Subgroups — #28507170

Order 15104.

498 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425852	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16837212	C₂	1	4	yes	maximal, complemented
3	C₄×#14273659	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#3147	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273667	C₂²	1	2	yes	—
6	C₂×#14273659	C₂²	1	2	yes	—
7	C₄×C₄⋊Dic₅₉	—	2	1	no	—
8	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
10	#16837212	—	2	8	no	—
11	#16837165	—	2	1	no	—
12	C₂×#135×C₅₉	—	2	1	no	—
13	C₂×#135×C₅₉	C₂²	1	2	yes	—
14	C₅₉×#532	—	2	1	no	—
15	C₅₉×#495	—	2	1	no	—
16	C₅₉×#495	C₄	1	2	yes	complemented
17	C₅₉×#494	C₄	1	1	yes	characteristic, complemented
18	C₅₉×#493	C₄	1	1	yes	characteristic, complemented
19	C₅₉×#492	—	2	1	no	—
20	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₅₉	—	2	2	no	—
22	C₂×C₄×Dic₅₉	—	4	1	no	—
23	#14273667	—	2	4	no	—
24	C₂×C₅₉⋊C₁₆	—	2	2	no	—
25	#14273659	—	2	2	no	—
26	#14273659	D₄	1	4	yes	complemented
27	C₅₉⋊#95	—	4	1	no	—
28	C₂×#41×C₅₉	—	2	2	no	—
29	#135×C₅₉	—	2	4	no	—
30	#135×C₅₉	—	2	4	no	—
31	#135×C₅₉	—	4	4	no	—
32	C₂×Q₁₆×C₅₉	—	2	1	no	—
33	C₂×Q₁₆×C₅₉	C₂×C₄	1	2	yes	—
34	C₂×SD₁₆×C₅₉	—	2	2	no	—
35	C₂×SD₁₆×C₅₉	—	2	2	no	—
36	C₂×D₈×C₅₉	—	2	1	no	—
37	C₂×D₈×C₅₉	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₅₉	—	2	1	no	—
41	#127×C₅₉	—	2	1	no	—
42	#124×C₅₉	—	2	2	no	—
43	#121×C₅₉	—	4	1	no	—
44	#121×C₅₉	—	4	1	no	—
45	C₄×D₄×C₅₉	—	4	1	no	—
46	C₄×D₄×C₅₉	—	4	1	no	—
47	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
48	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
52	#41×C₅₉	—	4	4	no	—
53	#41×C₅₉	—	4	4	no	—
54	C₂×Q₈×C₅₉	—	2	2	no	—
55	C₂×D₄×C₅₉	—	2	2	no	—
56	C₂×D₄×C₅₉	—	4	2	no	—
57	C₂×D₄×C₅₉	—	4	1	no	—
58	C₂×D₄×C₅₉	—	4	1	no	—
59	C₂²×C₄×C₅₉	—	4	2	no	—
60	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₅₉	—	4	2	no	—
64	Q₁₆×C₅₉	—	2	4	no	—
65	Q₁₆×C₅₉	—	4	2	no	—
66	SD₁₆×C₅₉	—	4	4	no	—
67	SD₁₆×C₅₉	—	4	2	no	—
68	SD₁₆×C₅₉	—	4	2	no	—
69	D₈×C₅₉	—	2	4	no	—
70	D₈×C₅₉	—	4	2	no	—
71	C₂×C₈×C₅₉	—	2	2	no	—
72	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₅₉	#41	1	2	yes	—
78	C₄⋊C₄×C₅₉	—	4	1	no	—
79	C₄⋊C₄×C₅₉	—	4	1	no	—
80	C₂²⋊C₄×C₅₉	—	4	2	no	—
81	C₂²⋊C₄×C₅₉	—	4	2	no	—
82	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
85	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
86	C₄⋊Dic₅₉	—	2	4	no	—
87	C₄⋊Dic₅₉	—	4	2	no	—
88	C₄×Dic₅₉	—	8	2	no	—
89	C₅₉⋊C₁₆	—	4	2	no	—
90	C₂³×C₅₉	—	4	2	no	—
91	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
92	Q₈×C₅₉	—	4	2	no	—
93	Q₈×C₅₉	—	4	2	no	—
94	D₄×C₅₉	—	4	2	no	—
95	D₄×C₅₉	—	4	2	no	—
96	D₄×C₅₉	—	8	2	no	—
97	D₄×C₅₉	—	8	4	no	—
98	D₄×C₅₉	—	8	2	no	—
99	C₂×C₄×C₅₉	—	2	2	no	—
100	C₂×C₄×C₅₉	—	4	1	no	—
101	C₂×C₄×C₅₉	—	4	2	no	—
102	C₂×C₄×C₅₉	—	8	4	no	—
103	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
104	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
105	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
106	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₉	#121	1	2	yes	—
109	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
111	C₂×Dic₅₉	—	4	4	no	—
112	C₂×Dic₅₉	—	8	2	no	—
113	C₂×Dic₅₉	—	8	2	no	—
114	C₈×C₅₉	—	2	1	no	—
115	C₈×C₅₉	—	2	1	no	—
116	C₈×C₅₉	C₄×D₄	1	2	yes	—
117	C₈×C₅₉	#124	1	1	yes	characteristic
118	C₈×C₅₉	#127	1	1	yes	characteristic
119	#13255	—	59	1	no	maximal, Sylow
120	C₂²×C₅₉	—	2	1	no	—
121	C₂²×C₅₉	—	2	1	no	—
122	C₂²×C₅₉	—	8	2	no	—
123	C₂²×C₅₉	—	8	2	no	—
124	C₂²×C₅₉	—	8	2	no	—
125	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
126	C₂²×C₅₉	#459	1	2	yes	—
127	C₄×C₅₉	—	2	1	no	—
128	C₄×C₅₉	—	2	1	no	—
129	C₄×C₅₉	—	2	1	no	—
130	C₄×C₅₉	—	8	2	no	—
131	C₄×C₅₉	#360	1	2	yes	complemented
132	C₄×C₅₉	#390	1	1	yes	characteristic
133	C₄×C₅₉	#418	1	1	yes	characteristic
134	C₄×C₅₉	#460	1	2	yes	—
135	Dic₅₉	—	8	4	no	—
136	#3147	D₅₉	1	1	yes	characteristic
137	#2150	—	59	1	no	—
138	C₂×#360	—	59	4	no	—
139	C₄×#106	—	59	1	no	—
140	C₂×C₅₉	—	2	2	no	—
141	C₂×C₅₉	—	16	2	no	—
142	C₂×C₅₉	#1982	1	1	yes	characteristic
143	C₂×C₅₉	C₂×#360	1	1	yes	characteristic
144	C₂×C₅₉	#2145	1	1	yes	characteristic
145	C₂×#135	—	2	1	no	—
146	C₂×#135	C₂×D₅₉	1	2	yes	—
147	#532	—	2	1	no	—
148	C₂×M₃₂	—	59	2	no	—
149	#495	—	2	1	no	—
150	#495	Dic₅₉	1	2	yes	complemented
151	#494	Dic₅₉	1	1	yes	characteristic, complemented
152	#493	Dic₅₉	1	1	yes	characteristic, complemented
153	#492	—	2	1	no	—
154	C₂×#106	—	59	2	no	—
155	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
156	C₄×C₄⋊C₄	—	118	1	no	—
157	C₄⋊C₁₆	—	118	1	no	—
158	C₄⋊C₁₆	—	118	1	no	—
159	#360	—	118	8	no	—
160	#340	—	118	1	no	—
161	C₅₉	#13255	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	—	2	2	no	—
163	#135	—	2	4	no	—
164	#135	—	2	4	no	—
165	#135	—	4	4	no	—
166	C₂×Q₁₆	—	2	1	no	—
167	C₂×Q₁₆	C₂×Dic₅₉	1	2	yes	—
168	C₂×SD₁₆	—	2	2	no	—
169	C₂×SD₁₆	—	2	2	no	—
170	C₂×D₈	—	2	1	no	—
171	C₂×D₈	C₂×Dic₅₉	1	2	yes	—
172	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
174	C₄⋊Q₈	—	2	1	no	—
175	#127	—	2	1	no	—
176	#124	—	2	2	no	—
177	#121	—	4	1	no	—
178	#121	—	4	1	no	—
179	C₄×D₄	—	4	1	no	—
180	C₄×D₄	—	4	1	no	—
181	C₂×C₄⋊C₄	—	118	2	no	—
182	C₂×C₄²	—	236	1	no	—
183	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
184	M₃₂	—	118	4	no	—
185	C₂×C₁₆	—	118	2	no	—
186	#106	—	59	4	no	—
187	#106	—	118	2	no	—
188	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
189	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
190	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
191	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
192	#95	—	236	1	no	—
193	#41	—	4	4	no	—
194	#41	—	4	4	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	4	2	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	236	2	no	—
202	C₂²×C₄	#11745669	1	1	yes	characteristic
203	C₂²×C₄	#11745669	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
205	Q₁₆	—	2	4	no	—
206	Q₁₆	—	4	2	no	—
207	SD₁₆	—	4	4	no	—
208	SD₁₆	—	4	2	no	—
209	SD₁₆	—	4	2	no	—
210	D₈	—	2	4	no	—
211	D₈	—	4	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
214	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
215	C₂×C₈	D₄×D₅₉	1	2	yes	—
216	C₂×C₈	#11745687	1	2	yes	—
217	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
218	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₄⋊C₄	—	118	4	no	—
222	C₄⋊C₄	—	236	2	no	—
223	C₂²⋊C₄	—	4	2	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₄²	—	472	2	no	—
226	C₄²	#11745670	1	1	yes	characteristic
227	C₄²	#11745670	1	1	yes	characteristic
228	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
229	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
230	C₁₆	—	236	2	no	—
231	C₂³	—	4	2	no	—
232	C₂³	C₂×#11745669	1	1	yes	characteristic
233	Q₈	—	4	2	no	—
234	Q₈	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	8	2	no	—
238	D₄	—	8	4	no	—
239	D₄	—	8	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	472	2	no	—
245	C₂×C₄	—	472	2	no	—
246	C₂×C₄	—	236	4	no	—
247	C₂×C₄	#14273648	1	1	yes	characteristic
248	C₂×C₄	#14273648	1	1	yes	characteristic
249	C₂×C₄	#14273648	1	1	yes	characteristic
250	C₂×C₄	#14273648	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
252	C₂×C₄	#14273698	1	2	yes	—
253	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
254	C₂×C₄	#14273753	1	2	yes	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	D₄×Dic₅₉	1	2	yes	—
258	C₈	#14273714	1	1	yes	characteristic
259	C₈	#14273723	1	1	yes	characteristic
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	8	2	no	—
263	C₂²	—	8	2	no	—
264	C₂²	—	8	2	no	—
265	C₂²	C₂×#14273648	1	1	yes	characteristic
266	C₂²	#16837594	1	2	yes	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	8	2	no	—
271	C₄	—	472	4	no	—
272	C₄	#16837212	1	2	yes	complemented
273	C₄	#16837305	1	1	yes	characteristic
274	C₄	#16837407	1	1	yes	characteristic
275	C₄	#16837599	1	2	yes	—
276	C₂	—	2	2	no	—
277	C₂	—	16	2	no	—
278	C₂	#21425024	1	1	yes	characteristic
279	C₂	C₂×#16837212	1	1	yes	characteristic
280	C₂	#21425832	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14273648
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting, prime radical	C₅₉×#3147	C₂
socle	C₂²×C₅₉	C₂×#102
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#3147	D₅₉
hypocenter	C₅₉	#13255
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #21425852	C₂	semidirect
2. C₂×#16837212	C₂	semidirect
3. C₄×#14273659	C₂	semidirect
16. C₅₉×#495	C₄	semidirect
17. C₅₉×#494	C₄	semidirect
18. C₅₉×#493	C₄	semidirect
26. #14273659	D₄	semidirect
131. C₄×C₅₉	#360	semidirect
150. #495	Dic₅₉	semidirect
151. #494	Dic₅₉	semidirect
152. #493	Dic₅₉	semidirect
161. C₅₉	#13255	semidirect
272. C₄	#16837212	semidirect

Maximal subgroups

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

subgroup	index
#21425852	2
C₂×#16837212	2
C₄×#14273659	2
C₅₉×#3147	2
#13255	59