Subgroups — #28481853

Order 15104.

454 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	#21424116	C₂	1	2	yes	maximal
2	C₄×#14273659	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422852	C₂	1	1	yes	maximal, characteristic, complemented
4	#21421998	C₂	1	2	yes	maximal, complemented
5	C₈×#41×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273659	—	2	1	no	—
7	#16837426	C₂²	1	2	yes	—
8	C₂×#14273654	—	2	2	no	—
9	C₄×C₄⋊Dic₅₉	C₂²	1	2	yes	—
10	#16837165	—	2	1	no	—
11	#16837161	—	2	2	no	—
12	#16837153	—	2	1	no	—
13	#16837153	—	2	1	no	—
14	C₄×#41×C₅₉	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₅₉	C₄	1	1	yes	characteristic, complemented
16	C₈×Q₈×C₅₉	C₄	1	1	yes	characteristic, complemented
17	C₈×D₄×C₅₉	—	2	1	no	—
18	C₈×D₄×C₅₉	—	2	1	no	—
19	C₈×D₄×C₅₉	C₄	1	1	yes	characteristic, complemented
20	C₈×D₄×C₅₉	C₄	1	1	yes	characteristic, complemented
21	C₅₉×#431	—	2	1	no	—
22	C₅₉×#431	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₉	—	2	1	no	—
24	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×#11745664	—	2	2	no	—
26	C₂×C₄⋊Dic₅₉	—	2	2	no	—
27	C₂×C₄×Dic₅₉	—	2	2	no	—
28	#14273659	—	2	4	no	—
29	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
30	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
31	#14273654	—	4	4	no	—
32	C₅₉⋊#95	—	2	2	no	—
33	C₂×#41×C₅₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₉	—	2	1	no	—
35	C₂²×C₈×C₅₉	—	2	1	no	—
36	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
38	C₄×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₅₉	Q₈	1	1	yes	characteristic
40	C₄×D₄×C₅₉	—	2	1	no	—
41	C₄×D₄×C₅₉	—	2	1	no	—
42	C₄×D₄×C₅₉	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₅₉	Q₈	1	1	yes	characteristic
44	#117×C₅₉	—	2	1	no	—
45	#117×C₅₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₉	—	2	1	no	—
47	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₅₉	—	2	1	no	—
49	C₄⋊C₈×C₅₉	—	2	1	no	—
50	C₄⋊C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
52	C₂²⋊C₈×C₅₉	—	2	1	no	—
53	C₂²⋊C₈×C₅₉	—	2	1	no	—
54	C₂²⋊C₈×C₅₉	—	2	1	no	—
55	C₄×C₈×C₅₉	—	2	1	no	—
56	C₄×C₈×C₅₉	—	2	1	no	—
57	C₄×C₈×C₅₉	—	2	1	no	—
58	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
61	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
62	#41×C₅₉	—	2	2	no	—
63	#41×C₅₉	SD₁₆	1	4	yes	—
64	C₂×Q₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂×D₄×C₅₉	—	2	1	no	—
66	C₂×D₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	—	2	1	no	—
68	C₂²×C₄×C₅₉	—	2	1	no	—
69	C₂²×C₄×C₅₉	—	2	1	no	—
70	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
73	C₂²×Dic₅₉	—	4	2	no	—
74	C₂×C₈×C₅₉	—	2	2	no	—
75	C₂×C₈×C₅₉	—	2	1	no	—
76	C₂×C₈×C₅₉	—	2	1	no	—
77	C₂×C₈×C₅₉	—	4	1	no	—
78	C₂×C₈×C₅₉	—	4	2	no	—
79	C₂×C₈×C₅₉	—	4	2	no	—
80	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₄⋊C₄×C₅₉	—	2	1	no	—
85	C₄⋊C₄×C₅₉	—	2	1	no	—
86	C₄⋊C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
88	C₂²⋊C₄×C₅₉	—	2	1	no	—
89	C₂²⋊C₄×C₅₉	—	2	1	no	—
90	C₂²⋊C₄×C₅₉	—	2	1	no	—
91	C₄²×C₅₉	—	2	1	no	—
92	C₄²×C₅₉	—	2	1	no	—
93	C₄²×C₅₉	—	2	1	no	—
94	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
97	C₄²×C₅₉	#41	1	1	yes	characteristic
98	C₄⋊Dic₅₉	—	2	4	no	—
99	C₄⋊Dic₅₉	—	4	2	no	—
100	C₄×Dic₅₉	—	4	2	no	—
101	C₄×Dic₅₉	—	4	2	no	—
102	#11745664	—	4	2	no	—
103	#11745664	—	4	2	no	—
104	C₂×C₅₉⋊C₈	—	4	2	no	—
105	C₂³×C₅₉	—	2	1	no	—
106	C₂³×C₅₉	#121	1	1	yes	characteristic
107	Q₈×C₅₉	—	2	1	no	—
108	Q₈×C₅₉	#106	1	2	yes	complemented
109	D₄×C₅₉	—	2	4	no	—
110	D₄×C₅₉	—	2	1	no	—
111	D₄×C₅₉	#106	1	2	yes	complemented
112	C₂×C₄×C₅₉	—	2	1	no	—
113	C₂×C₄×C₅₉	—	2	1	no	—
114	C₂×C₄×C₅₉	—	2	1	no	—
115	C₂×C₄×C₅₉	—	2	4	no	—
116	C₂×C₄×C₅₉	—	2	1	no	—
117	C₂×C₄×C₅₉	—	2	1	no	—
118	C₂×C₄×C₅₉	—	4	1	no	—
119	C₂×C₄×C₅₉	—	4	2	no	—
120	C₂×C₄×C₅₉	—	4	1	no	—
121	C₂×C₄×C₅₉	—	4	1	no	—
122	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
125	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
129	C₂×C₄×C₅₉	#135	1	2	yes	—
130	C₂×Dic₅₉	—	4	4	no	—
131	C₂×Dic₅₉	—	8	2	no	—
132	C₂×Dic₅₉	—	8	2	no	—
133	C₈×C₅₉	—	2	1	no	—
134	C₈×C₅₉	—	2	1	no	—
135	C₈×C₅₉	—	2	1	no	—
136	C₈×C₅₉	—	4	1	no	—
137	C₈×C₅₉	—	4	1	no	—
138	C₈×C₅₉	#104	1	2	yes	complemented
139	C₅₉⋊C₈	—	8	2	no	—
140	#8666	—	59	1	no	maximal, Sylow
141	C₂²×C₅₉	—	2	1	no	—
142	C₂²×C₅₉	—	2	1	no	—
143	C₂²×C₅₉	—	2	2	no	—
144	C₂²×C₅₉	—	4	1	no	—
145	C₂²×C₅₉	—	4	1	no	—
146	C₂²×C₅₉	#380	1	1	yes	characteristic
147	C₂²×C₅₉	#465	1	2	yes	—
148	C₄×C₅₉	—	2	1	no	—
149	C₄×C₅₉	—	2	2	no	—
150	C₄×C₅₉	—	2	1	no	—
151	C₄×C₅₉	—	2	1	no	—
152	C₄×C₅₉	—	4	1	no	—
153	C₄×C₅₉	—	4	1	no	—
154	C₄×C₅₉	C₂×#104	1	1	yes	characteristic
155	C₄×C₅₉	#421	1	1	yes	characteristic
156	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
157	C₄×C₅₉	#427	1	1	yes	characteristic
158	C₄×C₅₉	#465	1	2	yes	—
159	Dic₅₉	—	8	4	no	—
160	C₈×#41	D₅₉	1	1	yes	characteristic
161	#1806	—	59	2	no	—
162	C₄×#106	—	59	1	no	—
163	#1566	—	59	1	no	—
164	#1303	—	59	2	no	—
165	C₂×C₅₉	—	2	2	no	—
166	C₂×C₅₉	—	4	2	no	—
167	C₂×C₅₉	#1806	1	1	yes	characteristic
168	C₂×C₅₉	#1807	1	1	yes	characteristic
169	C₂×C₅₉	#1814	1	1	yes	characteristic
170	C₄×#41	C₂×D₅₉	1	1	yes	characteristic
171	C₈×Q₈	Dic₅₉	1	1	yes	characteristic, complemented
172	C₈×Q₈	Dic₅₉	1	1	yes	characteristic, complemented
173	C₈×D₄	—	2	1	no	—
174	C₈×D₄	—	2	1	no	—
175	C₈×D₄	Dic₅₉	1	1	yes	characteristic, complemented
176	C₈×D₄	Dic₅₉	1	1	yes	characteristic, complemented
177	#431	—	2	1	no	—
178	#431	C₂×D₅₉	1	1	yes	characteristic
179	C₂×#106	—	118	1	no	—
180	#423	—	59	2	no	—
181	C₂×#104	—	118	2	no	—
182	C₂×C₄×C₈	—	2	1	no	—
183	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
184	C₄×C₄⋊C₄	—	59	2	no	—
185	#340	—	118	1	no	—
186	#339	—	118	2	no	—
187	#334	—	118	1	no	—
188	#334	—	118	1	no	—
189	C₅₉	#8666	1	1	yes	characteristic, complemented, Sylow
190	C₂×#41	D₂₃₆	1	1	yes	characteristic
191	C₂×M₁₆	—	118	2	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	1	no	—
194	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
195	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
196	C₄×Q₈	Dic₁₁₈	1	1	yes	characteristic
197	C₄×Q₈	C₂×Dic₅₉	1	1	yes	characteristic
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	Dic₁₁₈	1	1	yes	characteristic
201	C₄×D₄	C₂×Dic₅₉	1	1	yes	characteristic
202	#117	—	2	1	no	—
203	#117	D₂₃₆	1	1	yes	characteristic
204	C₂×C₄⋊C₄	—	118	2	no	—
205	C₂×C₄²	—	2	1	no	—
206	C₂×C₄²	—	118	2	no	—
207	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
208	#106	—	118	4	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	118	2	no	—
212	C₄⋊C₈	—	118	2	no	—
213	C₄⋊C₈	C₂×Dic₅₉	1	1	yes	characteristic
214	C₄⋊C₈	C₂×Dic₅₉	1	1	yes	characteristic
215	#104	—	236	4	no	—
216	C₂²⋊C₈	—	2	1	no	—
217	C₂²⋊C₈	—	2	1	no	—
218	C₂²⋊C₈	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
223	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
224	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
225	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
226	#95	—	118	2	no	—
227	#41	—	2	2	no	—
228	#41	#11745660	1	4	yes	—
229	C₂×Q₈	C₄⋊Dic₅₉	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₄⋊Dic₅₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	236	2	no	—
236	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
237	C₂²×C₄	#11745685	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
239	M₁₆	—	236	2	no	—
240	M₁₆	—	236	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	236	2	no	—
248	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
250	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
251	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	118	4	no	—
255	C₄⋊C₄	—	236	2	no	—
256	C₄⋊C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
257	C₄⋊C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	236	2	no	—
265	C₄²	—	236	2	no	—
266	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
267	C₄²	#11745685	1	1	yes	characteristic
268	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
269	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	#14273701	1	1	yes	characteristic
272	Q₈	—	2	1	no	—
273	Q₈	#14273659	1	2	yes	complemented
274	D₄	—	2	4	no	—
275	D₄	—	2	1	no	—
276	D₄	#14273659	1	2	yes	complemented
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	472	2	no	—
288	C₂×C₄	—	472	2	no	—
289	C₂×C₄	—	236	4	no	—
290	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
291	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
292	C₂×C₄	#14273686	1	1	yes	characteristic
293	C₂×C₄	#14273701	1	1	yes	characteristic
294	C₂×C₄	#14273706	1	1	yes	characteristic
295	C₂×C₄	#14273706	1	1	yes	characteristic
296	C₂×C₄	C₂×#11745660	1	2	yes	—
297	C₂×C₄	#14273749	1	2	yes	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	4	1	no	—
302	C₈	—	4	1	no	—
303	C₈	—	472	2	no	—
304	C₈	#14273654	1	2	yes	complemented
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	#16837262	1	1	yes	characteristic
311	C₂²	#16837631	1	2	yes	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	472	4	no	—
319	C₄	C₂×#14273654	1	1	yes	characteristic
320	C₄	#16837419	1	1	yes	characteristic
321	C₄	C₂×#14273659	1	1	yes	characteristic
322	C₄	#16837440	1	1	yes	characteristic
323	C₄	#16837631	1	2	yes	—
324	C₂	—	2	2	no	—
325	C₂	—	4	2	no	—
326	C₂	#21424116	1	1	yes	characteristic
327	C₂	#21424121	1	1	yes	characteristic
328	C₂	#21424150	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273701
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₈×#41×C₅₉	C₂
socle	C₂²×C₅₉	#380
prime radical	C₂×#41×C₅₉	D₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₈×#41	D₅₉
hypocenter	C₅₉	#8666
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#14273659	C₂	semidirect
3. #21422852	C₂	semidirect
4. #21421998	C₂	semidirect
15. C₈×Q₈×C₅₉	C₄	semidirect
16. C₈×Q₈×C₅₉	C₄	semidirect
19. C₈×D₄×C₅₉	C₄	semidirect
20. C₈×D₄×C₅₉	C₄	semidirect
108. Q₈×C₅₉	#106	semidirect
111. D₄×C₅₉	#106	semidirect
138. C₈×C₅₉	#104	semidirect
171. C₈×Q₈	Dic₅₉	semidirect
172. C₈×Q₈	Dic₅₉	semidirect
175. C₈×D₄	Dic₅₉	semidirect
176. C₈×D₄	Dic₅₉	semidirect
189. C₅₉	#8666	semidirect
273. Q₈	#14273659	semidirect
276. D₄	#14273659	semidirect
304. C₈	#14273654	semidirect

Maximal subgroups

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

subgroup	index
#21424116	2
C₄×#14273659	2
#21422852	2
#21421998	2
C₈×#41×C₅₉	2
#8666	59