Subgroups — #11990539

Order 1280.

414 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₈²	C₂	1	1	yes	maximal, characteristic
2	C₄×#3949	C₂	1	1	yes	maximal, characteristic
3	#10600550	C₂	1	2	yes	maximal
4	C₂×#64677	C₂	1	1	yes	maximal, characteristic
5	#10598146	C₂	1	2	yes	maximal
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×#3949	C₄	1	2	yes	—
10	C₂×#3947	—	2	2	no	—
11	C₄×C₄⋊Dic₅	C₂²	1	2	yes	—
12	C₂×C₄⋊#168	C₂²	1	2	yes	—
13	C₅×C₈²	C₄	1	4	yes	complemented
14	#64771	C₄	1	2	yes	—
15	#64770	—	2	2	no	—
16	#64677	C₄	1	4	yes	complemented
17	#8283	—	5	1	no	maximal, Sylow
18	C₂²×C₅×C₈	—	2	1	no	—
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
25	C₂×C₄×Dic₅	—	2	2	no	—
26	C₂²×#168	—	2	2	no	—
27	C₄×C₅×C₈	—	2	1	no	—
28	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₄×C₅×C₈	Q₈	1	2	yes	—
33	#3963	—	2	2	no	—
34	#3949	—	2	4	no	—
35	#3947	—	4	4	no	—
36	C₄⋊#168	—	2	2	no	—
37	C₄⋊#168	C₂×C₄	1	4	yes	—
38	C₄×#106	—	5	1	no	—
39	#1770	—	5	2	no	—
40	C₂×#334	—	5	1	no	—
41	C₂×C₈²	D₅	1	1	yes	characteristic
42	#1279	—	5	2	no	—
43	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	#41	1	1	yes	characteristic
46	C₂²×Dic₅	—	2	4	no	—
47	C₂×C₅×C₈	—	2	4	no	—
48	C₂×C₅×C₈	—	2	2	no	—
49	C₂×C₅×C₈	—	2	2	no	—
50	C₂×C₅×C₈	—	2	2	no	—
51	C₂×C₅×C₈	C₄²	1	2	yes	—
52	C₂×C₅×C₈	C₄²	1	2	yes	—
53	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
54	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
55	C₂×C₅×C₈	M₁₆	1	4	yes	—
56	C₂×C₅×C₈	SD₁₆	1	4	yes	—
57	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₅	#41	1	1	yes	characteristic
61	C₄⋊Dic₅	—	2	4	no	—
62	C₄⋊Dic₅	—	2	4	no	—
63	C₄×Dic₅	—	4	2	no	—
64	C₄×Dic₅	—	4	2	no	—
65	C₂×#168	—	2	4	no	—
66	C₂×#168	—	4	2	no	—
67	C₂×#168	—	4	2	no	—
68	C₂×#106	—	5	2	no	—
69	C₂×C₄⋊C₈	—	5	2	no	—
70	C₂×#97	—	10	2	no	—
71	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
74	C₄×C₄⋊C₄	—	5	2	no	—
75	#340	—	5	2	no	—
76	#336	—	10	2	no	—
77	#334	—	5	4	no	—
78	C₈²	Dic₅	1	4	yes	complemented
79	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅	#117	1	1	yes	characteristic
84	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₅	#131	1	2	yes	—
89	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
90	C₂×C₄×C₅	#135	1	2	yes	—
91	C₂×Dic₅	—	4	4	no	—
92	C₂×Dic₅	—	4	4	no	—
93	C₂×Dic₅	—	4	4	no	—
94	C₅×C₈	—	2	4	no	—
95	C₅×C₈	—	2	2	no	—
96	C₅×C₈	—	2	2	no	—
97	C₅×C₈	#97	1	4	yes	complemented
98	C₅×C₈	#106	1	4	yes	complemented
99	#168	—	4	4	no	—
100	C₂²×C₈	—	2	1	no	—
101	C₂²×C₈	—	10	2	no	—
102	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
103	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
104	C₂²×C₈	D₂₀	1	1	yes	characteristic
105	C₂²×C₈	D₂₀	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	5	4	no	—
107	C₂×C₄²	—	10	2	no	—
108	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
109	#106	—	10	4	no	—
110	C₄⋊C₈	—	5	4	no	—
111	C₄⋊C₈	—	10	2	no	—
112	#97	—	20	4	no	—
113	C₄×C₈	—	2	1	no	—
114	C₄×C₈	Dic₁₀	1	2	yes	—
115	C₄×C₈	C₄×D₅	1	2	yes	—
116	C₄×C₈	C₂×Dic₅	1	2	yes	—
117	C₄×C₈	C₂×Dic₅	1	2	yes	—
118	C₄×C₈	C₂×Dic₅	1	2	yes	—
119	#95	—	10	2	no	—
120	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₅	#436	1	2	yes	—
122	C₂²×C₅	C₄×SD₁₆	1	2	yes	—
123	C₂²×C₅	#443	1	2	yes	—
124	C₄×C₅	—	2	1	no	—
125	C₄×C₅	—	2	1	no	—
126	C₄×C₅	C₂×#97	1	1	yes	characteristic
127	C₄×C₅	#405	1	1	yes	characteristic
128	C₄×C₅	C₂×#106	1	1	yes	characteristic
129	C₄×C₅	#427	1	1	yes	characteristic
130	C₄×C₅	C₄×SD₁₆	1	2	yes	—
131	C₄×C₅	#446	1	2	yes	—
132	Dic₅	—	8	4	no	—
133	C₂²×C₄	—	10	4	no	—
134	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
135	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
136	C₂²×C₄	#709	1	1	yes	characteristic
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	10	4	no	—
142	C₂×C₈	—	20	2	no	—
143	C₂×C₈	—	20	2	no	—
144	C₂×C₈	#676	1	4	yes	—
145	C₂×C₈	#677	1	4	yes	—
146	C₂×C₈	C₄×Dic₅	1	2	yes	—
147	C₂×C₈	C₄×Dic₅	1	2	yes	—
148	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
149	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
150	C₄⋊C₄	—	10	4	no	—
151	C₄⋊C₄	—	10	4	no	—
152	C₄²	—	20	2	no	—
153	C₄²	—	20	2	no	—
154	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
155	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
156	C₄²	#709	1	1	yes	characteristic
157	C₄²	C₂²×Dic₅	1	1	yes	characteristic
158	C₂×C₅	#1590	1	4	yes	—
159	C₂×C₅	#1770	1	1	yes	characteristic
160	C₂×C₅	#1772	1	1	yes	characteristic
161	C₂×C₅	C₄×#106	1	1	yes	characteristic
162	C₂³	C₄×D₂₀	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	20	4	no	—
165	C₂×C₄	—	20	4	no	—
166	C₂×C₄	—	20	4	no	—
167	C₂×C₄	C₄×Dic₁₀	1	1	yes	characteristic
168	C₂×C₄	C₄×Dic₁₀	1	1	yes	characteristic
169	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
170	C₂×C₄	C₂×#676	1	2	yes	—
171	C₂×C₄	#4047	1	2	yes	—
172	C₂×C₄	C₂×#677	1	2	yes	—
173	C₂×C₄	#4050	1	2	yes	—
174	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
176	C₂×C₄	#4072	1	1	yes	characteristic
177	C₈	—	2	4	no	—
178	C₈	—	2	2	no	—
179	C₈	—	2	2	no	—
180	C₈	—	20	4	no	—
181	C₈	#3947	1	4	yes	complemented
182	C₈	#3949	1	4	yes	complemented
183	C₅	#8283	1	1	yes	characteristic, complemented, Sylow
184	C₂²	#64977	1	2	yes	—
185	C₂²	C₄×#677	1	2	yes	—
186	C₂²	#64989	1	2	yes	—
187	C₂²	C₄×C₄⋊Dic₅	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	40	4	no	—
191	C₄	#64968	1	2	yes	—
192	C₄	C₄×#677	1	2	yes	—
193	C₄	C₂×#3947	1	1	yes	characteristic
194	C₄	#65390	1	1	yes	characteristic
195	C₄	C₂×#3949	1	1	yes	characteristic
196	C₄	#65395	1	1	yes	characteristic
197	C₂	#10598848	1	4	yes	—
198	C₂	#10600550	1	1	yes	characteristic
199	C₂	C₄×#3949	1	1	yes	characteristic
200	C₂	#10600572	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×C₈²	C₂
socle, prime radical	C₂³×C₅	C₄×D₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×C₈²	D₅
hypocenter	C₅	#8283
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
13. C₅×C₈²	C₄	semidirect
16. #64677	C₄	semidirect
78. C₈²	Dic₅	semidirect
97. C₅×C₈	#97	semidirect
98. C₅×C₈	#106	semidirect
181. C₈	#3947	semidirect
182. C₈	#3949	semidirect
183. C₅	#8283	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₈²	2
C₄×#3949	2
#10600550	2
C₂×#64677	2
#10598146	2
#8283	5