Subgroups — #28506734

Order 15104.

318 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837230	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837228	C₂	1	1	yes	maximal, characteristic
3	#21424829	C₂	1	2	yes	maximal
4	#21422188	C₂	1	2	yes	maximal
5	C₂×C₅₉×#346	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273661	C₂²	1	2	yes	—
7	C₂×#14273661	C₂²	1	2	yes	—
8	#16837300	—	2	2	no	—
9	#16837230	—	2	2	no	—
10	#16837228	C₄	1	4	yes	complemented
11	#16837165	—	2	2	no	—
12	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#346	—	2	2	no	—
16	C₂×C₄⋊Dic₅₉	—	2	2	no	—
17	C₂×C₄⋊Dic₅₉	—	2	2	no	—
18	C₂×Dic₅₉⋊C₄	—	4	2	no	—
19	#14273661	—	2	2	no	—
20	#14273661	—	2	2	no	—
21	#14273661	—	2	2	no	—
22	#14273661	C₂×C₄	1	4	yes	—
23	C₅₉⋊#95	—	4	2	no	—
24	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₉	—	2	1	no	—
28	C₂×C₁₆×C₅₉	—	2	1	no	—
29	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
31	C₂×C₁₆×C₅₉	Q₈	1	2	yes	—
32	C₂×C₁₆×C₅₉	Q₈	1	2	yes	—
33	C₄×C₈×C₅₉	—	2	1	no	—
34	C₄×C₈×C₅₉	—	2	1	no	—
35	C₂²×C₄×C₅₉	D₈	1	2	yes	—
36	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₅₉	—	4	2	no	—
38	C₂²×Dic₅₉	—	4	2	no	—
39	C₂×C₈×C₅₉	—	2	1	no	—
40	C₂×C₈×C₅₉	—	2	1	no	—
41	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
42	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
47	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
48	C₄²×C₅₉	—	2	1	no	—
49	C₄²×C₅₉	—	2	1	no	—
50	C₄⋊Dic₅₉	—	2	4	no	—
51	C₄⋊Dic₅₉	—	4	2	no	—
52	C₄⋊Dic₅₉	—	4	2	no	—
53	C₄⋊Dic₅₉	—	4	2	no	—
54	Dic₅₉⋊C₄	—	8	4	no	—
55	C₁₆×C₅₉	—	2	2	no	—
56	C₁₆×C₅₉	C₄⋊C₄	1	4	yes	complemented
57	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₅₉	—	2	2	no	—
59	C₂×C₄×C₅₉	—	2	2	no	—
60	C₂×C₄×C₅₉	—	2	2	no	—
61	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
67	C₂×Dic₅₉	—	4	4	no	—
68	C₂×Dic₅₉	—	8	2	no	—
69	C₂×Dic₅₉	—	8	2	no	—
70	C₂×Dic₅₉	—	8	2	no	—
71	C₂×Dic₅₉	—	8	2	no	—
72	C₂×Dic₅₉	—	8	2	no	—
73	C₈×C₅₉	—	2	2	no	—
74	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
76	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
77	C₈×C₅₉	#125	1	1	yes	characteristic
78	#13168	—	59	1	no	maximal, Sylow
79	C₂²×C₅₉	C₄×D₈	1	1	yes	characteristic
80	C₂²×C₅₉	#466	1	1	yes	characteristic
81	C₂²×C₅₉	#500	1	1	yes	characteristic
82	C₂²×C₅₉	#509	1	2	yes	—
83	C₂²×C₅₉	#510	1	2	yes	—
84	C₄×C₅₉	—	4	2	no	—
85	C₄×C₅₉	#389	1	1	yes	characteristic
86	C₄×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
87	C₄×C₅₉	#467	1	1	yes	characteristic
88	C₄×C₅₉	#499	1	1	yes	characteristic
89	Dic₅₉	—	8	4	no	—
90	Dic₅₉	—	16	2	no	—
91	C₂×#367	—	59	1	no	—
92	C₂×#366	—	59	1	no	—
93	C₂×#346	D₅₉	1	1	yes	characteristic
94	#1942	—	59	2	no	—
95	#1380	—	59	2	no	—
96	C₂×C₅₉	#1942	1	1	yes	characteristic
97	C₂×C₅₉	#2176	1	1	yes	characteristic
98	C₂×C₅₉	#2177	1	1	yes	characteristic
99	C₂×C₅₉	#2220	1	2	yes	—
100	C₂×C₅₉	#2255	1	2	yes	—
101	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
102	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
103	C₂×#107	—	59	2	no	—
104	C₂×#107	—	59	2	no	—
105	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
106	#389	—	118	2	no	—
107	#367	—	118	2	no	—
108	#366	—	59	4	no	—
109	#346	—	2	2	no	—
110	#340	—	118	2	no	—
111	C₅₉	#13168	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	118	2	no	—
115	C₂×C₄⋊C₄	—	118	2	no	—
116	C₂×C₄⋊C₄	—	236	2	no	—
117	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	Dic₁₁₈	1	2	yes	—
121	C₂×C₁₆	Dic₁₁₈	1	2	yes	—
122	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
123	C₂×C₁₆	C₂²⋊D₅₉	1	2	yes	—
124	#107	—	59	4	no	—
125	#107	—	118	2	no	—
126	#107	—	118	2	no	—
127	#107	—	118	2	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	—	2	1	no	—
130	#95	—	236	2	no	—
131	C₂²×C₄	—	236	2	no	—
132	C₂²×C₄	—	236	2	no	—
133	C₂²×C₄	D₄₇₂	1	2	yes	—
134	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	Dic₂₃₆	1	2	yes	—
138	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
139	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
140	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
141	C₂×C₈	#11745685	1	1	yes	characteristic
142	C₂×C₈	#11745685	1	1	yes	characteristic
143	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
144	C₄⋊C₄	—	118	4	no	—
145	C₄⋊C₄	—	472	4	no	—
146	C₄⋊C₄	—	236	2	no	—
147	C₄⋊C₄	—	236	2	no	—
148	C₄⋊C₄	—	236	2	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₁₆	—	2	2	no	—
152	C₁₆	Dic₅₉⋊C₄	1	4	yes	complemented
153	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	472	2	no	—
158	C₂×C₄	—	472	2	no	—
159	C₂×C₄	—	472	2	no	—
160	C₂×C₄	—	472	2	no	—
161	C₂×C₄	—	472	2	no	—
162	C₂×C₄	—	236	4	no	—
163	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
164	C₂×C₄	#14273701	1	1	yes	characteristic
165	C₂×C₄	#14273727	1	1	yes	characteristic
166	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
167	C₂×C₄	#14273749	1	1	yes	characteristic
168	C₂×C₄	#14273749	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
171	C₈	C₄×Dic₁₁₈	1	1	yes	characteristic
172	C₈	#14273706	1	1	yes	characteristic
173	C₈	#14273719	1	1	yes	characteristic
174	C₂²	C₄×D₄₇₂	1	1	yes	characteristic
175	C₂²	#16837638	1	1	yes	characteristic
176	C₂²	#16837807	1	1	yes	characteristic
177	C₂²	#16837840	1	2	yes	—
178	C₂²	#16837845	1	2	yes	—
179	C₄	—	4	2	no	—
180	C₄	—	944	2	no	—
181	C₄	—	472	4	no	—
182	C₄	#16837300	1	1	yes	characteristic
183	C₄	C₄×Dic₂₃₆	1	1	yes	characteristic
184	C₄	#16837643	1	1	yes	characteristic
185	C₄	#16837804	1	1	yes	characteristic
186	C₂	#21424829	1	1	yes	characteristic
187	C₂	#21425968	1	1	yes	characteristic
188	C₂	#21425973	1	1	yes	characteristic
189	C₂	#21426209	1	2	yes	—
190	C₂	#21426382	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#346	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#346	D₅₉
hypocenter	C₅₉	#13168
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
10. #16837228	C₄	semidirect
56. C₁₆×C₅₉	C₄⋊C₄	semidirect
111. C₅₉	#13168	semidirect
152. C₁₆	Dic₅₉⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837230	2
C₂×#16837228	2
#21424829	2
#21422188	2
C₂×C₅₉×#346	2
#13168	59