Subgroups — #28744220

Order 15104.

482 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837804	C₂	1	1	yes	maximal, characteristic
2	#21426379	C₂	1	4	yes	maximal, complemented
3	#21426372	C₂	1	4	yes	maximal, complemented
4	#21425885	C₂	1	4	yes	maximal
5	C₄×#14273667	C₂	1	1	yes	maximal, characteristic
6	C₅₉×#3160	C₂	1	1	yes	maximal, characteristic
7	C₂×#14273719	—	2	1	no	—
8	C₂×#14273667	C₂²	1	2	yes	—
9	#16837804	—	2	1	no	—
10	#16837804	—	2	1	no	—
11	#16837804	C₂²	1	2	yes	—
12	#16837804	C₂²	1	2	yes	—
13	C₂×#14273661	C₂²	1	2	yes	—
14	C₂×#14273659	—	2	1	no	—
15	#16837229	C₂²	1	8	yes	—
16	#16837227	C₂²	1	8	yes	—
17	#16837178	C₂²	1	2	yes	—
18	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	—
19	C₅₉×#533	—	2	1	no	—
20	C₅₉×#500	C₂²	1	2	yes	—
21	C₅₉×#499	—	2	1	no	—
22	C₅₉×#499	C₂²	1	2	yes	—
23	C₅₉×#498	—	2	1	no	—
24	C₅₉×#427	—	2	1	no	—
25	C₅₉×#427	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
27	#14273719	—	2	2	no	—
28	#14273719	—	2	2	no	—
29	#14273719	—	4	2	no	—
30	C₂×C₄⋊Dic₅₉	—	2	2	no	—
31	C₂×Dic₅₉⋊C₄	—	4	2	no	—
32	#14273667	Q₈	1	8	yes	complemented
33	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
34	#14273661	—	2	2	no	—
35	#14273661	C₂³	1	4	yes	—
36	#14273659	—	2	2	no	—
37	#14273659	—	2	2	no	—
38	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₅₉	—	2	2	no	—
41	#125×C₅₉	—	2	2	no	—
42	C₂²⋊Q₈×C₅₉	—	4	2	no	—
43	C₄×Q₈×C₅₉	—	4	2	no	—
44	#117×C₅₉	—	2	2	no	—
45	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
46	#107×C₅₉	—	2	2	no	—
47	#107×C₅₉	C₂³	1	4	yes	—
48	#106×C₅₉	—	2	2	no	—
49	#106×C₅₉	—	2	2	no	—
50	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
54	C₂×Q₈×C₅₉	—	4	2	no	—
55	C₂²×C₄×C₅₉	D₈	1	2	yes	—
56	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₅₉	—	4	2	no	—
58	C₂×C₈×C₅₉	Q₁₆	1	4	yes	—
59	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
63	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₄×C₅₉	—	2	4	no	—
65	C₄⋊C₄×C₅₉	—	4	4	no	—
66	C₄⋊C₄×C₅₉	—	4	2	no	—
67	C₂²⋊C₄×C₅₉	—	4	2	no	—
68	C₄²×C₅₉	—	4	2	no	—
69	C₄²×C₅₉	D₈	1	2	yes	—
70	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₄⋊Dic₅₉	—	2	4	no	—
73	C₄⋊Dic₅₉	—	4	2	no	—
74	Dic₅₉⋊C₄	—	4	4	no	—
75	Dic₅₉⋊C₄	—	4	4	no	—
76	C₅₉⋊C₁₆	C₂×Q₈	1	8	yes	—
77	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
78	Q₈×C₅₉	—	8	4	no	—
79	C₂×C₄×C₅₉	—	4	2	no	—
80	C₂×C₄×C₅₉	—	4	4	no	—
81	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
82	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
86	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
87	C₂×Dic₅₉	—	4	4	no	—
88	C₂×Dic₅₉	—	8	4	no	—
89	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
90	C₈×C₅₉	C₂×Q₁₆	1	2	yes	—
91	C₈×C₅₉	C₂×Q₁₆	1	2	yes	—
92	C₈×C₅₉	C₂²×Q₈	1	2	yes	—
93	#33732	—	59	1	no	maximal, Sylow
94	C₂²×C₅₉	—	2	2	no	—
95	C₂²×C₅₉	#500	1	2	yes	—
96	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₅₉	—	2	2	no	—
98	C₄×C₅₉	—	2	2	no	—
99	C₄×C₅₉	—	8	4	no	—
100	C₄×C₅₉	#500	1	2	yes	—
101	C₄×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
103	Dic₅₉	—	8	4	no	—
104	#3160	D₅₉	1	1	yes	characteristic
105	C₂×#499	—	59	1	no	—
106	#2255	—	59	4	no	—
107	#2253	—	59	4	no	—
108	#2159	—	59	4	no	—
109	C₄×M₃₂	—	59	1	no	—
110	C₂×C₅₉	—	2	2	no	—
111	C₂×C₅₉	C₂×#500	1	1	yes	characteristic
112	C₂×C₅₉	#3414	1	2	yes	—
113	#533	—	2	1	no	—
114	C₂×#125	—	118	1	no	—
115	C₂×M₃₂	—	59	2	no	—
116	#500	C₂×D₅₉	1	2	yes	—
117	#499	—	2	1	no	—
118	#499	—	59	2	no	—
119	#499	—	59	2	no	—
120	#499	—	118	1	no	—
121	#499	—	118	1	no	—
122	#499	C₂×D₅₉	1	2	yes	—
123	#498	—	2	1	no	—
124	#427	—	2	1	no	—
125	#427	C₂×D₅₉	1	2	yes	—
126	C₂×#107	—	59	2	no	—
127	C₂×#106	—	118	1	no	—
128	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
129	#367	—	59	8	no	—
130	#366	—	59	8	no	—
131	#346	—	59	2	no	—
132	C₄×C₁₆	—	59	2	no	—
133	C₅₉	#33732	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
136	C₄⋊Q₈	—	2	2	no	—
137	#125	—	2	2	no	—
138	#125	—	118	2	no	—
139	#125	—	118	2	no	—
140	#125	—	236	2	no	—
141	C₂²⋊Q₈	—	4	2	no	—
142	C₄×Q₈	—	4	2	no	—
143	#117	—	2	2	no	—
144	C₂×C₄⋊C₄	—	118	2	no	—
145	C₂×C₄⋊C₄	—	236	2	no	—
146	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
147	M₃₂	—	59	8	no	—
148	C₂×C₁₆	—	59	4	no	—
149	#107	—	2	2	no	—
150	#107	—	59	4	no	—
151	#107	—	118	2	no	—
152	#107	C₂²×D₅₉	1	4	yes	—
153	#106	—	2	2	no	—
154	#106	—	2	2	no	—
155	#106	—	118	2	no	—
156	#106	—	118	2	no	—
157	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
159	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
161	C₂×Q₈	—	4	2	no	—
162	C₂²×C₄	—	236	2	no	—
163	C₂²×C₄	#11745669	1	2	yes	—
164	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
165	C₂×C₈	#11745672	1	4	yes	—
166	C₂×C₈	Q₈×D₅₉	1	4	yes	—
167	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
168	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
169	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
170	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	118	4	no	—
175	C₄⋊C₄	—	236	2	no	—
176	C₄⋊C₄	—	236	4	no	—
177	C₄⋊C₄	—	236	4	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₄²	—	4	2	no	—
180	C₄²	#11745669	1	2	yes	—
181	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
182	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
183	C₁₆	—	59	8	no	—
184	C₂³	C₂×#11745669	1	1	yes	characteristic
185	Q₈	—	8	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	472	4	no	—
189	C₂×C₄	—	236	4	no	—
190	C₂×C₄	#14273725	1	2	yes	—
191	C₂×C₄	C₂×#11745669	1	2	yes	—
192	C₂×C₄	C₂×#11745669	1	2	yes	—
193	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
194	C₂×C₄	C₂×#11745672	1	2	yes	—
195	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
196	C₈	#14273725	1	2	yes	—
197	C₈	C₂×#11745672	1	2	yes	—
198	C₈	C₂×#11745672	1	2	yes	—
199	C₈	C₂×Q₈×D₅₉	1	2	yes	—
200	C₂²	—	2	2	no	—
201	C₂²	#16837805	1	2	yes	—
202	C₂²	C₂²×#11745669	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	8	4	no	—
206	C₄	—	472	4	no	—
207	C₄	#16837805	1	2	yes	—
208	C₄	C₂×#14273725	1	1	yes	characteristic
209	C₄	C₂²×#11745672	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	C₂×#16837805	1	1	yes	characteristic
212	C₂	#21437230	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	#33732
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. #21426379	C₂	semidirect
3. #21426372	C₂	semidirect
32. #14273667	Q₈	semidirect
133. C₅₉	#33732	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837804	2
#21426379	2
#21426372	2
#21425885	2
C₄×#14273667	2
C₅₉×#3160	2
#33732	59