Subgroups — #12005129

Order 1280.

814 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	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#65213	C₂	1	2	yes	maximal
3	C₂×#64772	C₂	1	4	yes	maximal
4	C₂³×C₅×C₈	C₂²	1	2	yes	—
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₂²×#681	C₂²	1	4	yes	—
7	C₂×C₄×C₅×C₈	—	2	4	no	—
8	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
9	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
10	#65213	—	2	4	no	—
11	#65213	C₄	1	8	yes	—
12	C₂×C₄⋊#168	—	2	4	no	—
13	#64772	—	2	8	no	—
14	#10844	—	5	1	no	maximal, Sylow
15	C₂³×C₄×C₅	Q₈	1	2	yes	—
16	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
17	C₂²×C₅×C₈	—	2	4	no	—
18	C₂²×C₅×C₈	—	2	4	no	—
19	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₅×C₈	D₄	1	4	yes	—
21	C₂²×C₅×C₈	Q₈	1	4	yes	—
22	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	D₄	1	2	yes	—
26	C₂×C₄²×C₅	D₄	1	4	yes	—
27	C₂×#681	—	2	4	no	—
28	C₂×#681	—	2	8	no	—
29	C₂×#681	—	2	8	no	—
30	C₂×#681	C₂×C₄	1	8	yes	—
31	C₂²×#168	—	2	4	no	—
32	C₄×C₅×C₈	—	2	4	no	—
33	C₄×C₅×C₈	—	2	4	no	—
34	C₄⋊#168	—	2	16	no	—
35	C₂×#423	—	5	2	no	—
36	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
37	C₂×#341	—	5	4	no	—
38	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₅	—	2	8	no	—
40	C₂²×C₄×C₅	C₄⋊C₄	1	8	yes	—
41	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
49	C₂²×C₄×C₅	#41	1	4	yes	—
50	C₂×C₅×C₈	—	2	4	no	—
51	C₂×C₅×C₈	—	2	4	no	—
52	C₂×C₅×C₈	—	2	8	no	—
53	C₂×C₅×C₈	—	2	8	no	—
54	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
55	C₄²×C₅	C₄²	1	2	yes	—
56	C₄²×C₅	C₄²	1	2	yes	—
57	C₄²×C₅	C₂²⋊C₄	1	8	yes	—
58	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
59	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
60	#681	—	2	16	no	—
61	#681	—	4	8	no	—
62	C₂×#168	—	2	8	no	—
63	C₂×#168	—	4	4	no	—
64	C₂×#168	—	4	4	no	—
65	C₂²×M₁₆	—	5	4	no	—
66	C₂³×C₈	C₂×D₅	1	2	yes	—
67	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
68	#423	—	5	8	no	—
69	#423	—	10	4	no	—
70	C₂×C₄⋊C₈	—	10	4	no	—
71	C₂×C₄×C₈	—	2	4	no	—
72	C₂×C₄×C₈	Dic₅	1	2	yes	—
73	C₂×C₄×C₈	Dic₅	1	2	yes	—
74	#341	—	10	8	no	—
75	C₂³×C₅	—	2	4	no	—
76	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
77	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂³×C₅	#125	1	2	yes	—
79	C₂³×C₅	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₅	—	2	4	no	—
81	C₂×C₄×C₅	—	2	16	no	—
82	C₂×C₄×C₅	#95	1	4	yes	—
83	C₂×C₄×C₅	#95	1	4	yes	—
84	C₂×C₄×C₅	#108	1	8	yes	—
85	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
88	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₅	#117	1	2	yes	—
90	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
92	C₂×C₄×C₅	#121	1	4	yes	—
93	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
94	C₂×C₄×C₅	#124	1	2	yes	—
95	C₂×C₄×C₅	#127	1	1	yes	characteristic
96	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
97	C₅×C₈	—	2	8	no	—
98	#168	—	4	8	no	—
99	C₂³×C₄	Dic₁₀	1	2	yes	—
100	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
101	C₂×M₁₆	—	5	8	no	—
102	C₂×M₁₆	—	10	4	no	—
103	C₂×M₁₆	—	10	8	no	—
104	C₂×M₁₆	—	10	8	no	—
105	C₂²×C₈	—	2	4	no	—
106	C₂²×C₈	—	2	4	no	—
107	C₂²×C₈	—	10	4	no	—
108	C₂²×C₈	Dic₁₀	1	4	yes	—
109	C₂²×C₈	D₂₀	1	4	yes	—
110	C₂²×C₈	C₂×Dic₅	1	4	yes	—
111	C₂×C₄²	C₄×D₅	1	4	yes	—
112	C₂×C₄²	D₂₀	1	2	yes	—
113	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
114	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
115	C₂×C₄²	C₂²⋊D₅	1	4	yes	—
116	C₄⋊C₈	—	10	16	no	—
117	C₄×C₈	—	2	4	no	—
118	C₄×C₈	—	2	4	no	—
119	C₂²×C₅	—	2	4	no	—
120	C₂²×C₅	—	2	4	no	—
121	C₂²×C₅	—	2	4	no	—
122	C₂²×C₅	C₂×#95	1	1	yes	characteristic
123	C₂²×C₅	#382	1	1	yes	characteristic
124	C₂²×C₅	#384	1	1	yes	characteristic
125	C₂²×C₅	#389	1	4	yes	—
126	C₂²×C₅	C₂×#108	1	4	yes	—
127	C₄×C₅	—	2	4	no	—
128	C₄×C₅	#341	1	8	yes	complemented
129	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
130	C₄×C₅	#380	1	2	yes	—
131	C₄×C₅	#390	1	2	yes	—
132	C₄×C₅	#391	1	2	yes	—
133	C₂⁴	C₂×Dic₁₀	1	1	yes	characteristic
134	C₂²×C₄	—	2	8	no	—
135	C₂²×C₄	#683	1	8	yes	—
136	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
137	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
138	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
139	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
140	C₂²×C₄	C₂×D₂₀	1	2	yes	—
141	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
142	C₂²×C₄	#709	1	4	yes	—
143	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
145	M₁₆	—	10	16	no	—
146	M₁₆	—	20	8	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	2	8	no	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	2	8	no	—
151	C₂×C₈	—	10	8	no	—
152	C₂×C₈	—	20	4	no	—
153	C₂×C₈	—	20	4	no	—
154	C₂×C₈	C₄⋊Dic₅	1	8	yes	—
155	C₄²	C₄×Dic₅	1	2	yes	—
156	C₄²	C₄×Dic₅	1	2	yes	—
157	C₄²	#685	1	8	yes	—
158	C₄²	C₂²⋊Dic₅	1	2	yes	—
159	C₄²	C₂²⋊Dic₅	1	2	yes	—
160	C₂×C₅	—	2	4	no	—
161	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
162	C₂×C₅	C₂×#341	1	2	yes	—
163	C₂×C₅	#1837	1	2	yes	—
164	C₂×C₅	#1848	1	2	yes	—
165	C₂³	—	2	4	no	—
166	C₂³	#4015	1	2	yes	—
167	C₂³	#4016	1	2	yes	—
168	C₂³	C₂×#683	1	2	yes	—
169	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	16	no	—
172	C₂×C₄	#3951	1	8	yes	—
173	C₂×C₄	#3963	1	4	yes	—
174	C₂×C₄	#3963	1	4	yes	—
175	C₂×C₄	C₄×Dic₁₀	1	4	yes	—
176	C₂×C₄	#4015	1	1	yes	characteristic
177	C₂×C₄	C₄×D₂₀	1	4	yes	—
178	C₂×C₄	#4020	1	1	yes	characteristic
179	C₂×C₄	#4021	1	2	yes	—
180	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
181	C₂×C₄	#4070	1	4	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
183	C₂×C₄	#4072	1	2	yes	—
184	C₂×C₄	C₂×#685	1	2	yes	—
185	C₂×C₄	#4076	1	4	yes	—
186	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
187	C₈	—	2	8	no	—
188	C₈	—	20	8	no	—
189	C₅	#10844	1	1	yes	characteristic, complemented, Sylow
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	#65217	1	4	yes	—
194	C₂²	#65224	1	1	yes	characteristic
195	C₂²	#65225	1	1	yes	characteristic
196	C₂²	C₂×#3951	1	4	yes	—
197	C₂²	C₂×#3963	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	#64772	1	8	yes	complemented
200	C₄	#65218	1	2	yes	—
201	C₄	C₄×C₄⋊Dic₅	1	2	yes	—
202	C₄	#65228	1	2	yes	—
203	C₄	#65501	1	2	yes	—
204	C₂	—	2	4	no	—
205	C₂	#10600584	1	2	yes	—
206	C₂	#10600585	1	2	yes	—
207	C₂	#10602328	1	1	yes	characteristic
208	C₂	C₂×#64772	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₂²×C₄×C₅×C₈	C₂
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	#10844
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
128. C₄×C₅	#341	semidirect
189. C₅	#10844	semidirect
199. C₄	#64772	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×C₈	2
C₂×#65213	2
C₂×#64772	2
#10844	5