Subgroups — #13128506

Order 1664.

1658 conjugacy classes of subgroups.

The subgroup lattice has 322 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	1	yes	maximal, characteristic, complemented
2	C₂×#41×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#520	C₂	1	1	yes	maximal, characteristic, complemented
4	#11719282	C₂	1	2	yes	maximal, complemented
5	#11719281	C₂	1	2	yes	maximal, complemented
6	#11719259	C₂	1	6	yes	maximal, complemented
7	D₄×D₅₂	C₂	1	6	yes	maximal, complemented
8	#11719230	C₂	1	3	yes	maximal, complemented
9	#11719229	C₂	1	3	yes	maximal, complemented
10	#11719215	C₂	1	3	yes	maximal, complemented
11	C₂×#88149	C₂	1	3	yes	maximal, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	#88256	—	2	8	no	—
14	#41×D₁₃	—	2	8	no	—
15	C₂×#6131	C₂²	1	2	yes	complemented
16	C₂×#6131	C₂²	1	1	yes	characteristic, complemented
17	C₂×Q₈×D₁₃	C₂²	1	1	yes	characteristic, complemented
18	C₂×#6129	C₂²	1	3	yes	complemented
19	C₂×D₄×D₁₃	C₂²	1	3	yes	complemented
20	C₂×D₄×D₁₃	C₂²	1	6	yes	complemented
21	C₂×#6127	C₂²	1	3	yes	complemented
22	C₂×#6127	C₂²	1	3	yes	complemented
23	C₂²×D₅₂	C₂²	1	3	yes	complemented
24	C₂²×D₅₂	C₂²	1	6	yes	complemented
25	C₂²×C₄×D₁₃	C₂²	1	3	yes	complemented
26	C₄×Q₈×C₁₃	C₂²	1	2	yes	complemented
27	C₄×D₄×C₁₃	C₂²	1	6	yes	complemented
28	C₁₃×#117	C₂²	1	3	yes	complemented
29	C₂×C₁₃×C₄⋊C₄	C₂²	1	3	yes	complemented
30	#88184	C₂²	1	12	yes	complemented
31	C₂×#6103	C₂²	1	6	yes	complemented
32	#88151	C₂²	1	3	yes	complemented
33	#88151	C₂²	1	6	yes	complemented
34	#88151	C₂²	1	1	yes	characteristic, complemented
35	#88149	C₂²	1	3	yes	complemented
36	#88149	C₂²	1	6	yes	complemented
37	#88149	C₂²	1	6	yes	complemented
38	#88149	C₂²	1	6	yes	complemented
39	#88149	C₂²	1	6	yes	complemented
40	#88149	C₂²	1	3	yes	complemented
41	#88140	C₂²	1	6	yes	complemented
42	#88136	C₂²	1	6	yes	complemented
43	#88136	C₂²	1	12	yes	complemented
44	#88129	C₂²	1	6	yes	complemented
45	#88128	C₂²	1	6	yes	complemented
46	C₄×D₅₂	C₂²	1	6	yes	complemented
47	C₄×D₅₂	C₂²	1	6	yes	complemented
48	C₂³×D₁₃	C₂³	1	3	yes	—
49	C₂³×D₁₃	C₂³	1	6	yes	—
50	C₁₃×#41	D₄	1	8	yes	complemented
51	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×D₄×C₁₃	C₂³	1	3	yes	—
53	C₂²×C₄×C₁₃	C₂³	1	6	yes	—
54	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
55	C₂×C₂²⋊D₁₃	C₂³	1	6	yes	—
56	C₂×C₂²⋊D₁₃	C₂³	1	6	yes	—
57	C₂²×Dic₁₃	C₂³	1	3	yes	—
58	#6131	—	2	4	no	—
59	#6131	—	2	8	no	—
60	Q₈×D₁₃	—	2	4	no	—
61	#6129	—	2	12	no	—
62	D₄×D₁₃	—	2	12	no	—
63	D₄×D₁₃	—	2	24	no	—
64	#6127	—	2	12	no	—
65	#6127	—	2	12	no	—
66	C₂×D₅₂	—	2	12	no	—
67	C₂×D₅₂	—	4	12	no	—
68	C₂×D₅₂	C₂³	1	6	yes	—
69	C₂×D₅₂	C₂³	1	3	yes	—
70	C₂×D₅₂	C₂³	1	6	yes	—
71	C₂×D₅₂	C₂³	1	3	yes	—
72	C₂×D₅₂	C₂³	1	6	yes	—
73	C₂×D₅₂	C₂³	1	12	yes	—
74	C₂×C₄×D₁₃	—	2	12	no	—
75	C₂×C₄×D₁₃	C₂³	1	3	yes	—
76	C₂×C₄×D₁₃	C₂³	1	3	yes	—
77	C₂×C₄×D₁₃	C₂³	1	6	yes	—
78	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
79	C₂×C₄×D₁₃	C₂³	1	3	yes	—
80	C₂×Dic₂₆	C₂³	1	3	yes	—
81	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
82	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
83	C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
84	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
85	C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
86	C₄²×C₁₃	C₂³	1	6	yes	—
87	#6103	C₂³	1	2	yes	—
88	#6103	C₂³	1	6	yes	—
89	#6103	C₂³	1	6	yes	—
90	#6103	C₂³	1	6	yes	—
91	#6103	C₂³	1	12	yes	—
92	C₄⋊Dic₁₃	C₂³	1	2	yes	—
93	C₄⋊Dic₁₃	C₂³	1	6	yes	—
94	#3488	—	13	1	no	maximal, Sylow
95	C₂³×C₁₃	C₂⁴	1	3	yes	—
96	C₂²×D₁₃	—	2	6	no	—
97	C₂²×D₁₃	—	2	12	no	—
98	C₂²×D₁₃	—	2	6	no	—
99	C₂²×D₁₃	—	4	3	no	—
100	C₂²×D₁₃	—	4	6	no	—
101	C₂²×D₁₃	—	4	6	no	—
102	C₂²×D₁₃	C₂⁴	1	6	yes	—
103	C₂²×D₁₃	C₂⁴	1	3	yes	—
104	C₂²×D₁₃	C₂⁴	1	1	yes	characteristic
105	Q₈×C₁₃	C₂×D₄	1	4	yes	complemented
106	D₄×C₁₃	C₂×D₄	1	12	yes	complemented
107	C₂×C₄×C₁₃	—	4	6	no	—
108	C₂×C₄×C₁₃	C₂×D₄	1	12	yes	complemented
109	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
110	C₂×C₄×C₁₃	C₂⁴	1	6	yes	—
111	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
112	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
113	C₂²⋊D₁₃	—	2	12	no	—
114	C₂²⋊D₁₃	—	2	12	no	—
115	C₂×Dic₁₃	—	2	6	no	—
116	C₂×Dic₁₃	C₂⁴	1	2	yes	—
117	C₂×Dic₁₃	C₂⁴	1	3	yes	—
118	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
119	D₅₂	—	2	12	no	—
120	D₅₂	—	2	6	no	—
121	D₅₂	—	2	12	no	—
122	D₅₂	—	2	6	no	—
123	D₅₂	—	4	12	no	—
124	D₅₂	—	4	6	no	—
125	C₄×D₁₃	—	2	2	no	—
126	C₄×D₁₃	—	2	6	no	—
127	C₄×D₁₃	—	2	6	no	—
128	C₄×D₁₃	—	2	6	no	—
129	C₄×D₁₃	—	2	12	no	—
130	Dic₂₆	—	2	2	no	—
131	Dic₂₆	—	2	6	no	—
132	C₂×H(5,𝔽₂)	—	13	1	no	—
133	C₂²×#41	—	13	1	no	—
134	#550	—	13	2	no	—
135	#548	—	13	2	no	—
136	#546	—	13	6	no	—
137	D₄²	—	13	6	no	—
138	#537	—	13	3	no	—
139	#535	—	13	3	no	—
140	#525	—	13	3	no	—
141	C₂×#121	—	13	3	no	—
142	#520	D₁₃	1	1	yes	characteristic, complemented
143	C₂²×C₁₃	—	2	3	no	—
144	C₂²×C₁₃	—	2	3	no	—
145	C₂²×C₁₃	C₂²×D₄	1	6	yes	—
146	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
147	C₂×D₁₃	—	2	1	no	—
148	C₂×D₁₃	—	2	1	no	—
149	C₂×D₁₃	—	2	3	no	—
150	C₂×D₁₃	—	2	6	no	—
151	C₂×D₁₃	—	2	6	no	—
152	C₂×D₁₃	—	2	3	no	—
153	C₂×D₁₃	—	2	3	no	—
154	C₂×D₁₃	—	2	6	no	—
155	C₂×D₁₃	—	4	12	no	—
156	C₂×D₁₃	—	8	12	no	—
157	C₂×D₁₃	H(5,𝔽₂)	1	2	yes	complemented
158	C₄×C₁₃	—	2	6	no	—
159	C₄×C₁₃	—	2	2	no	—
160	C₄×C₁₃	C₂²×D₄	1	2	yes	—
161	C₄×C₁₃	C₂²×D₄	1	6	yes	—
162	Dic₁₃	—	2	3	no	—
163	Dic₁₃	—	2	2	no	—
164	Dic₁₃	—	2	1	no	—
165	H(5,𝔽₂)	—	26	8	no	—
166	C₂×#41	—	13	2	no	—
167	C₂×#41	—	13	1	no	—
168	C₂×#41	—	13	3	no	—
169	C₂×#41	—	13	3	no	—
170	C₂×#41	—	13	3	no	—
171	C₂×#41	—	26	8	no	—
172	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
173	C₂²×Q₈	—	13	1	no	—
174	C₂²×D₄	—	13	3	no	—
175	C₂²×D₄	—	13	6	no	—
176	C₂²×D₄	—	13	3	no	—
177	C₂²×D₄	—	13	6	no	—
178	C₂³×C₄	—	13	3	no	—
179	#127	—	13	6	no	—
180	#124	—	13	6	no	—
181	#123	—	13	6	no	—
182	C₂²⋊Q₈	—	13	1	no	—
183	C₂²⋊Q₈	—	13	3	no	—
184	C₂²⋊Q₈	—	13	6	no	—
185	#121	—	13	12	no	—
186	#121	—	13	3	no	—
187	#121	—	13	6	no	—
188	#121	—	13	6	no	—
189	#121	—	13	6	no	—
190	#121	—	13	3	no	—
191	#121	—	13	6	no	—
192	#120	—	13	6	no	—
193	#120	—	13	12	no	—
194	C₄×Q₈	C₂×D₁₃	1	2	yes	complemented
195	C₄×D₄	—	13	6	no	—
196	C₄×D₄	—	13	6	no	—
197	C₄×D₄	C₂×D₁₃	1	6	yes	complemented
198	#117	C₂×D₁₃	1	3	yes	complemented
199	C₂×C₄⋊C₄	C₂×D₁₃	1	3	yes	complemented
200	C₂×C₂²⋊C₄	—	13	6	no	—
201	C₂×C₁₃	—	2	6	no	—
202	C₂×C₁₃	C₂³×D₄	1	1	yes	characteristic
203	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
204	C₂×C₁₃	#585	1	1	yes	characteristic
205	D₁₃	—	2	2	no	—
206	D₁₃	—	4	3	no	—
207	D₁₃	—	4	6	no	—
208	C₂⁴	—	13	3	no	—
209	C₂⁴	—	13	6	no	—
210	#41	—	26	4	no	—
211	#41	—	26	8	no	—
212	#41	—	26	12	no	—
213	#41	—	26	12	no	—
214	#41	—	26	12	no	—
215	#41	D₅₂	1	8	yes	complemented
216	C₂×Q₈	—	13	1	no	—
217	C₂×Q₈	—	13	3	no	—
218	C₂×Q₈	—	26	4	no	—
219	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
220	C₂×D₄	—	13	6	no	—
221	C₂×D₄	—	13	6	no	—
222	C₂×D₄	—	13	6	no	—
223	C₂×D₄	—	13	3	no	—
224	C₂×D₄	—	13	6	no	—
225	C₂×D₄	—	13	3	no	—
226	C₂×D₄	—	13	6	no	—
227	C₂×D₄	—	13	12	no	—
228	C₂×D₄	—	26	12	no	—
229	C₂×D₄	—	26	24	no	—
230	C₂×D₄	—	26	12	no	—
231	C₂×D₄	—	52	12	no	—
232	C₂×D₄	C₂²×D₁₃	1	3	yes	—
233	C₂²×C₄	—	13	3	no	—
234	C₂²×C₄	—	13	3	no	—
235	C₂²×C₄	—	13	3	no	—
236	C₂²×C₄	—	13	6	no	—
237	C₂²×C₄	—	13	1	no	—
238	C₂²×C₄	—	13	3	no	—
239	C₂²×C₄	—	26	12	no	—
240	C₂²×C₄	C₂²×D₁₃	1	6	yes	—
241	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
242	C₄⋊C₄	—	13	2	no	—
243	C₄⋊C₄	—	13	6	no	—
244	C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
245	C₄⋊C₄	C₂²×D₁₃	1	6	yes	—
246	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
247	C₂²⋊C₄	—	13	6	no	—
248	C₂²⋊C₄	—	13	12	no	—
249	C₂²⋊C₄	—	13	2	no	—
250	C₂²⋊C₄	—	13	6	no	—
251	C₂²⋊C₄	—	13	6	no	—
252	C₂²⋊C₄	C₂²×D₁₃	1	6	yes	—
253	C₄²	C₂²×D₁₃	1	6	yes	—
254	C₁₃	#3488	1	1	yes	characteristic, complemented, Sylow
255	C₂³	—	13	1	no	—
256	C₂³	—	13	6	no	—
257	C₂³	—	13	3	no	—
258	C₂³	—	26	6	no	—
259	C₂³	—	26	6	no	—
260	C₂³	—	26	12	no	—
261	C₂³	—	52	3	no	—
262	C₂³	—	52	6	no	—
263	C₂³	—	52	6	no	—
264	C₂³	C₂³×D₁₃	1	3	yes	—
265	Q₈	—	26	6	no	—
266	Q₈	—	26	2	no	—
267	Q₈	C₂×D₅₂	1	4	yes	complemented
268	D₄	—	26	12	no	—
269	D₄	—	26	12	no	—
270	D₄	—	26	12	no	—
271	D₄	—	26	6	no	—
272	D₄	—	26	12	no	—
273	D₄	—	26	6	no	—
274	D₄	—	52	6	no	—
275	D₄	—	52	12	no	—
276	D₄	C₂×D₅₂	1	12	yes	complemented
277	C₂×C₄	—	4	6	no	—
278	C₂×C₄	—	13	1	no	—
279	C₂×C₄	—	13	3	no	—
280	C₂×C₄	—	13	2	no	—
281	C₂×C₄	—	26	6	no	—
282	C₂×C₄	—	26	2	no	—
283	C₂×C₄	—	26	6	no	—
284	C₂×C₄	—	26	6	no	—
285	C₂×C₄	—	26	6	no	—
286	C₂×C₄	—	26	12	no	—
287	C₂×C₄	C₂×D₅₂	1	12	yes	complemented
288	C₂×C₄	C₂³×D₁₃	1	2	yes	—
289	C₂×C₄	C₂³×D₁₃	1	6	yes	—
290	C₂×C₄	C₂³×D₁₃	1	3	yes	—
291	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
292	C₂²	—	2	3	no	—
293	C₂²	—	2	3	no	—
294	C₂²	—	13	2	no	—
295	C₂²	—	26	1	no	—
296	C₂²	—	26	1	no	—
297	C₂²	—	26	3	no	—
298	C₂²	—	26	6	no	—
299	C₂²	—	26	6	no	—
300	C₂²	—	26	3	no	—
301	C₂²	—	26	3	no	—
302	C₂²	—	26	6	no	—
303	C₂²	—	52	12	no	—
304	C₂²	—	104	12	no	—
305	C₂²	C₂²×D₅₂	1	6	yes	—
306	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
307	C₄	—	2	6	no	—
308	C₄	—	2	2	no	—
309	C₄	—	26	1	no	—
310	C₄	—	26	2	no	—
311	C₄	—	26	3	no	—
312	C₄	C₂²×D₅₂	1	2	yes	—
313	C₄	C₂²×D₅₂	1	6	yes	—
314	C₂	—	2	6	no	—
315	C₂	—	26	2	no	—
316	C₂	—	52	3	no	—
317	C₂	—	52	6	no	—
318	C₂	C₂³×D₅₂	1	1	yes	characteristic
319	C₂	D₁₃×H(5,𝔽₂)	1	1	yes	characteristic
320	C₂	#11719652	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₁₃×#520	C₂
hypercenter	#520	D₁₃
hypocenter	C₁₃	#3488
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #520

Complements

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

N	H ≅ G/N	type
1. C₂×#88256	C₂	semidirect
2. C₂×#41×D₁₃	C₂	semidirect
3. C₁₃×#520	C₂	semidirect
4. #11719282	C₂	semidirect
5. #11719281	C₂	semidirect
6. #11719259	C₂	semidirect
7. D₄×D₅₂	C₂	semidirect
8. #11719230	C₂	semidirect
9. #11719229	C₂	semidirect
10. #11719215	C₂	semidirect
11. C₂×#88149	C₂	semidirect
15. C₂×#6131	C₂²	semidirect
16. C₂×#6131	C₂²	semidirect
17. C₂×Q₈×D₁₃	C₂²	semidirect
18. C₂×#6129	C₂²	semidirect
19. C₂×D₄×D₁₃	C₂²	semidirect
20. C₂×D₄×D₁₃	C₂²	semidirect
21. C₂×#6127	C₂²	semidirect
22. C₂×#6127	C₂²	semidirect
23. C₂²×D₅₂	C₂²	semidirect
24. C₂²×D₅₂	C₂²	semidirect
25. C₂²×C₄×D₁₃	C₂²	semidirect
26. C₄×Q₈×C₁₃	C₂²	semidirect
27. C₄×D₄×C₁₃	C₂²	semidirect
28. C₁₃×#117	C₂²	semidirect
29. C₂×C₁₃×C₄⋊C₄	C₂²	semidirect
30. #88184	C₂²	semidirect
31. C₂×#6103	C₂²	semidirect
32. #88151	C₂²	semidirect
33. #88151	C₂²	semidirect
34. #88151	C₂²	semidirect
35. #88149	C₂²	semidirect
36. #88149	C₂²	semidirect
37. #88149	C₂²	semidirect
38. #88149	C₂²	semidirect
39. #88149	C₂²	semidirect
40. #88149	C₂²	semidirect
41. #88140	C₂²	semidirect
42. #88136	C₂²	semidirect
43. #88136	C₂²	semidirect
44. #88129	C₂²	semidirect
45. #88128	C₂²	semidirect
46. C₄×D₅₂	C₂²	semidirect
47. C₄×D₅₂	C₂²	semidirect
50. C₁₃×#41	D₄	semidirect
105. Q₈×C₁₃	C₂×D₄	semidirect
106. D₄×C₁₃	C₂×D₄	semidirect
108. C₂×C₄×C₁₃	C₂×D₄	semidirect
142. #520	D₁₃	semidirect
157. C₂×D₁₃	H(5,𝔽₂)	semidirect
194. C₄×Q₈	C₂×D₁₃	semidirect
197. C₄×D₄	C₂×D₁₃	semidirect
198. #117	C₂×D₁₃	semidirect
199. C₂×C₄⋊C₄	C₂×D₁₃	semidirect
215. #41	D₅₂	semidirect
254. C₁₃	#3488	semidirect
267. Q₈	C₂×D₅₂	semidirect
276. D₄	C₂×D₅₂	semidirect
287. C₂×C₄	C₂×D₅₂	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
C₂×#41×D₁₃	2
C₁₃×#520	2
#11719282	2
#11719281	2
#11719259	2
D₄×D₅₂	2
#11719230	2
#11719229	2
#11719215	2
C₂×#88149	2
#3488	13