Subgroups — C₂×C₅₉×#2244 #28445808

Order 15104.

686 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	C₂×C₅₉×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₉×#493	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2244	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅₉×#357	C₂	1	4	yes	maximal, complemented
5	C₂×C₅₉×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₅₉	—	2	1	no	—
7	C₂²×D₈×C₅₉	C₂²	1	2	yes	—
8	C₂×#127×C₅₉	—	2	1	no	—
9	C₂×#125×C₅₉	—	2	1	no	—
10	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
11	C₅₉×#499	—	2	2	no	—
12	C₅₉×#499	C₂²	1	4	yes	complemented
13	C₅₉×#493	—	2	2	no	—
14	C₅₉×#493	C₂²	1	4	yes	—
15	C₂×#107×C₅₉	C₂²	1	2	yes	—
16	C₂×#106×C₅₉	—	2	1	no	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×#357	C₂²	1	16	yes	complemented
19	C₅₉×#346	C₂²	1	4	yes	complemented
20	C₂²×D₄×C₅₉	—	2	2	no	—
21	C₂²×D₄×C₅₉	—	4	2	no	—
22	C₂×D₈×C₅₉	—	2	2	no	—
23	C₂×D₈×C₅₉	—	2	4	no	—
24	C₂×D₈×C₅₉	—	2	8	no	—
25	C₂×D₈×C₅₉	—	4	4	no	—
26	C₂×D₈×C₅₉	C₂³	1	4	yes	—
27	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	#127×C₅₉	—	2	4	no	—
30	#127×C₅₉	—	4	2	no	—
31	#125×C₅₉	—	2	4	no	—
32	#125×C₅₉	—	4	2	no	—
33	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₉	—	4	2	no	—
35	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₅₉	—	2	4	no	—
37	C₂×C₁₆×C₅₉	C₂³	1	4	yes	—
38	#107×C₅₉	—	2	2	no	—
39	#107×C₅₉	C₂³	1	4	yes	—
40	#106×C₅₉	—	2	4	no	—
41	C₄×C₈×C₅₉	D₄	1	2	yes	—
42	C₄×C₈×C₅₉	C₂³	1	2	yes	—
43	C₂⁴×C₅₉	—	4	2	no	—
44	C₂×D₄×C₅₉	—	2	4	no	—
45	C₂×D₄×C₅₉	—	4	4	no	—
46	C₂×D₄×C₅₉	—	4	8	no	—
47	C₂×D₄×C₅₉	—	4	2	no	—
48	C₂×D₄×C₅₉	—	4	4	no	—
49	C₂×D₄×C₅₉	—	8	8	no	—
50	C₂²×C₄×C₅₉	—	4	2	no	—
51	C₂²×C₄×C₅₉	D₈	1	2	yes	—
52	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	D₈×C₅₉	—	2	8	no	—
54	D₈×C₅₉	—	4	4	no	—
55	D₈×C₅₉	—	4	8	no	—
56	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
57	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₅₉	#41	1	4	yes	—
60	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₅₉	—	2	4	no	—
62	C₄⋊C₄×C₅₉	—	4	2	no	—
63	C₄⋊C₄×C₅₉	—	4	8	no	—
64	C₄²×C₅₉	D₈	1	2	yes	—
65	C₄²×C₅₉	C₂×D₄	1	2	yes	—
66	C₁₆×C₅₉	—	2	4	no	—
67	C₂³×C₅₉	—	4	4	no	—
68	C₂³×C₅₉	—	8	4	no	—
69	C₂³×C₅₉	—	8	8	no	—
70	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
71	D₄×C₅₉	—	4	4	no	—
72	D₄×C₅₉	—	4	4	no	—
73	D₄×C₅₉	—	8	16	no	—
74	D₄×C₅₉	—	8	4	no	—
75	C₂×C₄×C₅₉	—	2	4	no	—
76	C₂×C₄×C₅₉	—	4	4	no	—
77	C₂×C₄×C₅₉	—	8	4	no	—
78	C₂×C₄×C₅₉	#124	1	2	yes	—
79	C₂×C₄×C₅₉	C₂×D₈	1	4	yes	—
80	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
83	C₈×C₅₉	#124	1	2	yes	—
84	C₈×C₅₉	C₂×SD₁₆	1	4	yes	—
85	C₈×C₅₉	C₂×#41	1	2	yes	—
86	C₂×#2244	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₅₉	—	8	4	no	—
88	C₂²×C₅₉	—	8	8	no	—
89	C₂²×C₅₉	—	16	8	no	—
90	C₂²×C₅₉	#486	1	2	yes	—
91	C₂²×C₅₉	#509	1	4	yes	—
92	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
93	C₄×C₅₉	—	2	4	no	—
94	C₄×C₅₉	—	8	4	no	—
95	C₄×C₅₉	#486	1	2	yes	—
96	C₄×C₅₉	C₂×#124	1	1	yes	characteristic
97	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
98	C₂×#499	C₂×C₅₉	1	1	yes	characteristic, complemented
99	C₂×#493	C₂×C₅₉	1	1	yes	characteristic
100	#2244	C₂×C₅₉	1	8	yes	direct product factor
101	C₂×#357	C₂×C₅₉	1	4	yes	complemented
102	C₂×#346	C₂×C₅₉	1	1	yes	characteristic, complemented
103	C₂×C₅₉	—	16	4	no	—
104	C₂×C₅₉	#2244	1	4	yes	direct product factor
105	C₂×C₅₉	C₂×#486	1	1	yes	characteristic
106	C₂×C₅₉	C₂×#509	1	2	yes	—
107	C₂²×D₈	—	2	1	no	—
108	C₂²×D₈	C₂²×C₅₉	1	2	yes	—
109	C₂×#127	—	2	1	no	—
110	C₂×#125	—	2	1	no	—
111	C₂²×C₁₆	C₂²×C₅₉	1	2	yes	—
112	#499	—	2	2	no	—
113	#499	C₂²×C₅₉	1	4	yes	complemented
114	#493	—	2	2	no	—
115	#493	C₂²×C₅₉	1	4	yes	—
116	C₂×#107	C₂²×C₅₉	1	2	yes	—
117	C₂×#106	—	2	1	no	—
118	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
119	#357	C₂²×C₅₉	1	16	yes	complemented
120	#346	C₂²×C₅₉	1	4	yes	complemented
121	C₅₉	C₂×#2244	1	1	yes	characteristic, direct product factor, Sylow
122	C₂²×D₄	—	2	2	no	—
123	C₂²×D₄	—	4	2	no	—
124	C₂×D₈	—	2	4	no	—
125	C₂×D₈	—	2	2	no	—
126	C₂×D₈	—	2	8	no	—
127	C₂×D₈	—	4	4	no	—
128	C₂×D₈	C₂³×C₅₉	1	4	yes	—
129	C₂²×C₈	D₄×C₅₉	1	1	yes	characteristic
130	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
131	#127	—	2	4	no	—
132	#127	—	4	2	no	—
133	#125	—	2	4	no	—
134	#125	—	4	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	4	2	no	—
137	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
138	C₂×C₁₆	—	2	4	no	—
139	C₂×C₁₆	C₂³×C₅₉	1	4	yes	—
140	#107	—	2	2	no	—
141	#107	C₂³×C₅₉	1	4	yes	—
142	#106	—	2	4	no	—
143	C₄×C₈	D₄×C₅₉	1	2	yes	—
144	C₄×C₈	C₂³×C₅₉	1	2	yes	—
145	C₂⁴	—	4	2	no	—
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	—	4	4	no	—
148	C₂×D₄	—	4	8	no	—
149	C₂×D₄	—	4	2	no	—
150	C₂×D₄	—	4	4	no	—
151	C₂×D₄	—	8	8	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	D₈×C₅₉	1	2	yes	—
154	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
155	D₈	—	2	8	no	—
156	D₈	—	4	4	no	—
157	D₈	—	4	8	no	—
158	C₂×C₈	SD₁₆×C₅₉	1	4	yes	—
159	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
160	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
161	C₂×C₈	#41×C₅₉	1	4	yes	—
162	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	8	no	—
166	C₄²	D₈×C₅₉	1	2	yes	—
167	C₄²	C₂×D₄×C₅₉	1	2	yes	—
168	C₁₆	—	2	4	no	—
169	C₂³	—	4	4	no	—
170	C₂³	—	8	4	no	—
171	C₂³	—	8	8	no	—
172	C₂³	C₂×D₈×C₅₉	1	1	yes	characteristic
173	D₄	—	4	4	no	—
174	D₄	—	4	4	no	—
175	D₄	—	8	16	no	—
176	D₄	—	8	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	4	no	—
180	C₂×C₄	#124×C₅₉	1	2	yes	—
181	C₂×C₄	C₂×D₈×C₅₉	1	4	yes	—
182	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
183	C₂×C₄	C₂×SD₁₆×C₅₉	1	2	yes	—
184	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
185	C₈	#124×C₅₉	1	2	yes	—
186	C₈	C₂×SD₁₆×C₅₉	1	4	yes	—
187	C₈	C₂×#41×C₅₉	1	2	yes	—
188	C₂²	—	8	4	no	—
189	C₂²	—	8	8	no	—
190	C₂²	—	16	8	no	—
191	C₂²	C₅₉×#486	1	2	yes	—
192	C₂²	C₅₉×#509	1	4	yes	—
193	C₂²	C₂²×D₈×C₅₉	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	8	4	no	—
196	C₄	C₅₉×#486	1	2	yes	—
197	C₄	C₂×#124×C₅₉	1	1	yes	characteristic
198	C₄	C₂²×SD₁₆×C₅₉	1	1	yes	characteristic
199	C₂	—	16	4	no	—
200	C₂	C₅₉×#2244	1	4	yes	direct product factor
201	C₂	C₂×C₅₉×#486	1	1	yes	characteristic
202	C₂	C₂×C₅₉×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₉
prime radical	C₂×C₅₉×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₂×C₄²×C₅₉ ◁ C₂×C₄×C₈×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₉×#499	C₂	semidirect
3. C₅₉×#2244	C₂	direct (N × H)
4. C₂×C₅₉×#357	C₂	semidirect
5. C₂×C₅₉×#346	C₂	semidirect
12. C₅₉×#499	C₂²	semidirect
18. C₅₉×#357	C₂²	semidirect
19. C₅₉×#346	C₂²	semidirect
86. C₂×#2244	C₅₉	direct (N × H)
98. C₂×#499	C₂×C₅₉	semidirect
100. #2244	C₂×C₅₉	direct (N × H)
101. C₂×#357	C₂×C₅₉	semidirect
102. C₂×#346	C₂×C₅₉	semidirect
104. C₂×C₅₉	#2244	direct (N × H)
113. #499	C₂²×C₅₉	semidirect
119. #357	C₂²×C₅₉	semidirect
120. #346	C₂²×C₅₉	semidirect
121. C₅₉	C₂×#2244	direct (N × H)
200. C₂	C₅₉×#2244	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₉×#499	2
C₂×C₅₉×#493	2
C₅₉×#2244	2
C₂×C₅₉×#357	2
C₂×C₅₉×#346	2
C₂×#2244	59

Subgroups — C2×C59×#2244 #28445808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₉×#2244 #28445808