Subgroups — #12068537

Order 1280.

702 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	C₅×C₈×#41	C₂	1	1	yes	maximal, characteristic
2	#10614243	C₂	1	2	yes	maximal
3	#10601543	C₂	1	6	yes	maximal, complemented
4	#10600524	C₂	1	3	yes	maximal, complemented
5	C₂×#64678	C₂	1	3	yes	maximal, complemented
6	C₄×C₅×#41	C₂²	1	1	yes	characteristic
7	C₂×#4094	—	2	2	no	—
8	C₅×C₈×Q₈	C₄	1	1	yes	characteristic
9	C₅×C₈×Q₈	C₄	1	1	yes	characteristic
10	C₅×C₈×D₄	C₄	1	3	yes	—
11	C₅×C₈×D₄	C₄	1	3	yes	—
12	C₅×#431	C₂²	1	3	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	3	yes	—
14	#65288	—	2	6	no	—
15	#65213	C₂²	1	6	yes	—
16	C₂×C₄⋊#168	C₂²	1	6	yes	—
17	#64678	C₂²	1	6	yes	—
18	#64678	C₂²	1	6	yes	—
19	#64678	C₂²	1	1	yes	characteristic
20	#64678	C₂²	1	3	yes	—
21	#17828	—	5	1	no	maximal, Sylow
22	C₂×C₅×#41	D₄	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	Q₈	1	1	yes	characteristic
27	C₄×C₅×D₄	C₂×C₄	1	3	yes	—
28	C₄×C₅×D₄	Q₈	1	3	yes	—
29	C₅×#117	D₄	1	3	yes	—
30	C₂×C₄²×C₅	C₂³	1	3	yes	—
31	#4094	—	4	8	no	—
32	C₂×#681	—	2	6	no	—
33	C₂²×#168	—	2	6	no	—
34	C₅×C₄⋊C₈	C₂×C₄	1	3	yes	—
35	C₅×C₄⋊C₈	C₂×C₄	1	3	yes	—
36	C₅×C₂²⋊C₈	C₂×C₄	1	6	yes	—
37	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
38	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
39	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
40	C₄×C₅×C₈	C₂³	1	3	yes	—
41	C₄⋊#168	—	2	6	no	—
42	C₄⋊#168	—	2	6	no	—
43	C₄⋊#168	C₂³	1	2	yes	—
44	C₄⋊#168	C₂³	1	6	yes	—
45	C₈×#41	D₅	1	1	yes	characteristic
46	#2906	—	5	2	no	—
47	#1571	—	5	6	no	—
48	#1567	—	5	3	no	—
49	C₂×#335	—	5	3	no	—
50	C₅×#41	D₈	1	4	yes	—
51	C₅×#41	Q₁₆	1	4	yes	—
52	C₂×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₅×D₄	C₄⋊C₄	1	3	yes	—
54	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
55	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₅	C₂×Q₈	1	3	yes	—
57	C₂×C₅×C₈	—	2	6	no	—
58	C₂×C₅×C₈	—	4	3	no	—
59	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
61	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
62	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₅×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
64	C₅×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
65	C₅×C₂²⋊C₄	C₄⋊C₄	1	6	yes	—
66	C₄²×C₅	C₂²×C₄	1	3	yes	—
67	C₄²×C₅	C₂×D₄	1	3	yes	—
68	C₄²×C₅	C₂×Q₈	1	3	yes	—
69	C₄²×C₅	C₂⁴	1	1	yes	characteristic
70	#681	—	4	12	no	—
71	C₂×#168	—	2	2	no	—
72	C₂×#168	—	2	6	no	—
73	C₂×#168	—	4	12	no	—
74	C₂×#131	—	10	2	no	—
75	C₄×#41	C₂×D₅	1	1	yes	characteristic
76	C₈×Q₈	Dic₅	1	1	yes	characteristic
77	C₈×Q₈	Dic₅	1	1	yes	characteristic
78	C₈×D₄	Dic₅	1	3	yes	—
79	C₈×D₄	Dic₅	1	3	yes	—
80	#431	C₂×D₅	1	3	yes	—
81	#424	—	10	6	no	—
82	#423	—	5	6	no	—
83	C₂×C₄⋊C₈	—	5	6	no	—
84	C₂×C₄×C₈	C₂×D₅	1	3	yes	—
85	#335	—	5	1	no	—
86	#335	—	5	3	no	—
87	#335	—	5	6	no	—
88	#335	—	5	6	no	—
89	C₂³×C₅	C₂×C₄⋊C₄	1	3	yes	—
90	C₅×Q₈	#107	1	4	yes	—
91	C₅×D₄	#107	1	12	yes	—
92	C₂×C₄×C₅	—	2	3	no	—
93	C₂×C₄×C₅	—	2	3	no	—
94	C₂×C₄×C₅	—	4	3	no	—
95	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
96	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
97	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
98	C₂×C₄×C₅	C₂×D₈	1	6	yes	—
99	C₂×C₄×C₅	C₂×Q₁₆	1	6	yes	—
100	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
103	C₅×C₈	—	2	3	no	—
104	C₅×C₈	—	2	1	no	—
105	C₅×C₈	—	2	3	no	—
106	C₅×C₈	#131	1	2	yes	complemented
107	#168	—	4	2	no	—
108	#168	—	4	6	no	—
109	C₂×#41	D₂₀	1	1	yes	characteristic
110	#131	—	20	8	no	—
111	C₂×M₁₆	—	10	6	no	—
112	C₂²×C₈	—	10	6	no	—
113	C₂²×C₈	C₂×Dic₅	1	3	yes	—
114	C₂²×C₈	C₂×Dic₅	1	3	yes	—
115	C₄×Q₈	Dic₁₀	1	1	yes	characteristic
116	C₄×Q₈	C₂×Dic₅	1	1	yes	characteristic
117	C₄×D₄	Dic₁₀	1	3	yes	—
118	C₄×D₄	C₂×Dic₅	1	3	yes	—
119	#117	D₂₀	1	3	yes	—
120	C₂×C₄²	C₂²×D₅	1	3	yes	—
121	C₄⋊C₈	—	5	2	no	—
122	C₄⋊C₈	—	5	6	no	—
123	C₄⋊C₈	—	10	6	no	—
124	C₄⋊C₈	—	10	6	no	—
125	C₄⋊C₈	C₂×Dic₅	1	3	yes	—
126	C₄⋊C₈	C₂×Dic₅	1	3	yes	—
127	C₂²⋊C₈	C₂×Dic₅	1	6	yes	—
128	C₄×C₈	C₂×Dic₅	1	3	yes	—
129	C₄×C₈	C₂×Dic₅	1	3	yes	—
130	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
131	C₄×C₈	C₂²×D₅	1	3	yes	—
132	C₂²×C₅	—	2	3	no	—
133	C₂²×C₅	—	2	3	no	—
134	C₂²×C₅	C₂×#107	1	6	yes	—
135	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₄×C₅	—	2	3	no	—
137	C₄×C₅	—	2	1	no	—
138	C₄×C₅	—	2	3	no	—
139	C₄×C₅	C₂×#107	1	6	yes	—
140	C₄×C₅	C₂×#131	1	1	yes	characteristic
141	C₄×C₅	#568	1	1	yes	characteristic
142	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
143	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
144	#41	D₄₀	1	4	yes	—
145	#41	Dic₂₀	1	4	yes	—
146	C₂×Q₈	C₄⋊Dic₅	1	1	yes	characteristic
147	C₂×D₄	C₄⋊Dic₅	1	3	yes	—
148	C₂²×C₄	C₂×Dic₁₀	1	3	yes	—
149	C₂²×C₄	C₂×D₂₀	1	3	yes	—
150	C₂²×C₄	C₂²×Dic₅	1	3	yes	—
151	M₁₆	—	20	12	no	—
152	C₂×C₈	—	2	6	no	—
153	C₂×C₈	—	4	3	no	—
154	C₂×C₈	—	10	2	no	—
155	C₂×C₈	—	10	6	no	—
156	C₂×C₈	—	20	12	no	—
157	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
158	C₂×C₈	C₂²×Dic₅	1	3	yes	—
159	C₂×C₈	C₂²×Dic₅	1	3	yes	—
160	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
161	C₄⋊C₄	C₄⋊Dic₅	1	3	yes	—
162	C₄⋊C₄	C₄⋊Dic₅	1	3	yes	—
163	C₂²⋊C₄	C₄⋊Dic₅	1	6	yes	—
164	C₄²	C₂×Dic₁₀	1	3	yes	—
165	C₄²	C₂×D₂₀	1	3	yes	—
166	C₄²	C₂²×Dic₅	1	3	yes	—
167	C₄²	C₂³×D₅	1	1	yes	characteristic
168	C₂×C₅	—	2	6	no	—
169	C₂×C₅	#2906	1	1	yes	characteristic
170	C₂×C₅	C₂²×#107	1	1	yes	characteristic
171	C₂×C₅	#2916	1	1	yes	characteristic
172	C₂³	C₂×C₄⋊Dic₅	1	3	yes	—
173	Q₈	#3950	1	4	yes	—
174	D₄	#3950	1	12	yes	—
175	C₂×C₄	—	2	3	no	—
176	C₂×C₄	—	2	3	no	—
177	C₂×C₄	—	4	3	no	—
178	C₂×C₄	C₂×D₄₀	1	6	yes	—
179	C₂×C₄	C₂×Dic₂₀	1	6	yes	—
180	C₂×C₄	C₂×C₄⋊Dic₅	1	3	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₅	1	3	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₅	1	3	yes	—
183	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
184	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
185	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
186	C₈	—	2	3	no	—
187	C₈	—	2	1	no	—
188	C₈	—	2	3	no	—
189	C₈	—	20	2	no	—
190	C₈	—	20	6	no	—
191	C₈	#4094	1	2	yes	complemented
192	C₅	#17828	1	1	yes	characteristic, complemented, Sylow
193	C₂²	—	2	3	no	—
194	C₂²	—	2	3	no	—
195	C₂²	C₂×#3950	1	6	yes	—
196	C₂²	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
197	C₄	—	2	1	no	—
198	C₄	—	2	3	no	—
199	C₄	—	2	3	no	—
200	C₄	C₂×#3950	1	6	yes	—
201	C₄	C₂²×D₄₀	1	1	yes	characteristic
202	C₄	C₂²×Dic₂₀	1	1	yes	characteristic
203	C₄	C₂×#4094	1	1	yes	characteristic
204	C₄	#66153	1	1	yes	characteristic
205	C₂	—	2	6	no	—
206	C₂	#10614243	1	1	yes	characteristic
207	C₂	C₂²×#3950	1	1	yes	characteristic
208	C₂	#10615520	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	D₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₈×#41	D₅
hypocenter	C₅	#17828
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. #10601543	C₂	semidirect
4. #10600524	C₂	semidirect
5. C₂×#64678	C₂	semidirect
106. C₅×C₈	#131	semidirect
191. C₈	#4094	semidirect
192. C₅	#17828	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
#10614243	2
#10601543	2
#10600524	2
C₂×#64678	2
#17828	5