Subgroups — C₂×#21422170 #28500270

Order 15104.

262 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837155	C₂	1	2	yes	maximal
2	#21422170	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
5	#16837155	—	2	2	no	—
6	#16837155	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₅₉	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₅₉	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₅₉	C₄	1	2	yes	—
11	C₄×C₁₆×C₅₉	C₂²	1	2	yes	—
12	C₂²×C₅₉⋊C₈	—	4	2	no	—
13	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
14	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
15	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₉	—	2	2	no	—
19	C₂×C₁₆×C₅₉	C₈	1	4	yes	complemented
20	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
24	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
28	C₂²×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
29	C₂×C₈×C₅₉	—	2	2	no	—
30	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
33	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
34	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₅₉	D₈	1	1	yes	characteristic
38	C₄²×C₅₉	Q₁₆	1	1	yes	characteristic
39	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₅₉⋊C₈	—	4	4	no	—
41	C₂×C₅₉⋊C₈	—	8	4	no	—
42	C₁₆×C₅₉	—	2	2	no	—
43	C₁₆×C₅₉	C₂×C₈	1	4	yes	complemented
44	C₂³×C₅₉	#107	1	1	yes	characteristic
45	C₂×C₄×C₅₉	—	2	2	no	—
46	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
47	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
48	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
49	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
50	C₂×C₄×C₅₉	#108	1	2	yes	—
51	C₂×C₄×C₅₉	SD₃₂	1	4	yes	—
52	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
55	C₈×C₅₉	—	2	1	no	—
56	C₈×C₅₉	—	2	1	no	—
57	C₈×C₅₉	C₄⋊C₈	1	2	yes	—
58	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
59	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
60	C₅₉⋊C₈	—	8	4	no	—
61	C₂×#1372	—	59	1	no	maximal, Sylow
62	C₂²×C₅₉	#335	1	2	yes	—
63	C₂²×C₅₉	#367	1	2	yes	—
64	C₂²×C₅₉	#368	1	2	yes	—
65	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
66	C₄×C₅₉	—	2	1	no	—
67	C₄×C₅₉	—	2	1	no	—
68	C₄×C₅₉	#335	1	2	yes	—
69	C₄×C₅₉	#367	1	2	yes	—
70	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₄×C₅₉	C₂×#108	1	1	yes	characteristic
72	C₄×C₅₉	C₂×SD₃₂	1	2	yes	—
73	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic
74	C₂×#335	—	59	2	no	—
75	#1372	—	59	4	no	—
76	C₂×C₅₉	#1372	1	4	yes	complemented
77	C₂×C₅₉	C₂×#335	1	1	yes	characteristic
78	C₂×C₅₉	C₂×#367	1	1	yes	characteristic
79	C₂×C₅₉	C₂×#368	1	1	yes	characteristic
80	C₂²×C₁₆	Dic₅₉	1	1	yes	characteristic
81	C₂²×C₁₆	Dic₅₉	1	1	yes	characteristic
82	C₂×C₄⋊C₈	—	118	2	no	—
83	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
84	C₄×C₁₆	Dic₅₉	1	2	yes	—
85	C₄×C₁₆	C₂×D₅₉	1	2	yes	—
86	#335	—	59	4	no	—
87	#335	—	118	2	no	—
88	C₅₉	C₂×#1372	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	—	236	2	no	—
90	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
91	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
92	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	C₅₉⋊C₈	1	4	yes	complemented
95	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
96	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
97	C₄⋊C₈	—	118	4	no	—
98	C₄⋊C₈	—	236	2	no	—
99	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
100	C₄×C₈	D₂₃₆	1	1	yes	characteristic
101	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
102	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
103	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
104	C₂²×C₄	Dic₂₃₆	1	1	yes	characteristic
105	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	472	4	no	—
108	C₂×C₈	—	236	4	no	—
109	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
110	C₂×C₈	#11745664	1	2	yes	—
111	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
112	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
113	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
114	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
115	C₄²	D₄₇₂	1	1	yes	characteristic
116	C₄²	Dic₂₃₆	1	1	yes	characteristic
117	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
118	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
119	C₁₆	—	2	2	no	—
120	C₁₆	C₂×C₅₉⋊C₈	1	4	yes	complemented
121	C₂³	#14273661	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
124	C₂×C₄	#14273661	1	1	yes	characteristic
125	C₂×C₄	#14273661	1	1	yes	characteristic
126	C₂×C₄	#14273661	1	1	yes	characteristic
127	C₂×C₄	#14273663	1	2	yes	—
128	C₂×C₄	#14273672	1	4	yes	—
129	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
130	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
131	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	472	4	no	—
135	C₈	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
136	C₈	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
137	C₈	C₂×#11745664	1	1	yes	characteristic
138	C₂²	#16837155	1	2	yes	—
139	C₂²	#16837230	1	2	yes	—
140	C₂²	#16837232	1	2	yes	—
141	C₂²	C₂×#14273661	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	#16837155	1	2	yes	—
145	C₄	#16837230	1	2	yes	—
146	C₄	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
147	C₄	C₂×#14273663	1	1	yes	characteristic
148	C₄	C₂×#14273672	1	2	yes	—
149	C₂	#21422170	1	4	yes	direct product factor
150	C₂	C₂×#16837155	1	1	yes	characteristic
151	C₂	C₂×#16837230	1	1	yes	characteristic
152	C₂	C₂×#16837232	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄₇₂
commutator	C₈×C₅₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	#107
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	C₂×#1372
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #21422170	C₂	direct (N × H)
19. C₂×C₁₆×C₅₉	C₈	semidirect
43. C₁₆×C₅₉	C₂×C₈	semidirect
76. C₂×C₅₉	#1372	semidirect
88. C₅₉	C₂×#1372	semidirect
94. C₂×C₁₆	C₅₉⋊C₈	semidirect
120. C₁₆	C₂×C₅₉⋊C₈	semidirect
149. C₂	#21422170	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×#16837155	2
#21422170	2
C₂×C₄×C₁₆×C₅₉	2
C₂×#1372	59

Subgroups — C2×#21422170 #28500270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21422170 #28500270