Subgroups — #28505597

Order 15104.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21437198	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16837525	C₂	1	1	yes	maximal, characteristic
3	#21425725	C₂	1	1	yes	maximal, characteristic, complemented
4	#21425723	C₂	1	1	yes	maximal, characteristic, complemented
5	#21425716	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422176	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1377	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273749	C₄	1	1	yes	characteristic
9	C₂×#14273733	C₂²	1	1	yes	characteristic
10	C₂×#14273733	C₂²	1	1	yes	characteristic
11	#16837820	—	2	2	no	—
12	#16837818	—	2	2	no	—
13	C₂×#14273667	C₂²	1	1	yes	characteristic
14	C₂×#14273667	C₂²	1	1	yes	characteristic
15	M₃₂×D₅₉	—	2	2	no	—
16	#16837525	—	2	2	no	—
17	#16837525	—	2	2	no	—
18	C₂×#14273663	C₄	1	1	yes	characteristic
19	C₂×#14273655	C₄	1	1	yes	characteristic
20	C₂×#14273655	C₄	1	1	yes	characteristic
21	#16837187	—	2	2	no	—
22	#16837185	—	2	2	no	—
23	#16837183	—	2	2	no	—
24	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
28	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
29	#14273749	—	2	2	no	—
30	#14273749	—	2	2	no	—
31	C₂×Dic₂₃₆	C₈	1	1	yes	characteristic
32	C₂×#11745660	C₈	1	1	yes	characteristic
33	C₂×#11745660	C₈	1	1	yes	characteristic
34	C₂×D₄₇₂	C₈	1	1	yes	characteristic
35	#14273733	—	2	1	no	—
36	#14273733	—	2	1	no	—
37	#14273733	—	2	1	no	—
38	#14273733	—	2	1	no	—
39	#14273733	—	2	1	no	—
40	#14273733	—	2	1	no	—
41	#14273733	D₄	1	2	yes	—
42	#14273733	D₄	1	2	yes	—
43	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
44	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
45	C₂×#11745659	—	2	1	no	—
46	C₂×#11745659	—	2	1	no	—
47	C₂×C₈×D₅₉	—	2	1	no	—
48	C₂×C₈×D₅₉	—	2	1	no	—
49	#14273667	—	2	1	no	—
50	#14273667	—	2	1	no	—
51	#14273667	—	2	1	no	—
52	#14273667	—	2	1	no	—
53	#14273666	—	2	4	no	—
54	C₂×C₅₉⋊C₁₆	—	2	1	no	—
55	C₂×C₅₉⋊C₁₆	D₄	1	2	yes	—
56	C₁₆×D₅₉	—	2	4	no	—
57	#14273663	—	2	2	no	—
58	#14273655	—	2	2	no	—
59	#14273655	—	2	2	no	—
60	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
63	M₃₂×C₅₉	—	2	1	no	—
64	M₃₂×C₅₉	—	2	1	no	—
65	M₃₂×C₅₉	—	2	1	no	—
66	M₃₂×C₅₉	D₄	1	2	yes	—
67	C₂×C₁₆×C₅₉	—	2	1	no	—
68	C₂×C₁₆×C₅₉	—	2	1	no	—
69	#97×C₅₉	—	2	1	no	—
70	#97×C₅₉	—	2	1	no	—
71	C₂²×C₄×C₅₉	—	2	1	no	—
72	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₂²⋊D₅₉	—	2	1	no	—
74	C₂×C₂²⋊D₅₉	—	2	1	no	—
75	#11745685	—	2	1	no	—
76	#11745685	—	2	1	no	—
77	#11745685	—	2	1	no	—
78	#11745685	—	2	1	no	—
79	#11745685	—	2	1	no	—
80	#11745685	—	2	1	no	—
81	#11745685	C₂²⋊C₄	1	2	yes	—
82	#11745685	M₁₆	1	2	yes	—
83	C₂×D₂₃₆	C₂×C₈	1	1	yes	characteristic
84	C₂×D₂₃₆	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₅₉	—	2	1	no	—
86	C₂×C₄×D₅₉	—	2	1	no	—
87	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
88	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
89	C₂×C₈×C₅₉	—	2	1	no	—
90	C₂×C₈×C₅₉	—	2	1	no	—
91	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
92	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
93	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
98	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
99	C₄²×C₅₉	—	2	1	no	—
100	C₄²×C₅₉	—	2	1	no	—
101	#11745664	—	2	1	no	—
102	#11745664	—	2	1	no	—
103	#11745664	—	2	1	no	—
104	#11745664	C₂²⋊C₄	1	2	yes	—
105	C₂×C₅₉⋊C₈	—	2	1	no	—
106	C₂×C₅₉⋊C₈	—	2	1	no	—
107	Dic₂₃₆	—	2	2	no	—
108	D₄₇₂	—	2	2	no	—
109	#11745660	—	2	2	no	—
110	#11745660	—	2	2	no	—
111	#11745659	—	2	2	no	—
112	#11745659	—	2	2	no	—
113	#11745659	—	4	1	no	—
114	#11745659	—	4	1	no	—
115	C₈×D₅₉	—	2	2	no	—
116	C₈×D₅₉	—	2	2	no	—
117	C₈×D₅₉	—	4	1	no	—
118	C₈×D₅₉	—	4	1	no	—
119	C₁₆×C₅₉	—	2	2	no	—
120	C₁₆×C₅₉	—	4	1	no	—
121	C₅₉⋊C₁₆	—	2	2	no	—
122	C₅₉⋊C₁₆	—	4	1	no	—
123	C₂³×C₅₉	#131	1	1	yes	characteristic
124	C₂²×D₅₉	—	2	1	no	—
125	C₂²×D₅₉	—	2	1	no	—
126	C₂×C₄×C₅₉	—	4	1	no	—
127	C₂×C₄×C₅₉	—	8	1	no	—
128	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
133	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
134	C₂²⋊D₅₉	—	2	2	no	—
135	C₂²⋊D₅₉	—	2	2	no	—
136	C₂²⋊D₅₉	—	4	1	no	—
137	C₂²⋊D₅₉	—	4	1	no	—
138	C₂×Dic₅₉	—	2	1	no	—
139	C₂×Dic₅₉	—	2	1	no	—
140	D₂₃₆	—	2	1	no	—
141	D₂₃₆	—	2	1	no	—
142	D₂₃₆	—	2	1	no	—
143	D₂₃₆	C₂²⋊C₈	1	2	yes	—
144	C₄×D₅₉	—	2	2	no	—
145	C₄×D₅₉	—	2	2	no	—
146	C₄×D₅₉	—	4	1	no	—
147	C₄×D₅₉	—	4	1	no	—
148	Dic₁₁₈	—	2	1	no	—
149	Dic₁₁₈	—	2	1	no	—
150	Dic₁₁₈	—	2	1	no	—
151	Dic₁₁₈	C₂²⋊C₈	1	2	yes	—
152	C₈×C₅₉	—	2	1	no	—
153	C₈×C₅₉	—	2	1	no	—
154	C₈×C₅₉	#120	1	1	yes	characteristic
155	C₈×C₅₉	#121	1	1	yes	characteristic
156	C₈×C₅₉	#121	1	1	yes	characteristic
157	C₈×C₅₉	#123	1	1	yes	characteristic
158	C₅₉⋊C₈	—	2	2	no	—
159	C₅₉⋊C₈	—	4	1	no	—
160	#12986	—	59	1	no	maximal, Sylow
161	C₂²×C₅₉	—	4	1	no	—
162	C₂²×C₅₉	#408	1	1	yes	characteristic
163	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
164	C₂²×C₅₉	#435	1	1	yes	characteristic
165	C₂×D₅₉	—	2	2	no	—
166	C₂×D₅₉	—	4	1	no	—
167	C₂×D₅₉	—	8	1	no	—
168	C₂×D₅₉	—	8	1	no	—
169	C₄×C₅₉	—	8	1	no	—
170	C₄×C₅₉	#386	1	1	yes	characteristic
171	C₄×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
172	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
173	C₄×C₅₉	#435	1	1	yes	characteristic
174	Dic₅₉	—	2	2	no	—
175	Dic₅₉	—	4	1	no	—
176	#3407	—	59	1	no	—
177	C₂×#444	—	59	1	no	—
178	#2117	—	59	1	no	—
179	#2117	—	59	1	no	—
180	#2115	—	59	1	no	—
181	#1377	D₅₉	1	1	yes	characteristic, complemented
182	#1375	—	59	1	no	—
183	C₂×C₅₉	—	2	1	no	—
184	C₂×C₅₉	—	2	1	no	—
185	C₂×C₅₉	—	2	1	no	—
186	C₂×C₅₉	#1869	1	1	yes	characteristic
187	D₅₉	—	4	2	no	—
188	D₅₉	—	8	1	no	—
189	C₂×#135	—	59	1	no	—
190	C₂×#131	—	59	1	no	—
191	C₂×#131	—	59	1	no	—
192	#504	—	118	2	no	—
193	#504	—	118	2	no	—
194	C₂×M₃₂	—	59	1	no	—
195	C₂×M₃₂	—	59	1	no	—
196	C₂×M₃₂	—	118	2	no	—
197	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
198	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
199	#444	—	118	2	no	—
200	#444	—	118	2	no	—
201	C₂×#108	—	59	1	no	—
202	C₂×#104	—	59	1	no	—
203	C₂×#104	—	59	1	no	—
204	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
205	#350	—	118	2	no	—
206	#350	—	118	2	no	—
207	#349	—	118	2	no	—
208	C₅₉	#12986	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	59	1	no	—
210	C₂×#41	—	59	1	no	—
211	#135	—	118	2	no	—
212	#135	—	118	2	no	—
213	C₂×Q₁₆	—	59	1	no	—
214	C₂×SD₁₆	—	59	1	no	—
215	C₂×SD₁₆	—	59	1	no	—
216	C₂×D₈	—	59	1	no	—
217	#131	—	59	2	no	—
218	#131	—	59	2	no	—
219	#131	—	118	1	no	—
220	#131	—	118	1	no	—
221	#131	—	118	1	no	—
222	#131	—	118	1	no	—
223	#131	—	118	1	no	—
224	#131	—	118	1	no	—
225	C₂×M₁₆	—	59	1	no	—
226	C₂×M₁₆	—	59	1	no	—
227	C₂×M₁₆	—	118	1	no	—
228	C₂×M₁₆	—	118	1	no	—
229	C₂²×C₈	—	118	1	no	—
230	C₂²×C₈	—	118	1	no	—
231	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
233	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
234	M₃₂	—	2	1	no	—
235	M₃₂	—	2	1	no	—
236	M₃₂	—	2	1	no	—
237	M₃₂	—	118	4	no	—
238	M₃₂	—	118	1	no	—
239	M₃₂	—	118	1	no	—
240	M₃₂	—	118	1	no	—
241	M₃₂	—	118	1	no	—
242	M₃₂	D₂₃₆	1	2	yes	—
243	C₂×C₁₆	—	2	1	no	—
244	C₂×C₁₆	—	2	1	no	—
245	C₂×C₁₆	—	59	2	no	—
246	C₂×C₁₆	—	118	4	no	—
247	C₂×C₁₆	—	118	1	no	—
248	#108	—	118	2	no	—
249	#104	—	118	2	no	—
250	#104	—	118	2	no	—
251	#97	—	2	1	no	—
252	#97	—	2	1	no	—
253	#41	—	59	2	no	—
254	#41	—	59	2	no	—
255	#41	—	118	1	no	—
256	#41	—	118	1	no	—
257	#41	—	118	1	no	—
258	#41	—	118	1	no	—
259	#41	—	118	1	no	—
260	#41	—	118	1	no	—
261	C₂×Q₈	—	59	1	no	—
262	C₂×Q₈	—	59	1	no	—
263	C₂×D₄	—	59	1	no	—
264	C₂×D₄	—	59	1	no	—
265	C₂×D₄	—	118	1	no	—
266	C₂×D₄	—	118	1	no	—
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	118	1	no	—
269	C₂²×C₄	—	118	1	no	—
270	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
271	Q₁₆	—	118	2	no	—
272	SD₁₆	—	118	2	no	—
273	SD₁₆	—	118	2	no	—
274	D₈	—	118	2	no	—
275	M₁₆	—	59	2	no	—
276	M₁₆	—	118	1	no	—
277	M₁₆	—	118	2	no	—
278	M₁₆	—	118	2	no	—
279	M₁₆	—	118	1	no	—
280	M₁₆	—	118	1	no	—
281	M₁₆	—	236	1	no	—
282	M₁₆	—	236	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	118	1	no	—
286	C₂×C₈	—	118	2	no	—
287	C₂×C₈	—	118	2	no	—
288	C₂×C₈	—	118	1	no	—
289	C₂×C₈	—	236	1	no	—
290	C₂×C₈	—	236	1	no	—
291	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
292	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
293	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
294	C₂×C₈	#11745685	1	1	yes	characteristic
295	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
296	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
297	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
298	C₂×C₈	#11745689	1	1	yes	characteristic
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₁₆	—	2	2	no	—
302	C₁₆	—	4	1	no	—
303	C₁₆	—	118	2	no	—
304	C₁₆	—	236	1	no	—
305	C₂³	—	118	1	no	—
306	C₂³	—	118	1	no	—
307	C₂³	#14273733	1	1	yes	characteristic
308	Q₈	—	59	2	no	—
309	Q₈	—	118	1	no	—
310	Q₈	—	118	1	no	—
311	Q₈	—	118	1	no	—
312	D₄	—	59	2	no	—
313	D₄	—	118	2	no	—
314	D₄	—	118	2	no	—
315	D₄	—	118	1	no	—
316	D₄	—	118	1	no	—
317	D₄	—	118	1	no	—
318	D₄	—	236	1	no	—
319	D₄	—	236	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	8	1	no	—
322	C₂×C₄	—	118	1	no	—
323	C₂×C₄	—	118	2	no	—
324	C₂×C₄	—	118	2	no	—
325	C₂×C₄	—	118	1	no	—
326	C₂×C₄	—	236	1	no	—
327	C₂×C₄	—	236	1	no	—
328	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
329	C₂×C₄	#14273689	1	1	yes	characteristic
330	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
331	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
332	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
333	C₂×C₄	#14273735	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	118	2	no	—
337	C₈	—	236	1	no	—
338	C₈	#14273697	1	1	yes	characteristic
339	C₈	#14273700	1	1	yes	characteristic
340	C₈	#14273702	1	1	yes	characteristic
341	C₈	#14273712	1	1	yes	characteristic
342	C₂²	—	4	1	no	—
343	C₂²	—	118	2	no	—
344	C₂²	—	472	1	no	—
345	C₂²	—	472	1	no	—
346	C₂²	—	236	1	no	—
347	C₂²	#16837367	1	1	yes	characteristic
348	C₂²	C₈×D₂₃₆	1	1	yes	characteristic
349	C₂²	#16837472	1	1	yes	characteristic
350	C₄	—	8	1	no	—
351	C₄	—	118	2	no	—
352	C₄	—	236	1	no	—
353	C₄	#16837280	1	1	yes	characteristic
354	C₄	C₂²⋊C₈×D₅₉	1	1	yes	characteristic
355	C₄	#16837467	1	1	yes	characteristic
356	C₄	#16837475	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	472	1	no	—
361	C₂	—	236	2	no	—
362	C₂	#21424387	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16837280
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1377	C₂
socle	C₂×C₅₉	#1869
prime radical	C₂×#14273749	C₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1377	D₅₉
hypocenter	C₅₉	#12986
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1377

Complements

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

N	H ≅ G/N	type
1. #21437198	C₂	semidirect
3. #21425725	C₂	semidirect
4. #21425723	C₂	semidirect
5. #21425716	C₂	semidirect
6. #21422176	C₂	semidirect
7. C₅₉×#1377	C₂	semidirect
181. #1377	D₅₉	semidirect
208. C₅₉	#12986	semidirect

Maximal subgroups

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

subgroup	index
#21437198	2
C₂×#16837525	2
#21425725	2
#21425723	2
#21425716	2
#21422176	2
C₅₉×#1377	2
#12986	59