	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
23	#88207	C₂²	1	4	yes	—
24	#88194	C₂²	1	2	yes	—
25	#88194	C₂²	1	4	yes	—
26	#88194	C₂²	1	2	yes	—
27	#88194	C₂²	1	2	yes	complemented
28	#88192	C₂²	1	4	yes	complemented
29	#88190	C₂²	1	2	yes	complemented
30	#88184	C₂²	1	8	yes	complemented
31	#88183	C₂²	1	8	yes	complemented
32	C₄×C₂²⋊D₁₃	C₂²	1	8	yes	complemented
33	#88180	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
35	#88149	C₂²	1	4	yes	complemented
36	#88147	C₂²	1	4	yes	complemented
37	#88147	C₂²	1	2	yes	complemented
38	#88144	C₂²	1	2	yes	complemented
39	#88142	C₂²	1	2	yes	complemented
40	#88141	C₂²	1	2	yes	complemented
41	#88139	C₂²	1	8	yes	complemented
42	#88138	C₂²	1	8	yes	complemented
43	#88138	C₂²	1	8	yes	complemented
44	#88137	C₂²	1	8	yes	complemented
45	#88136	C₂²	1	4	yes	—
46	#88135	C₂²	1	4	yes	—
47	C₂²⋊C₄×D₁₃	C₂²	1	4	yes	—
48	#88132	C₂²	1	4	yes	—
49	#88129	C₂²	1	2	yes	complemented
50	#88128	C₂²	1	2	yes	complemented
51	C₄×D₅₂	C₂²	1	4	yes	complemented
52	#88126	C₂²	1	2	yes	complemented
53	C₄²×D₁₃	C₂²	1	2	yes	complemented
54	C₄×Dic₂₆	C₂²	1	4	yes	complemented
55	C₂⁴×C₁₃	C₂³	1	2	yes	—
56	C₂³×D₁₃	C₂³	1	2	yes	—
57	C₂×D₄×C₁₃	—	2	4	no	—
58	C₂×D₄×C₁₃	C₂³	1	2	yes	—
59	C₂×D₄×C₁₃	C₂³	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂³	1	8	yes	—
61	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₃	—	4	8	no	—
63	C₂×C₂²⋊D₁₃	C₂³	1	2	yes	—
64	C₂×C₂²⋊D₁₃	C₂³	1	8	yes	—
65	C₂×C₂²⋊D₁₃	C₂³	1	8	yes	—
66	C₂×C₂²⋊D₁₃	C₂³	1	8	yes	—
67	C₂²×Dic₁₃	C₂³	1	2	yes	—
68	#6127	—	2	4	no	—
69	#6127	—	4	16	no	—
70	C₂×D₅₂	—	2	4	no	—
71	C₂×D₅₂	C₂³	1	2	yes	—
72	C₂×D₅₂	C₂³	1	1	yes	characteristic
73	C₂×D₅₂	C₂³	1	2	yes	—
74	C₂×D₅₂	C₂³	1	8	yes	—
75	C₂×C₄×D₁₃	C₂³	1	2	yes	—
76	C₂×C₄×D₁₃	C₂³	1	4	yes	—
77	C₂×C₄×D₁₃	C₂³	1	8	yes	—
78	C₂×Dic₂₆	C₂³	1	4	yes	—
79	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
80	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
81	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
82	C₁₃×C₂²⋊C₄	C₂³	1	2	yes	—
83	C₁₃×C₂²⋊C₄	C₂³	1	2	yes	—
84	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
85	C₄²×C₁₃	C₂³	1	2	yes	—
86	C₄²×C₁₃	C₂³	1	2	yes	—
87	C₂²⋊Dic₁₃	C₂³	1	2	yes	—
88	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
89	#6103	C₂³	1	4	yes	—
90	#6103	C₂³	1	8	yes	—
91	#6103	C₂³	1	8	yes	—
92	C₄⋊Dic₁₃	C₂³	1	4	yes	—
93	C₄⋊Dic₁₃	C₂³	1	2	yes	—
94	#6101	C₂³	1	4	yes	—
95	#6101	C₂³	1	4	yes	—
96	#6101	C₂³	1	4	yes	—
97	C₄×Dic₁₃	C₂³	1	2	yes	—
98	C₄×Dic₁₃	C₂³	1	2	yes	—
99	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
100	C₄×Dic₁₃	C₂³	1	4	yes	—
101	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
102	#3508	—	13	1	no	maximal, Sylow
103	C₂³×C₁₃	—	2	4	no	—
104	C₂³×C₁₃	—	4	4	no	—
105	C₂³×C₁₃	C₂⁴	1	4	yes	—
106	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
107	C₂²×D₁₃	—	2	4	no	—
108	C₂²×D₁₃	—	4	4	no	—
109	C₂²×D₁₃	C₂⁴	1	4	yes	—
110	C₂²×D₁₃	C₂⁴	1	2	yes	—
111	D₄×C₁₃	—	2	4	no	—
112	D₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	4	8	no	—
114	C₂×C₄×C₁₃	#41	1	4	yes	complemented
115	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
116	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
117	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₁₃	—	2	4	no	—
120	C₂²⋊D₁₃	—	2	16	no	—
121	C₂²⋊D₁₃	—	4	8	no	—
122	C₂²⋊D₁₃	—	4	8	no	—
123	C₂×Dic₁₃	—	2	4	no	—
124	C₂×Dic₁₃	C₂⁴	1	2	yes	—
125	C₂×Dic₁₃	C₂⁴	1	2	yes	—
126	C₂×Dic₁₃	C₂⁴	1	4	yes	—
127	C₂×Dic₁₃	C₂⁴	1	2	yes	—
128	D₅₂	—	2	4	no	—
129	D₅₂	—	2	4	no	—
130	D₅₂	—	2	2	no	—
131	D₅₂	—	4	8	no	—
132	C₄×D₁₃	—	2	4	no	—
133	C₄×D₁₃	—	2	8	no	—
134	C₄×D₁₃	—	4	8	no	—
135	Dic₂₆	—	2	2	no	—
136	Dic₂₆	—	2	8	no	—
137	#559	—	13	4	no	—
138	#550	—	13	4	no	—
139	#546	—	13	8	no	—
140	#540	—	13	4	no	—
141	#535	—	13	1	no	—
142	#535	—	13	1	no	—
143	#533	—	13	1	no	—
144	#532	—	13	4	no	—
145	C₂×#127	—	13	1	no	—
146	#518	—	13	1	no	—
147	#518	D₁₃	1	1	yes	characteristic, complemented
148	C₄×#41	—	13	1	no	—
149	C₂²×C₁₃	—	2	2	no	—
150	C₂²×C₁₃	—	2	4	no	—
151	C₂²×C₁₃	—	2	8	no	—
152	C₂²×C₁₃	—	8	4	no	—
153	C₂²×C₁₃	C₂×#41	1	2	yes	—
154	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
155	C₂×D₁₃	—	2	2	no	—
156	C₂×D₁₃	—	2	4	no	—
157	C₂×D₁₃	—	2	4	no	—
158	C₂×D₁₃	—	2	8	no	—
159	C₂×D₁₃	—	8	4	no	—
160	C₄×C₁₃	—	2	4	no	—
161	C₄×C₁₃	—	2	4	no	—
162	C₄×C₁₃	C₂×#41	1	2	yes	—
163	C₄×C₁₃	C₂×#41	1	2	yes	—
164	Dic₁₃	—	2	2	no	—
165	Dic₁₃	—	2	4	no	—
166	Dic₁₃	—	2	2	no	—
167	Dic₁₃	H(5,𝔽₂)	1	4	yes	—
168	C₂×#41	—	13	1	no	—
169	C₂×#41	—	13	8	no	—
170	C₂²×D₄	—	13	4	no	—
171	C₂²×D₄	—	13	1	no	—
172	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
173	C₄⋊Q₈	—	13	2	no	—
174	#127	—	13	2	no	—
175	#127	—	13	2	no	—
176	#127	—	13	4	no	—
177	#127	—	13	2	no	—
178	#127	—	13	2	no	—
179	#125	—	13	2	no	—
180	#124	—	13	2	no	—
181	#124	—	13	8	no	—
182	#124	—	13	2	no	—
183	#123	—	13	8	no	—
184	#123	—	13	8	no	—
185	C₂²⋊Q₈	—	13	4	no	—
186	#121	—	13	4	no	—
187	#121	—	13	8	no	—
188	#121	—	13	4	no	—
189	#121	—	13	8	no	—
190	#121	—	13	8	no	—
191	#120	—	13	4	no	—
192	#120	—	13	4	no	—
193	C₄×Q₈	—	13	2	no	—
194	C₄×Q₈	—	13	4	no	—
195	C₄×D₄	—	13	8	no	—
196	C₄×D₄	—	13	2	no	—
197	C₄×D₄	—	13	4	no	—
198	C₄×D₄	—	13	4	no	—
199	C₄×D₄	—	13	4	no	—
200	C₄×D₄	C₂×D₁₃	1	4	yes	complemented
201	C₄×D₄	C₂×D₁₃	1	4	yes	complemented
202	#117	—	13	1	no	—
203	#117	—	13	2	no	—
204	#117	C₂×D₁₃	1	1	yes	characteristic
205	#117	C₂×D₁₃	1	1	yes	characteristic
206	C₂×C₂²⋊C₄	—	13	4	no	—
207	C₂×C₂²⋊C₄	C₂×D₁₃	1	4	yes	—
208	C₂×C₄²	—	13	1	no	—
209	C₂×C₄²	—	13	2	no	—
210	C₂×C₁₃	—	2	2	no	—
211	C₂×C₁₃	—	4	4	no	—
212	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
213	C₂×C₁₃	C₂×H(5,𝔽₂)	1	2	yes	—
214	D₁₃	—	4	2	no	—
215	D₁₃	—	4	4	no	—
216	C₂⁴	—	13	2	no	—
217	C₂⁴	C₂²×D₁₃	1	2	yes	—
218	#41	—	26	4	no	—
219	#41	—	52	16	no	—
220	C₂×Q₈	—	13	1	no	—
221	C₂×Q₈	—	13	4	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	13	2	no	—
224	C₂×D₄	—	13	8	no	—
225	C₂×D₄	—	13	8	no	—
226	C₂×D₄	—	13	8	no	—
227	C₂×D₄	—	13	2	no	—
228	C₂×D₄	—	13	2	no	—
229	C₂×D₄	—	13	1	no	—
230	C₂×D₄	—	13	8	no	—
231	C₂×D₄	—	26	4	no	—
232	C₂×D₄	—	52	8	no	—
233	C₂×D₄	C₂²×D₁₃	1	2	yes	—
234	C₂×D₄	C₂²×D₁₃	1	2	yes	—
235	C₂²×C₄	—	13	2	no	—
236	C₂²×C₄	—	13	2	no	—
237	C₂²×C₄	—	13	4	no	—
238	C₂²×C₄	—	13	8	no	—
239	C₂²×C₄	C₂²×D₁₃	1	8	yes	—
240	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
241	C₄⋊C₄	—	13	2	no	—
242	C₄⋊C₄	—	13	4	no	—
243	C₄⋊C₄	—	13	4	no	—
244	C₄⋊C₄	—	13	4	no	—
245	C₄⋊C₄	—	13	4	no	—
246	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
247	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
248	C₂²⋊C₄	—	13	2	no	—
249	C₂²⋊C₄	—	13	8	no	—
250	C₂²⋊C₄	—	13	8	no	—
251	C₂²⋊C₄	—	13	4	no	—
252	C₂²⋊C₄	—	13	8	no	—
253	C₂²⋊C₄	C₂²×D₁₃	1	2	yes	—
254	C₂²⋊C₄	C₂²×D₁₃	1	2	yes	—
255	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
256	C₄²	—	13	1	no	—
257	C₄²	—	13	1	no	—
258	C₄²	—	13	4	no	—
259	C₄²	—	13	2	no	—
260	C₄²	—	13	2	no	—
261	C₄²	C₂²×D₁₃	1	2	yes	—
262	C₄²	C₂²×D₁₃	1	2	yes	—
263	C₁₃	#3508	1	1	yes	characteristic, complemented, Sylow
264	C₂³	—	2	4	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	13	4	no	—
267	C₂³	—	13	2	no	—
268	C₂³	—	26	4	no	—
269	C₂³	—	52	4	no	—
270	C₂³	C₂³×D₁₃	1	4	yes	—
271	C₂³	C₂³×D₁₃	1	1	yes	characteristic
272	Q₈	—	26	8	no	—
273	Q₈	—	26	2	no	—
274	D₄	—	2	4	no	—
275	D₄	—	2	4	no	—
276	D₄	—	26	16	no	—
277	D₄	—	26	4	no	—
278	D₄	—	26	4	no	—
279	D₄	—	26	4	no	—
280	D₄	—	26	2	no	—
281	D₄	—	52	8	no	—
282	D₄	—	52	8	no	—
283	D₄	—	52	8	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	13	2	no	—
286	C₂×C₄	—	13	2	no	—
287	C₂×C₄	—	13	2	no	—
288	C₂×C₄	—	13	4	no	—
289	C₂×C₄	—	26	8	no	—
290	C₂×C₄	—	26	4	no	—
291	C₂×C₄	—	26	4	no	—
292	C₂×C₄	—	52	8	no	—
293	C₂×C₄	#6127	1	4	yes	complemented
294	C₂×C₄	C₂³×D₁₃	1	4	yes	—
295	C₂×C₄	C₂³×D₁₃	1	4	yes	—
296	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	8	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	26	2	no	—
303	C₂²	—	26	4	no	—
304	C₂²	—	26	4	no	—
305	C₂²	—	26	8	no	—
306	C₂²	—	104	4	no	—
307	C₂²	C₂×#6127	1	2	yes	—
308	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
309	C₄	—	2	4	no	—
310	C₄	—	2	4	no	—
311	C₄	—	13	4	no	—
312	C₄	—	26	2	no	—
313	C₄	—	26	2	no	—
314	C₄	—	26	4	no	—
315	C₄	C₂×#6127	1	2	yes	—
316	C₄	C₂×#6127	1	2	yes	—
317	C₂	—	2	2	no	—
318	C₂	—	4	4	no	—
319	C₂	—	52	4	no	—
320	C₂	—	52	2	no	—
321	C₂	C₂²×#6127	1	1	yes	characteristic
322	C₂	D₁₃×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₁₃×#518	C₂
hypercenter	#518	D₁₃
hypocenter	C₁₃	#3508
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #518

Complements

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

N	H ≅ G/N	type
1. C₁₃×#518	C₂	semidirect
2. #11719510	C₂	semidirect
3. C₂×#88194	C₂	semidirect
4. #11719267	C₂	semidirect
5. #11719264	C₂	semidirect
6. #11719258	C₂	semidirect
7. #11719253	C₂	semidirect
8. #11719230	C₂	semidirect
9. #11719227	C₂	semidirect
10. #11719226	C₂	semidirect
11. #11719222	C₂	semidirect
12. #11719195	C₂	semidirect
16. C₂×#6127	C₂²	semidirect
18. C₄×D₄×C₁₃	C₂²	semidirect
19. C₄×D₄×C₁₃	C₂²	semidirect
27. #88194	C₂²	semidirect
28. #88192	C₂²	semidirect
29. #88190	C₂²	semidirect
30. #88184	C₂²	semidirect
31. #88183	C₂²	semidirect
32. C₄×C₂²⋊D₁₃	C₂²	semidirect
35. #88149	C₂²	semidirect
36. #88147	C₂²	semidirect
37. #88147	C₂²	semidirect
38. #88144	C₂²	semidirect
39. #88142	C₂²	semidirect
40. #88141	C₂²	semidirect
41. #88139	C₂²	semidirect
42. #88138	C₂²	semidirect
43. #88138	C₂²	semidirect
44. #88137	C₂²	semidirect
49. #88129	C₂²	semidirect
50. #88128	C₂²	semidirect
51. C₄×D₅₂	C₂²	semidirect
52. #88126	C₂²	semidirect
53. C₄²×D₁₃	C₂²	semidirect
54. C₄×Dic₂₆	C₂²	semidirect
114. C₂×C₄×C₁₃	#41	semidirect
147. #518	D₁₃	semidirect
200. C₄×D₄	C₂×D₁₃	semidirect
201. C₄×D₄	C₂×D₁₃	semidirect
263. C₁₃	#3508	semidirect
293. C₂×C₄	#6127	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#518	2
#11719510	2
C₂×#88194	2
#11719267	2
#11719264	2
#11719258	2
#11719253	2
#11719230	2
#11719227	2
#11719226	2
#11719222	2
#11719195	2
#3508	13