Subgroups — #21433112

Order 7552.

402 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11745687	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837526	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837480	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837478	C₂	1	2	yes	maximal, complemented
6	#16837477	C₂	1	2	yes	maximal, complemented
7	#16837458	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837456	C₂	1	2	yes	maximal, complemented
9	M₁₆×Dic₅₉	C₂	1	2	yes	maximal, complemented
10	C₄×#11745659	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11745687	C₄	1	1	yes	characteristic
13	C₂×#11745664	C₂²	1	2	yes	—
14	C₂×#11745659	C₂²	1	1	yes	characteristic
15	C₂×#11745659	C₂²	1	1	yes	characteristic
16	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
17	#14273692	C₄	1	1	yes	characteristic
18	D₄×Dic₅₉	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊D₅₉	C₂²	1	2	yes	—
20	(C₂²⋊D₅₉)⋊C₄	C₄	1	2	yes	—
21	#14273685	C₄	1	1	yes	characteristic
22	#14273683	C₄	1	2	yes	—
23	C₂×C₄×Dic₅₉	C₂²	1	2	yes	—
24	C₄²×D₅₉	C₂²	1	1	yes	characteristic
25	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
26	Dic₅₉⋊C₈	C₂²	1	2	yes	—
27	Dic₅₉⋊C₈	C₂²	1	2	yes	—
28	C₂²⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
29	(C₂×D₅₉)⋊C₈	C₂²	1	2	yes	—
30	#14273637	C₂²	1	1	yes	characteristic
31	#14273637	C₂²	1	2	yes	—
32	#14273637	C₂²	1	1	yes	characteristic
33	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
34	C₈×Dic₅₉	C₂²	1	2	yes	—
35	C₈×Dic₅₉	C₂²	1	2	yes	—
36	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
37	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₅₉	C₂²	1	2	yes	—
40	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
41	C₂×D₄×C₅₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₅₉	C₂×C₄	1	2	yes	—
44	C₂²×Dic₅₉	C₂×C₄	1	2	yes	—
45	C₂²×Dic₅₉	C₂×C₄	1	2	yes	—
46	#11745687	—	2	4	no	—
47	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
50	C₂×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
51	M₁₆×C₅₉	—	2	4	no	—
52	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₅₉	C₂³	1	2	yes	—
54	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
56	C₂²⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
57	C₄²×C₅₉	C₂³	1	1	yes	characteristic
58	C₂²⋊Dic₅₉	C₂×C₄	1	2	yes	—
59	(C₂×D₅₉)⋊C₄	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
61	Dic₅₉⋊C₄	C₂×C₄	1	2	yes	—
62	Dic₅₉⋊C₄	C₂×C₄	1	2	yes	—
63	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
66	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
67	C₄×Dic₅₉	C₂³	1	2	yes	—
68	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
69	#11745664	—	2	4	no	—
70	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
71	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
73	#11745659	—	2	2	no	—
74	#11745659	—	2	2	no	—
75	C₂³×C₅₉	C₂²×C₄	1	2	yes	—
76	C₂²×D₅₉	C₂²×C₄	1	1	yes	characteristic
77	D₄×C₅₉	—	2	2	no	—
78	C₂×C₄×C₅₉	—	2	2	no	—
79	C₂×C₄×C₅₉	—	2	2	no	—
80	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
84	C₂²⋊D₅₉	—	2	4	no	—
85	C₂×Dic₅₉	—	2	4	no	—
86	C₂×Dic₅₉	—	4	2	no	—
87	C₂×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
91	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
92	C₄×D₅₉	—	2	2	no	—
93	C₄×D₅₉	—	2	1	no	—
94	C₄×D₅₉	—	2	1	no	—
95	C₄×D₅₉	—	4	1	no	—
96	Dic₁₁₈	M₁₆	1	4	yes	complemented
97	C₈×C₅₉	—	2	1	no	—
98	C₈×C₅₉	—	2	2	no	—
99	C₈×C₅₉	#41	1	2	yes	complemented
100	C₅₉⋊C₈	—	2	1	no	—
101	C₅₉⋊C₈	—	2	2	no	—
102	C₅₉⋊C₈	#41	1	2	yes	complemented
103	C₂²×C₅₉	—	2	2	no	—
104	C₂²×C₅₉	—	2	2	no	—
105	C₂²×C₅₉	—	2	2	no	—
106	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
107	C₂×D₅₉	—	2	1	no	—
108	C₂×D₅₉	—	2	1	no	—
109	C₂×D₅₉	—	2	1	no	—
110	C₄×C₅₉	—	2	2	no	—
111	C₄×C₅₉	—	2	1	no	—
112	C₄×C₅₉	C₂×M₁₆	1	2	yes	—
113	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
114	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
115	Dic₅₉	—	2	1	no	—
116	Dic₅₉	—	2	1	no	—
117	Dic₅₉	—	2	1	no	—
118	Dic₅₉	—	2	2	no	—
119	Dic₅₉	C₂×M₁₆	1	4	yes	—
120	#2971	—	59	1	no	maximal, Sylow
121	C₂×C₅₉	—	4	2	no	—
122	C₂×C₅₉	C₄×#41	1	1	yes	characteristic
123	C₂×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
124	C₂×C₅₉	#568	1	1	yes	characteristic
125	D₅₉	—	4	1	no	—
126	C₄×#41	—	59	1	no	—
127	C₈×Q₈	—	59	1	no	—
128	C₈×Q₈	—	59	1	no	—
129	#436	—	59	2	no	—
130	#436	—	59	1	no	—
131	#436	—	59	2	no	—
132	#436	D₅₉	1	1	yes	characteristic, complemented
133	#432	—	59	1	no	—
134	#432	—	59	2	no	—
135	C₄×M₁₆	—	59	2	no	—
136	C₄×M₁₆	—	59	1	no	—
137	C₅₉	#2971	1	1	yes	characteristic, complemented, Sylow
138	C₂×#41	—	59	1	no	—
139	C₂×M₁₆	—	59	2	no	—
140	C₂×M₁₆	—	59	1	no	—
141	C₂×M₁₆	—	59	1	no	—
142	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
143	C₄×Q₈	—	59	1	no	—
144	C₄×Q₈	—	59	1	no	—
145	C₄×D₄	—	59	1	no	—
146	C₄×D₄	—	59	2	no	—
147	C₄×D₄	—	59	2	no	—
148	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
149	#117	—	59	1	no	—
150	#117	—	59	2	no	—
151	C₂×C₄²	—	59	2	no	—
152	C₂×C₄²	—	59	1	no	—
153	C₄⋊C₈	—	59	2	no	—
154	C₄⋊C₈	—	59	2	no	—
155	C₄⋊C₈	—	59	1	no	—
156	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
157	C₂²⋊C₈	—	59	2	no	—
158	C₂²⋊C₈	—	59	2	no	—
159	C₂²⋊C₈	C₂×D₅₉	1	2	yes	—
160	#97	—	59	1	no	—
161	#97	—	59	2	no	—
162	#97	—	59	1	no	—
163	C₄×C₈	—	59	2	no	—
164	C₄×C₈	—	59	2	no	—
165	C₄×C₈	—	59	1	no	—
166	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
167	#41	—	118	4	no	—
168	C₂×Q₈	—	59	1	no	—
169	C₂×D₄	—	59	2	no	—
170	C₂×D₄	C₄×D₅₉	1	1	yes	characteristic
171	C₂²×C₄	—	59	2	no	—
172	C₂²×C₄	—	59	2	no	—
173	C₂²×C₄	—	59	1	no	—
174	C₂²×C₄	—	59	1	no	—
175	C₂²×C₄	—	59	1	no	—
176	C₂²×C₄	C₂²×D₅₉	1	2	yes	—
177	M₁₆	—	2	4	no	—
178	M₁₆	—	118	2	no	—
179	M₁₆	—	118	4	no	—
180	M₁₆	—	118	2	no	—
181	C₂×C₈	—	59	1	no	—
182	C₂×C₈	—	59	2	no	—
183	C₂×C₈	—	59	1	no	—
184	C₂×C₈	C₂²×D₅₉	1	2	yes	—
185	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
186	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
187	C₄⋊C₄	—	59	2	no	—
188	C₄⋊C₄	—	59	2	no	—
189	C₄⋊C₄	—	59	1	no	—
190	C₄⋊C₄	C₄×D₅₉	1	1	yes	characteristic
191	C₂²⋊C₄	—	59	2	no	—
192	C₂²⋊C₄	—	59	2	no	—
193	C₂²⋊C₄	C₄×D₅₉	1	2	yes	—
194	C₄²	—	59	2	no	—
195	C₄²	—	59	1	no	—
196	C₄²	—	59	2	no	—
197	C₄²	—	59	1	no	—
198	C₄²	—	59	2	no	—
199	C₄²	—	59	1	no	—
200	C₄²	C₂²×D₅₉	1	1	yes	characteristic
201	C₂³	—	59	1	no	—
202	C₂³	C₂×C₄×D₅₉	1	2	yes	—
203	Q₈	—	59	4	no	—
204	D₄	—	2	2	no	—
205	D₄	—	118	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	59	2	no	—
209	C₂×C₄	—	59	1	no	—
210	C₂×C₄	—	59	1	no	—
211	C₂×C₄	—	59	2	no	—
212	C₂×C₄	—	59	1	no	—
213	C₂×C₄	—	118	4	no	—
214	C₂×C₄	—	118	2	no	—
215	C₂×C₄	—	118	1	no	—
216	C₂×C₄	—	118	1	no	—
217	C₂×C₄	—	236	1	no	—
218	C₂×C₄	—	236	2	no	—
219	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
220	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
222	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	2	no	—
225	C₈	—	59	2	no	—
226	C₈	—	118	2	no	—
227	C₈	—	118	1	no	—
228	C₈	#11745687	1	2	yes	complemented
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	118	1	no	—
233	C₂²	—	118	1	no	—
234	C₂²	—	118	1	no	—
235	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	59	4	no	—
239	C₄	—	118	1	no	—
240	C₄	—	118	1	no	—
241	C₄	—	118	2	no	—
242	C₄	—	118	1	no	—
243	C₄	M₁₆×D₅₉	1	2	yes	—
244	C₄	#41×D₅₉	1	1	yes	characteristic
245	C₄	C₂×#11745687	1	1	yes	characteristic
246	C₂	—	4	2	no	—
247	C₂	—	236	1	no	—
248	C₂	C₄×#11745687	1	1	yes	characteristic
249	C₂	C₂×M₁₆×D₅₉	1	1	yes	characteristic
250	C₂	#16838167	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₄
Fitting	C₅₉×#436	C₂
hypercenter	#436	D₅₉
hypocenter	C₅₉	#2971
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₁₁₈	C₂	semidirect
3. #16837526	C₂	semidirect
4. #16837480	C₂	semidirect
5. #16837478	C₂	semidirect
6. #16837477	C₂	semidirect
7. #16837458	C₂	semidirect
8. #16837456	C₂	semidirect
9. M₁₆×Dic₅₉	C₂	semidirect
10. C₄×#11745659	C₂	semidirect
11. C₅₉×#436	C₂	semidirect
96. Dic₁₁₈	M₁₆	semidirect
99. C₈×C₅₉	#41	semidirect
102. C₅₉⋊C₈	#41	semidirect
132. #436	D₅₉	semidirect
137. C₅₉	#2971	semidirect
228. C₈	#11745687	semidirect