Subgroups — #16776778

Order 3328.

10894 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13129647	C₂	1	1	yes	maximal, characteristic, complemented
2	#13129616	C₂	1	15	yes	maximal, complemented
3	#13129615	C₂	1	15	yes	maximal, complemented
4	#13129607	C₂	1	15	yes	maximal, complemented
5	C₂×#11719540	C₂	1	15	yes	maximal, complemented
6	#13129548	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×H(7,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#585	—	2	6	no	—
9	C₁₃×#585	C₂²	1	1	yes	characteristic
10	C₁₃×#585	C₂²	1	15	yes	complemented
11	C₂×C₁₃×H(5,𝔽₂)	—	2	10	no	—
12	C₂×C₁₃×H(5,𝔽₂)	C₂²	1	15	yes	complemented
13	#11719652	—	2	6	no	—
14	D₁₃×H(5,𝔽₂)	—	2	10	no	—
15	#11719648	C₂²	1	15	yes	complemented
16	C₂×#88256	C₂²	1	15	yes	complemented
17	#11719543	C₂²	1	20	yes	complemented
18	#11719542	C₂²	1	60	yes	complemented
19	#11719541	C₂²	1	60	yes	complemented
20	#11719540	C₂²	1	60	yes	complemented
21	#11719540	C₂²	1	120	yes	complemented
22	#11719527	C₂²	1	90	yes	complemented
23	C₂×#88204	C₂²	1	45	yes	complemented
24	C₂×#88203	C₂²	1	45	yes	complemented
25	#11719524	C₂²	1	15	yes	complemented
26	C₂×#88202	C₂²	1	15	yes	complemented
27	C₂×#88186	C₂²	1	45	yes	complemented
28	C₂²×#6104	C₂²	1	15	yes	complemented
29	C₁₃×2₋¹⁺⁴	—	2	15	no	—
30	C₁₃×2₋¹⁺⁴	D₄	1	6	yes	—
31	C₁₃×2₋¹⁺⁴	C₂³	1	20	yes	complemented
32	C₁₃×H(5,𝔽₂)	—	2	60	no	—
33	C₁₃×H(5,𝔽₂)	—	2	45	no	—
34	C₁₃×H(5,𝔽₂)	D₄	1	10	yes	—
35	C₁₃×H(5,𝔽₂)	C₂³	1	60	yes	complemented
36	C₂×C₁₃×#41	—	2	60	no	—
37	C₂×C₁₃×#41	—	2	15	no	—
38	C₂×C₁₃×#41	C₂³	1	15	yes	—
39	C₂×C₁₃×#41	C₂³	1	45	yes	complemented
40	C₂²×D₄×C₁₃	—	2	45	no	—
41	C₂²×D₄×C₁₃	C₂³	1	15	yes	complemented
42	#88257	C₂³	1	20	yes	complemented
43	#88256	—	2	60	no	—
44	#88256	C₂³	1	60	yes	complemented
45	#41×D₁₃	—	2	60	no	—
46	#88254	—	2	15	no	—
47	C₂×#6131	—	2	15	no	—
48	#88251	—	2	45	no	—
49	C₂×D₄×D₁₃	—	2	45	no	—
50	C₂×#6127	C₂³	1	45	yes	complemented
51	C₂²×D₅₂	C₂³	1	15	yes	complemented
52	#88205	C₂³	1	60	yes	complemented
53	#88204	C₂³	1	60	yes	complemented
54	#88204	C₂³	1	180	yes	complemented
55	#88203	C₂³	1	180	yes	complemented
56	#88202	C₂³	1	20	yes	complemented
57	#88202	C₂³	1	60	yes	complemented
58	C₂×#6107	C₂³	1	15	yes	complemented
59	#88196	C₂³	1	60	yes	complemented
60	C₂×#6106	C₂³	1	45	yes	complemented
61	C₂×#6105	C₂³	1	45	yes	complemented
62	#88186	C₂³	1	180	yes	complemented
63	C₂×#6104	C₂³	1	45	yes	complemented
64	C₂×#6104	C₂³	1	90	yes	complemented
65	C₂×#6099	C₂³	1	45	yes	complemented
66	C₂²×#1055	C₂³	1	15	yes	complemented
67	#63091	—	13	1	no	maximal, Sylow
68	C₂⁴×C₁₃	—	2	15	no	—
69	C₂³×D₁₃	—	2	15	no	—
70	C₁₃×#41	—	2	60	no	—
71	C₁₃×#41	—	2	60	no	—
72	C₁₃×#41	—	2	90	no	—
73	C₁₃×#41	C₂×D₄	1	60	yes	—
74	C₁₃×#41	C₂⁴	1	60	yes	—
75	C₁₃×#41	C₂⁴	1	20	yes	—
76	C₂×Q₈×C₁₃	—	2	20	no	—
77	C₂×Q₈×C₁₃	C₂×D₄	1	15	yes	—
78	C₂×Q₈×C₁₃	C₂⁴	1	15	yes	—
79	C₂×D₄×C₁₃	—	2	45	no	—
80	C₂×D₄×C₁₃	—	2	180	no	—
81	C₂×D₄×C₁₃	—	2	45	no	—
82	C₂×D₄×C₁₃	C₂×D₄	1	45	yes	—
83	C₂×D₄×C₁₃	C₂⁴	1	45	yes	—
84	C₂²×C₄×C₁₃	—	2	45	no	—
85	C₂²×C₄×C₁₃	C₂⁴	1	15	yes	—
86	C₂×C₂²⋊D₁₃	—	2	45	no	—
87	#6131	—	2	60	no	—
88	Q₈×D₁₃	—	2	20	no	—
89	#6129	—	2	60	no	—
90	D₄×D₁₃	—	2	180	no	—
91	#6127	—	2	90	no	—
92	#6127	C₂⁴	1	60	yes	—
93	C₂×D₅₂	—	2	45	no	—
94	C₂×D₅₂	C₂⁴	1	45	yes	—
95	C₂×C₄×D₁₃	—	2	45	no	—
96	C₂×Dic₂₆	C₂⁴	1	15	yes	—
97	#6107	C₂⁴	1	30	yes	—
98	#6107	C₂⁴	1	6	yes	—
99	#6106	C₂⁴	1	60	yes	—
100	#6105	C₂⁴	1	60	yes	—
101	#6104	C₂⁴	1	10	yes	—
102	#6104	C₂⁴	1	90	yes	—
103	#6099	C₂⁴	1	60	yes	—
104	C₂×#1055	C₂⁴	1	15	yes	—
105	C₂×#1055	C₂⁴	1	45	yes	—
106	H(7,𝔽₂)	—	13	1	no	—
107	H(7,𝔽₂)	D₁₃	1	1	yes	characteristic, complemented
108	#3585	—	13	15	no	—
109	#3585	—	13	15	no	—
110	#3584	—	13	15	no	—
111	C₂×#576	—	13	15	no	—
112	#3573	—	13	1	no	—
113	C₂³×C₁₃	—	2	45	no	—
114	C₂³×C₁₃	—	16	15	no	—
115	C₂³×C₁₃	C₂²×D₄	1	15	yes	—
116	C₂²×D₁₃	—	2	45	no	—
117	C₂²×D₁₃	—	16	15	no	—
118	Q₈×C₁₃	—	2	15	no	—
119	Q₈×C₁₃	C₂²×D₄	1	20	yes	—
120	Q₈×C₁₃	C₂⁵	1	6	yes	—
121	D₄×C₁₃	—	2	60	no	—
122	D₄×C₁₃	—	2	45	no	—
123	D₄×C₁₃	C₂²×D₄	1	60	yes	—
124	D₄×C₁₃	C₂⁵	1	10	yes	—
125	C₂×C₄×C₁₃	—	2	15	no	—
126	C₂×C₄×C₁₃	—	2	60	no	—
127	C₂×C₄×C₁₃	C₂²×D₄	1	45	yes	—
128	C₂×C₄×C₁₃	C₂⁵	1	15	yes	—
129	C₂²⋊D₁₃	—	2	60	no	—
130	C₂×Dic₁₃	—	2	15	no	—
131	D₅₂	—	2	45	no	—
132	D₅₂	C₂⁵	1	10	yes	—
133	C₄×D₁₃	—	2	60	no	—
134	Dic₂₆	—	2	15	no	—
135	Dic₂₆	C₂⁵	1	6	yes	—
136	#1055	C₂⁵	1	1	yes	characteristic
137	#1055	C₂⁵	1	15	yes	—
138	#585	—	2	6	no	—
139	#585	—	13	15	no	—
140	#585	—	26	6	no	—
141	#585	C₂×D₁₃	1	1	yes	characteristic
142	#585	C₂×D₁₃	1	15	yes	complemented
143	C₂×H(5,𝔽₂)	—	2	10	no	—
144	C₂×H(5,𝔽₂)	—	13	15	no	—
145	C₂×H(5,𝔽₂)	—	26	10	no	—
146	C₂×H(5,𝔽₂)	C₂×D₁₃	1	15	yes	complemented
147	#578	—	13	20	no	—
148	#577	—	13	60	no	—
149	#577	—	13	60	no	—
150	#576	—	13	60	no	—
151	#576	—	13	120	no	—
152	#575	—	13	90	no	—
153	C₂×AGL(1,ℤ/8)	—	13	45	no	—
154	C₂×AGL(1,ℤ/8)	—	13	45	no	—
155	C₂×#135	—	13	45	no	—
156	C₂²×D₈	—	13	15	no	—
157	#568	—	13	15	no	—
158	C₂×#131	—	13	15	no	—
159	C₂²×C₁₃	—	2	10	no	—
160	C₂²×C₁₃	—	4	15	no	—
161	C₂²×C₁₃	—	8	45	no	—
162	C₂²×C₁₃	C₂³×D₄	1	15	yes	—
163	C₂×D₁₃	—	2	10	no	—
164	C₂×D₁₃	—	8	45	no	—
165	C₄×C₁₃	—	2	6	no	—
166	C₄×C₁₃	C₂³×D₄	1	15	yes	—
167	C₄×C₁₃	C₂⁶	1	1	yes	characteristic
168	Dic₁₃	—	2	6	no	—
169	2₋¹⁺⁴	—	2	15	no	—
170	2₋¹⁺⁴	—	13	20	no	—
171	2₋¹⁺⁴	—	26	15	no	—
172	2₋¹⁺⁴	C₂²⋊D₁₃	1	6	yes	—
173	2₋¹⁺⁴	C₂²×D₁₃	1	20	yes	complemented
174	H(5,𝔽₂)	—	2	60	no	—
175	H(5,𝔽₂)	—	2	45	no	—
176	H(5,𝔽₂)	—	13	60	no	—
177	H(5,𝔽₂)	—	26	60	no	—
178	H(5,𝔽₂)	—	26	45	no	—
179	H(5,𝔽₂)	C₂²⋊D₁₃	1	10	yes	—
180	H(5,𝔽₂)	C₂²×D₁₃	1	60	yes	complemented
181	C₂×#41	—	2	60	no	—
182	C₂×#41	—	2	15	no	—
183	C₂×#41	—	13	45	no	—
184	C₂×#41	—	26	60	no	—
185	C₂×#41	—	26	15	no	—
186	C₂×#41	C₂²×D₁₃	1	15	yes	—
187	C₂×#41	C₂²×D₁₃	1	45	yes	complemented
188	C₂²×D₄	—	2	45	no	—
189	C₂²×D₄	—	13	15	no	—
190	C₂²×D₄	—	26	45	no	—
191	C₂²×D₄	C₂²×D₁₃	1	15	yes	complemented
192	#137	—	13	60	no	—
193	#137	—	13	60	no	—
194	AGL(1,ℤ/8)	—	13	180	no	—
195	AGL(1,ℤ/8)	—	13	180	no	—
196	#135	—	13	60	no	—
197	#135	—	13	180	no	—
198	C₂×Q₁₆	—	13	15	no	—
199	C₂×SD₁₆	—	13	45	no	—
200	C₂×SD₁₆	—	13	45	no	—
201	C₂×D₈	—	13	45	no	—
202	C₂×D₈	—	13	90	no	—
203	#131	—	13	20	no	—
204	#131	—	13	60	no	—
205	C₂×M₁₆	—	13	45	no	—
206	C₂²×C₈	—	13	15	no	—
207	C₂×C₁₃	—	2	15	no	—
208	C₂×C₁₃	—	4	10	no	—
209	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
210	D₁₃	—	4	10	no	—
211	C₂⁴	—	2	15	no	—
212	C₂⁴	—	26	15	no	—
213	#41	—	2	60	no	—
214	#41	—	2	60	no	—
215	#41	—	2	90	no	—
216	#41	—	13	60	no	—
217	#41	—	26	60	no	—
218	#41	—	26	60	no	—
219	#41	—	26	90	no	—
220	#41	C₂×C₂²⋊D₁₃	1	60	yes	—
221	#41	C₂³×D₁₃	1	60	yes	—
222	#41	C₂³×D₁₃	1	20	yes	—
223	C₂×Q₈	—	2	20	no	—
224	C₂×Q₈	—	13	15	no	—
225	C₂×Q₈	—	26	20	no	—
226	C₂×Q₈	C₂×C₂²⋊D₁₃	1	15	yes	—
227	C₂×Q₈	C₂³×D₁₃	1	15	yes	—
228	C₂×D₄	—	2	45	no	—
229	C₂×D₄	—	2	180	no	—
230	C₂×D₄	—	2	45	no	—
231	C₂×D₄	—	13	45	no	—
232	C₂×D₄	—	26	45	no	—
233	C₂×D₄	—	26	180	no	—
234	C₂×D₄	—	26	45	no	—
235	C₂×D₄	C₂×C₂²⋊D₁₃	1	45	yes	—
236	C₂×D₄	C₂³×D₁₃	1	45	yes	—
237	C₂²×C₄	—	2	45	no	—
238	C₂²×C₄	—	26	45	no	—
239	C₂²×C₄	C₂³×D₁₃	1	15	yes	—
240	Q₁₆	—	13	6	no	—
241	Q₁₆	—	13	30	no	—
242	SD₁₆	—	13	60	no	—
243	SD₁₆	—	13	60	no	—
244	D₈	—	13	10	no	—
245	D₈	—	13	90	no	—
246	M₁₆	—	13	60	no	—
247	C₂×C₈	—	13	15	no	—
248	C₂×C₈	—	13	45	no	—
249	C₁₃	#63091	1	1	yes	characteristic, complemented, Sylow
250	C₂³	—	2	45	no	—
251	C₂³	—	16	15	no	—
252	C₂³	—	26	45	no	—
253	C₂³	—	208	15	no	—
254	C₂³	C₂²×C₂²⋊D₁₃	1	15	yes	—
255	Q₈	—	2	15	no	—
256	Q₈	—	13	6	no	—
257	Q₈	—	26	15	no	—
258	Q₈	C₂²×C₂²⋊D₁₃	1	20	yes	—
259	Q₈	C₂⁴×D₁₃	1	6	yes	—
260	D₄	—	2	60	no	—
261	D₄	—	2	45	no	—
262	D₄	—	13	10	no	—
263	D₄	—	26	60	no	—
264	D₄	—	26	45	no	—
265	D₄	C₂²×C₂²⋊D₁₃	1	60	yes	—
266	D₄	C₂⁴×D₁₃	1	10	yes	—
267	C₂×C₄	—	2	15	no	—
268	C₂×C₄	—	2	60	no	—
269	C₂×C₄	—	26	15	no	—
270	C₂×C₄	—	26	60	no	—
271	C₂×C₄	C₂²×C₂²⋊D₁₃	1	45	yes	—
272	C₂×C₄	C₂⁴×D₁₃	1	15	yes	—
273	C₈	—	13	1	no	—
274	C₈	—	13	15	no	—
275	C₂²	—	2	10	no	—
276	C₂²	—	4	15	no	—
277	C₂²	—	8	45	no	—
278	C₂²	—	26	10	no	—
279	C₂²	—	104	45	no	—
280	C₂²	C₂³×C₂²⋊D₁₃	1	15	yes	—
281	C₄	—	2	6	no	—
282	C₄	—	26	6	no	—
283	C₄	C₂³×C₂²⋊D₁₃	1	15	yes	—
284	C₄	C₂⁵×D₁₃	1	1	yes	characteristic
285	C₂	—	2	15	no	—
286	C₂	—	4	10	no	—
287	C₂	—	52	10	no	—
288	C₂	C₂⁴×C₂²⋊D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂⁴×C₂²⋊D₁₃
commutator, cosocle	C₄×C₁₃	C₂⁶
Frattini	C₄	C₂⁵×D₁₃
Fitting	C₁₃×H(7,𝔽₂)	C₂
socle	C₂×C₁₃	C₂⁴×D₄
hypercenter	H(7,𝔽₂)	D₁₃
hypocenter	C₁₃	#63091
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ #585 ◁ H(7,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #13129647	C₂	semidirect
2. #13129616	C₂	semidirect
3. #13129615	C₂	semidirect
4. #13129607	C₂	semidirect
5. C₂×#11719540	C₂	semidirect
6. #13129548	C₂	semidirect
7. C₁₃×H(7,𝔽₂)	C₂	semidirect
10. C₁₃×#585	C₂²	semidirect
12. C₂×C₁₃×H(5,𝔽₂)	C₂²	semidirect
15. #11719648	C₂²	semidirect
16. C₂×#88256	C₂²	semidirect
17. #11719543	C₂²	semidirect
18. #11719542	C₂²	semidirect
19. #11719541	C₂²	semidirect
20. #11719540	C₂²	semidirect
21. #11719540	C₂²	semidirect
22. #11719527	C₂²	semidirect
23. C₂×#88204	C₂²	semidirect
24. C₂×#88203	C₂²	semidirect
25. #11719524	C₂²	semidirect
26. C₂×#88202	C₂²	semidirect
27. C₂×#88186	C₂²	semidirect
28. C₂²×#6104	C₂²	semidirect
31. C₁₃×2₋¹⁺⁴	C₂³	semidirect
35. C₁₃×H(5,𝔽₂)	C₂³	semidirect
39. C₂×C₁₃×#41	C₂³	semidirect
41. C₂²×D₄×C₁₃	C₂³	semidirect
42. #88257	C₂³	semidirect
44. #88256	C₂³	semidirect
50. C₂×#6127	C₂³	semidirect
51. C₂²×D₅₂	C₂³	semidirect
52. #88205	C₂³	semidirect
53. #88204	C₂³	semidirect
54. #88204	C₂³	semidirect
55. #88203	C₂³	semidirect
56. #88202	C₂³	semidirect
57. #88202	C₂³	semidirect
58. C₂×#6107	C₂³	semidirect
59. #88196	C₂³	semidirect
60. C₂×#6106	C₂³	semidirect
61. C₂×#6105	C₂³	semidirect
62. #88186	C₂³	semidirect
63. C₂×#6104	C₂³	semidirect
64. C₂×#6104	C₂³	semidirect
65. C₂×#6099	C₂³	semidirect
66. C₂²×#1055	C₂³	semidirect
107. H(7,𝔽₂)	D₁₃	semidirect
142. #585	C₂×D₁₃	semidirect
146. C₂×H(5,𝔽₂)	C₂×D₁₃	semidirect
173. 2₋¹⁺⁴	C₂²×D₁₃	semidirect
180. H(5,𝔽₂)	C₂²×D₁₃	semidirect
187. C₂×#41	C₂²×D₁₃	semidirect
191. C₂²×D₄	C₂²×D₁₃	semidirect
249. C₁₃	#63091	semidirect

Maximal subgroups

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

subgroup	index
#13129647	2
#13129616	2
#13129615	2
#13129607	2
C₂×#11719540	2
#13129548	2
C₁₃×H(7,𝔽₂)	2
#63091	13