Subgroups — #21432512

Order 7552.

564 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11745659	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×D₅₉	C₂	1	1	yes	maximal, characteristic
3	#117×D₅₉	C₂	1	1	yes	maximal, characteristic
4	#16837360	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837359	C₂	1	2	yes	maximal, complemented
6	#16837358	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837357	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837356	C₂	1	2	yes	maximal, complemented
9	#16837354	C₂	1	2	yes	maximal, complemented
10	C₄×#11745659	C₂	1	2	yes	maximal, complemented
11	C₅₉×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₅₉	C₂²	1	1	yes	characteristic
13	C₂×#11745664	C₂²	1	1	yes	characteristic
14	C₂×#11745659	C₄	1	8	yes	complemented
15	C₂×#11745659	C₂²	1	2	yes	—
16	C₂×#11745659	C₂²	1	1	yes	characteristic
17	C₂×#11745659	C₂²	1	1	yes	characteristic
18	C₂×#11745659	C₂²	1	1	yes	characteristic
19	C₂×#11745659	C₂²	1	1	yes	characteristic
20	M₁₆×D₅₉	C₄	1	8	yes	complemented
21	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₅₉	C₂²	1	2	yes	—
23	#14273686	C₂²	1	1	yes	characteristic
24	#14273685	C₄	1	2	yes	—
25	#14273684	C₂²	1	2	yes	—
26	#14273683	C₄	1	2	yes	—
27	C₄⋊C₄×D₅₉	C₄	1	2	yes	—
28	C₂²⋊C₄×D₅₉	C₄	1	2	yes	—
29	C₄²×D₅₉	C₂²	1	2	yes	—
30	#14273638	C₂²	1	2	yes	complemented
31	#14273637	C₂²	1	2	yes	complemented
32	#14273637	C₂²	1	2	yes	complemented
33	C₄×C₅₉⋊C₈	C₂²	1	2	yes	complemented
34	C₈×Dic₅₉	C₂²	1	2	yes	complemented
35	C₈×Dic₅₉	C₂²	1	2	yes	complemented
36	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
38	#117×C₅₉	C₂²	1	1	yes	characteristic
39	#97×C₅₉	C₂²	1	2	yes	complemented
40	C₄×C₈×C₅₉	C₂²	1	2	yes	complemented
41	C₂³×D₅₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
43	C₂²×Dic₅₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₅₉	C₂×C₄	1	4	yes	—
45	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₅₉	C₂×C₄	1	4	yes	—
48	C₂×C₄×D₅₉	C₂×C₄	1	4	yes	—
49	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
51	M₁₆×C₅₉	C₂×C₄	1	4	yes	complemented
52	C₂×C₈×C₅₉	C₂×C₄	1	4	yes	complemented
53	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₅₉	C₂³	1	2	yes	—
55	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
57	C₂²⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
58	C₄²×C₅₉	C₂³	1	2	yes	—
59	C₂²⋊Dic₅₉	C₂×C₄	1	2	yes	—
60	(C₂×D₅₉)⋊C₄	C₂×C₄	1	4	yes	—
61	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
62	Dic₅₉⋊C₄	C₂×C₄	1	4	yes	—
63	C₄×Dic₅₉	C₂×C₄	1	4	yes	—
64	C₄×Dic₅₉	C₂³	1	2	yes	—
65	#11745664	C₂×C₄	1	4	yes	complemented
66	C₂×C₅₉⋊C₈	C₂×C₄	1	4	yes	complemented
67	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
69	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
70	#11745659	C₂×C₄	1	8	yes	—
71	#11745659	C₂×C₄	1	4	yes	—
72	#11745659	C₂×C₄	1	4	yes	—
73	#11745659	C₂×C₄	1	4	yes	complemented
74	#11745659	C₂×C₄	1	4	yes	complemented
75	C₈×D₅₉	C₂×C₄	1	8	yes	complemented
76	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₅₉	—	4	2	no	—
78	C₂²×D₅₉	C₄²	1	4	yes	—
79	C₂²×D₅₉	C₂²×C₄	1	1	yes	characteristic
80	C₂²×D₅₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₉	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
86	C₂×Dic₅₉	C₄²	1	4	yes	—
87	C₂×Dic₅₉	C₂²×C₄	1	4	yes	—
88	C₂×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
90	C₄×D₅₉	—	4	4	no	—
91	C₄×D₅₉	C₄²	1	4	yes	—
92	C₄×D₅₉	C₄²	1	4	yes	—
93	C₄×D₅₉	C₂²×C₄	1	2	yes	—
94	C₄×D₅₉	C₂²×C₄	1	2	yes	—
95	C₄×D₅₉	C₂²×C₄	1	2	yes	—
96	C₄×D₅₉	C₂²×C₄	1	2	yes	—
97	C₈×C₅₉	C₂²×C₄	1	4	yes	—
98	C₈×C₅₉	C₂²×C₄	1	2	yes	—
99	C₈×C₅₉	C₂²×C₄	1	2	yes	—
100	C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
101	C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
102	C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
103	C₂²×C₅₉	—	2	2	no	—
104	C₂²×C₅₉	C₂×C₄²	1	2	yes	—
105	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
106	C₂×D₅₉	—	2	2	no	—
107	C₂×D₅₉	—	2	2	no	—
108	C₂×D₅₉	—	4	4	no	—
109	C₂×D₅₉	C₂×C₄²	1	2	yes	—
110	C₂×D₅₉	C₂×C₄²	1	2	yes	—
111	C₄×C₅₉	—	2	4	no	—
112	C₄×C₅₉	C₂×C₄²	1	2	yes	—
113	C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
114	C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
115	Dic₅₉	—	2	4	no	—
116	Dic₅₉	C₂×C₄²	1	2	yes	—
117	Dic₅₉	C₂×C₄²	1	2	yes	—
118	#2873	—	59	1	no	maximal, Sylow
119	C₂×C₅₉	—	2	2	no	—
120	C₂×C₅₉	C₂²×C₄²	1	1	yes	characteristic
121	C₂×C₅₉	#568	1	2	yes	—
122	D₅₉	—	2	2	no	—
123	D₅₉	—	2	2	no	—
124	C₂²×M₁₆	—	59	1	no	—
125	C₂²×M₁₆	—	59	1	no	—
126	C₂×#117	—	59	1	no	—
127	#405	—	59	1	no	—
128	#405	—	59	1	no	—
129	#405	—	59	1	no	—
130	#405	—	59	2	no	—
131	#405	—	59	2	no	—
132	#405	D₅₉	1	1	yes	characteristic, complemented
133	C₄×M₁₆	—	59	2	no	—
134	C₄×M₁₆	—	59	2	no	—
135	C₅₉	#2873	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₄	—	59	1	no	—
137	C₂×M₁₆	—	59	1	no	—
138	C₂×M₁₆	—	59	8	no	—
139	C₂×M₁₆	—	59	8	no	—
140	C₂×M₁₆	—	59	1	no	—
141	C₂×M₁₆	—	59	2	no	—
142	C₂×M₁₆	—	59	1	no	—
143	C₂×M₁₆	—	59	1	no	—
144	C₂×M₁₆	—	59	1	no	—
145	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
146	C₂²×C₈	—	59	1	no	—
147	C₂²×C₈	—	59	2	no	—
148	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
149	#117	—	59	1	no	—
150	#117	—	59	2	no	—
151	#117	—	59	2	no	—
152	#117	—	59	2	no	—
153	#117	C₂×D₅₉	1	1	yes	characteristic
154	C₂×C₄⋊C₄	—	59	2	no	—
155	C₂×C₂²⋊C₄	—	59	2	no	—
156	C₂×C₄²	—	59	2	no	—
157	#97	—	59	2	no	—
158	#97	—	59	2	no	—
159	#97	—	59	2	no	—
160	#97	C₂×D₅₉	1	2	yes	complemented
161	C₄×C₈	—	59	2	no	—
162	C₄×C₈	—	59	2	no	—
163	C₄×C₈	—	59	2	no	—
164	C₄×C₈	C₂×D₅₉	1	2	yes	complemented
165	C₂⁴	—	59	1	no	—
166	C₂²×C₄	—	59	1	no	—
167	C₂²×C₄	—	59	1	no	—
168	C₂²×C₄	—	59	1	no	—
169	C₂²×C₄	—	59	4	no	—
170	C₂²×C₄	—	59	4	no	—
171	C₂²×C₄	—	59	1	no	—
172	C₂²×C₄	—	59	1	no	—
173	C₂²×C₄	—	59	4	no	—
174	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
175	M₁₆	—	59	4	no	—
176	M₁₆	—	59	8	no	—
177	M₁₆	—	59	4	no	—
178	M₁₆	—	59	4	no	—
179	M₁₆	—	59	4	no	—
180	M₁₆	—	59	4	no	—
181	M₁₆	C₄×D₅₉	1	4	yes	complemented
182	C₂×C₈	—	59	4	no	—
183	C₂×C₈	—	59	1	no	—
184	C₂×C₈	—	59	2	no	—
185	C₂×C₈	—	59	1	no	—
186	C₂×C₈	—	59	8	no	—
187	C₂×C₈	C₄×D₅₉	1	4	yes	complemented
188	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
189	C₂×C₈	C₂²×D₅₉	1	2	yes	—
190	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
191	C₄⋊C₄	—	59	4	no	—
192	C₄⋊C₄	—	59	2	no	—
193	C₄⋊C₄	C₄×D₅₉	1	2	yes	—
194	C₂²⋊C₄	—	59	2	no	—
195	C₂²⋊C₄	—	59	4	no	—
196	C₂²⋊C₄	C₄×D₅₉	1	2	yes	—
197	C₄²	—	59	4	no	—
198	C₄²	—	59	2	no	—
199	C₄²	C₂²×D₅₉	1	2	yes	—
200	C₂³	—	59	1	no	—
201	C₂³	—	59	1	no	—
202	C₂³	—	59	4	no	—
203	C₂³	—	236	2	no	—
204	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
205	C₂×C₄	—	59	4	no	—
206	C₂×C₄	—	59	1	no	—
207	C₂×C₄	—	59	1	no	—
208	C₂×C₄	—	59	4	no	—
209	C₂×C₄	—	59	4	no	—
210	C₂×C₄	—	59	2	no	—
211	C₂×C₄	—	59	2	no	—
212	C₂×C₄	—	59	2	no	—
213	C₂×C₄	—	59	2	no	—
214	C₂×C₄	—	59	4	no	—
215	C₂×C₄	—	236	4	no	—
216	C₂×C₄	C₂×C₄×D₅₉	1	4	yes	—
217	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
218	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
219	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
220	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
221	C₈	—	59	4	no	—
222	C₈	—	59	2	no	—
223	C₈	—	59	2	no	—
224	C₈	C₂×C₄×D₅₉	1	4	yes	—
225	C₈	C₂×C₄×D₅₉	1	2	yes	—
226	C₈	C₂×C₄×D₅₉	1	2	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	59	2	no	—
229	C₂²	—	59	2	no	—
230	C₂²	—	118	2	no	—
231	C₂²	—	118	2	no	—
232	C₂²	—	236	4	no	—
233	C₂²	C₄²×D₅₉	1	2	yes	—
234	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
235	C₄	—	2	4	no	—
236	C₄	—	59	2	no	—
237	C₄	—	59	2	no	—
238	C₄	—	118	4	no	—
239	C₄	C₄²×D₅₉	1	2	yes	—
240	C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
241	C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	118	2	no	—
244	C₂	—	118	2	no	—
245	C₂	C₂×C₄²×D₅₉	1	1	yes	characteristic
246	C₂	#16838167	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator	C₂×C₅₉	C₂²×C₄²
Fitting	C₅₉×#405	C₂
socle	C₂²×C₅₉	C₂³×C₄
hypercenter	#405	D₅₉
hypocenter	C₅₉	#2873
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
4. #16837360	C₂	semidirect
5. #16837359	C₂	semidirect
6. #16837358	C₂	semidirect
7. #16837357	C₂	semidirect
8. #16837356	C₂	semidirect
9. #16837354	C₂	semidirect
10. C₄×#11745659	C₂	semidirect
11. C₅₉×#405	C₂	semidirect
14. C₂×#11745659	C₄	semidirect
20. M₁₆×D₅₉	C₄	semidirect
30. #14273638	C₂²	semidirect
31. #14273637	C₂²	semidirect
32. #14273637	C₂²	semidirect
33. C₄×C₅₉⋊C₈	C₂²	semidirect
34. C₈×Dic₅₉	C₂²	semidirect
35. C₈×Dic₅₉	C₂²	semidirect
39. #97×C₅₉	C₂²	semidirect
40. C₄×C₈×C₅₉	C₂²	semidirect
51. M₁₆×C₅₉	C₂×C₄	semidirect
52. C₂×C₈×C₅₉	C₂×C₄	semidirect
65. #11745664	C₂×C₄	semidirect
66. C₂×C₅₉⋊C₈	C₂×C₄	semidirect
73. #11745659	C₂×C₄	semidirect
74. #11745659	C₂×C₄	semidirect
75. C₈×D₅₉	C₂×C₄	semidirect
132. #405	D₅₉	semidirect
135. C₅₉	#2873	semidirect
160. #97	C₂×D₅₉	semidirect
164. C₄×C₈	C₂×D₅₉	semidirect
181. M₁₆	C₄×D₅₉	semidirect
187. C₂×C₈	C₄×D₅₉	semidirect