Subgroups — C₂×#21426375 #28744171

Order 15104.

494 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837804	C₂	1	2	yes	maximal
2	#21426375	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16837230	C₂	1	2	yes	maximal
4	C₂×#16837228	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273719	—	2	2	no	—
7	#16837804	—	2	4	no	—
8	#16837804	C₂²	1	8	yes	—
9	C₂×#14273661	C₂²	1	4	yes	—
10	C₂×#14273659	—	2	2	no	—
11	#16837230	C₂²	1	8	yes	—
12	#16837228	C₂²	1	8	yes	—
13	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₅₉	C₂²	1	4	yes	—
17	#14273719	—	2	8	no	—
18	#14273719	—	4	4	no	—
19	C₂×C₄⋊Dic₅₉	—	2	4	no	—
20	C₂×Dic₅₉⋊C₄	—	4	4	no	—
21	#14273661	—	2	4	no	—
22	#14273661	C₂³	1	8	yes	—
23	#14273659	—	2	8	no	—
24	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₉	Q₈	1	4	yes	—
28	C₂×C₁₆×C₅₉	Q₈	1	4	yes	—
29	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
30	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
31	C₄×C₈×C₅₉	D₄	1	2	yes	—
32	C₄×C₈×C₅₉	C₂³	1	2	yes	—
33	C₂²×C₄×C₅₉	D₈	1	2	yes	—
34	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₅₉	—	4	4	no	—
36	C₂×C₈×C₅₉	Q₁₆	1	4	yes	—
37	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
39	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
40	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
41	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
42	C₄²×C₅₉	D₈	1	2	yes	—
43	C₄²×C₅₉	C₂×D₄	1	2	yes	—
44	C₄⋊Dic₅₉	—	2	8	no	—
45	C₄⋊Dic₅₉	—	4	4	no	—
46	Dic₅₉⋊C₄	—	4	16	no	—
47	C₁₆×C₅₉	C₂×Q₈	1	4	yes	—
48	C₁₆×C₅₉	C₂×Q₈	1	4	yes	—
49	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₅₉	—	2	4	no	—
51	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₅₉	C₂×D₈	1	4	yes	—
53	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
55	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
56	C₂×Dic₅₉	—	4	8	no	—
57	C₂×Dic₅₉	—	8	8	no	—
58	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
59	C₈×C₅₉	C₂×Q₁₆	1	4	yes	—
60	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
61	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
62	C₂×#2253	—	59	1	no	maximal, Sylow
63	C₂²×C₅₉	#500	1	2	yes	—
64	C₂²×C₅₉	#508	1	4	yes	—
65	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
66	C₄×C₅₉	—	2	4	no	—
67	C₄×C₅₉	#500	1	2	yes	—
68	C₄×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
69	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
70	Dic₅₉	—	8	8	no	—
71	C₂×#499	—	59	2	no	—
72	#2253	—	59	8	no	—
73	C₂×#367	—	59	2	no	—
74	C₂×#366	—	59	2	no	—
75	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic
76	C₂×C₅₉	#2253	1	4	yes	complemented
77	C₂×C₅₉	C₂×#500	1	1	yes	characteristic
78	C₂×C₅₉	C₂×#508	1	2	yes	—
79	C₂×#125	—	118	2	no	—
80	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
81	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
82	#499	—	59	8	no	—
83	#499	—	118	4	no	—
84	C₂×#107	—	59	4	no	—
85	C₂×#106	—	118	2	no	—
86	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
87	#367	—	59	8	no	—
88	#366	—	59	8	no	—
89	C₄×C₁₆	C₂×D₅₉	1	4	yes	—
90	C₅₉	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
93	#125	—	118	8	no	—
94	#125	—	236	4	no	—
95	C₂×C₄⋊C₄	—	118	4	no	—
96	C₂×C₄⋊C₄	—	236	4	no	—
97	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
98	C₂×C₁₆	Dic₁₁₈	1	4	yes	—
99	C₂×C₁₆	Dic₁₁₈	1	4	yes	—
100	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
101	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
102	#107	—	59	8	no	—
103	#107	—	118	4	no	—
104	#106	—	118	8	no	—
105	C₄×C₈	D₂₃₆	1	2	yes	—
106	C₄×C₈	C₂²×D₅₉	1	2	yes	—
107	C₂²×C₄	—	236	4	no	—
108	C₂²×C₄	D₄₇₂	1	2	yes	—
109	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
110	C₂×C₈	Dic₂₃₆	1	4	yes	—
111	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
112	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
113	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
114	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
115	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
116	C₄⋊C₄	—	118	8	no	—
117	C₄⋊C₄	—	236	4	no	—
118	C₄⋊C₄	—	236	16	no	—
119	C₄²	D₄₇₂	1	2	yes	—
120	C₄²	C₂×D₂₃₆	1	2	yes	—
121	C₁₆	C₂×Dic₁₁₈	1	4	yes	—
122	C₁₆	C₂×Dic₁₁₈	1	4	yes	—
123	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	472	8	no	—
126	C₂×C₄	—	236	8	no	—
127	C₂×C₄	#14273727	1	2	yes	—
128	C₂×C₄	C₂×D₄₇₂	1	4	yes	—
129	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
130	C₂×C₄	C₂×Dic₂₃₆	1	2	yes	—
131	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
132	C₈	#14273727	1	2	yes	—
133	C₈	C₂×Dic₂₃₆	1	4	yes	—
134	C₈	C₂²×Dic₁₁₈	1	1	yes	characteristic
135	C₈	C₂²×Dic₁₁₈	1	1	yes	characteristic
136	C₂²	#16837807	1	2	yes	—
137	C₂²	#16837834	1	4	yes	—
138	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	472	8	no	—
141	C₄	#16837807	1	2	yes	—
142	C₄	C₂×#14273727	1	1	yes	characteristic
143	C₄	C₂²×Dic₂₃₆	1	1	yes	characteristic
144	C₂	#21426375	1	4	yes	direct product factor
145	C₂	C₂×#16837807	1	1	yes	characteristic
146	C₂	C₂×#16837834	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	C₂×#2253
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ 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. #21426375	C₂	direct (N × H)
76. C₂×C₅₉	#2253	semidirect
90. C₅₉	C₂×#2253	semidirect
144. C₂	#21426375	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×#16837804	2
#21426375	2
C₂×#16837230	2
C₂×#16837228	2
C₂×C₄×C₁₆×C₅₉	2
C₂×#2253	59

Subgroups — C2×#21426375 #28744171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426375 #28744171