Subgroups — #28487236

Order 15104.

382 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424814	C₂	1	2	yes	maximal
2	C₄×#14273661	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837153	C₂	1	1	yes	maximal, characteristic
4	#21421936	C₂	1	2	yes	maximal
5	C₂×C₅₉×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273661	C₄	1	1	yes	characteristic
7	C₂×#14273661	C₄	1	1	yes	characteristic
8	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×Dic₅₉⋊C₈	—	2	2	no	—
10	C₄×C₄⋊Dic₅₉	C₂²	1	2	yes	—
11	#16837165	C₄	1	2	yes	—
12	#16837156	—	2	2	no	—
13	#16837153	—	2	2	no	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#322	—	2	2	no	—
18	C₂²×C₅₉⋊C₈	—	2	2	no	—
19	C₂×C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₅₉	—	2	2	no	—
22	#14273661	—	2	2	no	—
23	#14273661	C₈	1	4	yes	complemented
24	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
26	Dic₅₉⋊C₈	—	4	4	no	—
27	C₅₉⋊#95	—	2	2	no	—
28	C₂²×C₈×C₅₉	—	2	1	no	—
29	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₉	D₄	1	2	yes	—
32	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₅₉	—	2	1	no	—
34	C₄×C₈×C₅₉	—	2	1	no	—
35	C₄×C₈×C₅₉	—	2	1	no	—
36	C₄×C₈×C₅₉	D₄	1	2	yes	—
37	C₄×C₈×C₅₉	Q₈	1	2	yes	—
38	C₄×C₈×C₅₉	Q₈	1	2	yes	—
39	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
42	C₂²×Dic₅₉	—	2	2	no	—
43	C₂²×Dic₅₉	—	2	2	no	—
44	C₂×C₈×C₅₉	—	2	1	no	—
45	C₂×C₈×C₅₉	—	2	1	no	—
46	C₂×C₈×C₅₉	—	2	2	no	—
47	C₂×C₈×C₅₉	—	2	2	no	—
48	C₂×C₈×C₅₉	—	2	2	no	—
49	C₂×C₈×C₅₉	—	2	2	no	—
50	C₂×C₈×C₅₉	—	4	1	no	—
51	C₂×C₈×C₅₉	—	4	1	no	—
52	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
56	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₅₉	#41	1	1	yes	characteristic
60	C₄⋊Dic₅₉	—	2	2	no	—
61	C₄⋊Dic₅₉	—	2	2	no	—
62	C₄⋊Dic₅₉	—	2	2	no	—
63	C₄⋊Dic₅₉	C₂×C₈	1	4	yes	—
64	C₄×Dic₅₉	—	4	2	no	—
65	C₄×Dic₅₉	—	4	2	no	—
66	C₂×C₅₉⋊C₈	—	4	2	no	—
67	C₂×C₅₉⋊C₈	—	4	2	no	—
68	C₂×C₅₉⋊C₈	—	4	2	no	—
69	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₉	—	2	1	no	—
71	C₂×C₄×C₅₉	—	2	1	no	—
72	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₅₉	#125	1	1	yes	characteristic
77	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
81	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
82	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₅₉	#137	1	2	yes	—
84	C₂×Dic₅₉	—	2	4	no	—
85	C₂×Dic₅₉	—	4	2	no	—
86	C₂×Dic₅₉	—	4	2	no	—
87	C₂×Dic₅₉	—	4	2	no	—
88	C₂×Dic₅₉	—	4	2	no	—
89	C₂×Dic₅₉	—	4	2	no	—
90	C₈×C₅₉	—	2	2	no	—
91	C₈×C₅₉	—	4	2	no	—
92	C₈×C₅₉	—	4	2	no	—
93	C₈×C₅₉	C₄⋊C₈	1	4	yes	complemented
94	C₅₉⋊C₈	—	8	2	no	—
95	#9595	—	59	1	no	maximal, Sylow
96	C₂²×C₅₉	#389	1	1	yes	characteristic
97	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
98	C₂²×C₅₉	#436	1	1	yes	characteristic
99	C₂²×C₅₉	#441	1	1	yes	characteristic
100	C₂²×C₅₉	#442	1	1	yes	characteristic
101	C₂²×C₅₉	#444	1	1	yes	characteristic
102	C₂²×C₅₉	#444	1	1	yes	characteristic
103	C₄×C₅₉	—	2	1	no	—
104	C₄×C₅₉	—	2	1	no	—
105	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₄×C₅₉	#424	1	1	yes	characteristic
107	C₄×C₅₉	C₈×Q₈	1	1	yes	characteristic
108	C₄×C₅₉	#446	1	1	yes	characteristic
109	C₄×C₅₉	#468	1	2	yes	—
110	C₄×C₅₉	#501	1	2	yes	—
111	Dic₅₉	—	4	4	no	—
112	Dic₅₉	—	8	2	no	—
113	#1939	—	59	2	no	—
114	C₄×#107	—	59	1	no	—
115	C₂×#334	—	59	1	no	—
116	C₂×#322	D₅₉	1	1	yes	characteristic
117	#1279	—	59	2	no	—
118	C₂×C₅₉	#1579	1	1	yes	characteristic
119	C₂×C₅₉	#1580	1	1	yes	characteristic
120	C₂×C₅₉	#1593	1	1	yes	characteristic
121	C₂×C₅₉	#1594	1	1	yes	characteristic
122	C₂×C₅₉	#1939	1	1	yes	characteristic
123	C₂×C₅₉	#1943	1	1	yes	characteristic
124	C₂×C₅₉	#1946	1	1	yes	characteristic
125	C₂×#107	—	59	1	no	—
126	C₂×#107	—	59	1	no	—
127	C₂×C₄⋊C₈	—	59	2	no	—
128	C₂×C₄⋊C₈	—	118	2	no	—
129	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
132	C₄×C₄⋊C₄	—	59	2	no	—
133	#340	—	59	2	no	—
134	#336	—	118	2	no	—
135	#334	—	118	2	no	—
136	#322	—	2	2	no	—
137	C₅₉	#9595	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₈	—	2	1	no	—
139	C₂²×C₈	—	118	2	no	—
140	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
141	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
142	C₂²×C₈	D₂₃₆	1	2	yes	—
143	C₂×C₄⋊C₄	—	59	2	no	—
144	C₂×C₄⋊C₄	—	59	2	no	—
145	C₂×C₄²	—	118	2	no	—
146	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
147	#107	—	59	4	no	—
148	#107	—	118	2	no	—
149	C₄⋊C₈	—	118	2	no	—
150	C₄⋊C₈	—	118	2	no	—
151	C₄⋊C₈	—	236	4	no	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	—	2	1	no	—
154	C₄×C₈	—	2	1	no	—
155	C₄×C₈	Dic₁₁₈	1	2	yes	—
156	C₄×C₈	Dic₁₁₈	1	2	yes	—
157	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
158	#95	—	118	2	no	—
159	C₂²×C₄	—	118	2	no	—
160	C₂²×C₄	—	118	2	no	—
161	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
162	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
163	C₂²×C₄	#11745685	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	4	1	no	—
172	C₂×C₈	—	236	2	no	—
173	C₂×C₈	—	236	2	no	—
174	C₂×C₈	—	236	2	no	—
175	C₂×C₈	C₈×D₅₉	1	2	yes	—
176	C₂×C₈	#11745659	1	2	yes	—
177	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
178	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
179	C₄⋊C₄	—	59	4	no	—
180	C₄⋊C₄	—	118	2	no	—
181	C₄⋊C₄	—	118	2	no	—
182	C₄⋊C₄	—	118	2	no	—
183	C₄²	—	236	2	no	—
184	C₄²	—	236	2	no	—
185	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
186	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
187	C₄²	#11745685	1	1	yes	characteristic
188	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
189	C₂³	C₄×D₂₃₆	1	1	yes	characteristic
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	118	4	no	—
193	C₂×C₄	—	236	2	no	—
194	C₂×C₄	—	236	2	no	—
195	C₂×C₄	—	236	2	no	—
196	C₂×C₄	—	236	2	no	—
197	C₂×C₄	—	236	2	no	—
198	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
199	C₂×C₄	C₄×Dic₁₁₈	1	1	yes	characteristic
200	C₂×C₄	#14273701	1	1	yes	characteristic
201	C₂×C₄	#14273706	1	1	yes	characteristic
202	C₂×C₄	#14273719	1	1	yes	characteristic
203	C₂×C₄	#14273727	1	1	yes	characteristic
204	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
205	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
206	C₂×C₄	#14273733	1	1	yes	characteristic
207	C₂×C₄	#14273733	1	1	yes	characteristic
208	C₂×C₄	#14273755	1	2	yes	—
209	C₂×C₄	#14273760	1	2	yes	—
210	C₈	—	2	2	no	—
211	C₈	—	4	2	no	—
212	C₈	—	4	2	no	—
213	C₈	—	472	2	no	—
214	C₈	Dic₅₉⋊C₈	1	4	yes	complemented
215	C₂²	#16837300	1	1	yes	characteristic
216	C₂²	C₈×D₂₃₆	1	1	yes	characteristic
217	C₂²	#16837481	1	1	yes	characteristic
218	C₂²	#16837510	1	1	yes	characteristic
219	C₂²	#16837515	1	1	yes	characteristic
220	C₂²	#16837525	1	1	yes	characteristic
221	C₂²	#16837525	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	472	2	no	—
225	C₄	—	236	4	no	—
226	C₄	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
227	C₄	#16837428	1	1	yes	characteristic
228	C₄	C₈×Dic₁₁₈	1	1	yes	characteristic
229	C₄	#16837532	1	1	yes	characteristic
230	C₄	#16837648	1	2	yes	—
231	C₄	#16837811	1	2	yes	—
232	C₂	#21422916	1	1	yes	characteristic
233	C₂	#21422921	1	1	yes	characteristic
234	C₂	#21423006	1	1	yes	characteristic
235	C₂	#21423011	1	1	yes	characteristic
236	C₂	#21424814	1	1	yes	characteristic
237	C₂	#21424834	1	1	yes	characteristic
238	C₂	#21424849	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#322	C₂
socle, prime radical	C₂³×C₅₉	C₄×D₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#322	D₅₉
hypocenter	C₅₉	#9595
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
23. #14273661	C₈	semidirect
93. C₈×C₅₉	C₄⋊C₈	semidirect
137. C₅₉	#9595	semidirect
214. C₈	Dic₅₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#21424814	2
C₄×#14273661	2
C₂×#16837153	2
#21421936	2
C₂×C₅₉×#322	2
#9595	59