Subgroups — #28505068

Order 15104.

342 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	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#21424256	C₂	1	1	yes	maximal, characteristic
3	#21422187	C₂	1	4	yes	maximal
4	C₅₉×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14273661	—	2	1	no	—
7	C₂×#14273661	C₂²	1	2	yes	—
8	#16837272	—	2	1	no	—
9	C₄×C₅₉⋊C₁₆	C₄	1	4	yes	complemented
10	C₂×#107×C₅₉	—	2	1	no	—
11	C₂×#107×C₅₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#384	—	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	C₂×C₅₉⋊C₁₆	—	2	2	no	—
18	C₂×C₅₉⋊C₁₆	—	2	2	no	—
19	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
20	#14273661	—	2	4	no	—
21	#14273661	—	4	2	no	—
22	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
26	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
27	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
28	#107×C₅₉	—	2	4	no	—
29	#107×C₅₉	—	4	2	no	—
30	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₅₉	Q₈	1	2	yes	—
32	C₂²×C₄×C₅₉	—	4	2	no	—
33	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₅₉	—	4	2	no	—
37	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
39	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
40	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₅₉	#41	1	2	yes	—
43	C₂×C₈×C₅₉	#41	1	2	yes	—
44	C₄⋊C₄×C₅₉	—	4	2	no	—
45	C₄⋊C₄×C₅₉	—	4	2	no	—
46	C₄⋊C₄×C₅₉	—	8	2	no	—
47	C₄⋊C₄×C₅₉	—	8	2	no	—
48	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
49	C₄²×C₅₉	Q₁₆	1	2	yes	—
50	C₄⋊Dic₅₉	—	4	2	no	—
51	C₄⋊Dic₅₉	—	4	2	no	—
52	C₄⋊Dic₅₉	—	8	2	no	—
53	C₄⋊Dic₅₉	—	8	2	no	—
54	C₅₉⋊C₁₆	—	2	4	no	—
55	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₅₉	—	8	2	no	—
57	C₂×C₄×C₅₉	—	8	2	no	—
58	C₂×C₄×C₅₉	—	8	2	no	—
59	C₂×C₄×C₅₉	#107	1	2	yes	—
60	C₂×C₄×C₅₉	#107	1	2	yes	—
61	C₂×C₄×C₅₉	D₁₆	1	2	yes	—
62	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
63	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
64	C₂×C₄×C₅₉	Q₃₂	1	2	yes	—
65	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×Dic₅₉	—	8	2	no	—
72	C₂×Dic₅₉	—	8	2	no	—
73	C₂×Dic₅₉	—	8	2	no	—
74	C₈×C₅₉	#106	1	4	yes	—
75	C₈×C₅₉	#117	1	2	yes	—
76	C₈×C₅₉	#125	1	2	yes	—
77	#12888	—	59	1	no	maximal, Sylow
78	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
79	C₂²×C₅₉	#486	1	1	yes	characteristic
80	C₂²×C₅₉	#486	1	1	yes	characteristic
81	C₂²×C₅₉	C₂×D₁₆	1	1	yes	characteristic
82	C₂²×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
83	C₂²×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
84	C₂²×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
85	C₄×C₅₉	—	16	2	no	—
86	C₄×C₅₉	#366	1	4	yes	—
87	C₄×C₅₉	#367	1	4	yes	—
88	C₄×C₅₉	#382	1	1	yes	characteristic
89	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
90	C₄×C₅₉	#487	1	1	yes	characteristic
91	C₄×C₅₉	#487	1	1	yes	characteristic
92	Dic₅₉	—	16	2	no	—
93	C₂×C₄×C₁₆	—	59	1	no	—
94	#1845	—	59	1	no	—
95	#1845	D₅₉	1	1	yes	characteristic
96	#1380	—	59	4	no	—
97	C₂×C₅₉	#1834	1	1	yes	characteristic
98	C₂×C₅₉	C₂×#366	1	1	yes	characteristic
99	C₂×C₅₉	C₂×#367	1	1	yes	characteristic
100	C₂×C₅₉	#2240	1	1	yes	characteristic
101	C₂×C₅₉	#2240	1	1	yes	characteristic
102	C₂×C₅₉	#2241	1	1	yes	characteristic
103	C₂×C₅₉	#2241	1	1	yes	characteristic
104	C₂²×C₁₆	—	59	2	no	—
105	C₂×#107	—	2	1	no	—
106	C₂×#107	—	59	2	no	—
107	C₂×#107	—	118	1	no	—
108	C₂×#107	C₂×D₅₉	1	2	yes	—
109	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
110	#384	—	2	1	no	—
111	#384	—	118	1	no	—
112	C₄×C₁₆	—	59	4	no	—
113	C₅₉	#12888	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	4	1	no	—
118	C₂×C₄⋊C₄	—	4	1	no	—
119	C₂×C₄⋊C₄	—	118	2	no	—
120	C₂×C₄⋊C₄	—	236	1	no	—
121	C₂×C₄⋊C₄	—	236	1	no	—
122	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
123	C₂×C₁₆	—	59	4	no	—
124	C₂×C₁₆	—	118	2	no	—
125	C₂×C₁₆	—	118	2	no	—
126	#107	—	2	4	no	—
127	#107	—	4	2	no	—
128	#107	—	118	4	no	—
129	#107	—	236	2	no	—
130	C₄×C₈	Dic₁₁₈	1	2	yes	—
131	C₄×C₈	C₄×D₅₉	1	2	yes	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	—	236	2	no	—
134	C₂²×C₄	#11745669	1	1	yes	characteristic
135	C₂²×C₄	#11745669	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
137	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
138	C₂×C₈	#11745670	1	2	yes	—
139	C₂×C₈	#11745671	1	2	yes	—
140	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
141	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
142	C₂×C₈	#11745687	1	2	yes	—
143	C₂×C₈	#11745689	1	2	yes	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	8	2	no	—
147	C₄⋊C₄	—	8	2	no	—
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	2	yes	—
153	C₄²	#11745672	1	2	yes	—
154	C₁₆	—	118	4	no	—
155	C₂³	C₂×#11745669	1	1	yes	characteristic
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	472	2	no	—
160	C₂×C₄	—	472	2	no	—
161	C₂×C₄	—	472	2	no	—
162	C₂×C₄	#14273660	1	2	yes	—
163	C₂×C₄	#14273660	1	2	yes	—
164	C₂×C₄	#14273668	1	2	yes	—
165	C₂×C₄	#14273670	1	2	yes	—
166	C₂×C₄	#14273671	1	2	yes	—
167	C₂×C₄	#14273673	1	2	yes	—
168	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
169	C₂×C₄	#14273714	1	1	yes	characteristic
170	C₂×C₄	#14273716	1	1	yes	characteristic
171	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
172	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
173	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
174	C₈	#14273658	1	4	yes	—
175	C₈	#14273685	1	2	yes	—
176	C₈	#14273718	1	2	yes	—
177	C₂²	C₂×#14273660	1	1	yes	characteristic
178	C₂²	#16837748	1	1	yes	characteristic
179	C₂²	#16837750	1	1	yes	characteristic
180	C₂²	C₂×#14273668	1	1	yes	characteristic
181	C₂²	C₂×#14273670	1	1	yes	characteristic
182	C₂²	C₂×#14273671	1	1	yes	characteristic
183	C₂²	C₂×#14273673	1	1	yes	characteristic
184	C₄	—	16	2	no	—
185	C₄	—	944	2	no	—
186	C₄	#16837227	1	4	yes	—
187	C₄	#16837229	1	4	yes	—
188	C₄	#16837267	1	1	yes	characteristic
189	C₄	C₂×#14273658	1	1	yes	characteristic
190	C₄	#16837752	1	1	yes	characteristic
191	C₄	#16837754	1	1	yes	characteristic
192	C₂	#21424223	1	1	yes	characteristic
193	C₂	C₂×#16837227	1	1	yes	characteristic
194	C₂	C₂×#16837229	1	1	yes	characteristic
195	C₂	#21426319	1	1	yes	characteristic
196	C₂	#21426321	1	1	yes	characteristic
197	C₂	#21426323	1	1	yes	characteristic
198	C₂	#21426325	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1845	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1845	D₅₉
hypocenter	C₅₉	#12888
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
9. C₄×C₅₉⋊C₁₆	C₄	semidirect
113. C₅₉	#12888	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅₉⋊C₁₆	2
#21424256	2
#21422187	2
C₅₉×#1845	2
#12888	59