Subgroups — #13128613

Order 1664.

1748 conjugacy classes of subgroups.

The subgroup lattice has 298 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×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719539	C₂	1	2	yes	maximal, complemented
4	#11719422	C₂	1	3	yes	maximal, complemented
5	D₁₃×#127	C₂	1	3	yes	maximal, complemented
6	#11719314	C₂	1	6	yes	maximal, complemented
7	#11719312	C₂	1	6	yes	maximal, complemented
8	#11719282	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#6131	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×D₅₂	C₂	1	3	yes	maximal, complemented
11	#11719253	C₂	1	3	yes	maximal, complemented
12	C₂×C₁₃×#41	C₂²	1	2	yes	complemented
13	#88256	—	2	16	no	—
14	C₂×#6131	C₂²	1	1	yes	characteristic
15	C₂×#6131	C₂²	1	2	yes	complemented
16	C₂×D₄×D₁₃	C₂²	1	6	yes	complemented
17	C₂×D₄×D₁₃	C₂²	1	6	yes	complemented
18	C₂×D₄×D₁₃	C₂²	1	6	yes	complemented
19	C₂×D₄×D₁₃	C₂²	1	6	yes	complemented
20	C₂×#6127	C₂²	1	6	yes	complemented
21	C₂²×D₅₂	C₂²	1	6	yes	complemented
22	C₁₃×#127	C₂²	1	3	yes	complemented
23	C₁₃×#121	C₂²	1	6	yes	complemented
24	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
25	C₄×D₄×C₁₃	C₂²	1	3	yes	complemented
26	Q₈×Dic₁₃	C₂²	1	1	yes	characteristic, complemented
27	#88194	C₂²	1	3	yes	complemented
28	#88194	C₂²	1	6	yes	complemented
29	#88194	C₂²	1	6	yes	complemented
30	#88192	C₂²	1	3	yes	complemented
31	#88192	C₂²	1	6	yes	complemented
32	#88191	C₂²	1	12	yes	complemented
33	#88184	C₂²	1	6	yes	complemented
34	C₄×C₂²⋊D₁₃	C₂²	1	6	yes	complemented
35	#88149	C₂²	1	3	yes	complemented
36	#88149	C₂²	1	6	yes	complemented
37	#88147	C₂²	1	3	yes	complemented
38	#88146	C₂²	1	3	yes	complemented
39	#88142	C₂²	1	1	yes	characteristic, complemented
40	#88139	C₂²	1	6	yes	complemented
41	#88138	C₂²	1	6	yes	complemented
42	#88136	C₂²	1	12	yes	complemented
43	#88128	C₂²	1	3	yes	complemented
44	C₄×D₅₂	C₂²	1	3	yes	complemented
45	C₄×D₅₂	C₂²	1	3	yes	complemented
46	C₄²×D₁₃	C₂²	1	3	yes	complemented
47	C₂³×D₁₃	C₂³	1	12	yes	—
48	C₁₃×#41	—	2	8	no	—
49	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₁₃	C₂³	1	3	yes	—
51	C₂×D₄×C₁₃	C₂³	1	6	yes	—
52	C₂×D₄×C₁₃	C₂³	1	6	yes	—
53	C₂²×C₄×C₁₃	C₂³	1	6	yes	—
54	C₂×C₂²⋊D₁₃	C₂³	1	6	yes	—
55	C₂×C₂²⋊D₁₃	C₂³	1	6	yes	—
56	C₂×C₂²⋊D₁₃	C₂³	1	6	yes	—
57	C₂×C₂²⋊D₁₃	C₂³	1	12	yes	—
58	#6131	—	2	8	no	—
59	#6131	D₄	1	8	yes	complemented
60	D₄×D₁₃	—	2	24	no	—
61	D₄×D₁₃	—	2	24	no	—
62	D₄×D₁₃	—	4	12	no	—
63	D₄×D₁₃	—	4	12	no	—
64	#6127	—	2	24	no	—
65	C₂×D₅₂	—	2	24	no	—
66	C₂×D₅₂	C₂³	1	3	yes	—
67	C₂×D₅₂	C₂³	1	6	yes	—
68	C₂×D₅₂	C₂³	1	6	yes	—
69	C₂×D₅₂	C₂³	1	6	yes	—
70	C₂×D₅₂	C₂³	1	3	yes	—
71	C₂×C₄×D₁₃	C₂³	1	3	yes	—
72	C₂×C₄×D₁₃	C₂³	1	3	yes	—
73	C₂×C₄×D₁₃	C₂³	1	6	yes	—
74	C₂×C₄×D₁₃	C₂³	1	3	yes	—
75	C₂×Dic₂₆	C₂³	1	2	yes	—
76	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
78	C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
79	C₄²×C₁₃	C₂³	1	3	yes	—
80	C₂²⋊Dic₁₃	C₂³	1	6	yes	—
81	#6103	C₂³	1	6	yes	—
82	#6103	C₂³	1	6	yes	—
83	#6103	C₂³	1	6	yes	—
84	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₁₃	C₂³	1	3	yes	—
86	#6101	C₂³	1	2	yes	—
87	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
88	C₄×Dic₁₃	C₂³	1	3	yes	—
89	C₄×Dic₁₃	C₂³	1	3	yes	—
90	#3498	—	13	1	no	maximal, Sylow
91	C₂³×C₁₃	C₂⁴	1	6	yes	—
92	C₂²×D₁₃	—	2	24	no	—
93	C₂²×D₁₃	—	4	12	no	—
94	C₂²×D₁₃	—	4	12	no	—
95	C₂²×D₁₃	C₂⁴	1	6	yes	—
96	C₂²×D₁₃	C₂⁴	1	3	yes	—
97	Q₈×C₁₃	C₂×D₄	1	4	yes	complemented
98	D₄×C₁₃	—	2	12	no	—
99	D₄×C₁₃	—	2	6	no	—
100	D₄×C₁₃	—	4	6	no	—
101	C₂×C₄×C₁₃	—	2	12	no	—
102	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
103	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
104	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
105	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₁₃	—	2	12	no	—
107	C₂²⋊D₁₃	—	2	12	no	—
108	C₂²⋊D₁₃	—	2	12	no	—
109	C₂²⋊D₁₃	—	4	12	no	—
110	C₂×Dic₁₃	C₂⁴	1	3	yes	—
111	C₂×Dic₁₃	C₂⁴	1	2	yes	—
112	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
113	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
114	D₅₂	—	2	12	no	—
115	D₅₂	—	2	12	no	—
116	D₅₂	—	2	6	no	—
117	D₅₂	—	4	6	no	—
118	D₅₂	C₂×D₄	1	12	yes	complemented
119	C₄×D₁₃	—	2	12	no	—
120	C₄×D₁₃	—	2	6	no	—
121	C₄×D₁₃	—	2	6	no	—
122	C₄×D₁₃	C₂×D₄	1	12	yes	complemented
123	Dic₂₆	—	2	4	no	—
124	C₂×H(5,𝔽₂)	—	13	2	no	—
125	#550	—	13	2	no	—
126	#550	—	13	1	no	—
127	#550	D₁₃	1	1	yes	characteristic, complemented
128	D₄²	—	13	3	no	—
129	D₄²	—	13	6	no	—
130	D₄²	—	13	3	no	—
131	#535	—	13	6	no	—
132	#532	—	13	3	no	—
133	C₂×#127	—	13	3	no	—
134	C₄×#41	—	13	1	no	—
135	C₂²×C₁₃	—	2	6	no	—
136	C₂²×C₁₃	—	2	6	no	—
137	C₂²×C₁₃	—	2	6	no	—
138	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
139	C₂×D₁₃	—	2	3	no	—
140	C₂×D₁₃	—	2	3	no	—
141	C₂×D₁₃	—	2	6	no	—
142	C₂×D₁₃	—	2	6	no	—
143	C₂×D₁₃	—	2	6	no	—
144	C₂×D₁₃	—	8	24	no	—
145	C₂×D₁₃	C₂²×D₄	1	6	yes	—
146	C₄×C₁₃	—	2	2	no	—
147	C₄×C₁₃	—	2	3	no	—
148	C₄×C₁₃	C₂²×D₄	1	6	yes	—
149	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	complemented
150	Dic₁₃	—	2	2	no	—
151	Dic₁₃	—	2	3	no	—
152	Dic₁₃	C₂²×D₄	1	2	yes	—
153	Dic₁₃	H(5,𝔽₂)	1	2	yes	complemented
154	H(5,𝔽₂)	—	26	16	no	—
155	C₂×#41	—	13	1	no	—
156	C₂×#41	—	13	2	no	—
157	C₂×#41	—	13	6	no	—
158	C₂×#41	C₂×D₁₃	1	2	yes	complemented
159	C₂²×D₄	—	13	6	no	—
160	C₂²×D₄	—	13	6	no	—
161	C₂²×D₄	—	13	6	no	—
162	C₂²×D₄	—	13	6	no	—
163	C₂²×D₄	—	13	6	no	—
164	C₄⋊Q₈	—	13	1	no	—
165	#127	—	13	3	no	—
166	#127	—	13	6	no	—
167	#127	—	13	6	no	—
168	#127	—	13	3	no	—
169	#127	C₂×D₁₃	1	3	yes	complemented
170	#124	—	13	6	no	—
171	#121	—	13	3	no	—
172	#121	—	13	6	no	—
173	#121	—	13	6	no	—
174	#121	—	13	3	no	—
175	#121	—	13	6	no	—
176	#121	—	13	6	no	—
177	#121	C₂×D₁₃	1	6	yes	complemented
178	#120	—	13	12	no	—
179	#120	—	13	12	no	—
180	C₄×Q₈	—	13	1	no	—
181	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic, complemented
182	C₄×D₄	—	13	6	no	—
183	C₄×D₄	—	13	3	no	—
184	C₄×D₄	—	13	3	no	—
185	C₄×D₄	—	13	3	no	—
186	C₄×D₄	C₂×D₁₃	1	3	yes	complemented
187	#117	—	13	3	no	—
188	C₂×C₄²	—	13	3	no	—
189	C₂×C₁₃	—	4	6	no	—
190	C₂×C₁₃	C₂³×D₄	1	1	yes	characteristic
191	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
192	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
193	D₁₃	—	2	6	no	—
194	D₁₃	—	4	6	no	—
195	C₂⁴	—	13	12	no	—
196	#41	—	2	8	no	—
197	#41	—	13	8	no	—
198	#41	—	26	24	no	—
199	#41	—	26	8	no	—
200	C₂×Q₈	—	13	2	no	—
201	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
202	C₂×D₄	—	13	6	no	—
203	C₂×D₄	—	13	6	no	—
204	C₂×D₄	—	13	6	no	—
205	C₂×D₄	—	13	12	no	—
206	C₂×D₄	—	13	3	no	—
207	C₂×D₄	—	13	3	no	—
208	C₂×D₄	—	13	6	no	—
209	C₂×D₄	—	13	6	no	—
210	C₂×D₄	—	13	6	no	—
211	C₂×D₄	—	26	24	no	—
212	C₂×D₄	—	26	24	no	—
213	C₂×D₄	—	26	24	no	—
214	C₂×D₄	—	52	12	no	—
215	C₂×D₄	—	52	12	no	—
216	C₂×D₄	C₂²×D₁₃	1	3	yes	—
217	C₂×D₄	C₂²×D₁₃	1	6	yes	—
218	C₂×D₄	C₂²×D₁₃	1	6	yes	—
219	C₂²×C₄	—	13	3	no	—
220	C₂²×C₄	—	13	3	no	—
221	C₂²×C₄	—	13	3	no	—
222	C₂²×C₄	—	13	6	no	—
223	C₂²×C₄	C₂²×D₁₃	1	6	yes	—
224	C₄⋊C₄	—	13	3	no	—
225	C₄⋊C₄	—	13	2	no	—
226	C₄⋊C₄	—	13	1	no	—
227	C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
228	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
229	C₂²⋊C₄	—	13	6	no	—
230	C₂²⋊C₄	—	13	6	no	—
231	C₂²⋊C₄	—	13	6	no	—
232	C₂²⋊C₄	—	13	6	no	—
233	C₂²⋊C₄	C₂²×D₁₃	1	6	yes	—
234	C₄²	—	13	1	no	—
235	C₄²	—	13	3	no	—
236	C₄²	—	13	3	no	—
237	C₄²	C₂²×D₁₃	1	3	yes	—
238	C₁₃	#3498	1	1	yes	characteristic, complemented, Sylow
239	C₂³	—	13	3	no	—
240	C₂³	—	13	6	no	—
241	C₂³	—	26	24	no	—
242	C₂³	—	52	12	no	—
243	C₂³	—	52	12	no	—
244	C₂³	C₂³×D₁₃	1	6	yes	—
245	Q₈	—	26	4	no	—
246	Q₈	D₄×D₁₃	1	4	yes	complemented
247	D₄	—	2	12	no	—
248	D₄	—	2	6	no	—
249	D₄	—	4	6	no	—
250	D₄	—	13	12	no	—
251	D₄	—	26	12	no	—
252	D₄	—	26	12	no	—
253	D₄	—	26	12	no	—
254	D₄	—	26	12	no	—
255	D₄	—	26	12	no	—
256	D₄	—	26	6	no	—
257	D₄	—	52	6	no	—
258	D₄	—	52	12	no	—
259	C₂×C₄	—	2	12	no	—
260	C₂×C₄	—	13	1	no	—
261	C₂×C₄	—	13	1	no	—
262	C₂×C₄	—	13	12	no	—
263	C₂×C₄	—	13	3	no	—
264	C₂×C₄	—	13	2	no	—
265	C₂×C₄	—	26	6	no	—
266	C₂×C₄	—	26	12	no	—
267	C₂×C₄	—	26	6	no	—
268	C₂×C₄	C₂³×D₁₃	1	3	yes	—
269	C₂×C₄	C₂³×D₁₃	1	2	yes	—
270	C₂×C₄	C₂³×D₁₃	1	3	yes	—
271	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
272	C₂²	—	2	6	no	—
273	C₂²	—	2	6	no	—
274	C₂²	—	2	6	no	—
275	C₂²	—	13	6	no	—
276	C₂²	—	26	3	no	—
277	C₂²	—	26	3	no	—
278	C₂²	—	26	6	no	—
279	C₂²	—	26	6	no	—
280	C₂²	—	26	6	no	—
281	C₂²	—	104	24	no	—
282	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	3	no	—
285	C₄	—	13	2	no	—
286	C₄	—	13	2	no	—
287	C₄	—	26	2	no	—
288	C₄	—	26	3	no	—
289	C₄	C₂×D₄×D₁₃	1	6	yes	—
290	C₄	#88256	1	2	yes	complemented
291	C₂	—	4	6	no	—
292	C₂	—	26	6	no	—
293	C₂	—	52	6	no	—
294	C₂	C₂²×D₄×D₁₃	1	1	yes	characteristic
295	C₂	C₂×#88256	1	1	yes	characteristic
296	C₂	D₁₃×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₁₃×#550	C₂
hypercenter	#550	D₁₃
hypocenter	C₁₃	#3498
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #550

Complements

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

N	H ≅ G/N	type
1. C₂×#88256	C₂	semidirect
2. C₁₃×#550	C₂	semidirect
3. #11719539	C₂	semidirect
4. #11719422	C₂	semidirect
5. D₁₃×#127	C₂	semidirect
6. #11719314	C₂	semidirect
7. #11719312	C₂	semidirect
8. #11719282	C₂	semidirect
9. C₄×#6131	C₂	semidirect
10. D₄×D₅₂	C₂	semidirect
11. #11719253	C₂	semidirect
12. C₂×C₁₃×#41	C₂²	semidirect
15. C₂×#6131	C₂²	semidirect
16. C₂×D₄×D₁₃	C₂²	semidirect
17. C₂×D₄×D₁₃	C₂²	semidirect
18. C₂×D₄×D₁₃	C₂²	semidirect
19. C₂×D₄×D₁₃	C₂²	semidirect
20. C₂×#6127	C₂²	semidirect
21. C₂²×D₅₂	C₂²	semidirect
22. C₁₃×#127	C₂²	semidirect
23. C₁₃×#121	C₂²	semidirect
24. C₄×Q₈×C₁₃	C₂²	semidirect
25. C₄×D₄×C₁₃	C₂²	semidirect
26. Q₈×Dic₁₃	C₂²	semidirect
27. #88194	C₂²	semidirect
28. #88194	C₂²	semidirect
29. #88194	C₂²	semidirect
30. #88192	C₂²	semidirect
31. #88192	C₂²	semidirect
32. #88191	C₂²	semidirect
33. #88184	C₂²	semidirect
34. C₄×C₂²⋊D₁₃	C₂²	semidirect
35. #88149	C₂²	semidirect
36. #88149	C₂²	semidirect
37. #88147	C₂²	semidirect
38. #88146	C₂²	semidirect
39. #88142	C₂²	semidirect
40. #88139	C₂²	semidirect
41. #88138	C₂²	semidirect
42. #88136	C₂²	semidirect
43. #88128	C₂²	semidirect
44. C₄×D₅₂	C₂²	semidirect
45. C₄×D₅₂	C₂²	semidirect
46. C₄²×D₁₃	C₂²	semidirect
59. #6131	D₄	semidirect
97. Q₈×C₁₃	C₂×D₄	semidirect
118. D₅₂	C₂×D₄	semidirect
122. C₄×D₁₃	C₂×D₄	semidirect
127. #550	D₁₃	semidirect
149. C₄×C₁₃	H(5,𝔽₂)	semidirect
153. Dic₁₃	H(5,𝔽₂)	semidirect
158. C₂×#41	C₂×D₁₃	semidirect
169. #127	C₂×D₁₃	semidirect
177. #121	C₂×D₁₃	semidirect
181. C₄×Q₈	C₂×D₁₃	semidirect
186. C₄×D₄	C₂×D₁₃	semidirect
238. C₁₃	#3498	semidirect
246. Q₈	D₄×D₁₃	semidirect
290. 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×#550	2
#11719539	2
#11719422	2
D₁₃×#127	2
#11719314	2
#11719312	2
#11719282	2
C₄×#6131	2
D₄×D₅₂	2
#11719253	2
#3498	13