Subgroups — #28741867

Order 15104.

682 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	C₂²×#14273666	C₂	1	1	yes	maximal, characteristic, complemented
2	#21432788	C₂	1	1	yes	maximal, characteristic
3	#21425926	C₂	1	4	yes	maximal, complemented
4	#21425921	C₂	1	4	yes	maximal, complemented
5	#21425856	C₂	1	1	yes	maximal, characteristic, complemented
6	#21425710	C₂	1	1	yes	maximal, characteristic, complemented
7	#21425706	C₂	1	1	yes	maximal, characteristic, complemented
8	#21425692	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅₉×#504	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₅₉	C₂²	1	1	yes	characteristic
11	#16837909	C₄	1	1	yes	characteristic
12	C₂×#14273667	C₂²	1	1	yes	characteristic
13	C₂×#14273666	—	2	4	no	—
14	C₂×#14273666	C₂²	1	2	yes	—
15	C₂×#14273666	C₂²	1	2	yes	—
16	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
17	#16837471	C₄	1	4	yes	—
18	C₈×C₂²⋊D₅₉	C₂²	1	4	yes	—
19	#16837430	C₄	1	1	yes	characteristic
20	#16837369	C₄	1	1	yes	characteristic
21	#16837365	C₄	1	1	yes	characteristic
22	#16837357	C₂²	1	1	yes	characteristic
23	Dic₅₉⋊C₁₆	C₂²	1	4	yes	complemented
24	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	4	yes	complemented
25	(C₂×D₅₉)⋊C₁₆	C₂²	1	4	yes	—
26	#16837177	C₂²	1	2	yes	complemented
27	C₁₆×Dic₅₉	C₂²	1	2	yes	complemented
28	C₂×#131×C₅₉	C₂²	1	1	yes	characteristic
29	C₅₉×#504	—	2	4	no	—
30	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
31	C₂×M₃₂×C₅₉	C₂²	1	2	yes	complemented
32	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
33	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	complemented
34	C₂²×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₅₉	—	2	2	no	—
38	C₂×C₈×D₅₉	—	2	2	no	—
39	C₂×C₈×D₅₉	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₅₉	C₂³	1	2	yes	—
41	#14273708	C₈	1	2	yes	—
42	#14273703	C₈	1	2	yes	—
43	#14273698	C₈	1	2	yes	—
44	#14273695	C₈	1	2	yes	—
45	C₄×C₂²⋊D₅₉	C₂×C₄	1	4	yes	—
46	#14273686	C₂×C₄	1	1	yes	characteristic
47	#14273667	—	2	2	no	—
48	#14273666	—	2	4	no	—
49	#14273666	—	2	4	no	—
50	C₂×C₅₉⋊C₁₆	—	2	2	no	—
51	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
52	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
53	Dic₅₉⋊C₈	C₂×C₄	1	4	yes	—
54	C₂²⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
55	(C₂×D₅₉)⋊C₈	C₂×C₄	1	4	yes	—
56	#14273637	C₂×C₄	1	2	yes	—
57	C₈×Dic₅₉	C₂³	1	2	yes	—
58	C₂×#41×C₅₉	C₂×C₄	1	1	yes	characteristic
59	#131×C₅₉	—	2	2	no	—
60	#131×C₅₉	—	2	2	no	—
61	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
62	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
63	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
64	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
65	M₃₂×C₅₉	—	2	2	no	—
66	M₃₂×C₅₉	—	2	4	no	—
67	C₂×C₁₆×C₅₉	—	2	4	no	—
68	C₂×C₁₆×C₅₉	D₄	1	4	yes	complemented
69	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
70	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
71	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
72	C₂³×D₅₉	C₂×C₈	1	1	yes	characteristic
73	#41×C₅₉	—	2	2	no	—
74	#41×C₅₉	—	2	2	no	—
75	C₂×Q₈×C₅₉	C₂×C₈	1	1	yes	characteristic
76	C₂×D₄×C₅₉	C₂×C₈	1	2	yes	—
77	C₂×D₄×C₅₉	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₂²⋊D₅₉	C₂×C₈	1	4	yes	—
81	C₂²×Dic₅₉	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₅₉	—	2	2	no	—
83	C₂×C₄×D₅₉	—	2	2	no	—
84	C₂×C₄×D₅₉	C₂×C₈	1	2	yes	—
85	C₂×C₄×D₅₉	C₂²×C₄	1	2	yes	—
86	M₁₆×C₅₉	—	2	2	no	—
87	M₁₆×C₅₉	—	2	4	no	—
88	C₂×C₈×C₅₉	—	2	2	no	—
89	C₂×C₈×C₅₉	—	2	2	no	—
90	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₅₉	#41	1	2	yes	—
94	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
95	C₂²⋊Dic₅₉	C₂×C₈	1	2	yes	—
96	C₂²⋊Dic₅₉	C₂×C₈	1	4	yes	—
97	(C₂×D₅₉)⋊C₄	C₂×C₈	1	4	yes	—
98	C₄⋊Dic₅₉	C₂×C₈	1	2	yes	—
99	Dic₅₉⋊C₄	C₂×C₈	1	4	yes	—
100	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
101	#11745664	—	2	2	no	—
102	C₂×C₅₉⋊C₈	—	2	2	no	—
103	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
104	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
105	C₈×D₅₉	—	2	4	no	—
106	C₈×D₅₉	—	2	2	no	—
107	C₈×D₅₉	—	2	2	no	—
108	C₁₆×C₅₉	—	2	2	no	—
109	C₁₆×C₅₉	C₂×D₄	1	2	yes	—
110	C₁₆×C₅₉	C₂×D₄	1	2	yes	—
111	C₅₉⋊C₁₆	—	2	2	no	—
112	C₅₉⋊C₁₆	—	2	2	no	—
113	C₂³×C₅₉	C₂²×C₈	1	2	yes	—
114	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
115	C₂²×D₅₉	—	2	2	no	—
116	C₂²×D₅₉	—	4	2	no	—
117	C₂²×D₅₉	C₂²×C₈	1	2	yes	—
118	Q₈×C₅₉	—	2	2	no	—
119	D₄×C₅₉	—	2	4	no	—
120	D₄×C₅₉	—	2	2	no	—
121	C₂×C₄×C₅₉	—	2	2	no	—
122	C₂×C₄×C₅₉	—	2	2	no	—
123	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₅₉	#131	1	2	yes	—
127	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
128	C₂²⋊D₅₉	—	4	4	no	—
129	C₂×Dic₅₉	—	2	2	no	—
130	C₂×Dic₅₉	C₂²×C₈	1	4	yes	—
131	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
132	C₄×D₅₉	—	2	2	no	—
133	C₄×D₅₉	—	2	2	no	—
134	C₄×D₅₉	—	2	4	no	—
135	C₈×C₅₉	—	2	2	no	—
136	C₈×C₅₉	C₄×D₄	1	2	yes	—
137	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
138	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
139	C₅₉⋊C₈	—	2	2	no	—
140	C₅₉⋊C₈	—	2	2	no	—
141	#33433	—	59	1	no	maximal, Sylow
142	C₂²×C₅₉	—	2	2	no	—
143	C₂²×C₅₉	—	2	4	no	—
144	C₂²×C₅₉	—	2	2	no	—
145	C₂²×C₅₉	C₈×D₄	1	2	yes	—
146	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
147	C₂×D₅₉	—	2	2	no	—
148	C₂×D₅₉	—	2	2	no	—
149	C₂×D₅₉	—	2	2	no	—
150	C₂×D₅₉	—	4	4	no	—
151	C₄×C₅₉	—	2	2	no	—
152	C₄×C₅₉	C₈×D₄	1	2	yes	—
153	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
154	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
155	Dic₅₉	—	2	4	no	—
156	Dic₅₉	—	2	2	no	—
157	C₂×#504	D₅₉	1	1	yes	characteristic, complemented
158	C₂²×M₃₂	—	59	1	no	—
159	#2929	—	59	1	no	—
160	#2169	—	59	4	no	—
161	#2168	—	59	4	no	—
162	#2151	—	59	1	no	—
163	#2113	—	59	1	no	—
164	#2112	—	59	1	no	—
165	#2108	—	59	1	no	—
166	C₂×C₅₉	—	2	2	no	—
167	C₂×C₅₉	—	4	2	no	—
168	C₂×C₅₉	C₂×C₈×D₄	1	1	yes	characteristic
169	C₂×C₅₉	#3407	1	2	yes	—
170	D₅₉	—	4	2	no	—
171	C₂×#131	C₂×D₅₉	1	1	yes	characteristic
172	C₂³×C₈	—	59	1	no	—
173	#525	—	59	1	no	—
174	#504	—	2	4	no	—
175	C₂×M₃₂	—	59	1	no	—
176	C₂×M₃₂	—	59	2	no	—
177	C₂×M₃₂	—	59	2	no	—
178	C₂×M₃₂	—	118	4	no	—
179	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
180	C₂×M₃₂	C₂×D₅₉	1	2	yes	complemented
181	C₂²×C₁₆	—	59	1	no	—
182	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
183	C₂²×C₁₆	C₂×D₅₉	1	2	yes	complemented
184	#435	—	59	4	no	—
185	C₈×D₄	—	59	4	no	—
186	#424	—	59	1	no	—
187	#408	—	59	1	no	—
188	#407	—	59	1	no	—
189	#405	—	59	1	no	—
190	C₄⋊C₁₆	—	59	4	no	—
191	C₂²⋊C₁₆	—	59	4	no	—
192	C₂²⋊C₁₆	—	59	4	no	—
193	#346	—	59	2	no	—
194	C₄×C₁₆	—	59	2	no	—
195	C₅₉	#33433	1	1	yes	characteristic, complemented, Sylow
196	C₂×#41	C₄×D₅₉	1	1	yes	characteristic
197	C₂³×C₄	—	59	1	no	—
198	#131	—	2	2	no	—
199	#131	—	2	2	no	—
200	C₂×M₁₆	—	59	1	no	—
201	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
202	C₂×M₁₆	C₄×D₅₉	1	2	yes	—
203	C₂²×C₈	—	59	1	no	—
204	C₂²×C₈	—	59	2	no	—
205	C₂²×C₈	—	59	2	no	—
206	C₂²×C₈	—	118	2	no	—
207	C₂²×C₈	—	118	2	no	—
208	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
209	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
210	#123	—	59	2	no	—
211	C₂²⋊Q₈	—	59	2	no	—
212	#121	—	59	2	no	—
213	#120	—	59	2	no	—
214	C₄×D₄	—	59	4	no	—
215	#117	—	59	1	no	—
216	M₃₂	—	2	2	no	—
217	M₃₂	—	2	4	no	—
218	M₃₂	—	118	2	no	—
219	M₃₂	—	118	4	no	—
220	M₃₂	—	118	4	no	—
221	C₂×C₁₆	—	2	4	no	—
222	C₂×C₁₆	—	59	2	no	—
223	C₂×C₁₆	—	59	2	no	—
224	C₂×C₁₆	—	118	2	no	—
225	C₂×C₁₆	C₂²⋊D₅₉	1	4	yes	complemented
226	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
227	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
228	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
229	C₄⋊C₈	—	59	4	no	—
230	C₂²⋊C₈	—	59	4	no	—
231	C₂²⋊C₈	—	59	4	no	—
232	#97	—	59	2	no	—
233	C₄×C₈	—	59	2	no	—
234	C₂⁴	—	59	1	no	—
235	#41	—	2	2	no	—
236	#41	—	2	2	no	—
237	C₂×Q₈	C₈×D₅₉	1	1	yes	characteristic
238	C₂×D₄	—	59	4	no	—
239	C₂×D₄	C₈×D₅₉	1	2	yes	—
240	C₂×D₄	C₈×D₅₉	1	1	yes	characteristic
241	C₂²×C₄	—	59	1	no	—
242	C₂²×C₄	—	59	2	no	—
243	C₂²×C₄	—	59	2	no	—
244	C₂²×C₄	—	118	2	no	—
245	C₂²×C₄	—	118	2	no	—
246	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
247	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
248	M₁₆	—	2	2	no	—
249	M₁₆	—	2	4	no	—
250	M₁₆	—	118	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	59	2	no	—
254	C₂×C₈	—	59	2	no	—
255	C₂×C₈	—	118	2	no	—
256	C₂×C₈	—	118	4	no	—
257	C₂×C₈	—	118	2	no	—
258	C₂×C₈	—	118	2	no	—
259	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
260	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
261	C₂×C₈	#11745685	1	2	yes	—
262	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
263	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
264	C₄⋊C₄	—	59	2	no	—
265	C₄⋊C₄	—	59	4	no	—
266	C₂²⋊C₄	—	59	2	no	—
267	C₂²⋊C₄	—	59	4	no	—
268	C₂²⋊C₄	—	59	4	no	—
269	C₄²	—	59	2	no	—
270	C₁₆	—	2	2	no	—
271	C₁₆	—	118	2	no	—
272	C₁₆	—	118	2	no	—
273	C₁₆	C₂×C₂²⋊D₅₉	1	2	yes	—
274	C₁₆	C₂×C₂²⋊D₅₉	1	2	yes	—
275	C₂³	—	59	2	no	—
276	C₂³	—	118	2	no	—
277	C₂³	—	236	2	no	—
278	C₂³	C₂×C₈×D₅₉	1	2	yes	—
279	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
280	Q₈	—	2	2	no	—
281	D₄	—	2	4	no	—
282	D₄	—	2	2	no	—
283	D₄	—	236	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	59	4	no	—
287	C₂×C₄	—	59	2	no	—
288	C₂×C₄	—	118	2	no	—
289	C₂×C₄	—	118	2	no	—
290	C₂×C₄	—	118	4	no	—
291	C₂×C₄	—	118	2	no	—
292	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
293	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
294	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
295	C₂×C₄	#14273733	1	2	yes	—
296	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	118	2	no	—
299	C₈	—	118	2	no	—
300	C₈	C₄×C₂²⋊D₅₉	1	2	yes	—
301	C₈	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
302	C₈	C₂×#11745685	1	1	yes	characteristic
303	C₂²	—	2	2	no	—
304	C₂²	—	2	4	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	118	2	no	—
307	C₂²	—	118	2	no	—
308	C₂²	—	118	2	no	—
309	C₂²	—	236	4	no	—
310	C₂²	C₈×C₂²⋊D₅₉	1	2	yes	—
311	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	118	4	no	—
314	C₄	—	118	2	no	—
315	C₄	C₈×C₂²⋊D₅₉	1	2	yes	—
316	C₄	C₂×C₄×C₂²⋊D₅₉	1	1	yes	characteristic
317	C₄	C₂×#14273733	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	2	no	—
320	C₂	—	236	2	no	—
321	C₂	C₂×C₈×C₂²⋊D₅₉	1	1	yes	characteristic
322	C₂	#21437197	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₈
Fitting	C₂×C₅₉×#504	C₂
prime radical	#16837909	C₄
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×#504	D₅₉
hypocenter	C₅₉	#33433
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂²×#14273666	C₂	semidirect
3. #21425926	C₂	semidirect
4. #21425921	C₂	semidirect
5. #21425856	C₂	semidirect
6. #21425710	C₂	semidirect
7. #21425706	C₂	semidirect
8. #21425692	C₂	semidirect
9. C₂×C₅₉×#504	C₂	semidirect
23. Dic₅₉⋊C₁₆	C₂²	semidirect
24. C₂²⋊(C₅₉⋊C₁₆)	C₂²	semidirect
26. #16837177	C₂²	semidirect
27. C₁₆×Dic₅₉	C₂²	semidirect
31. C₂×M₃₂×C₅₉	C₂²	semidirect
33. C₂²×C₁₆×C₅₉	C₂²	semidirect
68. C₂×C₁₆×C₅₉	D₄	semidirect
157. C₂×#504	D₅₉	semidirect
180. C₂×M₃₂	C₂×D₅₉	semidirect
183. C₂²×C₁₆	C₂×D₅₉	semidirect
195. C₅₉	#33433	semidirect
225. C₂×C₁₆	C₂²⋊D₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14273666	2
#21432788	2
#21425926	2
#21425921	2
#21425856	2
#21425710	2
#21425706	2
#21425692	2
C₂×C₅₉×#504	2
#33433	59