Subgroups — #28507299

Order 15104.

374 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837178	C₂	1	1	yes	maximal, characteristic
2	#21424256	C₂	1	1	yes	maximal, characteristic
3	#21422187	C₂	1	4	yes	maximal
4	C₂×C₅₉×#499	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	#16837178	—	2	2	no	—
10	C₂×#125×C₅₉	—	2	1	no	—
11	C₅₉×#499	—	2	2	no	—
12	C₅₉×#499	C₄	1	4	yes	complemented
13	C₂×#107×C₅₉	C₂²	1	2	yes	—
14	C₂×#106×C₅₉	—	2	1	no	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₅₉	—	2	2	no	—
17	C₂×C₄⋊Dic₅₉	—	4	2	no	—
18	C₂×C₅₉⋊C₁₆	—	2	2	no	—
19	C₂×C₅₉⋊C₁₆	—	2	2	no	—
20	C₂×C₅₉⋊C₁₆	Q₈	1	4	yes	—
21	#14273661	—	2	4	no	—
22	#14273661	—	4	2	no	—
23	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
25	#125×C₅₉	—	2	4	no	—
26	#125×C₅₉	—	4	2	no	—
27	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₉	—	4	2	no	—
29	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
30	#107×C₅₉	—	2	2	no	—
31	#107×C₅₉	C₂×C₄	1	4	yes	—
32	#106×C₅₉	—	2	4	no	—
33	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₉	D₄	1	2	yes	—
35	C₂²×C₄×C₅₉	—	4	2	no	—
36	C₂²×C₄×C₅₉	D₈	1	2	yes	—
37	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₅₉	—	4	2	no	—
39	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
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	2	yes	—
45	C₂×C₈×C₅₉	#41	1	2	yes	—
46	C₄⋊C₄×C₅₉	—	2	4	no	—
47	C₄⋊C₄×C₅₉	—	4	8	no	—
48	C₄⋊C₄×C₅₉	—	4	2	no	—
49	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
50	C₄²×C₅₉	SD₁₆	1	2	yes	—
51	C₄⋊Dic₅₉	—	4	2	no	—
52	C₄⋊Dic₅₉	—	4	2	no	—
53	C₄⋊Dic₅₉	—	8	4	no	—
54	C₅₉⋊C₁₆	—	2	4	no	—
55	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₅₉	—	2	2	no	—
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	—	4	4	no	—
59	C₂×C₄×C₅₉	—	8	2	no	—
60	C₂×C₄×C₅₉	—	8	2	no	—
61	C₂×C₄×C₅₉	#102	1	4	yes	—
62	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×Dic₅₉	—	8	2	no	—
69	C₂×Dic₅₉	—	8	4	no	—
70	C₈×C₅₉	#103	1	4	yes	—
71	C₈×C₅₉	C₄×Q₈	1	2	yes	—
72	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
73	#13284	—	59	1	no	maximal, Sylow
74	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
75	C₂²×C₅₉	#486	1	1	yes	characteristic
76	C₂²×C₅₉	#500	1	1	yes	characteristic
77	C₂²×C₅₉	#509	1	2	yes	—
78	C₂²×C₅₉	#510	1	2	yes	—
79	C₄×C₅₉	—	2	4	no	—
80	C₄×C₅₉	—	8	4	no	—
81	C₄×C₅₉	#391	1	1	yes	characteristic
82	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
83	C₄×C₅₉	#487	1	1	yes	characteristic
84	C₄×C₅₉	#498	1	1	yes	characteristic
85	Dic₅₉	—	16	2	no	—
86	C₂×#499	D₅₉	1	1	yes	characteristic
87	C₂×#346	—	59	1	no	—
88	#1845	—	59	1	no	—
89	#1380	—	59	4	no	—
90	C₂×C₅₉	#1990	1	1	yes	characteristic
91	C₂×C₅₉	#2139	1	2	yes	—
92	C₂×C₅₉	#2244	1	1	yes	characteristic
93	C₂×C₅₉	#2245	1	1	yes	characteristic
94	C₂×C₅₉	#2255	1	2	yes	—
95	C₂×#125	—	2	1	no	—
96	C₂²×C₁₆	—	59	2	no	—
97	#499	—	2	2	no	—
98	#499	Dic₅₉	1	4	yes	complemented
99	C₂×#107	—	59	2	no	—
100	C₂×#107	—	118	1	no	—
101	C₂×#107	C₂×D₅₉	1	2	yes	—
102	C₂×#106	—	2	1	no	—
103	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
104	#384	—	118	1	no	—
105	#346	—	118	2	no	—
106	C₅₉	#13284	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
109	#125	—	2	4	no	—
110	#125	—	4	2	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	4	2	no	—
113	C₂×C₄⋊C₄	—	118	2	no	—
114	C₂×C₄⋊C₄	—	236	2	no	—
115	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
116	C₂×C₁₆	—	59	4	no	—
117	C₂×C₁₆	—	118	2	no	—
118	C₂×C₁₆	—	118	2	no	—
119	#107	—	2	2	no	—
120	#107	—	118	4	no	—
121	#107	—	236	2	no	—
122	#107	C₂×Dic₅₉	1	4	yes	—
123	#106	—	2	4	no	—
124	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
125	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
126	C₂²×C₄	—	4	2	no	—
127	C₂²×C₄	—	236	2	no	—
128	C₂²×C₄	#11745669	1	2	yes	—
129	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
130	C₂×C₈	#11745671	1	2	yes	—
131	C₂×C₈	#11745672	1	2	yes	—
132	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
133	C₂×C₈	Q₈×D₅₉	1	2	yes	—
134	C₂×C₈	#11745689	1	2	yes	—
135	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
136	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
137	C₄⋊C₄	—	2	4	no	—
138	C₄⋊C₄	—	4	8	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	472	4	no	—
141	C₄⋊C₄	—	236	2	no	—
142	C₄⋊C₄	—	236	2	no	—
143	C₄²	#11745670	1	2	yes	—
144	C₄²	C₂²⋊Dic₅₉	1	2	yes	—
145	C₁₆	—	118	4	no	—
146	C₂³	C₂×#11745669	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	472	2	no	—
153	C₂×C₄	—	472	4	no	—
154	C₂×C₄	#14273648	1	4	yes	—
155	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
156	C₂×C₄	#14273716	1	1	yes	characteristic
157	C₂×C₄	#14273725	1	1	yes	characteristic
158	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
159	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
160	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
161	C₈	#14273651	1	4	yes	—
162	C₈	Q₈×Dic₅₉	1	2	yes	—
163	C₈	#14273703	1	2	yes	—
164	C₂²	C₂×#14273648	1	1	yes	characteristic
165	C₂²	#16837750	1	1	yes	characteristic
166	C₂²	#16837805	1	1	yes	characteristic
167	C₂²	#16837837	1	2	yes	—
168	C₂²	#16837842	1	2	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	8	4	no	—
171	C₄	—	944	2	no	—
172	C₄	#16837309	1	1	yes	characteristic
173	C₄	C₂×#14273651	1	1	yes	characteristic
174	C₄	#16837754	1	1	yes	characteristic
175	C₄	#16837798	1	1	yes	characteristic
176	C₂	#21425058	1	1	yes	characteristic
177	C₂	#21425808	1	2	yes	—
178	C₂	#21426339	1	1	yes	characteristic
179	C₂	#21426343	1	1	yes	characteristic
180	C₂	#21426383	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#499	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#499	D₅₉
hypocenter	C₅₉	#13284
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
12. C₅₉×#499	C₄	semidirect
98. #499	Dic₅₉	semidirect
106. C₅₉	#13284	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837178	2
#21424256	2
#21422187	2
C₂×C₅₉×#499	2
#13284	59