Subgroups — #21433247

Order 7552.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837936	C₂	1	1	yes	maximal, characteristic
2	#16837532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837481	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837477	C₂	1	2	yes	maximal, complemented
6	#16837476	C₂	1	2	yes	maximal, complemented
7	C₈×D₂₃₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837462	C₂	1	2	yes	maximal, complemented
9	#16837427	C₂	1	2	yes	maximal, complemented
10	#16837426	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11745664	C₂²	1	2	yes	—
13	C₂×#11745659	C₂²	1	2	yes	—
14	C₂×C₈×D₅₉	C₂²	1	2	yes	—
15	#14273721	C₄	1	2	yes	—
16	#14273719	C₄	1	1	yes	characteristic
17	#14273715	C₄	1	1	yes	characteristic
18	#14273710	C₄	1	2	yes	—
19	#14273706	C₄	1	1	yes	characteristic
20	#14273701	C₄	1	1	yes	characteristic
21	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
22	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₅₉	C₂²	1	2	yes	—
24	#14273684	C₂²	1	2	yes	—
25	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
26	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
28	Dic₅₉⋊C₈	C₂²	1	2	yes	—
29	Dic₅₉⋊C₈	C₂²	1	2	yes	—
30	(C₂×D₅₉)⋊C₈	C₂²	1	2	yes	—
31	(C₂×D₅₉)⋊C₈	C₂²	1	2	yes	—
32	#14273637	C₂²	1	2	yes	—
33	C₈×Dic₅₉	C₂²	1	2	yes	—
34	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
36	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
37	C₂²⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
38	C₂²⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₅₉	C₂×C₄	1	2	yes	—
45	C₂×D₂₃₆	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₅₉	C₂³	1	2	yes	—
47	C₂×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
52	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₅₉	C₂³	1	1	yes	characteristic
55	C₄²×C₅₉	C₂³	1	1	yes	characteristic
56	C₂²⋊Dic₅₉	C₂×C₄	1	2	yes	—
57	(C₂×D₅₉)⋊C₄	C₂×C₄	1	2	yes	—
58	(C₂×D₅₉)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₅₉)⋊C₄	C₂×C₄	1	2	yes	—
60	(C₂×D₅₉)⋊C₄	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
62	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
63	Dic₅₉⋊C₄	C₂×C₄	1	2	yes	—
64	Dic₅₉⋊C₄	C₂×C₄	1	2	yes	—
65	Dic₅₉⋊C₄	C₂×C₄	1	2	yes	—
66	Dic₅₉⋊C₄	C₂×C₄	1	2	yes	—
67	C₄×Dic₅₉	C₂³	1	2	yes	—
68	#11745664	—	4	2	no	—
69	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
70	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
71	#11745659	—	4	2	no	—
72	C₈×D₅₉	—	4	2	no	—
73	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×D₅₉	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₅₉	—	2	2	no	—
76	C₂×C₄×C₅₉	—	2	1	no	—
77	C₂×C₄×C₅₉	—	2	1	no	—
78	C₂×C₄×C₅₉	—	4	1	no	—
79	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
85	C₂²⋊D₅₉	—	4	2	no	—
86	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
87	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
88	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
89	D₂₃₆	—	2	2	no	—
90	C₄×D₅₉	—	2	2	no	—
91	C₄×D₅₉	—	2	2	no	—
92	Dic₁₁₈	—	2	2	no	—
93	C₈×C₅₉	—	2	1	no	—
94	C₈×C₅₉	—	2	1	no	—
95	C₈×C₅₉	—	2	1	no	—
96	C₈×C₅₉	—	2	1	no	—
97	C₅₉⋊C₈	—	2	2	no	—
98	C₅₉⋊C₈	—	2	2	no	—
99	C₂²×C₅₉	—	2	1	no	—
100	C₂²×C₅₉	—	2	1	no	—
101	C₂²×C₅₉	—	2	1	no	—
102	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₅₉	—	2	2	no	—
104	C₂×D₅₉	—	2	2	no	—
105	C₂×D₅₉	—	2	2	no	—
106	C₄×C₅₉	—	2	1	no	—
107	C₄×C₅₉	—	2	1	no	—
108	C₄×C₅₉	—	2	1	no	—
109	C₄×C₅₉	—	2	1	no	—
110	C₄×C₅₉	#131	1	2	yes	—
111	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
112	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
113	Dic₅₉	—	2	2	no	—
114	Dic₅₉	—	2	2	no	—
115	Dic₅₉	—	2	2	no	—
116	#2993	—	59	1	no	maximal, Sylow
117	C₂×C₅₉	—	4	1	no	—
118	C₂×C₅₉	#520	1	1	yes	characteristic
119	C₂×C₅₉	C₂×#131	1	1	yes	characteristic
120	C₂×C₅₉	#568	1	1	yes	characteristic
121	D₅₉	—	4	2	no	—
122	#529	—	59	1	no	—
123	#446	—	59	1	no	—
124	C₈×Q₈	—	59	1	no	—
125	#436	—	59	2	no	—
126	#436	—	59	1	no	—
127	#435	—	59	2	no	—
128	C₈×D₄	—	59	1	no	—
129	#433	—	59	2	no	—
130	#431	D₅₉	1	1	yes	characteristic, complemented
131	#424	—	59	2	no	—
132	#423	—	59	1	no	—
133	C₅₉	#2993	1	1	yes	characteristic, complemented, Sylow
134	C₂×M₁₆	—	59	2	no	—
135	C₂×M₁₆	—	59	2	no	—
136	C₂²×C₈	—	59	2	no	—
137	#126	—	59	2	no	—
138	#125	—	59	1	no	—
139	#124	—	59	1	no	—
140	#123	—	59	2	no	—
141	C₂²⋊Q₈	—	59	1	no	—
142	#121	—	59	1	no	—
143	C₄×Q₈	—	59	1	no	—
144	C₄×D₄	—	59	2	no	—
145	C₄×D₄	—	59	1	no	—
146	#117	—	59	2	no	—
147	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
148	C₄⋊C₈	—	59	2	no	—
149	C₄⋊C₈	—	59	2	no	—
150	C₄⋊C₈	—	59	2	no	—
151	C₄⋊C₈	—	59	1	no	—
152	C₄⋊C₈	—	59	1	no	—
153	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
154	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
155	C₂²⋊C₈	—	59	2	no	—
156	C₂²⋊C₈	—	59	2	no	—
157	C₂²⋊C₈	C₂×D₅₉	1	1	yes	characteristic
158	C₂²⋊C₈	C₂×D₅₉	1	1	yes	characteristic
159	#97	—	59	2	no	—
160	C₄×C₈	—	59	2	no	—
161	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
162	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
163	C₂×Q₈	—	59	1	no	—
164	C₂×D₄	—	59	2	no	—
165	C₂×D₄	—	59	1	no	—
166	C₂²×C₄	—	59	2	no	—
167	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
168	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
170	M₁₆	—	236	2	no	—
171	M₁₆	—	236	2	no	—
172	C₂×C₈	—	59	2	no	—
173	C₂×C₈	—	59	2	no	—
174	C₂×C₈	—	236	2	no	—
175	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
176	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
179	C₄⋊C₄	—	59	2	no	—
180	C₄⋊C₄	—	59	2	no	—
181	C₄⋊C₄	—	59	1	no	—
182	C₄⋊C₄	—	59	2	no	—
183	C₄⋊C₄	—	59	1	no	—
184	C₄⋊C₄	—	59	2	no	—
185	C₂²⋊C₄	—	59	2	no	—
186	C₂²⋊C₄	—	59	2	no	—
187	C₂²⋊C₄	—	59	2	no	—
188	C₂²⋊C₄	—	59	2	no	—
189	C₂²⋊C₄	—	59	2	no	—
190	C₄²	—	59	2	no	—
191	C₄²	C₄×D₅₉	1	1	yes	characteristic
192	C₄²	C₄×D₅₉	1	1	yes	characteristic
193	C₄²	C₂²×D₅₉	1	1	yes	characteristic
194	C₄²	C₂²×D₅₉	1	1	yes	characteristic
195	C₂³	—	59	2	no	—
196	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
197	Q₈	—	118	2	no	—
198	D₄	—	118	2	no	—
199	D₄	—	236	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	59	2	no	—
205	C₂×C₄	—	59	2	no	—
206	C₂×C₄	—	59	2	no	—
207	C₂×C₄	—	118	2	no	—
208	C₂×C₄	—	118	2	no	—
209	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	118	2	no	—
220	C₈	—	118	2	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	118	2	no	—
225	C₂²	—	118	2	no	—
226	C₂²	—	118	2	no	—
227	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	118	2	no	—
233	C₄	—	118	2	no	—
234	C₄	—	118	2	no	—
235	C₄	#14273733	1	2	yes	—
236	C₄	#14273773	1	1	yes	characteristic
237	C₄	#14273774	1	1	yes	characteristic
238	C₂	—	4	1	no	—
239	C₂	—	236	2	no	—
240	C₂	#16837882	1	1	yes	characteristic
241	C₂	C₂×#14273733	1	1	yes	characteristic
242	C₂	#16838168	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₄
Fitting	C₅₉×#431	C₂
hypercenter	#431	D₅₉
hypocenter	C₅₉	#2993
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #16837532	C₂	semidirect
3. C₈×Dic₁₁₈	C₂	semidirect
4. #16837481	C₂	semidirect
5. #16837477	C₂	semidirect
6. #16837476	C₂	semidirect
7. C₈×D₂₃₆	C₂	semidirect
8. #16837462	C₂	semidirect
9. #16837427	C₂	semidirect
10. #16837426	C₂	semidirect
11. C₅₉×#431	C₂	semidirect
130. #431	D₅₉	semidirect
133. C₅₉	#2993	semidirect