Subgroups — #28500274

Order 15104.

242 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21422844	C₂	1	2	yes	maximal
2	#21422170	C₂	1	2	yes	maximal, complemented
3	#21422168	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
5	#16837455	—	2	2	no	—
6	#16837155	C₂²	1	4	yes	—
7	#16837153	—	2	2	no	—
8	C₂²×C₁₆×C₅₉	C₄	1	1	yes	characteristic
9	C₂²×C₁₆×C₅₉	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₄×C₁₆×C₅₉	C₄	1	1	yes	characteristic
12	C₄×C₁₆×C₅₉	C₄	1	1	yes	characteristic
13	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
15	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
16	C₂²⋊(C₅₉⋊C₈)	—	4	2	no	—
17	C₄×C₅₉⋊C₈	—	4	2	no	—
18	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
20	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₅₉	—	2	2	no	—
22	C₂×C₁₆×C₅₉	C₈	1	4	yes	complemented
23	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
30	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
34	C₂×C₈×C₅₉	—	2	2	no	—
35	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
39	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
41	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₅₉	D₈	1	1	yes	characteristic
43	C₄²×C₅₉	Q₁₆	1	1	yes	characteristic
44	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₅₉⋊C₈	—	4	4	no	—
46	C₁₆×C₅₉	—	2	1	no	—
47	C₁₆×C₅₉	—	2	1	no	—
48	C₁₆×C₅₉	C₂×C₈	1	2	yes	—
49	C₁₆×C₅₉	C₂×C₈	1	2	yes	—
50	C₂³×C₅₉	#107	1	1	yes	characteristic
51	C₂×C₄×C₅₉	—	2	2	no	—
52	C₂×C₄×C₅₉	—	2	2	no	—
53	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
55	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
56	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
57	C₂×C₄×C₅₉	#108	1	2	yes	—
58	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
61	C₈×C₅₉	—	2	1	no	—
62	C₈×C₅₉	—	2	1	no	—
63	C₈×C₅₉	C₄⋊C₈	1	2	yes	—
64	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
65	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
66	C₅₉⋊C₈	—	8	4	no	—
67	#12008	—	59	1	no	maximal, Sylow
68	C₂²×C₅₉	—	2	1	no	—
69	C₂²×C₅₉	—	2	1	no	—
70	C₂²×C₅₉	#335	1	2	yes	—
71	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
72	C₄×C₅₉	—	2	1	no	—
73	C₄×C₅₉	—	2	1	no	—
74	C₄×C₅₉	#335	1	2	yes	—
75	C₄×C₅₉	#368	1	2	yes	—
76	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
77	C₄×C₅₉	C₂×#108	1	1	yes	characteristic
78	C₄×C₅₉	#508	1	2	yes	—
79	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic
80	#1564	—	59	2	no	—
81	#1372	—	59	2	no	—
82	#1371	—	59	2	no	—
83	C₂×C₅₉	—	2	2	no	—
84	C₂×C₅₉	C₂×#335	1	1	yes	characteristic
85	C₂×C₅₉	#2158	1	1	yes	characteristic
86	C₂×C₅₉	C₂×#368	1	1	yes	characteristic
87	C₂²×C₁₆	Dic₅₉	1	1	yes	characteristic
88	C₂²×C₁₆	Dic₅₉	1	1	yes	characteristic
89	#431	—	118	2	no	—
90	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
91	C₄×C₁₆	Dic₅₉	1	1	yes	characteristic
92	C₄×C₁₆	Dic₅₉	1	1	yes	characteristic
93	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
94	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
95	#335	—	59	4	no	—
96	#334	—	118	2	no	—
97	C₅₉	#12008	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
99	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
100	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	C₅₉⋊C₈	1	4	yes	complemented
103	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
104	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
105	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
106	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
107	C₄⋊C₈	—	118	4	no	—
108	C₂²⋊C₈	—	236	2	no	—
109	C₄×C₈	—	236	2	no	—
110	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
111	C₄×C₈	D₂₃₆	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
113	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
114	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
115	C₂²×C₄	Dic₂₃₆	1	1	yes	characteristic
116	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	236	4	no	—
119	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
120	C₂×C₈	#11745664	1	2	yes	—
121	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
122	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
123	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
124	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
125	C₄²	D₄₇₂	1	1	yes	characteristic
126	C₄²	Dic₂₃₆	1	1	yes	characteristic
127	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
128	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
129	C₁₆	—	2	1	no	—
130	C₁₆	—	2	1	no	—
131	C₁₆	C₂×C₅₉⋊C₈	1	2	yes	—
132	C₁₆	C₂×C₅₉⋊C₈	1	2	yes	—
133	C₂³	#14273661	1	1	yes	characteristic
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
137	C₂×C₄	#14273661	1	1	yes	characteristic
138	C₂×C₄	#14273661	1	1	yes	characteristic
139	C₂×C₄	#14273661	1	1	yes	characteristic
140	C₂×C₄	#14273663	1	2	yes	—
141	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
142	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
143	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	472	4	no	—
147	C₈	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
148	C₈	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
149	C₈	C₂×#11745664	1	1	yes	characteristic
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#16837155	1	2	yes	—
153	C₂²	C₂×#14273661	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#16837155	1	2	yes	—
157	C₄	#16837232	1	2	yes	—
158	C₄	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
159	C₄	C₂×#14273663	1	1	yes	characteristic
160	C₄	#16837834	1	2	yes	—
161	C₂	—	2	2	no	—
162	C₂	C₂×#16837155	1	1	yes	characteristic
163	C₂	#21425880	1	1	yes	characteristic
164	C₂	C₂×#16837232	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄₇₂
commutator	C₈×C₅₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂×#107
prime radical	C₂³×C₅₉	#107
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	#12008
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #21422170	C₂	semidirect
3. #21422168	C₂	semidirect
22. C₂×C₁₆×C₅₉	C₈	semidirect
97. C₅₉	#12008	semidirect
102. C₂×C₁₆	C₅₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#21422844	2
#21422170	2
#21422168	2
C₂×C₄×C₁₆×C₅₉	2
#12008	59