Subgroups — #28742191

Order 15104.

498 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	#21433025	C₂	1	1	yes	maximal, characteristic
2	#21426005	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426004	C₂	1	1	yes	maximal, characteristic, complemented
4	#21426004	C₂	1	2	yes	maximal, complemented
5	#21426003	C₂	1	2	yes	maximal, complemented
6	#21425998	C₂	1	1	yes	maximal, characteristic, complemented
7	#21425853	C₂	1	2	yes	maximal
8	C₄⋊C₁₆×D₅₉	C₂	1	1	yes	maximal, characteristic
9	#21425691	C₂	1	2	yes	maximal
10	C₄×#14273666	C₂	1	1	yes	maximal, characteristic
11	Q₈×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837956	C₄	1	1	yes	characteristic
13	C₂×#14273666	C₂²	1	1	yes	characteristic
14	C₂×#14273666	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₅₉	C₂²	1	2	yes	—
16	#16837533	C₄	1	1	yes	characteristic
17	#16837531	C₄	1	2	yes	—
18	#16837529	C₄	1	1	yes	characteristic
19	C₈×Dic₁₁₈	C₂²	1	1	yes	characteristic
20	C₈×Dic₁₁₈	C₂²	1	2	yes	—
21	#16837427	C₄	1	2	yes	—
22	#16837424	C₄	1	1	yes	characteristic
23	#16837356	C₂²	1	2	yes	—
24	C₄×C₈×D₅₉	C₂²	1	1	yes	characteristic
25	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
27	Dic₅₉⋊C₁₆	C₂²	1	2	yes	—
28	Dic₅₉⋊C₁₆	C₂²	1	2	yes	—
29	Dic₅₉⋊C₁₆	C₂²	1	2	yes	—
30	#16837178	C₂²	1	2	yes	—
31	#16837177	C₂²	1	1	yes	characteristic
32	#16837177	C₂²	1	1	yes	characteristic
33	#16837177	C₂²	1	2	yes	—
34	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₅₉	C₂²	1	2	yes	—
36	C₈×Q₈×C₅₉	C₂²	1	1	yes	characteristic
37	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
39	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₅₉	C₂²	1	2	yes	—
41	C₂×#11745659	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₅₉	C₂³	1	1	yes	characteristic
44	#14273726	C₈	1	1	yes	characteristic
45	#14273725	C₈	1	1	yes	characteristic
46	#14273717	C₈	1	2	yes	—
47	#14273707	C₈	1	1	yes	characteristic
48	#14273705	C₈	1	2	yes	—
49	#14273703	C₈	1	1	yes	characteristic
50	C₄×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
51	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
52	#14273684	C₂×C₄	1	2	yes	—
53	C₄²×D₅₉	C₂×C₄	1	1	yes	characteristic
54	#14273666	—	2	2	no	—
55	#14273666	Q₈	1	4	yes	complemented
56	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
58	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₁₆×D₅₉	—	2	4	no	—
60	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
61	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
62	Dic₅₉⋊C₈	C₂×C₄	1	2	yes	—
63	Dic₅₉⋊C₈	C₂×C₄	1	2	yes	—
64	Dic₅₉⋊C₈	C₂×C₄	1	2	yes	—
65	#14273638	C₂×C₄	1	2	yes	—
66	#14273637	C₂×C₄	1	2	yes	—
67	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₅₉	C₂³	1	2	yes	—
70	C₈×Dic₅₉	C₂³	1	1	yes	characteristic
71	C₄×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
72	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
73	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
76	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
77	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₅₉	C₂³	1	2	yes	—
79	C₂×Q₈×C₅₉	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₅₉	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₅₉	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₅₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
84	C₂×Dic₁₁₈	C₂×C₈	1	2	yes	—
85	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
89	C₄⋊C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
90	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
92	(C₂×D₅₉)⋊C₄	C₂×C₈	1	2	yes	—
93	(C₂×D₅₉)⋊C₄	C₂×C₈	1	2	yes	—
94	C₄⋊Dic₅₉	C₂×C₈	1	1	yes	characteristic
95	C₄⋊Dic₅₉	C₂×C₈	1	2	yes	—
96	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
97	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
98	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
99	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
100	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
101	C₄×Dic₅₉	C₂×C₈	1	1	yes	characteristic
102	C₄×Dic₅₉	C₂×C₈	1	1	yes	characteristic
103	C₄×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
104	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
105	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
107	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
108	#11745659	—	2	4	no	—
109	C₈×D₅₉	—	2	2	no	—
110	C₈×D₅₉	C₂×Q₈	1	2	yes	—
111	C₈×D₅₉	#41	1	2	yes	—
112	C₁₆×C₅₉	—	2	2	no	—
113	C₁₆×C₅₉	—	2	1	no	—
114	C₁₆×C₅₉	C₂×Q₈	1	2	yes	—
115	C₅₉⋊C₁₆	—	2	2	no	—
116	C₅₉⋊C₁₆	—	2	1	no	—
117	C₅₉⋊C₁₆	C₂×Q₈	1	2	yes	—
118	C₂²×D₅₉	C₂²×C₈	1	1	yes	characteristic
119	Q₈×C₅₉	—	2	2	no	—
120	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×Dic₅₉	C₂²×C₈	1	1	yes	characteristic
126	C₂×Dic₅₉	C₂²×C₈	1	1	yes	characteristic
127	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
128	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
129	C₂×Dic₅₉	C₂²×C₈	1	1	yes	characteristic
130	C₄×D₅₉	—	2	2	no	—
131	C₄×D₅₉	—	4	1	no	—
132	C₄×D₅₉	C₄×Q₈	1	2	yes	—
133	C₄×D₅₉	#131	1	2	yes	—
134	Dic₁₁₈	—	2	2	no	—
135	Dic₁₁₈	—	4	2	no	—
136	C₈×C₅₉	—	2	1	no	—
137	C₈×C₅₉	—	2	2	no	—
138	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
139	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
140	C₅₉⋊C₈	—	2	1	no	—
141	C₅₉⋊C₈	—	2	2	no	—
142	C₅₉⋊C₈	C₄×Q₈	1	2	yes	—
143	#33476	—	59	1	no	maximal, Sylow
144	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
145	C₂×D₅₉	—	2	1	no	—
146	C₂×D₅₉	—	2	1	no	—
147	C₂×D₅₉	C₈×Q₈	1	2	yes	—
148	C₄×C₅₉	—	2	2	no	—
149	C₄×C₅₉	—	2	2	no	—
150	C₄×C₅₉	—	2	1	no	—
151	C₄×C₅₉	#504	1	2	yes	—
152	C₄×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
153	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
154	Dic₅₉	—	2	2	no	—
155	Dic₅₉	—	2	1	no	—
156	Dic₅₉	—	2	1	no	—
157	Dic₅₉	—	2	2	no	—
158	Dic₅₉	C₈×Q₈	1	2	yes	—
159	#2960	—	59	1	no	—
160	#2183	—	59	1	no	—
161	#2183	—	59	2	no	—
162	#2183	—	59	1	no	—
163	#2183	—	59	2	no	—
164	Q₈×C₁₆	—	59	1	no	—
165	Q₈×C₁₆	D₅₉	1	1	yes	characteristic, complemented
166	#2151	—	59	2	no	—
167	C₂×C₄⋊C₁₆	—	59	1	no	—
168	#2108	—	59	2	no	—
169	C₄×M₃₂	—	59	1	no	—
170	C₂×C₅₉	C₂×C₈×Q₈	1	1	yes	characteristic
171	C₂×C₅₉	C₂×#504	1	1	yes	characteristic
172	C₂×C₅₉	#3407	1	1	yes	characteristic
173	D₅₉	—	2	2	no	—
174	#533	—	59	1	no	—
175	C₂×M₃₂	—	59	1	no	—
176	C₂×M₃₂	—	59	1	no	—
177	C₂²×C₁₆	—	59	2	no	—
178	#446	—	59	1	no	—
179	#446	—	59	1	no	—
180	#446	—	59	2	no	—
181	C₈×Q₈	—	59	1	no	—
182	C₈×Q₈	—	59	2	no	—
183	C₈×Q₈	C₂×D₅₉	1	1	yes	characteristic
184	#424	—	59	2	no	—
185	#423	—	59	1	no	—
186	#405	—	59	2	no	—
187	C₂×C₄×C₈	—	59	1	no	—
188	C₄⋊C₁₆	—	59	2	no	—
189	C₄⋊C₁₆	—	59	2	no	—
190	C₄⋊C₁₆	—	59	2	no	—
191	C₄⋊C₁₆	—	59	1	no	—
192	C₄⋊C₁₆	—	59	2	no	—
193	C₄⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
194	C₄⋊C₁₆	C₂×D₅₉	1	2	yes	—
195	#346	—	59	1	no	—
196	#346	—	59	1	no	—
197	#346	—	59	2	no	—
198	#346	—	59	2	no	—
199	C₄×C₁₆	—	59	2	no	—
200	C₄×C₁₆	—	59	1	no	—
201	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
202	C₄×C₁₆	C₂×D₅₉	1	2	yes	—
203	C₅₉	#33476	1	1	yes	characteristic, complemented, Sylow
204	C₂×M₁₆	—	59	2	no	—
205	C₂²×C₈	—	59	1	no	—
206	C₂²×C₈	—	59	1	no	—
207	C₄⋊Q₈	—	59	1	no	—
208	C₄⋊Q₈	—	59	1	no	—
209	#125	—	59	2	no	—
210	C₂²⋊Q₈	—	59	1	no	—
211	C₂²⋊Q₈	—	59	2	no	—
212	C₂²⋊Q₈	—	59	1	no	—
213	C₄×Q₈	—	59	2	no	—
214	C₄×Q₈	—	59	1	no	—
215	C₄×Q₈	C₄×D₅₉	1	1	yes	characteristic
216	#117	—	59	2	no	—
217	C₂×C₄²	—	59	1	no	—
218	M₃₂	—	59	4	no	—
219	M₃₂	—	118	2	no	—
220	C₂×C₁₆	—	59	1	no	—
221	C₂×C₁₆	—	59	1	no	—
222	C₂×C₁₆	—	59	2	no	—
223	C₂×C₁₆	—	118	4	no	—
224	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
225	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
226	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
227	C₄⋊C₈	—	59	2	no	—
228	C₄⋊C₈	—	59	2	no	—
229	C₄⋊C₈	—	59	2	no	—
230	C₄⋊C₈	—	59	1	no	—
231	C₄⋊C₈	—	59	2	no	—
232	C₄⋊C₈	C₄×D₅₉	1	1	yes	characteristic
233	C₄⋊C₈	C₄×D₅₉	1	2	yes	—
234	#97	—	59	2	no	—
235	#97	—	59	2	no	—
236	C₄×C₈	—	59	1	no	—
237	C₄×C₈	—	59	2	no	—
238	C₄×C₈	—	59	1	no	—
239	C₄×C₈	—	59	1	no	—
240	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₅₉	1	2	yes	—
242	C₂×Q₈	—	59	1	no	—
243	C₂×Q₈	—	59	2	no	—
244	C₂×Q₈	C₈×D₅₉	1	1	yes	characteristic
245	C₂²×C₄	—	59	1	no	—
246	C₂²×C₄	—	59	1	no	—
247	C₂²×C₄	—	59	1	no	—
248	M₁₆	—	118	4	no	—
249	C₂×C₈	—	59	1	no	—
250	C₂×C₈	—	59	2	no	—
251	C₂×C₈	—	59	2	no	—
252	C₂×C₈	—	59	1	no	—
253	C₂×C₈	—	59	2	no	—
254	C₂×C₈	—	118	2	no	—
255	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
257	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
258	C₄⋊C₄	—	59	2	no	—
259	C₄⋊C₄	—	59	2	no	—
260	C₄⋊C₄	—	59	2	no	—
261	C₄⋊C₄	—	59	2	no	—
262	C₄⋊C₄	—	59	2	no	—
263	C₄⋊C₄	—	59	2	no	—
264	C₄⋊C₄	—	59	1	no	—
265	C₄⋊C₄	C₈×D₅₉	1	2	yes	—
266	C₄⋊C₄	C₈×D₅₉	1	1	yes	characteristic
267	C₂²⋊C₄	—	59	2	no	—
268	C₂²⋊C₄	—	59	2	no	—
269	C₄²	—	59	1	no	—
270	C₄²	—	59	1	no	—
271	C₄²	—	59	1	no	—
272	C₄²	—	59	2	no	—
273	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
274	C₄²	C₂×C₄×D₅₉	1	2	yes	—
275	C₁₆	—	2	2	no	—
276	C₁₆	—	2	1	no	—
277	C₁₆	—	59	2	no	—
278	C₁₆	—	118	2	no	—
279	C₁₆	—	118	1	no	—
280	C₁₆	Q₈×D₅₉	1	2	yes	—
281	C₂³	—	59	1	no	—
282	Q₈	—	2	2	no	—
283	Q₈	—	118	2	no	—
284	Q₈	—	236	2	no	—
285	C₂×C₄	—	59	1	no	—
286	C₂×C₄	—	59	2	no	—
287	C₂×C₄	—	59	2	no	—
288	C₂×C₄	—	59	1	no	—
289	C₂×C₄	—	59	1	no	—
290	C₂×C₄	—	59	2	no	—
291	C₂×C₄	—	59	2	no	—
292	C₂×C₄	—	118	2	no	—
293	C₂×C₄	—	236	1	no	—
294	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
295	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
296	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
297	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	2	no	—
301	C₈	—	59	2	no	—
302	C₈	—	118	1	no	—
303	C₈	—	118	2	no	—
304	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
305	C₈	#41×D₅₉	1	1	yes	characteristic
306	C₂²	—	59	2	no	—
307	C₂²	—	118	1	no	—
308	C₂²	—	118	1	no	—
309	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	2	no	—
313	C₄	—	59	2	no	—
314	C₄	—	118	1	no	—
315	C₄	—	118	2	no	—
316	C₄	—	118	1	no	—
317	C₄	—	118	2	no	—
318	C₄	#16837818	1	2	yes	—
319	C₄	C₄×Q₈×D₅₉	1	1	yes	characteristic
320	C₄	#131×D₅₉	1	1	yes	characteristic
321	C₂	—	118	2	no	—
322	C₂	C₈×Q₈×D₅₉	1	1	yes	characteristic
323	C₂	C₂×#16837818	1	1	yes	characteristic
324	C₂	#21437197	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₈
Fitting	Q₈×C₁₆×C₅₉	C₂
prime radical	C₂²×D₅₉	C₂²×C₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	Q₈×C₁₆	D₅₉
hypocenter	C₅₉	#33476
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #21426005	C₂	semidirect
3. #21426004	C₂	semidirect
4. #21426004	C₂	semidirect
5. #21426003	C₂	semidirect
6. #21425998	C₂	semidirect
11. Q₈×C₁₆×C₅₉	C₂	semidirect
55. #14273666	Q₈	semidirect
165. Q₈×C₁₆	D₅₉	semidirect
203. C₅₉	#33476	semidirect

Maximal subgroups

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

subgroup	index
#21433025	2
#21426005	2
#21426004	2
#21426003	2
#21425998	2
#21425853	2
C₄⋊C₁₆×D₅₉	2
#21425691	2
C₄×#14273666	2
Q₈×C₁₆×C₅₉	2
#33476	59