Subgroups — #16776787

Order 3328.

10796 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13129649	C₂	1	1	yes	maximal, characteristic, complemented
2	#13129647	C₂	1	1	yes	maximal, characteristic, complemented
3	#13129624	C₂	1	15	yes	maximal, complemented
4	#13129617	C₂	1	15	yes	maximal, complemented
5	#13129610	C₂	1	15	yes	maximal, complemented
6	C₂×#11719456	C₂	1	15	yes	maximal, complemented
7	C₁₃×#3573	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#585	C₂²	1	1	yes	characteristic
9	#11719652	—	2	6	no	—
10	D₁₃×2₋¹⁺⁴	—	2	6	no	—
11	#11719650	—	2	10	no	—
12	D₁₃×H(5,𝔽₂)	—	2	10	no	—
13	#11719648	C₂²	1	15	yes	complemented
14	#11719648	C₂²	1	15	yes	complemented
15	C₂×#88257	C₂²	1	15	yes	complemented
16	C₂×#88256	C₂²	1	15	yes	complemented
17	C₁₃×#568	C₂²	1	15	yes	complemented
18	C₂×C₁₃×#131	C₂²	1	15	yes	complemented
19	#11719458	C₂²	1	60	yes	complemented
20	#11719457	C₂²	1	60	yes	complemented
21	#11719456	C₂²	1	60	yes	complemented
22	#11719456	C₂²	1	120	yes	complemented
23	#11719456	C₂²	1	20	yes	complemented
24	#11719453	C₂²	1	90	yes	complemented
25	C₂×#88163	C₂²	1	45	yes	complemented
26	C₂×#88162	C₂²	1	45	yes	complemented
27	C₂×#88158	C₂²	1	45	yes	complemented
28	C₂²×#6095	C₂²	1	15	yes	complemented
29	C₁₃×2₋¹⁺⁴	D₄	1	6	yes	—
30	C₁₃×H(5,𝔽₂)	D₄	1	10	yes	—
31	C₂×C₁₃×#41	C₂³	1	15	yes	—
32	#88257	—	2	60	no	—
33	#88257	C₂³	1	20	yes	—
34	#88257	C₂³	1	60	yes	complemented
35	#88256	—	2	60	no	—
36	#88256	C₂³	1	60	yes	—
37	#88256	C₂³	1	20	yes	complemented
38	#41×D₁₃	—	2	60	no	—
39	#41×D₁₃	—	2	60	no	—
40	#88254	—	2	15	no	—
41	#88254	—	2	15	no	—
42	C₂×#6131	—	2	15	no	—
43	C₂×Q₈×D₁₃	—	2	15	no	—
44	#88251	—	2	45	no	—
45	#88251	—	2	45	no	—
46	C₂×#6129	—	2	45	no	—
47	C₂×D₄×D₁₃	—	2	45	no	—
48	C₂×#6127	C₂³	1	45	yes	—
49	C₂×#6127	C₂³	1	45	yes	complemented
50	C₂²×D₅₂	C₂³	1	15	yes	—
51	C₂²×Dic₂₆	C₂³	1	15	yes	complemented
52	C₁₃×#131	C₂³	1	20	yes	complemented
53	C₁₃×#131	C₂³	1	60	yes	complemented
54	C₂×C₁₃×M₁₆	C₂³	1	45	yes	complemented
55	C₂²×C₈×C₁₃	C₂³	1	15	yes	complemented
56	#88163	C₂³	1	180	yes	complemented
57	#88163	C₂³	1	60	yes	complemented
58	#88162	C₂³	1	180	yes	complemented
59	#88162	C₂³	1	60	yes	complemented
60	C₂×Dic₅₂	C₂³	1	45	yes	complemented
61	#88158	C₂³	1	60	yes	complemented
62	#88158	C₂³	1	180	yes	complemented
63	C₂×D₁₀₄	C₂³	1	45	yes	complemented
64	C₂×#6095	C₂³	1	45	yes	complemented
65	C₂×#6095	C₂³	1	90	yes	complemented
66	C₂×#6095	C₂³	1	15	yes	complemented
67	#63092	—	13	1	no	maximal, Sylow
68	C₂³×D₁₃	—	2	15	no	—
69	C₁₃×#41	C₂×D₄	1	60	yes	—
70	C₁₃×#41	C₂⁴	1	20	yes	—
71	C₂×Q₈×C₁₃	C₂×D₄	1	15	yes	—
72	C₂×D₄×C₁₃	C₂×D₄	1	45	yes	—
73	C₂²×C₄×C₁₃	C₂⁴	1	15	yes	—
74	C₂×C₂²⋊D₁₃	—	2	45	no	—
75	C₂×C₂²⋊D₁₃	—	2	45	no	—
76	C₂²×Dic₁₃	—	2	15	no	—
77	#6131	—	2	60	no	—
78	#6131	—	2	20	no	—
79	Q₈×D₁₃	—	2	60	no	—
80	Q₈×D₁₃	—	2	20	no	—
81	#6129	—	2	180	no	—
82	#6129	—	2	60	no	—
83	D₄×D₁₃	—	2	180	no	—
84	D₄×D₁₃	—	2	60	no	—
85	#6127	—	2	90	no	—
86	#6127	—	2	90	no	—
87	#6127	C₂⁴	1	60	yes	—
88	#6127	C₂⁴	1	60	yes	—
89	C₂×D₅₂	—	2	45	no	—
90	C₂×D₅₂	C₂⁴	1	45	yes	—
91	C₂×D₅₂	C₂⁴	1	15	yes	—
92	C₂×C₄×D₁₃	—	2	45	no	—
93	C₂×C₄×D₁₃	—	2	45	no	—
94	C₂×Dic₂₆	—	2	45	no	—
95	C₂×Dic₂₆	C₂⁴	1	45	yes	—
96	C₂×Dic₂₆	C₂⁴	1	15	yes	—
97	C₁₃×M₁₆	C₂⁴	1	60	yes	—
98	C₂×C₈×C₁₃	C₂⁴	1	15	yes	—
99	C₂×C₈×C₁₃	C₂⁴	1	45	yes	—
100	Dic₅₂	C₂⁴	1	60	yes	—
101	D₁₀₄	C₂⁴	1	60	yes	—
102	#6095	C₂⁴	1	10	yes	—
103	#6095	C₂⁴	1	6	yes	—
104	#6095	C₂⁴	1	90	yes	—
105	#6095	C₂⁴	1	30	yes	—
106	2₋¹⁺⁶	—	13	1	no	—
107	H(7,𝔽₂)	—	13	1	no	—
108	#3586	—	13	15	no	—
109	#3585	—	13	15	no	—
110	#3584	—	13	15	no	—
111	C₂×#577	—	13	15	no	—
112	#3573	D₁₃	1	1	yes	characteristic, complemented
113	C₂³×C₁₃	C₂²×D₄	1	15	yes	—
114	C₂²×D₁₃	—	2	15	no	—
115	C₂²×D₁₃	—	2	45	no	—
116	C₂²×D₁₃	—	16	15	no	—
117	Q₈×C₁₃	C₂²×D₄	1	20	yes	—
118	D₄×C₁₃	C₂²×D₄	1	60	yes	—
119	C₂×C₄×C₁₃	C₂²×D₄	1	45	yes	—
120	C₂×C₄×C₁₃	C₂⁵	1	15	yes	—
121	C₂²⋊D₁₃	—	2	60	no	—
122	C₂²⋊D₁₃	—	2	60	no	—
123	C₂×Dic₁₃	—	2	45	no	—
124	C₂×Dic₁₃	—	2	15	no	—
125	D₅₂	—	2	45	no	—
126	D₅₂	—	2	15	no	—
127	D₅₂	C₂⁵	1	10	yes	—
128	D₅₂	C₂⁵	1	6	yes	—
129	C₄×D₁₃	—	2	60	no	—
130	C₄×D₁₃	—	2	60	no	—
131	Dic₂₆	—	2	45	no	—
132	Dic₂₆	—	2	15	no	—
133	Dic₂₆	C₂⁵	1	10	yes	—
134	Dic₂₆	C₂⁵	1	6	yes	—
135	C₈×C₁₃	C₂⁵	1	1	yes	characteristic
136	C₈×C₁₃	C₂⁵	1	15	yes	—
137	#585	—	13	15	no	—
138	#585	—	13	15	no	—
139	#585	—	26	6	no	—
140	#585	—	26	10	no	—
141	#585	C₂×D₁₃	1	1	yes	characteristic
142	C₂×2₋¹⁺⁴	—	13	15	no	—
143	C₂×2₋¹⁺⁴	—	26	6	no	—
144	C₂×H(5,𝔽₂)	—	13	15	no	—
145	C₂×H(5,𝔽₂)	—	26	10	no	—
146	#578	—	13	60	no	—
147	#577	—	13	20	no	—
148	#577	—	13	60	no	—
149	#577	—	13	120	no	—
150	#576	—	13	60	no	—
151	#575	—	13	90	no	—
152	C₂×#137	—	13	45	no	—
153	C₂×AGL(1,ℤ/8)	—	13	45	no	—
154	C₂×#135	—	13	45	no	—
155	C₂²×SD₁₆	—	13	15	no	—
156	#568	C₂×D₁₃	1	15	yes	complemented
157	C₂×#131	C₂×D₁₃	1	15	yes	complemented
158	C₂²×C₁₃	—	4	15	no	—
159	C₂²×C₁₃	C₂³×D₄	1	15	yes	—
160	C₂×D₁₃	—	2	6	no	—
161	C₂×D₁₃	—	2	10	no	—
162	C₂×D₁₃	—	8	15	no	—
163	C₂×D₁₃	—	8	45	no	—
164	C₄×C₁₃	C₂³×D₄	1	15	yes	—
165	C₄×C₁₃	C₂⁶	1	1	yes	characteristic
166	Dic₁₃	—	2	10	no	—
167	Dic₁₃	—	2	6	no	—
168	2₋¹⁺⁴	—	13	60	no	—
169	2₋¹⁺⁴	—	13	20	no	—
170	2₋¹⁺⁴	—	26	60	no	—
171	2₋¹⁺⁴	—	26	15	no	—
172	2₋¹⁺⁴	—	26	15	no	—
173	2₋¹⁺⁴	D₅₂	1	6	yes	—
174	H(5,𝔽₂)	—	13	20	no	—
175	H(5,𝔽₂)	—	13	60	no	—
176	H(5,𝔽₂)	—	26	60	no	—
177	H(5,𝔽₂)	—	26	45	no	—
178	H(5,𝔽₂)	—	26	45	no	—
179	H(5,𝔽₂)	D₅₂	1	10	yes	—
180	C₂×#41	—	13	45	no	—
181	C₂×#41	—	13	45	no	—
182	C₂×#41	—	26	60	no	—
183	C₂×#41	—	26	60	no	—
184	C₂×#41	—	26	15	no	—
185	C₂×#41	—	26	45	no	—
186	C₂×#41	C₂²×D₁₃	1	15	yes	—
187	C₂²×Q₈	—	13	15	no	—
188	C₂²×Q₈	—	26	15	no	—
189	C₂²×D₄	—	13	15	no	—
190	C₂²×D₄	—	26	45	no	—
191	#137	—	13	180	no	—
192	#137	—	13	60	no	—
193	AGL(1,ℤ/8)	—	13	180	no	—
194	AGL(1,ℤ/8)	—	13	60	no	—
195	#135	—	13	60	no	—
196	#135	—	13	180	no	—
197	C₂×Q₁₆	—	13	45	no	—
198	C₂×SD₁₆	—	13	45	no	—
199	C₂×SD₁₆	—	13	90	no	—
200	C₂×SD₁₆	—	13	15	no	—
201	C₂×D₈	—	13	45	no	—
202	#131	C₂²×D₁₃	1	20	yes	complemented
203	#131	C₂²×D₁₃	1	60	yes	complemented
204	C₂×M₁₆	C₂²×D₁₃	1	45	yes	complemented
205	C₂²×C₈	C₂²×D₁₃	1	15	yes	complemented
206	C₂×C₁₃	—	2	15	no	—
207	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
208	D₁₃	—	4	6	no	—
209	D₁₃	—	4	10	no	—
210	C₂⁴	—	26	15	no	—
211	#41	—	13	60	no	—
212	#41	—	13	60	no	—
213	#41	—	26	60	no	—
214	#41	—	26	20	no	—
215	#41	—	26	60	no	—
216	#41	—	26	180	no	—
217	#41	—	26	90	no	—
218	#41	—	26	90	no	—
219	#41	C₂×D₅₂	1	60	yes	—
220	#41	C₂³×D₁₃	1	20	yes	—
221	C₂×Q₈	—	13	45	no	—
222	C₂×Q₈	—	13	15	no	—
223	C₂×Q₈	—	26	60	no	—
224	C₂×Q₈	—	26	20	no	—
225	C₂×Q₈	—	26	45	no	—
226	C₂×Q₈	C₂×D₅₂	1	15	yes	—
227	C₂×D₄	—	13	15	no	—
228	C₂×D₄	—	13	45	no	—
229	C₂×D₄	—	26	45	no	—
230	C₂×D₄	—	26	45	no	—
231	C₂×D₄	—	26	180	no	—
232	C₂×D₄	—	26	60	no	—
233	C₂×D₄	—	26	45	no	—
234	C₂×D₄	C₂×D₅₂	1	45	yes	—
235	C₂²×C₄	—	26	15	no	—
236	C₂²×C₄	—	26	45	no	—
237	C₂²×C₄	—	26	45	no	—
238	C₂²×C₄	C₂³×D₁₃	1	15	yes	—
239	Q₁₆	—	13	60	no	—
240	SD₁₆	—	13	10	no	—
241	SD₁₆	—	13	6	no	—
242	SD₁₆	—	13	90	no	—
243	SD₁₆	—	13	30	no	—
244	D₈	—	13	60	no	—
245	M₁₆	C₂³×D₁₃	1	60	yes	—
246	C₂×C₈	C₂³×D₁₃	1	15	yes	—
247	C₂×C₈	C₂³×D₁₃	1	45	yes	—
248	C₁₃	#63092	1	1	yes	characteristic, complemented, Sylow
249	C₂³	—	26	15	no	—
250	C₂³	—	26	45	no	—
251	C₂³	—	208	15	no	—
252	C₂³	C₂²×D₅₂	1	15	yes	—
253	Q₈	—	13	10	no	—
254	Q₈	—	13	6	no	—
255	Q₈	—	26	45	no	—
256	Q₈	—	26	15	no	—
257	Q₈	C₂²×D₅₂	1	20	yes	—
258	D₄	—	13	6	no	—
259	D₄	—	13	10	no	—
260	D₄	—	26	60	no	—
261	D₄	—	26	60	no	—
262	D₄	—	26	45	no	—
263	D₄	—	26	15	no	—
264	D₄	C₂²×D₅₂	1	60	yes	—
265	C₂×C₄	—	26	15	no	—
266	C₂×C₄	—	26	45	no	—
267	C₂×C₄	—	26	60	no	—
268	C₂×C₄	—	26	60	no	—
269	C₂×C₄	C₂²×D₅₂	1	45	yes	—
270	C₂×C₄	C₂⁴×D₁₃	1	15	yes	—
271	C₈	C₂⁴×D₁₃	1	1	yes	characteristic
272	C₈	C₂⁴×D₁₃	1	15	yes	—
273	C₂²	—	4	15	no	—
274	C₂²	—	26	6	no	—
275	C₂²	—	26	10	no	—
276	C₂²	—	104	15	no	—
277	C₂²	—	104	45	no	—
278	C₂²	C₂³×D₅₂	1	15	yes	—
279	C₄	—	26	6	no	—
280	C₄	—	26	10	no	—
281	C₄	C₂³×D₅₂	1	15	yes	—
282	C₄	C₂⁵×D₁₃	1	1	yes	characteristic
283	C₂	—	2	15	no	—
284	C₂	—	52	10	no	—
285	C₂	—	52	6	no	—
286	C₂	C₂⁴×D₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂⁴×D₅₂
commutator, cosocle	C₄×C₁₃	C₂⁶
Frattini	C₄	C₂⁵×D₁₃
Fitting	C₁₃×#3573	C₂
socle	C₂×C₁₃	C₂⁴×D₄
hypercenter	#3573	D₁₃
hypocenter	C₁₃	#63092
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ #585 ◁ #3573

Complements

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

N	H ≅ G/N	type
1. #13129649	C₂	semidirect
2. #13129647	C₂	semidirect
3. #13129624	C₂	semidirect
4. #13129617	C₂	semidirect
5. #13129610	C₂	semidirect
6. C₂×#11719456	C₂	semidirect
7. C₁₃×#3573	C₂	semidirect
13. #11719648	C₂²	semidirect
14. #11719648	C₂²	semidirect
15. C₂×#88257	C₂²	semidirect
16. C₂×#88256	C₂²	semidirect
17. C₁₃×#568	C₂²	semidirect
18. C₂×C₁₃×#131	C₂²	semidirect
19. #11719458	C₂²	semidirect
20. #11719457	C₂²	semidirect
21. #11719456	C₂²	semidirect
22. #11719456	C₂²	semidirect
23. #11719456	C₂²	semidirect
24. #11719453	C₂²	semidirect
25. C₂×#88163	C₂²	semidirect
26. C₂×#88162	C₂²	semidirect
27. C₂×#88158	C₂²	semidirect
28. C₂²×#6095	C₂²	semidirect
34. #88257	C₂³	semidirect
37. #88256	C₂³	semidirect
49. C₂×#6127	C₂³	semidirect
51. C₂²×Dic₂₆	C₂³	semidirect
52. C₁₃×#131	C₂³	semidirect
53. C₁₃×#131	C₂³	semidirect
54. C₂×C₁₃×M₁₆	C₂³	semidirect
55. C₂²×C₈×C₁₃	C₂³	semidirect
56. #88163	C₂³	semidirect
57. #88163	C₂³	semidirect
58. #88162	C₂³	semidirect
59. #88162	C₂³	semidirect
60. C₂×Dic₅₂	C₂³	semidirect
61. #88158	C₂³	semidirect
62. #88158	C₂³	semidirect
63. C₂×D₁₀₄	C₂³	semidirect
64. C₂×#6095	C₂³	semidirect
65. C₂×#6095	C₂³	semidirect
66. C₂×#6095	C₂³	semidirect
112. #3573	D₁₃	semidirect
156. #568	C₂×D₁₃	semidirect
157. C₂×#131	C₂×D₁₃	semidirect
202. #131	C₂²×D₁₃	semidirect
203. #131	C₂²×D₁₃	semidirect
204. C₂×M₁₆	C₂²×D₁₃	semidirect
205. C₂²×C₈	C₂²×D₁₃	semidirect
248. C₁₃	#63092	semidirect

Maximal subgroups

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

subgroup	index
#13129649	2
#13129647	2
#13129624	2
#13129617	2
#13129610	2
C₂×#11719456	2
C₁₃×#3573	2
#63092	13