Subgroups — #28744284

Order 15104.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434735	C₂	1	2	yes	maximal, complemented
2	#21426371	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426357	C₂	1	1	yes	maximal, characteristic, complemented
4	#21426333	C₂	1	2	yes	maximal, complemented
5	#21426209	C₂	1	2	yes	maximal, complemented
6	#21426187	C₂	1	1	yes	maximal, characteristic, complemented
7	#21426182	C₂	1	1	yes	maximal, characteristic
8	#21425962	C₂	1	2	yes	maximal, complemented
9	C₄×Dic₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₉₄₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	#16838023	—	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	#16837807	C₂²	1	2	yes	—
17	#16837786	C₂²	1	2	yes	—
18	#16837762	—	2	2	no	—
19	#16837648	—	2	2	no	—
20	#16837643	C₂²	1	2	yes	—
21	#16837638	C₂²	1	2	yes	—
22	#16837504	—	2	2	no	—
23	C₄×Dic₂₃₆	C₂²	1	2	yes	—
24	C₄×D₄₇₂	C₂²	1	2	yes	—
25	#16837228	C₂²	1	1	yes	characteristic
26	#16837228	C₂²	1	1	yes	characteristic
27	#16837228	C₂²	1	2	yes	—
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	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
33	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
34	C₅₉×#405	C₂²	1	1	yes	characteristic
35	C₅₉×#346	C₂²	1	1	yes	characteristic
36	C₅₉×#346	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	#14273727	—	2	2	no	—
43	#14273722	—	4	2	no	—
44	#14273715	—	2	2	no	—
45	#14273713	—	4	2	no	—
46	#14273709	—	4	2	no	—
47	#14273706	—	2	2	no	—
48	#14273705	—	4	2	no	—
49	#14273701	—	2	2	no	—
50	C₄×Dic₁₁₈	—	2	2	no	—
51	C₄×D₂₃₆	—	2	2	no	—
52	Dic₄₇₂	—	2	2	no	—
53	#14273672	—	2	4	no	—
54	D₉₄₄	—	2	2	no	—
55	#14273661	—	2	2	no	—
56	#14273661	C₂³	1	2	yes	—
57	#14273661	C₂³	1	2	yes	—
58	#14273652	—	2	2	no	—
59	#14273652	—	2	2	no	—
60	#14273649	—	2	2	no	—
61	#14273649	—	2	2	no	—
62	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
64	#117×C₅₉	D₄	1	1	yes	characteristic
65	M₃₂×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	#97×C₅₉	D₄	1	1	yes	characteristic
72	#97×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	C₂×D₄	1	1	yes	characteristic
76	C₂×C₂²⋊D₅₉	—	4	2	no	—
77	C₂×D₂₃₆	—	2	2	no	—
78	C₂×C₄×D₅₉	—	4	2	no	—
79	C₂×Dic₁₁₈	—	2	2	no	—
80	C₂×Dic₁₁₈	—	4	2	no	—
81	M₁₆×C₅₉	—	2	2	no	—
82	C₂×C₈×C₅₉	—	2	1	no	—
83	C₂×C₈×C₅₉	—	2	1	no	—
84	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
89	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
90	C₂²⋊C₄×C₅₉	D₈	1	1	yes	characteristic
91	C₂²⋊C₄×C₅₉	D₈	1	1	yes	characteristic
92	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
94	C₂²⋊Dic₅₉	—	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	(C₂×D₅₉)⋊C₄	—	4	2	no	—
99	C₄⋊Dic₅₉	—	2	2	no	—
100	C₄⋊Dic₅₉	—	2	2	no	—
101	C₄⋊Dic₅₉	—	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	1	no	—
117	C₂×C₄×C₅₉	—	2	1	no	—
118	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
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	C₂²×D₄	1	1	yes	characteristic
124	C₂²⋊D₅₉	—	16	2	no	—
125	C₂×Dic₅₉	—	4	2	no	—
126	C₂×Dic₅₉	—	4	2	no	—
127	C₂×Dic₅₉	—	4	2	no	—
128	D₂₃₆	—	4	2	no	—
129	D₂₃₆	—	4	2	no	—
130	C₄×D₅₉	—	16	2	no	—
131	Dic₁₁₈	—	4	2	no	—
132	Dic₁₁₈	—	4	2	no	—
133	Dic₁₁₈	—	16	2	no	—
134	C₈×C₅₉	—	2	1	no	—
135	C₈×C₅₉	—	2	1	no	—
136	C₈×C₅₉	#135	1	2	yes	—
137	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
138	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
139	#33739	—	59	1	no	maximal, Sylow
140	C₂²×C₅₉	—	2	1	no	—
141	C₂²×C₅₉	—	2	1	no	—
142	C₂²×C₅₉	—	2	1	no	—
143	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
144	C₂×D₅₉	—	8	2	no	—
145	C₂×D₅₉	—	8	2	no	—
146	C₂×D₅₉	—	8	2	no	—
147	C₄×C₅₉	—	2	1	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	#532	1	1	yes	characteristic
153	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
154	Dic₅₉	—	8	2	no	—
155	Dic₅₉	—	8	2	no	—
156	Dic₅₉	—	8	2	no	—
157	#3175	—	59	2	no	—
158	#2252	—	59	1	no	—
159	#2249	—	59	1	no	—
160	#2243	—	59	2	no	—
161	#2220	—	59	2	no	—
162	#2216	—	59	1	no	—
163	#2215	—	59	1	no	—
164	#2175	—	59	2	no	—
165	C₄×Q₃₂	—	59	1	no	—
166	C₄×D₁₆	—	59	1	no	—
167	#2108	D₅₉	1	1	yes	characteristic, complemented
168	C₂×C₅₉	—	4	1	no	—
169	C₂×C₅₉	#3169	1	1	yes	characteristic
170	C₂×C₅₉	#3415	1	1	yes	characteristic
171	C₂×C₅₉	#3417	1	1	yes	characteristic
172	D₅₉	—	16	2	no	—
173	#542	—	118	2	no	—
174	C₂×Q₃₂	—	59	1	no	—
175	C₂×SD₃₂	—	59	2	no	—
176	C₂×D₁₆	—	59	1	no	—
177	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
178	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
179	#500	—	59	2	no	—
180	#495	—	59	2	no	—
181	#489	—	118	2	no	—
182	#468	—	118	2	no	—
183	#467	—	59	2	no	—
184	#466	—	59	2	no	—
185	#440	—	118	2	no	—
186	C₄×Q₁₆	—	59	2	no	—
187	C₄×D₈	—	59	2	no	—
188	#405	C₂×D₅₉	1	1	yes	characteristic
189	#366	—	59	1	no	—
190	#366	—	59	1	no	—
191	#366	—	59	2	no	—
192	#358	—	59	2	no	—
193	#358	—	59	2	no	—
194	#357	—	59	2	no	—
195	#357	—	59	2	no	—
196	#346	C₂×D₅₉	1	1	yes	characteristic
197	#346	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	C₅₉	#33739	1	1	yes	characteristic, complemented, Sylow
201	C₂×Q₁₆	—	59	2	no	—
202	C₂×SD₁₆	—	118	2	no	—
203	C₂×D₈	—	59	2	no	—
204	C₂×M₁₆	D₂₃₆	1	1	yes	characteristic
205	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
206	C₄⋊Q₈	—	118	2	no	—
207	#126	—	236	2	no	—
208	#124	—	118	2	no	—
209	#124	—	236	2	no	—
210	#123	—	236	2	no	—
211	C₂²⋊Q₈	—	118	2	no	—
212	C₂²⋊Q₈	—	236	2	no	—
213	#121	—	118	2	no	—
214	C₄×Q₈	—	118	2	no	—
215	C₄×D₄	—	118	2	no	—
216	#117	D₂₃₆	1	1	yes	characteristic
217	Q₃₂	—	118	2	no	—
218	SD₃₂	—	118	4	no	—
219	D₁₆	—	118	2	no	—
220	M₃₂	—	2	2	no	—
221	C₂×C₁₆	—	2	2	no	—
222	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
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	#107	—	59	2	no	—
227	#107	—	59	2	no	—
228	#107	—	118	2	no	—
229	#103	—	118	2	no	—
230	#103	—	118	2	no	—
231	#102	—	118	2	no	—
232	#102	—	118	2	no	—
233	#97	D₂₃₆	1	1	yes	characteristic
234	#97	D₂₃₆	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
237	C₂×Q₈	—	118	2	no	—
238	C₂×Q₈	—	236	2	no	—
239	C₂×D₄	—	118	2	no	—
240	C₂×D₄	—	236	2	no	—
241	C₂²×C₄	—	236	2	no	—
242	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
243	Q₁₆	—	118	2	no	—
244	Q₁₆	—	118	2	no	—
245	SD₁₆	—	236	4	no	—
246	D₈	—	118	2	no	—
247	D₈	—	118	2	no	—
248	M₁₆	—	2	2	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
252	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
253	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
254	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
255	C₄⋊C₄	—	118	2	no	—
256	C₄⋊C₄	—	118	2	no	—
257	C₄⋊C₄	—	236	2	no	—
258	C₄⋊C₄	—	236	2	no	—
259	C₄⋊C₄	—	236	2	no	—
260	C₄⋊C₄	D₄₇₂	1	1	yes	characteristic
261	C₄⋊C₄	D₄₇₂	1	1	yes	characteristic
262	C₂²⋊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	D₄₇₂	1	1	yes	characteristic
268	C₂²⋊C₄	D₄₇₂	1	1	yes	characteristic
269	C₄²	—	236	2	no	—
270	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
271	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
272	C₁₆	—	2	1	no	—
273	C₁₆	—	2	1	no	—
274	C₁₆	#11745685	1	2	yes	—
275	C₁₆	#11745685	1	2	yes	—
276	C₂³	—	236	2	no	—
277	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
278	Q₈	—	944	2	no	—
279	Q₈	—	236	2	no	—
280	Q₈	—	236	2	no	—
281	D₄	—	944	2	no	—
282	D₄	—	236	2	no	—
283	D₄	—	236	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	944	2	no	—
287	C₂×C₄	—	236	2	no	—
288	C₂×C₄	—	236	2	no	—
289	C₂×C₄	—	236	2	no	—
290	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
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	C₂²×D₂₃₆	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	#14273749	1	2	yes	—
299	C₈	C₂×#11745685	1	1	yes	characteristic
300	C₈	C₂×#11745685	1	1	yes	characteristic
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	472	2	no	—
305	C₂²	—	472	2	no	—
306	C₂²	—	472	2	no	—
307	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	472	2	no	—
314	C₄	—	472	2	no	—
315	C₄	—	472	2	no	—
316	C₄	#16837952	1	1	yes	characteristic
317	C₄	C₂×#14273749	1	1	yes	characteristic
318	C₂	—	4	1	no	—
319	C₂	—	944	2	no	—
320	C₂	#21434675	1	1	yes	characteristic
321	C₂	#21437237	1	1	yes	characteristic
322	C₂	#21437249	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2108	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	#21426182	C₂
hypercenter	#2108	D₅₉
hypocenter	C₅₉	#33739
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
1. #21434735	C₂	semidirect
2. #21426371	C₂	semidirect
3. #21426357	C₂	semidirect
4. #21426333	C₂	semidirect
5. #21426209	C₂	semidirect
6. #21426187	C₂	semidirect
8. #21425962	C₂	semidirect
9. C₄×Dic₄₇₂	C₂	semidirect
10. C₄×D₉₄₄	C₂	semidirect
11. C₅₉×#2108	C₂	semidirect
167. #2108	D₅₉	semidirect
200. C₅₉	#33739	semidirect

Maximal subgroups

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

subgroup	index
#21434735	2
#21426371	2
#21426357	2
#21426333	2
#21426209	2
#21426187	2
#21426182	2
#21425962	2
C₄×Dic₄₇₂	2
C₄×D₉₄₄	2
C₅₉×#2108	2
#33739	59