Subgroups — #28487985

Order 15104.

506 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837782	C₂	1	1	yes	maximal, characteristic
2	#21425048	C₂	1	2	yes	maximal
3	C₂×#16837153	C₂	1	1	yes	maximal, characteristic
4	C₂×#16837149	C₂	1	2	yes	maximal
5	C₅₉×#1770	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₃₆	—	2	1	no	—
7	C₂²×Dic₂₃₆	—	2	1	no	—
8	C₂×#14273727	C₂²	1	2	yes	—
9	#16837782	—	2	2	no	—
10	#16837782	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
12	#16837164	—	2	4	no	—
13	#16837153	—	2	2	no	—
14	#16837149	—	2	4	no	—
15	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×#97×C₅₉	—	2	1	no	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
19	C₅₉×#336	—	2	1	no	—
20	C₂²×Dic₁₁₈	—	2	2	no	—
21	C₂×Dic₂₃₆	—	2	4	no	—
22	C₂×Dic₂₃₆	—	2	4	no	—
23	C₂×Dic₂₃₆	—	4	2	no	—
24	C₂×Dic₂₃₆	—	4	2	no	—
25	C₂×Dic₂₃₆	—	4	2	no	—
26	C₂×Dic₂₃₆	—	4	2	no	—
27	C₂²×C₅₉⋊C₈	—	2	2	no	—
28	#14273727	—	2	2	no	—
29	#14273727	—	2	2	no	—
30	#14273727	—	2	2	no	—
31	#14273727	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊Dic₅₉	—	2	2	no	—
33	C₂×C₄⋊Dic₅₉	—	2	2	no	—
34	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
35	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
36	C₂²×C₈×C₅₉	—	2	1	no	—
37	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₅₉	D₄	1	2	yes	—
40	C₂×C₄²×C₅₉	—	2	1	no	—
41	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₅₉	—	2	1	no	—
43	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
44	#97×C₅₉	—	4	2	no	—
45	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₅₉	D₄	1	2	yes	—
47	#95×C₅₉	—	2	1	no	—
48	C₂²×C₄×C₅₉	—	2	2	no	—
49	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
52	C₂²×Dic₅₉	—	4	2	no	—
53	C₂×Dic₁₁₈	—	2	4	no	—
54	C₂×Dic₁₁₈	—	4	2	no	—
55	C₂×Dic₁₁₈	—	4	2	no	—
56	C₂×Dic₁₁₈	—	4	2	no	—
57	C₂×Dic₁₁₈	—	4	2	no	—
58	C₂×Dic₁₁₈	—	4	2	no	—
59	C₂×C₈×C₅₉	—	2	1	no	—
60	C₂×C₈×C₅₉	—	2	1	no	—
61	C₂×C₈×C₅₉	—	2	2	no	—
62	C₂×C₈×C₅₉	—	4	1	no	—
63	C₂×C₈×C₅₉	—	4	1	no	—
64	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₅₉	D₈	1	2	yes	—
67	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
68	C₄⋊C₄×C₅₉	—	2	4	no	—
69	C₄²×C₅₉	—	4	1	no	—
70	C₄²×C₅₉	—	4	1	no	—
71	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₉	#41	1	1	yes	characteristic
75	C₄⋊Dic₅₉	—	2	4	no	—
76	C₄⋊Dic₅₉	—	2	4	no	—
77	C₄⋊Dic₅₉	—	4	2	no	—
78	C₄⋊Dic₅₉	—	4	2	no	—
79	C₂×C₅₉⋊C₈	—	4	2	no	—
80	C₂×C₅₉⋊C₈	—	4	4	no	—
81	Dic₂₃₆	—	4	8	no	—
82	Dic₂₃₆	—	4	8	no	—
83	C₂³×C₅₉	#121	1	1	yes	characteristic
84	C₂×C₄×C₅₉	—	2	2	no	—
85	C₂×C₄×C₅₉	—	4	2	no	—
86	C₂×C₄×C₅₉	—	4	2	no	—
87	C₂×C₄×C₅₉	—	4	2	no	—
88	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
92	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
93	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
94	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₄×C₅₉	#137	1	4	yes	—
99	C₂×Dic₅₉	—	4	4	no	—
100	C₂×Dic₅₉	—	8	2	no	—
101	C₂×Dic₅₉	—	8	2	no	—
102	Dic₁₁₈	—	4	4	no	—
103	Dic₁₁₈	—	4	4	no	—
104	Dic₁₁₈	—	8	2	no	—
105	Dic₁₁₈	—	8	2	no	—
106	C₈×C₅₉	—	2	2	no	—
107	C₈×C₅₉	—	4	2	no	—
108	C₈×C₅₉	#102	1	4	yes	—
109	C₅₉⋊C₈	—	8	2	no	—
110	#9742	—	59	1	no	maximal, Sylow
111	C₂²×C₅₉	#390	1	1	yes	characteristic
112	C₂²×C₅₉	#459	1	1	yes	characteristic
113	C₂²×C₅₉	#460	1	1	yes	characteristic
114	C₂²×C₅₉	#470	1	2	yes	—
115	C₂²×C₅₉	#473	1	2	yes	—
116	C₄×C₅₉	—	2	1	no	—
117	C₄×C₅₉	—	2	1	no	—
118	C₄×C₅₉	—	8	2	no	—
119	C₄×C₅₉	C₂×#102	1	1	yes	characteristic
120	C₄×C₅₉	#418	1	1	yes	characteristic
121	C₄×C₅₉	#441	1	2	yes	—
122	C₄×C₅₉	#459	1	1	yes	characteristic
123	C₄×C₅₉	#460	1	1	yes	characteristic
124	C₄×C₅₉	#497	1	2	yes	—
125	Dic₅₉	—	8	4	no	—
126	C₂×#494	—	59	1	no	—
127	#1988	—	59	2	no	—
128	#1770	D₅₉	1	1	yes	characteristic
129	C₂×#334	—	59	1	no	—
130	C₂×#332	—	59	2	no	—
131	C₂×C₅₉	#1689	1	2	yes	—
132	C₂×C₅₉	#1690	1	2	yes	—
133	C₂×C₅₉	#1971	1	1	yes	characteristic
134	C₂×C₅₉	#1979	1	1	yes	characteristic
135	C₂×C₅₉	#1982	1	1	yes	characteristic
136	C₂²×Q₁₆	—	118	1	no	—
137	C₂²×Q₁₆	—	118	1	no	—
138	C₂×C₄⋊Q₈	—	59	2	no	—
139	#494	—	59	4	no	—
140	#494	—	118	2	no	—
141	C₂×C₄⋊C₈	—	59	2	no	—
142	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
143	C₂×#97	—	2	1	no	—
144	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
145	C₄×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
146	#340	—	118	4	no	—
147	#336	—	2	1	no	—
148	#334	—	118	2	no	—
149	#332	—	118	4	no	—
150	C₅₉	#9742	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₈	—	118	2	no	—
152	C₂×Q₁₆	—	118	4	no	—
153	C₂×Q₁₆	—	118	4	no	—
154	C₂×Q₁₆	—	236	2	no	—
155	C₂×Q₁₆	—	236	2	no	—
156	C₂×Q₁₆	—	236	2	no	—
157	C₂×Q₁₆	—	236	2	no	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	118	2	no	—
160	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
161	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
162	C₄⋊Q₈	—	59	4	no	—
163	C₄⋊Q₈	—	118	2	no	—
164	C₄⋊Q₈	—	118	2	no	—
165	C₄⋊Q₈	—	118	2	no	—
166	C₂×C₄⋊C₄	—	118	2	no	—
167	C₂×C₄⋊C₄	—	118	2	no	—
168	C₂×C₄⋊C₄	D₂₃₆	1	2	yes	—
169	C₂×C₄²	—	2	1	no	—
170	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	118	2	no	—
173	C₄⋊C₈	—	118	2	no	—
174	C₄⋊C₈	D₂₃₆	1	2	yes	—
175	#97	—	4	2	no	—
176	C₄×C₈	C₄×D₅₉	1	2	yes	—
177	C₄×C₈	D₂₃₆	1	2	yes	—
178	#95	—	2	1	no	—
179	C₂×Q₈	—	118	4	no	—
180	C₂×Q₈	—	236	2	no	—
181	C₂×Q₈	—	236	2	no	—
182	C₂×Q₈	—	236	2	no	—
183	C₂×Q₈	—	236	2	no	—
184	C₂×Q₈	—	236	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	236	2	no	—
187	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
188	C₂²×C₄	#11745685	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
190	Q₁₆	—	236	8	no	—
191	Q₁₆	—	236	8	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	236	2	no	—
198	C₂×C₈	—	236	4	no	—
199	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
200	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
201	C₂×C₈	#11745669	1	2	yes	—
202	C₂×C₈	#11745671	1	2	yes	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	118	4	no	—
205	C₄⋊C₄	—	118	4	no	—
206	C₄⋊C₄	—	236	2	no	—
207	C₄⋊C₄	—	236	2	no	—
208	C₄²	—	4	1	no	—
209	C₄²	—	4	1	no	—
210	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
211	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
212	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
213	C₄²	#11745685	1	1	yes	characteristic
214	C₂³	#14273701	1	1	yes	characteristic
215	Q₈	—	472	2	no	—
216	Q₈	—	472	2	no	—
217	Q₈	—	236	4	no	—
218	Q₈	—	236	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	472	2	no	—
224	C₂×C₄	—	472	2	no	—
225	C₂×C₄	—	236	4	no	—
226	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
227	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
228	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
229	C₂×C₄	#14273701	1	1	yes	characteristic
230	C₂×C₄	#14273715	1	1	yes	characteristic
231	C₂×C₄	#14273724	1	1	yes	characteristic
232	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
233	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
234	C₂×C₄	#14273752	1	1	yes	characteristic
235	C₂×C₄	#14273752	1	1	yes	characteristic
236	C₂×C₄	#14273760	1	4	yes	—
237	C₈	—	2	2	no	—
238	C₈	—	4	2	no	—
239	C₈	—	472	2	no	—
240	C₈	#14273647	1	4	yes	—
241	C₂²	#16837304	1	1	yes	characteristic
242	C₂²	#16837597	1	1	yes	characteristic
243	C₂²	#16837602	1	1	yes	characteristic
244	C₂²	#16837660	1	2	yes	—
245	C₂²	#16837677	1	2	yes	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	8	2	no	—
249	C₄	—	472	4	no	—
250	C₄	C₂×#14273647	1	1	yes	characteristic
251	C₄	#16837406	1	1	yes	characteristic
252	C₄	#16837510	1	2	yes	—
253	C₄	#16837597	1	1	yes	characteristic
254	C₄	#16837602	1	1	yes	characteristic
255	C₄	#16837796	1	2	yes	—
256	C₂	#21423585	1	2	yes	—
257	C₂	#21423590	1	2	yes	—
258	C₂	#21424967	1	1	yes	characteristic
259	C₂	#21425011	1	1	yes	characteristic
260	C₂	#21425023	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273701
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1770	C₂
socle, prime radical	C₂³×C₅₉	#121
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1770	D₅₉
hypocenter	C₅₉	#9742
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1770

Complements

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

N	H ≅ G/N	type
10. #16837782	C₄	semidirect
150. C₅₉	#9742	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837782	2
#21425048	2
C₂×#16837153	2
C₂×#16837149	2
C₅₉×#1770	2
#9742	59