Subgroups — #28744049

Order 15104.

578 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837782	C₂	1	1	yes	maximal, characteristic
2	#21426343	C₂	1	4	yes	maximal, complemented
3	#21426325	C₂	1	4	yes	maximal, complemented
4	#21425815	C₂	1	4	yes	maximal
5	C₄×#14273667	C₂	1	1	yes	maximal, characteristic
6	C₅₉×#3160	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₃₆	—	2	1	no	—
8	C₂²×Dic₂₃₆	C₂²	1	2	yes	—
9	C₂×#14273727	—	2	1	no	—
10	C₂×#14273667	C₂²	1	2	yes	—
11	#16837782	—	2	1	no	—
12	#16837782	—	2	1	no	—
13	#16837782	C₂²	1	2	yes	—
14	#16837782	C₂²	1	2	yes	—
15	#16837205	C₂²	1	16	yes	—
16	#16837178	C₂²	1	2	yes	—
17	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	—
18	C₅₉×#533	—	2	1	no	—
19	C₅₉×#500	C₂²	1	2	yes	—
20	C₅₉×#499	—	2	1	no	—
21	C₅₉×#499	C₂²	1	2	yes	—
22	C₅₉×#498	—	2	1	no	—
23	C₅₉×#427	—	2	1	no	—
24	C₅₉×#427	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₁₈	—	2	2	no	—
27	C₂×Dic₂₃₆	—	2	8	no	—
28	C₂×Dic₂₃₆	—	2	2	no	—
29	C₂×Dic₂₃₆	—	2	2	no	—
30	C₂×Dic₂₃₆	—	2	2	no	—
31	C₂×Dic₂₃₆	—	4	4	no	—
32	C₂×Dic₂₃₆	C₂³	1	4	yes	—
33	#14273727	—	2	2	no	—
34	#14273727	—	2	2	no	—
35	#14273727	—	4	2	no	—
36	C₂×C₄⋊Dic₅₉	—	4	2	no	—
37	#14273667	—	2	4	no	—
38	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
39	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₅₉	—	2	2	no	—
42	#125×C₅₉	—	2	2	no	—
43	C₂²⋊Q₈×C₅₉	—	4	2	no	—
44	C₄×Q₈×C₅₉	—	4	2	no	—
45	#117×C₅₉	—	2	2	no	—
46	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
47	#107×C₅₉	—	2	2	no	—
48	#107×C₅₉	C₂³	1	4	yes	—
49	#106×C₅₉	—	2	2	no	—
50	#106×C₅₉	—	2	2	no	—
51	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
55	C₂×Q₈×C₅₉	—	4	2	no	—
56	C₂²×C₄×C₅₉	D₈	1	2	yes	—
57	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₅₉	—	4	2	no	—
59	C₂×Dic₁₁₈	—	2	4	no	—
60	C₂×Dic₁₁₈	—	4	4	no	—
61	C₂×Dic₁₁₈	—	4	4	no	—
62	C₂×Dic₁₁₈	—	4	2	no	—
63	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
64	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₅₉	#41	1	4	yes	—
68	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₅₉	—	2	4	no	—
70	C₄⋊C₄×C₅₉	—	4	4	no	—
71	C₄⋊C₄×C₅₉	—	4	2	no	—
72	C₂²⋊C₄×C₅₉	—	4	2	no	—
73	C₄²×C₅₉	—	4	2	no	—
74	C₄²×C₅₉	D₈	1	2	yes	—
75	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₄⋊Dic₅₉	—	4	4	no	—
78	C₄⋊Dic₅₉	—	4	4	no	—
79	Dic₂₃₆	—	2	8	no	—
80	Dic₂₃₆	—	4	4	no	—
81	Dic₂₃₆	—	4	4	no	—
82	Dic₂₃₆	—	4	4	no	—
83	C₅₉⋊C₁₆	—	2	4	no	—
84	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
85	Q₈×C₅₉	—	8	4	no	—
86	C₂×C₄×C₅₉	—	2	4	no	—
87	C₂×C₄×C₅₉	—	4	4	no	—
88	C₂×C₄×C₅₉	#124	1	2	yes	—
89	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
93	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
94	C₂×Dic₅₉	—	4	4	no	—
95	C₂×Dic₅₉	—	8	4	no	—
96	Dic₁₁₈	—	4	4	no	—
97	Dic₁₁₈	—	4	4	no	—
98	Dic₁₁₈	—	8	4	no	—
99	C₈×C₅₉	#124	1	2	yes	—
100	C₈×C₅₉	C₂×SD₁₆	1	2	yes	—
101	C₈×C₅₉	C₂×SD₁₆	1	2	yes	—
102	C₈×C₅₉	C₂×#41	1	2	yes	—
103	#33705	—	59	1	no	maximal, Sylow
104	C₂²×C₅₉	—	2	2	no	—
105	C₂²×C₅₉	#486	1	2	yes	—
106	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
107	C₄×C₅₉	—	2	2	no	—
108	C₄×C₅₉	—	8	4	no	—
109	C₄×C₅₉	#486	1	2	yes	—
110	C₄×C₅₉	#510	1	4	yes	—
111	C₄×C₅₉	C₂×#124	1	1	yes	characteristic
112	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
113	Dic₅₉	—	8	4	no	—
114	#3160	D₅₉	1	1	yes	characteristic
115	C₂×#494	—	59	1	no	—
116	#2245	—	59	4	no	—
117	#2241	—	59	4	no	—
118	#2141	—	59	4	no	—
119	C₄×M₃₂	—	59	1	no	—
120	C₂×C₅₉	—	2	2	no	—
121	C₂×C₅₉	C₂×#486	1	1	yes	characteristic
122	C₂×C₅₉	C₂×#510	1	2	yes	—
123	C₂²×Q₁₆	—	59	2	no	—
124	C₂²×Q₁₆	—	118	1	no	—
125	#533	—	2	1	no	—
126	C₂×C₄⋊Q₈	—	118	1	no	—
127	C₂×M₃₂	—	59	2	no	—
128	#500	C₂×D₅₉	1	2	yes	—
129	#499	—	2	1	no	—
130	#499	C₂×D₅₉	1	2	yes	—
131	#498	—	2	1	no	—
132	#494	—	59	2	no	—
133	#494	—	59	2	no	—
134	#494	—	118	1	no	—
135	#494	—	118	1	no	—
136	#427	—	2	1	no	—
137	#427	C₂×D₅₉	1	2	yes	—
138	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
139	#358	—	59	16	no	—
140	#346	—	59	2	no	—
141	C₄×C₁₆	—	59	2	no	—
142	C₅₉	#33705	1	1	yes	characteristic, complemented, Sylow
143	C₂²×Q₈	—	118	2	no	—
144	C₂×Q₁₆	—	59	4	no	—
145	C₂×Q₁₆	—	118	8	no	—
146	C₂×Q₁₆	—	118	2	no	—
147	C₂×Q₁₆	—	118	2	no	—
148	C₂×Q₁₆	—	118	2	no	—
149	C₂×Q₁₆	—	236	4	no	—
150	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
151	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	—	118	2	no	—
154	C₄⋊Q₈	—	118	2	no	—
155	C₄⋊Q₈	—	236	2	no	—
156	#125	—	2	2	no	—
157	C₂²⋊Q₈	—	4	2	no	—
158	C₄×Q₈	—	4	2	no	—
159	#117	—	2	2	no	—
160	C₂×C₄⋊C₄	—	236	2	no	—
161	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
162	M₃₂	—	118	4	no	—
163	C₂×C₁₆	—	59	4	no	—
164	#107	—	2	2	no	—
165	#107	C₂²×D₅₉	1	4	yes	—
166	#106	—	2	2	no	—
167	#106	—	2	2	no	—
168	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
170	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
171	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	118	4	no	—
174	C₂×Q₈	—	236	4	no	—
175	C₂×Q₈	—	236	4	no	—
176	C₂×Q₈	—	236	2	no	—
177	C₂²×C₄	—	236	2	no	—
178	C₂²×C₄	#11745669	1	2	yes	—
179	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
180	Q₁₆	—	118	8	no	—
181	Q₁₆	—	236	4	no	—
182	Q₁₆	—	236	4	no	—
183	Q₁₆	—	236	4	no	—
184	C₂×C₈	#11745671	1	4	yes	—
185	C₂×C₈	#11745689	1	4	yes	—
186	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
187	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
188	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
189	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	236	4	no	—
194	C₄⋊C₄	—	236	4	no	—
195	C₂²⋊C₄	—	4	2	no	—
196	C₄²	—	4	2	no	—
197	C₄²	#11745669	1	2	yes	—
198	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
199	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
200	C₁₆	—	118	4	no	—
201	C₂³	C₂×#11745669	1	1	yes	characteristic
202	Q₈	—	8	4	no	—
203	Q₈	—	472	4	no	—
204	Q₈	—	236	4	no	—
205	Q₈	—	236	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	472	4	no	—
209	C₂×C₄	—	236	4	no	—
210	C₂×C₄	#14273716	1	2	yes	—
211	C₂×C₄	C₂×#11745669	1	2	yes	—
212	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
213	C₂×C₄	C₂×#11745669	1	2	yes	—
214	C₂×C₄	C₂×#11745671	1	2	yes	—
215	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
216	C₈	#14273716	1	2	yes	—
217	C₈	C₂×#11745671	1	2	yes	—
218	C₈	C₂×#11745671	1	2	yes	—
219	C₈	C₂×#11745689	1	2	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	#16837750	1	2	yes	—
222	C₂²	C₂²×#11745669	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	—	472	4	no	—
226	C₄	#16837750	1	2	yes	—
227	C₄	#16837842	1	4	yes	—
228	C₄	C₂×#14273716	1	1	yes	characteristic
229	C₄	C₂²×#11745671	1	1	yes	characteristic
230	C₂	—	2	2	no	—
231	C₂	C₂×#16837750	1	1	yes	characteristic
232	C₂	C₂×#16837842	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#3160	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂³×C₅₉	C₂×D₈
hypercenter	#3160	D₅₉
hypocenter	C₅₉	#33705
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #21426343	C₂	semidirect
3. #21426325	C₂	semidirect
142. C₅₉	#33705	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837782	2
#21426343	2
#21426325	2
#21425815	2
C₄×#14273667	2
C₅₉×#3160	2
#33705	59