Subgroups — #28746441

Order 15104.

474 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21437085	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426333	C₂	1	2	yes	maximal
4	#21426324	C₂	1	1	yes	maximal, characteristic, complemented
5	#21426244	C₂	1	1	yes	maximal, characteristic
6	#21426239	C₂	1	2	yes	maximal, complemented
7	#21426159	C₂	1	2	yes	maximal
8	#21426147	C₂	1	1	yes	maximal, characteristic, complemented
9	#21425968	C₂	1	2	yes	maximal, complemented
10	C₄×#14273672	C₂	1	1	yes	maximal, characteristic
11	C₅₉×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×Dic₁₁₈	—	2	1	no	—
13	#16838125	—	2	1	no	—
14	C₂×Dic₄₇₂	C₂²	1	2	yes	—
15	C₂×#14273672	C₂²	1	1	yes	characteristic
16	#16837807	C₂²	1	2	yes	—
17	#16837807	C₂²	1	1	yes	characteristic
18	#16837786	C₂²	1	2	yes	—
19	#16837782	C₂²	1	1	yes	characteristic
20	#16837698	—	2	1	no	—
21	#16837698	—	2	2	no	—
22	#16837625	—	2	2	no	—
23	#16837608	—	2	1	no	—
24	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
25	C₄×Dic₂₃₆	C₂²	1	2	yes	—
26	C₄×Dic₂₃₆	C₂²	1	2	yes	—
27	C₄×D₄₇₂	C₂²	1	1	yes	characteristic
28	#16837230	C₂²	1	1	yes	characteristic
29	#16837230	C₂²	1	2	yes	—
30	#16837207	C₂²	1	1	yes	characteristic
31	#16837207	C₂²	1	2	yes	—
32	#16837207	C₂²	1	2	yes	—
33	#16837207	C₂²	1	2	yes	—
34	#16837204	C₂²	1	1	yes	characteristic
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	—	2	1	no	—
42	C₂×Dic₂₃₆	C₂³	1	2	yes	—
43	C₂×Dic₂₃₆	C₂³	1	1	yes	characteristic
44	C₂×#11745660	—	2	2	no	—
45	C₂×D₄₇₂	C₂³	1	1	yes	characteristic
46	#14273727	—	2	1	no	—
47	#14273727	—	2	2	no	—
48	#14273727	—	2	1	no	—
49	#14273726	—	4	1	no	—
50	#14273726	—	4	2	no	—
51	#14273722	—	4	2	no	—
52	#14273715	—	2	2	no	—
53	#14273703	—	4	1	no	—
54	Q₈×Dic₅₉	—	4	1	no	—
55	C₄×Dic₁₁₈	—	2	1	no	—
56	C₄×Dic₁₁₈	—	2	2	no	—
57	C₄×Dic₁₁₈	—	2	2	no	—
58	C₄×D₂₃₆	—	2	1	no	—
59	#14273685	—	4	1	no	—
60	Dic₄₇₂	—	2	4	no	—
61	#14273672	—	2	2	no	—
62	#14273661	—	2	1	no	—
63	#14273661	—	2	2	no	—
64	#14273661	C₂³	1	1	yes	characteristic
65	#14273661	C₂³	1	2	yes	—
66	#14273661	C₂³	1	1	yes	characteristic
67	#14273652	—	2	1	no	—
68	#14273652	—	2	2	no	—
69	#14273652	—	2	2	no	—
70	#14273649	—	2	1	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	1	no	—
81	C₂×C₄×D₅₉	—	4	1	no	—
82	C₂×Dic₁₁₈	—	2	1	no	—
83	C₂×Dic₁₁₈	—	2	2	no	—
84	C₂×Dic₁₁₈	—	4	2	no	—
85	C₂×Dic₁₁₈	—	4	1	no	—
86	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
90	C₄⋊C₄×C₅₉	D₈	1	2	yes	—
91	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅₉	C₂×D₄	1	2	yes	—
93	(C₂×D₅₉)⋊C₄	—	4	2	no	—
94	(C₂×D₅₉)⋊C₄	—	4	1	no	—
95	(C₂×D₅₉)⋊C₄	—	4	2	no	—
96	C₄⋊Dic₅₉	—	2	1	no	—
97	C₄⋊Dic₅₉	—	2	1	no	—
98	C₄⋊Dic₅₉	—	2	2	no	—
99	C₄⋊Dic₅₉	—	4	1	no	—
100	C₄⋊Dic₅₉	—	4	2	no	—
101	C₄⋊Dic₅₉	—	4	1	no	—
102	C₄⋊Dic₅₉	—	4	1	no	—
103	Dic₅₉⋊C₄	—	4	2	no	—
104	Dic₅₉⋊C₄	—	4	2	no	—
105	Dic₅₉⋊C₄	—	4	1	no	—
106	C₄×Dic₅₉	—	4	2	no	—
107	C₄×Dic₅₉	—	4	2	no	—
108	C₄×Dic₅₉	—	4	1	no	—
109	Dic₂₃₆	—	2	2	no	—
110	Dic₂₃₆	—	2	1	no	—
111	Dic₂₃₆	—	2	2	no	—
112	Dic₂₃₆	—	8	1	no	—
113	Dic₂₃₆	#41	1	2	yes	—
114	D₄₇₂	—	2	1	no	—
115	D₄₇₂	—	2	1	no	—
116	#11745660	—	8	2	no	—
117	C₁₆×C₅₉	—	2	1	no	—
118	C₁₆×C₅₉	—	2	2	no	—
119	C₁₆×C₅₉	—	2	1	no	—
120	C₂²×D₅₉	—	4	1	no	—
121	Q₈×C₅₉	—	2	2	no	—
122	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
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	1	no	—
129	C₂×Dic₅₉	—	4	2	no	—
130	C₂×Dic₅₉	—	4	1	no	—
131	C₂×Dic₅₉	—	4	2	no	—
132	D₂₃₆	—	4	1	no	—
133	D₂₃₆	—	4	1	no	—
134	C₄×D₅₉	—	8	2	no	—
135	Dic₁₁₈	—	2	2	no	—
136	Dic₁₁₈	—	4	1	no	—
137	Dic₁₁₈	—	4	2	no	—
138	Dic₁₁₈	—	4	2	no	—
139	Dic₁₁₈	—	16	2	no	—
140	Dic₁₁₈	—	16	1	no	—
141	C₈×C₅₉	—	2	2	no	—
142	C₈×C₅₉	—	2	1	no	—
143	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
144	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
145	#33936	—	59	1	no	maximal, Sylow
146	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₅₉	—	8	1	no	—
148	C₂×D₅₉	—	8	1	no	—
149	C₂×D₅₉	—	8	1	no	—
150	C₄×C₅₉	—	2	2	no	—
151	C₄×C₅₉	—	2	2	no	—
152	C₄×C₅₉	—	2	1	no	—
153	C₄×C₅₉	#510	1	2	yes	—
154	C₄×C₅₉	#548	1	1	yes	characteristic
155	C₄×C₅₉	#577	1	1	yes	characteristic
156	Dic₅₉	—	4	2	no	—
157	Dic₅₉	—	8	2	no	—
158	Dic₅₉	—	8	1	no	—
159	Dic₅₉	—	8	1	no	—
160	Dic₅₉	—	8	2	no	—
161	#3393	—	59	1	no	—
162	Q₈×Q₁₆	—	59	1	no	—
163	#2243	—	59	2	no	—
164	#2241	—	59	1	no	—
165	#2226	—	59	1	no	—
166	#2225	—	59	2	no	—
167	#2211	—	59	2	no	—
168	#2209	—	59	1	no	—
169	#2183	D₅₉	1	1	yes	characteristic, complemented
170	#2176	—	59	2	no	—
171	C₄×SD₃₂	—	59	1	no	—
172	C₂×C₅₉	#3358	1	1	yes	characteristic
173	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
174	C₂×C₅₉	#3416	1	1	yes	characteristic
175	D₅₉	—	16	1	no	—
176	Q₈²	—	118	1	no	—
177	#556	—	118	1	no	—
178	C₂×Q₃₂	—	59	2	no	—
179	C₂×SD₃₂	—	59	1	no	—
180	#500	—	59	2	no	—
181	#500	—	59	1	no	—
182	#495	—	59	2	no	—
183	#494	—	59	1	no	—
184	#477	—	118	1	no	—
185	#477	—	118	2	no	—
186	#464	—	118	2	no	—
187	#461	—	118	1	no	—
188	C₈×Q₈	C₂×D₅₉	1	1	yes	characteristic
189	C₄×Q₁₆	—	59	1	no	—
190	C₄×Q₁₆	—	59	2	no	—
191	C₄×Q₁₆	—	59	2	no	—
192	C₄×D₈	—	59	1	no	—
193	#367	—	59	1	no	—
194	#367	—	59	2	no	—
195	C₄⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
196	C₄⋊C₁₆	C₂×D₅₉	1	2	yes	—
197	#358	—	59	1	no	—
198	#358	—	59	2	no	—
199	#358	—	59	2	no	—
200	#358	—	59	2	no	—
201	#357	—	59	1	no	—
202	#357	—	59	2	no	—
203	#346	C₂×D₅₉	1	2	yes	—
204	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
205	C₅₉	#33936	1	1	yes	characteristic, complemented, Sylow
206	C₂×Q₁₆	—	59	1	no	—
207	C₂×Q₁₆	—	59	2	no	—
208	C₂×Q₁₆	—	118	1	no	—
209	C₂×SD₁₆	—	118	2	no	—
210	C₂×D₈	—	59	1	no	—
211	C₄⋊Q₈	—	118	1	no	—
212	C₄⋊Q₈	—	118	2	no	—
213	C₄⋊Q₈	—	118	1	no	—
214	C₄⋊Q₈	—	236	1	no	—
215	C₄⋊Q₈	—	236	2	no	—
216	#126	—	236	2	no	—
217	#124	—	118	2	no	—
218	C₂²⋊Q₈	—	236	1	no	—
219	C₄×Q₈	—	118	2	no	—
220	C₄×Q₈	—	118	1	no	—
221	C₄×Q₈	—	118	2	no	—
222	C₄×Q₈	—	236	1	no	—
223	C₄×Q₈	D₂₃₆	1	1	yes	characteristic
224	C₄×D₄	—	118	1	no	—
225	#117	—	236	1	no	—
226	Q₃₂	—	118	4	no	—
227	SD₃₂	—	118	2	no	—
228	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
229	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
230	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
231	#107	—	59	1	no	—
232	#107	—	59	2	no	—
233	#107	—	59	1	no	—
234	#107	—	118	1	no	—
235	#107	—	118	2	no	—
236	C₄⋊C₈	D₂₃₆	1	1	yes	characteristic
237	C₄⋊C₈	D₂₃₆	1	2	yes	—
238	#103	—	118	2	no	—
239	#103	—	118	1	no	—
240	#103	—	118	2	no	—
241	#102	—	118	1	no	—
242	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₅₉	1	2	yes	—
244	C₂×Q₈	—	118	1	no	—
245	C₂×Q₈	—	118	2	no	—
246	C₂×Q₈	—	236	2	no	—
247	C₂×Q₈	—	236	1	no	—
248	C₂×Q₈	D₄₇₂	1	1	yes	characteristic
249	C₂×D₄	—	118	1	no	—
250	C₂²×C₄	—	236	1	no	—
251	Q₁₆	—	59	2	no	—
252	Q₁₆	—	118	2	no	—
253	Q₁₆	—	118	1	no	—
254	Q₁₆	—	118	2	no	—
255	Q₁₆	—	472	1	no	—
256	SD₁₆	—	472	2	no	—
257	D₈	—	118	1	no	—
258	D₈	—	118	1	no	—
259	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
260	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
261	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
262	C₄⋊C₄	—	118	1	no	—
263	C₄⋊C₄	—	118	1	no	—
264	C₄⋊C₄	—	118	2	no	—
265	C₄⋊C₄	—	236	1	no	—
266	C₄⋊C₄	—	236	2	no	—
267	C₄⋊C₄	—	236	2	no	—
268	C₄⋊C₄	—	236	1	no	—
269	C₄⋊C₄	—	236	2	no	—
270	C₄⋊C₄	—	236	1	no	—
271	C₄⋊C₄	—	236	1	no	—
272	C₄⋊C₄	D₄₇₂	1	1	yes	characteristic
273	C₄⋊C₄	D₄₇₂	1	2	yes	—
274	C₂²⋊C₄	—	236	2	no	—
275	C₂²⋊C₄	—	236	1	no	—
276	C₂²⋊C₄	—	236	2	no	—
277	C₄²	—	236	2	no	—
278	C₄²	—	236	2	no	—
279	C₄²	—	236	1	no	—
280	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
281	C₄²	C₂×D₂₃₆	1	2	yes	—
282	C₁₆	—	2	1	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	2	1	no	—
285	C₂³	—	236	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	118	2	no	—
288	Q₈	—	944	2	no	—
289	Q₈	—	944	1	no	—
290	Q₈	—	236	2	no	—
291	Q₈	—	236	2	no	—
292	Q₈	—	236	1	no	—
293	D₄	—	236	1	no	—
294	D₄	—	236	1	no	—
295	C₂×C₄	—	472	2	no	—
296	C₂×C₄	—	236	1	no	—
297	C₂×C₄	—	236	2	no	—
298	C₂×C₄	—	236	2	no	—
299	C₂×C₄	—	236	1	no	—
300	C₂×C₄	—	236	1	no	—
301	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
302	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
303	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	#41×D₅₉	1	1	yes	characteristic
309	C₈	#14273775	1	1	yes	characteristic
310	C₂²	—	472	1	no	—
311	C₂²	—	472	1	no	—
312	C₂²	—	472	1	no	—
313	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	472	1	no	—
318	C₄	—	472	2	no	—
319	C₄	—	472	2	no	—
320	C₄	—	472	1	no	—
321	C₄	—	236	2	no	—
322	C₄	#16837845	1	2	yes	—
323	C₄	#16838064	1	1	yes	characteristic
324	C₄	#16838214	1	1	yes	characteristic
325	C₂	—	944	1	no	—
326	C₂	#21436732	1	1	yes	characteristic
327	C₂	C₂×#16837845	1	1	yes	characteristic
328	C₂	#21437244	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	C₂×D₄₇₂	C₂³
hypercenter	#2183	D₅₉
hypocenter	C₅₉	#33936
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
4. #21426324	C₂	semidirect
6. #21426239	C₂	semidirect
8. #21426147	C₂	semidirect
9. #21425968	C₂	semidirect
11. C₅₉×#2183	C₂	semidirect
169. #2183	D₅₉	semidirect
205. C₅₉	#33936	semidirect

Maximal subgroups

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

subgroup	index
#21437085	2
Q₈×Dic₂₃₆	2
#21426333	2
#21426324	2
#21426244	2
#21426239	2
#21426159	2
#21426147	2
#21425968	2
C₄×#14273672	2
C₅₉×#2183	2
#33936	59