Subgroups — #28503786

Order 15104.

410 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837818	C₂	1	1	yes	maximal, characteristic
2	#21425716	C₂	1	1	yes	maximal, characteristic
3	#21425692	C₂	1	1	yes	maximal, characteristic, complemented
4	#21423897	C₂	1	1	yes	maximal, characteristic
5	#21422183	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422182	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273733	C₂²	1	1	yes	characteristic
9	#16837818	—	2	1	no	—
10	#16837818	—	2	1	no	—
11	#16837818	—	2	1	no	—
12	#16837818	—	2	1	no	—
13	C₂×#14273667	C₂²	1	1	yes	characteristic
14	C₂×#14273667	C₂²	1	1	yes	characteristic
15	C₂×#14273666	C₄	1	2	yes	complemented
16	C₂×C₁₆×D₅₉	C₄	1	2	yes	complemented
17	#16837384	C₄	1	1	yes	characteristic
18	#16837375	C₄	1	1	yes	characteristic
19	#16837357	C₂²	1	1	yes	characteristic
20	#16837183	C₄	1	4	yes	complemented
21	#16837177	—	2	1	no	—
22	C₁₆×Dic₅₉	—	2	1	no	—
23	#16837174	C₄	1	1	yes	characteristic
24	#16837172	C₄	1	1	yes	characteristic
25	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
26	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
27	C₅₉×#405	C₂²	1	1	yes	characteristic
28	C₅₉×#346	—	2	1	no	—
29	C₄×C₁₆×C₅₉	—	2	1	no	—
30	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
31	#14273733	—	2	1	no	—
32	#14273733	—	2	1	no	—
33	#14273733	—	2	1	no	—
34	#14273733	—	2	1	no	—
35	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
36	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
37	C₂×#11745659	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₅₉	C₂×C₄	1	2	yes	—
39	#14273686	C₂×C₄	1	1	yes	characteristic
40	#14273667	—	2	1	no	—
41	#14273667	—	2	1	no	—
42	#14273667	—	2	1	no	—
43	#14273667	C₂×C₄	1	2	yes	—
44	#14273666	—	2	2	no	—
45	#14273666	—	2	2	no	—
46	C₂×C₅₉⋊C₁₆	—	2	1	no	—
47	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
48	C₁₆×D₅₉	—	2	2	no	—
49	C₁₆×D₅₉	—	2	2	no	—
50	#14273645	C₈	1	2	yes	—
51	#14273643	C₈	1	2	yes	—
52	#14273641	C₈	1	4	yes	—
53	#14273637	—	2	1	no	—
54	C₈×Dic₅₉	—	2	1	no	—
55	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
57	#117×C₅₉	C₂×C₄	1	1	yes	characteristic
58	M₃₂×C₅₉	—	2	1	no	—
59	M₃₂×C₅₉	C₂×C₄	1	2	yes	—
60	C₂×C₁₆×C₅₉	—	2	1	no	—
61	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
62	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
63	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
64	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
65	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
66	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
67	#97×C₅₉	—	2	1	no	—
68	C₄×C₈×C₅₉	—	2	1	no	—
69	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊D₅₉	C₂×C₈	1	2	yes	—
71	#11745685	—	2	1	no	—
72	#11745685	—	2	1	no	—
73	#11745685	—	2	1	no	—
74	#11745685	—	2	1	no	—
75	C₂×D₂₃₆	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄×D₅₉	C₄²	1	2	yes	—
77	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
78	M₁₆×C₅₉	—	2	1	no	—
79	M₁₆×C₅₉	C₂×C₈	1	2	yes	—
80	C₂×C₈×C₅₉	—	2	1	no	—
81	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
86	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
87	C₄⋊C₄×C₅₉	—	2	1	no	—
88	C₂²⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
89	C₄²×C₅₉	—	2	1	no	—
90	C₂²⋊Dic₅₉	C₂×C₈	1	2	yes	—
91	C₄⋊Dic₅₉	—	2	1	no	—
92	C₄×Dic₅₉	—	2	1	no	—
93	#11745664	—	2	1	no	—
94	#11745664	—	2	1	no	—
95	#11745664	—	2	1	no	—
96	#11745664	C₂×C₈	1	2	yes	—
97	C₂×C₅₉⋊C₈	—	2	1	no	—
98	C₂×C₅₉⋊C₈	C₄²	1	2	yes	—
99	#11745659	—	2	2	no	—
100	#11745659	—	2	2	no	—
101	C₈×D₅₉	—	2	2	no	—
102	C₈×D₅₉	—	2	2	no	—
103	C₁₆×C₅₉	—	2	2	no	—
104	C₁₆×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
105	C₁₆×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
106	C₁₆×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
107	C₁₆×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
108	C₅₉⋊C₁₆	—	2	2	no	—
109	C₅₉⋊C₁₆	—	2	2	no	—
110	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
111	C₂²×D₅₉	C₄×C₈	1	2	yes	—
112	C₂×C₄×C₅₉	—	2	2	no	—
113	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
118	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
119	C₂²⋊D₅₉	—	2	2	no	—
120	C₂²⋊D₅₉	—	2	2	no	—
121	C₂×Dic₅₉	—	2	2	no	—
122	C₂×Dic₅₉	C₄×C₈	1	2	yes	—
123	D₂₃₆	—	2	1	no	—
124	D₂₃₆	—	2	1	no	—
125	C₄×D₅₉	—	2	2	no	—
126	C₄×D₅₉	—	2	2	no	—
127	Dic₁₁₈	—	2	1	no	—
128	Dic₁₁₈	—	2	1	no	—
129	C₈×C₅₉	—	2	2	no	—
130	C₈×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₈×C₅₉	#117	1	1	yes	characteristic
132	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
133	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
134	C₅₉⋊C₈	—	2	2	no	—
135	C₅₉⋊C₈	—	2	2	no	—
136	#12613	—	59	1	no	maximal, Sylow
137	C₂²×C₅₉	—	4	1	no	—
138	C₂²×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
139	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
140	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
141	C₂×D₅₉	—	2	2	no	—
142	C₂×D₅₉	—	4	2	no	—
143	C₄×C₅₉	—	4	2	no	—
144	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₄×C₅₉	#405	1	1	yes	characteristic
146	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
147	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
148	Dic₅₉	—	2	2	no	—
149	Dic₅₉	—	4	2	no	—
150	C₂×#504	—	59	1	no	—
151	#2115	—	59	1	no	—
152	#2108	—	59	1	no	—
153	#2108	D₅₉	1	1	yes	characteristic, complemented
154	#1757	—	59	1	no	—
155	#1378	—	59	1	no	—
156	#1378	—	59	1	no	—
157	C₂×C₅₉	—	2	1	no	—
158	C₂×C₅₉	—	2	1	no	—
159	C₂×C₅₉	—	2	1	no	—
160	C₂×C₅₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
161	D₅₉	—	4	2	no	—
162	C₂×#131	—	59	1	no	—
163	#504	—	118	1	no	—
164	#504	—	118	1	no	—
165	#504	—	118	1	no	—
166	#504	—	118	1	no	—
167	C₂×M₃₂	—	59	1	no	—
168	C₂×M₃₂	—	59	2	no	—
169	C₂×M₃₂	—	59	1	no	—
170	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
171	C₂²×C₁₆	—	59	2	no	—
172	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
173	#413	—	59	1	no	—
174	#410	—	59	1	no	—
175	#405	—	59	1	no	—
176	#405	C₂×D₅₉	1	1	yes	characteristic
177	#349	—	59	4	no	—
178	#346	—	2	1	no	—
179	#346	—	118	1	no	—
180	C₄×C₁₆	—	2	1	no	—
181	C₄×C₁₆	—	118	1	no	—
182	#344	—	59	1	no	—
183	#344	—	59	1	no	—
184	C₅₉	#12613	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	59	1	no	—
186	#131	—	118	1	no	—
187	#131	—	118	1	no	—
188	#131	—	118	1	no	—
189	#131	—	118	1	no	—
190	C₂×M₁₆	—	59	1	no	—
191	C₂×M₁₆	—	59	2	no	—
192	C₂×M₁₆	—	59	1	no	—
193	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
194	C₂²×C₈	—	59	2	no	—
195	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
196	#117	—	59	1	no	—
197	#117	C₄×D₅₉	1	1	yes	characteristic
198	M₃₂	—	2	1	no	—
199	M₃₂	—	59	2	no	—
200	M₃₂	—	118	1	no	—
201	M₃₂	—	118	1	no	—
202	M₃₂	—	118	1	no	—
203	M₃₂	—	118	2	no	—
204	M₃₂	—	118	2	no	—
205	M₃₂	C₄×D₅₉	1	2	yes	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	59	2	no	—
208	C₂×C₁₆	—	118	1	no	—
209	C₂×C₁₆	—	118	2	no	—
210	C₂×C₁₆	—	118	2	no	—
211	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
212	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
213	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
214	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
215	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
216	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
217	#101	—	59	2	no	—
218	#100	—	59	2	no	—
219	#99	—	59	4	no	—
220	#97	—	2	1	no	—
221	#97	—	118	1	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	118	1	no	—
224	#41	—	118	1	no	—
225	#41	—	118	1	no	—
226	#41	—	118	1	no	—
227	#41	—	118	1	no	—
228	C₂×Q₈	—	59	1	no	—
229	C₂×D₄	—	59	2	no	—
230	C₂×D₄	—	59	1	no	—
231	C₂²×C₄	—	59	2	no	—
232	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
233	M₁₆	—	2	1	no	—
234	M₁₆	—	59	2	no	—
235	M₁₆	—	118	1	no	—
236	M₁₆	—	118	1	no	—
237	M₁₆	—	118	1	no	—
238	M₁₆	—	118	2	no	—
239	M₁₆	—	118	2	no	—
240	M₁₆	C₈×D₅₉	1	2	yes	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	59	2	no	—
243	C₂×C₈	—	118	1	no	—
244	C₂×C₈	—	118	2	no	—
245	C₂×C₈	—	118	2	no	—
246	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
248	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
249	C₂×C₈	#11745685	1	1	yes	characteristic
250	C₂×C₈	#11745685	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	118	1	no	—
254	C₂²⋊C₄	—	59	2	no	—
255	C₂²⋊C₄	C₈×D₅₉	1	2	yes	—
256	C₄²	—	2	1	no	—
257	C₄²	—	118	1	no	—
258	C₁₆	—	2	2	no	—
259	C₁₆	—	118	2	no	—
260	C₁₆	—	118	2	no	—
261	C₁₆	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
262	C₁₆	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
263	C₁₆	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
264	C₁₆	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
265	C₂³	—	59	2	no	—
266	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
267	Q₈	—	118	1	no	—
268	Q₈	—	118	1	no	—
269	D₄	—	118	2	no	—
270	D₄	—	118	2	no	—
271	D₄	—	118	1	no	—
272	D₄	—	118	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	59	2	no	—
275	C₂×C₄	—	118	2	no	—
276	C₂×C₄	—	118	2	no	—
277	C₂×C₄	—	118	2	no	—
278	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
279	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
280	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
281	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
282	C₂×C₄	#14273733	1	1	yes	characteristic
283	C₂×C₄	#14273733	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	118	2	no	—
286	C₈	—	118	2	no	—
287	C₈	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
288	C₈	#14273684	1	1	yes	characteristic
289	C₈	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
290	C₈	C₄×D₂₃₆	1	1	yes	characteristic
291	C₂²	—	4	1	no	—
292	C₂²	—	118	2	no	—
293	C₂²	—	236	2	no	—
294	C₂²	C₄×C₈×D₅₉	1	1	yes	characteristic
295	C₂²	C₈×C₂²⋊D₅₉	1	1	yes	characteristic
296	C₂²	C₈×D₂₃₆	1	1	yes	characteristic
297	C₄	—	4	2	no	—
298	C₄	—	118	2	no	—
299	C₄	—	236	2	no	—
300	C₄	C₄×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
301	C₄	#16837356	1	1	yes	characteristic
302	C₄	C₈×C₂²⋊D₅₉	1	1	yes	characteristic
303	C₄	C₈×D₂₃₆	1	1	yes	characteristic
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	236	2	no	—
308	C₂	C₈×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	(C₂×D₅₉)⋊C₄
commutator	C₂²×C₅₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2108	C₂
socle	C₂×C₅₉	C₈×C₂²⋊C₄
prime radical	C₂×#11745685	C₂×C₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2108	D₅₉
hypocenter	C₅₉	#12613
LCS	G ▷ C₂²×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
3. #21425692	C₂	semidirect
5. #21422183	C₂	semidirect
6. #21422182	C₂	semidirect
7. C₅₉×#2108	C₂	semidirect
15. C₂×#14273666	C₄	semidirect
16. C₂×C₁₆×D₅₉	C₄	semidirect
20. #16837183	C₄	semidirect
153. #2108	D₅₉	semidirect
184. C₅₉	#12613	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837818	2
#21425716	2
#21425692	2
#21423897	2
#21422183	2
#21422182	2
C₅₉×#2108	2
#12613	59