Subgroups — #28481918

Order 15104.

350 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424227	C₂	1	1	yes	maximal, characteristic
2	#21422844	C₂	1	1	yes	maximal, characteristic
3	#21422660	C₂	1	2	yes	maximal
4	#21421956	C₂	1	2	yes	maximal
5	C₅₉×#2960	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	#16837268	C₂²	1	2	yes	—
8	#16837165	—	2	1	no	—
9	#16837165	—	2	1	no	—
10	#16837155	—	2	1	no	—
11	#16837153	—	2	1	no	—
12	#16837139	—	2	4	no	—
13	C₅₉×#533	C₂²	1	1	yes	characteristic
14	C₅₉×#446	—	2	1	no	—
15	C₅₉×#446	C₄	1	2	yes	complemented
16	C₈×Q₈×C₅₉	—	2	1	no	—
17	C₈×Q₈×C₅₉	C₄	1	2	yes	complemented
18	C₅₉×#424	—	2	1	no	—
19	C₅₉×#423	C₂²	1	1	yes	characteristic
20	C₅₉×#405	—	2	1	no	—
21	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₅₉	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
25	C₄×C₅₉⋊C₈	—	2	2	no	—
26	C₅₉⋊#95	—	2	2	no	—
27	C₅₉⋊#95	—	2	2	no	—
28	C₂×M₁₆×C₅₉	—	2	1	no	—
29	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₅₉	Q₈	1	1	yes	characteristic
32	C₄⋊Q₈×C₅₉	Q₈	1	1	yes	characteristic
33	#125×C₅₉	—	2	1	no	—
34	C₂²⋊Q₈×C₅₉	—	2	1	no	—
35	C₂²⋊Q₈×C₅₉	D₄	1	2	yes	—
36	C₄×Q₈×C₅₉	—	2	1	no	—
37	C₄×Q₈×C₅₉	C₂×C₄	1	2	yes	—
38	#117×C₅₉	—	2	1	no	—
39	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₅₉	—	2	2	no	—
41	C₄⋊C₈×C₅₉	—	2	2	no	—
42	C₄⋊C₈×C₅₉	—	2	1	no	—
43	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
44	#97×C₅₉	—	2	2	no	—
45	C₄×C₈×C₅₉	—	2	2	no	—
46	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
50	C₂×Q₈×C₅₉	—	2	1	no	—
51	C₂×Q₈×C₅₉	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
55	C₂²×Dic₅₉	—	4	2	no	—
56	M₁₆×C₅₉	—	4	2	no	—
57	C₂×C₈×C₅₉	—	2	2	no	—
58	C₂×C₈×C₅₉	—	2	2	no	—
59	C₂×C₈×C₅₉	—	2	2	no	—
60	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₄⋊C₄×C₅₉	—	2	2	no	—
65	C₄⋊C₄×C₅₉	—	2	2	no	—
66	C₄⋊C₄×C₅₉	—	2	1	no	—
67	C₄⋊C₄×C₅₉	—	2	1	no	—
68	C₄⋊C₄×C₅₉	—	2	1	no	—
69	C₄⋊C₄×C₅₉	C₄⋊C₄	1	2	yes	—
70	C₂²⋊C₄×C₅₉	—	2	2	no	—
71	C₄²×C₅₉	—	2	2	no	—
72	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
75	C₄²×C₅₉	#41	1	1	yes	characteristic
76	C₄⋊Dic₅₉	—	4	2	no	—
77	C₄⋊Dic₅₉	—	4	2	no	—
78	C₂×C₅₉⋊C₈	—	4	2	no	—
79	C₂³×C₅₉	#121	1	1	yes	characteristic
80	Q₈×C₅₉	—	4	2	no	—
81	Q₈×C₅₉	—	4	2	no	—
82	C₂×C₄×C₅₉	—	2	2	no	—
83	C₂×C₄×C₅₉	—	2	2	no	—
84	C₂×C₄×C₅₉	—	4	1	no	—
85	C₂×C₄×C₅₉	—	4	1	no	—
86	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
89	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
90	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₅₉	#135	1	2	yes	—
93	C₂×C₄×C₅₉	#137	1	2	yes	—
94	C₂×Dic₅₉	—	8	2	no	—
95	C₂×Dic₅₉	—	8	2	no	—
96	C₂×Dic₅₉	—	8	2	no	—
97	C₈×C₅₉	—	2	1	no	—
98	C₈×C₅₉	—	2	1	no	—
99	C₈×C₅₉	—	2	1	no	—
100	C₈×C₅₉	—	2	1	no	—
101	C₈×C₅₉	—	4	2	no	—
102	C₅₉⋊C₈	—	8	2	no	—
103	#8679	—	59	1	no	maximal, Sylow
104	C₂²×C₅₉	—	2	1	no	—
105	C₂²×C₅₉	—	2	1	no	—
106	C₂²×C₅₉	#380	1	1	yes	characteristic
107	C₂²×C₅₉	#464	1	2	yes	—
108	C₄×C₅₉	—	2	1	no	—
109	C₄×C₅₉	—	2	1	no	—
110	C₄×C₅₉	—	2	1	no	—
111	C₄×C₅₉	—	2	1	no	—
112	C₄×C₅₉	—	4	2	no	—
113	C₄×C₅₉	—	4	2	no	—
114	C₄×C₅₉	#416	1	1	yes	characteristic
115	C₄×C₅₉	#419	1	1	yes	characteristic
116	C₄×C₅₉	#479	1	2	yes	—
117	Dic₅₉	—	16	2	no	—
118	#2960	D₅₉	1	1	yes	characteristic
119	#1835	—	59	1	no	—
120	#1564	—	59	1	no	—
121	#1518	—	59	2	no	—
122	#1286	—	59	2	no	—
123	C₂×C₅₉	—	2	2	no	—
124	C₂×C₅₉	#1805	1	1	yes	characteristic
125	C₂×C₅₉	#1812	1	1	yes	characteristic
126	C₂×C₅₉	#1814	1	1	yes	characteristic
127	#533	C₂×D₅₉	1	1	yes	characteristic
128	#446	—	2	1	no	—
129	#446	Dic₅₉	1	2	yes	complemented
130	C₈×Q₈	—	2	1	no	—
131	C₈×Q₈	Dic₅₉	1	2	yes	complemented
132	#431	—	59	2	no	—
133	#424	—	2	1	no	—
134	#423	C₂×D₅₉	1	1	yes	characteristic
135	#405	—	2	1	no	—
136	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
137	#382	—	59	2	no	—
138	#340	—	118	1	no	—
139	#340	—	118	1	no	—
140	#335	—	118	1	no	—
141	#334	—	118	1	no	—
142	#328	—	118	4	no	—
143	C₅₉	#8679	1	1	yes	characteristic, complemented, Sylow
144	C₂×M₁₆	—	2	1	no	—
145	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
147	C₄⋊Q₈	Dic₁₁₈	1	1	yes	characteristic
148	C₄⋊Q₈	Dic₁₁₈	1	1	yes	characteristic
149	#125	—	2	1	no	—
150	C₂²⋊Q₈	—	2	1	no	—
151	C₂²⋊Q₈	D₂₃₆	1	2	yes	—
152	C₄×Q₈	—	2	1	no	—
153	C₄×Q₈	C₂×Dic₅₉	1	2	yes	—
154	#117	—	2	1	no	—
155	C₂×C₄⋊C₄	—	118	2	no	—
156	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	1	no	—
160	C₄⋊C₈	—	118	2	no	—
161	C₄⋊C₈	C₂×Dic₅₉	1	2	yes	—
162	C₂²⋊C₈	—	118	2	no	—
163	#97	—	2	2	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	118	2	no	—
166	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
167	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
170	#95	—	118	2	no	—
171	#95	—	118	2	no	—
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	C₄⋊Dic₅₉	1	2	yes	—
174	C₂²×C₄	—	236	2	no	—
175	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
176	C₂²×C₄	#11745685	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
178	M₁₆	—	4	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	236	2	no	—
183	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
184	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
185	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
186	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	236	2	no	—
193	C₄⋊C₄	—	236	2	no	—
194	C₄⋊C₄	C₄⋊Dic₅₉	1	2	yes	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
198	C₄²	#11745685	1	1	yes	characteristic
199	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
200	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
201	C₂³	#14273701	1	1	yes	characteristic
202	Q₈	—	4	2	no	—
203	Q₈	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	472	2	no	—
209	C₂×C₄	—	472	2	no	—
210	C₂×C₄	—	472	2	no	—
211	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
213	C₂×C₄	#14273686	1	1	yes	characteristic
214	C₂×C₄	#14273701	1	1	yes	characteristic
215	C₂×C₄	#14273706	1	1	yes	characteristic
216	C₂×C₄	#14273706	1	1	yes	characteristic
217	C₂×C₄	#14273747	1	2	yes	—
218	C₂×C₄	#14273758	1	2	yes	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	4	2	no	—
224	C₈	—	472	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	#16837262	1	1	yes	characteristic
228	C₂²	#16837624	1	2	yes	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	944	2	no	—
236	C₄	#16837398	1	1	yes	characteristic
237	C₄	#16837411	1	1	yes	characteristic
238	C₄	#16837709	1	2	yes	—
239	C₂	—	2	2	no	—
240	C₂	#21424111	1	1	yes	characteristic
241	C₂	#21424142	1	1	yes	characteristic
242	C₂	#21424150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837262
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#2960	C₂
socle	C₂²×C₅₉	#380
prime radical	C₂³×C₅₉	#121
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#2960	D₅₉
hypocenter	C₅₉	#8679
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
15. C₅₉×#446	C₄	semidirect
17. C₈×Q₈×C₅₉	C₄	semidirect
129. #446	Dic₅₉	semidirect
131. C₈×Q₈	Dic₅₉	semidirect
143. C₅₉	#8679	semidirect

Maximal subgroups

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

subgroup	index
#21424227	2
#21422844	2
#21422660	2
#21421956	2
C₅₉×#2960	2
#8679	59