Subgroups — #28597303

Order 15104.

1046 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434524	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837520	C₂	1	4	yes	maximal, complemented
3	#21432809	C₂	1	2	yes	maximal, complemented
4	#21424984	C₂	1	4	yes	maximal, complemented
5	#21424924	C₂	1	2	yes	maximal, complemented
6	#21424268	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273749	C₂²	1	4	yes	—
9	C₂×#14273733	C₂²	1	4	yes	—
10	#16837952	C₂²	1	2	yes	—
11	#16837786	C₄	1	4	yes	complemented
12	#16837782	C₄	1	1	yes	characteristic, complemented
13	#16837779	C₄	1	1	yes	characteristic, complemented
14	#16837775	C₄	1	2	yes	complemented
15	#16837520	—	2	16	no	—
16	#16837481	—	2	4	no	—
17	C₈×D₂₃₆	—	2	4	no	—
18	C₂×#14273663	C₂²	1	4	yes	—
19	#16837426	C₂²	1	2	yes	—
20	C₂×#14273655	C₂²	1	8	yes	—
21	#16837368	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
26	C₄³×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#11745685	C₂³	1	4	yes	—
28	#14273749	—	2	8	no	—
29	#14273749	—	2	8	no	—
30	C₂×Dic₂₃₆	C₂×C₄	1	4	yes	—
31	C₂×#11745660	C₂×C₄	1	8	yes	—
32	C₂×D₄₇₂	C₂×C₄	1	4	yes	—
33	#14273733	—	2	16	no	—
34	C₂×#11745664	C₂³	1	4	yes	—
35	C₂×#11745659	—	2	4	no	—
36	C₂×C₈×D₅₉	—	2	4	no	—
37	#14273727	C₂×C₄	1	2	yes	—
38	#14273724	C₂×C₄	1	2	yes	—
39	#14273715	C₂×C₄	1	4	yes	—
40	#14273701	—	2	4	no	—
41	C₄×D₂₃₆	—	2	4	no	—
42	#14273663	—	2	8	no	—
43	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
44	#14273655	—	2	16	no	—
45	(C₂×D₅₉)⋊C₈	—	2	8	no	—
46	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
48	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₅₉	—	2	2	no	—
50	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
51	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₅₉	—	2	4	no	—
53	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₅₉	D₄	1	2	yes	—
57	C₄×C₈×C₅₉	D₄	1	2	yes	—
58	C₄×C₈×C₅₉	D₄	1	8	yes	complemented
59	C₂²×C₄×C₅₉	—	2	2	no	—
60	C₂²×C₄×C₅₉	#41	1	2	yes	—
61	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₉	—	2	4	no	—
63	#11745685	—	2	8	no	—
64	#11745685	—	2	8	no	—
65	C₂×D₂₃₆	—	2	4	no	—
66	C₂×D₂₃₆	C₂²×C₄	1	4	yes	—
67	C₂×C₄×D₅₉	—	2	4	no	—
68	C₂×Dic₁₁₈	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₅₉	—	2	2	no	—
70	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
74	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
76	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₅₉	#41	1	4	yes	—
78	C₄²×C₅₉	—	2	8	no	—
79	C₄²×C₅₉	—	2	2	no	—
80	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅₉	C₂×D₄	1	4	yes	—
83	C₄²×C₅₉	C₂×D₄	1	2	yes	—
84	(C₂×D₅₉)⋊C₄	—	2	8	no	—
85	C₄⋊Dic₅₉	—	2	4	no	—
86	#11745664	—	2	8	no	—
87	C₂×C₅₉⋊C₈	—	2	4	no	—
88	Dic₂₃₆	—	2	8	no	—
89	D₄₇₂	—	2	8	no	—
90	#11745660	—	2	16	no	—
91	#11745659	—	4	8	no	—
92	C₈×D₅₉	—	4	8	no	—
93	C₂³×C₅₉	C₂×#41	1	1	yes	characteristic
94	C₂²×D₅₉	—	2	4	no	—
95	C₂×C₄×C₅₉	—	2	8	no	—
96	C₂×C₄×C₅₉	—	2	4	no	—
97	C₂×C₄×C₅₉	—	2	4	no	—
98	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
102	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
104	C₂²⋊D₅₉	—	4	8	no	—
105	C₂×Dic₅₉	—	2	4	no	—
106	D₂₃₆	—	2	8	no	—
107	D₂₃₆	—	4	8	no	—
108	C₄×D₅₉	—	4	4	no	—
109	C₄×D₅₉	—	4	4	no	—
110	Dic₁₁₈	—	2	8	no	—
111	C₈×C₅₉	—	2	2	no	—
112	C₈×C₅₉	C₄×D₄	1	4	yes	—
113	C₈×C₅₉	C₄×D₄	1	4	yes	—
114	C₈×C₅₉	#127	1	2	yes	—
115	C₈×C₅₉	#127	1	2	yes	—
116	C₅₉⋊C₈	—	4	4	no	—
117	#21705	—	59	1	no	maximal, Sylow
118	C₂²×C₅₉	—	2	2	no	—
119	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
120	C₂²×C₅₉	#532	1	2	yes	—
121	C₂×D₅₉	—	4	4	no	—
122	C₂×D₅₉	—	4	8	no	—
123	C₄×C₅₉	—	2	8	no	—
124	C₄×C₅₉	—	2	2	no	—
125	C₄×C₅₉	#443	1	4	yes	complemented
126	C₄×C₅₉	C₂×C₄×D₄	1	2	yes	—
127	C₄×C₅₉	C₂×#127	1	1	yes	characteristic
128	C₄×C₅₉	#532	1	1	yes	characteristic
129	Dic₅₉	—	4	4	no	—
130	#3147	—	59	1	no	—
131	C₂×#443	—	59	4	no	—
132	#2931	—	59	2	no	—
133	#1974	—	59	4	no	—
134	#1962	—	59	2	no	—
135	#1848	—	59	1	no	—
136	C₄²×C₈	D₅₉	1	1	yes	characteristic, complemented
137	C₂×C₅₉	—	2	2	no	—
138	C₂×C₅₉	C₄×#127	1	1	yes	characteristic
139	C₂×C₅₉	C₂×#443	1	2	yes	—
140	D₅₉	—	8	4	no	—
141	C₂×#135	—	59	4	no	—
142	C₂×#131	—	59	4	no	—
143	#532	—	59	2	no	—
144	#495	—	59	4	no	—
145	#494	—	59	1	no	—
146	#493	—	59	1	no	—
147	#492	—	59	2	no	—
148	#443	—	118	16	no	—
149	#436	—	118	4	no	—
150	C₈×D₄	—	118	4	no	—
151	C₂×#108	—	59	4	no	—
152	#423	—	59	2	no	—
153	C₂×#104	—	59	8	no	—
154	#408	—	59	4	no	—
155	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
156	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
158	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
159	C₄³	C₂×D₅₉	1	1	yes	characteristic
160	C₅₉	#21705	1	1	yes	characteristic, complemented, Sylow
161	C₂×#41	—	59	4	no	—
162	#135	—	118	8	no	—
163	#135	—	118	8	no	—
164	C₂×Q₁₆	—	59	4	no	—
165	C₂×SD₁₆	—	59	8	no	—
166	C₂×D₈	—	59	4	no	—
167	#131	—	118	16	no	—
168	C₂×M₁₆	—	59	4	no	—
169	C₂×M₁₆	—	118	4	no	—
170	C₂²×C₈	—	118	4	no	—
171	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
173	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
174	C₄⋊Q₈	—	59	2	no	—
175	#127	—	59	2	no	—
176	#124	—	59	4	no	—
177	#121	—	118	4	no	—
178	C₄×D₄	—	118	4	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	C₂²×D₅₉	1	2	yes	—
181	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
182	#108	—	118	8	no	—
183	C₄⋊C₈	—	118	4	no	—
184	#104	—	118	16	no	—
185	C₂²⋊C₈	—	118	8	no	—
186	C₄×C₈	—	2	4	no	—
187	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
188	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
189	C₄×C₈	C₄×D₅₉	1	2	yes	—
190	C₄×C₈	D₂₃₆	1	2	yes	—
191	C₄×C₈	D₂₃₆	1	2	yes	—
192	C₄×C₈	D₂₃₆	1	8	yes	complemented
193	#41	—	118	8	no	—
194	#41	—	118	8	no	—
195	C₂×Q₈	—	59	4	no	—
196	C₂×D₄	—	59	4	no	—
197	C₂×D₄	—	118	4	no	—
198	C₂×D₄	—	118	4	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	118	4	no	—
201	C₂²×C₄	#11745685	1	2	yes	—
202	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
203	Q₁₆	—	118	8	no	—
204	SD₁₆	—	118	16	no	—
205	D₈	—	118	8	no	—
206	M₁₆	—	118	8	no	—
207	M₁₆	—	236	8	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	118	4	no	—
210	C₂×C₈	—	236	8	no	—
211	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
212	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
213	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
214	C₂×C₈	C₂×D₂₃₆	1	4	yes	—
215	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
216	C₂×C₈	C₂×D₂₃₆	1	4	yes	—
217	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
218	C₂×C₈	#11745685	1	4	yes	—
219	C₄⋊C₄	—	118	4	no	—
220	C₂²⋊C₄	—	118	8	no	—
221	C₄²	—	2	8	no	—
222	C₄²	—	2	2	no	—
223	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
224	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
225	C₄²	C₂×D₂₃₆	1	4	yes	—
226	C₄²	C₂×D₂₃₆	1	2	yes	—
227	C₂³	—	118	4	no	—
228	C₂³	C₂×#11745685	1	1	yes	characteristic
229	Q₈	—	118	8	no	—
230	D₄	—	118	8	no	—
231	D₄	—	236	8	no	—
232	D₄	—	236	8	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	118	4	no	—
237	C₂×C₄	—	236	4	no	—
238	C₂×C₄	—	236	4	no	—
239	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
240	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
241	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
242	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₂₃₆	1	2	yes	—
244	C₂×C₄	C₂×#11745685	1	2	yes	—
245	C₈	—	2	2	no	—
246	C₈	—	236	4	no	—
247	C₈	C₄×D₂₃₆	1	4	yes	—
248	C₈	C₄×D₂₃₆	1	4	yes	—
249	C₈	#14273724	1	2	yes	—
250	C₈	#14273724	1	2	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	236	8	no	—
253	C₂²	—	236	4	no	—
254	C₂²	C₂×C₄×D₂₃₆	1	1	yes	characteristic
255	C₂²	#16837952	1	2	yes	—
256	C₄	—	2	8	no	—
257	C₄	—	2	2	no	—
258	C₄	—	236	4	no	—
259	C₄	#16837520	1	4	yes	complemented
260	C₄	C₂×C₄×D₂₃₆	1	2	yes	—
261	C₄	C₂×#14273724	1	1	yes	characteristic
262	C₄	#16837952	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	472	4	no	—
265	C₂	C₄×#14273724	1	1	yes	characteristic
266	C₂	C₂×#16837520	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₄²×C₈×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	#21434524	C₂
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₄²×C₈	D₅₉
hypocenter	C₅₉	#21705
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16837520	C₂	semidirect
3. #21432809	C₂	semidirect
4. #21424984	C₂	semidirect
5. #21424924	C₂	semidirect
6. #21424268	C₂	semidirect
7. C₄²×C₈×C₅₉	C₂	semidirect
11. #16837786	C₄	semidirect
12. #16837782	C₄	semidirect
13. #16837779	C₄	semidirect
14. #16837775	C₄	semidirect
58. C₄×C₈×C₅₉	D₄	semidirect
125. C₄×C₅₉	#443	semidirect
136. C₄²×C₈	D₅₉	semidirect
160. C₅₉	#21705	semidirect
192. C₄×C₈	D₂₃₆	semidirect
259. C₄	#16837520	semidirect

Maximal subgroups

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

subgroup	index
#21434524	2
C₂×#16837520	2
#21432809	2
#21424984	2
#21424924	2
#21424268	2
C₄²×C₈×C₅₉	2
#21705	59