Subgroups — #28607220

Order 15104.

808 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433014	C₂	1	2	yes	maximal
2	C₂×#16837525	C₂	1	1	yes	maximal, characteristic
3	#21432704	C₂	1	2	yes	maximal, complemented
4	#21424886	C₂	1	2	yes	maximal, complemented
5	#21424337	C₂	1	4	yes	maximal, complemented
6	#21424271	C₂	1	1	yes	maximal, characteristic, complemented
7	#21424147	C₂	1	2	yes	maximal, complemented
8	C₅₉×#1773	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14273749	C₂²	1	1	yes	characteristic
10	#16838168	C₂²	1	4	yes	—
11	C₂×#14273733	C₂²	1	2	yes	—
12	#16837525	—	2	4	no	—
13	#16837525	C₄	1	8	yes	complemented
14	#16837520	—	2	4	no	—
15	#16837450	C₂²	1	4	yes	—
16	#16837449	—	2	4	no	—
17	C₂×#14273663	C₂²	1	2	yes	—
18	C₂×#14273663	C₂²	1	2	yes	—
19	C₂×#14273663	C₂²	1	1	yes	characteristic
20	#108×D₅₉	—	2	4	no	—
21	#16837426	C₂²	1	2	yes	—
22	#16837420	C₂²	1	4	yes	—
23	C₂×#14273655	C₂²	1	2	yes	—
24	#16837172	C₂²	1	4	yes	—
25	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
26	C₅₉×#427	C₂²	1	1	yes	characteristic
27	C₅₉×#405	C₂²	1	2	yes	—
28	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
29	C₅₉×#339	C₂²	1	2	yes	—
30	C₂×#11745685	C₂³	1	2	yes	—
31	#14273749	C₂×C₄	1	4	yes	—
32	#14273749	D₄	1	4	yes	—
33	C₂×Dic₂₃₆	Q₈	1	1	yes	characteristic
34	C₂×#11745660	Q₈	1	2	yes	—
35	C₂×D₄₇₂	Q₈	1	1	yes	characteristic
36	#14273735	—	4	4	no	—
37	#14273733	—	2	2	no	—
38	#14273733	—	2	2	no	—
39	#14273733	—	2	8	no	—
40	#14273733	C₂×C₄	1	4	yes	—
41	#14273733	C₂×C₄	1	4	yes	—
42	C₂×#11745664	C₂³	1	4	yes	—
43	C₂×#11745664	C₂³	1	2	yes	—
44	C₂×#11745659	—	2	2	no	—
45	M₁₆×D₅₉	—	2	8	no	—
46	C₂×C₈×D₅₉	—	2	2	no	—
47	#14273663	—	2	2	no	—
48	#14273663	—	2	2	no	—
49	#14273663	—	2	2	no	—
50	#14273663	—	2	2	no	—
51	#14273663	C₂×C₄	1	4	yes	—
52	#14273662	—	2	8	no	—
53	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
54	#14273655	—	2	8	no	—
55	#14273655	C₂×C₄	1	8	yes	—
56	C₂×M₁₆×C₅₉	C₂³	1	2	yes	—
57	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
59	#117×C₅₉	C₂³	1	2	yes	—
60	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
61	#108×C₅₉	D₄	1	4	yes	—
62	#107×C₅₉	—	2	1	no	—
63	#106×C₅₉	—	2	1	no	—
64	#97×C₅₉	—	2	2	no	—
65	#97×C₅₉	C₂×C₄	1	4	yes	—
66	C₄×C₈×C₅₉	—	2	2	no	—
67	C₂²×C₄×C₅₉	—	2	1	no	—
68	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
69	C₂×C₂²⋊D₅₉	—	2	2	no	—
70	#11745685	—	2	2	no	—
71	#11745685	—	2	2	no	—
72	#11745685	C₂²×C₄	1	4	yes	—
73	#11745685	C₂×D₄	1	4	yes	—
74	C₂×D₂₃₆	C₂×Q₈	1	2	yes	—
75	C₂×C₄×D₅₉	—	2	2	no	—
76	C₂×Dic₁₁₈	C₂×Q₈	1	2	yes	—
77	M₁₆×C₅₉	—	2	2	no	—
78	M₁₆×C₅₉	C₂×D₄	1	4	yes	—
79	C₂×C₈×C₅₉	—	2	2	no	—
80	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
87	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
88	C₄⋊C₄×C₅₉	—	2	2	no	—
89	C₂²⋊C₄×C₅₉	—	2	2	no	—
90	C₄²×C₅₉	—	2	1	no	—
91	C₄²×C₅₉	—	2	2	no	—
92	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
93	#11745664	—	2	2	no	—
94	#11745664	—	2	4	no	—
95	#11745664	—	2	4	no	—
96	#11745664	C₂²×C₄	1	4	yes	—
97	C₂×C₅₉⋊C₈	—	2	2	no	—
98	C₂×C₅₉⋊C₈	—	2	4	no	—
99	Dic₂₃₆	C₄⋊C₄	1	4	yes	—
100	D₄₇₂	C₄⋊C₄	1	4	yes	—
101	#11745660	C₄⋊C₄	1	8	yes	—
102	#11745659	—	2	4	no	—
103	#11745659	—	2	4	no	—
104	#11745659	—	4	8	no	—
105	C₈×D₅₉	—	2	4	no	—
106	C₈×D₅₉	—	2	4	no	—
107	C₈×D₅₉	—	4	8	no	—
108	C₂³×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂²×D₅₉	—	2	2	no	—
110	C₂×C₄×C₅₉	—	2	2	no	—
111	C₂×C₄×C₅₉	—	4	2	no	—
112	C₂×C₄×C₅₉	—	8	1	no	—
113	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂²⋊D₅₉	—	2	4	no	—
120	C₂²⋊D₅₉	—	2	4	no	—
121	C₂×Dic₅₉	—	2	2	no	—
122	D₂₃₆	—	2	2	no	—
123	D₂₃₆	C₂×C₄⋊C₄	1	4	yes	—
124	C₄×D₅₉	—	2	4	no	—
125	C₄×D₅₉	—	2	4	no	—
126	Dic₁₁₈	—	2	2	no	—
127	Dic₁₁₈	C₂×C₄⋊C₄	1	4	yes	—
128	C₈×C₅₉	—	2	4	no	—
129	C₈×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
130	C₈×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
131	C₈×C₅₉	C₄×D₄	1	2	yes	—
132	C₈×C₅₉	C₄×D₄	1	2	yes	—
133	C₅₉⋊C₈	—	2	4	no	—
134	C₅₉⋊C₈	—	4	4	no	—
135	#22695	—	59	1	no	maximal, Sylow
136	C₂²×C₅₉	—	4	1	no	—
137	C₂²×C₅₉	#520	1	1	yes	characteristic
138	C₂²×C₅₉	#547	1	1	yes	characteristic
139	C₂²×C₅₉	D₄×Q₈	1	1	yes	characteristic
140	C₂×D₅₉	—	2	4	no	—
141	C₂×D₅₉	—	8	2	no	—
142	C₄×C₅₉	—	4	2	no	—
143	C₄×C₅₉	—	8	2	no	—
144	C₄×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
146	C₄×C₅₉	D₄²	1	1	yes	characteristic
147	C₄×C₅₉	#554	1	1	yes	characteristic
148	Dic₅₉	—	2	4	no	—
149	#2957	—	59	2	no	—
150	C₂×#444	—	59	1	no	—
151	#2916	—	59	2	no	—
152	#1953	—	59	2	no	—
153	#1861	—	59	4	no	—
154	#1849	—	59	1	no	—
155	#1814	—	59	2	no	—
156	#1773	D₅₉	1	1	yes	characteristic, complemented
157	C₂×C₅₉	—	2	1	no	—
158	C₂×C₅₉	—	2	1	no	—
159	C₂×C₅₉	—	2	1	no	—
160	C₂×C₅₉	D₄×C₄⋊C₄	1	1	yes	characteristic
161	D₅₉	—	4	4	no	—
162	C₂×#135	—	59	1	no	—
163	#568	—	59	4	no	—
164	C₂×#131	—	59	2	no	—
165	#444	—	59	8	no	—
166	#444	—	118	4	no	—
167	#443	—	118	4	no	—
168	#430	—	59	4	no	—
169	#430	—	118	4	no	—
170	C₂×#108	—	59	1	no	—
171	C₂×#108	—	59	2	no	—
172	C₂×#108	—	59	2	no	—
173	C₂×#108	—	118	4	no	—
174	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
175	#427	C₂×D₅₉	1	1	yes	characteristic
176	#423	—	59	2	no	—
177	#421	—	59	4	no	—
178	C₂×#104	—	59	2	no	—
179	#405	C₂×D₅₉	1	2	yes	—
180	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
181	#344	—	59	4	no	—
182	#339	C₂×D₅₉	1	2	yes	—
183	C₅₉	#22695	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	59	2	no	—
185	#135	—	59	4	no	—
186	#135	—	59	4	no	—
187	C₂×Q₁₆	—	59	1	no	—
188	C₂×SD₁₆	—	59	2	no	—
189	C₂×D₈	—	59	1	no	—
190	#131	—	59	4	no	—
191	#131	—	59	4	no	—
192	#131	—	118	8	no	—
193	#131	—	118	2	no	—
194	#131	—	118	2	no	—
195	#131	—	236	4	no	—
196	C₂×M₁₆	—	59	2	no	—
197	C₂×M₁₆	—	59	4	no	—
198	C₂×M₁₆	—	118	2	no	—
199	C₂×M₁₆	—	118	8	no	—
200	C₂×M₁₆	C₂²×D₅₉	1	2	yes	—
201	C₂²×C₈	—	118	2	no	—
202	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
204	#117	C₂²×D₅₉	1	2	yes	—
205	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
206	#108	—	59	4	no	—
207	#108	—	118	8	no	—
208	#108	—	118	2	no	—
209	#108	—	118	2	no	—
210	#108	—	118	2	no	—
211	#108	—	118	2	no	—
212	#108	D₂₃₆	1	4	yes	—
213	#107	—	2	1	no	—
214	#106	—	2	1	no	—
215	C₄⋊C₈	—	236	2	no	—
216	#104	—	59	8	no	—
217	#104	—	118	8	no	—
218	#97	—	2	2	no	—
219	#97	C₄×D₅₉	1	4	yes	—
220	C₄×C₈	—	2	2	no	—
221	#41	—	59	4	no	—
222	#41	—	59	4	no	—
223	#41	—	118	2	no	—
224	#41	—	118	2	no	—
225	C₂×Q₈	—	59	2	no	—
226	C₂×D₄	—	59	2	no	—
227	C₂×D₄	—	118	2	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	118	2	no	—
230	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
231	Q₁₆	—	59	4	no	—
232	SD₁₆	—	59	8	no	—
233	D₈	—	59	4	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	59	4	no	—
236	M₁₆	—	118	2	no	—
237	M₁₆	—	118	4	no	—
238	M₁₆	—	118	4	no	—
239	M₁₆	—	118	4	no	—
240	M₁₆	—	118	4	no	—
241	M₁₆	—	236	8	no	—
242	M₁₆	C₂×D₂₃₆	1	4	yes	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	118	2	no	—
245	C₂×C₈	—	118	4	no	—
246	C₂×C₈	—	118	4	no	—
247	C₂×C₈	—	118	4	no	—
248	C₂×C₈	—	236	8	no	—
249	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
250	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
251	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
252	C₂×C₈	#11745685	1	1	yes	characteristic
253	C₂×C₈	#11745685	1	1	yes	characteristic
254	C₂×C₈	D₄×D₅₉	1	2	yes	—
255	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
256	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	2	no	—
261	C₄²	C₂×C₄×D₅₉	1	2	yes	—
262	C₂³	—	118	2	no	—
263	C₂³	#14273774	1	1	yes	characteristic
264	Q₈	—	59	4	no	—
265	Q₈	—	118	2	no	—
266	D₄	—	59	4	no	—
267	D₄	—	118	4	no	—
268	D₄	—	118	4	no	—
269	D₄	—	118	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	118	2	no	—
274	C₂×C₄	—	118	4	no	—
275	C₂×C₄	—	118	4	no	—
276	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
279	C₂×C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
280	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
281	C₂×C₄	#14273773	1	1	yes	characteristic
282	C₈	—	2	4	no	—
283	C₈	—	118	4	no	—
284	C₈	—	236	4	no	—
285	C₈	C₄⋊C₄×D₅₉	1	2	yes	—
286	C₈	C₄⋊C₄×D₅₉	1	2	yes	—
287	C₈	C₄×D₂₃₆	1	2	yes	—
288	C₈	C₄×D₂₃₆	1	2	yes	—
289	C₂²	—	4	1	no	—
290	C₂²	—	118	4	no	—
291	C₂²	—	472	2	no	—
292	C₂²	#16837882	1	1	yes	characteristic
293	C₂²	#16838057	1	1	yes	characteristic
294	C₂²	Q₈×D₂₃₆	1	1	yes	characteristic
295	C₄	—	4	2	no	—
296	C₄	—	8	2	no	—
297	C₄	—	118	4	no	—
298	C₄	C₂×C₄⋊C₄×D₅₉	1	1	yes	characteristic
299	C₄	C₂×C₄×D₂₃₆	1	1	yes	characteristic
300	C₄	D₄×D₂₃₆	1	1	yes	characteristic
301	C₄	#16838112	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	236	4	no	—
306	C₂	C₄⋊C₄×D₂₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₅₉×#1773	C₂
socle	C₂×C₅₉	D₄×C₄⋊C₄
prime radical	C₂×#14273749	C₂²
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	#1773	D₅₉
hypocenter	C₅₉	#22695
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1773

Complements

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

N	H ≅ G/N	type
3. #21432704	C₂	semidirect
4. #21424886	C₂	semidirect
5. #21424337	C₂	semidirect
6. #21424271	C₂	semidirect
7. #21424147	C₂	semidirect
8. C₅₉×#1773	C₂	semidirect
13. #16837525	C₄	semidirect
156. #1773	D₅₉	semidirect
183. C₅₉	#22695	semidirect

Maximal subgroups

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

subgroup	index
#21433014	2
C₂×#16837525	2
#21432704	2
#21424886	2
#21424337	2
#21424271	2
#21424147	2
C₅₉×#1773	2
#22695	59