Subgroups — #28478055

Order 15104.

526 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21423842	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837124	C₂	1	2	yes	maximal
4	C₂×#16837123	C₂	1	2	yes	maximal
5	C₅₉×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₅₉⋊C₈	C₄	1	4	yes	—
7	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	#16837157	C₄	1	4	yes	—
11	#16837124	C₄	1	8	yes	complemented
12	#16837123	C₄	1	8	yes	complemented
13	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
14	C₄³×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#336	C₄	1	4	yes	—
16	C₂²×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
17	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
18	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
19	C₄×C₅₉⋊C₈	—	2	8	no	—
20	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
21	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
22	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
23	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₅₉	C₄²	1	4	yes	—
32	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₉	—	2	8	no	—
36	C₂×C₈×C₅₉	C₄²	1	8	yes	—
37	C₄²×C₅₉	—	2	4	no	—
38	C₄²×C₅₉	—	2	4	no	—
39	C₄²×C₅₉	M₁₆	1	8	yes	—
40	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
41	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
42	C₂×C₅₉⋊C₈	—	2	4	no	—
43	C₂×C₅₉⋊C₈	—	2	4	no	—
44	C₂×C₅₉⋊C₈	—	2	8	no	—
45	C₂×C₅₉⋊C₈	C₄²	1	4	yes	—
46	C₂×C₅₉⋊C₈	C₄²	1	4	yes	—
47	C₂×C₅₉⋊C₈	C₄²	1	8	yes	—
48	C₂³×C₅₉	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₅₉	—	2	4	no	—
50	C₂×C₄×C₅₉	—	2	4	no	—
51	C₂×C₄×C₅₉	#97	1	8	yes	—
52	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
55	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
56	C₂×C₄×C₅₉	C₂×M₁₆	1	4	yes	—
57	C₂×C₄×C₅₉	#131	1	4	yes	—
58	C₂×C₄×C₅₉	#131	1	4	yes	—
59	C₈×C₅₉	—	2	8	no	—
60	C₅₉⋊C₈	—	2	4	no	—
61	C₅₉⋊C₈	—	2	4	no	—
62	C₅₉⋊C₈	—	2	8	no	—
63	#7843	—	59	1	no	maximal, Sylow
64	C₂²×C₅₉	C₄³	1	1	yes	characteristic
65	C₂²×C₅₉	C₂×#97	1	2	yes	—
66	C₂²×C₅₉	#405	1	2	yes	—
67	C₂²×C₅₉	#405	1	2	yes	—
68	C₄×C₅₉	—	2	8	no	—
69	C₄×C₅₉	C₄×M₁₆	1	2	yes	—
70	C₄×C₅₉	C₄×M₁₆	1	2	yes	—
71	C₄×C₅₉	#405	1	4	yes	—
72	C₄×C₅₉	#405	1	4	yes	—
73	#1744	—	59	1	no	—
74	#1744	D₅₉	1	1	yes	characteristic
75	C₄²×C₈	—	59	1	no	—
76	C₂×#322	—	59	2	no	—
77	C₂×#322	—	59	2	no	—
78	C₂×C₅₉	#1452	1	4	yes	—
79	C₂×C₅₉	C₄×#97	1	1	yes	characteristic
80	C₂×C₅₉	#1726	1	1	yes	characteristic
81	C₂×C₅₉	#1726	1	1	yes	characteristic
82	C₂×C₄×C₈	—	59	1	no	—
83	C₂×C₄×C₈	—	59	1	no	—
84	C₂×C₄×C₈	—	59	2	no	—
85	C₂×C₄×C₈	—	59	4	no	—
86	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
87	C₄³	C₂×D₅₉	1	1	yes	characteristic
88	#336	—	59	4	no	—
89	#336	Dic₅₉	1	4	yes	—
90	#322	—	59	8	no	—
91	#322	—	59	8	no	—
92	C₅₉	#7843	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₈	—	59	4	no	—
94	C₂²×C₈	—	59	2	no	—
95	C₂²×C₈	—	59	2	no	—
96	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
97	C₂×C₄²	C₄×D₅₉	1	2	yes	—
98	C₂×C₄²	C₄×D₅₉	1	2	yes	—
99	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
100	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
101	C₄×C₈	—	59	4	no	—
102	C₄×C₈	—	59	4	no	—
103	C₄×C₈	—	59	4	no	—
104	C₄×C₈	—	59	4	no	—
105	C₄×C₈	—	118	8	no	—
106	C₄×C₈	C₄×D₅₉	1	4	yes	—
107	C₄×C₈	C₄×D₅₉	1	4	yes	—
108	C₂²×C₄	C₄×Dic₅₉	1	4	yes	—
109	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
110	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
111	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
112	C₂×C₈	—	2	8	no	—
113	C₂×C₈	—	59	4	no	—
114	C₂×C₈	—	59	4	no	—
115	C₂×C₈	—	59	8	no	—
116	C₂×C₈	—	118	4	no	—
117	C₂×C₈	—	118	4	no	—
118	C₂×C₈	—	118	8	no	—
119	C₂×C₈	C₄×Dic₅₉	1	8	yes	—
120	C₄²	—	2	4	no	—
121	C₄²	—	2	4	no	—
122	C₄²	#11745659	1	8	yes	—
123	C₄²	C₂×C₄×D₅₉	1	2	yes	—
124	C₄²	C₂×C₄×D₅₉	1	2	yes	—
125	C₂³	C₂×C₄×Dic₅₉	1	1	yes	characteristic
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	#14273637	1	8	yes	—
129	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
130	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
131	C₂×C₄	C₄²×D₅₉	1	2	yes	—
132	C₂×C₄	C₂×C₄×Dic₅₉	1	2	yes	—
133	C₂×C₄	C₂×#11745659	1	4	yes	—
134	C₂×C₄	#14273734	1	4	yes	—
135	C₂×C₄	#14273735	1	4	yes	—
136	C₈	—	2	8	no	—
137	C₈	—	118	4	no	—
138	C₈	—	118	4	no	—
139	C₈	—	118	8	no	—
140	C₂²	C₄²×Dic₅₉	1	1	yes	characteristic
141	C₂²	C₂×#14273637	1	2	yes	—
142	C₂²	#16837358	1	2	yes	—
143	C₂²	#16837360	1	2	yes	—
144	C₄	—	2	8	no	—
145	C₄	C₄×#11745659	1	2	yes	—
146	C₄	C₄×#11745659	1	2	yes	—
147	C₄	#16837358	1	4	yes	—
148	C₄	#16837359	1	4	yes	—
149	C₂	#21422389	1	4	yes	—
150	C₂	C₄×#14273637	1	1	yes	characteristic
151	C₂	#21423783	1	1	yes	characteristic
152	C₂	#21423784	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₉
commutator	C₂²×C₅₉	C₄³
Fitting	C₅₉×#1744	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₄²
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1744	D₅₉
hypocenter	C₅₉	#7843
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄² ◁ #1744

Complements

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

N	H ≅ G/N	type
11. #16837124	C₄	semidirect
12. #16837123	C₄	semidirect
92. C₅₉	#7843	semidirect

Maximal subgroups

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

subgroup	index
#21423842	2
C₄²×C₅₉⋊C₈	2
C₂×#16837124	2
C₂×#16837123	2
C₅₉×#1744	2
#7843	59