Subgroups — C₂×#21422231 #28508873

Order 15104.

222 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837236	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837177	C₂	1	1	yes	maximal, characteristic
3	#21422231	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₉×M₆₄	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₈×Dic₅₉	C₄	1	1	yes	characteristic
7	#16837236	C₄	1	2	yes	complemented
8	#16837236	C₂²	1	2	yes	—
9	C₂×C₅₉⋊C₃₂	C₄	1	2	yes	complemented
10	#16837177	C₄	1	2	yes	—
11	#16837177	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×M₆₄	C₄	1	2	yes	complemented
14	C₅₉×M₆₄	C₂²	1	2	yes	—
15	C₂×C₃₂×C₅₉	C₄	1	2	yes	complemented
16	C₂²×C₅₉⋊C₈	C₈	1	1	yes	characteristic
17	C₂×C₄×Dic₅₉	C₈	1	1	yes	characteristic
18	C₂×C₅₉⋊C₁₆	C₈	1	2	yes	—
19	C₂×C₅₉⋊C₁₆	C₈	1	2	yes	—
20	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₈×Dic₅₉	C₈	1	2	yes	—
22	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
23	C₅₉⋊C₃₂	C₂×C₄	1	4	yes	complemented
24	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
31	C₃₂×C₅₉	C₂×C₄	1	4	yes	complemented
32	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
33	C₂²×Dic₅₉	—	2	1	no	—
34	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	C₄×Dic₅₉	—	2	1	no	—
41	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
42	C₂×C₅₉⋊C₈	—	2	1	no	—
43	C₂×C₅₉⋊C₈	—	2	1	no	—
44	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
45	C₁₆×C₅₉	C₄²	1	1	yes	characteristic
46	C₁₆×C₅₉	C₄²	1	1	yes	characteristic
47	C₁₆×C₅₉	C₂²×C₄	1	1	yes	characteristic
48	C₁₆×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	C₅₉⋊C₁₆	C₂×C₈	1	4	yes	—
50	C₂³×C₅₉	M₃₂	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₅₉	M₃₂	1	1	yes	characteristic
54	C₂×C₄×C₅₉	M₃₂	1	1	yes	characteristic
55	C₂×C₄×C₅₉	M₃₂	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
57	C₂×Dic₅₉	—	2	2	no	—
58	C₂×Dic₅₉	—	4	2	no	—
59	C₈×C₅₉	C₄×C₈	1	1	yes	characteristic
60	C₈×C₅₉	C₄×C₈	1	1	yes	characteristic
61	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
62	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
63	C₅₉⋊C₈	—	2	2	no	—
64	C₂×#1398	—	59	1	no	maximal, Sylow
65	C₂²×C₅₉	—	2	2	no	—
66	C₂²×C₅₉	#346	1	1	yes	characteristic
67	C₂²×C₅₉	#346	1	1	yes	characteristic
68	C₂²×C₅₉	C₂×M₃₂	1	1	yes	characteristic
69	C₄×C₅₉	#346	1	1	yes	characteristic
70	C₄×C₅₉	#346	1	1	yes	characteristic
71	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
72	C₄×C₅₉	C₂×M₃₂	1	1	yes	characteristic
73	Dic₅₉	—	4	2	no	—
74	C₂×M₆₄	—	59	1	no	—
75	C₂×M₆₄	D₅₉	1	1	yes	characteristic
76	C₂×#346	—	59	1	no	—
77	#1398	—	59	4	no	—
78	C₂×C₅₉	—	2	1	no	—
79	C₂×C₅₉	—	2	1	no	—
80	C₂×C₅₉	#1398	1	2	yes	complemented
81	C₂×C₅₉	C₂×#346	1	1	yes	characteristic
82	C₂²×C₁₆	—	59	1	no	—
83	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
84	C₂×C₄×C₈	—	59	1	no	—
85	M₆₄	—	59	2	no	—
86	M₆₄	—	59	2	no	—
87	M₆₄	Dic₅₉	1	2	yes	complemented
88	M₆₄	C₂×D₅₉	1	2	yes	—
89	C₂×C₃₂	—	59	2	no	—
90	C₂×C₃₂	Dic₅₉	1	2	yes	complemented
91	#346	—	59	2	no	—
92	#346	—	59	2	no	—
93	C₅₉	C₂×#1398	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	—	59	1	no	—
95	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
96	C₂×C₄²	—	59	1	no	—
97	C₂×C₁₆	—	59	2	no	—
98	C₂×C₁₆	—	59	2	no	—
99	C₂×C₁₆	—	59	2	no	—
100	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
101	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
102	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
103	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
104	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
105	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
106	C₄×C₈	—	59	2	no	—
107	C₄×C₈	—	59	2	no	—
108	C₃₂	—	59	4	no	—
109	C₃₂	C₂×Dic₅₉	1	4	yes	complemented
110	C₂²×C₄	—	118	1	no	—
111	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
112	C₂×C₈	—	59	2	no	—
113	C₂×C₈	—	118	1	no	—
114	C₂×C₈	—	118	1	no	—
115	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
116	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
117	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
118	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
119	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
120	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
121	C₄²	—	59	2	no	—
122	C₄²	—	118	1	no	—
123	C₁₆	—	59	4	no	—
124	C₁₆	C₄×Dic₅₉	1	1	yes	characteristic
125	C₁₆	C₄×Dic₅₉	1	1	yes	characteristic
126	C₁₆	C₂×C₄×D₅₉	1	1	yes	characteristic
127	C₁₆	C₂²×Dic₅₉	1	1	yes	characteristic
128	C₂³	#14273666	1	1	yes	characteristic
129	C₂×C₄	—	118	2	no	—
130	C₂×C₄	—	236	2	no	—
131	C₂×C₄	C₈×Dic₅₉	1	1	yes	characteristic
132	C₂×C₄	C₈×Dic₅₉	1	1	yes	characteristic
133	C₂×C₄	#14273666	1	1	yes	characteristic
134	C₂×C₄	#14273666	1	1	yes	characteristic
135	C₂×C₄	#14273666	1	1	yes	characteristic
136	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
137	C₈	—	118	2	no	—
138	C₈	C₈×Dic₅₉	1	1	yes	characteristic
139	C₈	C₈×Dic₅₉	1	1	yes	characteristic
140	C₈	C₂×C₄×Dic₅₉	1	1	yes	characteristic
141	C₈	C₂×C₈×D₅₉	1	1	yes	characteristic
142	C₂²	—	2	2	no	—
143	C₂²	#16837177	1	1	yes	characteristic
144	C₂²	#16837177	1	1	yes	characteristic
145	C₂²	C₂×#14273666	1	1	yes	characteristic
146	C₄	—	236	2	no	—
147	C₄	#16837177	1	1	yes	characteristic
148	C₄	#16837177	1	1	yes	characteristic
149	C₄	C₂×C₈×Dic₅₉	1	1	yes	characteristic
150	C₄	C₂×#14273666	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	2	1	no	—
153	C₂	#21422231	1	2	yes	direct product factor
154	C₂	C₂×#16837177	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₅₉
commutator	C₄×C₅₉	C₂×C₄×C₈
Frattini	C₂×C₁₆	C₂²×D₅₉
Fitting	C₂×C₅₉×M₆₄	C₂
socle	C₂²×C₅₉	#346
prime radical	C₂³×C₅₉	M₃₂
cosocle	C₂×C₁₆×C₅₉	C₂³
hypercenter	C₂×M₆₄	D₅₉
hypocenter	C₅₉	C₂×#1398
LCS	G ▷ C₄×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
3. #21422231	C₂	direct (N × H)
7. #16837236	C₄	semidirect
9. C₂×C₅₉⋊C₃₂	C₄	semidirect
13. C₅₉×M₆₄	C₄	semidirect
15. C₂×C₃₂×C₅₉	C₄	semidirect
23. C₅₉⋊C₃₂	C₂×C₄	semidirect
31. C₃₂×C₅₉	C₂×C₄	semidirect
80. C₂×C₅₉	#1398	semidirect
87. M₆₄	Dic₅₉	semidirect
90. C₂×C₃₂	Dic₅₉	semidirect
93. C₅₉	C₂×#1398	semidirect
109. C₃₂	C₂×Dic₅₉	semidirect
153. C₂	#21422231	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×#16837236	2
C₂×#16837177	2
#21422231	2
C₂×C₅₉×M₆₄	2
C₂×#1398	59

Subgroups — C2×#21422231 #28508873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21422231 #28508873