Subgroups — #28746515

Order 15104.

486 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21437109	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426333	C₂	1	2	yes	maximal, complemented
4	#21426320	C₂	1	1	yes	maximal, characteristic
5	#21426249	C₂	1	1	yes	maximal, characteristic, complemented
6	#21426234	C₂	1	2	yes	maximal, complemented
7	#21426159	C₂	1	2	yes	maximal, complemented
8	#21426137	C₂	1	1	yes	maximal, characteristic
9	#21425962	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	#16838137	—	2	1	no	—
13	Q₈×Dic₁₁₈	—	2	1	no	—
14	C₂×Dic₄₇₂	C₂²	1	1	yes	characteristic
15	C₂×#14273672	C₂²	1	2	yes	—
16	#16837807	C₂²	1	2	yes	—
17	#16837807	C₂²	1	1	yes	characteristic
18	#16837786	C₂²	1	2	yes	—
19	#16837779	C₂²	1	1	yes	characteristic
20	#16837698	—	2	1	no	—
21	#16837698	—	2	2	no	—
22	#16837618	—	2	2	no	—
23	#16837603	—	2	1	no	—
24	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
25	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₃₆	C₂²	1	2	yes	—
27	C₄×D₄₇₂	C₂²	1	2	yes	—
28	#16837228	C₂²	1	1	yes	characteristic
29	#16837228	C₂²	1	2	yes	—
30	#16837207	C₂²	1	1	yes	characteristic
31	#16837207	C₂²	1	1	yes	characteristic
32	#16837207	C₂²	1	2	yes	—
33	#16837204	C₂²	1	2	yes	—
34	#16837204	C₂²	1	2	yes	—
35	#16837204	C₂²	1	2	yes	—
36	C₈×Q₈×C₅₉	C₂²	1	1	yes	characteristic
37	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
38	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₅₉×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₃₆	C₂³	1	1	yes	characteristic
42	C₂×Dic₂₃₆	C₂³	1	1	yes	characteristic
43	C₂×#11745660	—	2	2	no	—
44	C₂×D₄₇₂	—	2	1	no	—
45	C₂×D₄₇₂	C₂³	1	2	yes	—
46	#14273727	—	2	2	no	—
47	#14273727	—	2	1	no	—
48	#14273726	—	4	1	no	—
49	#14273726	—	4	2	no	—
50	#14273724	—	2	1	no	—
51	#14273716	—	4	1	no	—
52	#14273715	—	2	2	no	—
53	#14273712	—	4	2	no	—
54	Q₈×Dic₅₉	—	4	1	no	—
55	C₄×Dic₁₁₈	—	2	2	no	—
56	C₄×Dic₁₁₈	—	2	1	no	—
57	C₄×Dic₁₁₈	—	2	1	no	—
58	C₄×D₂₃₆	—	2	2	no	—
59	#14273689	—	4	1	no	—
60	Dic₄₇₂	—	2	2	no	—
61	#14273672	—	2	4	no	—
62	#14273661	—	2	2	no	—
63	#14273661	—	2	1	no	—
64	#14273661	C₂³	1	2	yes	—
65	#14273661	C₂³	1	1	yes	characteristic
66	#14273661	C₂³	1	1	yes	characteristic
67	#14273652	—	2	2	no	—
68	#14273652	—	2	1	no	—
69	#14273652	—	2	1	no	—
70	#14273649	—	2	2	no	—
71	C₄×Q₈×C₅₉	D₄	1	1	yes	characteristic
72	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
75	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
76	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
77	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₅₉	C₂³	1	2	yes	—
79	C₂×Q₈×C₅₉	D₈	1	1	yes	characteristic
80	C₂×D₂₃₆	—	2	2	no	—
81	C₂×D₂₃₆	—	4	1	no	—
82	C₂×D₂₃₆	—	4	1	no	—
83	C₂×C₄×D₅₉	—	4	2	no	—
84	C₂×Dic₁₁₈	—	2	1	no	—
85	C₂×Dic₁₁₈	—	2	1	no	—
86	C₂×Dic₁₁₈	—	4	1	no	—
87	C₂×Dic₁₁₈	—	4	2	no	—
88	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
92	C₄⋊C₄×C₅₉	D₈	1	2	yes	—
93	C₄²×C₅₉	C₂×D₄	1	2	yes	—
94	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
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	1	no	—
99	C₄⋊Dic₅₉	—	2	2	no	—
100	C₄⋊Dic₅₉	—	2	1	no	—
101	C₄⋊Dic₅₉	—	4	2	no	—
102	C₄⋊Dic₅₉	—	4	1	no	—
103	Dic₅₉⋊C₄	—	4	1	no	—
104	Dic₅₉⋊C₄	—	4	1	no	—
105	Dic₅₉⋊C₄	—	4	2	no	—
106	C₄×Dic₅₉	—	4	1	no	—
107	C₄×Dic₅₉	—	4	2	no	—
108	C₄×Dic₅₉	—	4	1	no	—
109	Dic₂₃₆	—	2	1	no	—
110	Dic₂₃₆	—	2	1	no	—
111	Dic₂₃₆	—	2	1	no	—
112	Dic₂₃₆	#41	1	2	yes	—
113	D₄₇₂	—	2	2	no	—
114	D₄₇₂	—	2	2	no	—
115	D₄₇₂	—	8	1	no	—
116	#11745660	—	8	2	no	—
117	C₁₆×C₅₉	—	2	2	no	—
118	C₁₆×C₅₉	—	2	1	no	—
119	C₁₆×C₅₉	—	2	1	no	—
120	C₂²×D₅₉	—	4	2	no	—
121	Q₈×C₅₉	—	2	2	no	—
122	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₅₉	—	4	1	no	—
128	C₂×Dic₅₉	—	4	2	no	—
129	C₂×Dic₅₉	—	4	1	no	—
130	C₂×Dic₅₉	—	4	1	no	—
131	C₂×Dic₅₉	—	4	1	no	—
132	D₂₃₆	—	4	2	no	—
133	D₂₃₆	—	4	2	no	—
134	D₂₃₆	—	8	2	no	—
135	D₂₃₆	—	16	1	no	—
136	C₄×D₅₉	—	16	2	no	—
137	Dic₁₁₈	—	2	2	no	—
138	Dic₁₁₈	—	4	1	no	—
139	Dic₁₁₈	—	4	1	no	—
140	Dic₁₁₈	—	4	1	no	—
141	Dic₁₁₈	—	16	2	no	—
142	Dic₁₁₈	—	16	1	no	—
143	C₈×C₅₉	—	2	2	no	—
144	C₈×C₅₉	—	2	1	no	—
145	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
146	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
147	#33942	—	59	1	no	maximal, Sylow
148	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₅₉	—	8	2	no	—
150	C₂×D₅₉	—	8	2	no	—
151	C₂×D₅₉	—	8	2	no	—
152	C₄×C₅₉	—	2	2	no	—
153	C₄×C₅₉	—	2	2	no	—
154	C₄×C₅₉	—	2	1	no	—
155	C₄×C₅₉	#510	1	2	yes	—
156	C₄×C₅₉	#548	1	1	yes	characteristic
157	C₄×C₅₉	#577	1	1	yes	characteristic
158	Dic₅₉	—	4	2	no	—
159	Dic₅₉	—	8	1	no	—
160	Dic₅₉	—	8	1	no	—
161	Dic₅₉	—	8	2	no	—
162	Dic₅₉	—	8	1	no	—
163	#3396	—	59	1	no	—
164	Q₈×Q₁₆	—	59	1	no	—
165	#2243	—	59	2	no	—
166	#2240	—	59	1	no	—
167	#2227	—	59	1	no	—
168	#2224	—	59	2	no	—
169	#2211	—	59	2	no	—
170	#2207	—	59	1	no	—
171	#2183	D₅₉	1	1	yes	characteristic, complemented
172	#2175	—	59	2	no	—
173	C₄×Q₃₂	—	59	1	no	—
174	C₂×C₅₉	#3358	1	1	yes	characteristic
175	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
176	C₂×C₅₉	#3415	1	1	yes	characteristic
177	D₅₉	—	16	2	no	—
178	#559	—	118	1	no	—
179	Q₈²	—	118	1	no	—
180	C₂×Q₃₂	—	59	1	no	—
181	C₂×SD₃₂	—	59	2	no	—
182	#500	—	59	2	no	—
183	#500	—	59	1	no	—
184	#495	—	59	2	no	—
185	#493	—	59	1	no	—
186	#477	—	118	1	no	—
187	#477	—	118	2	no	—
188	#463	—	118	2	no	—
189	#460	—	118	1	no	—
190	C₈×Q₈	C₂×D₅₉	1	1	yes	characteristic
191	C₄×Q₁₆	—	59	1	no	—
192	C₄×Q₁₆	—	59	2	no	—
193	C₄×Q₁₆	—	59	1	no	—
194	C₄×D₈	—	59	2	no	—
195	#366	—	59	1	no	—
196	#366	—	59	2	no	—
197	C₄⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
198	C₄⋊C₁₆	C₂×D₅₉	1	2	yes	—
199	#358	—	59	1	no	—
200	#358	—	59	1	no	—
201	#358	—	59	2	no	—
202	#357	—	59	2	no	—
203	#357	—	59	2	no	—
204	#357	—	59	2	no	—
205	#346	C₂×D₅₉	1	2	yes	—
206	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
207	C₅₉	#33942	1	1	yes	characteristic, complemented, Sylow
208	C₂×Q₁₆	—	59	1	no	—
209	C₂×Q₁₆	—	59	1	no	—
210	C₂×SD₁₆	—	118	2	no	—
211	C₂×D₈	—	59	2	no	—
212	C₂×D₈	—	118	1	no	—
213	C₄⋊Q₈	—	118	2	no	—
214	C₄⋊Q₈	—	118	1	no	—
215	C₄⋊Q₈	—	236	1	no	—
216	C₄⋊Q₈	—	236	2	no	—
217	#127	—	118	1	no	—
218	#124	—	118	2	no	—
219	#124	—	236	1	no	—
220	#123	—	236	2	no	—
221	C₄×Q₈	—	118	2	no	—
222	C₄×Q₈	—	118	1	no	—
223	C₄×Q₈	—	118	1	no	—
224	C₄×Q₈	—	236	1	no	—
225	C₄×Q₈	D₂₃₆	1	1	yes	characteristic
226	C₄×D₄	—	118	2	no	—
227	C₄×D₄	—	236	1	no	—
228	Q₃₂	—	118	2	no	—
229	SD₃₂	—	118	4	no	—
230	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
231	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
232	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
233	#107	—	59	2	no	—
234	#107	—	59	1	no	—
235	#107	—	59	1	no	—
236	#107	—	118	2	no	—
237	#107	—	118	1	no	—
238	C₄⋊C₈	D₂₃₆	1	1	yes	characteristic
239	C₄⋊C₈	D₂₃₆	1	2	yes	—
240	#103	—	118	1	no	—
241	#103	—	118	2	no	—
242	#103	—	118	1	no	—
243	#102	—	118	2	no	—
244	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₅₉	1	2	yes	—
246	C₂×Q₈	—	118	1	no	—
247	C₂×Q₈	—	118	1	no	—
248	C₂×Q₈	—	236	2	no	—
249	C₂×Q₈	—	236	1	no	—
250	C₂×Q₈	D₄₇₂	1	1	yes	characteristic
251	C₂×D₄	—	118	2	no	—
252	C₂×D₄	—	236	1	no	—
253	C₂×D₄	—	236	1	no	—
254	C₂²×C₄	—	236	2	no	—
255	Q₁₆	—	59	2	no	—
256	Q₁₆	—	118	1	no	—
257	Q₁₆	—	118	1	no	—
258	Q₁₆	—	118	1	no	—
259	SD₁₆	—	472	2	no	—
260	D₈	—	118	2	no	—
261	D₈	—	118	2	no	—
262	D₈	—	472	1	no	—
263	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
264	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
265	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
266	C₄⋊C₄	—	118	1	no	—
267	C₄⋊C₄	—	118	2	no	—
268	C₄⋊C₄	—	118	1	no	—
269	C₄⋊C₄	—	236	1	no	—
270	C₄⋊C₄	—	236	2	no	—
271	C₄⋊C₄	—	236	1	no	—
272	C₄⋊C₄	—	236	2	no	—
273	C₄⋊C₄	—	236	1	no	—
274	C₄⋊C₄	D₄₇₂	1	1	yes	characteristic
275	C₄⋊C₄	D₄₇₂	1	2	yes	—
276	C₂²⋊C₄	—	236	2	no	—
277	C₂²⋊C₄	—	236	2	no	—
278	C₂²⋊C₄	—	236	2	no	—
279	C₄²	—	236	1	no	—
280	C₄²	—	236	2	no	—
281	C₄²	—	236	1	no	—
282	C₄²	C₂×D₂₃₆	1	2	yes	—
283	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
284	C₁₆	—	2	2	no	—
285	C₁₆	—	2	1	no	—
286	C₁₆	—	2	1	no	—
287	C₂³	—	236	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	118	2	no	—
290	Q₈	—	944	2	no	—
291	Q₈	—	944	1	no	—
292	Q₈	—	236	1	no	—
293	Q₈	—	236	1	no	—
294	Q₈	—	236	1	no	—
295	D₄	—	944	1	no	—
296	D₄	—	472	2	no	—
297	D₄	—	236	2	no	—
298	D₄	—	236	2	no	—
299	C₂×C₄	—	944	2	no	—
300	C₂×C₄	—	236	1	no	—
301	C₂×C₄	—	236	1	no	—
302	C₂×C₄	—	236	2	no	—
303	C₂×C₄	—	236	1	no	—
304	C₂×C₄	—	236	1	no	—
305	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
306	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
307	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	2	no	—
312	C₈	#41×D₅₉	1	1	yes	characteristic
313	C₈	#14273775	1	1	yes	characteristic
314	C₂²	—	472	2	no	—
315	C₂²	—	472	2	no	—
316	C₂²	—	472	2	no	—
317	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	1	no	—
321	C₄	—	472	1	no	—
322	C₄	—	472	1	no	—
323	C₄	—	472	2	no	—
324	C₄	—	472	1	no	—
325	C₄	—	236	2	no	—
326	C₄	#16837845	1	2	yes	—
327	C₄	#16838064	1	1	yes	characteristic
328	C₄	#16838214	1	1	yes	characteristic
329	C₂	—	944	2	no	—
330	C₂	#21436732	1	1	yes	characteristic
331	C₂	C₂×#16837845	1	1	yes	characteristic
332	C₂	#21437237	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2183	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	#16837779	C₂²
hypercenter	#2183	D₅₉
hypocenter	C₅₉	#33942
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₂₃₆	C₂	semidirect
3. #21426333	C₂	semidirect
5. #21426249	C₂	semidirect
6. #21426234	C₂	semidirect
7. #21426159	C₂	semidirect
9. #21425962	C₂	semidirect
10. C₄×Dic₄₇₂	C₂	semidirect
11. C₅₉×#2183	C₂	semidirect
171. #2183	D₅₉	semidirect
207. C₅₉	#33942	semidirect

Maximal subgroups

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

subgroup	index
#21437109	2
Q₈×Dic₂₃₆	2
#21426333	2
#21426320	2
#21426249	2
#21426234	2
#21426159	2
#21426137	2
#21425962	2
C₄×Dic₄₇₂	2
C₅₉×#2183	2
#33942	59