Subgroups — #28504056

Order 15104.

326 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425695	C₂	1	2	yes	maximal
2	C₂×#16837178	C₂	1	1	yes	maximal, characteristic
3	#21423975	C₂	1	1	yes	maximal, characteristic
4	#21422186	C₂	1	2	yes	maximal
5	C₂×C₅₉×#364	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#14273663	C₄	1	1	yes	characteristic
9	#16837440	C₄	1	1	yes	characteristic
10	#16837357	C₂²	1	2	yes	—
11	#16837178	C₄	1	4	yes	complemented
12	#16837177	—	2	2	no	—
13	C₁₆×Dic₅₉	—	2	2	no	—
14	#16837162	C₄	1	2	yes	—
15	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#364	C₄	1	4	yes	complemented
18	C₂×#11745664	C₂×C₄	1	2	yes	—
19	#14273686	C₂×C₄	1	2	yes	—
20	C₂×C₅₉⋊C₁₆	—	2	1	no	—
21	C₂×C₅₉⋊C₁₆	—	2	1	no	—
22	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₅₉⋊C₁₆	D₄	1	2	yes	—
25	C₂×C₅₉⋊C₁₆	Q₈	1	2	yes	—
26	#14273663	C₈	1	4	yes	—
27	#14273661	C₈	1	2	yes	—
28	#14273659	C₈	1	2	yes	—
29	#14273637	—	2	2	no	—
30	C₈×Dic₅₉	—	2	2	no	—
31	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
34	M₃₂×C₅₉	—	2	2	no	—
35	M₃₂×C₅₉	C₂×C₄	1	4	yes	—
36	C₂×C₁₆×C₅₉	—	2	2	no	—
37	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₅₉	—	2	1	no	—
40	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₄²	1	2	yes	—
42	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
50	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
51	C₄²×C₅₉	—	2	1	no	—
52	C₄²×C₅₉	C₄²	1	2	yes	—
53	C₂²⋊Dic₅₉	—	2	2	no	—
54	C₄⋊Dic₅₉	C₂×C₈	1	4	yes	—
55	C₄×Dic₅₉	—	2	2	no	—
56	#11745664	—	2	2	no	—
57	#11745664	C₂×C₈	1	4	yes	—
58	C₂×C₅₉⋊C₈	—	2	2	no	—
59	C₁₆×C₅₉	—	2	4	no	—
60	C₅₉⋊C₁₆	—	2	2	no	—
61	C₅₉⋊C₁₆	C₄⋊C₄	1	4	yes	—
62	C₂³×C₅₉	#131	1	1	yes	characteristic
63	C₂×C₄×C₅₉	—	2	2	no	—
64	C₂×C₄×C₅₉	—	4	2	no	—
65	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
71	C₂×Dic₅₉	—	2	4	no	—
72	C₈×C₅₉	C₄×C₈	1	2	yes	—
73	C₈×C₅₉	C₄×C₈	1	2	yes	—
74	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₅₉	#117	1	1	yes	characteristic
76	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
77	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
78	C₅₉⋊C₈	—	2	4	no	—
79	#12665	—	59	1	no	maximal, Sylow
80	C₂²×C₅₉	—	2	2	no	—
81	C₂²×C₅₉	#405	1	1	yes	characteristic
82	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
83	C₂²×C₅₉	C₈×Q₈	1	1	yes	characteristic
84	C₄×C₅₉	—	4	2	no	—
85	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
86	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
87	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
88	C₄×C₅₉	C₈×Q₈	1	1	yes	characteristic
89	Dic₅₉	—	4	4	no	—
90	C₂×#364	D₅₉	1	1	yes	characteristic
91	#2108	—	59	2	no	—
92	C₂×#346	—	59	1	no	—
93	#1772	—	59	1	no	—
94	#1379	—	59	2	no	—
95	C₂×C₅₉	—	2	1	no	—
96	C₂×C₅₉	—	2	1	no	—
97	C₂×C₅₉	C₈×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₅₉	#2171	1	2	yes	—
99	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
100	C₂²×C₁₆	—	59	1	no	—
101	C₂²×C₁₆	—	59	1	no	—
102	C₂×#108	—	59	1	no	—
103	#427	—	59	1	no	—
104	#405	—	59	2	no	—
105	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
106	#364	Dic₅₉	1	4	yes	complemented
107	#346	—	59	4	no	—
108	#346	—	118	2	no	—
109	C₄×C₁₆	—	118	2	no	—
110	#339	—	59	2	no	—
111	C₅₉	#12665	1	1	yes	characteristic, complemented, Sylow
112	C₂×M₁₆	—	59	2	no	—
113	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
115	#117	—	59	2	no	—
116	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
117	M₃₂	—	2	2	no	—
118	M₃₂	C₂×Dic₅₉	1	4	yes	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	59	2	no	—
121	C₂×C₁₆	—	59	2	no	—
122	C₂×C₁₆	—	59	2	no	—
123	C₂×C₁₆	—	59	2	no	—
124	C₂×C₁₆	—	118	1	no	—
125	C₂×C₁₆	—	118	1	no	—
126	#108	—	59	4	no	—
127	#107	—	59	2	no	—
128	#106	—	59	2	no	—
129	#97	—	118	2	no	—
130	C₄×C₈	—	118	2	no	—
131	C₄×C₈	C₄×D₅₉	1	2	yes	—
132	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
135	M₁₆	—	59	4	no	—
136	M₁₆	—	118	2	no	—
137	C₂×C₈	—	118	2	no	—
138	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
139	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
140	C₂×C₈	C₈×D₅₉	1	2	yes	—
141	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
142	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
143	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
144	C₂×C₈	#11745687	1	1	yes	characteristic
145	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
146	C₂×C₈	#11745689	1	1	yes	characteristic
147	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
148	C₄⋊C₄	—	59	4	no	—
149	C₂²⋊C₄	—	118	2	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	118	2	no	—
152	C₄²	C₄×Dic₅₉	1	2	yes	—
153	C₁₆	—	2	4	no	—
154	C₁₆	—	59	4	no	—
155	C₁₆	—	118	2	no	—
156	C₂³	#14273735	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	118	4	no	—
160	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
161	C₂×C₄	#14273689	1	1	yes	characteristic
162	C₂×C₄	#14273692	1	1	yes	characteristic
163	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
164	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
165	C₂×C₄	#14273735	1	1	yes	characteristic
166	C₈	—	118	4	no	—
167	C₈	C₈×Dic₅₉	1	2	yes	—
168	C₈	C₈×Dic₅₉	1	2	yes	—
169	C₈	C₄⋊C₄×D₅₉	1	1	yes	characteristic
170	C₈	#14273685	1	1	yes	characteristic
171	C₈	D₄×Dic₅₉	1	1	yes	characteristic
172	C₈	Q₈×Dic₅₉	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	#16837360	1	1	yes	characteristic
175	C₂²	#16837467	1	1	yes	characteristic
176	C₂²	#16837526	1	1	yes	characteristic
177	C₄	—	4	2	no	—
178	C₄	—	236	4	no	—
179	C₄	C₄⋊C₄×Dic₅₉	1	1	yes	characteristic
180	C₄	C₂×C₈×Dic₅₉	1	1	yes	characteristic
181	C₄	#16837467	1	1	yes	characteristic
182	C₄	#16837526	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	#21423961	1	1	yes	characteristic
186	C₂	#21425937	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₅₉
commutator	C₄×C₅₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#364	C₂
socle	C₂²×C₅₉	C₈×D₄
prime radical	C₂³×C₅₉	#131
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#364	D₅₉
hypocenter	C₅₉	#12665
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
11. #16837178	C₄	semidirect
17. C₅₉×#364	C₄	semidirect
106. #364	Dic₅₉	semidirect
111. C₅₉	#12665	semidirect

Maximal subgroups

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

subgroup	index
#21425695	2
C₂×#16837178	2
#21423975	2
#21422186	2
C₂×C₅₉×#364	2
#12665	59