Subgroups — #28744242

Order 15104.

526 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434666	C₂	1	2	yes	maximal, complemented
2	#21426375	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426357	C₂	1	1	yes	maximal, characteristic, complemented
4	#21426328	C₂	1	2	yes	maximal, complemented
5	#21426192	C₂	1	2	yes	maximal, complemented
6	#21426187	C₂	1	1	yes	maximal, characteristic, complemented
7	#21426182	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14273672	C₂	1	2	yes	maximal, complemented
10	C₄×D₉₄₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837936	—	2	2	no	—
13	C₂×Dic₄₇₂	C₂²	1	1	yes	characteristic
14	C₂×#14273672	C₂²	1	2	yes	—
15	C₂×D₉₄₄	C₂²	1	1	yes	characteristic
16	#16837804	C₂²	1	2	yes	—
17	#16837786	C₂²	1	2	yes	—
18	#16837757	—	2	2	no	—
19	#16837643	C₂²	1	2	yes	—
20	#16837638	C₂²	1	2	yes	—
21	#16837631	—	2	2	no	—
22	C₄×Dic₂₃₆	C₂²	1	2	yes	—
23	C₄×#11745660	—	2	2	no	—
24	C₄×D₄₇₂	C₂²	1	2	yes	—
25	#16837230	C₂²	1	2	yes	—
26	#16837228	C₂²	1	1	yes	characteristic
27	#16837228	C₂²	1	1	yes	characteristic
28	#16837207	C₂²	1	2	yes	—
29	#16837207	C₂²	1	2	yes	—
30	#16837204	C₂²	1	2	yes	—
31	#16837204	C₂²	1	2	yes	—
32	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
33	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
35	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
36	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
37	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
39	C₂×Dic₂₃₆	C₂³	1	2	yes	—
40	C₂×#11745660	—	2	2	no	—
41	C₂×D₄₇₂	C₂³	1	2	yes	—
42	#14273721	—	4	2	no	—
43	#14273719	—	2	2	no	—
44	#14273715	—	2	2	no	—
45	#14273710	—	4	2	no	—
46	#14273706	—	2	2	no	—
47	#14273701	—	2	2	no	—
48	C₄×Dic₁₁₈	—	2	2	no	—
49	C₄×D₂₃₆	—	2	2	no	—
50	C₄×C₂²⋊D₅₉	—	4	2	no	—
51	#14273684	—	4	2	no	—
52	Dic₄₇₂	—	2	2	no	—
53	#14273672	—	2	4	no	—
54	D₉₄₄	—	2	2	no	—
55	#14273661	C₂³	1	2	yes	—
56	#14273661	C₂³	1	2	yes	—
57	#14273659	—	2	2	no	—
58	#14273652	—	2	2	no	—
59	#14273652	—	2	2	no	—
60	#14273649	—	2	2	no	—
61	#14273649	—	2	2	no	—
62	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
65	C₂×C₁₆×C₅₉	—	2	2	no	—
66	C₂×C₁₆×C₅₉	—	2	2	no	—
67	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
69	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
72	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
73	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₅₉	—	4	2	no	—
79	C₂×D₂₃₆	—	2	2	no	—
80	C₂×C₄×D₅₉	—	4	2	no	—
81	C₂×Dic₁₁₈	—	2	2	no	—
82	C₂×C₈×C₅₉	—	2	2	no	—
83	C₂×C₈×C₅₉	—	2	1	no	—
84	C₂×C₈×C₅₉	—	2	1	no	—
85	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
89	C₄²×C₅₉	D₈	1	1	yes	characteristic
90	C₄²×C₅₉	D₈	1	1	yes	characteristic
91	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₂²⋊Dic₅₉	—	4	2	no	—
94	(C₂×D₅₉)⋊C₄	—	4	2	no	—
95	(C₂×D₅₉)⋊C₄	—	4	2	no	—
96	(C₂×D₅₉)⋊C₄	—	4	2	no	—
97	(C₂×D₅₉)⋊C₄	—	4	2	no	—
98	C₄⋊Dic₅₉	—	2	2	no	—
99	C₄⋊Dic₅₉	—	2	2	no	—
100	Dic₅₉⋊C₄	—	4	2	no	—
101	Dic₅₉⋊C₄	—	4	2	no	—
102	Dic₅₉⋊C₄	—	4	2	no	—
103	Dic₅₉⋊C₄	—	4	2	no	—
104	C₄×Dic₅₉	—	4	2	no	—
105	Dic₂₃₆	—	2	2	no	—
106	Dic₂₃₆	—	2	2	no	—
107	D₄₇₂	—	2	2	no	—
108	D₄₇₂	—	2	2	no	—
109	#11745660	—	4	4	no	—
110	C₁₆×C₅₉	—	2	1	no	—
111	C₁₆×C₅₉	—	2	1	no	—
112	C₁₆×C₅₉	#41	1	2	yes	—
113	C₁₆×C₅₉	#41	1	2	yes	—
114	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₅₉	—	4	2	no	—
116	C₂×C₄×C₅₉	—	2	2	no	—
117	C₂×C₄×C₅₉	—	2	1	no	—
118	C₂×C₄×C₅₉	—	2	1	no	—
119	C₂×C₄×C₅₉	—	2	2	no	—
120	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
126	C₂²⋊D₅₉	—	8	4	no	—
127	C₂×Dic₅₉	—	4	2	no	—
128	C₂×Dic₅₉	—	4	2	no	—
129	C₂×Dic₅₉	—	4	2	no	—
130	D₂₃₆	—	4	2	no	—
131	D₂₃₆	—	4	2	no	—
132	C₄×D₅₉	—	8	4	no	—
133	Dic₁₁₈	—	4	2	no	—
134	Dic₁₁₈	—	4	2	no	—
135	C₈×C₅₉	—	2	1	no	—
136	C₈×C₅₉	—	2	1	no	—
137	C₈×C₅₉	#135	1	2	yes	—
138	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
139	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
140	#33735	—	59	1	no	maximal, Sylow
141	C₂²×C₅₉	—	2	1	no	—
142	C₂²×C₅₉	—	2	1	no	—
143	C₂²×C₅₉	#508	1	2	yes	—
144	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₅₉	—	8	2	no	—
146	C₂×D₅₉	—	8	2	no	—
147	C₂×D₅₉	—	8	2	no	—
148	C₄×C₅₉	—	2	1	no	—
149	C₄×C₅₉	—	2	1	no	—
150	C₄×C₅₉	—	2	1	no	—
151	C₄×C₅₉	—	2	1	no	—
152	C₄×C₅₉	#508	1	2	yes	—
153	C₄×C₅₉	#532	1	1	yes	characteristic
154	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
155	Dic₅₉	—	8	2	no	—
156	Dic₅₉	—	8	2	no	—
157	Dic₅₉	—	8	2	no	—
158	#3168	—	59	2	no	—
159	#2253	—	59	1	no	—
160	#2249	—	59	1	no	—
161	#2242	—	59	2	no	—
162	#2217	—	59	2	no	—
163	#2216	—	59	1	no	—
164	#2215	—	59	1	no	—
165	C₄×Q₃₂	—	59	1	no	—
166	C₄×SD₃₂	—	59	2	no	—
167	C₄×D₁₆	—	59	1	no	—
168	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic, complemented
169	C₂×C₅₉	—	2	2	no	—
170	C₂×C₅₉	#3169	1	1	yes	characteristic
171	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
172	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
173	D₅₉	—	16	2	no	—
174	#529	—	118	2	no	—
175	C₂×Q₃₂	—	59	1	no	—
176	C₂×SD₃₂	—	59	2	no	—
177	C₂×D₁₆	—	59	1	no	—
178	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
179	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
180	#499	—	59	2	no	—
181	#495	—	59	2	no	—
182	#488	—	118	2	no	—
183	#467	—	59	2	no	—
184	#466	—	59	2	no	—
185	#465	—	118	2	no	—
186	C₄×Q₁₆	—	59	2	no	—
187	C₄×SD₁₆	—	118	2	no	—
188	C₄×D₈	—	59	2	no	—
189	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
190	#367	—	59	2	no	—
191	#366	—	59	1	no	—
192	#366	—	59	1	no	—
193	#358	—	59	2	no	—
194	#358	—	59	2	no	—
195	#357	—	59	2	no	—
196	#357	—	59	2	no	—
197	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
198	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
199	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
200	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
201	C₅₉	#33735	1	1	yes	characteristic, complemented, Sylow
202	C₂×Q₁₆	—	59	2	no	—
203	C₂×SD₁₆	—	118	2	no	—
204	C₂×D₈	—	59	2	no	—
205	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
207	#126	—	236	2	no	—
208	#125	—	118	2	no	—
209	#124	—	118	2	no	—
210	#123	—	236	2	no	—
211	C₂²⋊Q₈	—	118	2	no	—
212	#121	—	118	2	no	—
213	C₄×Q₈	—	118	2	no	—
214	C₄×D₄	—	118	2	no	—
215	C₄×D₄	—	236	2	no	—
216	#117	—	236	2	no	—
217	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
218	Q₃₂	—	118	2	no	—
219	SD₃₂	—	118	4	no	—
220	D₁₆	—	118	2	no	—
221	C₂×C₁₆	—	2	2	no	—
222	C₂×C₁₆	—	2	2	no	—
223	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
224	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
225	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
226	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
227	#107	—	59	2	no	—
228	#107	—	59	2	no	—
229	#106	—	118	2	no	—
230	#103	—	118	2	no	—
231	#103	—	118	2	no	—
232	#102	—	118	2	no	—
233	#102	—	118	2	no	—
234	C₄×C₈	D₂₃₆	1	1	yes	characteristic
235	C₄×C₈	D₂₃₆	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
238	C₂×Q₈	—	118	2	no	—
239	C₂×D₄	—	118	2	no	—
240	C₂×D₄	—	236	2	no	—
241	C₂²×C₄	—	236	2	no	—
242	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
243	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
245	Q₁₆	—	118	2	no	—
246	Q₁₆	—	118	2	no	—
247	SD₁₆	—	236	4	no	—
248	D₈	—	118	2	no	—
249	D₈	—	118	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
254	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
255	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
256	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
257	C₄⋊C₄	—	118	2	no	—
258	C₄⋊C₄	—	118	2	no	—
259	C₄⋊C₄	—	236	2	no	—
260	C₄⋊C₄	—	236	2	no	—
261	C₄⋊C₄	—	236	2	no	—
262	C₄⋊C₄	—	236	2	no	—
263	C₂²⋊C₄	—	236	2	no	—
264	C₂²⋊C₄	—	236	2	no	—
265	C₂²⋊C₄	—	236	2	no	—
266	C₂²⋊C₄	—	236	2	no	—
267	C₂²⋊C₄	—	236	2	no	—
268	C₄²	—	236	2	no	—
269	C₄²	D₄₇₂	1	1	yes	characteristic
270	C₄²	D₄₇₂	1	1	yes	characteristic
271	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
272	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
273	C₁₆	—	2	1	no	—
274	C₁₆	—	2	1	no	—
275	C₁₆	#11745685	1	2	yes	—
276	C₁₆	#11745685	1	2	yes	—
277	C₂³	—	236	2	no	—
278	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
279	Q₈	—	236	2	no	—
280	Q₈	—	236	2	no	—
281	D₄	—	472	4	no	—
282	D₄	—	236	2	no	—
283	D₄	—	236	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	472	4	no	—
289	C₂×C₄	—	236	2	no	—
290	C₂×C₄	—	236	2	no	—
291	C₂×C₄	—	236	2	no	—
292	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
294	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	#14273749	1	2	yes	—
301	C₈	C₂×#11745685	1	1	yes	characteristic
302	C₈	C₂×#11745685	1	1	yes	characteristic
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	472	2	no	—
306	C₂²	—	472	2	no	—
307	C₂²	—	472	2	no	—
308	C₂²	#16837834	1	2	yes	—
309	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	472	2	no	—
315	C₄	—	472	2	no	—
316	C₄	—	472	2	no	—
317	C₄	#16837834	1	2	yes	—
318	C₄	#16837952	1	1	yes	characteristic
319	C₄	C₂×#14273749	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	944	2	no	—
322	C₂	#21434675	1	1	yes	characteristic
323	C₂	C₂×#16837834	1	1	yes	characteristic
324	C₂	C₂×#16837834	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	#21426182	C₂
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	#33735
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. #21434666	C₂	semidirect
2. #21426375	C₂	semidirect
3. #21426357	C₂	semidirect
4. #21426328	C₂	semidirect
5. #21426192	C₂	semidirect
6. #21426187	C₂	semidirect
8. C₄×Dic₄₇₂	C₂	semidirect
9. C₄×#14273672	C₂	semidirect
10. C₄×D₉₄₄	C₂	semidirect
11. C₂×C₄×C₁₆×C₅₉	C₂	semidirect
168. C₂×C₄×C₁₆	D₅₉	semidirect
201. C₅₉	#33735	semidirect

Maximal subgroups

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

subgroup	index
#21434666	2
#21426375	2
#21426357	2
#21426328	2
#21426192	2
#21426187	2
#21426182	2
C₄×Dic₄₇₂	2
C₄×#14273672	2
C₄×D₉₄₄	2
C₂×C₄×C₁₆×C₅₉	2
#33735	59