Subgroups — #28475888

Order 15104.

246 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425852	C₂	1	1	yes	maximal, characteristic
2	#21424253	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273667	—	2	1	no	—
5	C₂×#14273659	—	2	1	no	—
6	C₂×#14273659	C₄	1	2	yes	—
7	#16837272	—	2	1	no	—
8	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₅₉⋊C₁₆)	C₄	1	2	yes	complemented
10	C₅₉×#431	—	2	1	no	—
11	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₅₉×#335	C₄	1	2	yes	complemented
13	C₅₉×#334	—	2	1	no	—
14	C₂×C₄⋊Dic₅₉	—	2	2	no	—
15	C₂×C₄⋊Dic₅₉	—	4	1	no	—
16	C₂×C₄⋊Dic₅₉	—	4	1	no	—
17	#14273667	—	4	1	no	—
18	#14273667	—	4	1	no	—
19	C₂×C₅₉⋊C₁₆	—	2	2	no	—
20	#14273659	—	4	1	no	—
21	#14273659	—	4	1	no	—
22	#14273659	—	4	2	no	—
23	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
26	C₄⋊C₈×C₅₉	—	2	2	no	—
27	C₂²⋊C₈×C₅₉	—	4	1	no	—
28	C₄×C₈×C₅₉	—	4	1	no	—
29	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
36	C₂²×Dic₅₉	—	4	2	no	—
37	C₂×C₈×C₅₉	—	2	1	no	—
38	C₂×C₈×C₅₉	—	2	1	no	—
39	C₂×C₈×C₅₉	—	4	2	no	—
40	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
45	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₅₉	Q₁₆	1	1	yes	characteristic
48	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
51	C₄²×C₅₉	Q₁₆	1	1	yes	characteristic
52	C₄⋊Dic₅₉	—	4	2	no	—
53	C₄⋊Dic₅₉	—	4	2	no	—
54	C₄⋊Dic₅₉	—	8	2	no	—
55	C₄⋊Dic₅₉	—	8	2	no	—
56	C₅₉⋊C₁₆	—	4	2	no	—
57	C₂³×C₅₉	#102	1	1	yes	characteristic
58	C₂×C₄×C₅₉	—	2	2	no	—
59	C₂×C₄×C₅₉	—	2	2	no	—
60	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#100	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#101	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
68	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
69	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
70	C₂×Dic₅₉	—	8	2	no	—
71	C₂×Dic₅₉	—	8	2	no	—
72	C₂×Dic₅₉	—	8	2	no	—
73	C₈×C₅₉	—	2	1	no	—
74	C₈×C₅₉	—	2	1	no	—
75	C₈×C₅₉	—	8	2	no	—
76	C₈×C₅₉	#102	1	1	yes	characteristic
77	C₈×C₅₉	#103	1	1	yes	characteristic
78	C₈×C₅₉	#106	1	1	yes	characteristic
79	C₈×C₅₉	#107	1	1	yes	characteristic
80	#7278	—	59	1	no	maximal, Sylow
81	C₂²×C₅₉	—	2	1	no	—
82	C₂²×C₅₉	—	2	1	no	—
83	C₂²×C₅₉	#331	1	1	yes	characteristic
84	C₂²×C₅₉	#332	1	1	yes	characteristic
85	C₂²×C₅₉	#340	1	1	yes	characteristic
86	C₄×C₅₉	—	2	1	no	—
87	C₄×C₅₉	—	2	1	no	—
88	C₄×C₅₉	#332	1	1	yes	characteristic
89	C₄×C₅₉	#333	1	1	yes	characteristic
90	C₄×C₅₉	#340	1	1	yes	characteristic
91	C₄×C₅₉	#343	1	1	yes	characteristic
92	C₄×C₅₉	#360	1	2	yes	—
93	C₄×C₅₉	#365	1	2	yes	—
94	Dic₅₉	—	16	2	no	—
95	#2150	—	59	1	no	—
96	#1844	—	59	1	no	—
97	#1564	D₅₉	1	1	yes	characteristic
98	C₂×C₅₉	—	2	1	no	—
99	C₂×C₅₉	—	2	1	no	—
100	C₂×C₅₉	#1296	1	1	yes	characteristic
101	C₂×C₅₉	#1383	1	1	yes	characteristic
102	C₂×C₅₉	#1386	1	1	yes	characteristic
103	C₂×M₃₂	—	118	1	no	—
104	#431	—	2	1	no	—
105	C₂×#106	—	59	2	no	—
106	C₂×#106	—	118	1	no	—
107	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
108	#384	—	118	1	no	—
109	C₄⋊C₁₆	—	59	2	no	—
110	C₄⋊C₁₆	—	118	1	no	—
111	#335	Dic₅₉	1	2	yes	complemented
112	#334	—	2	1	no	—
113	C₅₉	#7278	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
115	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	118	2	no	—
117	C₂×C₄⋊C₄	—	236	1	no	—
118	C₂×C₄⋊C₄	—	236	1	no	—
119	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
120	M₃₂	—	236	1	no	—
121	M₃₂	—	236	1	no	—
122	C₂×C₁₆	—	118	2	no	—
123	#106	—	236	1	no	—
124	#106	—	236	1	no	—
125	#106	—	236	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₂²⋊C₈	—	4	1	no	—
128	C₄×C₈	—	4	1	no	—
129	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
130	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
131	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
132	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
133	C₂²×C₄	—	236	2	no	—
134	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
135	C₂²×C₄	#11745669	1	1	yes	characteristic
136	C₂²×C₄	#11745671	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	#11745660	1	1	yes	characteristic
141	C₂×C₈	#11745660	1	1	yes	characteristic
142	C₂×C₈	D₄₇₂	1	1	yes	characteristic
143	C₂×C₈	Dic₂₃₆	1	1	yes	characteristic
144	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
145	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
146	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
147	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
148	C₄⋊C₄	—	472	2	no	—
149	C₄⋊C₄	—	472	2	no	—
150	C₄⋊C₄	—	236	2	no	—
151	C₄⋊C₄	—	236	2	no	—
152	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
153	C₄²	#11745670	1	1	yes	characteristic
154	C₄²	#11745672	1	1	yes	characteristic
155	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
156	C₁₆	—	236	2	no	—
157	C₂³	#14273647	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	472	2	no	—
161	C₂×C₄	—	472	2	no	—
162	C₂×C₄	—	472	2	no	—
163	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
164	C₂×C₄	#14273643	1	1	yes	characteristic
165	C₂×C₄	#14273645	1	1	yes	characteristic
166	C₂×C₄	#14273648	1	1	yes	characteristic
167	C₂×C₄	#14273649	1	1	yes	characteristic
168	C₂×C₄	#14273650	1	1	yes	characteristic
169	C₂×C₄	#14273651	1	1	yes	characteristic
170	C₂×C₄	#14273652	1	1	yes	characteristic
171	C₂×C₄	#14273658	1	1	yes	characteristic
172	C₂×C₄	#14273660	1	1	yes	characteristic
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	8	2	no	—
176	C₈	#14273649	1	1	yes	characteristic
177	C₈	#14273652	1	1	yes	characteristic
178	C₈	#14273659	1	1	yes	characteristic
179	C₈	#14273661	1	1	yes	characteristic
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	#16837145	1	1	yes	characteristic
183	C₂²	#16837149	1	1	yes	characteristic
184	C₂²	#16837164	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	944	2	no	—
188	C₄	#16837147	1	1	yes	characteristic
189	C₄	#16837150	1	1	yes	characteristic
190	C₄	#16837165	1	1	yes	characteristic
191	C₄	#16837170	1	1	yes	characteristic
192	C₄	#16837212	1	2	yes	—
193	C₄	#16837226	1	2	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#21421980	1	1	yes	characteristic
197	C₂	#21422196	1	1	yes	characteristic
198	C₂	#21422201	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837164
commutator	C₂×C₈×C₅₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₉
Fitting	C₅₉×#1564	C₂
socle	C₂²×C₅₉	#340
prime radical	C₂³×C₅₉	#102
cosocle	C₂×C₄×C₈×C₅₉	C₂²
hypercenter	#1564	D₅₉
hypocenter	C₅₉	#7278
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
9. C₄⋊(C₅₉⋊C₁₆)	C₄	semidirect
12. C₅₉×#335	C₄	semidirect
111. #335	Dic₅₉	semidirect
113. C₅₉	#7278	semidirect

Maximal subgroups

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

subgroup	index
#21425852	2
#21424253	2
C₅₉×#1564	2
#7278	59