Subgroups — #28503809

Order 15104.

382 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425717	C₂	1	1	yes	maximal, characteristic
2	#21425700	C₂	1	2	yes	maximal, complemented
3	#21423896	C₂	1	1	yes	maximal, characteristic
4	#21422184	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₉×#504	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₂²	1	2	yes	—
7	#16837386	C₄	1	1	yes	characteristic
8	#16837376	C₄	1	1	yes	characteristic
9	#16837357	C₂²	1	2	yes	—
10	#16837184	C₄	1	4	yes	complemented
11	#16837179	—	2	4	no	—
12	#16837173	C₄	1	2	yes	—
13	C₂×#131×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×#504	—	2	4	no	—
15	C₂×M₃₂×C₅₉	C₄	1	2	yes	complemented
16	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₅₉	C₄	1	2	yes	complemented
18	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×#11745664	C₂×C₄	1	2	yes	—
20	#14273686	C₂×C₄	1	2	yes	—
21	#14273667	—	2	2	no	—
22	#14273667	C₂×C₄	1	4	yes	—
23	C₂×C₅₉⋊C₁₆	—	2	2	no	—
24	#14273646	—	2	1	no	—
25	#14273644	—	2	1	no	—
26	#14273642	—	2	2	no	—
27	#14273637	—	2	2	no	—
28	C₈×Dic₅₉	—	2	2	no	—
29	C₂×#41×C₅₉	C₂×C₄	1	1	yes	characteristic
30	#131×C₅₉	—	2	2	no	—
31	#131×C₅₉	—	2	2	no	—
32	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
36	M₃₂×C₅₉	—	2	4	no	—
37	M₃₂×C₅₉	D₄	1	4	yes	—
38	C₂×C₁₆×C₅₉	—	2	2	no	—
39	C₂×C₁₆×C₅₉	—	2	4	no	—
40	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
43	#41×C₅₉	—	2	2	no	—
44	#41×C₅₉	—	2	2	no	—
45	C₂×Q₈×C₅₉	M₁₆	1	1	yes	characteristic
46	C₂×D₄×C₅₉	M₁₆	1	2	yes	—
47	C₂×D₄×C₅₉	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₅₉	C₄²	1	2	yes	—
49	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	M₁₆×C₅₉	—	2	2	no	—
51	M₁₆×C₅₉	—	2	4	no	—
52	C₂×C₈×C₅₉	—	2	2	no	—
53	C₂×C₈×C₅₉	—	2	2	no	—
54	C₂×C₈×C₅₉	C₄²	1	2	yes	—
55	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₅₉	#41	1	2	yes	—
59	C₂²⋊Dic₅₉	—	2	2	no	—
60	C₄⋊Dic₅₉	—	2	2	no	—
61	C₄×Dic₅₉	—	2	2	no	—
62	#11745664	—	2	2	no	—
63	#11745664	—	2	2	no	—
64	C₂×C₅₉⋊C₈	—	2	2	no	—
65	C₁₆×C₅₉	—	2	1	no	—
66	C₁₆×C₅₉	—	2	1	no	—
67	C₁₆×C₅₉	C₂²⋊C₄	1	4	yes	—
68	C₅₉⋊C₁₆	—	2	4	no	—
69	C₂³×C₅₉	#97	1	2	yes	—
70	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
71	Q₈×C₅₉	—	2	2	no	—
72	D₄×C₅₉	—	2	4	no	—
73	D₄×C₅₉	—	2	2	no	—
74	C₂×C₄×C₅₉	—	2	2	no	—
75	C₂×C₄×C₅₉	—	2	2	no	—
76	C₂×C₄×C₅₉	#97	1	2	yes	—
77	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅₉	#131	1	2	yes	—
81	C₂×Dic₅₉	—	4	2	no	—
82	C₈×C₅₉	—	2	2	no	—
83	C₈×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₈×C₅₉	#117	1	1	yes	characteristic
85	C₈×C₅₉	C₄×D₄	1	2	yes	—
86	C₅₉⋊C₈	—	4	2	no	—
87	#12617	—	59	1	no	maximal, Sylow
88	C₂²×C₅₉	—	2	2	no	—
89	C₂²×C₅₉	—	4	1	no	—
90	C₂²×C₅₉	—	4	2	no	—
91	C₂²×C₅₉	C₂×#97	1	1	yes	characteristic
92	C₂²×C₅₉	#435	1	2	yes	—
93	C₄×C₅₉	—	2	2	no	—
94	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₅₉	#405	1	1	yes	characteristic
96	C₄×C₅₉	#435	1	2	yes	—
97	Dic₅₉	—	8	2	no	—
98	C₂×#504	D₅₉	1	1	yes	characteristic
99	#2115	—	59	1	no	—
100	#2110	—	59	2	no	—
101	#1757	—	59	1	no	—
102	#1379	—	59	2	no	—
103	C₂×C₅₉	—	2	2	no	—
104	C₂×C₅₉	—	2	1	no	—
105	C₂×C₅₉	—	4	2	no	—
106	C₂×C₅₉	#1752	1	1	yes	characteristic
107	C₂×#131	C₂×D₅₉	1	1	yes	characteristic
108	#504	—	2	4	no	—
109	C₂×M₃₂	—	59	2	no	—
110	C₂×M₃₂	Dic₅₉	1	2	yes	complemented
111	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
112	C₂²×C₁₆	Dic₅₉	1	2	yes	complemented
113	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
114	#413	—	59	1	no	—
115	#410	—	59	1	no	—
116	#405	—	59	2	no	—
117	#349	—	59	4	no	—
118	#347	—	118	4	no	—
119	#344	—	59	2	no	—
120	C₅₉	#12617	1	1	yes	characteristic, complemented, Sylow
121	C₂×#41	C₄×D₅₉	1	1	yes	characteristic
122	#131	—	2	2	no	—
123	#131	—	2	2	no	—
124	C₂×M₁₆	—	59	2	no	—
125	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
126	C₂×M₁₆	C₄×D₅₉	1	2	yes	—
127	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
128	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
129	#117	—	59	2	no	—
130	M₃₂	—	2	4	no	—
131	M₃₂	—	59	4	no	—
132	M₃₂	—	118	2	no	—
133	M₃₂	C₂²⋊D₅₉	1	4	yes	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	4	no	—
136	C₂×C₁₆	—	118	2	no	—
137	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
138	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
139	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
140	#101	—	118	1	no	—
141	#100	—	118	1	no	—
142	#99	—	118	2	no	—
143	#97	—	118	2	no	—
144	C₄×C₈	—	118	2	no	—
145	#41	—	2	2	no	—
146	#41	—	2	2	no	—
147	C₂×Q₈	#11745659	1	1	yes	characteristic
148	C₂×D₄	#11745659	1	2	yes	—
149	C₂×D₄	#11745659	1	1	yes	characteristic
150	C₂²×C₄	C₄×Dic₅₉	1	2	yes	—
151	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	2	4	no	—
154	M₁₆	—	118	2	no	—
155	M₁₆	—	118	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	118	2	no	—
159	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
160	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
161	C₂×C₈	#11745685	1	2	yes	—
162	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
163	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
164	C₄⋊C₄	—	118	2	no	—
165	C₂²⋊C₄	—	118	2	no	—
166	C₄²	—	118	2	no	—
167	C₁₆	—	2	1	no	—
168	C₁₆	—	2	1	no	—
169	C₁₆	—	118	4	no	—
170	C₁₆	C₂²⋊Dic₅₉	1	4	yes	—
171	C₂³	#14273637	1	2	yes	—
172	C₂³	C₂×#11745659	1	1	yes	characteristic
173	Q₈	—	2	2	no	—
174	D₄	—	2	4	no	—
175	D₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	236	2	no	—
179	C₂×C₄	#14273637	1	2	yes	—
180	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
181	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
182	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
183	C₂×C₄	#14273733	1	2	yes	—
184	C₈	—	2	2	no	—
185	C₈	—	236	2	no	—
186	C₈	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
187	C₈	#14273686	1	1	yes	characteristic
188	C₈	C₄×C₂²⋊D₅₉	1	2	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	1	no	—
191	C₂²	—	4	2	no	—
192	C₂²	C₂×#14273637	1	1	yes	characteristic
193	C₂²	#16837473	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	472	2	no	—
196	C₄	C₄×C₂²⋊Dic₅₉	1	1	yes	characteristic
197	C₄	#16837357	1	1	yes	characteristic
198	C₄	#16837473	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	2	1	no	—
201	C₂	—	4	2	no	—
202	C₂	#21423872	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14273686
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#504	C₂
socle	C₂×C₅₉	#1752
prime radical	C₂×#41×C₅₉	C₂×C₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#504	D₅₉
hypocenter	C₅₉	#12617
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
2. #21425700	C₂	semidirect
4. #21422184	C₂	semidirect
10. #16837184	C₄	semidirect
15. C₂×M₃₂×C₅₉	C₄	semidirect
17. C₂²×C₁₆×C₅₉	C₄	semidirect
110. C₂×M₃₂	Dic₅₉	semidirect
112. C₂²×C₁₆	Dic₅₉	semidirect
120. C₅₉	#12617	semidirect

Maximal subgroups

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

subgroup	index
#21425717	2
#21425700	2
#21423896	2
#21422184	2
C₂×C₅₉×#504	2
#12617	59