Subgroups — #28505284

Order 15104.

490 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837228	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16837204	C₂	1	1	yes	maximal, characteristic
4	C₂×#16837204	C₂	1	1	yes	maximal, characteristic
5	#21424829	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422188	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄₇₂	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₂₃₆	—	2	1	no	—
10	C₄×D₄₇₂	—	2	2	no	—
11	C₄×D₄₇₂	—	2	2	no	—
12	C₂×#14273661	C₂²	1	1	yes	characteristic
13	C₂×#14273661	C₂²	1	1	yes	characteristic
14	C₂×#14273661	C₂²	1	1	yes	characteristic
15	C₂×#14273649	—	2	1	no	—
16	#16837300	—	2	1	no	—
17	#16837228	—	2	2	no	—
18	#16837204	—	2	2	no	—
19	#16837204	C₄	1	4	yes	complemented
20	#16837165	—	2	1	no	—
21	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂²×D₂₃₆	—	2	1	no	—
25	C₂²×C₄×D₅₉	—	4	1	no	—
26	C₂×D₄₇₂	—	2	1	no	—
27	C₂×D₄₇₂	—	2	1	no	—
28	C₂×D₄₇₂	—	2	1	no	—
29	C₂×D₄₇₂	C₂×C₄	1	2	yes	—
30	C₂×D₄₇₂	D₄	1	2	yes	—
31	C₂×D₄₇₂	Q₈	1	2	yes	—
32	C₄×D₂₃₆	—	2	4	no	—
33	C₄×D₂₃₆	—	4	2	no	—
34	C₂×C₄⋊Dic₅₉	—	2	1	no	—
35	C₂×C₄⋊Dic₅₉	—	2	1	no	—
36	C₂×C₄⋊Dic₅₉	—	2	1	no	—
37	C₂×Dic₅₉⋊C₄	—	4	1	no	—
38	C₂×(C₂×D₅₉)⋊C₄	—	4	1	no	—
39	#14273661	—	2	1	no	—
40	#14273661	—	2	1	no	—
41	#14273661	—	2	1	no	—
42	#14273661	—	2	1	no	—
43	#14273661	C₂×C₄	1	2	yes	complemented
44	#14273661	D₄	1	2	yes	—
45	#14273649	—	2	4	no	—
46	C₅₉⋊#95	—	4	1	no	—
47	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
50	C₂×C₁₆×C₅₉	—	2	1	no	—
51	C₂×C₁₆×C₅₉	—	2	1	no	—
52	C₂×C₁₆×C₅₉	—	2	1	no	—
53	C₂×C₁₆×C₅₉	—	2	1	no	—
54	C₂×C₁₆×C₅₉	—	2	1	no	—
55	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	complemented
56	C₄×C₈×C₅₉	—	2	1	no	—
57	C₄×C₈×C₅₉	—	2	1	no	—
58	C₂³×D₅₉	—	4	1	no	—
59	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₅₉	—	4	1	no	—
63	C₂²×Dic₅₉	—	4	1	no	—
64	C₂²×Dic₅₉	—	4	1	no	—
65	C₂×D₂₃₆	—	2	2	no	—
66	C₂×D₂₃₆	—	2	2	no	—
67	C₂×D₂₃₆	—	2	2	no	—
68	C₂×D₂₃₆	—	4	1	no	—
69	C₂×D₂₃₆	—	4	1	no	—
70	C₂×D₂₃₆	—	4	1	no	—
71	C₂×C₄×D₅₉	—	4	4	no	—
72	C₂×C₄×D₅₉	—	8	2	no	—
73	C₂×C₄×D₅₉	—	8	2	no	—
74	C₂×C₈×C₅₉	—	2	1	no	—
75	C₂×C₈×C₅₉	—	2	1	no	—
76	C₂×C₈×C₅₉	—	2	1	no	—
77	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
82	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
83	C₄²×C₅₉	—	2	1	no	—
84	C₄²×C₅₉	—	2	1	no	—
85	(C₂×D₅₉)⋊C₄	—	4	4	no	—
86	C₄⋊Dic₅₉	—	2	2	no	—
87	C₄⋊Dic₅₉	—	2	2	no	—
88	C₄⋊Dic₅₉	—	4	1	no	—
89	C₄⋊Dic₅₉	—	4	1	no	—
90	C₄⋊Dic₅₉	—	4	1	no	—
91	C₄⋊Dic₅₉	—	4	1	no	—
92	Dic₅₉⋊C₄	—	8	2	no	—
93	D₄₇₂	—	2	2	no	—
94	D₄₇₂	—	2	2	no	—
95	D₄₇₂	—	2	2	no	—
96	D₄₇₂	C₄⋊C₄	1	4	yes	complemented
97	C₁₆×C₅₉	—	2	2	no	—
98	C₁₆×C₅₉	—	4	1	no	—
99	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
100	C₂²×D₅₉	—	4	2	no	—
101	C₂²×D₅₉	—	4	2	no	—
102	C₂²×D₅₉	—	4	2	no	—
103	C₂²×D₅₉	—	8	1	no	—
104	C₂²×D₅₉	—	8	1	no	—
105	C₂²×D₅₉	—	8	1	no	—
106	C₂²×D₅₉	—	8	1	no	—
107	C₂×C₄×C₅₉	—	2	1	no	—
108	C₂×C₄×C₅₉	—	2	1	no	—
109	C₂×C₄×C₅₉	—	2	1	no	—
110	C₂×C₄×C₅₉	—	2	1	no	—
111	C₂×C₄×C₅₉	—	2	1	no	—
112	C₂×C₄×C₅₉	D₁₆	1	2	yes	—
113	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
117	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
119	C₂×Dic₅₉	—	4	2	no	—
120	C₂×Dic₅₉	—	4	2	no	—
121	C₂×Dic₅₉	—	8	1	no	—
122	C₂×Dic₅₉	—	8	1	no	—
123	C₂×Dic₅₉	—	8	1	no	—
124	C₂×Dic₅₉	—	8	1	no	—
125	C₂×Dic₅₉	—	8	1	no	—
126	C₂×Dic₅₉	—	8	1	no	—
127	C₂×Dic₅₉	—	8	1	no	—
128	D₂₃₆	—	2	4	no	—
129	D₂₃₆	—	4	2	no	—
130	D₂₃₆	—	4	2	no	—
131	D₂₃₆	—	4	2	no	—
132	C₄×D₅₉	—	8	8	no	—
133	C₈×C₅₉	—	4	1	no	—
134	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
136	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
137	C₈×C₅₉	#123	1	1	yes	characteristic
138	#12935	—	59	1	no	maximal, Sylow
139	C₂²×C₅₉	C₄×D₈	1	1	yes	characteristic
140	C₂²×C₅₉	#447	1	1	yes	characteristic
141	C₂²×C₅₉	#474	1	1	yes	characteristic
142	C₂²×C₅₉	C₂×D₁₆	1	1	yes	characteristic
143	C₂²×C₅₉	#508	1	1	yes	characteristic
144	C₂²×C₅₉	#509	1	1	yes	characteristic
145	C₂²×C₅₉	#510	1	1	yes	characteristic
146	C₂×D₅₉	—	4	4	no	—
147	C₂×D₅₉	—	8	2	no	—
148	C₂×D₅₉	—	8	2	no	—
149	C₂×D₅₉	—	8	2	no	—
150	C₂×D₅₉	—	8	2	no	—
151	C₂×D₅₉	—	8	2	no	—
152	C₂×D₅₉	—	8	2	no	—
153	C₄×C₅₉	—	2	2	no	—
154	C₄×C₅₉	—	4	1	no	—
155	C₄×C₅₉	#385	1	1	yes	characteristic
156	C₄×C₅₉	#440	1	1	yes	characteristic
157	C₄×C₅₉	#452	1	1	yes	characteristic
158	C₄×C₅₉	#478	1	1	yes	characteristic
159	Dic₅₉	—	8	2	no	—
160	Dic₅₉	—	8	2	no	—
161	Dic₅₉	—	16	1	no	—
162	C₂×C₄×D₈	—	59	1	no	—
163	C₂×#366	—	59	1	no	—
164	C₂×#357	—	59	1	no	—
165	C₂×#357	—	59	1	no	—
166	#1942	—	59	1	no	—
167	#1380	—	59	1	no	—
168	#1374	D₅₉	1	1	yes	characteristic, complemented
169	C₂×C₅₉	#1862	1	1	yes	characteristic
170	C₂×C₅₉	C₄×D₁₆	1	1	yes	characteristic
171	C₂×C₅₉	#2175	1	1	yes	characteristic
172	C₂×C₅₉	#2184	1	1	yes	characteristic
173	C₂×C₅₉	#2189	1	1	yes	characteristic
174	C₂×C₅₉	#2224	1	1	yes	characteristic
175	C₂×C₅₉	#2228	1	1	yes	characteristic
176	D₅₉	—	8	4	no	—
177	C₂²×D₈	—	59	1	no	—
178	C₂×C₄×D₄	—	118	1	no	—
179	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
180	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
181	C₄×D₈	—	118	2	no	—
182	C₄×D₈	—	118	2	no	—
183	C₂×#107	—	59	1	no	—
184	C₂×#107	—	59	1	no	—
185	C₂×#107	—	59	1	no	—
186	C₂×#102	—	118	1	no	—
187	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
188	#389	—	118	1	no	—
189	#366	—	118	2	no	—
190	#357	—	59	4	no	—
191	#357	—	118	2	no	—
192	#340	—	118	1	no	—
193	C₅₉	#12935	1	1	yes	characteristic, complemented, Sylow
194	C₂²×D₄	—	118	1	no	—
195	C₂³×C₄	—	236	1	no	—
196	C₂×D₈	—	59	2	no	—
197	C₂×D₈	—	59	2	no	—
198	C₂×D₈	—	59	2	no	—
199	C₂×D₈	—	118	1	no	—
200	C₂×D₈	—	118	1	no	—
201	C₂×D₈	—	118	1	no	—
202	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
204	C₄×D₄	—	118	4	no	—
205	C₄×D₄	—	236	2	no	—
206	C₂×C₄⋊C₄	—	118	1	no	—
207	C₂×C₄⋊C₄	—	118	1	no	—
208	C₂×C₄⋊C₄	—	118	1	no	—
209	C₂×C₄⋊C₄	—	236	1	no	—
210	C₂×C₂²⋊C₄	—	236	1	no	—
211	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	C₄×D₅₉	1	2	yes	complemented
218	#107	—	59	2	no	—
219	#107	—	59	2	no	—
220	#107	—	118	1	no	—
221	#107	—	118	1	no	—
222	#107	—	118	1	no	—
223	#107	—	118	1	no	—
224	#102	—	118	4	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	#95	—	236	1	no	—
228	C₂⁴	—	236	1	no	—
229	C₂×D₄	—	118	2	no	—
230	C₂×D₄	—	118	2	no	—
231	C₂×D₄	—	118	2	no	—
232	C₂×D₄	—	236	1	no	—
233	C₂×D₄	—	236	1	no	—
234	C₂×D₄	—	236	1	no	—
235	C₂²×C₄	—	472	2	no	—
236	C₂²×C₄	—	472	2	no	—
237	C₂²×C₄	—	236	4	no	—
238	C₂²×C₄	—	236	1	no	—
239	C₂²×C₄	—	236	1	no	—
240	C₂²×C₄	—	236	1	no	—
241	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
242	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
244	D₈	—	59	4	no	—
245	D₈	—	118	2	no	—
246	D₈	—	118	2	no	—
247	D₈	—	118	2	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
252	C₂×C₈	#11745685	1	1	yes	characteristic
253	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
254	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
255	C₂×C₈	#11745687	1	1	yes	characteristic
256	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
257	C₄⋊C₄	—	118	2	no	—
258	C₄⋊C₄	—	118	2	no	—
259	C₄⋊C₄	—	472	2	no	—
260	C₄⋊C₄	—	236	1	no	—
261	C₄⋊C₄	—	236	1	no	—
262	C₄⋊C₄	—	236	1	no	—
263	C₄⋊C₄	—	236	1	no	—
264	C₂²⋊C₄	—	236	4	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₁₆	—	2	2	no	—
268	C₁₆	—	4	1	no	—
269	C₂³	—	472	1	no	—
270	C₂³	—	472	1	no	—
271	C₂³	—	472	1	no	—
272	C₂³	—	472	1	no	—
273	C₂³	—	236	2	no	—
274	C₂³	—	236	2	no	—
275	C₂³	—	236	2	no	—
276	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
277	D₄	—	118	4	no	—
278	D₄	—	236	2	no	—
279	D₄	—	236	2	no	—
280	D₄	—	236	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	472	8	no	—
287	C₂×C₄	—	472	1	no	—
288	C₂×C₄	—	472	1	no	—
289	C₂×C₄	—	472	1	no	—
290	C₂×C₄	—	472	1	no	—
291	C₂×C₄	—	472	1	no	—
292	C₂×C₄	—	472	1	no	—
293	C₂×C₄	—	472	1	no	—
294	C₂×C₄	—	236	2	no	—
295	C₂×C₄	—	236	2	no	—
296	C₂×C₄	D₉₄₄	1	2	yes	—
297	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
298	C₂×C₄	#14273697	1	1	yes	characteristic
299	C₂×C₄	#14273707	1	1	yes	characteristic
300	C₂×C₄	#14273749	1	1	yes	characteristic
301	C₂×C₄	#14273755	1	1	yes	characteristic
302	C₂×C₄	#14273760	1	1	yes	characteristic
303	C₈	—	4	1	no	—
304	C₈	C₄⋊C₄×D₅₉	1	1	yes	characteristic
305	C₈	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
306	C₈	#14273705	1	1	yes	characteristic
307	C₈	#14273709	1	1	yes	characteristic
308	C₂²	—	472	2	no	—
309	C₂²	—	472	2	no	—
310	C₂²	—	472	2	no	—
311	C₂²	—	472	2	no	—
312	C₂²	—	472	2	no	—
313	C₂²	—	472	2	no	—
314	C₂²	—	236	4	no	—
315	C₂²	C₄×D₄₇₂	1	1	yes	characteristic
316	C₂²	#16837538	1	1	yes	characteristic
317	C₂²	#16837683	1	1	yes	characteristic
318	C₂²	C₂×D₉₄₄	1	1	yes	characteristic
319	C₂²	#16837834	1	1	yes	characteristic
320	C₂²	#16837840	1	1	yes	characteristic
321	C₂²	#16837845	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	4	1	no	—
324	C₄	—	944	1	no	—
325	C₄	—	472	2	no	—
326	C₄	—	472	2	no	—
327	C₄	#16837276	1	1	yes	characteristic
328	C₄	#16837504	1	1	yes	characteristic
329	C₄	#16837566	1	1	yes	characteristic
330	C₄	#16837703	1	1	yes	characteristic
331	C₂	—	472	4	no	—
332	C₂	#21424346	1	1	yes	characteristic
333	C₂	C₄×D₉₄₄	1	1	yes	characteristic
334	C₂	#21425962	1	1	yes	characteristic
335	C₂	#21426010	1	1	yes	characteristic
336	C₂	#21426038	1	1	yes	characteristic
337	C₂	#21426234	1	1	yes	characteristic
338	C₂	#21426254	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1374	C₂
socle	C₂³×C₅₉	C₂×D₈
prime radical	C₂²×D₄₇₂	C₂²
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1374	D₅₉
hypocenter	C₅₉	#12935
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
2. C₂×#16837228	C₂	semidirect
5. #21424829	C₂	semidirect
6. #21422188	C₂	semidirect
7. C₅₉×#1374	C₂	semidirect
19. #16837204	C₄	semidirect
43. #14273661	C₂×C₄	semidirect
55. C₂×C₁₆×C₅₉	C₂×C₄	semidirect
96. D₄₇₂	C₄⋊C₄	semidirect
168. #1374	D₅₉	semidirect
193. C₅₉	#12935	semidirect
217. C₂×C₁₆	C₄×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄₇₂	2
C₂×#16837228	2
C₂×#16837204	2
#21424829	2
#21422188	2
C₅₉×#1374	2
#12935	59