Subgroups — #28481730

Order 15104.

542 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×C₈×D₂₃₆	C₂	1	1	yes	maximal, characteristic
2	#21424093	C₂	1	2	yes	maximal
3	C₂×#16837153	C₂	1	1	yes	maximal, characteristic, complemented
4	#21421979	C₂	1	2	yes	maximal, complemented
5	C₅₉×#1844	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₅₉	—	2	1	no	—
7	C₂×C₄×D₂₃₆	C₂²	1	1	yes	characteristic
8	C₈×D₂₃₆	—	2	2	no	—
9	C₈×D₂₃₆	C₄	1	4	yes	complemented
10	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#14273647	—	2	2	no	—
13	C₂×(C₂×D₅₉)⋊C₈	—	2	1	no	—
14	#16837164	—	2	2	no	—
15	#16837153	—	2	2	no	—
16	C₂×#106×C₅₉	—	2	1	no	—
17	C₂×#106×C₅₉	—	2	1	no	—
18	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₅₉×#384	C₂²	1	2	yes	—
20	C₂²×D₂₃₆	D₄	1	1	yes	characteristic
21	C₂²×C₄×D₅₉	—	2	1	no	—
22	C₂²×C₅₉⋊C₈	—	2	1	no	—
23	C₂²×C₅₉⋊C₈	—	2	2	no	—
24	C₂×C₈×D₅₉	—	2	4	no	—
25	C₂×C₈×D₅₉	—	4	2	no	—
26	C₂×C₈×D₅₉	—	4	2	no	—
27	C₄×D₂₃₆	—	2	1	no	—
28	C₄×D₂₃₆	—	2	1	no	—
29	C₄×D₂₃₆	C₂×C₄	1	2	yes	—
30	C₄×D₂₃₆	Q₈	1	2	yes	—
31	C₂×C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
32	C₂×(C₂×D₅₉)⋊C₄	—	2	1	no	—
33	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
34	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
35	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
36	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
37	#14273647	—	4	4	no	—
38	(C₂×D₅₉)⋊C₈	—	2	4	no	—
39	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
44	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
45	#106×C₅₉	—	4	2	no	—
46	#106×C₅₉	—	4	2	no	—
47	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₅₉	D₄	1	2	yes	—
49	C₂³×D₅₉	—	2	1	no	—
50	C₂²×C₄×C₅₉	—	4	2	no	—
51	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
54	C₂²×Dic₅₉	—	2	1	no	—
55	C₂×D₂₃₆	—	2	1	no	—
56	C₂×D₂₃₆	—	2	2	no	—
57	C₂×D₂₃₆	C₄⋊C₄	1	2	yes	—
58	C₂×D₂₃₆	SD₁₆	1	4	yes	—
59	C₂×C₄×D₅₉	—	2	2	no	—
60	C₂×C₄×D₅₉	—	2	2	no	—
61	C₂×C₄×D₅₉	—	4	1	no	—
62	C₂×C₄×D₅₉	—	4	1	no	—
63	C₂×C₄×D₅₉	—	4	1	no	—
64	C₂×C₄×D₅₉	—	4	1	no	—
65	C₂×C₈×C₅₉	—	2	1	no	—
66	C₂×C₈×C₅₉	—	2	1	no	—
67	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₅₉	D₈	1	2	yes	—
70	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
71	C₄⋊C₄×C₅₉	—	4	2	no	—
72	C₄⋊C₄×C₅₉	—	4	2	no	—
73	C₄⋊C₄×C₅₉	—	4	2	no	—
74	C₄⋊C₄×C₅₉	—	4	2	no	—
75	C₄⋊C₄×C₅₉	—	8	2	no	—
76	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
79	C₄²×C₅₉	#41	1	1	yes	characteristic
80	(C₂×D₅₉)⋊C₄	—	2	2	no	—
81	(C₂×D₅₉)⋊C₄	—	2	2	no	—
82	C₄⋊Dic₅₉	—	2	1	no	—
83	C₄⋊Dic₅₉	C₄⋊C₄	1	2	yes	—
84	C₂×C₅₉⋊C₈	—	2	2	no	—
85	C₂×C₅₉⋊C₈	—	4	1	no	—
86	C₂×C₅₉⋊C₈	—	4	1	no	—
87	C₂×C₅₉⋊C₈	—	4	2	no	—
88	C₂×C₅₉⋊C₈	—	4	2	no	—
89	C₂×C₅₉⋊C₈	—	4	2	no	—
90	C₈×D₅₉	—	4	8	no	—
91	C₂³×C₅₉	#121	1	1	yes	characteristic
92	C₂²×D₅₉	—	2	2	no	—
93	C₂²×D₅₉	—	2	4	no	—
94	C₂²×D₅₉	—	4	1	no	—
95	C₂²×D₅₉	—	4	1	no	—
96	C₂²×D₅₉	—	4	2	no	—
97	C₂×C₄×C₅₉	—	2	2	no	—
98	C₂×C₄×C₅₉	—	8	2	no	—
99	C₂×C₄×C₅₉	—	8	2	no	—
100	C₂×C₄×C₅₉	—	8	2	no	—
101	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
104	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
109	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅₉	#135	1	2	yes	—
111	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
113	C₂×Dic₅₉	—	2	2	no	—
114	C₂×Dic₅₉	—	4	1	no	—
115	C₂×Dic₅₉	—	4	1	no	—
116	D₂₃₆	—	2	2	no	—
117	D₂₃₆	—	2	4	no	—
118	D₂₃₆	#106	1	4	yes	complemented
119	C₄×D₅₉	—	4	2	no	—
120	C₄×D₅₉	—	4	2	no	—
121	C₄×D₅₉	—	4	2	no	—
122	C₄×D₅₉	—	4	2	no	—
123	C₈×C₅₉	—	2	2	no	—
124	C₈×C₅₉	#102	1	4	yes	complemented
125	C₅₉⋊C₈	—	4	2	no	—
126	C₅₉⋊C₈	—	8	2	no	—
127	#8642	—	59	1	no	maximal, Sylow
128	C₂²×C₅₉	#380	1	1	yes	characteristic
129	C₂²×C₅₉	#459	1	1	yes	characteristic
130	C₂²×C₅₉	#461	1	1	yes	characteristic
131	C₂²×C₅₉	#465	1	2	yes	—
132	C₂²×C₅₉	#471	1	2	yes	—
133	C₂×D₅₉	—	2	4	no	—
134	C₂×D₅₉	—	4	2	no	—
135	C₂×D₅₉	—	4	2	no	—
136	C₂×D₅₉	—	4	4	no	—
137	C₂×D₅₉	—	4	4	no	—
138	C₄×C₅₉	—	2	1	no	—
139	C₄×C₅₉	—	2	1	no	—
140	C₄×C₅₉	—	16	2	no	—
141	C₄×C₅₉	C₂×#102	1	1	yes	characteristic
142	C₄×C₅₉	#417	1	1	yes	characteristic
143	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
144	C₄×C₅₉	#427	1	1	yes	characteristic
145	C₄×C₅₉	#465	1	2	yes	—
146	C₄×C₅₉	#474	1	1	yes	characteristic
147	C₄×C₅₉	#476	1	1	yes	characteristic
148	Dic₅₉	—	4	2	no	—
149	C₂×C₈×D₄	—	59	1	no	—
150	#1844	D₅₉	1	1	yes	characteristic, complemented
151	#1802	—	59	2	no	—
152	C₂×#334	—	59	1	no	—
153	#1296	—	59	2	no	—
154	C₂×C₅₉	#1665	1	2	yes	—
155	C₂×C₅₉	#1671	1	2	yes	—
156	C₂×C₅₉	#1802	1	1	yes	characteristic
157	C₂×C₅₉	#1807	1	1	yes	characteristic
158	C₂×C₅₉	#1815	1	1	yes	characteristic
159	D₅₉	—	4	4	no	—
160	C₂³×C₈	—	118	1	no	—
161	C₂×C₄×D₄	—	59	1	no	—
162	C₈×D₄	—	59	4	no	—
163	C₈×D₄	—	118	2	no	—
164	C₂×#106	—	2	1	no	—
165	C₂×#106	—	2	1	no	—
166	C₂×C₄⋊C₈	—	59	1	no	—
167	C₂×C₄⋊C₈	—	59	2	no	—
168	C₂×#102	—	118	2	no	—
169	C₂×C₂²⋊C₈	—	118	1	no	—
170	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
171	#384	C₂×D₅₉	1	2	yes	—
172	#340	—	118	2	no	—
173	#334	—	118	2	no	—
174	C₅₉	#8642	1	1	yes	characteristic, complemented, Sylow
175	C₂²×D₄	—	59	1	no	—
176	C₂³×C₄	—	118	1	no	—
177	C₂²×C₈	—	118	1	no	—
178	C₂²×C₈	—	118	4	no	—
179	C₂²×C₈	—	118	2	no	—
180	C₂²×C₈	—	236	2	no	—
181	C₂²×C₈	—	236	2	no	—
182	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
183	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
184	C₄×D₄	—	59	2	no	—
185	C₄×D₄	—	59	2	no	—
186	C₄×D₄	—	118	1	no	—
187	C₄×D₄	—	118	1	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	59	1	no	—
192	C₂×C₂²⋊C₄	—	118	1	no	—
193	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
194	#106	—	4	2	no	—
195	#106	—	4	2	no	—
196	C₄⋊C₈	—	59	2	no	—
197	C₄⋊C₈	—	118	1	no	—
198	C₄⋊C₈	—	118	2	no	—
199	C₄⋊C₈	—	118	2	no	—
200	#102	—	236	4	no	—
201	C₂²⋊C₈	—	118	4	no	—
202	C₄×C₈	C₄×D₅₉	1	2	yes	—
203	C₄×C₈	D₂₃₆	1	2	yes	—
204	C₂⁴	—	118	1	no	—
205	C₂×D₄	—	59	2	no	—
206	C₂×D₄	—	59	4	no	—
207	C₂×D₄	—	118	1	no	—
208	C₂×D₄	—	118	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	118	1	no	—
211	C₂²×C₄	—	118	2	no	—
212	C₂²×C₄	—	118	2	no	—
213	C₂²×C₄	—	236	1	no	—
214	C₂²×C₄	—	236	1	no	—
215	C₂²×C₄	—	236	1	no	—
216	C₂²×C₄	—	236	1	no	—
217	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
218	C₂²×C₄	#11745687	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	118	2	no	—
223	C₂×C₈	—	236	2	no	—
224	C₂×C₈	—	236	2	no	—
225	C₂×C₈	—	236	2	no	—
226	C₂×C₈	—	236	1	no	—
227	C₂×C₈	—	236	1	no	—
228	C₂×C₈	—	236	8	no	—
229	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
230	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
231	C₂×C₈	#11745669	1	2	yes	—
232	C₂×C₈	#11745671	1	2	yes	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	8	2	no	—
238	C₄⋊C₄	—	59	2	no	—
239	C₄⋊C₄	—	118	1	no	—
240	C₂²⋊C₄	—	118	2	no	—
241	C₂²⋊C₄	—	118	2	no	—
242	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
243	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
244	C₄²	Q₈×D₅₉	1	1	yes	characteristic
245	C₄²	#11745689	1	1	yes	characteristic
246	C₂³	—	118	2	no	—
247	C₂³	—	118	4	no	—
248	C₂³	—	236	2	no	—
249	C₂³	—	236	1	no	—
250	C₂³	—	236	1	no	—
251	C₂³	#14273698	1	1	yes	characteristic
252	D₄	—	59	4	no	—
253	D₄	—	118	4	no	—
254	D₄	—	118	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	118	2	no	—
260	C₂×C₄	—	236	1	no	—
261	C₂×C₄	—	236	1	no	—
262	C₂×C₄	—	236	2	no	—
263	C₂×C₄	—	236	2	no	—
264	C₂×C₄	—	236	2	no	—
265	C₂×C₄	—	236	2	no	—
266	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
267	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
268	C₂×C₄	#14273685	1	1	yes	characteristic
269	C₂×C₄	#14273702	1	1	yes	characteristic
270	C₂×C₄	#14273703	1	1	yes	characteristic
271	C₂×C₄	#14273707	1	1	yes	characteristic
272	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
273	C₂×C₄	SD₁₆×D₅₉	1	2	yes	—
274	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
275	C₂×C₄	#14273748	1	2	yes	—
276	C₂×C₄	#14273753	1	1	yes	characteristic
277	C₂×C₄	#14273758	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	472	2	no	—
280	C₈	—	236	2	no	—
281	C₈	#14273647	1	4	yes	complemented
282	C₂²	—	118	4	no	—
283	C₂²	—	236	2	no	—
284	C₂²	—	236	4	no	—
285	C₂²	—	236	4	no	—
286	C₂²	—	236	2	no	—
287	C₂²	#16837259	1	1	yes	characteristic
288	C₂²	#16837594	1	1	yes	characteristic
289	C₂²	#16837604	1	1	yes	characteristic
290	C₂²	#16837628	1	2	yes	—
291	C₂²	#16837662	1	2	yes	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	16	2	no	—
295	C₄	—	236	2	no	—
296	C₄	C₂×#14273647	1	1	yes	characteristic
297	C₄	#16837401	1	1	yes	characteristic
298	C₄	#106×D₅₉	1	1	yes	characteristic
299	C₄	#16837437	1	1	yes	characteristic
300	C₄	#16837632	1	2	yes	—
301	C₄	#16837679	1	1	yes	characteristic
302	C₄	#16837689	1	1	yes	characteristic
303	C₂	—	236	4	no	—
304	C₂	#21423434	1	2	yes	—
305	C₂	#21423468	1	2	yes	—
306	C₂	#21424093	1	1	yes	characteristic
307	C₂	#21424118	1	1	yes	characteristic
308	C₂	#21424156	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273698
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1844	C₂
socle	C₂³×C₅₉	#121
prime radical	C₂²×D₂₃₆	D₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1844	D₅₉
hypocenter	C₅₉	#8642
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
3. C₂×#16837153	C₂	semidirect
4. #21421979	C₂	semidirect
5. C₅₉×#1844	C₂	semidirect
9. C₈×D₂₃₆	C₄	semidirect
118. D₂₃₆	#106	semidirect
124. C₈×C₅₉	#102	semidirect
150. #1844	D₅₉	semidirect
174. C₅₉	#8642	semidirect
281. C₈	#14273647	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₂₃₆	2
#21424093	2
C₂×#16837153	2
#21421979	2
C₅₉×#1844	2
#8642	59