Subgroups — #12068598

Order 1280.

612 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₈×#41	C₂	1	1	yes	maximal, characteristic
2	#10614254	C₂	1	2	yes	maximal
3	#10601546	C₂	1	6	yes	maximal, complemented
4	#10600523	C₂	1	3	yes	maximal, complemented
5	#10600522	C₂	1	3	yes	maximal, complemented
6	C₄×C₅×#41	C₂²	1	1	yes	characteristic
7	C₅×C₈×Q₈	C₄	1	1	yes	characteristic
8	C₅×C₈×Q₈	C₄	1	1	yes	characteristic
9	C₅×C₈×D₄	C₄	1	3	yes	—
10	C₅×C₈×D₄	C₄	1	3	yes	—
11	C₅×#431	C₂²	1	3	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	3	yes	—
13	Q₈×#168	—	2	2	no	—
14	#65301	—	2	6	no	—
15	#65214	C₂²	1	6	yes	—
16	#65213	C₂²	1	6	yes	—
17	#64678	C₂²	1	6	yes	—
18	#64677	C₂²	1	1	yes	characteristic
19	#64677	C₂²	1	3	yes	—
20	#64677	C₂²	1	6	yes	—
21	#17835	—	5	1	no	maximal, Sylow
22	C₂×C₅×#41	Q₈	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
24	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
25	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
27	C₄×C₅×D₄	C₂×C₄	1	3	yes	—
28	C₄×C₅×D₄	D₄	1	3	yes	—
29	C₅×#117	Q₈	1	3	yes	—
30	C₂×C₄²×C₅	C₂³	1	3	yes	—
31	C₂×#681	—	2	6	no	—
32	C₅×C₄⋊C₈	C₂×C₄	1	3	yes	—
33	C₅×C₄⋊C₈	C₂×C₄	1	3	yes	—
34	C₅×C₂²⋊C₈	C₂×C₄	1	6	yes	—
35	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
36	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
37	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₄×C₅×C₈	C₂³	1	3	yes	—
39	C₂²⋊#168	—	2	6	no	—
40	C₄⋊#168	—	2	6	no	—
41	C₄⋊#168	C₂³	1	6	yes	—
42	C₄⋊#168	C₂³	1	2	yes	—
43	C₄×#168	—	2	6	no	—
44	#2991	—	5	2	no	—
45	C₈×#41	D₅	1	1	yes	characteristic
46	#1574	—	5	6	no	—
47	#1566	—	5	3	no	—
48	#1564	—	5	3	no	—
49	C₅×#41	—	2	4	no	—
50	C₂×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₅×D₄	C₄⋊C₄	1	3	yes	—
52	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
54	C₂²×C₄×C₅	C₂×Q₈	1	3	yes	—
55	C₂×C₅×C₈	—	2	6	no	—
56	C₂×C₅×C₈	—	4	3	no	—
57	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
59	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
60	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₅×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
62	C₅×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
63	C₅×C₂²⋊C₄	C₄⋊C₄	1	6	yes	—
64	C₄²×C₅	C₂²×C₄	1	3	yes	—
65	C₄²×C₅	C₂×D₄	1	3	yes	—
66	C₄²×C₅	C₂×Q₈	1	3	yes	—
67	C₄²×C₅	C₂⁴	1	1	yes	characteristic
68	#681	—	8	6	no	—
69	C₂×#168	—	2	2	no	—
70	C₂×#168	—	2	6	no	—
71	C₄×#41	C₂×D₅	1	1	yes	characteristic
72	C₈×Q₈	—	10	2	no	—
73	C₈×Q₈	Dic₅	1	1	yes	characteristic
74	C₈×Q₈	Dic₅	1	1	yes	characteristic
75	#436	—	10	6	no	—
76	C₈×D₄	Dic₅	1	3	yes	—
77	C₈×D₄	Dic₅	1	3	yes	—
78	#431	—	5	6	no	—
79	#431	C₂×D₅	1	3	yes	—
80	#423	—	5	6	no	—
81	C₂×C₄×C₈	C₂×D₅	1	3	yes	—
82	#335	—	5	6	no	—
83	#334	—	5	1	no	—
84	#334	—	5	3	no	—
85	#334	—	5	6	no	—
86	C₂³×C₅	C₂×C₄⋊C₄	1	3	yes	—
87	C₅×Q₈	#108	1	4	yes	—
88	C₅×D₄	—	2	6	no	—
89	C₂×C₄×C₅	—	2	3	no	—
90	C₂×C₄×C₅	—	2	3	no	—
91	C₂×C₄×C₅	—	2	6	no	—
92	C₂×C₄×C₅	—	2	6	no	—
93	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
94	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
95	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
96	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
99	C₅×C₈	—	2	3	no	—
100	C₅×C₈	—	2	1	no	—
101	C₅×C₈	—	2	3	no	—
102	C₅×C₈	#131	1	2	yes	—
103	#168	—	4	2	no	—
104	#168	—	4	6	no	—
105	C₂×#41	Dic₁₀	1	1	yes	characteristic
106	C₂×M₁₆	—	10	6	no	—
107	C₂²×C₈	C₂×Dic₅	1	3	yes	—
108	C₂²×C₈	C₂×Dic₅	1	3	yes	—
109	C₄×Q₈	D₂₀	1	1	yes	characteristic
110	C₄×Q₈	C₂×Dic₅	1	1	yes	characteristic
111	C₄×D₄	D₂₀	1	3	yes	—
112	C₄×D₄	C₂×Dic₅	1	3	yes	—
113	#117	Dic₁₀	1	3	yes	—
114	C₂×C₄²	C₂²×D₅	1	3	yes	—
115	C₄⋊C₈	—	5	6	no	—
116	C₄⋊C₈	—	5	2	no	—
117	C₄⋊C₈	—	10	6	no	—
118	C₄⋊C₈	C₂×Dic₅	1	3	yes	—
119	C₄⋊C₈	C₂×Dic₅	1	3	yes	—
120	C₂²⋊C₈	—	10	6	no	—
121	C₂²⋊C₈	C₂×Dic₅	1	6	yes	—
122	C₄×C₈	—	10	6	no	—
123	C₄×C₈	C₂×Dic₅	1	3	yes	—
124	C₄×C₈	C₂×Dic₅	1	3	yes	—
125	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
126	C₄×C₈	C₂²×D₅	1	3	yes	—
127	C₂²×C₅	—	2	3	no	—
128	C₂²×C₅	—	2	3	no	—
129	C₂²×C₅	—	2	3	no	—
130	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₅	—	2	3	no	—
132	C₄×C₅	—	2	3	no	—
133	C₄×C₅	—	2	1	no	—
134	C₄×C₅	C₂×#108	1	6	yes	—
135	C₄×C₅	C₂×#131	1	1	yes	characteristic
136	C₄×C₅	#568	1	1	yes	characteristic
137	C₄×C₅	#577	1	2	yes	—
138	#41	—	2	4	no	—
139	C₂×Q₈	C₄⋊Dic₅	1	1	yes	characteristic
140	C₂×D₄	C₄⋊Dic₅	1	3	yes	—
141	C₂²×C₄	C₂×Dic₁₀	1	3	yes	—
142	C₂²×C₄	C₂×D₂₀	1	3	yes	—
143	C₂²×C₄	C₂²×Dic₅	1	3	yes	—
144	M₁₆	—	40	6	no	—
145	C₂×C₈	—	2	6	no	—
146	C₂×C₈	—	4	3	no	—
147	C₂×C₈	—	10	2	no	—
148	C₂×C₈	—	10	6	no	—
149	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
150	C₂×C₈	C₂²×Dic₅	1	3	yes	—
151	C₂×C₈	C₂²×Dic₅	1	3	yes	—
152	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
153	C₄⋊C₄	C₄⋊Dic₅	1	3	yes	—
154	C₄⋊C₄	C₄⋊Dic₅	1	3	yes	—
155	C₂²⋊C₄	C₄⋊Dic₅	1	6	yes	—
156	C₄²	C₂×Dic₁₀	1	3	yes	—
157	C₄²	C₂×D₂₀	1	3	yes	—
158	C₄²	C₂²×Dic₅	1	3	yes	—
159	C₄²	C₂³×D₅	1	1	yes	characteristic
160	C₂×C₅	—	4	3	no	—
161	C₂×C₅	#2906	1	1	yes	characteristic
162	C₂×C₅	#2912	1	1	yes	characteristic
163	C₂×C₅	C₂²×#108	1	1	yes	characteristic
164	C₂³	C₂×C₄⋊Dic₅	1	3	yes	—
165	Q₈	#3951	1	4	yes	—
166	D₄	—	2	6	no	—
167	C₂×C₄	—	2	6	no	—
168	C₂×C₄	—	2	3	no	—
169	C₂×C₄	—	2	3	no	—
170	C₂×C₄	—	2	6	no	—
171	C₂×C₄	C₂×C₄⋊Dic₅	1	3	yes	—
172	C₂×C₄	C₂×C₄⋊Dic₅	1	3	yes	—
173	C₂×C₄	C₂×C₄⋊Dic₅	1	3	yes	—
174	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
175	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
176	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
177	C₈	—	2	3	no	—
178	C₈	—	2	1	no	—
179	C₈	—	2	3	no	—
180	C₈	—	20	6	no	—
181	C₈	—	20	2	no	—
182	C₈	#4094	1	2	yes	—
183	C₅	#17835	1	1	yes	characteristic, complemented, Sylow
184	C₂²	—	2	3	no	—
185	C₂²	—	2	3	no	—
186	C₂²	—	2	3	no	—
187	C₂²	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	2	3	no	—
190	C₄	—	2	3	no	—
191	C₄	C₂×#3951	1	6	yes	—
192	C₄	#66085	1	2	yes	—
193	C₄	C₂×#4094	1	1	yes	characteristic
194	C₄	#66153	1	1	yes	characteristic
195	C₂	—	4	3	no	—
196	C₂	#10614243	1	1	yes	characteristic
197	C₂	C₂²×#3951	1	1	yes	characteristic
198	C₂	#10615515	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×C₈×#41	C₂
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂×C₅×#41	Q₈
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₈×#41	D₅
hypocenter	C₅	#17835
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
3. #10601546	C₂	semidirect
4. #10600523	C₂	semidirect
5. #10600522	C₂	semidirect
183. C₅	#17835	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₈×#41	2
#10614254	2
#10601546	2
#10600523	2
#10600522	2
#17835	5