Subgroups — #28505638

Order 15104.

574 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273661	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837206	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#21422187	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₉×#467	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₅₉	—	2	1	no	—
7	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
8	C₂×#14273661	—	2	4	no	—
9	C₂×#14273661	—	2	1	no	—
10	C₂×#14273661	C₂²	1	2	yes	—
11	#16837206	—	2	4	no	—
12	C₂²⋊(C₅₉⋊C₁₆)	—	2	2	no	—
13	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₅₉	—	2	1	no	—
16	C₅₉×#467	—	2	2	no	—
17	C₅₉×#467	C₄	1	4	yes	complemented
18	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×#103×C₅₉	—	2	1	no	—
20	C₂³×Dic₅₉	—	4	1	no	—
21	C₂×C₄⋊Dic₅₉	—	2	4	no	—
22	C₂×C₄⋊Dic₅₉	—	2	2	no	—
23	C₂×C₄⋊Dic₅₉	—	4	2	no	—
24	C₂×C₄⋊Dic₅₉	—	4	1	no	—
25	C₂×C₅₉⋊C₁₆	—	2	2	no	—
26	C₂×C₅₉⋊C₁₆	—	2	2	no	—
27	C₂×C₅₉⋊C₁₆	—	2	2	no	—
28	#14273661	—	2	4	no	—
29	#14273661	—	2	2	no	—
30	#14273661	D₄	1	4	yes	—
31	C₂²×Q₈×C₅₉	—	2	1	no	—
32	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
33	C₂×Q₁₆×C₅₉	—	2	2	no	—
34	C₂×Q₁₆×C₅₉	—	2	1	no	—
35	C₂×Q₁₆×C₅₉	—	2	2	no	—
36	C₂×Q₁₆×C₅₉	C₂×C₄	1	2	yes	complemented
37	C₂²×C₈×C₅₉	—	2	2	no	—
38	C₂²×C₈×C₅₉	—	2	2	no	—
39	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₉	D₄	1	2	yes	—
42	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
43	C₂²⋊Q₈×C₅₉	—	2	4	no	—
44	C₂²⋊Q₈×C₅₉	—	4	2	no	—
45	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
47	C₂×C₂²⋊C₄×C₅₉	—	4	1	no	—
48	#107×C₅₉	—	2	1	no	—
49	#107×C₅₉	C₂×C₄	1	2	yes	complemented
50	#103×C₅₉	—	2	4	no	—
51	C₂⁴×C₅₉	D₈	1	1	yes	characteristic
52	C₂×Q₈×C₅₉	—	2	2	no	—
53	C₂×Q₈×C₅₉	—	4	1	no	—
54	C₂×Q₈×C₅₉	—	4	1	no	—
55	C₂×Q₈×C₅₉	—	4	1	no	—
56	C₂×Q₈×C₅₉	—	4	1	no	—
57	C₂×Q₈×C₅₉	—	4	1	no	—
58	C₂²×C₄×C₅₉	—	2	1	no	—
59	C₂²×C₄×C₅₉	—	2	1	no	—
60	C₂²×C₄×C₅₉	—	2	1	no	—
61	C₂²×C₄×C₅₉	—	2	1	no	—
62	C₂²×C₄×C₅₉	—	4	1	no	—
63	C₂²×C₄×C₅₉	—	4	1	no	—
64	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₉	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₅₉	—	4	4	no	—
69	C₂²×Dic₅₉	—	4	2	no	—
70	C₂²×Dic₅₉	—	8	2	no	—
71	C₂²×Dic₅₉	—	8	2	no	—
72	Q₁₆×C₅₉	—	2	4	no	—
73	Q₁₆×C₅₉	—	4	2	no	—
74	C₂×C₈×C₅₉	—	2	1	no	—
75	C₂×C₈×C₅₉	—	2	1	no	—
76	C₂×C₈×C₅₉	—	2	4	no	—
77	C₂×C₈×C₅₉	—	2	4	no	—
78	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₅₉	#41	1	2	yes	—
83	C₄⋊C₄×C₅₉	—	2	2	no	—
84	C₄⋊C₄×C₅₉	—	4	4	no	—
85	C₄⋊C₄×C₅₉	—	4	1	no	—
86	C₂²⋊C₄×C₅₉	—	4	4	no	—
87	C₄⋊Dic₅₉	—	2	4	no	—
88	C₄⋊Dic₅₉	—	4	4	no	—
89	C₄⋊Dic₅₉	—	4	2	no	—
90	C₅₉⋊C₁₆	—	4	2	no	—
91	C₂³×C₅₉	—	2	1	no	—
92	C₂³×C₅₉	—	2	1	no	—
93	C₂³×C₅₉	—	2	1	no	—
94	C₂³×C₅₉	—	2	1	no	—
95	C₂³×C₅₉	#102	1	2	yes	—
96	C₂³×C₅₉	SD₃₂	1	2	yes	—
97	C₂³×C₅₉	Q₃₂	1	2	yes	—
98	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
99	Q₈×C₅₉	—	4	2	no	—
100	Q₈×C₅₉	—	4	2	no	—
101	Q₈×C₅₉	—	8	1	no	—
102	Q₈×C₅₉	—	8	1	no	—
103	C₂×C₄×C₅₉	—	2	1	no	—
104	C₂×C₄×C₅₉	—	2	1	no	—
105	C₂×C₄×C₅₉	—	2	2	no	—
106	C₂×C₄×C₅₉	—	2	2	no	—
107	C₂×C₄×C₅₉	—	2	2	no	—
108	C₂×C₄×C₅₉	—	2	2	no	—
109	C₂×C₄×C₅₉	—	4	2	no	—
110	C₂×C₄×C₅₉	—	4	2	no	—
111	C₂×C₄×C₅₉	—	8	1	no	—
112	C₂×C₄×C₅₉	—	8	1	no	—
113	C₂×C₄×C₅₉	—	8	1	no	—
114	C₂×C₄×C₅₉	—	8	1	no	—
115	C₂×C₄×C₅₉	#102	1	2	yes	—
116	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
118	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
122	C₂×Dic₅₉	—	4	4	no	—
123	C₂×Dic₅₉	—	8	8	no	—
124	C₂×Dic₅₉	—	8	2	no	—
125	C₂×Dic₅₉	—	8	2	no	—
126	C₈×C₅₉	—	2	2	no	—
127	C₈×C₅₉	C₄×D₄	1	2	yes	—
128	C₈×C₅₉	#121	1	2	yes	—
129	#12993	—	59	1	no	maximal, Sylow
130	C₂²×C₅₉	—	2	2	no	—
131	C₂²×C₅₉	—	2	2	no	—
132	C₂²×C₅₉	—	2	2	no	—
133	C₂²×C₅₉	—	2	2	no	—
134	C₂²×C₅₉	—	2	2	no	—
135	C₂²×C₅₉	—	2	2	no	—
136	C₂²×C₅₉	#358	1	4	yes	complemented
137	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
138	C₂²×C₅₉	#447	1	1	yes	characteristic
139	C₂²×C₅₉	#480	1	1	yes	characteristic
140	C₂²×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
141	C₂²×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
142	C₂²×C₅₉	#509	1	1	yes	characteristic
143	C₂²×C₅₉	#510	1	1	yes	characteristic
144	C₄×C₅₉	—	2	2	no	—
145	C₄×C₅₉	—	8	2	no	—
146	C₄×C₅₉	—	8	2	no	—
147	C₄×C₅₉	#386	1	1	yes	characteristic
148	C₄×C₅₉	#417	1	1	yes	characteristic
149	C₄×C₅₉	#448	1	1	yes	characteristic
150	C₄×C₅₉	#481	1	1	yes	characteristic
151	Dic₅₉	—	8	4	no	—
152	C₂×#467	D₅₉	1	1	yes	characteristic
153	C₂²×#107	—	59	1	no	—
154	C₂×#358	—	59	2	no	—
155	C₂×C₂²⋊C₁₆	—	59	1	no	—
156	#1380	—	59	2	no	—
157	C₂×C₅₉	—	2	4	no	—
158	C₂×C₅₉	#1874	1	1	yes	characteristic
159	C₂×C₅₉	C₂×#358	1	1	yes	characteristic
160	C₂×C₅₉	#2139	1	1	yes	characteristic
161	C₂×C₅₉	#2187	1	1	yes	characteristic
162	C₂×C₅₉	#2188	1	1	yes	characteristic
163	C₂×C₅₉	#2235	1	1	yes	characteristic
164	C₂×C₅₉	#2236	1	1	yes	characteristic
165	C₂²×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
166	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
167	C₂×C₂²⋊Q₈	—	2	1	no	—
168	C₂²×C₄⋊C₄	—	118	1	no	—
169	C₂²×C₁₆	—	59	2	no	—
170	#467	—	2	2	no	—
171	#467	Dic₅₉	1	4	yes	complemented
172	C₂×#107	—	59	2	no	—
173	C₂×#107	—	118	4	no	—
174	C₂×#107	—	118	1	no	—
175	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
176	C₂×#103	—	2	1	no	—
177	#358	—	118	4	no	—
178	C₂²⋊C₁₆	—	118	2	no	—
179	C₅₉	#12993	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	—	2	1	no	—
181	C₂³×C₄	—	236	1	no	—
182	C₂³×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	2	2	no	—
185	C₂×Q₁₆	—	2	1	no	—
186	C₂×Q₁₆	C₂×Dic₅₉	1	2	yes	complemented
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
190	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
191	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
192	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
193	C₂²⋊Q₈	—	2	4	no	—
194	C₂²⋊Q₈	—	4	2	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	4	1	no	—
197	C₂×C₄⋊C₄	—	118	4	no	—
198	C₂×C₄⋊C₄	—	118	2	no	—
199	C₂×C₄⋊C₄	—	236	2	no	—
200	C₂×C₄⋊C₄	—	236	1	no	—
201	C₂×C₂²⋊C₄	—	4	1	no	—
202	C₂×C₁₆	—	118	2	no	—
203	C₂×C₁₆	—	118	2	no	—
204	C₂×C₁₆	—	118	2	no	—
205	#107	—	2	1	no	—
206	#107	—	59	4	no	—
207	#107	—	118	4	no	—
208	#107	—	118	2	no	—
209	#107	C₂×Dic₅₉	1	2	yes	complemented
210	#103	—	2	4	no	—
211	C₂⁴	#11745669	1	1	yes	characteristic
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	4	1	no	—
214	C₂×Q₈	—	4	1	no	—
215	C₂×Q₈	—	4	1	no	—
216	C₂×Q₈	—	4	1	no	—
217	C₂×Q₈	—	4	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	472	2	no	—
225	C₂²×C₄	—	472	2	no	—
226	C₂²×C₄	—	236	4	no	—
227	C₂²×C₄	—	236	2	no	—
228	C₂²×C₄	#11745669	1	1	yes	characteristic
229	C₂²×C₄	#11745670	1	2	yes	—
230	C₂²×C₄	C₂²⋊Dic₅₉	1	2	yes	—
231	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
232	Q₁₆	—	2	4	no	—
233	Q₁₆	—	4	2	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
239	C₂×C₈	D₄×D₅₉	1	2	yes	—
240	C₂×C₈	#11745687	1	2	yes	—
241	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
242	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₄⋊C₄	—	4	1	no	—
246	C₄⋊C₄	—	118	4	no	—
247	C₄⋊C₄	—	236	4	no	—
248	C₄⋊C₄	—	236	2	no	—
249	C₂²⋊C₄	—	4	4	no	—
250	C₁₆	—	236	2	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	#14273648	1	2	yes	—
256	C₂³	#14273670	1	2	yes	—
257	C₂³	#14273673	1	2	yes	—
258	C₂³	C₂×#11745669	1	1	yes	characteristic
259	Q₈	—	4	2	no	—
260	Q₈	—	4	2	no	—
261	Q₈	—	8	1	no	—
262	Q₈	—	8	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	472	8	no	—
276	C₂×C₄	—	472	2	no	—
277	C₂×C₄	—	472	2	no	—
278	C₂×C₄	—	236	4	no	—
279	C₂×C₄	#14273648	1	2	yes	—
280	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
281	C₂×C₄	#14273695	1	1	yes	characteristic
282	C₂×C₄	#14273708	1	1	yes	characteristic
283	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
284	C₂×C₄	#14273752	1	1	yes	characteristic
285	C₂×C₄	#14273757	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	D₄×Dic₅₉	1	2	yes	—
288	C₈	#14273699	1	2	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	#16837206	1	4	yes	complemented
296	C₂²	C₂×#14273648	1	1	yes	characteristic
297	C₂²	#16837535	1	1	yes	characteristic
298	C₂²	#16837713	1	1	yes	characteristic
299	C₂²	C₂×#14273670	1	1	yes	characteristic
300	C₂²	C₂×#14273673	1	1	yes	characteristic
301	C₂²	#16837837	1	1	yes	characteristic
302	C₂²	#16837842	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	2	no	—
306	C₄	—	472	4	no	—
307	C₄	#16837282	1	1	yes	characteristic
308	C₄	#16837403	1	1	yes	characteristic
309	C₄	#16837540	1	1	yes	characteristic
310	C₄	#16837718	1	1	yes	characteristic
311	C₂	—	2	4	no	—
312	C₂	#21424414	1	1	yes	characteristic
313	C₂	C₂×#16837206	1	1	yes	characteristic
314	C₂	#21425808	1	1	yes	characteristic
315	C₂	#21426024	1	1	yes	characteristic
316	C₂	#21426029	1	1	yes	characteristic
317	C₂	#21426291	1	1	yes	characteristic
318	C₂	#21426296	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#467	C₂
socle	C₂³×C₅₉	C₂×D₈
prime radical	C₂⁴×C₅₉	D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#467	D₅₉
hypocenter	C₅₉	#12993
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
2. C₂×#16837206	C₂	semidirect
4. #21422187	C₂	semidirect
17. C₅₉×#467	C₄	semidirect
36. C₂×Q₁₆×C₅₉	C₂×C₄	semidirect
49. #107×C₅₉	C₂×C₄	semidirect
136. C₂²×C₅₉	#358	semidirect
171. #467	Dic₅₉	semidirect
179. C₅₉	#12993	semidirect
186. C₂×Q₁₆	C₂×Dic₅₉	semidirect
209. #107	C₂×Dic₅₉	semidirect
295. C₂²	#16837206	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14273661	2
C₂×#16837206	2
C₂×C₂²⋊(C₅₉⋊C₁₆)	2
#21422187	2
C₂×C₅₉×#467	2
#12993	59