Subgroups — #28504764

Order 15104.

474 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837484	C₂	1	1	yes	maximal, characteristic
2	#21425851	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16837216	C₂	1	1	yes	maximal, characteristic
4	C₂×#16837214	C₂	1	1	yes	maximal, characteristic
5	#21424266	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422204	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1387	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄₇₂	C₂²	1	1	yes	characteristic
9	C₂×#14273689	—	2	1	no	—
10	C₂×#14273667	C₂²	1	1	yes	characteristic
11	#16837484	—	2	1	no	—
12	#16837484	—	2	1	no	—
13	#16837484	C₄	1	2	yes	—
14	#16837484	C₄	1	2	yes	—
15	C₂×#14273663	C₂²	1	1	yes	characteristic
16	C₂×#14273662	—	2	1	no	—
17	#16837425	—	2	1	no	—
18	C₂×#14273647	—	2	1	no	—
19	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
20	Dic₅₉⋊C₁₆	—	2	1	no	—
21	Dic₅₉⋊C₁₆	—	2	1	no	—
22	#16837216	—	2	2	no	—
23	#16837214	—	2	2	no	—
24	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
27	C₂²×D₂₃₆	—	2	1	no	—
28	C₂²×C₄×D₅₉	—	4	1	no	—
29	C₂×D₄₇₂	—	2	1	no	—
30	C₂×D₄₇₂	—	2	1	no	—
31	C₂×D₄₇₂	—	2	1	no	—
32	C₂×D₄₇₂	C₂×C₄	1	2	yes	—
33	C₂×D₄₇₂	D₄	1	2	yes	—
34	C₂×D₄₇₂	Q₈	1	2	yes	—
35	C₂×#11745664	—	2	1	no	—
36	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
37	#14273689	—	2	2	no	—
38	#14273689	—	2	2	no	—
39	#14273689	—	4	1	no	—
40	#14273689	—	4	1	no	—
41	C₂×(C₂×D₅₉)⋊C₄	—	4	1	no	—
42	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
43	#14273667	—	2	1	no	—
44	#14273667	—	2	1	no	—
45	C₂×C₅₉⋊C₁₆	—	2	1	no	—
46	#14273663	—	2	1	no	—
47	#14273663	—	2	1	no	—
48	#14273662	—	4	2	no	—
49	Dic₅₉⋊C₈	—	4	1	no	—
50	Dic₅₉⋊C₈	—	4	1	no	—
51	#14273647	—	2	2	no	—
52	#14273647	—	2	2	no	—
53	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
54	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₅₉	D₄	1	1	yes	characteristic
56	C₈×Dic₅₉	D₄	1	1	yes	characteristic
57	C₂×M₁₆×C₅₉	—	2	1	no	—
58	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
60	M₃₂×C₅₉	—	2	1	no	—
61	M₃₂×C₅₉	—	2	1	no	—
62	C₂×C₁₆×C₅₉	—	2	1	no	—
63	#108×C₅₉	—	2	1	no	—
64	#108×C₅₉	—	2	1	no	—
65	#107×C₅₉	—	2	1	no	—
66	#107×C₅₉	C₂×C₄	1	2	yes	—
67	C₂³×D₅₉	—	4	1	no	—
68	C₂²×C₄×C₅₉	—	4	1	no	—
69	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₅₉	SD₁₆	1	1	yes	characteristic
71	C₂²×Dic₅₉	SD₁₆	1	1	yes	characteristic
72	C₂×D₂₃₆	—	2	2	no	—
73	C₂×D₂₃₆	—	2	2	no	—
74	C₂×D₂₃₆	—	2	2	no	—
75	C₂×D₂₃₆	—	4	1	no	—
76	C₂×D₂₃₆	—	4	1	no	—
77	C₂×D₂₃₆	—	4	1	no	—
78	C₂×C₄×D₅₉	—	4	2	no	—
79	C₂×C₄×D₅₉	—	4	2	no	—
80	C₂×C₄×D₅₉	—	8	4	no	—
81	M₁₆×C₅₉	—	4	1	no	—
82	M₁₆×C₅₉	—	4	1	no	—
83	C₂×C₈×C₅₉	—	4	1	no	—
84	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
89	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
90	C₄⋊C₄×C₅₉	—	2	2	no	—
91	C₄⋊C₄×C₅₉	—	4	1	no	—
92	(C₂×D₅₉)⋊C₄	—	4	2	no	—
93	(C₂×D₅₉)⋊C₄	—	4	2	no	—
94	C₄×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
96	C₄×Dic₅₉	D₈	1	1	yes	characteristic
97	C₄×Dic₅₉	D₈	1	1	yes	characteristic
98	#11745664	—	4	1	no	—
99	#11745664	—	4	1	no	—
100	C₂×C₅₉⋊C₈	—	2	1	no	—
101	C₂×C₅₉⋊C₈	—	2	1	no	—
102	C₂×C₅₉⋊C₈	—	4	1	no	—
103	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
105	D₄₇₂	—	2	2	no	—
106	D₄₇₂	—	2	2	no	—
107	D₄₇₂	—	2	2	no	—
108	D₄₇₂	C₄⋊C₄	1	4	yes	—
109	C₁₆×C₅₉	—	4	1	no	—
110	C₅₉⋊C₁₆	—	4	1	no	—
111	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
112	C₂²×D₅₉	—	4	2	no	—
113	C₂²×D₅₉	—	4	2	no	—
114	C₂²×D₅₉	—	4	2	no	—
115	C₂²×D₅₉	—	8	2	no	—
116	C₂²×D₅₉	—	8	2	no	—
117	C₂×C₄×C₅₉	—	4	2	no	—
118	C₂×C₄×C₅₉	—	8	2	no	—
119	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
125	C₂×Dic₅₉	—	2	2	no	—
126	C₂×Dic₅₉	—	4	1	no	—
127	C₂×Dic₅₉	#102	1	1	yes	characteristic
128	C₂×Dic₅₉	#102	1	1	yes	characteristic
129	C₂×Dic₅₉	#102	1	1	yes	characteristic
130	C₂×Dic₅₉	#102	1	1	yes	characteristic
131	D₂₃₆	—	2	4	no	—
132	D₂₃₆	—	4	2	no	—
133	D₂₃₆	—	4	2	no	—
134	D₂₃₆	—	4	2	no	—
135	C₄×D₅₉	—	8	4	no	—
136	C₄×D₅₉	—	8	4	no	—
137	C₈×C₅₉	—	8	1	no	—
138	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₈×C₅₉	#117	1	1	yes	characteristic
140	C₈×C₅₉	#121	1	1	yes	characteristic
141	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
142	C₅₉⋊C₈	—	2	1	no	—
143	C₅₉⋊C₈	—	2	1	no	—
144	C₅₉⋊C₈	—	8	1	no	—
145	#12818	—	59	1	no	maximal, Sylow
146	C₂²×C₅₉	—	2	1	no	—
147	C₂²×C₅₉	—	2	1	no	—
148	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
149	C₂²×C₅₉	#461	1	1	yes	characteristic
150	C₂²×C₅₉	#476	1	1	yes	characteristic
151	C₂×D₅₉	—	4	4	no	—
152	C₂×D₅₉	—	8	4	no	—
153	C₂×D₅₉	—	8	4	no	—
154	C₂×D₅₉	—	8	2	no	—
155	C₂×D₅₉	—	8	2	no	—
156	C₄×C₅₉	—	8	2	no	—
157	C₄×C₅₉	#380	1	1	yes	characteristic
158	C₄×C₅₉	#417	1	1	yes	characteristic
159	C₄×C₅₉	#459	1	1	yes	characteristic
160	C₄×C₅₉	#474	1	1	yes	characteristic
161	Dic₅₉	—	2	1	no	—
162	Dic₅₉	—	2	1	no	—
163	Dic₅₉	—	2	1	no	—
164	Dic₅₉	#361	1	2	yes	complemented
165	C₂×C₄×D₈	—	59	1	no	—
166	#2150	—	59	1	no	—
167	C₂×#361	—	59	1	no	—
168	C₂×#361	—	59	1	no	—
169	#1848	—	59	1	no	—
170	#1387	—	59	1	no	—
171	#1387	D₅₉	1	1	yes	characteristic, complemented
172	C₂×C₅₉	—	2	1	no	—
173	C₂×C₅₉	—	2	1	no	—
174	C₂×C₅₉	#1802	1	1	yes	characteristic
175	C₂×C₅₉	C₂×#361	1	1	yes	characteristic
176	C₂×C₅₉	#2148	1	1	yes	characteristic
177	D₅₉	—	8	4	no	—
178	C₂²×D₈	—	59	1	no	—
179	C₂×C₄×D₄	—	118	1	no	—
180	C₂×M₃₂	—	59	1	no	—
181	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
182	C₄×D₈	—	59	2	no	—
183	C₄×D₈	—	59	2	no	—
184	C₄×D₈	—	118	1	no	—
185	C₄×D₈	—	118	1	no	—
186	C₂×#108	—	59	1	no	—
187	C₂×#108	—	118	1	no	—
188	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
189	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
190	#423	—	118	1	no	—
191	C₂×#102	—	118	1	no	—
192	C₂×C₄×C₈	—	59	1	no	—
193	C₄⋊C₁₆	—	118	1	no	—
194	C₄⋊C₁₆	—	118	1	no	—
195	#361	—	118	2	no	—
196	#361	—	118	2	no	—
197	C₅₉	#12818	1	1	yes	characteristic, complemented, Sylow
198	C₂²×D₄	—	118	1	no	—
199	C₂³×C₄	—	236	1	no	—
200	C₂×D₈	—	59	2	no	—
201	C₂×D₈	—	59	2	no	—
202	C₂×D₈	—	59	2	no	—
203	C₂×D₈	—	118	1	no	—
204	C₂×D₈	—	118	1	no	—
205	C₂×D₈	—	118	1	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂×M₁₆	—	118	1	no	—
208	C₂²×C₈	—	59	1	no	—
209	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
210	C₄×D₄	—	118	2	no	—
211	C₄×D₄	—	118	2	no	—
212	C₄×D₄	—	236	1	no	—
213	C₄×D₄	—	236	1	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₂²⋊C₄	—	236	1	no	—
216	C₂×C₄²	—	59	1	no	—
217	M₃₂	—	2	1	no	—
218	M₃₂	—	2	1	no	—
219	M₃₂	—	118	1	no	—
220	M₃₂	—	118	1	no	—
221	C₂×C₁₆	—	2	1	no	—
222	C₂×C₁₆	—	118	1	no	—
223	#108	—	2	1	no	—
224	#108	—	2	1	no	—
225	#108	—	118	1	no	—
226	#108	—	118	1	no	—
227	#108	—	236	2	no	—
228	#107	—	2	1	no	—
229	#107	C₄×D₅₉	1	2	yes	—
230	C₄⋊C₈	—	236	1	no	—
231	C₄⋊C₈	—	236	1	no	—
232	#102	—	118	2	no	—
233	#102	—	118	2	no	—
234	C₄×C₈	—	59	1	no	—
235	C₄×C₈	—	59	1	no	—
236	C₄×C₈	—	59	1	no	—
237	C₄×C₈	—	59	1	no	—
238	C₂⁴	—	236	1	no	—
239	C₂×D₄	—	118	2	no	—
240	C₂×D₄	—	118	2	no	—
241	C₂×D₄	—	118	2	no	—
242	C₂×D₄	—	236	1	no	—
243	C₂×D₄	—	236	1	no	—
244	C₂×D₄	—	236	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	59	1	no	—
247	C₂²×C₄	—	59	1	no	—
248	C₂²×C₄	—	472	4	no	—
249	C₂²×C₄	—	236	2	no	—
250	C₂²×C₄	—	236	2	no	—
251	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
252	D₈	—	59	4	no	—
253	D₈	—	118	2	no	—
254	D₈	—	118	2	no	—
255	D₈	—	118	2	no	—
256	M₁₆	—	4	1	no	—
257	M₁₆	—	4	1	no	—
258	M₁₆	—	236	1	no	—
259	M₁₆	—	236	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	59	1	no	—
262	C₂×C₈	—	59	1	no	—
263	C₂×C₈	—	118	1	no	—
264	C₂×C₈	—	118	1	no	—
265	C₂×C₈	—	236	1	no	—
266	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
267	C₂×C₈	#11745685	1	1	yes	characteristic
268	C₂×C₈	#11745685	1	1	yes	characteristic
269	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
270	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
271	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	236	2	no	—
275	C₂²⋊C₄	—	236	2	no	—
276	C₄²	—	59	1	no	—
277	C₄²	—	59	1	no	—
278	C₄²	—	59	1	no	—
279	C₄²	—	59	1	no	—
280	C₁₆	—	4	1	no	—
281	C₁₆	—	236	1	no	—
282	C₂³	—	472	2	no	—
283	C₂³	—	472	2	no	—
284	C₂³	—	236	2	no	—
285	C₂³	—	236	2	no	—
286	C₂³	—	236	2	no	—
287	C₂³	SD₁₆×D₅₉	1	1	yes	characteristic
288	D₄	—	118	4	no	—
289	D₄	—	236	2	no	—
290	D₄	—	236	2	no	—
291	D₄	—	236	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	8	2	no	—
294	C₂×C₄	—	59	1	no	—
295	C₂×C₄	—	59	1	no	—
296	C₂×C₄	—	59	1	no	—
297	C₂×C₄	—	59	1	no	—
298	C₂×C₄	—	118	2	no	—
299	C₂×C₄	—	472	4	no	—
300	C₂×C₄	—	472	4	no	—
301	C₂×C₄	—	236	1	no	—
302	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
303	C₂×C₄	#14273700	1	1	yes	characteristic
304	C₂×C₄	#14273705	1	1	yes	characteristic
305	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
306	C₂×C₄	#14273754	1	1	yes	characteristic
307	C₂×C₄	#14273759	1	1	yes	characteristic
308	C₈	—	8	1	no	—
309	C₈	—	118	1	no	—
310	C₈	—	118	1	no	—
311	C₈	—	472	1	no	—
312	C₈	C₄⋊C₄×D₅₉	1	1	yes	characteristic
313	C₈	#14273684	1	1	yes	characteristic
314	C₈	#14273700	1	1	yes	characteristic
315	C₈	#14273705	1	1	yes	characteristic
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	472	4	no	—
319	C₂²	—	472	4	no	—
320	C₂²	—	472	2	no	—
321	C₂²	—	472	2	no	—
322	C₂²	—	236	4	no	—
323	C₂²	#102×D₅₉	1	1	yes	characteristic
324	C₂²	#16837606	1	1	yes	characteristic
325	C₂²	#16837691	1	1	yes	characteristic
326	C₄	—	8	2	no	—
327	C₄	—	59	2	no	—
328	C₄	—	118	1	no	—
329	C₄	—	118	1	no	—
330	C₄	—	118	1	no	—
331	C₄	#16837261	1	1	yes	characteristic
332	C₄	#16837402	1	1	yes	characteristic
333	C₄	#16837596	1	1	yes	characteristic
334	C₄	#16837681	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	472	4	no	—
338	C₂	#21424095	1	1	yes	characteristic
339	C₂	#361×D₅₉	1	1	yes	characteristic
340	C₂	#21425844	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#102×D₅₉
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1387	C₂
socle	C₂²×C₅₉	C₂×#102
prime radical	C₂²×D₄₇₂	C₂²
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1387	D₅₉
hypocenter	C₅₉	#12818
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1387

Complements

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

N	H ≅ G/N	type
2. #21425851	C₂	semidirect
5. #21424266	C₂	semidirect
6. #21422204	C₂	semidirect
7. C₅₉×#1387	C₂	semidirect
164. Dic₅₉	#361	semidirect
171. #1387	D₅₉	semidirect
197. C₅₉	#12818	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837484	2
#21425851	2
C₂×#16837216	2
C₂×#16837214	2
#21424266	2
#21422204	2
C₅₉×#1387	2
#12818	59