Subgroups — #28493401

Order 15104.

366 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425032	C₂	1	1	yes	maximal, characteristic
2	#21424256	C₂	1	1	yes	maximal, characteristic
3	#21424102	C₂	1	2	yes	maximal
4	#21421980	C₂	1	2	yes	maximal
5	C₅₉×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
7	C₂×#14273661	—	2	1	no	—
8	C₂×#14273661	—	2	1	no	—
9	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#14273652	—	2	2	no	—
11	#16837272	C₂²	1	2	yes	—
12	#16837165	—	2	2	no	—
13	#16837156	—	2	1	no	—
14	#16837156	—	2	1	no	—
15	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₁₈	D₄	1	1	yes	characteristic
18	C₂²×C₅₉⋊C₈	—	2	1	no	—
19	C₄×Dic₁₁₈	—	2	2	no	—
20	C₄×Dic₁₁₈	—	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	C₂×C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
25	C₂×Dic₅₉⋊C₄	—	2	1	no	—
26	C₂×C₄×Dic₅₉	—	2	1	no	—
27	#14273661	—	4	2	no	—
28	#14273661	—	4	2	no	—
29	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
30	#14273652	—	4	4	no	—
31	C₂²×C₈×C₅₉	—	2	2	no	—
32	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₅₉	—	2	2	no	—
36	C₄⋊C₈×C₅₉	—	2	2	no	—
37	C₄×C₈×C₅₉	—	2	2	no	—
38	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
41	C₂²×Dic₅₉	—	2	1	no	—
42	C₂²×Dic₅₉	—	2	1	no	—
43	C₂²×Dic₅₉	—	4	2	no	—
44	C₂×Dic₁₁₈	—	2	2	no	—
45	C₂×Dic₁₁₈	—	2	2	no	—
46	C₂×Dic₁₁₈	—	2	1	no	—
47	C₂×Dic₁₁₈	D₈	1	2	yes	—
48	C₂×C₈×C₅₉	—	2	1	no	—
49	C₂×C₈×C₅₉	—	2	1	no	—
50	C₂×C₈×C₅₉	—	2	1	no	—
51	C₂×C₈×C₅₉	—	2	1	no	—
52	C₂×C₈×C₅₉	—	4	2	no	—
53	C₂×C₈×C₅₉	—	4	2	no	—
54	C₂×C₈×C₅₉	—	4	2	no	—
55	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
56	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
57	C₄²×C₅₉	#41	1	2	yes	—
58	C₄²×C₅₉	#41	1	2	yes	—
59	C₄⋊Dic₅₉	—	2	2	no	—
60	C₄⋊Dic₅₉	—	4	2	no	—
61	C₄⋊Dic₅₉	—	4	2	no	—
62	C₄⋊Dic₅₉	—	4	2	no	—
63	C₄⋊Dic₅₉	—	4	2	no	—
64	C₄⋊Dic₅₉	—	8	2	no	—
65	Dic₅₉⋊C₄	—	4	2	no	—
66	C₄×Dic₅₉	—	4	1	no	—
67	C₄×Dic₅₉	—	4	1	no	—
68	C₂×C₅₉⋊C₈	—	4	1	no	—
69	C₂×C₅₉⋊C₈	—	4	1	no	—
70	C₂×C₅₉⋊C₈	—	4	1	no	—
71	C₂³×C₅₉	#121	1	1	yes	characteristic
72	C₂×C₄×C₅₉	—	2	1	no	—
73	C₂×C₄×C₅₉	—	2	1	no	—
74	C₂×C₄×C₅₉	#123	1	2	yes	—
75	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
76	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
77	C₂×C₄×C₅₉	#126	1	2	yes	—
78	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
82	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₄×C₅₉	#137	1	2	yes	—
84	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
85	C₂×Dic₅₉	—	2	2	no	—
86	C₂×Dic₅₉	—	4	1	no	—
87	C₂×Dic₅₉	—	4	1	no	—
88	C₂×Dic₅₉	—	4	1	no	—
89	C₂×Dic₅₉	—	4	1	no	—
90	C₂×Dic₅₉	—	4	1	no	—
91	C₂×Dic₅₉	—	8	2	no	—
92	C₂×Dic₅₉	—	8	2	no	—
93	C₂×Dic₅₉	—	8	2	no	—
94	Dic₁₁₈	—	2	4	no	—
95	Dic₁₁₈	—	4	2	no	—
96	C₈×C₅₉	—	2	2	no	—
97	C₈×C₅₉	—	2	2	no	—
98	C₈×C₅₉	—	8	2	no	—
99	C₅₉⋊C₈	—	8	1	no	—
100	#10617	—	59	1	no	maximal, Sylow
101	C₂²×C₅₉	#397	1	1	yes	characteristic
102	C₂²×C₅₉	#460	1	1	yes	characteristic
103	C₂²×C₅₉	#462	1	1	yes	characteristic
104	C₂²×C₅₉	#466	1	1	yes	characteristic
105	C₂²×C₅₉	#470	1	1	yes	characteristic
106	C₂²×C₅₉	#472	1	1	yes	characteristic
107	C₂²×C₅₉	#473	1	1	yes	characteristic
108	C₄×C₅₉	—	2	2	no	—
109	C₄×C₅₉	#480	1	2	yes	—
110	C₄×C₅₉	#483	1	1	yes	characteristic
111	C₄×C₅₉	#484	1	1	yes	characteristic
112	C₄×C₅₉	#485	1	2	yes	—
113	C₄×C₅₉	#487	1	1	yes	characteristic
114	C₄×C₅₉	#489	1	1	yes	characteristic
115	Dic₅₉	—	4	2	no	—
116	Dic₅₉	—	8	1	no	—
117	Dic₅₉	—	16	2	no	—
118	#1984	—	59	1	no	—
119	#1845	—	59	1	no	—
120	#1803	—	59	2	no	—
121	#1296	—	59	2	no	—
122	#1277	D₅₉	1	1	yes	characteristic
123	C₂×C₅₉	#1668	1	1	yes	characteristic
124	C₂×C₅₉	#1674	1	1	yes	characteristic
125	C₂×C₅₉	#1690	1	1	yes	characteristic
126	C₂×C₅₉	#1696	1	1	yes	characteristic
127	C₂×C₅₉	#2072	1	1	yes	characteristic
128	C₂×C₅₉	#2075	1	1	yes	characteristic
129	C₂×C₅₉	#2078	1	1	yes	characteristic
130	C₂×C₄×Q₈	—	59	1	no	—
131	C₂×#107	—	118	1	no	—
132	C₂×#107	—	118	1	no	—
133	C₂×C₄⋊C₈	—	59	1	no	—
134	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
135	C₂×#103	—	118	2	no	—
136	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
137	#384	—	59	2	no	—
138	#340	—	118	2	no	—
139	#336	—	118	1	no	—
140	#336	—	118	1	no	—
141	C₅₉	#10617	1	1	yes	characteristic, complemented, Sylow
142	C₂²×Q₈	—	59	1	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	118	1	no	—
145	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
146	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
147	C₄×Q₈	—	118	2	no	—
148	C₄×Q₈	—	118	2	no	—
149	C₂×C₄⋊C₄	—	59	1	no	—
150	C₂×C₄⋊C₄	—	118	2	no	—
151	C₂×C₄⋊C₄	—	118	1	no	—
152	C₂×C₄⋊C₄	—	118	2	no	—
153	C₂×C₄⋊C₄	—	118	2	no	—
154	C₂×C₄²	—	118	1	no	—
155	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
156	#107	—	236	2	no	—
157	#107	—	236	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	118	2	no	—
161	#103	—	236	4	no	—
162	C₄×C₈	—	2	2	no	—
163	C₂×Q₈	—	59	2	no	—
164	C₂×Q₈	—	118	1	no	—
165	C₂×Q₈	—	118	2	no	—
166	C₂×Q₈	—	118	2	no	—
167	C₂²×C₄	—	118	1	no	—
168	C₂²×C₄	—	118	1	no	—
169	C₂²×C₄	—	236	2	no	—
170	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
171	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
172	C₂²×C₄	#11745689	1	1	yes	characteristic
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	236	1	no	—
181	C₂×C₈	—	236	1	no	—
182	C₂×C₈	—	236	1	no	—
183	C₂×C₈	#11745660	1	2	yes	—
184	C₂×C₈	Dic₂₃₆	1	2	yes	—
185	C₄⋊C₄	—	118	2	no	—
186	C₄⋊C₄	—	472	2	no	—
187	C₄⋊C₄	—	236	2	no	—
188	C₄⋊C₄	—	236	2	no	—
189	C₄⋊C₄	—	236	2	no	—
190	C₄⋊C₄	—	236	2	no	—
191	C₄⋊C₄	—	236	2	no	—
192	C₄²	—	236	1	no	—
193	C₄²	—	236	1	no	—
194	C₄²	#11745685	1	2	yes	—
195	C₄²	#11745687	1	2	yes	—
196	C₂³	#14273702	1	1	yes	characteristic
197	Q₈	—	118	4	no	—
198	Q₈	—	236	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	118	2	no	—
202	C₂×C₄	—	472	2	no	—
203	C₂×C₄	—	472	2	no	—
204	C₂×C₄	—	472	2	no	—
205	C₂×C₄	—	236	1	no	—
206	C₂×C₄	—	236	1	no	—
207	C₂×C₄	—	236	1	no	—
208	C₂×C₄	—	236	1	no	—
209	C₂×C₄	—	236	1	no	—
210	C₂×C₄	#14273709	1	2	yes	—
211	C₂×C₄	#14273711	1	1	yes	characteristic
212	C₂×C₄	#14273715	1	1	yes	characteristic
213	C₂×C₄	#14273722	1	2	yes	—
214	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
215	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
216	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
217	C₂×C₄	#14273750	1	1	yes	characteristic
218	C₂×C₄	#14273755	1	1	yes	characteristic
219	C₂×C₄	#14273756	1	2	yes	—
220	C₂×C₄	#14273760	1	1	yes	characteristic
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	8	2	no	—
224	C₈	—	472	1	no	—
225	C₂²	#16837329	1	1	yes	characteristic
226	C₂²	#16837603	1	1	yes	characteristic
227	C₂²	#16837613	1	1	yes	characteristic
228	C₂²	#16837639	1	1	yes	characteristic
229	C₂²	#16837661	1	1	yes	characteristic
230	C₂²	#16837673	1	1	yes	characteristic
231	C₂²	#16837678	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	944	2	no	—
234	C₄	—	472	1	no	—
235	C₄	—	236	2	no	—
236	C₄	#16837714	1	2	yes	—
237	C₄	#16837733	1	1	yes	characteristic
238	C₄	#16837738	1	1	yes	characteristic
239	C₄	#16837742	1	2	yes	—
240	C₄	#16837753	1	1	yes	characteristic
241	C₄	#16837762	1	1	yes	characteristic
242	C₂	#21423457	1	1	yes	characteristic
243	C₂	#21423491	1	1	yes	characteristic
244	C₂	#21423591	1	1	yes	characteristic
245	C₂	#21423625	1	1	yes	characteristic
246	C₂	#21425506	1	1	yes	characteristic
247	C₂	#21425521	1	1	yes	characteristic
248	C₂	#21425538	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273702
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1277	C₂
socle, prime radical	C₂³×C₅₉	#121
hypercenter	#1277	D₅₉
hypocenter	C₅₉	#10617
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
141. C₅₉	#10617	semidirect

Maximal subgroups

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

subgroup	index
#21425032	2
#21424256	2
#21424102	2
#21421980	2
C₅₉×#1277	2
#10617	59