Subgroups — #28481759

Order 15104.

438 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	#21424224	C₂	1	1	yes	maximal, characteristic
3	#21424102	C₂	1	2	yes	maximal
4	#21421980	C₂	1	2	yes	maximal
5	C₂×C₅₉×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₁₈	—	2	2	no	—
8	C₈×Dic₁₁₈	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×Dic₅₉⋊C₈	—	2	1	no	—
11	C₂×#14273652	—	2	2	no	—
12	C₂×C₈×Dic₅₉	—	2	1	no	—
13	#16837272	C₂²	1	2	yes	—
14	#16837165	—	2	2	no	—
15	#16837165	—	2	1	no	—
16	#16837165	—	2	1	no	—
17	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₅₉×#334	—	2	2	no	—
20	C₂²×Dic₁₁₈	D₄	1	1	yes	characteristic
21	C₂²×C₅₉⋊C₈	—	2	1	no	—
22	C₄×Dic₁₁₈	—	2	1	no	—
23	C₄×Dic₁₁₈	—	2	1	no	—
24	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₁₈	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₅₉	—	2	2	no	—
27	C₂×C₄⋊Dic₅₉	—	2	2	no	—
28	C₂×C₄⋊Dic₅₉	—	2	2	no	—
29	C₂×C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
30	C₂×Dic₅₉⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₅₉	—	2	1	no	—
32	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
33	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
34	Dic₅₉⋊C₈	—	2	4	no	—
35	#14273652	—	4	4	no	—
36	C₈×Dic₅₉	—	2	4	no	—
37	C₂²×C₈×C₅₉	—	2	2	no	—
38	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₉	—	2	2	no	—
42	C₄⋊C₈×C₅₉	—	2	2	no	—
43	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₉	D₄	1	2	yes	—
45	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
48	C₂²×Dic₅₉	—	2	1	no	—
49	C₂²×Dic₅₉	—	2	1	no	—
50	C₂²×Dic₅₉	—	4	2	no	—
51	C₂×Dic₁₁₈	—	2	1	no	—
52	C₂×Dic₁₁₈	—	2	2	no	—
53	C₂×Dic₁₁₈	C₄⋊C₄	1	2	yes	—
54	C₂×Dic₁₁₈	SD₁₆	1	4	yes	—
55	C₂×C₈×C₅₉	—	2	2	no	—
56	C₂×C₈×C₅₉	—	2	2	no	—
57	C₂×C₈×C₅₉	—	4	2	no	—
58	C₂×C₈×C₅₉	—	4	2	no	—
59	C₂×C₈×C₅₉	—	4	2	no	—
60	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₅₉	#41	1	1	yes	characteristic
66	C₄⋊Dic₅₉	—	2	1	no	—
67	C₄⋊Dic₅₉	—	4	2	no	—
68	C₄⋊Dic₅₉	—	4	2	no	—
69	C₄⋊Dic₅₉	—	4	2	no	—
70	C₄⋊Dic₅₉	—	4	2	no	—
71	C₄⋊Dic₅₉	—	8	2	no	—
72	C₄⋊Dic₅₉	C₄⋊C₄	1	2	yes	—
73	Dic₅₉⋊C₄	—	2	2	no	—
74	Dic₅₉⋊C₄	—	2	2	no	—
75	C₄×Dic₅₉	—	2	2	no	—
76	C₄×Dic₅₉	—	2	2	no	—
77	C₂×C₅₉⋊C₈	—	2	2	no	—
78	C₂×C₅₉⋊C₈	—	4	2	no	—
79	C₂³×C₅₉	#121	1	1	yes	characteristic
80	C₂×C₄×C₅₉	—	2	1	no	—
81	C₂×C₄×C₅₉	—	2	1	no	—
82	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
85	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
86	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
89	C₂×C₄×C₅₉	#135	1	2	yes	—
90	C₂×C₄×C₅₉	#135	1	2	yes	—
91	C₂×C₄×C₅₉	#137	1	2	yes	—
92	C₂×Dic₅₉	—	2	2	no	—
93	C₂×Dic₅₉	—	2	4	no	—
94	C₂×Dic₅₉	—	2	2	no	—
95	C₂×Dic₅₉	—	4	1	no	—
96	C₂×Dic₅₉	—	4	1	no	—
97	C₂×Dic₅₉	—	8	2	no	—
98	C₂×Dic₅₉	—	8	2	no	—
99	C₂×Dic₅₉	—	8	2	no	—
100	Dic₁₁₈	—	2	2	no	—
101	Dic₁₁₈	—	2	4	no	—
102	Dic₁₁₈	#106	1	4	yes	—
103	C₈×C₅₉	—	2	2	no	—
104	C₈×C₅₉	—	2	2	no	—
105	C₈×C₅₉	—	8	2	no	—
106	C₅₉⋊C₈	—	4	2	no	—
107	#8647	—	59	1	no	maximal, Sylow
108	C₂²×C₅₉	#380	1	1	yes	characteristic
109	C₂²×C₅₉	#464	1	2	yes	—
110	C₂²×C₅₉	#465	1	2	yes	—
111	C₂²×C₅₉	#472	1	1	yes	characteristic
112	C₂²×C₅₉	#473	1	1	yes	characteristic
113	C₄×C₅₉	—	2	1	no	—
114	C₄×C₅₉	—	2	1	no	—
115	C₄×C₅₉	#416	1	1	yes	characteristic
116	C₄×C₅₉	#419	1	1	yes	characteristic
117	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
118	C₄×C₅₉	#427	1	1	yes	characteristic
119	C₄×C₅₉	#465	1	2	yes	—
120	C₄×C₅₉	#479	1	2	yes	—
121	Dic₅₉	—	2	4	no	—
122	Dic₅₉	—	4	2	no	—
123	Dic₅₉	—	16	2	no	—
124	C₂×C₈×Q₈	—	59	1	no	—
125	#1834	—	59	1	no	—
126	#1803	—	59	2	no	—
127	C₂×#334	D₅₉	1	1	yes	characteristic
128	#1296	—	59	2	no	—
129	C₂×C₅₉	#1678	1	2	yes	—
130	C₂×C₅₉	#1680	1	2	yes	—
131	C₂×C₅₉	#1805	1	1	yes	characteristic
132	C₂×C₅₉	#1807	1	1	yes	characteristic
133	C₂×C₅₉	#1816	1	1	yes	characteristic
134	C₂×C₄×Q₈	—	59	1	no	—
135	C₈×Q₈	—	59	4	no	—
136	C₈×Q₈	—	118	2	no	—
137	C₂×C₄⋊C₈	—	59	1	no	—
138	C₂×C₄⋊C₈	—	118	1	no	—
139	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
140	C₂×#103	—	118	2	no	—
141	C₂×C₄×C₈	—	118	1	no	—
142	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
143	#384	—	59	2	no	—
144	#340	—	118	2	no	—
145	#340	—	118	1	no	—
146	#340	—	118	1	no	—
147	#334	—	2	2	no	—
148	C₅₉	#8647	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₈	—	59	1	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	118	1	no	—
152	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
153	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
154	C₄×Q₈	—	59	2	no	—
155	C₄×Q₈	—	59	2	no	—
156	C₄×Q₈	—	118	1	no	—
157	C₄×Q₈	—	118	1	no	—
158	C₂×C₄⋊C₄	—	59	1	no	—
159	C₂×C₄⋊C₄	—	118	1	no	—
160	C₂×C₄⋊C₄	—	118	2	no	—
161	C₂×C₄⋊C₄	—	118	2	no	—
162	C₂×C₄⋊C₄	—	118	2	no	—
163	C₂×C₄²	—	118	1	no	—
164	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	59	2	no	—
168	C₄⋊C₈	—	118	4	no	—
169	C₄⋊C₈	—	118	1	no	—
170	#103	—	236	4	no	—
171	C₄×C₈	—	118	4	no	—
172	C₄×C₈	C₄×D₅₉	1	2	yes	—
173	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
174	C₂×Q₈	—	59	2	no	—
175	C₂×Q₈	—	59	4	no	—
176	C₂×Q₈	—	118	1	no	—
177	C₂×Q₈	—	118	2	no	—
178	C₂²×C₄	—	118	1	no	—
179	C₂²×C₄	—	118	1	no	—
180	C₂²×C₄	—	236	2	no	—
181	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
182	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
183	C₂²×C₄	#11745689	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	118	2	no	—
190	C₂×C₈	—	236	2	no	—
191	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
192	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
193	C₄⋊C₄	—	59	2	no	—
194	C₄⋊C₄	—	118	2	no	—
195	C₄⋊C₄	—	118	1	no	—
196	C₄⋊C₄	—	118	2	no	—
197	C₄⋊C₄	—	472	2	no	—
198	C₄⋊C₄	—	236	2	no	—
199	C₄⋊C₄	—	236	2	no	—
200	C₄⋊C₄	—	236	2	no	—
201	C₄⋊C₄	—	236	2	no	—
202	C₄²	—	118	2	no	—
203	C₄²	—	118	2	no	—
204	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
205	C₄²	#11745687	1	1	yes	characteristic
206	C₄²	Q₈×D₅₉	1	1	yes	characteristic
207	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
208	C₂³	#14273702	1	1	yes	characteristic
209	Q₈	—	59	4	no	—
210	Q₈	—	118	2	no	—
211	Q₈	—	118	4	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	118	4	no	—
215	C₂×C₄	—	118	2	no	—
216	C₂×C₄	—	118	2	no	—
217	C₂×C₄	—	472	2	no	—
218	C₂×C₄	—	472	2	no	—
219	C₂×C₄	—	472	2	no	—
220	C₂×C₄	—	236	1	no	—
221	C₂×C₄	—	236	1	no	—
222	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
223	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
224	C₂×C₄	#14273685	1	1	yes	characteristic
225	C₂×C₄	#14273698	1	1	yes	characteristic
226	C₂×C₄	#14273703	1	1	yes	characteristic
227	C₂×C₄	#14273707	1	1	yes	characteristic
228	C₂×C₄	SD₁₆×D₅₉	1	2	yes	—
229	C₂×C₄	#14273748	1	2	yes	—
230	C₂×C₄	#14273749	1	2	yes	—
231	C₂×C₄	#14273760	1	2	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	2	2	no	—
234	C₈	—	8	2	no	—
235	C₈	—	236	2	no	—
236	C₂²	#16837259	1	1	yes	characteristic
237	C₂²	#16837625	1	2	yes	—
238	C₂²	#16837632	1	2	yes	—
239	C₂²	#16837673	1	1	yes	characteristic
240	C₂²	#16837678	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	118	4	no	—
244	C₄	—	944	2	no	—
245	C₄	—	236	2	no	—
246	C₄	#16837399	1	1	yes	characteristic
247	C₄	#16837412	1	1	yes	characteristic
248	C₄	#106×D₅₉	1	1	yes	characteristic
249	C₄	#16837437	1	1	yes	characteristic
250	C₄	#16837628	1	2	yes	—
251	C₄	#16837710	1	2	yes	—
252	C₂	#21423513	1	2	yes	—
253	C₂	#21423527	1	2	yes	—
254	C₂	#21424112	1	1	yes	characteristic
255	C₂	#21424118	1	1	yes	characteristic
256	C₂	#21424157	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273702
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#334	C₂
socle, prime radical	C₂³×C₅₉	#121
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#334	D₅₉
hypocenter	C₅₉	#8647
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. C₈×Dic₁₁₈	C₄	semidirect
148. C₅₉	#8647	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₁₈	2
#21424224	2
#21424102	2
#21421980	2
C₂×C₅₉×#334	2
#8647	59