Subgroups — #28481908

Order 15104.

366 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424260	C₂	1	1	yes	maximal, characteristic
2	#21422844	C₂	1	1	yes	maximal, characteristic
3	#21422660	C₂	1	2	yes	maximal
4	#21421956	C₂	1	2	yes	maximal
5	C₅₉×#2960	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	C₂×#14273661	—	2	1	no	—
8	C₂×#14273659	—	2	1	no	—
9	#16837268	C₂²	1	2	yes	—
10	#16837155	—	2	1	no	—
11	#16837153	—	2	1	no	—
12	#16837139	—	2	4	no	—
13	C₅₉×#533	C₂²	1	1	yes	characteristic
14	C₅₉×#446	—	2	1	no	—
15	C₅₉×#446	C₄	1	2	yes	complemented
16	C₈×Q₈×C₅₉	—	2	1	no	—
17	C₈×Q₈×C₅₉	C₄	1	2	yes	complemented
18	C₅₉×#424	—	2	1	no	—
19	C₅₉×#423	C₂²	1	1	yes	characteristic
20	C₅₉×#405	—	2	1	no	—
21	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₅₉	—	2	2	no	—
23	#14273661	—	4	2	no	—
24	#14273659	—	4	2	no	—
25	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
26	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
27	C₄×C₅₉⋊C₈	—	2	2	no	—
28	C₅₉⋊#95	—	2	2	no	—
29	C₅₉⋊#95	—	2	2	no	—
30	C₂×M₁₆×C₅₉	—	2	1	no	—
31	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₄⋊Q₈×C₅₉	Q₈	1	1	yes	characteristic
34	C₄⋊Q₈×C₅₉	Q₈	1	1	yes	characteristic
35	#125×C₅₉	—	2	1	no	—
36	C₂²⋊Q₈×C₅₉	—	2	1	no	—
37	C₂²⋊Q₈×C₅₉	D₄	1	2	yes	—
38	C₄×Q₈×C₅₉	—	2	1	no	—
39	C₄×Q₈×C₅₉	C₂×C₄	1	2	yes	—
40	#117×C₅₉	—	2	1	no	—
41	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₅₉	—	2	2	no	—
43	C₄⋊C₈×C₅₉	—	2	2	no	—
44	C₄⋊C₈×C₅₉	—	2	1	no	—
45	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
46	#97×C₅₉	—	2	2	no	—
47	C₄×C₈×C₅₉	—	2	2	no	—
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	C₂×Q₈×C₅₉	—	2	1	no	—
53	C₂×Q₈×C₅₉	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
57	C₂²×Dic₅₉	—	4	2	no	—
58	M₁₆×C₅₉	—	4	1	no	—
59	M₁₆×C₅₉	—	4	1	no	—
60	C₂×C₈×C₅₉	—	2	1	no	—
61	C₂×C₈×C₅₉	—	2	1	no	—
62	C₂×C₈×C₅₉	—	2	2	no	—
63	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
68	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
69	C₄⋊C₄×C₅₉	—	2	2	no	—
70	C₄⋊C₄×C₅₉	—	2	2	no	—
71	C₄⋊C₄×C₅₉	—	2	1	no	—
72	C₄⋊C₄×C₅₉	—	2	1	no	—
73	C₄⋊C₄×C₅₉	—	2	1	no	—
74	C₄⋊C₄×C₅₉	C₄⋊C₄	1	2	yes	—
75	C₂²⋊C₄×C₅₉	—	2	2	no	—
76	C₄²×C₅₉	—	2	2	no	—
77	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
80	C₄²×C₅₉	#41	1	1	yes	characteristic
81	C₄⋊Dic₅₉	—	4	2	no	—
82	C₄⋊Dic₅₉	—	4	2	no	—
83	C₂×C₅₉⋊C₈	—	4	2	no	—
84	C₂³×C₅₉	#121	1	1	yes	characteristic
85	Q₈×C₅₉	—	4	2	no	—
86	Q₈×C₅₉	—	4	2	no	—
87	C₂×C₄×C₅₉	—	2	2	no	—
88	C₂×C₄×C₅₉	—	2	2	no	—
89	C₂×C₄×C₅₉	—	4	1	no	—
90	C₂×C₄×C₅₉	—	4	1	no	—
91	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
94	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
95	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
101	C₂×Dic₅₉	—	8	2	no	—
102	C₂×Dic₅₉	—	8	2	no	—
103	C₂×Dic₅₉	—	8	2	no	—
104	C₈×C₅₉	—	2	1	no	—
105	C₈×C₅₉	—	2	1	no	—
106	C₈×C₅₉	—	4	2	no	—
107	C₈×C₅₉	#103	1	2	yes	—
108	C₈×C₅₉	#103	1	2	yes	—
109	C₅₉⋊C₈	—	8	2	no	—
110	#8677	—	59	1	no	maximal, Sylow
111	C₂²×C₅₉	—	2	1	no	—
112	C₂²×C₅₉	—	2	1	no	—
113	C₂²×C₅₉	#380	1	1	yes	characteristic
114	C₂²×C₅₉	#460	1	1	yes	characteristic
115	C₂²×C₅₉	#462	1	1	yes	characteristic
116	C₄×C₅₉	—	2	1	no	—
117	C₄×C₅₉	—	2	1	no	—
118	C₄×C₅₉	—	2	1	no	—
119	C₄×C₅₉	—	2	1	no	—
120	C₄×C₅₉	—	4	2	no	—
121	C₄×C₅₉	—	4	2	no	—
122	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
123	C₄×C₅₉	#417	1	1	yes	characteristic
124	C₄×C₅₉	#475	1	1	yes	characteristic
125	C₄×C₅₉	#477	1	1	yes	characteristic
126	Dic₅₉	—	16	2	no	—
127	#2960	D₅₉	1	1	yes	characteristic
128	#1846	—	59	1	no	—
129	#1564	—	59	1	no	—
130	#1518	—	59	2	no	—
131	#1286	—	59	2	no	—
132	C₂×C₅₉	—	2	1	no	—
133	C₂×C₅₉	—	2	1	no	—
134	C₂×C₅₉	#1803	1	1	yes	characteristic
135	C₂×C₅₉	#1812	1	1	yes	characteristic
136	C₂×C₅₉	#1814	1	1	yes	characteristic
137	#533	C₂×D₅₉	1	1	yes	characteristic
138	#446	—	2	1	no	—
139	#446	Dic₅₉	1	2	yes	complemented
140	C₈×Q₈	—	2	1	no	—
141	C₈×Q₈	Dic₅₉	1	2	yes	complemented
142	#431	—	59	2	no	—
143	C₂×#107	—	118	1	no	—
144	C₂×#106	—	118	1	no	—
145	#424	—	2	1	no	—
146	#423	C₂×D₅₉	1	1	yes	characteristic
147	#405	—	2	1	no	—
148	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
149	#382	—	59	2	no	—
150	#335	—	118	1	no	—
151	#334	—	118	1	no	—
152	#328	—	118	4	no	—
153	C₅₉	#8677	1	1	yes	characteristic, complemented, Sylow
154	C₂×M₁₆	—	2	1	no	—
155	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
156	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
157	C₄⋊Q₈	Dic₁₁₈	1	1	yes	characteristic
158	C₄⋊Q₈	Dic₁₁₈	1	1	yes	characteristic
159	#125	—	2	1	no	—
160	C₂²⋊Q₈	—	2	1	no	—
161	C₂²⋊Q₈	D₂₃₆	1	2	yes	—
162	C₄×Q₈	—	2	1	no	—
163	C₄×Q₈	C₂×Dic₅₉	1	2	yes	—
164	#117	—	2	1	no	—
165	C₂×C₄⋊C₄	—	118	2	no	—
166	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
167	#107	—	236	2	no	—
168	#106	—	236	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	118	2	no	—
173	C₄⋊C₈	C₂×Dic₅₉	1	2	yes	—
174	C₂²⋊C₈	—	118	2	no	—
175	#97	—	2	2	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	118	2	no	—
178	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
179	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
182	#95	—	118	2	no	—
183	#95	—	118	2	no	—
184	C₂×Q₈	—	2	1	no	—
185	C₂×Q₈	C₄⋊Dic₅₉	1	2	yes	—
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	M₁₆	—	4	1	no	—
191	M₁₆	—	4	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	236	2	no	—
196	C₂×C₈	#11745671	1	2	yes	—
197	C₂×C₈	#11745672	1	2	yes	—
198	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
199	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
201	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	236	2	no	—
208	C₄⋊C₄	—	236	2	no	—
209	C₄⋊C₄	C₄⋊Dic₅₉	1	2	yes	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
213	C₄²	#11745685	1	1	yes	characteristic
214	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
215	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
216	C₂³	#14273701	1	1	yes	characteristic
217	Q₈	—	4	2	no	—
218	Q₈	—	4	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	472	2	no	—
224	C₂×C₄	—	472	2	no	—
225	C₂×C₄	—	472	2	no	—
226	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
228	C₂×C₄	#14273686	1	1	yes	characteristic
229	C₂×C₄	#14273701	1	1	yes	characteristic
230	C₂×C₄	#14273706	1	1	yes	characteristic
231	C₂×C₄	#14273706	1	1	yes	characteristic
232	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
233	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
234	C₂×C₄	#14273752	1	1	yes	characteristic
235	C₂×C₄	#14273757	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	4	2	no	—
239	C₈	—	472	2	no	—
240	C₈	#14273651	1	2	yes	—
241	C₈	#14273651	1	2	yes	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	#16837262	1	1	yes	characteristic
245	C₂²	#16837602	1	1	yes	characteristic
246	C₂²	#16837612	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	4	2	no	—
252	C₄	—	4	2	no	—
253	C₄	—	944	2	no	—
254	C₄	C₂×#14273651	1	1	yes	characteristic
255	C₄	#16837403	1	1	yes	characteristic
256	C₄	#16837687	1	1	yes	characteristic
257	C₄	#16837697	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	#21424101	1	1	yes	characteristic
261	C₂	#21424142	1	1	yes	characteristic
262	C₂	#21424150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837262
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#2960	C₂
socle	C₂²×C₅₉	#380
prime radical	C₂³×C₅₉	#121
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#2960	D₅₉
hypocenter	C₅₉	#8677
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
15. C₅₉×#446	C₄	semidirect
17. C₈×Q₈×C₅₉	C₄	semidirect
139. #446	Dic₅₉	semidirect
141. C₈×Q₈	Dic₅₉	semidirect
153. C₅₉	#8677	semidirect

Maximal subgroups

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

subgroup	index
#21424260	2
#21422844	2
#21422660	2
#21421956	2
C₅₉×#2960	2
#8677	59