Subgroups — #28478050

Order 15104.

558 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21423841	C₂	1	1	yes	maximal, characteristic
2	C₄×#14273637	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#16837124	C₂	1	1	yes	maximal, characteristic
5	C₂×#16837124	C₂	1	1	yes	maximal, characteristic
6	C₂×#16837124	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅₉×#322	C₂	1	1	yes	maximal, characteristic
8	C₂×#14273637	C₄	1	4	yes	—
9	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₅₉	C₄	1	4	yes	—
13	C₄²×Dic₅₉	C₂²	1	1	yes	characteristic
14	#16837156	C₄	1	4	yes	—
15	#16837124	C₄	1	4	yes	complemented
16	#16837124	C₄	1	4	yes	complemented
17	#16837124	C₄	1	4	yes	complemented
18	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₅₉×#322	C₄	1	4	yes	complemented
22	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
26	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
27	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
28	#14273637	—	2	8	no	—
29	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
30	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
31	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
32	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
33	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
34	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
35	C₈×Dic₅₉	—	2	8	no	—
36	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₅₉	C₄²	1	4	yes	—
50	C₂×C₈×C₅₉	—	2	4	no	—
51	C₂×C₈×C₅₉	—	2	4	no	—
52	C₂×C₈×C₅₉	—	2	4	no	—
53	C₂×C₈×C₅₉	C₄²	1	4	yes	—
54	C₂×C₈×C₅₉	C₄²	1	4	yes	—
55	C₂×C₈×C₅₉	C₄²	1	4	yes	—
56	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
60	C₄×Dic₅₉	—	2	4	no	—
61	C₄×Dic₅₉	—	2	4	no	—
62	C₄×Dic₅₉	M₁₆	1	8	yes	—
63	C₂×C₅₉⋊C₈	—	2	4	no	—
64	C₂×C₅₉⋊C₈	—	2	4	no	—
65	C₂×C₅₉⋊C₈	C₄²	1	4	yes	—
66	C₂×C₅₉⋊C₈	C₄²	1	4	yes	—
67	C₂×C₅₉⋊C₈	C₄²	1	4	yes	—
68	C₂×C₅₉⋊C₈	M₁₆	1	8	yes	—
69	C₂³×C₅₉	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂×M₁₆	1	4	yes	—
77	C₂×C₄×C₅₉	C₂×M₁₆	1	4	yes	—
78	C₂×C₄×C₅₉	#131	1	4	yes	—
79	C₂×Dic₅₉	—	2	4	no	—
80	C₂×Dic₅₉	—	2	4	no	—
81	C₂×Dic₅₉	#97	1	8	yes	—
82	C₈×C₅₉	—	2	4	no	—
83	C₈×C₅₉	—	2	4	no	—
84	C₈×C₅₉	—	2	4	no	—
85	C₅₉⋊C₈	—	2	4	no	—
86	C₅₉⋊C₈	—	2	4	no	—
87	C₅₉⋊C₈	#97	1	8	yes	complemented
88	#7842	—	59	1	no	maximal, Sylow
89	C₂²×C₅₉	C₄³	1	1	yes	characteristic
90	C₂²×C₅₉	C₂×#97	1	2	yes	—
91	C₂²×C₅₉	C₄×M₁₆	1	2	yes	—
92	C₂²×C₅₉	#405	1	2	yes	—
93	C₄×C₅₉	C₂×#97	1	2	yes	—
94	C₄×C₅₉	C₄×M₁₆	1	2	yes	—
95	C₄×C₅₉	C₄×M₁₆	1	2	yes	—
96	C₄×C₅₉	C₄×M₁₆	1	2	yes	—
97	C₄×C₅₉	#405	1	2	yes	—
98	C₄×C₅₉	#405	1	2	yes	—
99	Dic₅₉	—	2	8	no	—
100	#1744	—	59	1	no	—
101	C₄×#97	—	59	1	no	—
102	C₄²×C₈	—	59	1	no	—
103	C₂×#322	—	59	1	no	—
104	C₂×#322	—	59	1	no	—
105	C₂×#322	—	59	1	no	—
106	C₂×#322	D₅₉	1	1	yes	characteristic
107	C₂×C₅₉	#1451	1	4	yes	—
108	C₂×C₅₉	C₄×#97	1	1	yes	characteristic
109	C₂×C₅₉	C₄×#97	1	1	yes	characteristic
110	C₂×C₅₉	#1726	1	1	yes	characteristic
111	C₂×#97	—	59	4	no	—
112	C₂×C₄×C₈	—	59	1	no	—
113	C₂×C₄×C₈	—	59	1	no	—
114	C₂×C₄×C₈	—	59	1	no	—
115	C₂×C₄×C₈	—	59	4	no	—
116	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
119	C₄³	—	59	1	no	—
120	#336	—	59	4	no	—
121	#322	—	59	4	no	—
122	#322	—	59	4	no	—
123	#322	—	59	4	no	—
124	#322	Dic₅₉	1	4	yes	complemented
125	C₅₉	#7842	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	59	2	no	—
127	C₂²×C₈	—	59	2	no	—
128	C₂²×C₈	—	59	2	no	—
129	C₂²×C₈	C₄×D₅₉	1	2	yes	—
130	C₂²×C₈	C₄×D₅₉	1	2	yes	—
131	C₂²×C₈	C₄×D₅₉	1	2	yes	—
132	C₂×C₄²	—	59	2	no	—
133	C₂×C₄²	—	59	2	no	—
134	C₂×C₄²	—	59	2	no	—
135	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
136	#97	—	118	8	no	—
137	C₄×C₈	—	59	2	no	—
138	C₄×C₈	—	59	2	no	—
139	C₄×C₈	—	59	2	no	—
140	C₄×C₈	—	59	2	no	—
141	C₄×C₈	—	59	2	no	—
142	C₄×C₈	—	59	2	no	—
143	C₄×C₈	—	118	8	no	—
144	C₄×C₈	C₄×D₅₉	1	2	yes	—
145	C₄×C₈	C₄×D₅₉	1	2	yes	—
146	C₄×C₈	C₄×D₅₉	1	2	yes	—
147	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
148	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
149	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
150	C₂²×C₄	—	59	4	no	—
151	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	59	4	no	—
158	C₂×C₈	—	59	8	no	—
159	C₂×C₈	—	59	4	no	—
160	C₂×C₈	—	59	4	no	—
161	C₂×C₈	—	118	4	no	—
162	C₂×C₈	—	118	4	no	—
163	C₂×C₈	C₄×Dic₅₉	1	4	yes	—
164	C₂×C₈	C₄×Dic₅₉	1	4	yes	—
165	C₂×C₈	C₄×Dic₅₉	1	4	yes	—
166	C₄²	—	59	8	no	—
167	C₄²	—	118	4	no	—
168	C₄²	—	118	4	no	—
169	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
170	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
171	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
172	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
173	C₂³	C₄²×D₅₉	1	1	yes	characteristic
174	C₂×C₄	—	59	8	no	—
175	C₂×C₄	—	118	4	no	—
176	C₂×C₄	—	118	4	no	—
177	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
178	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
179	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
183	C₂×C₄	M₁₆×D₅₉	1	4	yes	—
184	C₂×C₄	M₁₆×D₅₉	1	4	yes	—
185	C₂×C₄	#14273733	1	4	yes	—
186	C₈	—	2	4	no	—
187	C₈	—	2	4	no	—
188	C₈	—	2	4	no	—
189	C₈	—	59	8	no	—
190	C₈	—	118	4	no	—
191	C₈	—	118	4	no	—
192	C₂²	C₄²×Dic₅₉	1	1	yes	characteristic
193	C₂²	#97×D₅₉	1	2	yes	—
194	C₂²	#16837354	1	2	yes	—
195	C₂²	#16837356	1	2	yes	—
196	C₄	—	118	8	no	—
197	C₄	#97×D₅₉	1	2	yes	—
198	C₄	M₁₆×Dic₅₉	1	2	yes	—
199	C₄	M₁₆×Dic₅₉	1	2	yes	—
200	C₄	#16837354	1	2	yes	—
201	C₄	#16837356	1	2	yes	—
202	C₄	#16837357	1	2	yes	—
203	C₂	#21422384	1	4	yes	—
204	C₂	#97×Dic₅₉	1	1	yes	characteristic
205	C₂	#97×Dic₅₉	1	1	yes	characteristic
206	C₂	#21423782	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₉
commutator	C₂²×C₅₉	C₄³
Fitting	C₂×C₅₉×#322	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₄²
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#322	D₅₉
hypocenter	C₅₉	#7842
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
15. #16837124	C₄	semidirect
16. #16837124	C₄	semidirect
17. #16837124	C₄	semidirect
21. C₅₉×#322	C₄	semidirect
87. C₅₉⋊C₈	#97	semidirect
124. #322	Dic₅₉	semidirect
125. C₅₉	#7842	semidirect

Maximal subgroups

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

subgroup	index
#21423841	2
C₄×#14273637	2
C₄×C₈×Dic₅₉	2
C₂×#16837124	2
C₂×C₅₉×#322	2
#7842	59