Subgroups — #28481610

Order 15104.

398 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×C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	#21424089	C₂	1	2	yes	maximal
3	#21422852	C₂	1	1	yes	maximal, characteristic
4	#21421970	C₂	1	2	yes	maximal
5	C₅₉×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₁₈	—	2	1	no	—
8	C₈×Dic₁₁₈	—	2	1	no	—
9	C₈×Dic₁₁₈	—	2	1	no	—
10	C₈×Dic₁₁₈	—	2	1	no	—
11	#16837426	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₅₉⋊C₈	C₄	1	2	yes	—
14	C₂×#14273645	—	2	2	no	—
15	C₂×C₈×Dic₅₉	C₄	1	2	yes	—
16	#16837170	—	2	2	no	—
17	#16837153	—	2	1	no	—
18	#16837153	—	2	1	no	—
19	C₅₉×#423	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₅₉×#334	—	2	1	no	—
22	C₅₉×#334	—	2	1	no	—
23	C₂²×Dic₁₁₈	D₄	1	1	yes	characteristic
24	C₂×#11745664	—	2	2	no	—
25	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₁₈	—	2	1	no	—
27	C₄×Dic₁₁₈	—	2	1	no	—
28	C₄×Dic₁₁₈	—	2	1	no	—
29	C₄×Dic₁₁₈	—	2	1	no	—
30	C₂×C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
31	C₂×Dic₅₉⋊C₄	Q₈	1	2	yes	—
32	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
33	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
34	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
35	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
36	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
37	Dic₅₉⋊C₈	—	2	2	no	—
38	Dic₅₉⋊C₈	—	2	2	no	—
39	#14273645	—	4	4	no	—
40	C₈×Dic₅₉	—	2	2	no	—
41	C₈×Dic₅₉	—	2	2	no	—
42	C₂×M₁₆×C₅₉	—	2	2	no	—
43	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₅₉	—	2	2	no	—
47	C₄⋊C₈×C₅₉	—	2	2	no	—
48	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
55	C₂²×Dic₅₉	C₄⋊C₄	1	2	yes	—
56	C₂²×Dic₅₉	C₄⋊C₄	1	2	yes	—
57	C₂×Dic₁₁₈	—	2	1	no	—
58	C₂×Dic₁₁₈	—	2	1	no	—
59	C₂×Dic₁₁₈	—	2	2	no	—
60	C₂×Dic₁₁₈	SD₁₆	1	4	yes	—
61	M₁₆×C₅₉	—	4	2	no	—
62	M₁₆×C₅₉	—	4	2	no	—
63	C₂×C₈×C₅₉	—	2	2	no	—
64	C₂×C₈×C₅₉	—	4	1	no	—
65	C₂×C₈×C₅₉	—	4	2	no	—
66	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₅₉	#41	1	1	yes	characteristic
73	C₄²×C₅₉	#41	1	1	yes	characteristic
74	C₄⋊Dic₅₉	—	2	1	no	—
75	C₄⋊Dic₅₉	—	2	1	no	—
76	Dic₅₉⋊C₄	—	2	2	no	—
77	Dic₅₉⋊C₄	—	2	2	no	—
78	C₄×Dic₅₉	—	2	2	no	—
79	C₄×Dic₅₉	—	2	2	no	—
80	#11745664	—	4	2	no	—
81	#11745664	—	4	2	no	—
82	C₂×C₅₉⋊C₈	—	2	2	no	—
83	C₂×C₅₉⋊C₈	—	4	2	no	—
84	C₂×C₅₉⋊C₈	—	4	2	no	—
85	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₅₉	—	2	1	no	—
87	C₂×C₄×C₅₉	—	2	1	no	—
88	C₂×C₄×C₅₉	—	4	1	no	—
89	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
91	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
92	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
93	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
96	C₂×C₄×C₅₉	#135	1	2	yes	—
97	C₂×Dic₅₉	—	2	2	no	—
98	C₂×Dic₅₉	—	2	2	no	—
99	C₂×Dic₅₉	—	2	2	no	—
100	C₂×Dic₅₉	—	4	2	no	—
101	C₂×Dic₅₉	#106	1	4	yes	—
102	Dic₁₁₈	—	2	4	no	—
103	Dic₁₁₈	—	2	4	no	—
104	C₈×C₅₉	—	2	1	no	—
105	C₈×C₅₉	—	2	1	no	—
106	C₈×C₅₉	—	2	1	no	—
107	C₈×C₅₉	—	8	2	no	—
108	C₈×C₅₉	#101	1	2	yes	complemented
109	C₅₉⋊C₈	—	4	2	no	—
110	C₅₉⋊C₈	—	8	2	no	—
111	#8615	—	59	1	no	maximal, Sylow
112	C₂²×C₅₉	—	2	1	no	—
113	C₂²×C₅₉	—	2	1	no	—
114	C₂²×C₅₉	#380	1	1	yes	characteristic
115	C₂²×C₅₉	C₂×#106	1	1	yes	characteristic
116	C₂²×C₅₉	#427	1	1	yes	characteristic
117	C₄×C₅₉	—	2	1	no	—
118	C₄×C₅₉	—	2	1	no	—
119	C₄×C₅₉	—	2	1	no	—
120	C₄×C₅₉	C₂×#101	1	1	yes	characteristic
121	C₄×C₅₉	#413	1	1	yes	characteristic
122	C₄×C₅₉	#465	1	2	yes	—
123	C₄×C₅₉	#465	1	2	yes	—
124	Dic₅₉	—	2	4	no	—
125	Dic₅₉	—	4	2	no	—
126	C₂×C₈×Q₈	—	59	1	no	—
127	#1801	—	59	2	no	—
128	#1566	—	59	1	no	—
129	#1566	D₅₉	1	1	yes	characteristic
130	#1292	—	59	2	no	—
131	C₂×C₅₉	—	2	1	no	—
132	C₂×C₅₉	—	2	1	no	—
133	C₂×C₅₉	#1801	1	1	yes	characteristic
134	C₂×C₅₉	#1807	1	1	yes	characteristic
135	C₂×C₅₉	#1813	1	1	yes	characteristic
136	C₂×C₄×Q₈	—	59	1	no	—
137	C₈×Q₈	—	118	1	no	—
138	C₈×Q₈	—	118	1	no	—
139	C₈×Q₈	—	118	1	no	—
140	C₈×Q₈	—	118	1	no	—
141	#423	—	59	2	no	—
142	#423	C₂×D₅₉	1	2	yes	—
143	C₂×C₄⋊C₈	—	59	2	no	—
144	C₂×C₄⋊C₈	—	59	1	no	—
145	C₂×#101	—	118	2	no	—
146	C₂×C₄×C₈	—	59	2	no	—
147	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
148	#343	—	118	2	no	—
149	#334	—	2	1	no	—
150	#334	—	2	1	no	—
151	#334	—	118	1	no	—
152	#334	—	118	1	no	—
153	C₅₉	#8615	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	—	59	1	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	118	2	no	—
157	C₂²×C₈	—	59	2	no	—
158	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
159	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
160	C₄×Q₈	—	118	1	no	—
161	C₄×Q₈	—	118	1	no	—
162	C₄×Q₈	—	118	1	no	—
163	C₄×Q₈	—	118	1	no	—
164	C₂×C₄⋊C₄	—	59	1	no	—
165	C₂×C₄⋊C₄	—	59	2	no	—
166	C₂×C₄²	—	59	2	no	—
167	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	118	2	no	—
171	C₄⋊C₈	—	118	2	no	—
172	C₄⋊C₈	—	118	1	no	—
173	C₄⋊C₈	—	118	2	no	—
174	C₄⋊C₈	—	118	2	no	—
175	C₄⋊C₈	—	118	1	no	—
176	#101	—	236	4	no	—
177	C₄×C₈	—	118	2	no	—
178	C₄×C₈	—	118	2	no	—
179	C₄×C₈	D₂₃₆	1	1	yes	characteristic
180	C₄×C₈	D₂₃₆	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
183	C₂×Q₈	—	59	4	no	—
184	C₂×Q₈	—	118	1	no	—
185	C₂×Q₈	—	118	1	no	—
186	C₂×Q₈	—	118	2	no	—
187	C₂²×C₄	—	59	2	no	—
188	C₂²×C₄	—	59	2	no	—
189	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
190	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
191	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	M₁₆	—	236	2	no	—
195	M₁₆	—	236	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	118	2	no	—
200	C₂×C₈	—	236	2	no	—
201	C₂×C₈	—	236	2	no	—
202	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
203	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
204	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
205	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
206	C₄⋊C₄	—	118	1	no	—
207	C₄⋊C₄	—	118	1	no	—
208	C₄⋊C₄	—	118	2	no	—
209	C₄⋊C₄	—	118	2	no	—
210	C₄²	—	118	2	no	—
211	C₄²	—	118	2	no	—
212	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
213	C₄²	#11745687	1	1	yes	characteristic
214	C₄²	#11745689	1	1	yes	characteristic
215	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
216	C₂³	C₄⋊C₄×D₅₉	1	1	yes	characteristic
217	Q₈	—	118	4	no	—
218	Q₈	—	118	4	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	59	4	no	—
223	C₂×C₄	—	118	2	no	—
224	C₂×C₄	—	118	2	no	—
225	C₂×C₄	—	118	2	no	—
226	C₂×C₄	—	236	2	no	—
227	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
228	C₂×C₄	#14273685	1	1	yes	characteristic
229	C₂×C₄	#14273698	1	1	yes	characteristic
230	C₂×C₄	#14273702	1	1	yes	characteristic
231	C₂×C₄	#14273703	1	1	yes	characteristic
232	C₂×C₄	#14273707	1	1	yes	characteristic
233	C₂×C₄	SD₁₆×D₅₉	1	2	yes	—
234	C₂×C₄	#14273748	1	2	yes	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	8	2	no	—
239	C₈	—	472	2	no	—
240	C₈	—	236	2	no	—
241	C₈	#14273645	1	2	yes	complemented
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	#16837259	1	1	yes	characteristic
245	C₂²	#106×D₅₉	1	1	yes	characteristic
246	C₂²	#16837437	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	118	4	no	—
251	C₄	—	236	2	no	—
252	C₄	C₂×#14273645	1	1	yes	characteristic
253	C₄	#16837384	1	1	yes	characteristic
254	C₄	#16837628	1	2	yes	—
255	C₄	#16837632	1	2	yes	—
256	C₂	—	2	1	no	—
257	C₂	—	2	1	no	—
258	C₂	#21424089	1	1	yes	characteristic
259	C₂	#21424118	1	1	yes	characteristic
260	C₂	#21424143	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837259
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1566	C₂
socle	C₂²×C₅₉	#380
prime radical	C₂³×C₅₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1566	D₅₉
hypocenter	C₅₉	#8615
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
108. C₈×C₅₉	#101	semidirect
153. C₅₉	#8615	semidirect
241. C₈	#14273645	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
#21424089	2
#21422852	2
#21421970	2
C₅₉×#1566	2
#8615	59