Subgroups — #28503927

Order 15104.

438 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837520	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837208	C₂	1	1	yes	maximal, characteristic
3	#21425801	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#21422191	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422189	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1772	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273749	C₂²	1	1	yes	characteristic
9	C₂×#14273733	—	2	1	no	—
10	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	#16837520	—	2	2	no	—
13	#16837520	C₄	1	4	yes	complemented
14	C₂×#14273663	C₂²	1	1	yes	characteristic
15	C₂×#14273655	—	2	1	no	—
16	#16837208	C₄	1	4	yes	complemented
17	#16837205	C₄	1	2	yes	complemented
18	#16837202	C₄	1	2	yes	complemented
19	C₄×C₅₉⋊C₁₆	—	2	2	no	—
20	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
21	C₅₉×#427	C₂²	1	1	yes	characteristic
22	C₅₉×#405	—	2	1	no	—
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₅₉×#339	—	2	1	no	—
25	C₂×#11745685	—	2	1	no	—
26	#14273749	—	2	1	no	—
27	#14273749	—	2	1	no	—
28	#14273749	C₂×C₄	1	2	yes	—
29	#14273749	C₂×C₄	1	2	yes	—
30	C₂×Dic₂₃₆	C₂×C₄	1	1	yes	characteristic
31	C₂×#11745660	—	2	1	no	—
32	C₂×D₄₇₂	C₂×C₄	1	1	yes	characteristic
33	#14273733	—	2	4	no	—
34	#14273733	—	4	2	no	—
35	C₂×#11745664	—	2	1	no	—
36	C₂×#11745659	—	4	1	no	—
37	C₂×C₈×D₅₉	—	4	1	no	—
38	C₂×C₅₉⋊C₁₆	—	2	1	no	—
39	C₂×C₅₉⋊C₁₆	—	2	1	no	—
40	C₂×C₅₉⋊C₁₆	—	2	1	no	—
41	C₂×C₅₉⋊C₁₆	—	2	1	no	—
42	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
43	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
44	#14273663	—	2	1	no	—
45	#14273663	C₂×C₄	1	2	yes	—
46	#14273655	—	2	4	no	—
47	C₂×M₁₆×C₅₉	—	2	1	no	—
48	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
50	#117×C₅₉	—	2	1	no	—
51	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
52	#108×C₅₉	—	2	1	no	—
53	#108×C₅₉	C₂×C₄	1	2	yes	—
54	#107×C₅₉	C₂×C₄	1	2	yes	—
55	#106×C₅₉	—	2	1	no	—
56	#97×C₅₉	—	4	1	no	—
57	C₄×C₈×C₅₉	—	4	1	no	—
58	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₅₉	D₄	1	2	yes	—
60	C₂²×C₄×C₅₉	—	2	1	no	—
61	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₉	—	4	1	no	—
63	#11745685	—	2	2	no	—
64	#11745685	—	2	2	no	—
65	#11745685	—	4	1	no	—
66	#11745685	—	4	1	no	—
67	C₂×D₂₃₆	—	2	1	no	—
68	C₂×C₄×D₅₉	—	4	1	no	—
69	C₂×Dic₁₁₈	—	2	1	no	—
70	M₁₆×C₅₉	—	2	2	no	—
71	M₁₆×C₅₉	—	4	1	no	—
72	C₂×C₈×C₅₉	—	4	1	no	—
73	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
75	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
76	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
77	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
82	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
83	C₄⋊C₄×C₅₉	—	2	2	no	—
84	C₂²⋊C₄×C₅₉	—	4	1	no	—
85	C₄²×C₅₉	—	2	1	no	—
86	C₄²×C₅₉	—	4	1	no	—
87	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
88	#11745664	—	2	2	no	—
89	#11745664	—	4	1	no	—
90	C₂×C₅₉⋊C₈	—	4	1	no	—
91	Dic₂₃₆	—	2	1	no	—
92	Dic₂₃₆	C₄²	1	2	yes	—
93	D₄₇₂	—	2	1	no	—
94	D₄₇₂	C₄²	1	2	yes	—
95	#11745660	—	2	2	no	—
96	#11745660	—	2	2	no	—
97	#11745659	—	4	4	no	—
98	C₈×D₅₉	—	4	4	no	—
99	C₅₉⋊C₁₆	—	2	2	no	—
100	C₅₉⋊C₁₆	—	2	2	no	—
101	C₂³×C₅₉	#135	1	1	yes	characteristic
102	C₂²×D₅₉	—	4	1	no	—
103	C₂×C₄×C₅₉	—	2	4	no	—
104	C₂×C₄×C₅₉	—	2	2	no	—
105	C₂×C₄×C₅₉	—	4	2	no	—
106	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
112	C₂²⋊D₅₉	—	4	2	no	—
113	C₂²⋊D₅₉	—	4	2	no	—
114	C₂×Dic₅₉	—	4	1	no	—
115	D₂₃₆	—	2	2	no	—
116	D₂₃₆	—	4	1	no	—
117	C₄×D₅₉	—	4	2	no	—
118	C₄×D₅₉	—	4	2	no	—
119	Dic₁₁₈	—	2	2	no	—
120	Dic₁₁₈	—	4	1	no	—
121	C₈×C₅₉	—	4	2	no	—
122	C₈×C₅₉	#102	1	2	yes	—
123	C₈×C₅₉	#102	1	2	yes	—
124	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
125	C₈×C₅₉	#117	1	1	yes	characteristic
126	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
127	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
128	C₅₉⋊C₈	—	4	2	no	—
129	#12641	—	59	1	no	maximal, Sylow
130	C₂²×C₅₉	—	2	2	no	—
131	C₂²×C₅₉	#416	1	1	yes	characteristic
132	C₂²×C₅₉	C₄×D₈	1	1	yes	characteristic
133	C₂²×C₅₉	C₄×SD₁₆	1	1	yes	characteristic
134	C₂×D₅₉	—	4	2	no	—
135	C₂×D₅₉	—	8	2	no	—
136	C₄×C₅₉	—	2	4	no	—
137	C₄×C₅₉	—	8	2	no	—
138	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₅₉	C₂×#102	1	1	yes	characteristic
140	C₄×C₅₉	C₄×D₈	1	1	yes	characteristic
141	C₄×C₅₉	C₄×SD₁₆	1	1	yes	characteristic
142	Dic₅₉	—	4	2	no	—
143	C₂×#443	—	59	1	no	—
144	C₂×#359	—	59	1	no	—
145	#2138	—	59	1	no	—
146	C₂×C₄×C₁₆	—	59	1	no	—
147	#1772	D₅₉	1	1	yes	characteristic, complemented
148	#1382	—	59	1	no	—
149	#1381	—	59	1	no	—
150	C₂×C₅₉	—	2	1	no	—
151	C₂×C₅₉	—	2	1	no	—
152	C₂×C₅₉	C₄×#102	1	1	yes	characteristic
153	C₂×C₅₉	#2178	1	2	yes	—
154	D₅₉	—	8	2	no	—
155	C₂×#135	—	59	1	no	—
156	C₂×#131	—	118	1	no	—
157	C₂²×C₁₆	—	59	1	no	—
158	C₂²×C₁₆	—	59	1	no	—
159	#443	—	59	4	no	—
160	#443	—	118	2	no	—
161	C₂×#108	—	59	1	no	—
162	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
163	#427	C₂×D₅₉	1	1	yes	characteristic
164	C₂×#104	—	118	1	no	—
165	#405	—	2	1	no	—
166	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
167	#359	—	59	4	no	—
168	#358	—	59	2	no	—
169	#357	—	59	2	no	—
170	C₄×C₁₆	—	118	2	no	—
171	#339	—	2	1	no	—
172	C₅₉	#12641	1	1	yes	characteristic, complemented, Sylow
173	C₂×#41	—	118	1	no	—
174	#135	—	59	2	no	—
175	#135	—	59	2	no	—
176	#135	—	118	1	no	—
177	#135	—	118	1	no	—
178	C₂×Q₁₆	—	59	1	no	—
179	C₂×SD₁₆	—	118	1	no	—
180	C₂×D₈	—	59	1	no	—
181	#131	—	118	4	no	—
182	#131	—	236	2	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	118	1	no	—
185	C₂×M₁₆	—	236	1	no	—
186	C₂²×C₈	—	236	1	no	—
187	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
189	#117	—	2	1	no	—
190	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
191	C₂×C₁₆	—	59	2	no	—
192	C₂×C₁₆	—	59	2	no	—
193	C₂×C₁₆	—	118	1	no	—
194	C₂×C₁₆	—	118	1	no	—
195	C₂×C₁₆	—	118	1	no	—
196	C₂×C₁₆	—	118	1	no	—
197	#108	—	2	1	no	—
198	#108	—	59	2	no	—
199	#108	—	118	1	no	—
200	#108	C₄×D₅₉	1	2	yes	—
201	#107	C₄×D₅₉	1	2	yes	—
202	#106	—	2	1	no	—
203	#104	—	118	4	no	—
204	#97	—	4	1	no	—
205	C₄×C₈	—	4	1	no	—
206	C₄×C₈	C₄×D₅₉	1	2	yes	—
207	C₄×C₈	D₂₃₆	1	2	yes	—
208	#41	—	118	2	no	—
209	#41	—	118	2	no	—
210	#41	—	236	1	no	—
211	#41	—	236	1	no	—
212	C₂×Q₈	—	118	1	no	—
213	C₂×D₄	—	118	1	no	—
214	C₂×D₄	—	236	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	236	1	no	—
217	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
218	Q₁₆	—	59	2	no	—
219	Q₁₆	—	118	1	no	—
220	SD₁₆	—	118	2	no	—
221	SD₁₆	—	118	2	no	—
222	D₈	—	59	2	no	—
223	D₈	—	118	1	no	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	4	1	no	—
226	M₁₆	—	118	2	no	—
227	M₁₆	—	236	1	no	—
228	M₁₆	—	236	4	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	236	1	no	—
231	C₂×C₈	—	236	4	no	—
232	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
233	C₂×C₈	#11745669	1	1	yes	characteristic
234	C₂×C₈	#11745669	1	1	yes	characteristic
235	C₂×C₈	#11745671	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
239	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
240	C₂×C₈	#11745685	1	1	yes	characteristic
241	C₂×C₈	#11745685	1	1	yes	characteristic
242	C₄⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	4	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
247	C₁₆	—	118	2	no	—
248	C₁₆	—	118	2	no	—
249	C₂³	—	236	1	no	—
250	C₂³	#14273747	1	1	yes	characteristic
251	Q₈	—	118	2	no	—
252	Q₈	—	236	1	no	—
253	D₄	—	118	2	no	—
254	D₄	—	236	2	no	—
255	D₄	—	236	2	no	—
256	D₄	—	236	1	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	236	1	no	—
261	C₂×C₄	—	236	2	no	—
262	C₂×C₄	—	236	2	no	—
263	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
264	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
265	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
266	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
267	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
268	C₂×C₄	#14273747	1	1	yes	characteristic
269	C₈	—	4	2	no	—
270	C₈	—	236	2	no	—
271	C₈	#14273647	1	2	yes	—
272	C₈	#14273647	1	2	yes	—
273	C₈	C₄²×D₅₉	1	1	yes	characteristic
274	C₈	#14273684	1	1	yes	characteristic
275	C₈	C₄×D₂₃₆	1	1	yes	characteristic
276	C₈	C₄×D₂₃₆	1	1	yes	characteristic
277	C₂²	—	2	2	no	—
278	C₂²	—	472	2	no	—
279	C₂²	—	236	2	no	—
280	C₂²	#16837396	1	1	yes	characteristic
281	C₂²	C₄×#11745669	1	1	yes	characteristic
282	C₂²	C₄×#11745671	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	8	2	no	—
285	C₄	—	236	2	no	—
286	C₄	C₄×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
287	C₄	C₂×#14273647	1	1	yes	characteristic
288	C₄	C₄×#11745669	1	1	yes	characteristic
289	C₄	C₄×#11745671	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	472	2	no	—
293	C₂	C₄×#14273647	1	1	yes	characteristic
294	C₂	#21425975	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#11745669
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1772	C₂
socle	C₂²×C₅₉	C₄×D₈
prime radical	C₂×#14273749	C₂²
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1772	D₅₉
hypocenter	C₅₉	#12641
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
4. C₂×C₄×C₅₉⋊C₁₆	C₂	semidirect
5. #21422191	C₂	semidirect
6. #21422189	C₂	semidirect
7. C₅₉×#1772	C₂	semidirect
13. #16837520	C₄	semidirect
16. #16837208	C₄	semidirect
17. #16837205	C₄	semidirect
18. #16837202	C₄	semidirect
147. #1772	D₅₉	semidirect
172. C₅₉	#12641	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837520	2
C₂×#16837208	2
#21425801	2
C₂×C₄×C₅₉⋊C₁₆	2
#21422191	2
#21422189	2
C₅₉×#1772	2
#12641	59