Subgroups — #28505203

Order 15104.

404 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108×D₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837231	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×(C₂×D₅₉)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#21424131	C₂	1	1	yes	maximal, characteristic
5	#21422190	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422189	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₉×#368	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₅₉	—	2	1	no	—
9	C₂²×C₈×D₅₉	C₂²	1	1	yes	characteristic
10	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×#14273663	C₂²	1	1	yes	characteristic
12	C₂×#14273663	C₂²	1	1	yes	characteristic
13	C₂×#14273662	—	2	1	no	—
14	#108×D₅₉	—	2	2	no	—
15	#108×D₅₉	—	2	2	no	—
16	#16837365	—	2	1	no	—
17	#16837231	—	2	2	no	—
18	(C₂×D₅₉)⋊C₁₆	C₄	1	2	yes	—
19	(C₂×D₅₉)⋊C₁₆	C₄	1	2	yes	—
20	#16837166	—	2	1	no	—
21	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
24	C₅₉×#368	—	2	2	no	—
25	C₂²×C₄×D₅₉	D₄	1	1	yes	characteristic
26	C₂×#11745664	—	2	1	no	—
27	C₂×#11745664	—	2	1	no	—
28	C₂×#11745659	—	4	1	no	—
29	M₁₆×D₅₉	—	2	4	no	—
30	M₁₆×D₅₉	—	4	2	no	—
31	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
32	C₂×C₈×D₅₉	—	2	1	no	—
33	C₂×C₈×D₅₉	—	2	1	no	—
34	C₂×C₈×D₅₉	—	2	1	no	—
35	C₂×C₈×D₅₉	—	2	1	no	—
36	C₂×C₈×D₅₉	—	4	1	no	—
37	C₂×C₈×D₅₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×D₅₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₅₉	Q₈	1	1	yes	characteristic
40	C₂×C₈×D₅₉	Q₈	1	1	yes	characteristic
41	C₂×C₅₉⋊C₁₆	—	2	1	no	—
42	C₂×C₅₉⋊C₁₆	—	2	1	no	—
43	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
44	#14273663	—	2	1	no	—
45	#14273663	—	2	1	no	—
46	#14273663	—	2	1	no	—
47	#14273663	D₄	1	2	yes	—
48	#14273662	—	2	4	no	—
49	(C₂×D₅₉)⋊C₈	—	4	1	no	—
50	(C₂×D₅₉)⋊C₈	—	4	1	no	—
51	C₂×M₁₆×C₅₉	—	2	1	no	—
52	C₂×M₁₆×C₅₉	—	2	1	no	—
53	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₅₉	—	2	1	no	—
55	C₂×C₁₆×C₅₉	—	2	1	no	—
56	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
57	#108×C₅₉	—	2	1	no	—
58	#108×C₅₉	—	2	1	no	—
59	#108×C₅₉	—	2	1	no	—
60	#108×C₅₉	D₄	1	2	yes	—
61	C₂³×D₅₉	Q₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₅₉	Q₁₆	1	1	yes	characteristic
64	C₂×C₄×D₅₉	—	2	1	no	—
65	C₂×C₄×D₅₉	—	2	1	no	—
66	C₂×C₄×D₅₉	—	2	1	no	—
67	C₂×C₄×D₅₉	—	2	1	no	—
68	C₂×C₄×D₅₉	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×D₅₉	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×D₅₉	D₈	1	1	yes	characteristic
71	C₂×C₄×D₅₉	D₈	1	1	yes	characteristic
72	M₁₆×C₅₉	—	2	2	no	—
73	M₁₆×C₅₉	—	4	1	no	—
74	M₁₆×C₅₉	—	4	1	no	—
75	M₁₆×C₅₉	—	4	1	no	—
76	C₂×C₈×C₅₉	—	4	1	no	—
77	C₂×C₈×C₅₉	—	4	1	no	—
78	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
83	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
84	#11745664	—	2	2	no	—
85	#11745664	—	4	1	no	—
86	#11745664	—	4	1	no	—
87	#11745664	—	4	1	no	—
88	C₂×C₅₉⋊C₈	—	2	1	no	—
89	C₂×C₅₉⋊C₈	—	2	1	no	—
90	C₂×C₅₉⋊C₈	—	4	1	no	—
91	C₂×C₅₉⋊C₈	—	4	1	no	—
92	C₂×C₅₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₅₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
94	#11745659	—	4	4	no	—
95	C₈×D₅₉	—	2	1	no	—
96	C₈×D₅₉	—	2	1	no	—
97	C₈×D₅₉	—	2	1	no	—
98	C₈×D₅₉	—	2	1	no	—
99	C₈×D₅₉	—	2	1	no	—
100	C₈×D₅₉	—	2	1	no	—
101	C₈×D₅₉	—	2	1	no	—
102	C₈×D₅₉	—	2	1	no	—
103	C₈×D₅₉	—	4	4	no	—
104	C₁₆×C₅₉	—	2	2	no	—
105	C₅₉⋊C₁₆	—	2	2	no	—
106	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
107	C₂²×D₅₉	—	2	1	no	—
108	C₂²×D₅₉	—	2	1	no	—
109	C₂²×D₅₉	—	2	2	no	—
110	C₂²×D₅₉	—	4	1	no	—
111	C₂²×D₅₉	#107	1	1	yes	characteristic
112	C₂²×D₅₉	#107	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
115	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂×D₈	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	—	2	1	no	—
120	C₂×Dic₅₉	—	2	1	no	—
121	C₂×Dic₅₉	#107	1	1	yes	characteristic
122	C₂×Dic₅₉	#107	1	1	yes	characteristic
123	C₄×D₅₉	—	2	1	no	—
124	C₄×D₅₉	—	2	1	no	—
125	C₄×D₅₉	—	2	1	no	—
126	C₄×D₅₉	—	2	1	no	—
127	C₄×D₅₉	—	2	1	no	—
128	C₄×D₅₉	—	2	1	no	—
129	C₄×D₅₉	—	2	1	no	—
130	C₄×D₅₉	—	2	1	no	—
131	C₈×C₅₉	—	4	2	no	—
132	C₈×C₅₉	—	8	1	no	—
133	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
134	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
135	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
136	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
137	C₅₉⋊C₈	—	2	1	no	—
138	C₅₉⋊C₈	—	2	1	no	—
139	C₅₉⋊C₈	—	4	2	no	—
140	C₅₉⋊C₈	—	8	1	no	—
141	#12922	—	59	1	no	maximal, Sylow
142	C₂²×C₅₉	—	2	1	no	—
143	C₂²×C₅₉	—	2	1	no	—
144	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
145	C₂²×C₅₉	#451	1	1	yes	characteristic
146	C₂²×C₅₉	#484	1	1	yes	characteristic
147	C₂×D₅₉	—	2	1	no	—
148	C₂×D₅₉	—	2	1	no	—
149	C₂×D₅₉	—	2	1	no	—
150	C₂×D₅₉	—	2	1	no	—
151	C₂×D₅₉	—	2	1	no	—
152	C₂×D₅₉	—	4	2	no	—
153	C₂×D₅₉	—	4	2	no	—
154	C₂×D₅₉	#368	1	2	yes	complemented
155	C₄×C₅₉	#385	1	1	yes	characteristic
156	C₄×C₅₉	#428	1	1	yes	characteristic
157	C₄×C₅₉	#447	1	1	yes	characteristic
158	C₄×C₅₉	#480	1	1	yes	characteristic
159	Dic₅₉	—	2	1	no	—
160	Dic₅₉	—	2	1	no	—
161	C₂²×#108	—	59	1	no	—
162	C₂×#368	—	59	1	no	—
163	C₂×#368	D₅₉	1	1	yes	characteristic, complemented
164	C₂×C₂²⋊C₁₆	—	59	1	no	—
165	#1810	—	59	1	no	—
166	#1381	—	59	1	no	—
167	#1381	—	59	1	no	—
168	C₂×C₅₉	—	2	1	no	—
169	C₂×C₅₉	—	2	1	no	—
170	C₂×C₅₉	#1852	1	1	yes	characteristic
171	C₂×C₅₉	C₂×#368	1	1	yes	characteristic
172	C₂×C₅₉	#2161	1	1	yes	characteristic
173	D₅₉	—	2	2	no	—
174	D₅₉	—	4	1	no	—
175	C₂²×M₁₆	—	118	1	no	—
176	C₂³×C₈	—	59	1	no	—
177	C₂²×C₁₆	—	59	1	no	—
178	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
179	C₂×#108	—	59	1	no	—
180	C₂×#108	—	59	1	no	—
181	C₂×#108	—	118	2	no	—
182	C₂×#108	—	118	1	no	—
183	C₂×#108	—	118	2	no	—
184	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
185	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
186	#407	—	118	1	no	—
187	#368	—	2	2	no	—
188	#368	—	118	2	no	—
189	C₂²⋊C₁₆	—	59	2	no	—
190	C₂²⋊C₁₆	—	59	2	no	—
191	#341	—	118	1	no	—
192	C₅₉	#12922	1	1	yes	characteristic, complemented, Sylow
193	C₂³×C₄	—	59	1	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₂×M₁₆	—	2	1	no	—
196	C₂×M₁₆	—	118	4	no	—
197	C₂×M₁₆	—	118	1	no	—
198	C₂×M₁₆	—	118	1	no	—
199	C₂×M₁₆	—	236	1	no	—
200	C₂×M₁₆	—	236	2	no	—
201	C₂²×C₈	—	59	1	no	—
202	C₂²×C₈	—	59	1	no	—
203	C₂²×C₈	—	59	1	no	—
204	C₂²×C₈	—	59	1	no	—
205	C₂²×C₈	—	59	1	no	—
206	C₂²×C₈	—	118	1	no	—
207	C₂²×C₈	—	118	1	no	—
208	C₂²×C₈	—	118	1	no	—
209	C₂²×C₈	—	118	1	no	—
210	C₂²×C₈	—	236	1	no	—
211	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	59	2	no	—
215	C₂×C₁₆	—	118	1	no	—
216	C₂×C₁₆	—	118	1	no	—
217	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
218	#108	—	2	1	no	—
219	#108	—	2	1	no	—
220	#108	—	2	1	no	—
221	#108	—	59	2	no	—
222	#108	—	118	4	no	—
223	#108	—	118	1	no	—
224	#108	—	118	1	no	—
225	#108	—	118	1	no	—
226	#108	C₂²⋊D₅₉	1	2	yes	—
227	C₂²⋊C₈	—	236	1	no	—
228	C₂²⋊C₈	—	236	1	no	—
229	C₂⁴	—	59	1	no	—
230	C₂²×C₄	—	59	1	no	—
231	C₂²×C₄	—	59	1	no	—
232	C₂²×C₄	—	59	1	no	—
233	C₂²×C₄	—	59	1	no	—
234	C₂²×C₄	—	59	1	no	—
235	C₂²×C₄	—	118	1	no	—
236	C₂²×C₄	—	118	1	no	—
237	C₂²×C₄	—	118	1	no	—
238	C₂²×C₄	—	118	1	no	—
239	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
240	M₁₆	—	2	2	no	—
241	M₁₆	—	4	1	no	—
242	M₁₆	—	4	1	no	—
243	M₁₆	—	4	1	no	—
244	M₁₆	—	118	2	no	—
245	M₁₆	—	236	4	no	—
246	M₁₆	—	236	1	no	—
247	M₁₆	—	236	1	no	—
248	M₁₆	—	236	1	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	59	1	no	—
252	C₂×C₈	—	59	1	no	—
253	C₂×C₈	—	118	1	no	—
254	C₂×C₈	—	118	1	no	—
255	C₂×C₈	—	118	1	no	—
256	C₂×C₈	—	118	1	no	—
257	C₂×C₈	—	118	1	no	—
258	C₂×C₈	—	118	1	no	—
259	C₂×C₈	—	118	1	no	—
260	C₂×C₈	—	118	1	no	—
261	C₂×C₈	—	118	1	no	—
262	C₂×C₈	—	118	1	no	—
263	C₂×C₈	—	236	4	no	—
264	C₂×C₈	—	236	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	D₄×D₅₉	1	1	yes	characteristic
269	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
270	C₂×C₈	#11745689	1	1	yes	characteristic
271	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
272	C₁₆	—	2	2	no	—
273	C₁₆	—	118	2	no	—
274	C₂³	—	59	1	no	—
275	C₂³	—	59	1	no	—
276	C₂³	—	118	2	no	—
277	C₂³	—	118	1	no	—
278	C₂³	—	118	1	no	—
279	C₂³	—	236	1	no	—
280	C₂³	Q₁₆×D₅₉	1	1	yes	characteristic
281	C₂×C₄	—	59	1	no	—
282	C₂×C₄	—	59	1	no	—
283	C₂×C₄	—	118	1	no	—
284	C₂×C₄	—	118	1	no	—
285	C₂×C₄	—	118	1	no	—
286	C₂×C₄	—	118	1	no	—
287	C₂×C₄	—	118	1	no	—
288	C₂×C₄	—	118	1	no	—
289	C₂×C₄	—	118	1	no	—
290	C₂×C₄	—	118	1	no	—
291	C₂×C₄	—	118	1	no	—
292	C₂×C₄	—	118	1	no	—
293	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
294	C₂×C₄	#14273695	1	1	yes	characteristic
295	C₂×C₄	#14273712	1	1	yes	characteristic
296	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
297	C₂×C₄	#14273751	1	1	yes	characteristic
298	C₂×C₄	#14273759	1	1	yes	characteristic
299	C₈	—	4	2	no	—
300	C₈	—	8	1	no	—
301	C₈	—	118	1	no	—
302	C₈	—	118	1	no	—
303	C₈	—	472	1	no	—
304	C₈	—	236	2	no	—
305	C₈	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
306	C₈	#14273689	1	1	yes	characteristic
307	C₈	#14273703	1	1	yes	characteristic
308	C₈	#14273705	1	1	yes	characteristic
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	59	2	no	—
312	C₂²	—	118	1	no	—
313	C₂²	—	118	1	no	—
314	C₂²	—	118	1	no	—
315	C₂²	—	118	1	no	—
316	C₂²	—	118	1	no	—
317	C₂²	—	236	2	no	—
318	C₂²	—	236	2	no	—
319	C₂²	#107×D₅₉	1	1	yes	characteristic
320	C₂²	#16837556	1	1	yes	characteristic
321	C₂²	#16837739	1	1	yes	characteristic
322	C₄	—	118	1	no	—
323	C₄	—	118	1	no	—
324	C₄	#16837273	1	1	yes	characteristic
325	C₄	#16837443	1	1	yes	characteristic
326	C₄	#16837534	1	1	yes	characteristic
327	C₄	#16837717	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	118	2	no	—
331	C₂	—	236	1	no	—
332	C₂	#21424282	1	1	yes	characteristic
333	C₂	#368×D₅₉	1	1	yes	characteristic
334	C₂	#21425889	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#368	C₂
socle	C₂²×C₅₉	C₂×#107
prime radical	C₂³×D₅₉	Q₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#368	D₅₉
hypocenter	C₅₉	#12922
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
2. C₂×#16837231	C₂	semidirect
5. #21422190	C₂	semidirect
6. #21422189	C₂	semidirect
7. C₂×C₅₉×#368	C₂	semidirect
154. C₂×D₅₉	#368	semidirect
163. C₂×#368	D₅₉	semidirect
192. C₅₉	#12922	semidirect

Maximal subgroups

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

subgroup	index
C₂×#108×D₅₉	2
C₂×#16837231	2
C₂×(C₂×D₅₉)⋊C₁₆	2
#21424131	2
#21422190	2
#21422189	2
C₂×C₅₉×#368	2
#12922	59