Subgroups — #13129169

Order 1664.

1260 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719511	C₂	1	4	yes	maximal, complemented
3	#11719509	C₂	1	4	yes	maximal, complemented
4	D₄×C₂²⋊D₁₃	C₂	1	4	yes	maximal, complemented
5	#11719425	C₂	1	4	yes	maximal, complemented
6	#11719424	C₂	1	4	yes	maximal, complemented
7	#11719423	C₂	1	2	yes	maximal, complemented
8	#11719422	C₂	1	1	yes	maximal, characteristic, complemented
9	#11719420	C₂	1	4	yes	maximal, complemented
10	#11719418	C₂	1	1	yes	maximal, characteristic, complemented
11	#11719184	C₂	1	2	yes	maximal, complemented
12	C₂²×D₄×C₁₃	C₂²	1	4	yes	complemented
13	C₂²×D₄×C₁₃	C₂²	1	2	yes	complemented
14	C₂²×C₂²⋊D₁₃	C₂²	1	4	yes	complemented
15	C₂×#6129	C₂²	1	4	yes	complemented
16	C₂×D₄×D₁₃	C₂²	1	4	yes	complemented
17	C₁₃×#127	C₂²	1	1	yes	characteristic
18	C₁₃×#127	C₂²	1	4	yes	complemented
19	C₁₃×#127	C₂²	1	2	yes	complemented
20	C₁₃×#127	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic, complemented
22	#88207	C₂²	1	8	yes	complemented
23	C₂×C₂²⋊Dic₁₃	C₂²	1	4	yes	complemented
24	#88194	C₂²	1	4	yes	complemented
25	#88193	C₂²	1	8	yes	complemented
26	#88193	C₂²	1	8	yes	complemented
27	#88193	C₂²	1	8	yes	complemented
28	#88193	C₂²	1	8	yes	complemented
29	#88192	C₂²	1	4	yes	complemented
30	#88192	C₂²	1	4	yes	complemented
31	#88191	C₂²	1	8	yes	complemented
32	#88191	C₂²	1	4	yes	complemented
33	#88191	C₂²	1	4	yes	complemented
34	#88190	C₂²	1	4	yes	complemented
35	#88189	C₂²	1	4	yes	complemented
36	#88189	C₂²	1	8	yes	complemented
37	#88189	C₂²	1	4	yes	complemented
38	D₄×Dic₁₃	C₂²	1	4	yes	complemented
39	D₄×Dic₁₃	C₂²	1	4	yes	complemented
40	#88184	C₂²	1	2	yes	—
41	#88183	C₂²	1	4	yes	complemented
42	C₄×C₂²⋊D₁₃	C₂²	1	4	yes	complemented
43	#88178	C₂²	1	2	yes	complemented
44	#88130	C₂²	1	4	yes	complemented
45	#88129	C₂²	1	2	yes	complemented
46	C₄×D₅₂	C₂²	1	2	yes	complemented
47	#88126	C₂²	1	4	yes	complemented
48	#88124	C₂²	1	2	yes	complemented
49	C₄×Dic₂₆	C₂²	1	2	yes	complemented
50	C₂⁴×C₁₃	C₂³	1	4	yes	—
51	C₂³×D₁₃	C₂³	1	4	yes	—
52	C₂×D₄×C₁₃	—	4	8	no	—
53	C₂×D₄×C₁₃	—	4	4	no	—
54	C₂×D₄×C₁₃	C₂³	1	4	yes	—
55	C₂×D₄×C₁₃	C₂³	1	4	yes	—
56	C₂×D₄×C₁₃	C₂³	1	2	yes	—
57	C₂×D₄×C₁₃	C₂³	1	4	yes	—
58	C₂×D₄×C₁₃	C₂³	1	8	yes	—
59	C₂×D₄×C₁₃	C₂³	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₃	—	4	8	no	—
63	C₂×C₂²⋊D₁₃	C₂³	1	4	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	4	yes	—
68	C₂²×Dic₁₃	C₂³	1	8	yes	—
69	#6129	—	4	8	no	—
70	D₄×D₁₃	—	4	8	no	—
71	C₂×D₅₂	C₂³	1	2	yes	—
72	C₂×C₄×D₁₃	C₂³	1	4	yes	—
73	C₂×Dic₂₆	C₂³	1	2	yes	—
74	C₄²×C₁₃	C₂³	1	1	yes	characteristic
75	C₄²×C₁₃	C₂³	1	2	yes	—
76	C₄²×C₁₃	C₂³	1	1	yes	characteristic
77	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
78	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
79	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
80	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
81	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
82	#6103	C₂³	1	4	yes	—
83	#6103	C₂³	1	4	yes	—
84	#6103	C₂³	1	4	yes	—
85	#6103	C₂³	1	4	yes	—
86	C₄⋊Dic₁₃	C₂³	1	2	yes	—
87	C₄⋊Dic₁₃	C₂³	1	2	yes	—
88	#6101	C₂³	1	4	yes	—
89	#6101	C₂³	1	4	yes	—
90	#6101	C₂³	1	4	yes	—
91	#6101	C₂³	1	4	yes	—
92	C₄×Dic₁₃	C₂³	1	4	yes	—
93	#3543	—	13	1	no	maximal, Sylow
94	C₂³×C₁₃	—	2	8	no	—
95	C₂³×C₁₃	—	4	8	no	—
96	C₂³×C₁₃	C₂⁴	1	4	yes	—
97	C₂³×C₁₃	C₂⁴	1	4	yes	—
98	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
99	C₂²×D₁₃	—	4	8	no	—
100	C₂²×D₁₃	—	4	4	no	—
101	C₂²×D₁₃	C₂⁴	1	4	yes	—
102	D₄×C₁₃	—	2	8	no	—
103	D₄×C₁₃	—	2	8	no	—
104	D₄×C₁₃	—	4	8	no	—
105	D₄×C₁₃	—	4	2	no	—
106	D₄×C₁₃	—	4	4	no	—
107	D₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
113	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
114	C₂²⋊D₁₃	—	2	8	no	—
115	C₂²⋊D₁₃	—	4	8	no	—
116	C₂²⋊D₁₃	—	4	8	no	—
117	C₂²⋊D₁₃	—	4	8	no	—
118	C₂×Dic₁₃	—	4	8	no	—
119	C₂×Dic₁₃	—	4	4	no	—
120	C₂×Dic₁₃	C₂⁴	1	4	yes	—
121	C₂×Dic₁₃	C₂⁴	1	4	yes	—
122	C₂×Dic₁₃	C₂⁴	1	4	yes	—
123	D₅₂	—	2	4	no	—
124	C₄×D₁₃	—	2	8	no	—
125	Dic₂₆	—	2	4	no	—
126	#560	—	13	4	no	—
127	#559	—	13	1	no	—
128	#550	—	13	2	no	—
129	#546	—	13	4	no	—
130	D₄²	—	13	4	no	—
131	D₄²	—	13	1	no	—
132	#540	—	13	4	no	—
133	#535	—	13	4	no	—
134	#534	—	13	4	no	—
135	C₂×#127	D₁₃	1	1	yes	characteristic, complemented
136	#529	—	13	2	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	—	2	8	no	—
139	C₂²×C₁₃	—	2	4	no	—
140	C₂²×C₁₃	—	2	4	no	—
141	C₂²×C₁₃	—	2	8	no	—
142	C₂²×C₁₃	—	8	8	no	—
143	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
144	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
145	C₂×D₁₃	—	2	4	no	—
146	C₂×D₁₃	—	2	4	no	—
147	C₂×D₁₃	—	2	4	no	—
148	C₂×D₁₃	—	8	8	no	—
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	H(5,𝔽₂)	1	4	yes	—
153	Dic₁₃	—	2	4	no	—
154	Dic₁₃	—	2	4	no	—
155	Dic₁₃	—	2	4	no	—
156	C₂×#41	—	13	4	no	—
157	C₂²×D₄	—	13	4	no	—
158	C₂²×D₄	—	13	4	no	—
159	C₂²×D₄	C₂×D₁₃	1	2	yes	complemented
160	C₂²×D₄	C₂×D₁₃	1	4	yes	complemented
161	#127	—	13	4	no	—
162	#127	C₂×D₁₃	1	1	yes	characteristic
163	#127	C₂×D₁₃	1	4	yes	complemented
164	#127	C₂×D₁₃	1	1	yes	characteristic, complemented
165	#127	C₂×D₁₃	1	2	yes	complemented
166	#126	—	13	4	no	—
167	#125	—	13	2	no	—
168	#124	—	13	4	no	—
169	#124	—	13	2	no	—
170	#123	—	13	4	no	—
171	#123	—	13	4	no	—
172	#123	—	13	8	no	—
173	#123	—	13	4	no	—
174	C₂²⋊Q₈	—	13	2	no	—
175	#121	—	13	8	no	—
176	#121	—	13	8	no	—
177	#121	—	13	8	no	—
178	#121	—	13	4	no	—
179	#121	—	13	4	no	—
180	#121	—	13	8	no	—
181	#121	—	13	2	no	—
182	#120	—	13	8	no	—
183	#120	—	13	4	no	—
184	#120	—	13	4	no	—
185	#120	—	13	8	no	—
186	C₄×Q₈	—	13	2	no	—
187	C₄×D₄	—	13	4	no	—
188	C₄×D₄	—	13	4	no	—
189	C₄×D₄	—	13	4	no	—
190	C₄×D₄	—	13	2	no	—
191	#117	—	13	4	no	—
192	C₂×C₂²⋊C₄	—	13	4	no	—
193	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic, complemented
194	C₂×C₁₃	—	2	2	no	—
195	C₂×C₁₃	—	4	4	no	—
196	C₂×C₁₃	—	4	4	no	—
197	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
198	C₂×C₁₃	C₂×H(5,𝔽₂)	1	2	yes	—
199	D₁₃	—	4	4	no	—
200	C₂⁴	—	13	4	no	—
201	C₂⁴	C₂²×D₁₃	1	4	yes	—
202	#41	—	52	8	no	—
203	C₂×Q₈	—	13	2	no	—
204	C₂×D₄	—	4	8	no	—
205	C₂×D₄	—	4	4	no	—
206	C₂×D₄	—	13	4	no	—
207	C₂×D₄	—	13	8	no	—
208	C₂×D₄	—	13	8	no	—
209	C₂×D₄	—	13	8	no	—
210	C₂×D₄	—	13	2	no	—
211	C₂×D₄	—	52	8	no	—
212	C₂×D₄	—	52	8	no	—
213	C₂×D₄	C₂²×D₁₃	1	4	yes	—
214	C₂×D₄	C₂²×D₁₃	1	4	yes	—
215	C₂×D₄	C₂²×D₁₃	1	2	yes	—
216	C₂×D₄	C₂²×D₁₃	1	4	yes	—
217	C₂×D₄	C₂²×D₁₃	1	8	yes	—
218	C₂×D₄	C₂²×D₁₃	1	2	yes	—
219	C₂²×C₄	—	13	8	no	—
220	C₂²×C₄	—	13	4	no	—
221	C₂²×C₄	—	13	4	no	—
222	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
223	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
224	C₄⋊C₄	—	13	4	no	—
225	C₄⋊C₄	—	13	4	no	—
226	C₄⋊C₄	—	13	2	no	—
227	C₄⋊C₄	—	13	2	no	—
228	C₄⋊C₄	—	13	4	no	—
229	C₄⋊C₄	—	13	4	no	—
230	C₂²⋊C₄	—	13	4	no	—
231	C₂²⋊C₄	—	13	4	no	—
232	C₂²⋊C₄	—	13	4	no	—
233	C₂²⋊C₄	—	13	8	no	—
234	C₂²⋊C₄	—	13	8	no	—
235	C₂²⋊C₄	—	13	4	no	—
236	C₂²⋊C₄	—	13	4	no	—
237	C₂²⋊C₄	—	13	4	no	—
238	C₂²⋊C₄	—	13	4	no	—
239	C₄²	—	13	4	no	—
240	C₄²	C₂²×D₁₃	1	2	yes	—
241	C₄²	C₂²×D₁₃	1	1	yes	characteristic
242	C₄²	C₂²×D₁₃	1	1	yes	characteristic
243	C₁₃	#3543	1	1	yes	characteristic, complemented, Sylow
244	C₂³	—	2	8	no	—
245	C₂³	—	4	8	no	—
246	C₂³	—	13	4	no	—
247	C₂³	—	52	4	no	—
248	C₂³	—	52	8	no	—
249	C₂³	C₂³×D₁₃	1	4	yes	—
250	C₂³	C₂³×D₁₃	1	4	yes	—
251	C₂³	C₂³×D₁₃	1	1	yes	characteristic
252	Q₈	—	26	4	no	—
253	D₄	—	2	8	no	—
254	D₄	—	2	8	no	—
255	D₄	—	4	8	no	—
256	D₄	—	4	2	no	—
257	D₄	—	4	4	no	—
258	D₄	—	4	2	no	—
259	D₄	—	26	8	no	—
260	D₄	—	26	4	no	—
261	D₄	—	52	8	no	—
262	D₄	—	52	8	no	—
263	D₄	—	52	8	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	13	4	no	—
267	C₂×C₄	—	13	4	no	—
268	C₂×C₄	—	13	4	no	—
269	C₂×C₄	—	26	8	no	—
270	C₂×C₄	—	52	4	no	—
271	C₂×C₄	—	52	8	no	—
272	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
273	C₂×C₄	C₂³×D₁₃	1	2	yes	—
274	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
275	C₂×C₄	C₂³×D₁₃	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	8	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	8	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	8	8	no	—
282	C₂²	—	26	4	no	—
283	C₂²	—	26	4	no	—
284	C₂²	—	26	4	no	—
285	C₂²	—	104	8	no	—
286	C₂²	#88251	1	2	yes	—
287	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	26	4	no	—
292	C₄	—	26	4	no	—
293	C₄	—	26	4	no	—
294	C₄	#88251	1	4	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	4	4	no	—
297	C₂	—	4	4	no	—
298	C₂	—	52	4	no	—
299	C₂	C₂×#88251	1	1	yes	characteristic
300	C₂	C₂×#88251	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₂×C₁₃×#127	C₂
hypercenter	C₂×#127	D₁₃
hypocenter	C₁₃	#3543
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#127	C₂	semidirect
2. #11719511	C₂	semidirect
3. #11719509	C₂	semidirect
4. D₄×C₂²⋊D₁₃	C₂	semidirect
5. #11719425	C₂	semidirect
6. #11719424	C₂	semidirect
7. #11719423	C₂	semidirect
8. #11719422	C₂	semidirect
9. #11719420	C₂	semidirect
10. #11719418	C₂	semidirect
11. #11719184	C₂	semidirect
12. C₂²×D₄×C₁₃	C₂²	semidirect
13. C₂²×D₄×C₁₃	C₂²	semidirect
14. C₂²×C₂²⋊D₁₃	C₂²	semidirect
15. C₂×#6129	C₂²	semidirect
16. C₂×D₄×D₁₃	C₂²	semidirect
18. C₁₃×#127	C₂²	semidirect
19. C₁₃×#127	C₂²	semidirect
20. C₁₃×#127	C₂²	semidirect
21. C₂×C₄²×C₁₃	C₂²	semidirect
22. #88207	C₂²	semidirect
23. C₂×C₂²⋊Dic₁₃	C₂²	semidirect
24. #88194	C₂²	semidirect
25. #88193	C₂²	semidirect
26. #88193	C₂²	semidirect
27. #88193	C₂²	semidirect
28. #88193	C₂²	semidirect
29. #88192	C₂²	semidirect
30. #88192	C₂²	semidirect
31. #88191	C₂²	semidirect
32. #88191	C₂²	semidirect
33. #88191	C₂²	semidirect
34. #88190	C₂²	semidirect
35. #88189	C₂²	semidirect
36. #88189	C₂²	semidirect
37. #88189	C₂²	semidirect
38. D₄×Dic₁₃	C₂²	semidirect
39. D₄×Dic₁₃	C₂²	semidirect
41. #88183	C₂²	semidirect
42. C₄×C₂²⋊D₁₃	C₂²	semidirect
43. #88178	C₂²	semidirect
44. #88130	C₂²	semidirect
45. #88129	C₂²	semidirect
46. C₄×D₅₂	C₂²	semidirect
47. #88126	C₂²	semidirect
48. #88124	C₂²	semidirect
49. C₄×Dic₂₆	C₂²	semidirect
135. C₂×#127	D₁₃	semidirect
159. C₂²×D₄	C₂×D₁₃	semidirect
160. C₂²×D₄	C₂×D₁₃	semidirect
163. #127	C₂×D₁₃	semidirect
164. #127	C₂×D₁₃	semidirect
165. #127	C₂×D₁₃	semidirect
193. C₂×C₄²	C₂×D₁₃	semidirect
243. C₁₃	#3543	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#127	2
#11719511	2
#11719509	2
D₄×C₂²⋊D₁₃	2
#11719425	2
#11719424	2
#11719423	2
#11719422	2
#11719420	2
#11719418	2
#11719184	2
#3543	13