Subgroups — #28576395

Order 15104.

706 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434683	C₂	1	2	yes	maximal
2	#21434680	C₂	1	2	yes	maximal
3	#21434607	C₂	1	2	yes	maximal
4	#21423714	C₂	1	1	yes	maximal, characteristic, complemented
5	#21423705	C₂	1	2	yes	maximal, complemented
6	#21423694	C₂	1	2	yes	maximal, complemented
7	#21423690	C₂	1	2	yes	maximal, complemented
8	C₈²×D₅₉	C₂	1	1	yes	maximal, characteristic
9	C₅₉×#1713	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837957	C₂²	1	4	yes	—
11	#16837807	C₂²	1	2	yes	—
12	#16837807	C₂²	1	2	yes	—
13	#16837803	—	2	4	no	—
14	#16837798	C₂²	1	4	yes	—
15	#16837782	C₂²	1	2	yes	—
16	#16837754	C₂²	1	4	yes	—
17	#16837752	C₂²	1	4	yes	—
18	#16837644	—	2	4	no	—
19	#16837640	—	2	4	no	—
20	Q₁₆×Dic₅₉	—	2	4	no	—
21	#16837496	—	2	4	no	—
22	#16837439	—	2	4	no	—
23	C₄×C₈×D₅₉	C₂²	1	1	yes	characteristic
24	C₄×C₈×D₅₉	C₂²	1	1	yes	characteristic
25	C₄×C₈×D₅₉	C₂²	1	1	yes	characteristic
26	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
27	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
28	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
29	C₅₉×#500	C₂²	1	2	yes	—
30	C₅₉×#500	C₂²	1	2	yes	—
31	C₅₉×#494	C₂²	1	2	yes	—
32	C₈²×C₅₉	C₂²	1	1	yes	characteristic
33	C₂×Dic₂₃₆	—	2	4	no	—
34	C₂×C₈×D₅₉	D₄	1	2	yes	—
35	C₂×C₈×D₅₉	D₄	1	2	yes	—
36	C₂×C₈×D₅₉	D₄	1	2	yes	—
37	#14273727	C₂³	1	4	yes	—
38	#14273718	—	2	4	no	—
39	#14273707	—	2	4	no	—
40	#14273703	—	2	4	no	—
41	Q₈×Dic₅₉	—	2	4	no	—
42	#14273692	—	2	4	no	—
43	#14273685	—	2	4	no	—
44	C₄²×D₅₉	C₂³	1	1	yes	characteristic
45	#14273661	—	2	4	no	—
46	#14273661	—	2	4	no	—
47	#14273658	—	2	8	no	—
48	#14273651	—	2	8	no	—
49	#14273650	—	2	8	no	—
50	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
51	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
52	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
53	C₈×Dic₅₉	D₄	1	2	yes	—
54	C₈×Dic₅₉	D₄	1	2	yes	—
55	C₈×Dic₅₉	D₄	1	2	yes	—
56	C₂×Q₁₆×C₅₉	—	2	4	no	—
57	C₄⋊Q₈×C₅₉	C₂³	1	4	yes	—
58	#107×C₅₉	—	2	4	no	—
59	#107×C₅₉	—	2	4	no	—
60	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₅₉	—	2	4	no	—
64	C₂×C₄×D₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×D₅₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×D₅₉	C₂×D₄	1	1	yes	characteristic
67	C₂×Dic₁₁₈	—	2	4	no	—
68	Q₁₆×C₅₉	—	4	8	no	—
69	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
72	C₄⋊C₄×C₅₉	—	2	4	no	—
73	C₄⋊C₄×C₅₉	—	2	4	no	—
74	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
75	(C₂×D₅₉)⋊C₄	—	4	4	no	—
76	C₄⋊Dic₅₉	—	2	4	no	—
77	C₄⋊Dic₅₉	—	2	4	no	—
78	C₄⋊Dic₅₉	—	4	4	no	—
79	Dic₅₉⋊C₄	—	4	4	no	—
80	C₄×Dic₅₉	—	4	4	no	—
81	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
82	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
83	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
85	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
86	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
87	Dic₂₃₆	—	4	8	no	—
88	C₈×D₅₉	—	2	4	no	—
89	C₈×D₅₉	—	2	4	no	—
90	C₈×D₅₉	Q₁₆	1	8	yes	complemented
91	C₂²×D₅₉	#127	1	1	yes	characteristic
92	Q₈×C₅₉	—	4	8	no	—
93	C₂×C₄×C₅₉	—	4	4	no	—
94	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
97	C₂×Dic₅₉	—	4	4	no	—
98	C₂×Dic₅₉	#127	1	1	yes	characteristic
99	C₂×Dic₅₉	#127	1	1	yes	characteristic
100	C₂×Dic₅₉	#127	1	1	yes	characteristic
101	C₄×D₅₉	—	2	2	no	—
102	C₄×D₅₉	—	2	2	no	—
103	C₄×D₅₉	C₂×Q₁₆	1	4	yes	—
104	Dic₁₁₈	—	4	8	no	—
105	C₈×C₅₉	C₂×Q₁₆	1	4	yes	—
106	C₈×C₅₉	#135	1	4	yes	—
107	C₈×C₅₉	#135	1	4	yes	—
108	C₅₉⋊C₈	—	2	2	no	—
109	C₅₉⋊C₈	—	2	2	no	—
110	C₅₉⋊C₈	C₂×Q₁₆	1	4	yes	—
111	#19604	—	59	1	no	maximal, Sylow
112	C₂²×C₅₉	C₂×#127	1	1	yes	characteristic
113	C₂×D₅₉	—	2	1	no	—
114	C₂×D₅₉	—	2	1	no	—
115	C₂×D₅₉	#494	1	2	yes	—
116	C₄×C₅₉	—	8	4	no	—
117	C₄×C₅₉	C₂²×Q₁₆	1	2	yes	—
118	C₄×C₅₉	C₂×#135	1	2	yes	—
119	C₄×C₅₉	C₂×#135	1	2	yes	—
120	Dic₅₉	—	2	1	no	—
121	Dic₅₉	—	2	1	no	—
122	Dic₅₉	—	8	4	no	—
123	Dic₅₉	#494	1	2	yes	—
124	#3170	—	59	2	no	—
125	#3170	—	59	2	no	—
126	#3160	—	59	2	no	—
127	#1713	—	59	1	no	—
128	#1713	D₅₉	1	1	yes	characteristic, complemented
129	#1710	—	59	2	no	—
130	#1707	—	59	2	no	—
131	#1707	—	59	2	no	—
132	C₂×C₈²	—	59	1	no	—
133	C₂×C₅₉	C₂×#494	1	1	yes	characteristic
134	C₂×C₅₉	#3147	1	1	yes	characteristic
135	C₂×C₅₉	#3147	1	1	yes	characteristic
136	D₅₉	—	2	2	no	—
137	#533	—	59	4	no	—
138	#500	—	59	2	no	—
139	#500	—	59	2	no	—
140	#500	C₂×D₅₉	1	2	yes	—
141	#500	C₂×D₅₉	1	2	yes	—
142	#499	—	118	4	no	—
143	#498	—	59	4	no	—
144	#494	—	59	2	no	—
145	#494	C₂×D₅₉	1	2	yes	—
146	#487	—	59	4	no	—
147	#487	—	59	4	no	—
148	#467	—	118	4	no	—
149	#467	—	118	4	no	—
150	C₄×Q₁₆	—	118	4	no	—
151	C₄×Q₁₆	—	118	4	no	—
152	#427	—	118	4	no	—
153	C₂×C₄×C₈	—	59	1	no	—
154	C₂×C₄×C₈	—	59	1	no	—
155	C₂×C₄×C₈	—	59	1	no	—
156	C₈²	—	59	1	no	—
157	C₈²	—	59	1	no	—
158	C₈²	—	59	1	no	—
159	C₈²	C₂×D₅₉	1	1	yes	characteristic
160	C₅₉	#19604	1	1	yes	characteristic, complemented, Sylow
161	C₂×Q₁₆	—	2	4	no	—
162	C₂×Q₁₆	—	118	4	no	—
163	C₂²×C₈	—	59	2	no	—
164	C₂²×C₈	—	59	2	no	—
165	C₂²×C₈	—	59	2	no	—
166	C₄⋊Q₈	—	59	4	no	—
167	C₄⋊Q₈	C₂²×D₅₉	1	4	yes	—
168	#125	—	118	4	no	—
169	C₂²⋊Q₈	—	118	4	no	—
170	C₂²⋊Q₈	—	118	4	no	—
171	C₄×Q₈	—	118	4	no	—
172	C₄×Q₈	—	118	4	no	—
173	#117	—	118	4	no	—
174	C₂×C₄²	—	59	1	no	—
175	#107	—	2	4	no	—
176	#107	—	2	4	no	—
177	#107	—	118	4	no	—
178	#107	—	118	4	no	—
179	#106	—	118	8	no	—
180	#103	—	118	8	no	—
181	#103	—	118	8	no	—
182	C₄×C₈	—	59	2	no	—
183	C₄×C₈	—	59	2	no	—
184	C₄×C₈	—	59	2	no	—
185	C₄×C₈	—	59	1	no	—
186	C₄×C₈	—	59	1	no	—
187	C₄×C₈	—	59	1	no	—
188	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
189	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
190	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	—	118	4	no	—
193	C₂²×C₄	—	59	1	no	—
194	C₂²×C₄	—	59	1	no	—
195	C₂²×C₄	—	59	1	no	—
196	Q₁₆	—	4	8	no	—
197	Q₁₆	—	236	8	no	—
198	C₂×C₈	—	59	2	no	—
199	C₂×C₈	—	59	8	no	—
200	C₂×C₈	—	59	2	no	—
201	C₂×C₈	—	59	2	no	—
202	C₂×C₈	—	118	4	no	—
203	C₂×C₈	—	118	4	no	—
204	C₂×C₈	D₄×D₅₉	1	2	yes	—
205	C₂×C₈	D₄×D₅₉	1	2	yes	—
206	C₂×C₈	D₄×D₅₉	1	2	yes	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	118	4	no	—
210	C₄⋊C₄	—	118	4	no	—
211	C₄⋊C₄	—	236	4	no	—
212	C₄⋊C₄	—	236	4	no	—
213	C₂²⋊C₄	—	236	4	no	—
214	C₄²	—	59	1	no	—
215	C₄²	—	59	1	no	—
216	C₄²	—	59	1	no	—
217	C₄²	—	236	4	no	—
218	C₄²	C₂³×D₅₉	1	1	yes	characteristic
219	C₂³	—	59	1	no	—
220	Q₈	—	4	8	no	—
221	Q₈	—	236	8	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	59	1	no	—
224	C₂×C₄	—	59	4	no	—
225	C₂×C₄	—	59	1	no	—
226	C₂×C₄	—	59	1	no	—
227	C₂×C₄	—	118	2	no	—
228	C₂×C₄	—	118	2	no	—
229	C₂×C₄	—	236	4	no	—
230	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
231	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
232	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
233	C₈	—	59	4	no	—
234	C₈	—	118	2	no	—
235	C₈	—	118	2	no	—
236	C₈	Q₁₆×D₅₉	1	4	yes	—
237	C₈	#14273746	1	4	yes	—
238	C₈	#14273750	1	4	yes	—
239	C₂²	—	59	2	no	—
240	C₂²	—	118	1	no	—
241	C₂²	—	118	1	no	—
242	C₂²	#127×D₅₉	1	1	yes	characteristic
243	C₄	—	8	4	no	—
244	C₄	—	59	2	no	—
245	C₄	—	118	1	no	—
246	C₄	—	118	1	no	—
247	C₄	—	472	4	no	—
248	C₄	C₂×Q₁₆×D₅₉	1	2	yes	—
249	C₄	C₂×#14273746	1	2	yes	—
250	C₄	C₂×#14273750	1	2	yes	—
251	C₂	—	118	2	no	—
252	C₂	#494×D₅₉	1	1	yes	characteristic
253	C₂	#21434523	1	1	yes	characteristic
254	C₂	#21434525	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₅₉
commutator, cosocle	C₄²×C₅₉	C₂⁴
Frattini	C₄²	C₂³×D₅₉
Fitting	C₅₉×#1713	C₂
socle	C₂²×C₅₉	C₂×#127
prime radical	C₂²×D₅₉	#127
hypercenter	#1713	D₅₉
hypocenter	C₅₉	#19604
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1713

Complements

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

N	H ≅ G/N	type
4. #21423714	C₂	semidirect
5. #21423705	C₂	semidirect
6. #21423694	C₂	semidirect
7. #21423690	C₂	semidirect
9. C₅₉×#1713	C₂	semidirect
90. C₈×D₅₉	Q₁₆	semidirect
128. #1713	D₅₉	semidirect
160. C₅₉	#19604	semidirect

Maximal subgroups

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

subgroup	index
#21434683	2
#21434680	2
#21434607	2
#21423714	2
#21423705	2
#21423694	2
#21423690	2
C₈²×D₅₉	2
C₅₉×#1713	2
#19604	59