Subgroups — #28497708

Order 15104.

290 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425895	C₂	1	1	yes	maximal, characteristic
2	C₈×#11745664	C₂	1	1	yes	maximal, characteristic
3	#21422015	C₂	1	2	yes	maximal, complemented
4	#21422012	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₄	1	1	yes	characteristic
7	C₄×#11745664	C₄	1	1	yes	characteristic
8	C₄⋊(C₅₉⋊C₁₆)	C₄	1	2	yes	—
9	C₂²⋊(C₅₉⋊C₁₆)	C₄	1	2	yes	—
10	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	—
11	#16837124	C₄	1	2	yes	—
12	C₈×C₅₉⋊C₈	C₂²	1	2	yes	—
13	C₂²×C₁₆×C₅₉	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₅₉	C₄	1	2	yes	—
16	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×#11745664	C₈	1	2	yes	—
19	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
20	C₄⋊(C₅₉⋊C₈)	C₈	1	2	yes	—
21	C₂²⋊(C₅₉⋊C₈)	C₈	1	2	yes	—
22	#14273638	C₈	1	2	yes	—
23	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₉	—	2	4	no	—
29	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
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	1	yes	characteristic
34	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₅₉	—	2	2	no	—
39	C₂×C₈×C₅₉	—	2	2	no	—
40	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₅₉	C₄²	1	1	yes	characteristic
45	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
48	#11745664	—	2	4	no	—
49	C₂×C₅₉⋊C₈	C₂×C₈	1	4	yes	—
50	C₂×C₅₉⋊C₈	C₂×C₈	1	4	yes	—
51	C₁₆×C₅₉	—	2	2	no	—
52	C₁₆×C₅₉	M₁₆	1	4	yes	complemented
53	C₅₉⋊C₁₆	—	2	4	no	—
54	C₂³×C₅₉	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄×C₅₉	—	2	1	no	—
56	C₂×C₄×C₅₉	—	2	1	no	—
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	C₄×C₈	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₄×C₈	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₄×C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₉	M₃₂	1	4	yes	—
62	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
65	C₈×C₅₉	—	2	1	no	—
66	C₈×C₅₉	—	2	1	no	—
67	C₈×C₅₉	C₂×M₁₆	1	2	yes	—
68	C₈×C₅₉	#131	1	2	yes	—
69	C₅₉⋊C₈	—	2	4	no	—
70	C₅₉⋊C₈	—	2	4	no	—
71	#11408	—	59	1	no	maximal, Sylow
72	C₂²×C₅₉	—	2	1	no	—
73	C₂²×C₅₉	—	2	1	no	—
74	C₂²×C₅₉	#346	1	2	yes	—
75	C₂²×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
76	C₄×C₅₉	—	2	1	no	—
77	C₄×C₅₉	—	2	1	no	—
78	C₄×C₅₉	#346	1	2	yes	—
79	C₄×C₅₉	C₄×M₁₆	1	1	yes	characteristic
80	C₄×C₅₉	#405	1	1	yes	characteristic
81	C₄×C₅₉	C₂×M₃₂	1	2	yes	—
82	C₄×C₅₉	#504	1	2	yes	—
83	#2162	—	59	1	no	—
84	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic
85	C₈×M₁₆	—	59	1	no	—
86	#1312	—	59	2	no	—
87	#1311	—	59	2	no	—
88	C₂×C₅₉	—	2	2	no	—
89	C₂×C₅₉	C₈×M₁₆	1	1	yes	characteristic
90	C₂×C₅₉	C₂×#346	1	1	yes	characteristic
91	C₂×C₅₉	#2108	1	1	yes	characteristic
92	C₂²×C₁₆	Dic₅₉	1	2	yes	—
93	#431	—	59	1	no	—
94	C₄×M₁₆	—	59	1	no	—
95	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
96	C₄⋊C₁₆	—	59	2	no	—
97	C₂²⋊C₁₆	—	59	2	no	—
98	C₄×C₁₆	—	59	2	no	—
99	C₄×C₁₆	Dic₅₉	1	2	yes	—
100	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
101	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
102	#322	—	59	2	no	—
103	C₈²	—	59	2	no	—
104	C₅₉	#11408	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₁₆	—	59	2	no	—
106	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
108	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
109	C₂×C₁₆	—	2	4	no	—
110	C₂×C₁₆	—	59	4	no	—
111	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
112	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
113	C₄⋊C₈	—	59	2	no	—
114	C₂²⋊C₈	—	59	2	no	—
115	#97	—	59	2	no	—
116	C₄×C₈	—	59	2	no	—
117	C₄×C₈	—	59	2	no	—
118	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
119	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
121	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
122	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
123	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
124	C₂²×C₄	C₄×Dic₅₉	1	1	yes	characteristic
125	M₁₆	—	118	4	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	59	4	no	—
129	C₂×C₈	—	59	4	no	—
130	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
131	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
133	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
134	C₄²	C₈×D₅₉	1	1	yes	characteristic
135	C₄²	C₈×D₅₉	1	1	yes	characteristic
136	C₄²	C₄×Dic₅₉	1	1	yes	characteristic
137	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
138	C₁₆	—	2	2	no	—
139	C₁₆	—	118	4	no	—
140	C₁₆	#11745664	1	4	yes	complemented
141	C₂³	C₈×Dic₅₉	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	C₈×Dic₅₉	1	1	yes	characteristic
146	C₂×C₄	C₈×Dic₅₉	1	1	yes	characteristic
147	C₂×C₄	C₈×Dic₅₉	1	1	yes	characteristic
148	C₂×C₄	#14273666	1	4	yes	—
149	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
150	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
151	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	118	4	no	—
155	C₈	—	118	4	no	—
156	C₈	C₂×#11745664	1	2	yes	—
157	C₈	#14273733	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#16837177	1	2	yes	—
161	C₂²	C₂×C₈×Dic₅₉	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	#16837177	1	2	yes	—
165	C₄	C₄×#11745664	1	1	yes	characteristic
166	C₄	#16837357	1	1	yes	characteristic
167	C₄	C₂×#14273666	1	2	yes	—
168	C₄	#16837818	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	C₈×#11745664	1	1	yes	characteristic
171	C₂	C₂×#16837177	1	1	yes	characteristic
172	C₂	#21425692	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₅₉
commutator, socle	C₂²×C₅₉	C₂×C₄×C₈
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
prime radical	C₂³×C₅₉	C₄×C₈
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	#11408
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #21422015	C₂	semidirect
4. #21422012	C₂	semidirect
52. C₁₆×C₅₉	M₁₆	semidirect
104. C₅₉	#11408	semidirect
140. C₁₆	#11745664	semidirect

Maximal subgroups

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

subgroup	index
#21425895	2
C₈×#11745664	2
#21422015	2
#21422012	2
C₂×C₄×C₁₆×C₅₉	2
#11408	59