Subgroups — C₂×#21426372 #28744168

Order 15104.

494 conjugacy classes of subgroups.

The subgroup lattice has 188 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	1	yes	maximal, characteristic
2	#21426372	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16837229	C₂	1	2	yes	maximal
4	C₂×#16837227	C₂	1	2	yes	maximal
5	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₉×#499	C₂	1	1	yes	maximal, characteristic
7	C₂×#14273719	—	2	1	no	—
8	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	#16837804	—	2	2	no	—
11	#16837804	C₂²	1	4	yes	—
12	C₂×#14273661	C₂²	1	2	yes	—
13	C₂×#14273659	—	2	1	no	—
14	#16837229	C₂²	1	8	yes	—
15	#16837227	C₂²	1	8	yes	—
16	C₄×C₅₉⋊C₁₆	C₂²	1	4	yes	—
17	C₂×#125×C₅₉	—	2	1	no	—
18	C₅₉×#499	—	2	2	no	—
19	C₅₉×#499	C₂²	1	4	yes	—
20	C₂×#107×C₅₉	C₂²	1	2	yes	—
21	C₂×#106×C₅₉	—	2	1	no	—
22	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	#14273719	—	2	4	no	—
24	#14273719	—	4	2	no	—
25	C₂×C₄⋊Dic₅₉	—	2	2	no	—
26	C₂×Dic₅₉⋊C₄	—	4	2	no	—
27	C₂×C₅₉⋊C₁₆	Q₈	1	4	yes	—
28	C₂×C₅₉⋊C₁₆	Q₈	1	4	yes	—
29	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
30	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
31	#14273661	—	2	2	no	—
32	#14273661	C₂³	1	4	yes	—
33	#14273659	—	2	4	no	—
34	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
36	#125×C₅₉	—	2	4	no	—
37	#125×C₅₉	—	4	2	no	—
38	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₉	—	4	2	no	—
40	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
41	#107×C₅₉	—	2	2	no	—
42	#107×C₅₉	C₂³	1	4	yes	—
43	#106×C₅₉	—	2	4	no	—
44	C₄×C₈×C₅₉	D₄	1	2	yes	—
45	C₄×C₈×C₅₉	C₂³	1	2	yes	—
46	C₂²×C₄×C₅₉	—	4	2	no	—
47	C₂²×C₄×C₅₉	D₈	1	2	yes	—
48	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₅₉	—	4	2	no	—
50	C₂×C₈×C₅₉	Q₁₆	1	4	yes	—
51	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₅₉	—	2	4	no	—
57	C₄⋊C₄×C₅₉	—	4	2	no	—
58	C₄⋊C₄×C₅₉	—	4	8	no	—
59	C₄²×C₅₉	D₈	1	2	yes	—
60	C₄²×C₅₉	C₂×D₄	1	2	yes	—
61	C₄⋊Dic₅₉	—	2	4	no	—
62	C₄⋊Dic₅₉	—	4	2	no	—
63	Dic₅₉⋊C₄	—	4	8	no	—
64	C₅₉⋊C₁₆	C₂×Q₈	1	4	yes	—
65	C₅₉⋊C₁₆	C₂×Q₈	1	4	yes	—
66	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₅₉	—	2	4	no	—
68	C₂×C₄×C₅₉	—	4	4	no	—
69	C₂×C₄×C₅₉	—	8	4	no	—
70	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
71	C₂×C₄×C₅₉	C₂×D₈	1	4	yes	—
72	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
74	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
75	C₂×Dic₅₉	—	4	4	no	—
76	C₂×Dic₅₉	—	8	4	no	—
77	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
78	C₈×C₅₉	C₂×Q₁₆	1	4	yes	—
79	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
80	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
81	C₂×#2253	—	59	1	no	maximal, Sylow
82	C₂²×C₅₉	#500	1	2	yes	—
83	C₂²×C₅₉	#508	1	4	yes	—
84	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
85	C₄×C₅₉	—	2	4	no	—
86	C₄×C₅₉	—	8	4	no	—
87	C₄×C₅₉	#500	1	2	yes	—
88	C₄×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
89	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
90	Dic₅₉	—	8	4	no	—
91	C₂×#499	—	59	1	no	—
92	C₂×#499	D₅₉	1	1	yes	characteristic
93	#2253	—	59	8	no	—
94	C₂×#367	—	59	2	no	—
95	C₂×#366	—	59	2	no	—
96	C₂×C₄×C₁₆	—	59	1	no	—
97	C₂×C₅₉	#2253	1	4	yes	complemented
98	C₂×C₅₉	C₂×#500	1	1	yes	characteristic
99	C₂×C₅₉	C₂×#508	1	2	yes	—
100	C₂×#125	—	2	1	no	—
101	C₂×#125	—	118	1	no	—
102	C₂²×C₁₆	—	59	1	no	—
103	C₂²×C₁₆	—	59	1	no	—
104	#499	—	2	2	no	—
105	#499	—	59	4	no	—
106	#499	—	118	2	no	—
107	#499	C₂×D₅₉	1	4	yes	—
108	C₂×#107	—	59	2	no	—
109	C₂×#107	C₂×D₅₉	1	2	yes	—
110	C₂×#106	—	2	1	no	—
111	C₂×#106	—	118	1	no	—
112	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
113	#367	—	59	8	no	—
114	#366	—	59	8	no	—
115	C₄×C₁₆	—	59	4	no	—
116	C₅₉	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
119	#125	—	2	4	no	—
120	#125	—	4	2	no	—
121	#125	—	118	4	no	—
122	#125	—	236	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	4	2	no	—
125	C₂×C₄⋊C₄	—	118	2	no	—
126	C₂×C₄⋊C₄	—	236	2	no	—
127	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
128	C₂×C₁₆	—	59	4	no	—
129	C₂×C₁₆	—	59	4	no	—
130	C₂×C₁₆	—	59	2	no	—
131	C₂×C₁₆	—	59	2	no	—
132	#107	—	2	2	no	—
133	#107	—	59	4	no	—
134	#107	—	118	2	no	—
135	#107	C₂²×D₅₉	1	4	yes	—
136	#106	—	2	4	no	—
137	#106	—	118	4	no	—
138	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
139	C₄×C₈	C₂²×D₅₉	1	2	yes	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	—	236	2	no	—
142	C₂²×C₄	#11745669	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
144	C₂×C₈	#11745672	1	4	yes	—
145	C₂×C₈	Q₈×D₅₉	1	2	yes	—
146	C₂×C₈	Q₈×D₅₉	1	2	yes	—
147	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
148	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
149	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	8	no	—
153	C₄⋊C₄	—	118	4	no	—
154	C₄⋊C₄	—	236	2	no	—
155	C₄⋊C₄	—	236	8	no	—
156	C₄²	#11745669	1	2	yes	—
157	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
158	C₁₆	—	59	4	no	—
159	C₁₆	—	59	4	no	—
160	C₂³	C₂×#11745669	1	1	yes	characteristic
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	—	472	4	no	—
165	C₂×C₄	—	236	4	no	—
166	C₂×C₄	#14273725	1	2	yes	—
167	C₂×C₄	C₂×#11745669	1	4	yes	—
168	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
169	C₂×C₄	C₂×#11745672	1	2	yes	—
170	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
171	C₈	#14273725	1	2	yes	—
172	C₈	C₂×#11745672	1	4	yes	—
173	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
174	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
175	C₂²	#16837805	1	2	yes	—
176	C₂²	#16837832	1	4	yes	—
177	C₂²	C₂²×#11745669	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	8	4	no	—
180	C₄	—	472	4	no	—
181	C₄	#16837805	1	2	yes	—
182	C₄	C₂×#14273725	1	1	yes	characteristic
183	C₄	C₂²×#11745672	1	1	yes	characteristic
184	C₂	#21426372	1	4	yes	direct product factor
185	C₂	C₂×#16837805	1	1	yes	characteristic
186	C₂	C₂×#16837832	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#499	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
hypercenter	C₂×#499	D₅₉
hypocenter	C₅₉	C₂×#2253
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. #21426372	C₂	direct (N × H)
97. C₂×C₅₉	#2253	semidirect
116. C₅₉	C₂×#2253	semidirect
184. C₂	#21426372	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
#21426372	2
C₂×#16837229	2
C₂×#16837227	2
C₂×C₄×C₅₉⋊C₁₆	2
C₂×C₅₉×#499	2
C₂×#2253	59

Subgroups — C2×#21426372 #28744168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426372 #28744168