Subgroups — #28744252

Order 15104.

558 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434675	C₂	1	1	yes	maximal, characteristic
2	#21426376	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426353	C₂	1	1	yes	maximal, characteristic, complemented
4	#21426323	C₂	1	1	yes	maximal, characteristic, complemented
5	#21426321	C₂	1	1	yes	maximal, characteristic, complemented
6	#21426194	C₂	1	4	yes	maximal, complemented
7	C₄×#14273670	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#3170	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837952	—	2	1	no	—
11	C₂×#14273670	C₂²	1	4	yes	—
12	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
13	#16837807	C₂²	1	1	yes	characteristic
14	#16837779	C₂²	1	1	yes	characteristic
15	#16837749	—	2	1	no	—
16	#16837638	—	2	1	no	—
17	#16837638	C₂²	1	2	yes	—
18	C₄×D₄₇₂	—	2	1	no	—
19	C₄×D₄₇₂	C₂²	1	2	yes	—
20	#16837229	C₂²	1	4	yes	—
21	#16837206	C₂²	1	4	yes	—
22	#16837202	C₂²	1	4	yes	—
23	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	—
24	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₅₉×#533	—	2	1	no	—
27	C₅₉×#500	C₂²	1	1	yes	characteristic
28	C₅₉×#494	C₂²	1	1	yes	characteristic
29	C₅₉×#487	—	2	1	no	—
30	C₅₉×#467	—	2	1	no	—
31	C₅₉×#467	C₂²	1	2	yes	—
32	C₄×Q₁₆×C₅₉	—	2	1	no	—
33	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
35	C₂×#11745685	—	4	1	no	—
36	C₂×D₄₇₂	—	2	1	no	—
37	C₂×D₄₇₂	C₂³	1	2	yes	—
38	#14273727	—	2	1	no	—
39	#14273724	—	2	1	no	—
40	#14273715	—	4	1	no	—
41	#14273701	—	2	2	no	—
42	#14273701	—	4	1	no	—
43	C₄×D₂₃₆	—	2	2	no	—
44	C₄×D₂₃₆	—	4	1	no	—
45	#14273670	—	2	8	no	—
46	C₂×C₅₉⋊C₁₆	—	2	4	no	—
47	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
48	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
49	#14273661	—	2	1	no	—
50	#14273661	C₂³	1	2	yes	—
51	#14273649	—	2	2	no	—
52	#14273649	—	2	2	no	—
53	C₂×Q₁₆×C₅₉	—	2	1	no	—
54	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
55	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈×C₅₉	—	2	1	no	—
58	C₄⋊Q₈×C₅₉	—	2	1	no	—
59	#125×C₅₉	—	4	1	no	—
60	C₂²⋊Q₈×C₅₉	—	2	2	no	—
61	C₂²⋊Q₈×C₅₉	—	4	1	no	—
62	C₄×Q₈×C₅₉	—	2	2	no	—
63	C₄×Q₈×C₅₉	—	4	1	no	—
64	#117×C₅₉	—	4	1	no	—
65	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
66	#107×C₅₉	—	2	1	no	—
67	#107×C₅₉	C₂³	1	2	yes	—
68	#103×C₅₉	—	2	2	no	—
69	#103×C₅₉	—	2	2	no	—
70	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
71	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
72	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
74	C₂×Q₈×C₅₉	—	2	2	no	—
75	C₂×Q₈×C₅₉	—	4	1	no	—
76	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₂²⋊D₅₉	—	4	2	no	—
80	#11745685	—	8	4	no	—
81	C₂×D₂₃₆	—	2	2	no	—
82	C₂×D₂₃₆	—	4	1	no	—
83	C₂×D₂₃₆	—	4	1	no	—
84	C₂×C₄×D₅₉	—	4	2	no	—
85	C₂×Dic₁₁₈	—	4	1	no	—
86	Q₁₆×C₅₉	—	2	4	no	—
87	Q₁₆×C₅₉	—	4	2	no	—
88	C₂×C₈×C₅₉	—	2	2	no	—
89	C₂×C₈×C₅₉	—	2	2	no	—
90	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₅₉	—	2	2	no	—
95	C₄⋊C₄×C₅₉	—	4	1	no	—
96	C₄⋊C₄×C₅₉	—	4	1	no	—
97	C₄⋊C₄×C₅₉	—	4	2	no	—
98	C₄⋊C₄×C₅₉	—	4	2	no	—
99	C₄⋊C₄×C₅₉	—	4	1	no	—
100	C₂²⋊C₄×C₅₉	—	4	2	no	—
101	C₄²×C₅₉	—	4	2	no	—
102	C₄²×C₅₉	D₈	1	1	yes	characteristic
103	C₄²×C₅₉	D₈	1	1	yes	characteristic
104	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
106	(C₂×D₅₉)⋊C₄	—	4	2	no	—
107	(C₂×D₅₉)⋊C₄	—	4	2	no	—
108	C₄⋊Dic₅₉	—	2	2	no	—
109	C₄⋊Dic₅₉	—	4	1	no	—
110	D₄₇₂	—	2	4	no	—
111	D₄₇₂	—	4	2	no	—
112	C₅₉⋊C₁₆	—	2	2	no	—
113	C₅₉⋊C₁₆	#41	1	4	yes	complemented
114	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₅₉	—	4	2	no	—
116	Q₈×C₅₉	—	4	2	no	—
117	Q₈×C₅₉	—	4	2	no	—
118	Q₈×C₅₉	—	8	2	no	—
119	C₂×C₄×C₅₉	—	2	1	no	—
120	C₂×C₄×C₅₉	—	2	1	no	—
121	C₂×C₄×C₅₉	—	2	2	no	—
122	C₂×C₄×C₅₉	—	4	2	no	—
123	C₂×C₄×C₅₉	—	4	2	no	—
124	C₂×C₄×C₅₉	SD₃₂	1	4	yes	complemented
125	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
131	C₂²⋊D₅₉	—	8	4	no	—
132	C₂×Dic₅₉	—	4	2	no	—
133	D₂₃₆	—	4	2	no	—
134	D₂₃₆	—	4	2	no	—
135	D₂₃₆	—	8	2	no	—
136	D₂₃₆	—	8	2	no	—
137	C₄×D₅₉	—	8	4	no	—
138	Dic₁₁₈	—	8	2	no	—
139	C₈×C₅₉	—	2	1	no	—
140	C₈×C₅₉	—	2	1	no	—
141	C₈×C₅₉	#135	1	2	yes	—
142	C₈×C₅₉	C₂×#41	1	2	yes	—
143	#33736	—	59	1	no	maximal, Sylow
144	C₂²×C₅₉	—	2	1	no	—
145	C₂²×C₅₉	—	2	1	no	—
146	C₂²×C₅₉	C₂×SD₃₂	1	2	yes	—
147	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₅₉	—	8	2	no	—
149	C₂×D₅₉	—	8	2	no	—
150	C₂×D₅₉	—	8	2	no	—
151	C₄×C₅₉	—	2	1	no	—
152	C₄×C₅₉	—	2	1	no	—
153	C₄×C₅₉	—	8	2	no	—
154	C₄×C₅₉	—	8	2	no	—
155	C₄×C₅₉	C₂×SD₃₂	1	2	yes	—
156	C₄×C₅₉	C₂×SD₃₂	1	2	yes	—
157	C₄×C₅₉	#508	1	2	yes	—
158	C₄×C₅₉	#532	1	1	yes	characteristic
159	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
160	Dic₅₉	—	8	2	no	—
161	#3170	D₅₉	1	1	yes	characteristic, complemented
162	#3169	—	59	1	no	—
163	#2254	—	59	1	no	—
164	#2248	—	59	1	no	—
165	#2241	—	59	1	no	—
166	#2240	—	59	1	no	—
167	#2217	—	59	4	no	—
168	C₄×SD₃₂	—	59	4	no	—
169	C₂×C₄×C₁₆	—	59	1	no	—
170	C₂×C₅₉	—	2	2	no	—
171	C₂×C₅₉	#3169	1	1	yes	characteristic
172	C₂×C₅₉	C₂²×SD₃₂	1	1	yes	characteristic
173	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
174	D₅₉	—	16	2	no	—
175	#533	—	2	1	no	—
176	#532	—	118	1	no	—
177	C₂×SD₃₂	—	59	4	no	—
178	C₂²×C₁₆	—	59	2	no	—
179	#500	—	59	1	no	—
180	#500	C₂×D₅₉	1	1	yes	characteristic
181	#494	C₂×D₅₉	1	1	yes	characteristic
182	#493	—	59	1	no	—
183	#487	—	2	1	no	—
184	#486	—	118	1	no	—
185	#467	—	2	1	no	—
186	#467	C₂×D₅₉	1	2	yes	—
187	#466	—	59	2	no	—
188	#466	—	118	1	no	—
189	C₄×Q₁₆	—	2	1	no	—
190	C₄×Q₁₆	C₂×D₅₉	1	2	yes	—
191	C₄×D₈	—	59	2	no	—
192	C₄×D₈	—	118	1	no	—
193	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
194	#367	—	59	4	no	—
195	#358	—	59	4	no	—
196	#357	—	59	4	no	—
197	C₄×C₁₆	—	59	2	no	—
198	C₄×C₁₆	—	59	1	no	—
199	C₄×C₁₆	—	59	1	no	—
200	C₅₉	#33736	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	236	1	no	—
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	C₂²×D₅₉	1	2	yes	—
204	C₂×D₈	—	59	2	no	—
205	C₂×D₈	—	118	1	no	—
206	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
207	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	118	1	no	—
211	#127	—	118	1	no	—
212	#125	—	4	1	no	—
213	#124	—	236	1	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	4	1	no	—
216	#121	—	118	2	no	—
217	#121	—	236	1	no	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	4	1	no	—
220	C₄×D₄	—	118	2	no	—
221	C₄×D₄	—	236	1	no	—
222	#117	—	4	1	no	—
223	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
224	SD₃₂	—	118	8	no	—
225	C₂×C₁₆	—	59	2	no	—
226	C₂×C₁₆	—	59	2	no	—
227	C₂×C₁₆	—	118	4	no	—
228	#107	—	2	1	no	—
229	#107	—	59	2	no	—
230	#107	—	118	1	no	—
231	#107	C₂²×D₅₉	1	2	yes	—
232	#103	—	2	2	no	—
233	#103	—	2	2	no	—
234	#102	—	118	2	no	—
235	#102	—	118	2	no	—
236	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
237	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
240	#41	—	472	4	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	236	1	no	—
244	C₂×D₄	—	118	2	no	—
245	C₂×D₄	—	236	2	no	—
246	C₂×D₄	—	236	1	no	—
247	C₂×D₄	—	236	1	no	—
248	C₂²×C₄	—	236	2	no	—
249	C₂²×C₄	#11745669	1	1	yes	characteristic
250	C₂²×C₄	#11745669	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
252	Q₁₆	—	2	4	no	—
253	Q₁₆	—	4	2	no	—
254	D₈	—	118	4	no	—
255	D₈	—	236	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
259	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
261	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	118	2	no	—
269	C₄⋊C₄	—	236	1	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	236	2	no	—
272	C₂²⋊C₄	—	236	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	#11745669	1	1	yes	characteristic
275	C₄²	#11745669	1	1	yes	characteristic
276	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
277	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
278	C₁₆	—	59	4	no	—
279	C₁₆	—	118	2	no	—
280	C₂³	—	236	2	no	—
281	C₂³	C₂×#11745669	1	1	yes	characteristic
282	Q₈	—	4	2	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	8	2	no	—
285	Q₈	—	472	2	no	—
286	D₄	—	472	4	no	—
287	D₄	—	472	2	no	—
288	D₄	—	472	2	no	—
289	D₄	—	236	2	no	—
290	D₄	—	236	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	472	4	no	—
297	C₂×C₄	—	236	2	no	—
298	C₂×C₄	#14273670	1	4	yes	complemented
299	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
300	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
301	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
302	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
303	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	#14273747	1	2	yes	—
308	C₈	#41×D₅₉	1	2	yes	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	472	2	no	—
312	C₂²	—	472	2	no	—
313	C₂²	—	472	2	no	—
314	C₂²	C₂×#14273670	1	2	yes	—
315	C₂²	C₂²×#11745669	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	472	2	no	—
321	C₄	C₂×#14273670	1	2	yes	—
322	C₄	C₂×#14273670	1	2	yes	—
323	C₄	#16837832	1	2	yes	—
324	C₄	#16837948	1	1	yes	characteristic
325	C₄	C₂×#14273747	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	944	2	no	—
328	C₂	#21434671	1	1	yes	characteristic
329	C₂	C₂²×#14273670	1	1	yes	characteristic
330	C₂	C₂×#16837832	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#3170	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	#21434675	C₂
hypercenter	#3170	D₅₉
hypocenter	C₅₉	#33736
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #21426376	C₂	semidirect
3. #21426353	C₂	semidirect
4. #21426323	C₂	semidirect
5. #21426321	C₂	semidirect
6. #21426194	C₂	semidirect
7. C₄×#14273670	C₂	semidirect
8. C₂×C₄×C₅₉⋊C₁₆	C₂	semidirect
9. C₅₉×#3170	C₂	semidirect
113. C₅₉⋊C₁₆	#41	semidirect
124. C₂×C₄×C₅₉	SD₃₂	semidirect
161. #3170	D₅₉	semidirect
200. C₅₉	#33736	semidirect
298. C₂×C₄	#14273670	semidirect

Maximal subgroups

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

subgroup	index
#21434675	2
#21426376	2
#21426353	2
#21426323	2
#21426321	2
#21426194	2
C₄×#14273670	2
C₂×C₄×C₅₉⋊C₁₆	2
C₅₉×#3170	2
#33736	59