Subgroups — #12017331

Order 1280.

326 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#366	C₂	1	1	yes	maximal, characteristic
2	C₂×#64724	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic
4	#10601405	C₂	1	2	yes	maximal
5	#10598213	C₂	1	2	yes	maximal
6	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₅×#107	C₂²	1	2	yes	—
8	C₂×#3950	C₂²	1	2	yes	—
9	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
10	C₂²×#672	C₂²	1	1	yes	characteristic
11	#65274	—	2	2	no	—
12	C₅×#366	C₄	1	4	yes	complemented
13	#64749	—	2	2	no	—
14	#64724	—	2	2	no	—
15	C₁₆×Dic₅	—	2	2	no	—
16	#13165	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₅×C₄⋊C₄	—	2	2	no	—
19	C₂×C₄⋊Dic₅	—	2	2	no	—
20	C₂×C₄⋊Dic₅	—	4	2	no	—
21	C₂×C₄×Dic₅	D₄	1	1	yes	characteristic
22	C₂²×#168	D₄	1	1	yes	characteristic
23	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₅×C₁₆	D₄	1	2	yes	—
25	C₂×C₅×C₁₆	Q₈	1	2	yes	—
26	C₅×#107	—	2	2	no	—
27	C₅×#107	C₂×C₄	1	4	yes	—
28	#3963	—	4	2	no	—
29	#3950	—	2	2	no	—
30	#3950	—	2	2	no	—
31	C₈×Dic₅	—	2	1	no	—
32	C₈×Dic₅	—	2	1	no	—
33	C₂×#672	—	2	1	no	—
34	C₂×#672	—	2	1	no	—
35	C₂×#672	Q₈	1	2	yes	—
36	C₂×#366	—	5	1	no	—
37	C₂×#366	D₅	1	1	yes	characteristic
38	C₂×C₄×C₁₆	—	5	1	no	—
39	#1942	—	5	2	no	—
40	#1380	—	5	2	no	—
41	C₂²×C₄×C₅	—	4	2	no	—
42	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅	—	4	2	no	—
44	C₂²×Dic₅	D₈	1	1	yes	characteristic
45	C₂²×Dic₅	D₈	1	1	yes	characteristic
46	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
48	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₅×C₈	#41	1	1	yes	characteristic
51	C₂×C₅×C₈	#41	1	1	yes	characteristic
52	C₅×C₄⋊C₄	—	2	4	no	—
53	C₅×C₄⋊C₄	—	4	2	no	—
54	C₄⋊Dic₅	—	4	2	no	—
55	C₄⋊Dic₅	—	4	2	no	—
56	C₄⋊Dic₅	—	8	4	no	—
57	C₄×Dic₅	—	2	1	no	—
58	C₄×Dic₅	—	2	1	no	—
59	C₂×#168	—	2	1	no	—
60	C₂×#168	—	2	1	no	—
61	C₂×#168	Q₁₆	1	2	yes	—
62	C₅×C₁₆	C₄⋊C₄	1	4	yes	complemented
63	#672	—	2	2	no	—
64	C₂²×C₁₆	—	5	1	no	—
65	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
66	C₂×#107	—	5	2	no	—
67	C₂×#107	C₂×D₅	1	2	yes	—
68	C₂×C₄×C₈	—	5	1	no	—
69	#389	—	10	2	no	—
70	#366	—	10	2	no	—
71	#366	Dic₅	1	4	yes	complemented
72	C₄×C₁₆	—	10	2	no	—
73	#340	—	10	2	no	—
74	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₅	—	4	4	no	—
76	C₂×C₄×C₅	—	8	2	no	—
77	C₂×C₄×C₅	—	8	2	no	—
78	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅	#121	1	1	yes	characteristic
80	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅	#135	1	1	yes	characteristic
83	C₂×C₄×C₅	#135	1	1	yes	characteristic
84	C₂×Dic₅	—	2	1	no	—
85	C₂×Dic₅	—	2	1	no	—
86	C₂×Dic₅	—	2	1	no	—
87	C₂×Dic₅	—	2	1	no	—
88	C₂×Dic₅	—	8	2	no	—
89	C₂×Dic₅	—	8	2	no	—
90	C₂×Dic₅	—	8	2	no	—
91	C₂×Dic₅	D₁₆	1	2	yes	—
92	C₂×Dic₅	Q₃₂	1	2	yes	—
93	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
95	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
96	C₅×C₈	#125	1	1	yes	characteristic
97	#168	—	2	2	no	—
98	C₂²×C₈	—	5	1	no	—
99	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
100	C₂×C₄⋊C₄	—	2	2	no	—
101	C₂×C₄⋊C₄	—	10	2	no	—
102	C₂×C₄⋊C₄	—	20	2	no	—
103	C₂×C₄²	—	5	1	no	—
104	C₂×C₁₆	—	5	2	no	—
105	C₂×C₁₆	—	10	1	no	—
106	C₂×C₁₆	—	10	1	no	—
107	C₂×C₁₆	Dic₁₀	1	2	yes	—
108	C₂×C₁₆	D₂₀	1	2	yes	—
109	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
110	#107	—	2	2	no	—
111	#107	—	10	2	no	—
112	#107	—	10	2	no	—
113	#107	C₂×Dic₅	1	4	yes	—
114	C₄×C₈	—	10	1	no	—
115	C₄×C₈	—	10	1	no	—
116	#95	—	20	2	no	—
117	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
118	C₂²×C₅	#466	1	1	yes	characteristic
119	C₂²×C₅	#500	1	1	yes	characteristic
120	C₂²×C₅	C₂×D₁₆	1	1	yes	characteristic
121	C₂²×C₅	C₂×Q₃₂	1	1	yes	characteristic
122	C₂²×C₅	#508	1	1	yes	characteristic
123	C₂²×C₅	#508	1	1	yes	characteristic
124	C₄×C₅	—	8	4	no	—
125	C₄×C₅	#389	1	1	yes	characteristic
126	C₄×C₅	C₄×Q₁₆	1	1	yes	characteristic
127	C₄×C₅	#467	1	1	yes	characteristic
128	C₄×C₅	#499	1	1	yes	characteristic
129	Dic₅	—	2	2	no	—
130	Dic₅	—	2	2	no	—
131	Dic₅	—	16	2	no	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	—	5	1	no	—
134	C₂²×C₄	—	5	1	no	—
135	C₂²×C₄	—	20	2	no	—
136	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
137	C₂×C₈	—	5	2	no	—
138	C₂×C₈	—	10	1	no	—
139	C₂×C₈	—	10	1	no	—
140	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
141	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
142	C₂×C₈	#711	1	1	yes	characteristic
143	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
144	C₂×C₈	#713	1	1	yes	characteristic
145	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	20	2	no	—
149	C₄⋊C₄	—	20	2	no	—
150	C₄⋊C₄	—	40	4	no	—
151	C₄²	—	10	1	no	—
152	C₄²	—	10	1	no	—
153	C₁₆	—	10	2	no	—
154	C₁₆	C₄⋊Dic₅	1	4	yes	complemented
155	C₂×C₅	#1942	1	1	yes	characteristic
156	C₂×C₅	C₄×D₁₆	1	1	yes	characteristic
157	C₂×C₅	C₄×Q₃₂	1	1	yes	characteristic
158	C₂×C₅	#2215	1	1	yes	characteristic
159	C₂×C₅	#2216	1	1	yes	characteristic
160	C₂×C₅	#2252	1	1	yes	characteristic
161	C₂×C₅	#2253	1	1	yes	characteristic
162	C₂³	D₅×D₈	1	1	yes	characteristic
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	—	5	2	no	—
165	C₂×C₄	—	5	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	10	1	no	—
169	C₂×C₄	—	10	1	no	—
170	C₂×C₄	—	10	1	no	—
171	C₂×C₄	—	10	1	no	—
172	C₂×C₄	—	40	2	no	—
173	C₂×C₄	—	40	2	no	—
174	C₂×C₄	—	40	2	no	—
175	C₂×C₄	#4034	1	1	yes	characteristic
176	C₂×C₄	#4041	1	1	yes	characteristic
177	C₂×C₄	#4058	1	1	yes	characteristic
178	C₂×C₄	D₅×Q₁₆	1	1	yes	characteristic
179	C₂×C₄	#4065	1	1	yes	characteristic
180	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
181	C₈	—	10	2	no	—
182	C₈	#4036	1	1	yes	characteristic
183	C₈	#4043	1	1	yes	characteristic
184	C₈	C₂×C₄⋊Dic₅	1	1	yes	characteristic
185	C₈	Q₈×Dic₅	1	1	yes	characteristic
186	C₅	#13165	1	1	yes	characteristic, complemented, Sylow
187	C₂²	#65163	1	1	yes	characteristic
188	C₂²	#65172	1	1	yes	characteristic
189	C₂²	D₅×D₁₆	1	1	yes	characteristic
190	C₂²	#65201	1	1	yes	characteristic
191	C₂²	D₅×Q₃₂	1	1	yes	characteristic
192	C₂²	#65208	1	1	yes	characteristic
193	C₂²	D₈×Dic₅	1	1	yes	characteristic
194	C₄	—	8	4	no	—
195	C₄	—	10	2	no	—
196	C₄	—	10	2	no	—
197	C₄	—	80	2	no	—
198	C₄	#65167	1	1	yes	characteristic
199	C₄	#65176	1	1	yes	characteristic
200	C₄	#65274	1	1	yes	characteristic
201	C₄	Q₁₆×Dic₅	1	1	yes	characteristic
202	C₂	#10600113	1	1	yes	characteristic
203	C₂	#10600114	1	1	yes	characteristic
204	C₂	#10600120	1	1	yes	characteristic
205	C₂	#10600121	1	1	yes	characteristic
206	C₂	#10601405	1	1	yes	characteristic
207	C₂	Dic₅×D₁₆	1	1	yes	characteristic
208	C₂	Dic₅×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₅×D₈
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#366	C₂
socle, prime radical	C₂³×C₅	C₂×D₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#366	D₅
hypocenter	C₅	#13165
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
12. C₅×#366	C₄	semidirect
62. C₅×C₁₆	C₄⋊C₄	semidirect
71. #366	Dic₅	semidirect
154. C₁₆	C₄⋊Dic₅	semidirect
186. C₅	#13165	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#366	2
C₂×#64724	2
C₂×C₁₆×Dic₅	2
#10601405	2
#10598213	2
#13165	5