Subgroups — #28506715

Order 15104.

284 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425862	C₂	1	2	yes	maximal
2	C₂×#16837223	C₂	1	1	yes	maximal, characteristic
3	C₄×#14273667	C₂	1	1	yes	maximal, characteristic
4	#21422182	C₂	1	2	yes	maximal
5	C₅₉×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₂²	1	2	yes	—
7	C₂×#14273667	C₂²	1	1	yes	characteristic
8	C₂×#14273667	C₂²	1	1	yes	characteristic
9	#16837223	—	2	4	no	—
10	#16837223	—	2	1	no	—
11	#16837223	—	2	1	no	—
12	#16837178	—	2	1	no	—
13	C₄×C₅₉⋊C₁₆	—	2	1	no	—
14	C₂×#108×C₅₉	C₄	1	1	yes	characteristic
15	C₅₉×#427	C₄	1	1	yes	characteristic
16	C₅₉×#405	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×#339	C₄	1	2	yes	—
19	#14273667	—	2	1	no	—
20	#14273667	—	2	1	no	—
21	#14273667	—	2	2	no	—
22	#14273667	—	2	2	no	—
23	#14273667	D₄	1	2	yes	—
24	#14273667	Q₈	1	2	yes	—
25	C₂×C₅₉⋊C₁₆	—	2	1	no	—
26	C₂×C₅₉⋊C₁₆	—	2	1	no	—
27	C₂×C₅₉⋊C₁₆	—	2	2	no	—
28	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
31	#117×C₅₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
33	#108×C₅₉	C₈	1	2	yes	—
34	#108×C₅₉	C₈	1	2	yes	—
35	#107×C₅₉	—	2	1	no	—
36	#106×C₅₉	—	2	1	no	—
37	#97×C₅₉	—	2	2	no	—
38	C₄×C₈×C₅₉	—	2	2	no	—
39	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
40	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
42	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	—	2	1	no	—
44	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	M₁₆×C₅₉	—	2	2	no	—
46	M₁₆×C₅₉	C₂×C₈	1	4	yes	—
47	C₂×C₈×C₅₉	—	2	2	no	—
48	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
52	C₂×C₈×C₅₉	M₁₆	1	1	yes	characteristic
53	C₂×C₈×C₅₉	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
59	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₅₉	—	2	2	no	—
61	C₂²⋊C₄×C₅₉	—	2	2	no	—
62	C₄²×C₅₉	—	2	1	no	—
63	C₄²×C₅₉	—	2	2	no	—
64	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
65	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
66	C₅₉⋊C₁₆	—	2	2	no	—
67	C₅₉⋊C₁₆	—	2	2	no	—
68	C₅₉⋊C₁₆	—	4	2	no	—
69	C₂³×C₅₉	#131	1	1	yes	characteristic
70	C₂×C₄×C₅₉	—	2	1	no	—
71	C₂×C₄×C₅₉	—	2	1	no	—
72	C₂×C₄×C₅₉	—	4	2	no	—
73	C₂×C₄×C₅₉	—	4	2	no	—
74	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
80	C₈×C₅₉	—	2	4	no	—
81	C₈×C₅₉	C₄⋊C₈	1	1	yes	characteristic
82	C₈×C₅₉	C₄⋊C₈	1	1	yes	characteristic
83	C₈×C₅₉	C₄⋊C₈	1	1	yes	characteristic
84	C₈×C₅₉	C₄⋊C₈	1	1	yes	characteristic
85	C₈×C₅₉	#121	1	1	yes	characteristic
86	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
87	C₈×C₅₉	#125	1	1	yes	characteristic
88	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
89	#13164	—	59	1	no	maximal, Sylow
90	C₂²×C₅₉	—	4	1	no	—
91	C₂²×C₅₉	#424	1	1	yes	characteristic
92	C₂²×C₅₉	#436	1	1	yes	characteristic
93	C₂²×C₅₉	C₈×Q₈	1	1	yes	characteristic
94	C₄×C₅₉	—	2	2	no	—
95	C₄×C₅₉	—	4	1	no	—
96	C₄×C₅₉	—	8	2	no	—
97	C₄×C₅₉	#389	1	1	yes	characteristic
98	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
100	C₄×C₅₉	#446	1	1	yes	characteristic
101	#2153	—	59	2	no	—
102	C₂×#364	—	59	1	no	—
103	C₄×M₃₂	—	59	1	no	—
104	#1772	D₅₉	1	1	yes	characteristic
105	#1378	—	59	2	no	—
106	C₂×C₅₉	—	2	1	no	—
107	C₂×C₅₉	—	2	1	no	—
108	C₂×C₅₉	—	2	1	no	—
109	C₂×C₅₉	#1939	1	1	yes	characteristic
110	C₂×M₃₂	—	59	1	no	—
111	C₂×M₃₂	—	59	1	no	—
112	C₂×M₃₂	—	59	2	no	—
113	C₂×#108	Dic₅₉	1	1	yes	characteristic
114	#427	Dic₅₉	1	1	yes	characteristic
115	#405	C₂×D₅₉	1	2	yes	—
116	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
117	#364	—	118	4	no	—
118	#364	—	118	1	no	—
119	#364	—	118	1	no	—
120	#346	—	118	1	no	—
121	C₄×C₁₆	—	118	1	no	—
122	#339	Dic₅₉	1	2	yes	—
123	C₅₉	#13164	1	1	yes	characteristic, complemented, Sylow
124	C₂×M₁₆	C₂×Dic₅₉	1	2	yes	—
125	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
127	#117	C₂×Dic₅₉	1	2	yes	—
128	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
129	M₃₂	—	59	2	no	—
130	M₃₂	—	59	2	no	—
131	M₃₂	—	118	1	no	—
132	M₃₂	—	118	1	no	—
133	M₃₂	—	118	2	no	—
134	M₃₂	—	118	2	no	—
135	C₂×C₁₆	—	118	1	no	—
136	C₂×C₁₆	—	118	1	no	—
137	C₂×C₁₆	—	118	2	no	—
138	#108	C₅₉⋊C₈	1	2	yes	—
139	#108	C₅₉⋊C₈	1	2	yes	—
140	#107	—	2	1	no	—
141	#106	—	2	1	no	—
142	#97	—	2	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
145	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
146	C₄×C₈	D₂₃₆	1	1	yes	characteristic
147	C₄×C₈	D₂₃₆	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	C₂×C₅₉⋊C₈	1	4	yes	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
154	C₂×C₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
155	C₂×C₈	#11745664	1	1	yes	characteristic
156	C₂×C₈	#11745664	1	1	yes	characteristic
157	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
158	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
159	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
160	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
161	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
162	C₂×C₈	#11745687	1	1	yes	characteristic
163	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
164	C₂×C₈	#11745689	1	1	yes	characteristic
165	C₄⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	2	no	—
169	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
170	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
171	C₁₆	—	118	2	no	—
172	C₁₆	—	118	2	no	—
173	C₁₆	—	236	2	no	—
174	C₂³	#14273734	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
180	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
181	C₂×C₄	Q₈×Dic₅₉	1	1	yes	characteristic
182	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
183	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
184	C₂×C₄	#14273734	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
187	C₈	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
188	C₈	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
189	C₈	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
190	C₈	#14273702	1	1	yes	characteristic
191	C₈	#14273707	1	1	yes	characteristic
192	C₈	#14273718	1	1	yes	characteristic
193	C₈	#14273726	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	#16837429	1	1	yes	characteristic
196	C₂²	#16837480	1	1	yes	characteristic
197	C₂²	Q₈×C₅₉⋊C₈	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	8	2	no	—
201	C₄	#16837301	1	1	yes	characteristic
202	C₄	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
203	C₄	D₄×C₅₉⋊C₈	1	1	yes	characteristic
204	C₄	#16837533	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#21424815	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14273702
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1772	C₂
socle	C₂×C₅₉	#1939
prime radical	C₂³×C₅₉	#131
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1772	D₅₉
hypocenter	C₅₉	#13164
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
123. C₅₉	#13164	semidirect

Maximal subgroups

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

subgroup	index
#21425862	2
C₂×#16837223	2
C₄×#14273667	2
#21422182	2
C₅₉×#1772	2
#13164	59