Subgroups — #28479128

Order 15104.

354 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21423782	C₂	1	1	yes	maximal, characteristic
2	#21422379	C₂	1	1	yes	maximal, characteristic
3	#21421926	C₂	1	2	yes	maximal
4	#21421924	C₂	1	2	yes	maximal
5	C₅₉×#1450	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	C₂×#14273638	C₂²	1	1	yes	characteristic
8	C₂×#14273637	C₄	1	2	yes	—
9	C₂×C₈×Dic₅₉	C₄	1	2	yes	—
10	#16837248	C₂²	1	1	yes	characteristic
11	#16837123	C₄	1	2	yes	complemented
12	C₈×C₅₉⋊C₈	C₄	1	2	yes	complemented
13	C₅₉×#431	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#322	C₄	1	2	yes	complemented
16	C₈²×C₅₉	C₄	1	2	yes	complemented
17	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
19	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
20	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
21	#14273638	—	2	2	no	—
22	#14273637	—	2	4	no	—
23	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
24	C₈×Dic₅₉	C₈	1	8	yes	complemented
25	C₅₉⋊#95	—	2	2	no	—
26	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₅₉	—	2	2	no	—
30	C₂²⋊C₈×C₅₉	—	2	2	no	—
31	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	#41	1	2	yes	—
38	C₂²×Dic₅₉	—	2	2	no	—
39	C₂×C₈×C₅₉	—	2	4	no	—
40	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₅₉	M₁₆	1	4	yes	—
46	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₅₉	#41	1	2	yes	—
49	C₄×Dic₅₉	—	2	2	no	—
50	C₄×Dic₅₉	C₂×C₈	1	4	yes	—
51	C₂×C₅₉⋊C₈	—	2	2	no	—
52	C₂×C₅₉⋊C₈	—	2	2	no	—
53	C₂×C₅₉⋊C₈	—	2	4	no	—
54	C₂×C₅₉⋊C₈	C₂×C₈	1	4	yes	—
55	C₂³×C₅₉	#117	1	1	yes	characteristic
56	C₂×C₄×C₅₉	—	4	2	no	—
57	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₅₉	#117	1	2	yes	—
59	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
60	C₂×C₄×C₅₉	#117	1	2	yes	—
61	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
62	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
63	C₂×Dic₅₉	—	2	4	no	—
64	C₂×Dic₅₉	—	4	4	no	—
65	C₈×C₅₉	—	4	4	no	—
66	C₈×C₅₉	C₄×C₈	1	2	yes	—
67	C₈×C₅₉	C₄×C₈	1	2	yes	—
68	C₈×C₅₉	#97	1	2	yes	—
69	C₈×C₅₉	#97	1	2	yes	—
70	C₅₉⋊C₈	—	2	4	no	—
71	C₅₉⋊C₈	—	4	4	no	—
72	#8137	—	59	1	no	maximal, Sylow
73	C₂²×C₅₉	—	2	2	no	—
74	C₂²×C₅₉	#376	1	1	yes	characteristic
75	C₂²×C₅₉	#431	1	2	yes	—
76	C₄×C₅₉	—	2	2	no	—
77	C₄×C₅₉	—	2	2	no	—
78	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
79	C₄×C₅₉	C₂×#97	1	1	yes	characteristic
80	C₄×C₅₉	#431	1	2	yes	—
81	Dic₅₉	—	4	4	no	—
82	#1726	—	59	1	no	—
83	#1450	—	59	1	no	—
84	#1450	D₅₉	1	1	yes	characteristic
85	#1275	—	59	2	no	—
86	#1274	—	59	2	no	—
87	C₂×C₅₉	—	2	2	no	—
88	C₂×C₅₉	#1744	1	1	yes	characteristic
89	C₂×C₅₉	#1747	1	2	yes	—
90	#431	—	59	2	no	—
91	#431	C₂×D₅₉	1	2	yes	—
92	C₂×#97	—	59	2	no	—
93	C₂×#97	—	59	1	no	—
94	C₂×C₄×C₈	—	59	2	no	—
95	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
96	#376	—	59	1	no	—
97	#322	—	59	2	no	—
98	#322	Dic₅₉	1	2	yes	complemented
99	C₈²	—	59	2	no	—
100	C₈²	Dic₅₉	1	2	yes	complemented
101	C₅₉	#8137	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₈	—	59	2	no	—
103	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
104	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
105	C₂×C₄²	—	59	2	no	—
106	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	118	2	no	—
109	C₂²⋊C₈	—	2	2	no	—
110	C₂²⋊C₈	—	118	2	no	—
111	#97	—	118	4	no	—
112	#97	—	118	2	no	—
113	C₄×C₈	—	59	8	no	—
114	C₄×C₈	—	59	4	no	—
115	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
116	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
117	C₄×C₈	C₂×Dic₅₉	1	4	yes	—
118	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
119	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
120	#95	—	118	2	no	—
121	C₂²×C₄	—	118	2	no	—
122	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
123	C₂²×C₄	#11745685	1	2	yes	—
124	C₂×C₈	—	2	4	no	—
125	C₂×C₈	—	59	4	no	—
126	C₂×C₈	—	118	2	no	—
127	C₂×C₈	—	118	2	no	—
128	C₂×C₈	—	118	4	no	—
129	C₂×C₈	C₈×D₅₉	1	4	yes	—
130	C₂×C₈	#11745659	1	4	yes	—
131	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
132	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
133	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
134	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
135	C₄²	—	59	4	no	—
136	C₄²	—	118	2	no	—
137	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
138	C₄²	#11745685	1	2	yes	—
139	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
140	C₂³	#14273684	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	118	4	no	—
143	C₂×C₄	—	236	4	no	—
144	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
145	C₂×C₄	#14273684	1	2	yes	—
146	C₂×C₄	#14273684	1	1	yes	characteristic
147	C₂×C₄	#14273686	1	2	yes	—
148	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
149	C₂×C₄	C₂×#11745659	1	2	yes	—
150	C₈	—	4	4	no	—
151	C₈	—	118	4	no	—
152	C₈	—	236	4	no	—
153	C₈	C₈×Dic₅₉	1	2	yes	—
154	C₈	C₈×Dic₅₉	1	2	yes	—
155	C₈	#14273637	1	2	yes	—
156	C₈	#14273637	1	2	yes	—
157	C₂²	—	2	2	no	—
158	C₂²	#16837248	1	1	yes	characteristic
159	C₂²	#16837453	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	236	4	no	—
163	C₄	C₂×C₈×Dic₅₉	1	1	yes	characteristic
164	C₄	C₂×#14273637	1	1	yes	characteristic
165	C₄	#16837453	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#21423841	1	1	yes	characteristic
168	C₂	#21423851	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₅₉
commutator	C₄×C₅₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1450	C₂
socle	C₂²×C₅₉	#376
prime radical	C₂³×C₅₉	#117
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1450	D₅₉
hypocenter	C₅₉	#8137
LCS	G ▷ C₄×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
11. #16837123	C₄	semidirect
12. C₈×C₅₉⋊C₈	C₄	semidirect
15. C₅₉×#322	C₄	semidirect
16. C₈²×C₅₉	C₄	semidirect
24. C₈×Dic₅₉	C₈	semidirect
98. #322	Dic₅₉	semidirect
100. C₈²	Dic₅₉	semidirect
101. C₅₉	#8137	semidirect

Maximal subgroups

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

subgroup	index
#21423782	2
#21422379	2
#21421926	2
#21421924	2
C₅₉×#1450	2
#8137	59