Subgroups — #13128609

Order 1664.

1748 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×#88256	C₂	1	2	yes	maximal, complemented
2	C₁₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719535	C₂	1	2	yes	maximal, complemented
4	#11719430	C₂	1	2	yes	maximal, complemented
5	D₁₃×#127	C₂	1	2	yes	maximal, complemented
6	#11719379	C₂	1	4	yes	maximal, complemented
7	#11719314	C₂	1	8	yes	maximal, complemented
8	#11719258	C₂	1	4	yes	maximal, complemented
9	D₄×D₅₂	C₂	1	4	yes	maximal, complemented
10	C₂×#88128	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#6127	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	2	yes	complemented
13	#88256	—	2	16	no	—
14	C₂×#6131	C₂²	1	4	yes	complemented
15	C₂×D₄×D₁₃	C₂²	1	8	yes	complemented
16	C₂×D₄×D₁₃	C₂²	1	4	yes	complemented
17	C₂×D₄×D₁₃	C₂²	1	8	yes	complemented
18	C₂×#6127	C₂²	1	1	yes	characteristic
19	C₂×#6127	C₂²	1	4	yes	complemented
20	C₂²×D₅₂	C₂²	1	2	yes	complemented
21	C₂²×D₅₂	C₂²	1	4	yes	complemented
22	C₂²×D₅₂	C₂²	1	4	yes	complemented
23	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₁₃×#127	C₂²	1	1	yes	characteristic, complemented
25	C₁₃×#124	C₂²	1	2	yes	complemented
26	C₁₃×#121	C₂²	1	4	yes	complemented
27	C₄×D₄×C₁₃	C₂²	1	4	yes	complemented
28	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic, complemented
29	#88201	C₂²	1	4	yes	complemented
30	#88194	C₂²	1	8	yes	complemented
31	#88194	C₂²	1	4	yes	complemented
32	#88191	C₂²	1	8	yes	complemented
33	#88184	C₂²	1	8	yes	complemented
34	C₄×C₂²⋊D₁₃	C₂²	1	4	yes	complemented
35	#88180	C₂²	1	1	yes	characteristic, complemented
36	#88149	C₂²	1	8	yes	complemented
37	#88147	C₂²	1	8	yes	complemented
38	#88138	C₂²	1	16	yes	complemented
39	#88136	C₂²	1	16	yes	complemented
40	#88128	C₂²	1	2	yes	complemented
41	#88128	C₂²	1	4	yes	complemented
42	#88128	C₂²	1	2	yes	complemented
43	C₄×D₅₂	C₂²	1	2	yes	complemented
44	#88126	C₂²	1	2	yes	complemented
45	C₄²×D₁₃	C₂²	1	2	yes	complemented
46	C₄×Dic₂₆	C₂²	1	2	yes	complemented
47	C₂³×D₁₃	C₂³	1	8	yes	—
48	C₂³×D₁₃	C₂³	1	4	yes	—
49	C₁₃×#41	—	2	8	no	—
50	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
51	C₂×D₄×C₁₃	C₂³	1	4	yes	—
52	C₂×D₄×C₁₃	C₂³	1	4	yes	—
53	C₂×D₄×C₁₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
56	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊D₁₃	C₂³	1	8	yes	—
58	C₂×C₂²⋊D₁₃	C₂³	1	16	yes	—
59	C₂×C₂²⋊D₁₃	C₂³	1	2	yes	—
60	#6131	—	2	16	no	—
61	D₄×D₁₃	—	2	32	no	—
62	D₄×D₁₃	—	2	16	no	—
63	D₄×D₁₃	—	4	16	no	—
64	#6127	—	2	16	no	—
65	#6127	D₄	1	8	yes	complemented
66	C₂×D₅₂	—	2	8	no	—
67	C₂×D₅₂	—	2	16	no	—
68	C₂×D₅₂	—	4	8	no	—
69	C₂×D₅₂	C₂³	1	8	yes	—
70	C₂×D₅₂	C₂³	1	4	yes	—
71	C₂×D₅₂	C₂³	1	4	yes	—
72	C₂×D₅₂	C₂³	1	8	yes	—
73	C₂×D₅₂	C₂³	1	8	yes	—
74	C₂×D₅₂	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₁₃	C₂³	1	4	yes	—
76	C₂×C₄×D₁₃	C₂³	1	8	yes	—
77	C₂×C₄×D₁₃	C₂³	1	2	yes	—
78	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
79	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
80	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
81	C₄²×C₁₃	C₂³	1	2	yes	—
82	C₄²×C₁₃	C₂³	1	1	yes	characteristic
83	C₄²×C₁₃	C₂³	1	1	yes	characteristic
84	C₂²⋊Dic₁₃	C₂³	1	2	yes	—
85	#6103	C₂³	1	4	yes	—
86	#6103	C₂³	1	16	yes	—
87	C₄⋊Dic₁₃	C₂³	1	2	yes	—
88	#6101	C₂³	1	4	yes	—
89	C₄×Dic₁₃	C₂³	1	4	yes	—
90	C₄×Dic₁₃	C₂³	1	2	yes	—
91	#3498	—	13	1	no	maximal, Sylow
92	C₂³×C₁₃	C₂⁴	1	4	yes	—
93	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₂²×D₁₃	—	2	8	no	—
95	C₂²×D₁₃	—	2	16	no	—
96	C₂²×D₁₃	—	4	8	no	—
97	C₂²×D₁₃	—	4	16	no	—
98	C₂²×D₁₃	C₂⁴	1	8	yes	—
99	C₂²×D₁₃	C₂⁴	1	2	yes	—
100	Q₈×C₁₃	—	2	4	no	—
101	D₄×C₁₃	—	2	8	no	—
102	D₄×C₁₃	—	2	4	no	—
103	D₄×C₁₃	—	2	8	no	—
104	C₂×C₄×C₁₃	—	2	8	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	complemented
107	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
108	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
109	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
112	C₂²⋊D₁₃	—	2	16	no	—
113	C₂²⋊D₁₃	—	4	16	no	—
114	C₂²⋊D₁₃	C₂×D₄	1	8	yes	complemented
115	C₂×Dic₁₃	C₂⁴	1	4	yes	—
116	C₂×Dic₁₃	C₂⁴	1	2	yes	—
117	D₅₂	—	2	16	no	—
118	D₅₂	—	2	8	no	—
119	D₅₂	—	2	8	no	—
120	D₅₂	—	2	16	no	—
121	D₅₂	—	4	8	no	—
122	D₅₂	C₂×D₄	1	4	yes	complemented
123	C₄×D₁₃	—	2	8	no	—
124	C₄×D₁₃	—	2	16	no	—
125	C₄×D₁₃	C₂×D₄	1	8	yes	complemented
126	Dic₂₆	C₂×D₄	1	4	yes	complemented
127	C₂×H(5,𝔽₂)	—	13	2	no	—
128	#550	—	13	4	no	—
129	D₄²	—	13	8	no	—
130	D₄²	—	13	4	no	—
131	#535	—	13	2	no	—
132	#535	—	13	4	no	—
133	#532	—	13	2	no	—
134	#532	D₁₃	1	1	yes	characteristic, complemented
135	C₂×#127	—	13	2	no	—
136	C₂×#127	—	13	1	no	—
137	C₄×#41	—	13	1	no	—
138	C₂²×C₁₃	—	2	2	no	—
139	C₂²×C₁₃	—	2	4	no	—
140	C₂²×C₁₃	—	2	8	no	—
141	C₂²×C₁₃	C₂²×D₄	1	2	yes	—
142	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
143	C₂×D₁₃	—	2	4	no	—
144	C₂×D₁₃	—	2	8	no	—
145	C₂×D₁₃	—	2	16	no	—
146	C₂×D₁₃	—	8	16	no	—
147	C₂×D₁₃	—	8	8	no	—
148	C₂×D₁₃	C₂²×D₄	1	4	yes	—
149	C₄×C₁₃	—	2	4	no	—
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	C₂²×D₄	1	2	yes	—
152	C₄×C₁₃	C₂²×D₄	1	2	yes	—
153	C₄×C₁₃	H(5,𝔽₂)	1	4	yes	complemented
154	Dic₁₃	—	2	4	no	—
155	Dic₁₃	C₂²×D₄	1	4	yes	—
156	H(5,𝔽₂)	—	26	16	no	—
157	C₂×#41	—	13	4	no	—
158	C₂×#41	—	13	1	no	—
159	C₂×#41	—	13	4	no	—
160	C₂×#41	C₂×D₁₃	1	2	yes	complemented
161	C₂²×D₄	—	13	4	no	—
162	C₂²×D₄	—	13	8	no	—
163	C₂²×D₄	—	13	8	no	—
164	C₂²×D₄	—	13	2	no	—
165	C₂²×D₄	—	13	4	no	—
166	C₂²×D₄	—	13	4	no	—
167	C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic, complemented
168	#127	—	13	4	no	—
169	#127	—	13	8	no	—
170	#127	—	13	2	no	—
171	#127	—	13	2	no	—
172	#127	—	13	4	no	—
173	#127	C₂×D₁₃	1	1	yes	characteristic, complemented
174	#124	—	13	4	no	—
175	#124	C₂×D₁₃	1	2	yes	complemented
176	#121	—	13	8	no	—
177	#121	—	13	8	no	—
178	#121	—	13	16	no	—
179	#121	C₂×D₁₃	1	4	yes	complemented
180	#120	—	13	8	no	—
181	#120	—	13	16	no	—
182	C₄×Q₈	—	13	2	no	—
183	C₄×D₄	—	13	4	no	—
184	C₄×D₄	—	13	8	no	—
185	C₄×D₄	—	13	2	no	—
186	C₄×D₄	C₂×D₁₃	1	4	yes	complemented
187	#117	—	13	1	no	—
188	#117	—	13	2	no	—
189	C₂×C₄²	—	13	2	no	—
190	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic, complemented
191	C₂×C₁₃	—	2	2	no	—
192	C₂×C₁₃	—	4	4	no	—
193	C₂×C₁₃	C₂³×D₄	1	1	yes	characteristic
194	C₂×C₁₃	C₂×H(5,𝔽₂)	1	2	yes	—
195	D₁₃	—	2	4	no	—
196	D₁₃	—	4	8	no	—
197	C₂⁴	—	13	8	no	—
198	C₂⁴	—	13	4	no	—
199	#41	—	2	8	no	—
200	#41	—	13	8	no	—
201	#41	—	26	16	no	—
202	#41	—	26	16	no	—
203	C₂×Q₈	—	13	1	no	—
204	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
205	C₂×D₄	—	13	2	no	—
206	C₂×D₄	—	13	8	no	—
207	C₂×D₄	—	13	16	no	—
208	C₂×D₄	—	13	1	no	—
209	C₂×D₄	—	13	8	no	—
210	C₂×D₄	—	13	8	no	—
211	C₂×D₄	—	13	4	no	—
212	C₂×D₄	—	13	4	no	—
213	C₂×D₄	—	13	8	no	—
214	C₂×D₄	—	26	16	no	—
215	C₂×D₄	—	26	16	no	—
216	C₂×D₄	—	26	32	no	—
217	C₂×D₄	—	26	8	no	—
218	C₂×D₄	—	52	8	no	—
219	C₂×D₄	—	52	16	no	—
220	C₂×D₄	C₂²×D₁₃	1	4	yes	—
221	C₂×D₄	C₂²×D₁₃	1	4	yes	—
222	C₂×D₄	C₂²×D₁₃	1	2	yes	—
223	C₂²×C₄	—	13	2	no	—
224	C₂²×C₄	—	13	4	no	—
225	C₂²×C₄	—	13	8	no	—
226	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
227	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
228	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
229	C₄⋊C₄	—	13	4	no	—
230	C₄⋊C₄	—	13	2	no	—
231	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
232	C₂²⋊C₄	—	13	2	no	—
233	C₂²⋊C₄	—	13	4	no	—
234	C₂²⋊C₄	—	13	16	no	—
235	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
236	C₄²	—	13	2	no	—
237	C₄²	—	13	4	no	—
238	C₄²	C₂²×D₁₃	1	1	yes	characteristic
239	C₄²	C₂²×D₁₃	1	1	yes	characteristic
240	C₄²	C₂²×D₁₃	1	2	yes	—
241	C₁₃	#3498	1	1	yes	characteristic, complemented, Sylow
242	C₂³	—	13	2	no	—
243	C₂³	—	13	8	no	—
244	C₂³	—	26	16	no	—
245	C₂³	—	26	8	no	—
246	C₂³	—	52	8	no	—
247	C₂³	—	52	16	no	—
248	C₂³	C₂³×D₁₃	1	4	yes	—
249	C₂³	C₂³×D₁₃	1	1	yes	characteristic
250	Q₈	—	2	4	no	—
251	Q₈	—	13	4	no	—
252	D₄	—	2	8	no	—
253	D₄	—	2	4	no	—
254	D₄	—	2	8	no	—
255	D₄	—	13	8	no	—
256	D₄	—	13	4	no	—
257	D₄	—	26	16	no	—
258	D₄	—	26	16	no	—
259	D₄	—	26	8	no	—
260	D₄	—	26	8	no	—
261	D₄	—	26	16	no	—
262	D₄	—	52	8	no	—
263	D₄	—	52	16	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	13	2	no	—
267	C₂×C₄	—	13	8	no	—
268	C₂×C₄	—	13	4	no	—
269	C₂×C₄	—	26	16	no	—
270	C₂×C₄	—	26	8	no	—
271	C₂×C₄	D₄×D₁₃	1	4	yes	complemented
272	C₂×C₄	C₂³×D₁₃	1	2	yes	—
273	C₂×C₄	C₂³×D₁₃	1	4	yes	—
274	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
275	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
276	C₂×C₄	C₂³×D₁₃	1	2	yes	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	8	no	—
280	C₂²	—	13	4	no	—
281	C₂²	—	26	4	no	—
282	C₂²	—	26	8	no	—
283	C₂²	—	26	16	no	—
284	C₂²	—	104	8	no	—
285	C₂²	—	104	16	no	—
286	C₂²	C₂×D₄×D₁₃	1	2	yes	—
287	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
288	C₄	—	2	4	no	—
289	C₄	—	2	2	no	—
290	C₄	—	13	4	no	—
291	C₄	—	26	4	no	—
292	C₄	C₂×D₄×D₁₃	1	2	yes	—
293	C₄	C₂×D₄×D₁₃	1	2	yes	—
294	C₄	#88256	1	4	yes	complemented
295	C₂	—	2	2	no	—
296	C₂	—	4	4	no	—
297	C₂	—	26	4	no	—
298	C₂	—	52	8	no	—
299	C₂	C₂²×D₄×D₁₃	1	1	yes	characteristic
300	C₂	C₂×#88256	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₁₃×#532	C₂
hypercenter	#532	D₁₃
hypocenter	C₁₃	#3498
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #532

Complements

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

N	H ≅ G/N	type
1. C₂×#88256	C₂	semidirect
2. C₁₃×#532	C₂	semidirect
3. #11719535	C₂	semidirect
4. #11719430	C₂	semidirect
5. D₁₃×#127	C₂	semidirect
6. #11719379	C₂	semidirect
7. #11719314	C₂	semidirect
8. #11719258	C₂	semidirect
9. D₄×D₅₂	C₂	semidirect
10. C₂×#88128	C₂	semidirect
11. C₄×#6127	C₂	semidirect
12. C₂×C₁₃×#41	C₂²	semidirect
14. C₂×#6131	C₂²	semidirect
15. C₂×D₄×D₁₃	C₂²	semidirect
16. C₂×D₄×D₁₃	C₂²	semidirect
17. C₂×D₄×D₁₃	C₂²	semidirect
19. C₂×#6127	C₂²	semidirect
20. C₂²×D₅₂	C₂²	semidirect
21. C₂²×D₅₂	C₂²	semidirect
22. C₂²×D₅₂	C₂²	semidirect
23. C₁₃×C₄⋊Q₈	C₂²	semidirect
24. C₁₃×#127	C₂²	semidirect
25. C₁₃×#124	C₂²	semidirect
26. C₁₃×#121	C₂²	semidirect
27. C₄×D₄×C₁₃	C₂²	semidirect
28. C₂×C₄²×C₁₃	C₂²	semidirect
29. #88201	C₂²	semidirect
30. #88194	C₂²	semidirect
31. #88194	C₂²	semidirect
32. #88191	C₂²	semidirect
33. #88184	C₂²	semidirect
34. C₄×C₂²⋊D₁₃	C₂²	semidirect
35. #88180	C₂²	semidirect
36. #88149	C₂²	semidirect
37. #88147	C₂²	semidirect
38. #88138	C₂²	semidirect
39. #88136	C₂²	semidirect
40. #88128	C₂²	semidirect
41. #88128	C₂²	semidirect
42. #88128	C₂²	semidirect
43. C₄×D₅₂	C₂²	semidirect
44. #88126	C₂²	semidirect
45. C₄²×D₁₃	C₂²	semidirect
46. C₄×Dic₂₆	C₂²	semidirect
65. #6127	D₄	semidirect
106. C₂×C₄×C₁₃	C₂×D₄	semidirect
114. C₂²⋊D₁₃	C₂×D₄	semidirect
122. D₅₂	C₂×D₄	semidirect
125. C₄×D₁₃	C₂×D₄	semidirect
126. Dic₂₆	C₂×D₄	semidirect
134. #532	D₁₃	semidirect
153. C₄×C₁₃	H(5,𝔽₂)	semidirect
160. C₂×#41	C₂×D₁₃	semidirect
167. C₄⋊Q₈	C₂×D₁₃	semidirect
173. #127	C₂×D₁₃	semidirect
175. #124	C₂×D₁₃	semidirect
179. #121	C₂×D₁₃	semidirect
186. C₄×D₄	C₂×D₁₃	semidirect
190. C₂×C₄²	C₂×D₁₃	semidirect
241. C₁₃	#3498	semidirect
271. C₂×C₄	D₄×D₁₃	semidirect
294. C₄	#88256	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88256	2
C₁₃×#532	2
#11719535	2
#11719430	2
D₁₃×#127	2
#11719379	2
#11719314	2
#11719258	2
D₄×D₅₂	2
C₂×#88128	2
C₄×#6127	2
#3498	13