Subgroups — #28506431

Order 15104.

326 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837223	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16837185	C₂	1	1	yes	maximal, characteristic
3	#21425709	C₂	1	1	yes	maximal, characteristic
4	#21425692	C₂	1	1	yes	maximal, characteristic, complemented
5	#21423933	C₂	1	1	yes	maximal, characteristic
6	#21422184	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273733	C₂²	1	1	yes	characteristic
9	C₂×#14273667	C₂²	1	1	yes	characteristic
10	C₂×#14273667	C₂²	1	1	yes	characteristic
11	C₂×#14273655	C₄	1	1	yes	characteristic
12	#16837399	C₄	1	1	yes	characteristic
13	#16837357	C₂²	1	1	yes	characteristic
14	#16837223	—	2	2	no	—
15	#16837185	—	2	2	no	—
16	(C₂×D₅₉)⋊C₁₆	—	2	1	no	—
17	(C₂×D₅₉)⋊C₁₆	—	2	1	no	—
18	#16837177	—	2	1	no	—
19	C₁₆×Dic₅₉	—	2	1	no	—
20	#16837167	C₄	1	1	yes	characteristic
21	#16837162	C₄	1	1	yes	characteristic
22	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
26	#14273733	—	2	1	no	—
27	#14273733	—	2	1	no	—
28	#14273733	—	2	1	no	—
29	#14273733	—	2	1	no	—
30	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
31	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
32	C₂×#11745659	—	2	1	no	—
33	C₂×C₈×D₅₉	—	2	1	no	—
34	#14273686	C₂×C₄	1	1	yes	characteristic
35	#14273667	—	2	1	no	—
36	#14273667	—	2	1	no	—
37	#14273667	—	2	1	no	—
38	#14273667	D₄	1	2	yes	—
39	C₂×C₅₉⋊C₁₆	—	2	1	no	—
40	C₂×C₅₉⋊C₁₆	—	2	1	no	—
41	#14273655	—	2	2	no	—
42	#14273652	C₈	1	1	yes	characteristic
43	#14273652	C₈	1	1	yes	characteristic
44	#14273649	C₈	1	1	yes	characteristic
45	#14273649	C₈	1	1	yes	characteristic
46	#14273637	—	2	1	no	—
47	C₈×Dic₅₉	—	2	1	no	—
48	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
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	—	2	1	no	—
56	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
57	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
58	C₄×C₈×C₅₉	—	2	1	no	—
59	C₄×C₈×C₅₉	—	2	1	no	—
60	C₂²×C₄×C₅₉	—	2	1	no	—
61	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₉	—	2	1	no	—
63	#11745685	—	2	1	no	—
64	#11745685	—	2	1	no	—
65	#11745685	—	2	1	no	—
66	#11745685	—	2	1	no	—
67	C₂×D₂₃₆	C₂×C₈	1	1	yes	characteristic
68	C₂×C₄×D₅₉	—	2	1	no	—
69	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
70	C₂×C₈×C₅₉	—	2	1	no	—
71	C₂×C₈×C₅₉	—	2	1	no	—
72	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
73	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
74	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
77	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
78	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
79	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
80	C₄²×C₅₉	—	2	1	no	—
81	C₄²×C₅₉	—	2	1	no	—
82	C₂²⋊Dic₅₉	—	2	1	no	—
83	C₄⋊Dic₅₉	C₂×C₈	1	1	yes	characteristic
84	C₄⋊Dic₅₉	C₂×C₈	1	1	yes	characteristic
85	C₄×Dic₅₉	—	2	1	no	—
86	#11745664	—	2	1	no	—
87	#11745664	—	2	1	no	—
88	#11745664	—	2	1	no	—
89	#11745664	M₁₆	1	2	yes	—
90	C₂×C₅₉⋊C₈	—	2	1	no	—
91	C₂×C₅₉⋊C₈	—	2	1	no	—
92	#11745659	—	4	1	no	—
93	#11745659	—	4	1	no	—
94	C₈×D₅₉	—	4	1	no	—
95	C₈×D₅₉	—	4	1	no	—
96	C₁₆×C₅₉	—	2	1	no	—
97	C₁₆×C₅₉	—	2	1	no	—
98	C₁₆×C₅₉	—	4	1	no	—
99	C₅₉⋊C₁₆	—	2	2	no	—
100	C₅₉⋊C₁₆	—	4	1	no	—
101	C₂³×C₅₉	#131	1	1	yes	characteristic
102	C₂²×D₅₉	—	2	1	no	—
103	C₂×C₄×C₅₉	—	2	2	no	—
104	C₂×C₄×C₅₉	—	4	1	no	—
105	C₂×C₄×C₅₉	—	4	1	no	—
106	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
112	C₂²⋊D₅₉	—	4	1	no	—
113	C₂²⋊D₅₉	—	4	1	no	—
114	C₂×Dic₅₉	—	2	1	no	—
115	C₂×Dic₅₉	—	2	1	no	—
116	C₂×Dic₅₉	—	2	1	no	—
117	D₂₃₆	—	2	1	no	—
118	D₂₃₆	—	2	1	no	—
119	C₄×D₅₉	—	4	1	no	—
120	C₄×D₅₉	—	4	1	no	—
121	Dic₁₁₈	—	2	1	no	—
122	Dic₁₁₈	—	2	1	no	—
123	C₈×C₅₉	—	2	1	no	—
124	C₈×C₅₉	—	2	1	no	—
125	C₈×C₅₉	#121	1	1	yes	characteristic
126	C₈×C₅₉	#123	1	1	yes	characteristic
127	C₈×C₅₉	#124	1	1	yes	characteristic
128	C₈×C₅₉	#126	1	1	yes	characteristic
129	C₅₉⋊C₈	—	2	2	no	—
130	C₅₉⋊C₈	—	4	1	no	—
131	#13122	—	59	1	no	maximal, Sylow
132	C₂²×C₅₉	—	2	1	no	—
133	C₂²×C₅₉	—	2	1	no	—
134	C₂²×C₅₉	#433	1	1	yes	characteristic
135	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
136	C₂²×C₅₉	#436	1	1	yes	characteristic
137	C₂×D₅₉	—	4	1	no	—
138	C₂×D₅₉	—	4	1	no	—
139	C₂×D₅₉	—	4	1	no	—
140	C₄×C₅₉	—	4	2	no	—
141	C₄×C₅₉	#388	1	1	yes	characteristic
142	C₄×C₅₉	#431	1	1	yes	characteristic
143	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
144	C₄×C₅₉	#435	1	1	yes	characteristic
145	Dic₅₉	—	4	1	no	—
146	Dic₅₉	—	4	1	no	—
147	Dic₅₉	—	4	1	no	—
148	C₂×#364	—	59	1	no	—
149	C₂×#350	—	59	1	no	—
150	#2113	—	59	1	no	—
151	#2108	—	59	1	no	—
152	#1764	—	59	1	no	—
153	#1379	—	59	1	no	—
154	#1374	D₅₉	1	1	yes	characteristic, complemented
155	C₂×C₅₉	—	2	1	no	—
156	C₂×C₅₉	—	2	1	no	—
157	C₂×C₅₉	#1923	1	1	yes	characteristic
158	C₂×C₅₉	#2170	1	1	yes	characteristic
159	C₂×C₅₉	#2171	1	1	yes	characteristic
160	D₅₉	—	8	1	no	—
161	C₂×#131	—	59	1	no	—
162	C₂×M₃₂	—	59	1	no	—
163	C₂×M₃₂	—	59	1	no	—
164	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
165	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
166	C₂×#104	—	59	1	no	—
167	#416	—	59	1	no	—
168	#405	—	59	1	no	—
169	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
170	#364	—	118	2	no	—
171	#350	—	118	2	no	—
172	C₂²⋊C₁₆	—	118	1	no	—
173	C₂²⋊C₁₆	—	118	1	no	—
174	#346	—	118	1	no	—
175	C₄×C₁₆	—	118	1	no	—
176	#341	—	59	1	no	—
177	#339	—	59	1	no	—
178	C₅₉	#13122	1	1	yes	characteristic, complemented, Sylow
179	C₂×#41	—	59	1	no	—
180	#131	—	118	1	no	—
181	#131	—	118	1	no	—
182	#131	—	118	1	no	—
183	#131	—	118	1	no	—
184	C₂×M₁₆	—	59	1	no	—
185	C₂×M₁₆	—	59	1	no	—
186	C₂×M₁₆	—	118	1	no	—
187	C₂²×C₈	—	118	1	no	—
188	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
190	#117	—	59	1	no	—
191	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
192	M₃₂	—	59	2	no	—
193	M₃₂	—	118	1	no	—
194	M₃₂	—	118	1	no	—
195	M₃₂	—	118	1	no	—
196	C₂×C₁₆	—	2	1	no	—
197	C₂×C₁₆	—	2	1	no	—
198	C₂×C₁₆	—	2	1	no	—
199	C₂×C₁₆	—	2	1	no	—
200	C₂×C₁₆	—	2	1	no	—
201	C₂×C₁₆	—	118	1	no	—
202	C₂×C₁₆	—	118	1	no	—
203	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
204	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
205	#104	—	118	2	no	—
206	#103	—	59	1	no	—
207	#103	—	59	1	no	—
208	#102	—	59	1	no	—
209	#102	—	59	1	no	—
210	#97	—	118	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	118	1	no	—
214	#41	—	118	1	no	—
215	#41	—	118	1	no	—
216	#41	—	118	1	no	—
217	#41	—	118	1	no	—
218	C₂×Q₈	—	59	1	no	—
219	C₂×D₄	—	59	1	no	—
220	C₂×D₄	—	118	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	118	1	no	—
223	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
224	M₁₆	—	59	2	no	—
225	M₁₆	—	118	1	no	—
226	M₁₆	—	118	1	no	—
227	M₁₆	—	118	1	no	—
228	M₁₆	—	236	1	no	—
229	M₁₆	—	236	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	118	1	no	—
233	C₂×C₈	—	118	1	no	—
234	C₂×C₈	—	236	1	no	—
235	C₂×C₈	—	236	1	no	—
236	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
237	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
238	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
239	C₂×C₈	#11745685	1	1	yes	characteristic
240	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
241	C₂×C₈	#11745687	1	1	yes	characteristic
242	C₂×C₈	#11745687	1	1	yes	characteristic
243	C₂×C₈	#11745689	1	1	yes	characteristic
244	C₄⋊C₄	—	59	1	no	—
245	C₄⋊C₄	—	59	1	no	—
246	C₂²⋊C₄	—	118	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	118	1	no	—
250	C₁₆	—	2	1	no	—
251	C₁₆	—	2	1	no	—
252	C₁₆	—	4	1	no	—
253	C₁₆	—	118	2	no	—
254	C₁₆	—	236	1	no	—
255	C₂³	—	118	1	no	—
256	C₂³	#14273733	1	1	yes	characteristic
257	Q₈	—	118	1	no	—
258	Q₈	—	118	1	no	—
259	D₄	—	118	1	no	—
260	D₄	—	118	1	no	—
261	D₄	—	236	1	no	—
262	D₄	—	236	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	118	1	no	—
267	C₂×C₄	—	118	1	no	—
268	C₂×C₄	—	118	1	no	—
269	C₂×C₄	—	236	1	no	—
270	C₂×C₄	—	236	1	no	—
271	C₂×C₄	#14273685	1	1	yes	characteristic
272	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
274	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
275	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
276	C₂×C₄	#14273735	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	118	2	no	—
280	C₈	—	236	1	no	—
281	C₈	#14273702	1	1	yes	characteristic
282	C₈	#14273711	1	1	yes	characteristic
283	C₈	#14273713	1	1	yes	characteristic
284	C₈	#14273722	1	1	yes	characteristic
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	236	1	no	—
288	C₂²	—	236	1	no	—
289	C₂²	—	236	1	no	—
290	C₂²	#16837462	1	1	yes	characteristic
291	C₂²	C₈×D₂₃₆	1	1	yes	characteristic
292	C₂²	#16837477	1	1	yes	characteristic
293	C₄	—	4	2	no	—
294	C₄	—	236	1	no	—
295	C₄	—	236	1	no	—
296	C₄	—	236	1	no	—
297	C₄	#16837294	1	1	yes	characteristic
298	C₄	#16837454	1	1	yes	characteristic
299	C₄	(C₂²⋊D₅₉)⋊C₈	1	1	yes	characteristic
300	C₄	#16837475	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	472	1	no	—
304	C₂	#21424703	1	1	yes	characteristic
305	C₂	#21425934	1	1	yes	characteristic
306	C₂	#21425939	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1374	C₂
socle	C₂²×C₅₉	C₈×D₄
prime radical	C₂×#11745685	C₂×C₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1374	D₅₉
hypocenter	C₅₉	#13122
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ #1374

Complements

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

N	H ≅ G/N	type
1. C₂×#16837223	C₂	semidirect
4. #21425692	C₂	semidirect
6. #21422184	C₂	semidirect
7. C₅₉×#1374	C₂	semidirect
154. #1374	D₅₉	semidirect
178. C₅₉	#13122	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837223	2
C₂×#16837185	2
#21425709	2
#21425692	2
#21423933	2
#21422184	2
C₅₉×#1374	2
#13122	59