Subgroups — #28481748

Order 15104.

454 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424256	C₂	1	1	yes	maximal, characteristic
2	#21424101	C₂	1	2	yes	maximal
3	C₂×#16837155	C₂	1	1	yes	maximal, characteristic
4	#21421980	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273661	—	2	1	no	—
7	C₂×#14273661	—	2	1	no	—
8	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×#14273651	—	2	2	no	—
10	#16837272	C₂²	1	2	yes	—
11	#16837164	—	2	2	no	—
12	#16837155	—	2	2	no	—
13	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₅₉	—	2	2	no	—
15	C₈×Q₈×C₅₉	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₈×C₅₉	—	2	1	no	—
17	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₉	—	2	1	no	—
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₅₉⋊C₈	—	2	2	no	—
21	C₂×C₄⋊Dic₅₉	—	2	2	no	—
22	C₂×C₄⋊Dic₅₉	—	2	2	no	—
23	C₂×C₄⋊Dic₅₉	—	2	2	no	—
24	#14273661	—	4	2	no	—
25	#14273661	—	4	2	no	—
26	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
27	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
28	#14273651	—	4	4	no	—
29	C₂²×Q₈×C₅₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×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	—	2	1	no	—
34	C₄×Q₈×C₅₉	—	2	1	no	—
35	C₄×Q₈×C₅₉	C₂×C₄	1	2	yes	—
36	C₄×Q₈×C₅₉	Q₈	1	2	yes	—
37	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₉	—	2	1	no	—
40	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₉	—	2	4	no	—
42	C₄⋊C₈×C₅₉	—	2	1	no	—
43	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₉	—	2	4	no	—
45	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₅₉	D₄	1	2	yes	—
47	C₂×Q₈×C₅₉	—	2	1	no	—
48	C₂×Q₈×C₅₉	—	2	1	no	—
49	C₂×Q₈×C₅₉	—	2	1	no	—
50	C₂×Q₈×C₅₉	C₄⋊C₄	1	2	yes	—
51	C₂×Q₈×C₅₉	D₈	1	2	yes	—
52	C₂×Q₈×C₅₉	Q₁₆	1	2	yes	—
53	C₂²×C₄×C₅₉	—	2	1	no	—
54	C₂²×C₄×C₅₉	—	2	1	no	—
55	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
58	C₂²×Dic₅₉	—	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	SD₁₆	1	2	yes	—
67	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
68	C₄⋊C₄×C₅₉	—	2	2	no	—
69	C₄⋊C₄×C₅₉	—	2	1	no	—
70	C₄⋊C₄×C₅₉	—	2	2	no	—
71	C₄⋊C₄×C₅₉	C₄⋊C₄	1	2	yes	—
72	C₄²×C₅₉	—	2	2	no	—
73	C₄²×C₅₉	—	2	2	no	—
74	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
77	C₄²×C₅₉	#41	1	1	yes	characteristic
78	C₄⋊Dic₅₉	—	4	2	no	—
79	C₄⋊Dic₅₉	—	4	2	no	—
80	C₄⋊Dic₅₉	—	4	2	no	—
81	C₄⋊Dic₅₉	—	4	2	no	—
82	C₄⋊Dic₅₉	—	8	2	no	—
83	C₂×C₅₉⋊C₈	—	4	2	no	—
84	C₂×C₅₉⋊C₈	—	4	2	no	—
85	C₂×C₅₉⋊C₈	—	4	2	no	—
86	C₂³×C₅₉	#121	1	1	yes	characteristic
87	Q₈×C₅₉	—	2	2	no	—
88	Q₈×C₅₉	—	2	2	no	—
89	Q₈×C₅₉	—	2	2	no	—
90	Q₈×C₅₉	#107	1	4	yes	complemented
91	C₂×C₄×C₅₉	—	2	2	no	—
92	C₂×C₄×C₅₉	—	2	2	no	—
93	C₂×C₄×C₅₉	—	2	2	no	—
94	C₂×C₄×C₅₉	—	2	1	no	—
95	C₂×C₄×C₅₉	—	2	1	no	—
96	C₂×C₄×C₅₉	—	2	2	no	—
97	C₂×C₄×C₅₉	—	4	1	no	—
98	C₂×C₄×C₅₉	—	4	1	no	—
99	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
102	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
103	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
110	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
111	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
113	C₂×Dic₅₉	—	8	2	no	—
114	C₂×Dic₅₉	—	8	2	no	—
115	C₂×Dic₅₉	—	8	2	no	—
116	C₈×C₅₉	—	2	2	no	—
117	C₈×C₅₉	—	4	2	no	—
118	C₈×C₅₉	#103	1	4	yes	complemented
119	C₅₉⋊C₈	—	8	2	no	—
120	#8645	—	59	1	no	maximal, Sylow
121	C₂²×C₅₉	#380	1	1	yes	characteristic
122	C₂²×C₅₉	#460	1	1	yes	characteristic
123	C₂²×C₅₉	#462	1	1	yes	characteristic
124	C₂²×C₅₉	#466	1	1	yes	characteristic
125	C₂²×C₅₉	#467	1	1	yes	characteristic
126	C₂²×C₅₉	#472	1	1	yes	characteristic
127	C₂²×C₅₉	#473	1	1	yes	characteristic
128	C₄×C₅₉	—	2	4	no	—
129	C₄×C₅₉	—	2	1	no	—
130	C₄×C₅₉	—	2	1	no	—
131	C₄×C₅₉	—	4	2	no	—
132	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
133	C₄×C₅₉	#417	1	1	yes	characteristic
134	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
135	C₄×C₅₉	#427	1	1	yes	characteristic
136	C₄×C₅₉	#466	1	1	yes	characteristic
137	C₄×C₅₉	#467	1	1	yes	characteristic
138	C₄×C₅₉	#475	1	1	yes	characteristic
139	C₄×C₅₉	#477	1	1	yes	characteristic
140	Dic₅₉	—	16	2	no	—
141	C₂×C₈×Q₈	D₅₉	1	1	yes	characteristic
142	#1845	—	59	1	no	—
143	#1803	—	59	2	no	—
144	C₂×#335	—	59	1	no	—
145	#1296	—	59	2	no	—
146	C₂×C₅₉	#1668	1	1	yes	characteristic
147	C₂×C₅₉	#1670	1	1	yes	characteristic
148	C₂×C₅₉	#1674	1	1	yes	characteristic
149	C₂×C₅₉	#1676	1	1	yes	characteristic
150	C₂×C₅₉	#1803	1	1	yes	characteristic
151	C₂×C₅₉	#1808	1	1	yes	characteristic
152	C₂×C₅₉	#1816	1	1	yes	characteristic
153	C₂×C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
154	C₈×Q₈	—	2	2	no	—
155	C₈×Q₈	Dic₅₉	1	4	yes	complemented
156	C₂×#107	—	118	1	no	—
157	C₂×#107	—	118	1	no	—
158	C₂×C₄⋊C₈	—	2	1	no	—
159	C₂×C₄⋊C₈	—	59	2	no	—
160	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
161	C₂×#103	—	118	2	no	—
162	C₂×C₄×C₈	—	2	1	no	—
163	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
164	#384	—	59	2	no	—
165	#340	—	118	2	no	—
166	#335	—	118	2	no	—
167	C₅₉	#8645	1	1	yes	characteristic, complemented, Sylow
168	C₂²×Q₈	D₂₃₆	1	1	yes	characteristic
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	—	118	2	no	—
171	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
172	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
173	C₄×Q₈	—	2	1	no	—
174	C₄×Q₈	—	2	1	no	—
175	C₄×Q₈	Dic₁₁₈	1	2	yes	—
176	C₄×Q₈	C₂×Dic₅₉	1	2	yes	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	118	2	no	—
179	C₂×C₄⋊C₄	—	118	2	no	—
180	C₂×C₄⋊C₄	—	118	2	no	—
181	C₂×C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
182	C₂×C₄²	—	2	1	no	—
183	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
184	#107	—	236	2	no	—
185	#107	—	236	2	no	—
186	C₄⋊C₈	—	2	4	no	—
187	C₄⋊C₈	—	2	1	no	—
188	C₄⋊C₈	—	118	2	no	—
189	C₄⋊C₈	—	118	2	no	—
190	C₄⋊C₈	C₂×Dic₅₉	1	2	yes	—
191	#103	—	236	4	no	—
192	C₄×C₈	—	2	4	no	—
193	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
194	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
195	C₂×Q₈	—	2	1	no	—
196	C₂×Q₈	—	2	1	no	—
197	C₂×Q₈	—	2	1	no	—
198	C₂×Q₈	D₄₇₂	1	2	yes	—
199	C₂×Q₈	Dic₂₃₆	1	2	yes	—
200	C₂×Q₈	C₄⋊Dic₅₉	1	2	yes	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	236	2	no	—
204	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
205	C₂²×C₄	#11745685	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	4	1	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	—	236	2	no	—
213	C₂×C₈	—	236	2	no	—
214	C₂×C₈	—	236	2	no	—
215	C₂×C₈	#11745671	1	2	yes	—
216	C₂×C₈	#11745672	1	2	yes	—
217	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
218	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	472	2	no	—
223	C₄⋊C₄	—	236	2	no	—
224	C₄⋊C₄	—	236	2	no	—
225	C₄⋊C₄	—	236	2	no	—
226	C₄⋊C₄	—	236	2	no	—
227	C₄⋊C₄	C₄⋊Dic₅₉	1	2	yes	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
231	C₄²	#11745685	1	1	yes	characteristic
232	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
233	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
234	C₂³	#14273701	1	1	yes	characteristic
235	Q₈	—	2	2	no	—
236	Q₈	—	2	2	no	—
237	Q₈	—	2	2	no	—
238	Q₈	#14273661	1	4	yes	complemented
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	472	2	no	—
248	C₂×C₄	—	472	2	no	—
249	C₂×C₄	—	472	2	no	—
250	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
252	C₂×C₄	#14273686	1	1	yes	characteristic
253	C₂×C₄	#14273701	1	1	yes	characteristic
254	C₂×C₄	#14273706	1	1	yes	characteristic
255	C₂×C₄	#14273706	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
257	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
258	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
259	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
260	C₂×C₄	#14273749	1	1	yes	characteristic
261	C₂×C₄	#14273749	1	1	yes	characteristic
262	C₂×C₄	#14273752	1	1	yes	characteristic
263	C₂×C₄	#14273757	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₈	—	472	2	no	—
267	C₈	#14273651	1	4	yes	complemented
268	C₂²	#16837262	1	1	yes	characteristic
269	C₂²	#16837602	1	1	yes	characteristic
270	C₂²	#16837612	1	1	yes	characteristic
271	C₂²	#16837638	1	1	yes	characteristic
272	C₂²	#16837643	1	1	yes	characteristic
273	C₂²	#16837672	1	1	yes	characteristic
274	C₂²	#16837677	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	4	no	—
278	C₄	—	4	2	no	—
279	C₄	—	944	2	no	—
280	C₄	C₂×#14273651	1	1	yes	characteristic
281	C₄	#16837403	1	1	yes	characteristic
282	C₄	C₂×#14273661	1	1	yes	characteristic
283	C₄	#16837440	1	1	yes	characteristic
284	C₄	#16837638	1	1	yes	characteristic
285	C₄	#16837643	1	1	yes	characteristic
286	C₄	#16837687	1	1	yes	characteristic
287	C₄	#16837697	1	1	yes	characteristic
288	C₂	#21423456	1	1	yes	characteristic
289	C₂	#21423466	1	1	yes	characteristic
290	C₂	#21423490	1	1	yes	characteristic
291	C₂	#21423500	1	1	yes	characteristic
292	C₂	#21424101	1	1	yes	characteristic
293	C₂	#21424125	1	1	yes	characteristic
294	C₂	#21424160	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273701
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₈×Q₈×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	#121
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×C₈×Q₈	D₅₉
hypocenter	C₅₉	#8645
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₅₉	C₄	semidirect
90. Q₈×C₅₉	#107	semidirect
118. C₈×C₅₉	#103	semidirect
155. C₈×Q₈	Dic₅₉	semidirect
167. C₅₉	#8645	semidirect
238. Q₈	#14273661	semidirect
267. C₈	#14273651	semidirect

Maximal subgroups

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

subgroup	index
#21424256	2
#21424101	2
C₂×#16837155	2
#21421980	2
C₂×C₈×Q₈×C₅₉	2
#8645	59