Subgroups — #28506730

Order 15104.

318 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837228	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837177	C₂	1	1	yes	maximal, characteristic
3	#21424830	C₂	1	2	yes	maximal
4	#21422187	C₂	1	2	yes	maximal
5	C₂×C₅₉×#367	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14273661	C₂²	1	2	yes	—
8	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
9	#16837301	—	2	2	no	—
10	#16837228	—	2	2	no	—
11	#16837177	—	2	2	no	—
12	#16837164	—	2	2	no	—
13	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×#107×C₅₉	C₂²	1	2	yes	—
15	C₅₉×#367	C₄	1	4	yes	complemented
16	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₅₉	—	2	2	no	—
18	C₂×C₄⋊Dic₅₉	—	4	2	no	—
19	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
20	C₂×C₅₉⋊C₁₆	—	2	1	no	—
21	C₂×C₅₉⋊C₁₆	—	2	1	no	—
22	C₂×C₅₉⋊C₁₆	Q₈	1	2	yes	—
23	#14273661	—	2	2	no	—
24	#14273661	—	2	2	no	—
25	C₈×Dic₅₉	—	2	1	no	—
26	C₈×Dic₅₉	—	2	1	no	—
27	C₅₉⋊#95	—	4	2	no	—
28	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
30	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
32	C₂×C₁₆×C₅₉	Q₈	1	2	yes	—
33	#107×C₅₉	—	2	2	no	—
34	#107×C₅₉	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₅₉	—	4	2	no	—
36	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₅₉	—	4	2	no	—
38	C₂²×Dic₅₉	D₈	1	1	yes	characteristic
39	C₂²×Dic₅₉	D₈	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
45	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
46	C₄⋊C₄×C₅₉	—	2	4	no	—
47	C₄⋊C₄×C₅₉	—	4	2	no	—
48	C₄⋊Dic₅₉	—	4	2	no	—
49	C₄⋊Dic₅₉	—	4	2	no	—
50	C₄⋊Dic₅₉	—	8	4	no	—
51	C₄×Dic₅₉	—	2	1	no	—
52	C₄×Dic₅₉	—	2	1	no	—
53	C₂×C₅₉⋊C₈	—	2	1	no	—
54	C₂×C₅₉⋊C₈	—	2	1	no	—
55	C₂×C₅₉⋊C₈	Q₁₆	1	2	yes	—
56	C₁₆×C₅₉	C₄⋊C₄	1	4	yes	complemented
57	C₅₉⋊C₁₆	—	2	2	no	—
58	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₅₉	—	4	4	no	—
60	C₂×C₄×C₅₉	—	8	2	no	—
61	C₂×C₄×C₅₉	—	8	2	no	—
62	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
68	C₂×Dic₅₉	—	2	1	no	—
69	C₂×Dic₅₉	—	2	1	no	—
70	C₂×Dic₅₉	—	2	2	no	—
71	C₂×Dic₅₉	—	2	2	no	—
72	C₂×Dic₅₉	—	8	2	no	—
73	C₂×Dic₅₉	—	8	2	no	—
74	C₂×Dic₅₉	—	8	2	no	—
75	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
77	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
78	C₈×C₅₉	#125	1	1	yes	characteristic
79	C₅₉⋊C₈	—	2	2	no	—
80	#13167	—	59	1	no	maximal, Sylow
81	C₂²×C₅₉	C₄×D₈	1	1	yes	characteristic
82	C₂²×C₅₉	#466	1	1	yes	characteristic
83	C₂²×C₅₉	#500	1	1	yes	characteristic
84	C₂²×C₅₉	#509	1	2	yes	—
85	C₂²×C₅₉	#510	1	2	yes	—
86	C₄×C₅₉	—	8	4	no	—
87	C₄×C₅₉	#389	1	1	yes	characteristic
88	C₄×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
89	C₄×C₅₉	#467	1	1	yes	characteristic
90	C₄×C₅₉	#499	1	1	yes	characteristic
91	Dic₅₉	—	4	1	no	—
92	Dic₅₉	—	4	1	no	—
93	Dic₅₉	—	16	2	no	—
94	C₂×#367	D₅₉	1	1	yes	characteristic
95	C₂×#366	—	59	1	no	—
96	C₂×#346	—	59	1	no	—
97	#1942	—	59	2	no	—
98	#1380	—	59	2	no	—
99	C₂×C₅₉	#1942	1	1	yes	characteristic
100	C₂×C₅₉	#2175	1	2	yes	—
101	C₂×C₅₉	#2218	1	1	yes	characteristic
102	C₂×C₅₉	#2219	1	1	yes	characteristic
103	C₂×C₅₉	#2255	1	2	yes	—
104	C₂²×C₁₆	—	59	1	no	—
105	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
106	C₂×#107	—	59	2	no	—
107	C₂×#107	C₂×D₅₉	1	2	yes	—
108	C₂×C₄×C₈	—	59	1	no	—
109	#389	—	118	2	no	—
110	#367	Dic₅₉	1	4	yes	complemented
111	#366	—	118	2	no	—
112	#346	—	118	2	no	—
113	#340	—	118	2	no	—
114	C₅₉	#13167	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	59	1	no	—
116	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	118	2	no	—
119	C₂×C₄⋊C₄	—	236	2	no	—
120	C₂×C₄²	—	59	1	no	—
121	C₂×C₁₆	—	59	2	no	—
122	C₂×C₁₆	—	118	1	no	—
123	C₂×C₁₆	—	118	1	no	—
124	C₂×C₁₆	Dic₁₁₈	1	2	yes	—
125	C₂×C₁₆	D₂₃₆	1	2	yes	—
126	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
127	#107	—	2	2	no	—
128	#107	—	118	2	no	—
129	#107	—	118	2	no	—
130	#107	C₂×Dic₅₉	1	4	yes	—
131	C₄×C₈	—	118	1	no	—
132	C₄×C₈	—	118	1	no	—
133	#95	—	236	2	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	59	1	no	—
136	C₂²×C₄	—	59	1	no	—
137	C₂²×C₄	—	236	2	no	—
138	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
139	C₂×C₈	—	59	2	no	—
140	C₂×C₈	—	118	1	no	—
141	C₂×C₈	—	118	1	no	—
142	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
143	C₂×C₈	C₂×D₂₃₆	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	—	2	4	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	472	4	no	—
151	C₄⋊C₄	—	236	2	no	—
152	C₄⋊C₄	—	236	2	no	—
153	C₄²	—	118	1	no	—
154	C₄²	—	118	1	no	—
155	C₁₆	—	118	2	no	—
156	C₁₆	C₄⋊Dic₅₉	1	4	yes	complemented
157	C₂³	D₈×D₅₉	1	1	yes	characteristic
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	118	1	no	—
162	C₂×C₄	—	118	1	no	—
163	C₂×C₄	—	118	2	no	—
164	C₂×C₄	—	118	2	no	—
165	C₂×C₄	—	472	2	no	—
166	C₂×C₄	—	472	2	no	—
167	C₂×C₄	—	472	2	no	—
168	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
169	C₂×C₄	#14273702	1	1	yes	characteristic
170	C₂×C₄	#14273726	1	1	yes	characteristic
171	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
172	C₂×C₄	#14273746	1	1	yes	characteristic
173	C₂×C₄	#14273750	1	1	yes	characteristic
174	C₈	—	118	2	no	—
175	C₈	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
176	C₈	Q₈×Dic₅₉	1	1	yes	characteristic
177	C₈	#14273707	1	1	yes	characteristic
178	C₈	#14273718	1	1	yes	characteristic
179	C₂²	D₈×Dic₅₉	1	1	yes	characteristic
180	C₂²	#16837639	1	1	yes	characteristic
181	C₂²	#16837806	1	1	yes	characteristic
182	C₂²	#16837839	1	2	yes	—
183	C₂²	#16837841	1	2	yes	—
184	C₄	—	8	4	no	—
185	C₄	—	944	2	no	—
186	C₄	—	236	1	no	—
187	C₄	—	236	1	no	—
188	C₄	#16837301	1	1	yes	characteristic
189	C₄	Q₁₆×Dic₅₉	1	1	yes	characteristic
190	C₄	#16837644	1	1	yes	characteristic
191	C₄	#16837803	1	1	yes	characteristic
192	C₂	#21424830	1	1	yes	characteristic
193	C₂	#21425961	1	2	yes	—
194	C₂	#21426200	1	1	yes	characteristic
195	C₂	#21426205	1	1	yes	characteristic
196	C₂	#21426381	1	2	yes	—

	subgroup	G/H
center	C₂³	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#367	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#367	D₅₉
hypocenter	C₅₉	#13167
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
15. C₅₉×#367	C₄	semidirect
56. C₁₆×C₅₉	C₄⋊C₄	semidirect
110. #367	Dic₅₉	semidirect
114. C₅₉	#13167	semidirect
156. C₁₆	C₄⋊Dic₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837228	2
C₂×#16837177	2
#21424830	2
#21422187	2
C₂×C₅₉×#367	2
#13167	59