Subgroups — #28741970

Order 15104.

690 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837834	C₂	1	1	yes	maximal, characteristic
2	#21432867	C₂	1	2	yes	maximal, complemented
3	#21425954	C₂	1	2	yes	maximal, complemented
4	#21425947	C₂	1	4	yes	maximal, complemented
5	#21425942	C₂	1	2	yes	maximal, complemented
6	#21425801	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#2158	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14273749	C₂²	1	2	yes	—
10	#16837870	—	2	2	no	—
11	#16837834	C₄	1	8	yes	complemented
12	C₂×Dic₄₇₂	C₂²	1	1	yes	characteristic
13	C₂×#14273672	C₂²	1	2	yes	—
14	C₂×D₉₄₄	C₂²	1	1	yes	characteristic
15	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	#16837496	C₂²	1	4	yes	—
17	#16837489	—	2	4	no	—
18	#16837484	C₂²	1	4	yes	—
19	#16837396	—	2	2	no	—
20	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
21	#16837205	C₂²	1	4	yes	—
22	#16837202	C₂²	1	4	yes	—
23	C₁₆×Dic₅₉	C₂²	1	2	yes	—
24	C₁₆×Dic₅₉	C₂²	1	2	yes	—
25	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
26	C₅₉×#427	C₂²	1	2	yes	—
27	C₅₉×#367	C₂²	1	2	yes	—
28	C₅₉×#366	C₂²	1	2	yes	—
29	C₂×#11745685	—	2	2	no	—
30	#14273749	—	2	4	no	—
31	#14273749	C₂×C₄	1	8	yes	complemented
32	C₂×Dic₂₃₆	C₂³	1	2	yes	—
33	C₂×#11745660	—	2	2	no	—
34	C₂×D₄₇₂	C₂³	1	2	yes	—
35	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
36	#14273692	—	2	4	no	—
37	#14273689	—	2	4	no	—
38	(C₂²⋊D₅₉)⋊C₄	—	4	4	no	—
39	#14273684	—	4	2	no	—
40	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
41	C₄²×D₅₉	—	4	2	no	—
42	Dic₄₇₂	C₂×C₄	1	4	yes	complemented
43	#14273672	C₂×C₄	1	8	yes	complemented
44	D₉₄₄	C₂×C₄	1	4	yes	complemented
45	C₂×C₅₉⋊C₁₆	—	2	2	no	—
46	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
47	#14273650	—	2	4	no	—
48	#14273647	—	2	4	no	—
49	C₈×Dic₅₉	D₄	1	2	yes	—
50	C₈×Dic₅₉	C₂³	1	2	yes	—
51	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
52	#117×C₅₉	—	2	2	no	—
53	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	complemented
54	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
55	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
56	#107×C₅₉	C₂³	1	4	yes	—
57	#106×C₅₉	—	2	2	no	—
58	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₂²⋊D₅₉	—	4	2	no	—
60	C₂²×Dic₅₉	D₈	1	2	yes	—
61	#11745685	—	2	8	no	—
62	#11745685	—	4	4	no	—
63	C₂×D₂₃₆	—	2	2	no	—
64	C₂×C₄×D₅₉	—	4	2	no	—
65	C₂×C₄×D₅₉	—	4	4	no	—
66	C₂×Dic₁₁₈	—	2	2	no	—
67	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₅₉	#41	1	2	yes	—
70	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
71	C₄⋊C₄×C₅₉	—	2	4	no	—
72	C₂²⋊C₄×C₅₉	—	4	2	no	—
73	C₄²×C₅₉	—	4	2	no	—
74	(C₂×D₅₉)⋊C₄	—	4	4	no	—
75	Dic₅₉⋊C₄	—	4	4	no	—
76	C₄×Dic₅₉	—	4	4	no	—
77	C₄×Dic₅₉	D₈	1	1	yes	characteristic
78	C₄×Dic₅₉	D₈	1	1	yes	characteristic
79	C₄×Dic₅₉	C₂×D₄	1	2	yes	—
80	C₂×C₅₉⋊C₈	—	2	2	no	—
81	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
82	Dic₂₃₆	—	2	2	no	—
83	Dic₂₃₆	C₂²×C₄	1	4	yes	—
84	D₄₇₂	—	2	2	no	—
85	D₄₇₂	C₂²×C₄	1	4	yes	—
86	#11745660	—	2	8	no	—
87	C₁₆×C₅₉	C₂²×C₄	1	2	yes	—
88	C₁₆×C₅₉	C₂²×C₄	1	2	yes	—
89	C₅₉⋊C₁₆	—	2	2	no	—
90	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
91	C₂²×D₅₉	—	4	2	no	—
92	C₂×C₄×C₅₉	—	4	4	no	—
93	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₅₉	#135	1	2	yes	—
96	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
97	C₂²⋊D₅₉	—	4	8	no	—
98	C₂×Dic₅₉	—	2	4	no	—
99	C₂×Dic₅₉	—	4	2	no	—
100	C₂×Dic₅₉	C₂×D₈	1	2	yes	—
101	C₂×Dic₅₉	C₂×D₈	1	2	yes	—
102	D₂₃₆	—	2	4	no	—
103	D₂₃₆	—	4	2	no	—
104	C₄×D₅₉	—	4	8	no	—
105	C₄×D₅₉	—	8	8	no	—
106	Dic₁₁₈	—	2	4	no	—
107	Dic₁₁₈	—	4	2	no	—
108	C₈×C₅₉	C₄×D₄	1	2	yes	—
109	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
111	C₅₉⋊C₈	—	2	2	no	—
112	C₄×#508	—	59	1	no	maximal, Sylow
113	C₂²×C₅₉	—	2	2	no	—
114	C₂²×C₅₉	C₄×D₈	1	2	yes	—
115	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
116	C₂×D₅₉	—	4	4	no	—
117	C₂×D₅₉	—	8	4	no	—
118	C₄×C₅₉	—	8	4	no	—
119	C₄×C₅₉	C₄×D₈	1	2	yes	—
120	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
121	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
122	Dic₅₉	—	2	2	no	—
123	Dic₅₉	—	4	4	no	—
124	Dic₅₉	#508	1	4	yes	complemented
125	C₂×#508	—	59	1	no	—
126	C₄×#135	—	59	2	no	—
127	C₄×Q₃₂	—	59	2	no	—
128	C₄×SD₃₂	—	59	4	no	—
129	C₄×D₁₆	—	59	2	no	—
130	#2158	D₅₉	1	1	yes	characteristic, complemented
131	#2138	—	59	2	no	—
132	C₂×C₄×C₁₆	—	59	1	no	—
133	C₂×C₅₉	—	2	2	no	—
134	C₂×C₅₉	C₂×C₄×D₈	1	1	yes	characteristic
135	C₂×C₅₉	C₂×#508	1	2	yes	—
136	D₅₉	—	8	4	no	—
137	C₂×#135	—	59	2	no	—
138	C₄×#41	—	118	2	no	—
139	#508	—	59	8	no	—
140	C₂×Q₃₂	—	59	1	no	—
141	C₂×SD₃₂	—	59	2	no	—
142	C₂×D₁₆	—	59	1	no	—
143	C₂²×C₁₆	—	59	1	no	—
144	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
145	C₄×Q₁₆	—	59	4	no	—
146	C₄×SD₁₆	—	118	4	no	—
147	C₄×D₈	—	59	4	no	—
148	#427	C₂×D₅₉	1	2	yes	—
149	#416	—	118	2	no	—
150	C₂×C₄×C₈	—	59	1	no	—
151	#367	C₂×D₅₉	1	2	yes	—
152	#366	C₂×D₅₉	1	2	yes	—
153	#358	—	59	4	no	—
154	#357	—	59	4	no	—
155	C₄×C₁₆	—	59	2	no	—
156	C₄×C₁₆	—	59	2	no	—
157	C₅₉	C₄×#508	1	1	yes	characteristic, complemented, Sylow
158	C₂×#41	—	118	2	no	—
159	#135	—	59	8	no	—
160	#135	—	118	4	no	—
161	C₂×Q₁₆	—	59	2	no	—
162	C₂×SD₁₆	—	118	2	no	—
163	C₂×D₈	—	59	2	no	—
164	C₂²×C₈	—	59	1	no	—
165	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
166	C₄×Q₈	—	118	4	no	—
167	C₄×D₄	—	118	4	no	—
168	C₄×D₄	—	236	4	no	—
169	#117	—	2	2	no	—
170	#117	—	236	2	no	—
171	C₂×C₄²	—	59	1	no	—
172	C₂×C₄²	—	236	2	no	—
173	Q₃₂	—	59	4	no	—
174	SD₃₂	—	59	8	no	—
175	D₁₆	—	59	4	no	—
176	C₂×C₁₆	—	59	2	no	—
177	C₂×C₁₆	—	118	2	no	—
178	C₂×C₁₆	C₄×D₅₉	1	4	yes	complemented
179	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
180	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
181	#107	C₂²×D₅₉	1	4	yes	—
182	#106	—	2	2	no	—
183	#103	—	118	4	no	—
184	#102	—	118	4	no	—
185	C₄×C₈	—	59	2	no	—
186	C₄×C₈	—	59	2	no	—
187	#41	—	118	8	no	—
188	#41	—	236	4	no	—
189	C₂×Q₈	—	118	2	no	—
190	C₂×D₄	—	118	2	no	—
191	C₂×D₄	—	236	2	no	—
192	C₂²×C₄	—	59	2	no	—
193	C₂²×C₄	—	236	4	no	—
194	C₂²×C₄	—	236	2	no	—
195	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
196	Q₁₆	—	59	4	no	—
197	Q₁₆	—	118	2	no	—
198	SD₁₆	—	118	8	no	—
199	D₈	—	59	4	no	—
200	D₈	—	118	2	no	—
201	C₂×C₈	—	59	2	no	—
202	C₂×C₈	—	118	2	no	—
203	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
204	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
205	C₂×C₈	#11745689	1	2	yes	—
206	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	236	4	no	—
209	C₂²⋊C₄	—	4	2	no	—
210	C₂²⋊C₄	—	236	4	no	—
211	C₄²	—	4	2	no	—
212	C₄²	—	59	1	no	—
213	C₄²	—	59	1	no	—
214	C₄²	—	59	2	no	—
215	C₄²	—	236	4	no	—
216	C₁₆	—	118	2	no	—
217	C₁₆	C₂×C₄×D₅₉	1	2	yes	—
218	C₁₆	C₂×C₄×D₅₉	1	2	yes	—
219	C₂³	—	236	2	no	—
220	C₂³	D₈×D₅₉	1	1	yes	characteristic
221	Q₈	—	118	4	no	—
222	Q₈	—	236	2	no	—
223	D₄	—	118	4	no	—
224	D₄	—	236	8	no	—
225	D₄	—	236	2	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	59	2	no	—
228	C₂×C₄	—	59	2	no	—
229	C₂×C₄	—	118	4	no	—
230	C₂×C₄	—	472	8	no	—
231	C₂×C₄	—	236	2	no	—
232	C₂×C₄	—	236	8	no	—
233	C₂×C₄	#14273689	1	2	yes	—
234	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
235	C₂×C₄	#14273750	1	2	yes	—
236	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
237	C₈	—	118	2	no	—
238	C₈	#14273689	1	2	yes	—
239	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
240	C₈	C₂×#11745689	1	1	yes	characteristic
241	C₂²	—	2	2	no	—
242	C₂²	—	472	4	no	—
243	C₂²	—	236	4	no	—
244	C₂²	#16837484	1	2	yes	—
245	C₂²	C₂×D₈×D₅₉	1	1	yes	characteristic
246	C₄	—	8	4	no	—
247	C₄	—	59	4	no	—
248	C₄	—	118	2	no	—
249	C₄	—	236	4	no	—
250	C₄	#16837484	1	2	yes	—
251	C₄	C₂×#14273689	1	1	yes	characteristic
252	C₄	C₂×#14273750	1	1	yes	characteristic
253	C₂	—	2	2	no	—
254	C₂	—	472	4	no	—
255	C₂	C₂×#16837484	1	1	yes	characteristic
256	C₂	#508×D₅₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₉
commutator	C₈×C₅₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2158	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂×#16837834	C₂
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	#2158	D₅₉
hypocenter	C₅₉	C₄×#508
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
2. #21432867	C₂	semidirect
3. #21425954	C₂	semidirect
4. #21425947	C₂	semidirect
5. #21425942	C₂	semidirect
6. #21425801	C₂	semidirect
7. C₂×C₁₆×Dic₅₉	C₂	semidirect
8. C₅₉×#2158	C₂	semidirect
11. #16837834	C₄	semidirect
31. #14273749	C₂×C₄	semidirect
42. Dic₄₇₂	C₂×C₄	semidirect
43. #14273672	C₂×C₄	semidirect
44. D₉₄₄	C₂×C₄	semidirect
53. C₂×C₁₆×C₅₉	C₂×C₄	semidirect
124. Dic₅₉	#508	semidirect
130. #2158	D₅₉	semidirect
157. C₅₉	C₄×#508	semidirect
178. C₂×C₁₆	C₄×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837834	2
#21432867	2
#21425954	2
#21425947	2
#21425942	2
#21425801	2
C₂×C₁₆×Dic₅₉	2
C₅₉×#2158	2
C₄×#508	59