Subgroups — #28500272

Order 15104.

242 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#21422844	C₂	1	1	yes	maximal, characteristic
3	#21422169	C₂	1	2	yes	maximal, complemented
4	#21422167	C₂	1	2	yes	maximal, complemented
5	C₅₉×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₅₉⋊C₁₆	C₄	1	1	yes	characteristic
8	#16837455	—	2	1	no	—
9	C₄×C₅₉⋊C₁₆	C₄	1	1	yes	characteristic
10	C₄×C₅₉⋊C₁₆	C₄	1	1	yes	characteristic
11	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	#16837155	C₂²	1	2	yes	—
14	#16837153	—	2	1	no	—
15	C₅₉×#431	—	2	1	no	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#335	C₂²	1	2	yes	—
18	C₅₉×#334	—	2	1	no	—
19	C₂×C₅₉⋊C₁₆	—	2	2	no	—
20	C₂×C₅₉⋊C₁₆	C₈	1	4	yes	complemented
21	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
26	C₂²⋊(C₅₉⋊C₈)	—	4	1	no	—
27	C₄×C₅₉⋊C₈	—	4	1	no	—
28	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
31	C₄⋊C₈×C₅₉	—	2	2	no	—
32	C₂²⋊C₈×C₅₉	—	4	1	no	—
33	C₄×C₈×C₅₉	—	4	1	no	—
34	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
36	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
37	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
41	C₂×C₈×C₅₉	—	2	2	no	—
42	C₂×C₈×C₅₉	—	4	2	no	—
43	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
47	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₅₉	D₈	1	1	yes	characteristic
51	C₄²×C₅₉	Q₁₆	1	1	yes	characteristic
52	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₅₉⋊C₈	—	4	2	no	—
54	C₅₉⋊C₁₆	—	2	1	no	—
55	C₅₉⋊C₁₆	—	2	1	no	—
56	C₅₉⋊C₁₆	C₂×C₈	1	2	yes	—
57	C₅₉⋊C₁₆	C₂×C₈	1	2	yes	—
58	C₂³×C₅₉	#107	1	1	yes	characteristic
59	C₂×C₄×C₅₉	—	2	2	no	—
60	C₂×C₄×C₅₉	—	2	2	no	—
61	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
62	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#108	1	2	yes	—
66	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
69	C₈×C₅₉	—	2	1	no	—
70	C₈×C₅₉	—	2	1	no	—
71	C₈×C₅₉	—	8	2	no	—
72	C₈×C₅₉	C₄⋊C₈	1	2	yes	—
73	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
74	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
75	C₅₉⋊C₈	—	8	2	no	—
76	#12008	—	59	1	no	maximal, Sylow
77	C₂²×C₅₉	—	2	1	no	—
78	C₂²×C₅₉	—	2	1	no	—
79	C₂²×C₅₉	#335	1	2	yes	—
80	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
81	C₄×C₅₉	—	2	1	no	—
82	C₄×C₅₉	—	2	1	no	—
83	C₄×C₅₉	#335	1	2	yes	—
84	C₄×C₅₉	#368	1	2	yes	—
85	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₄×C₅₉	C₂×#108	1	1	yes	characteristic
87	C₄×C₅₉	#508	1	2	yes	—
88	C₂×C₄×C₁₆	—	59	1	no	—
89	#1564	—	59	1	no	—
90	#1564	D₅₉	1	1	yes	characteristic
91	#1372	—	59	2	no	—
92	#1371	—	59	2	no	—
93	C₂×C₅₉	—	2	2	no	—
94	C₂×C₅₉	C₂×#335	1	1	yes	characteristic
95	C₂×C₅₉	#2158	1	1	yes	characteristic
96	C₂×C₅₉	C₂×#368	1	1	yes	characteristic
97	C₂²×C₁₆	—	59	1	no	—
98	C₂²×C₁₆	—	59	1	no	—
99	#431	—	2	1	no	—
100	#431	—	118	1	no	—
101	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
102	C₄×C₁₆	—	59	1	no	—
103	C₄×C₁₆	—	59	1	no	—
104	C₄×C₁₆	—	59	1	no	—
105	C₄×C₁₆	—	59	1	no	—
106	#335	—	59	2	no	—
107	#335	C₂×D₅₉	1	2	yes	—
108	#334	—	2	1	no	—
109	#334	—	118	1	no	—
110	C₅₉	#12008	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
112	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
113	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
114	C₂×C₁₆	—	59	4	no	—
115	C₂×C₁₆	—	59	1	no	—
116	C₂×C₁₆	—	59	1	no	—
117	C₂×C₁₆	—	59	1	no	—
118	C₂×C₁₆	—	59	1	no	—
119	C₂×C₁₆	—	118	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	118	2	no	—
122	C₂²⋊C₈	—	4	1	no	—
123	C₂²⋊C₈	—	236	1	no	—
124	C₄×C₈	—	4	1	no	—
125	C₄×C₈	—	236	1	no	—
126	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
127	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
128	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
129	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
130	C₂²×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
131	C₂²×C₄	#11745669	1	1	yes	characteristic
132	C₂²×C₄	#11745672	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	236	2	no	—
136	C₂×C₈	C₈×D₅₉	1	2	yes	—
137	C₂×C₈	#11745659	1	2	yes	—
138	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
139	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
140	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
141	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
142	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
143	C₄²	#11745669	1	1	yes	characteristic
144	C₄²	#11745672	1	1	yes	characteristic
145	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
146	C₁₆	—	59	2	no	—
147	C₁₆	—	59	2	no	—
148	C₁₆	—	118	1	no	—
149	C₁₆	—	118	1	no	—
150	C₂³	#14273660	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	Dic₅₉⋊C₈	1	2	yes	—
154	C₂×C₄	#14273660	1	1	yes	characteristic
155	C₂×C₄	#14273660	1	1	yes	characteristic
156	C₂×C₄	#14273660	1	1	yes	characteristic
157	C₂×C₄	#14273662	1	2	yes	—
158	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
159	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
160	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	8	2	no	—
164	C₈	—	472	2	no	—
165	C₈	Dic₅₉⋊C₈	1	2	yes	—
166	C₈	C₂×C₈×D₅₉	1	1	yes	characteristic
167	C₈	C₂×#11745659	1	1	yes	characteristic
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#16837154	1	2	yes	—
171	C₂²	C₂×#14273660	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	#16837154	1	2	yes	—
175	C₄	#16837231	1	2	yes	—
176	C₄	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
177	C₄	C₂×#14273662	1	1	yes	characteristic
178	C₄	#16837832	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	C₂×#16837154	1	1	yes	characteristic
181	C₂	#21425878	1	1	yes	characteristic
182	C₂	C₂×#16837231	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	#11745669
commutator	C₈×C₅₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1564	C₂
socle	C₂²×C₅₉	C₂×#107
prime radical	C₂³×C₅₉	#107
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#1564	D₅₉
hypocenter	C₅₉	#12008
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #21422169	C₂	semidirect
4. #21422167	C₂	semidirect
20. C₂×C₅₉⋊C₁₆	C₈	semidirect
110. C₅₉	#12008	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅₉⋊C₁₆	2
#21422844	2
#21422169	2
#21422167	2
C₅₉×#1564	2
#12008	59