Subgroups — #28500291

Order 15104.

250 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425899	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#21422165	C₂	1	2	yes	maximal, complemented
4	#21422159	C₂	1	2	yes	maximal, complemented
5	C₅₉×#2163	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
8	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
9	C₂²⋊(C₅₉⋊C₁₆)	—	2	1	no	—
10	#16837178	—	2	1	no	—
11	C₈×C₅₉⋊C₈	C₄	1	1	yes	characteristic
12	C₈×C₅₉⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₅₉×C₂²⋊C₁₆	—	2	1	no	—
18	C₅₉×#346	—	2	1	no	—
19	C₂²×C₅₉⋊C₈	—	2	1	no	—
20	C₂²×C₅₉⋊C₈	C₈	1	1	yes	characteristic
21	C₂²×C₅₉⋊C₈	C₈	1	1	yes	characteristic
22	C₂×C₅₉⋊C₁₆	—	2	2	no	—
23	C₄×C₅₉⋊C₈	—	2	1	no	—
24	C₄×C₅₉⋊C₈	C₈	1	1	yes	characteristic
25	C₄×C₅₉⋊C₈	C₈	1	1	yes	characteristic
26	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₅₉	—	2	2	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	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	M₁₆	1	1	yes	characteristic
41	C₂×C₈×C₅₉	—	2	2	no	—
42	C₂×C₈×C₅₉	—	2	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×D₄	1	1	yes	characteristic
46	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₅₉	M₁₆	1	1	yes	characteristic
50	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₅₉⋊C₈	—	2	4	no	—
52	C₂×C₅₉⋊C₈	—	2	1	no	—
53	C₂×C₅₉⋊C₈	—	2	1	no	—
54	C₂×C₅₉⋊C₈	—	2	2	no	—
55	C₂×C₅₉⋊C₈	—	2	2	no	—
56	C₂×C₅₉⋊C₈	C₂×C₈	1	1	yes	characteristic
57	C₂×C₅₉⋊C₈	C₂×C₈	1	1	yes	characteristic
58	C₂×C₅₉⋊C₈	C₂×C₈	1	1	yes	characteristic
59	C₂×C₅₉⋊C₈	C₂×C₈	1	1	yes	characteristic
60	C₁₆×C₅₉	—	4	2	no	—
61	C₅₉⋊C₁₆	—	4	2	no	—
62	C₂³×C₅₉	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₅₉	—	2	2	no	—
64	C₂×C₄×C₅₉	—	2	1	no	—
65	C₂×C₄×C₅₉	—	2	1	no	—
66	C₂×C₄×C₅₉	—	2	1	no	—
67	C₂×C₄×C₅₉	—	2	1	no	—
68	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
74	C₈×C₅₉	—	2	1	no	—
75	C₈×C₅₉	—	2	1	no	—
76	C₈×C₅₉	#108	1	2	yes	—
77	C₈×C₅₉	#135	1	2	yes	—
78	C₅₉⋊C₈	—	2	2	no	—
79	C₅₉⋊C₈	—	2	2	no	—
80	C₅₉⋊C₈	—	2	1	no	—
81	C₅₉⋊C₈	—	2	1	no	—
82	C₅₉⋊C₈	M₃₂	1	2	yes	—
83	C₅₉⋊C₈	M₃₂	1	2	yes	—
84	#12013	—	59	1	no	maximal, Sylow
85	C₂²×C₅₉	—	2	1	no	—
86	C₂²×C₅₉	—	2	1	no	—
87	C₂²×C₅₉	#364	1	2	yes	—
88	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₄×C₅₉	—	2	1	no	—
90	C₄×C₅₉	—	2	1	no	—
91	C₄×C₅₉	—	2	1	no	—
92	C₄×C₅₉	—	2	1	no	—
93	C₄×C₅₉	#427	1	1	yes	characteristic
94	C₄×C₅₉	C₂×#108	1	1	yes	characteristic
95	C₄×C₅₉	C₂×M₃₂	1	1	yes	characteristic
96	C₄×C₅₉	C₂×M₃₂	1	1	yes	characteristic
97	#2163	—	59	1	no	—
98	#2163	D₅₉	1	1	yes	characteristic
99	C₂×C₈²	—	59	1	no	—
100	#1370	—	59	2	no	—
101	#1367	—	59	2	no	—
102	C₂×C₅₉	—	2	2	no	—
103	C₂×C₅₉	#1564	1	1	yes	characteristic
104	C₂×C₅₉	#2150	1	1	yes	characteristic
105	C₂×C₅₉	C₂×#364	1	1	yes	characteristic
106	C₂×C₄×C₈	—	59	1	no	—
107	C₂×C₄×C₈	—	59	1	no	—
108	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
109	C₄⋊C₁₆	—	59	2	no	—
110	C₄⋊C₁₆	C₂×D₅₉	1	2	yes	—
111	C₂²⋊C₁₆	—	2	1	no	—
112	C₂²⋊C₁₆	—	118	1	no	—
113	#346	—	2	1	no	—
114	#346	—	118	1	no	—
115	C₈²	—	59	1	no	—
116	C₈²	—	59	1	no	—
117	C₈²	—	59	1	no	—
118	C₈²	—	59	1	no	—
119	C₅₉	#12013	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₈	—	59	1	no	—
121	C₂²×C₈	—	59	1	no	—
122	C₂²×C₈	—	118	1	no	—
123	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
124	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
125	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	118	2	no	—
128	C₄×C₈	—	59	1	no	—
129	C₄×C₈	—	59	1	no	—
130	C₄×C₈	—	59	1	no	—
131	C₄×C₈	—	59	1	no	—
132	C₄×C₈	—	59	1	no	—
133	C₄×C₈	—	59	1	no	—
134	C₄×C₈	—	118	1	no	—
135	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
136	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
138	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
139	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
140	C₂²×C₄	#11745659	1	1	yes	characteristic
141	C₂²×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	59	1	no	—
145	C₂×C₈	—	59	1	no	—
146	C₂×C₈	—	59	1	no	—
147	C₂×C₈	—	59	1	no	—
148	C₂×C₈	—	118	2	no	—
149	C₂×C₈	—	118	2	no	—
150	C₂×C₈	—	118	4	no	—
151	C₂×C₈	—	118	1	no	—
152	C₂×C₈	—	118	1	no	—
153	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
154	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
155	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
156	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
157	C₄²	C₈×D₅₉	1	1	yes	characteristic
158	C₄²	#11745659	1	1	yes	characteristic
159	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
160	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
161	C₁₆	—	4	2	no	—
162	C₁₆	—	236	2	no	—
163	C₂³	Dic₅₉⋊C₈	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	Dic₅₉⋊C₈	1	1	yes	characteristic
170	C₂×C₄	Dic₅₉⋊C₈	1	1	yes	characteristic
171	C₂×C₄	Dic₅₉⋊C₈	1	1	yes	characteristic
172	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
173	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
174	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	59	2	no	—
178	C₈	—	59	2	no	—
179	C₈	—	118	2	no	—
180	C₈	—	118	2	no	—
181	C₈	—	118	1	no	—
182	C₈	—	118	1	no	—
183	C₈	#14273662	1	2	yes	—
184	C₈	#14273747	1	2	yes	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	#16837222	1	2	yes	—
188	C₂²	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	#16837438	1	1	yes	characteristic
194	C₄	C₂×#14273662	1	1	yes	characteristic
195	C₄	M₃₂×D₅₉	1	1	yes	characteristic
196	C₄	M₃₂×D₅₉	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	#21422842	1	1	yes	characteristic
199	C₂	#21425851	1	1	yes	characteristic
200	C₂	C₂×#16837222	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#2163	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₈
prime radical	C₂³×C₅₉	C₄⋊C₈
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#2163	D₅₉
hypocenter	C₅₉	#12013
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
3. #21422165	C₂	semidirect
4. #21422159	C₂	semidirect
119. C₅₉	#12013	semidirect

Maximal subgroups

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

subgroup	index
#21425899	2
C₂×C₈×C₅₉⋊C₈	2
#21422165	2
#21422159	2
C₅₉×#2163	2
#12013	59