Subgroups — #28741936

Order 15104.

642 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	#21437198	C₂	1	2	yes	maximal, complemented
2	#21433000	C₂	1	1	yes	maximal, characteristic
3	#21425937	C₂	1	4	yes	maximal, complemented
4	#21425932	C₂	1	4	yes	maximal, complemented
5	C₂×#16837187	C₂	1	2	yes	maximal, complemented
6	#21425693	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	#16838202	C₄	1	1	yes	characteristic
9	C₂×#14273733	C₂²	1	2	yes	—
10	#16837820	C₂²	1	8	yes	complemented
11	#16837818	—	2	4	no	—
12	C₂×#14273667	C₂²	1	1	yes	characteristic
13	M₃₂×D₅₉	—	2	4	no	—
14	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
15	#16837525	C₄	1	4	yes	—
16	#16837520	C₂²	1	4	yes	—
17	#16837451	C₄	1	1	yes	characteristic
18	#16837420	C₄	1	2	yes	—
19	#16837187	C₂²	1	4	yes	complemented
20	#16837187	C₂²	1	4	yes	complemented
21	#16837178	C₂²	1	2	yes	complemented
22	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	complemented
23	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
24	C₅₉×#405	C₂²	1	1	yes	characteristic
25	C₅₉×#364	C₂²	1	4	yes	complemented
26	C₂×#11745685	C₂×C₄	1	2	yes	—
27	#14273760	C₈	1	4	yes	—
28	#14273755	C₈	1	4	yes	—
29	#14273749	C₂×C₄	1	4	yes	—
30	#14273733	—	2	2	no	—
31	#14273733	—	2	2	no	—
32	#14273733	C₂×C₄	1	4	yes	—
33	#14273733	C₂³	1	4	yes	—
34	C₂×#11745664	C₂×C₄	1	2	yes	—
35	C₂×#11745659	—	2	2	no	—
36	C₂×C₈×D₅₉	—	2	2	no	—
37	#14273667	D₄	1	4	yes	—
38	#14273666	—	2	8	no	—
39	C₂×C₅₉⋊C₁₆	—	2	2	no	—
40	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
42	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
43	C₁₆×D₅₉	—	2	8	no	—
44	#14273663	C₂×C₄	1	4	yes	—
45	#14273655	C₂×C₄	1	8	yes	—
46	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
48	#117×C₅₉	C₂×C₄	1	1	yes	characteristic
49	M₃₂×C₅₉	—	2	2	no	—
50	M₃₂×C₅₉	C₂³	1	4	yes	—
51	C₂×C₁₆×C₅₉	—	2	2	no	—
52	#97×C₅₉	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₅₉	C₂³	1	2	yes	—
54	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₂²⋊D₅₉	—	2	2	no	—
56	#11745685	—	2	2	no	—
57	#11745685	—	2	2	no	—
58	#11745685	C₂×C₈	1	4	yes	—
59	#11745685	C₂²×C₄	1	4	yes	—
60	C₂×D₂₃₆	C₂×C₈	1	2	yes	—
61	C₂×C₄×D₅₉	—	2	2	no	—
62	C₂×Dic₁₁₈	C₂×C₈	1	2	yes	—
63	M₁₆×C₅₉	C₂×C₈	1	4	yes	—
64	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₅₉	#41	1	2	yes	—
68	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₅₉	—	2	1	no	—
70	C₂²⋊C₄×C₅₉	—	2	1	no	—
71	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
72	#11745664	—	2	2	no	—
73	#11745664	C₂²×C₄	1	4	yes	—
74	C₂×C₅₉⋊C₈	—	2	2	no	—
75	Dic₂₃₆	C₂×C₈	1	4	yes	—
76	D₄₇₂	C₂×C₈	1	4	yes	—
77	#11745660	C₂×C₈	1	8	yes	—
78	#11745659	—	2	4	no	—
79	#11745659	—	2	4	no	—
80	C₈×D₅₉	—	2	4	no	—
81	C₈×D₅₉	—	2	4	no	—
82	C₁₆×C₅₉	—	2	4	no	—
83	C₅₉⋊C₁₆	—	2	1	no	—
84	C₅₉⋊C₁₆	—	2	1	no	—
85	C₅₉⋊C₁₆	C₂×D₄	1	4	yes	—
86	C₂³×C₅₉	#131	1	1	yes	characteristic
87	C₂²×D₅₉	—	2	2	no	—
88	C₂×C₄×C₅₉	—	2	2	no	—
89	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
91	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
92	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
93	C₂²⋊D₅₉	—	2	4	no	—
94	C₂²⋊D₅₉	—	2	4	no	—
95	C₂×Dic₅₉	—	2	2	no	—
96	D₂₃₆	—	2	2	no	—
97	D₂₃₆	C₂²×C₈	1	4	yes	—
98	C₄×D₅₉	—	2	4	no	—
99	C₄×D₅₉	—	2	4	no	—
100	Dic₁₁₈	—	2	2	no	—
101	Dic₁₁₈	C₂²×C₈	1	4	yes	—
102	C₈×C₅₉	C₄×D₄	1	2	yes	—
103	C₈×C₅₉	C₂²×C₈	1	4	yes	—
104	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
105	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
106	C₅₉⋊C₈	—	2	4	no	—
107	#33442	—	59	1	no	maximal, Sylow
108	C₂²×C₅₉	—	4	1	no	—
109	C₂²×C₅₉	C₈×D₄	1	2	yes	—
110	C₂²×C₅₉	C₂×#131	1	1	yes	characteristic
111	C₂×D₅₉	—	2	4	no	—
112	C₂×D₅₉	—	8	2	no	—
113	C₄×C₅₉	—	4	2	no	—
114	C₄×C₅₉	C₈×D₄	1	2	yes	—
115	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
116	C₄×C₅₉	C₂³×C₈	1	1	yes	characteristic
117	Dic₅₉	—	2	4	no	—
118	#3407	—	59	2	no	—
119	#2955	—	59	1	no	—
120	#2171	—	59	4	no	—
121	#2170	—	59	4	no	—
122	#2153	D₅₉	1	1	yes	characteristic, complemented
123	C₂×#350	—	59	2	no	—
124	#2108	—	59	1	no	—
125	C₂×C₅₉	—	2	1	no	—
126	C₂×C₅₉	—	2	2	no	—
127	C₂×C₅₉	C₂×C₈×D₄	1	1	yes	characteristic
128	D₅₉	—	4	4	no	—
129	#575	—	59	1	no	—
130	C₂×#131	—	59	2	no	—
131	#504	—	59	8	no	—
132	#504	—	118	4	no	—
133	C₂×M₃₂	—	59	1	no	—
134	C₂×M₃₂	—	118	4	no	—
135	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
136	C₂²×C₁₆	—	59	1	no	—
137	#444	—	59	4	no	—
138	#443	—	59	4	no	—
139	#430	—	59	1	no	—
140	#421	—	59	2	no	—
141	#405	C₂×D₅₉	1	1	yes	characteristic
142	#364	C₂×D₅₉	1	4	yes	complemented
143	#350	—	59	4	no	—
144	#350	—	59	4	no	—
145	#346	—	59	2	no	—
146	C₄×C₁₆	—	59	2	no	—
147	C₅₉	#33442	1	1	yes	characteristic, complemented, Sylow
148	C₂×#41	—	59	2	no	—
149	#137	—	59	4	no	—
150	AGL(1,ℤ/8)	—	59	4	no	—
151	#135	—	59	4	no	—
152	#131	—	59	4	no	—
153	#131	—	59	4	no	—
154	#131	—	118	2	no	—
155	#131	—	118	2	no	—
156	C₂×M₁₆	—	59	2	no	—
157	C₂×M₁₆	—	118	2	no	—
158	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
159	C₂²×C₈	—	118	2	no	—
160	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
161	#117	C₄×D₅₉	1	1	yes	characteristic
162	M₃₂	—	2	2	no	—
163	M₃₂	—	59	4	no	—
164	M₃₂	—	118	8	no	—
165	M₃₂	C₂²×D₅₉	1	4	yes	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	59	2	no	—
168	C₂×C₁₆	—	59	1	no	—
169	C₂×C₁₆	—	59	1	no	—
170	C₂×C₁₆	—	118	8	no	—
171	C₂×C₁₆	—	118	2	no	—
172	#108	—	59	4	no	—
173	#104	—	59	8	no	—
174	#97	C₄×D₅₉	1	2	yes	—
175	C₄×C₈	C₂²×D₅₉	1	2	yes	—
176	#41	—	59	4	no	—
177	#41	—	59	4	no	—
178	#41	—	118	2	no	—
179	#41	—	118	2	no	—
180	C₂×Q₈	—	59	2	no	—
181	C₂×D₄	—	59	2	no	—
182	C₂×D₄	—	118	2	no	—
183	C₂²×C₄	—	118	2	no	—
184	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
185	Q₁₆	—	59	4	no	—
186	SD₁₆	—	59	8	no	—
187	D₈	—	59	4	no	—
188	M₁₆	—	59	4	no	—
189	M₁₆	—	118	2	no	—
190	M₁₆	—	118	4	no	—
191	M₁₆	—	118	4	no	—
192	M₁₆	C₈×D₅₉	1	4	yes	—
193	C₂×C₈	—	118	2	no	—
194	C₂×C₈	—	118	4	no	—
195	C₂×C₈	—	118	4	no	—
196	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
197	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
198	C₂×C₈	D₄×D₅₉	1	2	yes	—
199	C₂×C₈	#11745689	1	2	yes	—
200	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
201	C₄⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₄²	C₂×C₄×D₅₉	1	2	yes	—
204	C₁₆	—	2	4	no	—
205	C₁₆	—	59	4	no	—
206	C₁₆	—	118	1	no	—
207	C₁₆	—	118	1	no	—
208	C₂³	—	118	2	no	—
209	C₂³	#14273735	1	1	yes	characteristic
210	Q₈	—	59	4	no	—
211	Q₈	—	118	2	no	—
212	D₄	—	59	4	no	—
213	D₄	—	118	4	no	—
214	D₄	—	118	4	no	—
215	D₄	—	118	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	118	2	no	—
218	C₂×C₄	—	118	4	no	—
219	C₂×C₄	—	118	4	no	—
220	C₂×C₄	#14273689	1	2	yes	—
221	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
222	C₂×C₄	#14273735	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
224	C₈	—	118	4	no	—
225	C₈	#14273689	1	2	yes	—
226	C₈	C₂×C₈×D₅₉	1	4	yes	—
227	C₈	C₂×D₄×D₅₉	1	1	yes	characteristic
228	C₈	C₂×#11745689	1	1	yes	characteristic
229	C₂²	—	4	1	no	—
230	C₂²	—	118	4	no	—
231	C₂²	—	472	2	no	—
232	C₂²	#16837467	1	2	yes	—
233	C₂²	C₂×#14273735	1	1	yes	characteristic
234	C₄	—	4	2	no	—
235	C₄	—	118	4	no	—
236	C₄	#16837467	1	2	yes	—
237	C₄	C₂×#14273689	1	1	yes	characteristic
238	C₄	C₂²×C₈×D₅₉	1	1	yes	characteristic
239	C₂	—	2	1	no	—
240	C₂	—	2	2	no	—
241	C₂	—	236	4	no	—
242	C₂	C₂×#16837467	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₅₉
commutator	C₄×C₅₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2153	C₂
socle	C₂×C₅₉	C₂×C₈×D₄
prime radical	#16838202	C₄
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	#2153	D₅₉
hypocenter	C₅₉	#33442
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

Complements

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

N	H ≅ G/N	type
1. #21437198	C₂	semidirect
3. #21425937	C₂	semidirect
4. #21425932	C₂	semidirect
5. C₂×#16837187	C₂	semidirect
6. #21425693	C₂	semidirect
7. C₅₉×#2153	C₂	semidirect
10. #16837820	C₂²	semidirect
19. #16837187	C₂²	semidirect
20. #16837187	C₂²	semidirect
21. #16837178	C₂²	semidirect
22. C₄×C₅₉⋊C₁₆	C₂²	semidirect
25. C₅₉×#364	C₂²	semidirect
122. #2153	D₅₉	semidirect
142. #364	C₂×D₅₉	semidirect
147. C₅₉	#33442	semidirect

Maximal subgroups

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

subgroup	index
#21437198	2
#21433000	2
#21425937	2
#21425932	2
C₂×#16837187	2
#21425693	2
C₅₉×#2153	2
#33442	59