Subgroups — #28503700

Order 15104.

354 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14273667	C₂	1	1	yes	maximal, characteristic
2	C₄×#14273637	C₂	1	1	yes	maximal, characteristic
3	#21422175	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273667	C₂²	1	2	yes	—
6	C₂×#14273637	C₄	1	2	yes	—
7	C₂×#14273637	C₂²	1	2	yes	—
8	C₂×C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
9	C₄²×Dic₅₉	C₄	1	1	yes	characteristic
10	#16837178	C₄	1	2	yes	complemented
11	C₄×C₅₉⋊C₁₆	C₄	1	2	yes	complemented
12	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×#346	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₅₉	C₄	1	2	yes	complemented
16	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄×Dic₅₉	—	2	2	no	—
18	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
19	#14273667	—	2	4	no	—
20	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
21	#14273637	—	2	4	no	—
22	#14273637	C₈	1	8	yes	—
23	C₄×C₅₉⋊C₈	—	2	2	no	—
24	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₅₉	—	2	4	no	—
28	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	M₁₆	1	2	yes	—
34	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
35	C₂²×Dic₅₉	—	2	2	no	—
36	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	—
39	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	#41	1	4	yes	—
42	C₄²×C₅₉	C₄²	1	1	yes	characteristic
43	C₄²×C₅₉	C₄²	1	1	yes	characteristic
44	C₄²×C₅₉	M₁₆	1	2	yes	—
45	C₄×Dic₅₉	—	2	2	no	—
46	C₄×Dic₅₉	—	4	4	no	—
47	C₄×Dic₅₉	C₂×C₈	1	4	yes	—
48	C₂×C₅₉⋊C₈	—	2	2	no	—
49	C₂×C₅₉⋊C₈	—	2	2	no	—
50	C₂×C₅₉⋊C₈	C₂×C₈	1	4	yes	—
51	C₁₆×C₅₉	—	2	4	no	—
52	C₅₉⋊C₁₆	—	2	4	no	—
53	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
54	C₂×C₄×C₅₉	—	2	4	no	—
55	C₂×C₄×C₅₉	#97	1	2	yes	—
56	C₂×C₄×C₅₉	#97	1	2	yes	—
57	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₅₉	#117	1	2	yes	—
59	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
60	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
61	C₂×Dic₅₉	—	2	4	no	—
62	C₂×Dic₅₉	—	4	4	no	—
63	C₈×C₅₉	C₄×C₈	1	2	yes	—
64	C₈×C₅₉	C₄×C₈	1	2	yes	—
65	C₈×C₅₉	#117	1	2	yes	—
66	C₈×C₅₉	#117	1	2	yes	—
67	C₅₉⋊C₈	—	2	4	no	—
68	#12594	—	59	1	no	maximal, Sylow
69	C₂²×C₅₉	—	2	2	no	—
70	C₂²×C₅₉	C₂×#97	1	1	yes	characteristic
71	C₂²×C₅₉	#431	1	2	yes	—
72	C₄×C₅₉	—	2	2	no	—
73	C₄×C₅₉	#347	1	4	yes	—
74	C₄×C₅₉	#376	1	1	yes	characteristic
75	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
76	C₄×C₅₉	#431	1	2	yes	—
77	Dic₅₉	—	4	4	no	—
78	C₄×M₃₂	—	59	1	no	—
79	C₄×M₃₂	D₅₉	1	1	yes	characteristic
80	C₄×#97	—	59	1	no	—
81	#1375	—	59	4	no	—
82	C₂×C₅₉	—	2	2	no	—
83	C₂×C₅₉	#1744	1	1	yes	characteristic
84	C₂×C₅₉	C₂×#347	1	2	yes	—
85	C₂×M₃₂	—	59	2	no	—
86	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
87	C₂×#97	—	59	2	no	—
88	C₂×#97	—	59	2	no	—
89	C₂×C₄×C₈	—	59	1	no	—
90	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
91	C₄³	—	59	1	no	—
92	#346	—	59	2	no	—
93	#346	Dic₅₉	1	2	yes	complemented
94	C₄×C₁₆	—	59	2	no	—
95	C₄×C₁₆	Dic₅₉	1	2	yes	complemented
96	C₅₉	#12594	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₈	—	59	2	no	—
98	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
99	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
100	C₂×C₄²	—	59	2	no	—
101	C₂×C₄²	—	118	2	no	—
102	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
103	M₃₂	—	2	4	no	—
104	M₃₂	—	118	4	no	—
105	C₂×C₁₆	—	59	4	no	—
106	C₂×C₁₆	C₂×Dic₅₉	1	4	yes	—
107	#97	—	59	8	no	—
108	#97	—	118	4	no	—
109	C₄×C₈	—	118	2	no	—
110	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
111	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
113	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
114	C₂²×C₄	—	118	2	no	—
115	C₂²×C₄	#11745659	1	2	yes	—
116	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
117	C₂×C₈	—	59	4	no	—
118	C₂×C₈	—	118	2	no	—
119	C₂×C₈	—	118	2	no	—
120	C₂×C₈	C₈×D₅₉	1	4	yes	—
121	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
122	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
123	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
124	C₂×C₈	#11745685	1	4	yes	—
125	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
126	C₄²	—	59	4	no	—
127	C₄²	—	118	2	no	—
128	C₄²	—	236	4	no	—
129	C₄²	#11745659	1	2	yes	—
130	C₄²	C₄×Dic₅₉	1	1	yes	characteristic
131	C₄²	C₄×Dic₅₉	1	1	yes	characteristic
132	C₁₆	—	2	4	no	—
133	C₁₆	—	118	4	no	—
134	C₂³	C₂×#11745659	1	1	yes	characteristic
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	118	4	no	—
137	C₂×C₄	—	236	4	no	—
138	C₂×C₄	#14273637	1	2	yes	—
139	C₂×C₄	#14273637	1	2	yes	—
140	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
141	C₂×C₄	#14273684	1	2	yes	—
142	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
143	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
144	C₈	—	118	4	no	—
145	C₈	C₈×Dic₅₉	1	2	yes	—
146	C₈	C₈×Dic₅₉	1	2	yes	—
147	C₈	#14273684	1	2	yes	—
148	C₈	#14273686	1	2	yes	—
149	C₂²	—	2	2	no	—
150	C₂²	C₂×#14273637	1	1	yes	characteristic
151	C₂²	#16837453	1	2	yes	—
152	C₄	—	2	2	no	—
153	C₄	—	236	4	no	—
154	C₄	#16837179	1	4	yes	—
155	C₄	#16837248	1	1	yes	characteristic
156	C₄	C₂×C₈×Dic₅₉	1	1	yes	characteristic
157	C₄	#16837453	1	2	yes	—
158	C₂	—	2	2	no	—
159	C₂	#21423841	1	1	yes	characteristic
160	C₂	C₂×#16837179	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₅₉
commutator	C₄×C₅₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₄×M₃₂×C₅₉	C₂
socle	C₂²×C₅₉	C₂×#97
prime radical	C₂³×C₅₉	C₂×M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₄×M₃₂	D₅₉
hypocenter	C₅₉	#12594
LCS	G ▷ C₄×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
10. #16837178	C₄	semidirect
11. C₄×C₅₉⋊C₁₆	C₄	semidirect
14. C₅₉×#346	C₄	semidirect
15. C₄×C₁₆×C₅₉	C₄	semidirect
93. #346	Dic₅₉	semidirect
95. C₄×C₁₆	Dic₅₉	semidirect
96. C₅₉	#12594	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14273667	2
C₄×#14273637	2
#21422175	2
C₄×M₃₂×C₅₉	2
#12594	59