Subgroups — #10611640

Order 640.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65914	C₂	1	1	yes	maximal, characteristic
2	C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#65313	C₂	1	1	yes	maximal, characteristic, complemented
4	#65128	C₂	1	2	yes	maximal, complemented
5	#65125	C₂	1	2	yes	maximal, complemented
6	#65123	C₂	1	1	yes	maximal, characteristic, complemented
7	D₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#64979	C₂	1	2	yes	maximal, complemented
9	#64977	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×D₂₀	C₂	1	2	yes	maximal, complemented
11	#64974	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
13	#4093	C₄	1	1	yes	characteristic
14	#4092	C₄	1	1	yes	characteristic
15	C₂×#676	C₂²	1	2	yes	—
16	C₂×C₈×D₅	C₂²	1	2	yes	—
17	C₂×C₈×D₅	C₂²	1	2	yes	—
18	#4044	C₄	1	2	yes	—
19	#4041	C₄	1	1	yes	characteristic
20	#4040	C₄	1	2	yes	—
21	#4036	C₄	1	1	yes	characteristic
22	C₄×D₂₀	C₂²	1	1	yes	characteristic
23	C₄×D₂₀	C₂²	1	2	yes	—
24	#4018	C₂²	1	2	yes	—
25	C₄²×D₅	C₂²	1	1	yes	characteristic
26	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₅×C₄⋊C₈	C₂²	1	2	yes	—
28	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
29	C₄×C₅×C₈	C₂²	1	2	yes	—
30	#3952	C₂²	1	2	yes	—
31	#3952	C₂²	1	1	yes	characteristic
32	#3952	C₂²	1	1	yes	characteristic
33	#3952	C₂²	1	2	yes	—
34	#3946	C₂²	1	2	yes	—
35	#3946	C₂²	1	1	yes	characteristic
36	#3946	C₂²	1	1	yes	characteristic
37	C₄⋊#168	C₂²	1	1	yes	characteristic
38	C₄⋊#168	C₂²	1	2	yes	—
39	#3935	C₂²	1	2	yes	—
40	C₄×#168	C₂²	1	1	yes	characteristic
41	#2994	—	5	1	no	maximal, Sylow
42	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
43	C₂×D₂₀	C₂×C₄	1	2	yes	—
44	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₅	C₂³	1	2	yes	—
49	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₅×C₈	C₂³	1	2	yes	—
51	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
52	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
53	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
54	C₄²×C₅	C₂³	1	1	yes	characteristic
55	C₄²×C₅	C₂³	1	2	yes	—
56	#685	C₂×C₄	1	2	yes	—
57	#685	C₂×C₄	1	2	yes	—
58	#685	C₂×C₄	1	2	yes	—
59	#685	C₂×C₄	1	2	yes	—
60	#685	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
62	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
63	#683	C₂×C₄	1	2	yes	—
64	#683	C₂×C₄	1	2	yes	—
65	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
67	C₄×Dic₅	C₂³	1	1	yes	characteristic
68	C₂×#168	C₂³	1	1	yes	characteristic
69	C₂×#168	C₂³	1	1	yes	characteristic
70	C₂×#168	C₂³	1	2	yes	—
71	#676	—	4	2	no	—
72	C₈×D₅	—	4	2	no	—
73	C₈×D₅	—	4	2	no	—
74	#529	—	5	1	no	—
75	#446	—	5	1	no	—
76	C₈×Q₈	D₅	1	1	yes	characteristic, complemented
77	#436	—	5	1	no	—
78	#435	—	5	2	no	—
79	C₈×D₄	—	5	1	no	—
80	C₈×D₄	—	5	2	no	—
81	#433	—	5	2	no	—
82	#431	—	5	1	no	—
83	#424	—	5	2	no	—
84	C₂×C₄⋊C₈	—	5	1	no	—
85	C₂²×D₅	C₂²×C₄	1	2	yes	—
86	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
87	C₅×Q₈	—	2	2	no	—
88	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
93	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₅	C₂²×C₄	1	2	yes	—
97	D₂₀	—	2	4	no	—
98	D₂₀	—	2	2	no	—
99	C₄×D₅	—	2	2	no	—
100	C₄×D₅	—	2	2	no	—
101	C₄×D₅	—	2	1	no	—
102	C₄×D₅	—	2	2	no	—
103	C₄×D₅	—	2	1	no	—
104	C₄×D₅	—	2	2	no	—
105	C₅×C₈	—	2	2	no	—
106	C₅×C₈	—	2	1	no	—
107	C₅×C₈	—	2	1	no	—
108	#168	—	2	1	no	—
109	#168	—	2	2	no	—
110	#168	—	2	1	no	—
111	C₂×M₁₆	—	5	2	no	—
112	C₂²×C₈	—	5	2	no	—
113	C₂²×C₈	—	5	2	no	—
114	#126	—	5	2	no	—
115	#125	—	5	1	no	—
116	#124	—	5	1	no	—
117	#123	—	5	2	no	—
118	C₂²⋊Q₈	—	5	1	no	—
119	#121	—	5	1	no	—
120	C₄×Q₈	C₂×D₅	1	1	yes	characteristic
121	C₄×D₄	—	5	2	no	—
122	C₄×D₄	—	5	1	no	—
123	#117	—	5	2	no	—
124	C₂×C₄²	—	5	1	no	—
125	C₄⋊C₈	—	5	2	no	—
126	C₄⋊C₈	—	5	1	no	—
127	C₄⋊C₈	—	5	1	no	—
128	C₄⋊C₈	—	5	1	no	—
129	C₄⋊C₈	—	5	2	no	—
130	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
131	C₄⋊C₈	C₂×D₅	1	2	yes	—
132	C₂²⋊C₈	—	5	1	no	—
133	C₂²⋊C₈	—	5	1	no	—
134	C₂²⋊C₈	—	5	2	no	—
135	C₂²⋊C₈	—	5	2	no	—
136	#97	—	5	2	no	—
137	C₄×C₈	—	5	1	no	—
138	C₄×C₈	C₂×D₅	1	1	yes	characteristic
139	C₄×C₈	C₂×D₅	1	2	yes	—
140	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
141	C₂×D₅	—	2	1	no	—
142	C₂×D₅	—	2	1	no	—
143	C₂×D₅	—	2	2	no	—
144	C₂×D₅	—	2	2	no	—
145	C₂×D₅	—	2	2	no	—
146	C₂×D₅	#131	1	2	yes	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	—	2	2	no	—
149	C₄×C₅	—	2	2	no	—
150	C₄×C₅	#131	1	2	yes	—
151	C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
152	C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
153	Dic₅	—	2	1	no	—
154	Dic₅	—	2	2	no	—
155	Dic₅	—	2	1	no	—
156	Dic₅	—	2	1	no	—
157	C₂×Q₈	C₄×D₅	1	1	yes	characteristic
158	C₂×D₄	—	5	2	no	—
159	C₂×D₄	—	5	1	no	—
160	C₂²×C₄	—	5	1	no	—
161	C₂²×C₄	—	5	1	no	—
162	C₂²×C₄	—	5	1	no	—
163	C₂²×C₄	—	5	2	no	—
164	M₁₆	—	20	2	no	—
165	C₂×C₈	—	5	1	no	—
166	C₂×C₈	—	5	1	no	—
167	C₂×C₈	—	5	2	no	—
168	C₂×C₈	—	20	2	no	—
169	C₂×C₈	—	20	2	no	—
170	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₅	1	2	yes	—
172	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
173	C₄⋊C₄	—	5	2	no	—
174	C₄⋊C₄	—	5	2	no	—
175	C₄⋊C₄	—	5	1	no	—
176	C₄⋊C₄	—	5	2	no	—
177	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
178	C₄⋊C₄	C₄×D₅	1	2	yes	—
179	C₂²⋊C₄	—	5	2	no	—
180	C₂²⋊C₄	—	5	2	no	—
181	C₂²⋊C₄	—	5	2	no	—
182	C₂²⋊C₄	—	5	2	no	—
183	C₂²⋊C₄	—	5	2	no	—
184	C₄²	—	5	1	no	—
185	C₄²	—	5	1	no	—
186	C₄²	—	5	1	no	—
187	C₄²	C₂²×D₅	1	1	yes	characteristic
188	C₄²	C₂²×D₅	1	2	yes	—
189	C₂×C₅	#520	1	1	yes	characteristic
190	C₂×C₅	C₂×#131	1	1	yes	characteristic
191	C₂×C₅	C₂×#131	1	1	yes	characteristic
192	D₅	—	2	2	no	—
193	D₅	—	4	2	no	—
194	C₂³	—	5	1	no	—
195	C₂³	—	5	2	no	—
196	Q₈	—	2	2	no	—
197	D₄	—	10	4	no	—
198	D₄	—	10	2	no	—
199	C₂×C₄	—	5	1	no	—
200	C₂×C₄	—	5	1	no	—
201	C₂×C₄	—	5	1	no	—
202	C₂×C₄	—	5	2	no	—
203	C₂×C₄	—	10	2	no	—
204	C₂×C₄	—	10	1	no	—
205	C₂×C₄	—	10	2	no	—
206	C₂×C₄	—	10	1	no	—
207	C₂×C₄	—	10	2	no	—
208	C₂×C₄	—	10	2	no	—
209	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
211	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
213	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
214	C₈	—	2	1	no	—
215	C₈	—	2	2	no	—
216	C₈	—	2	1	no	—
217	C₈	—	10	1	no	—
218	C₈	—	10	1	no	—
219	C₈	—	10	2	no	—
220	C₅	#2994	1	1	yes	characteristic, complemented, Sylow
221	C₂²	—	5	2	no	—
222	C₂²	—	10	1	no	—
223	C₂²	—	10	1	no	—
224	C₂²	—	10	2	no	—
225	C₂²	—	10	2	no	—
226	C₂²	—	10	2	no	—
227	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	10	1	no	—
232	C₄	—	10	1	no	—
233	C₄	—	10	1	no	—
234	C₄	—	10	2	no	—
235	C₄	#4047	1	2	yes	—
236	C₄	#4147	1	1	yes	characteristic
237	C₄	#4149	1	1	yes	characteristic
238	C₂	—	10	2	no	—
239	C₂	—	20	2	no	—
240	C₂	#65908	1	1	yes	characteristic
241	C₂	C₂×#4047	1	1	yes	characteristic
242	C₂	D₅×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×C₈×Q₈	C₂
prime radical	#4041	C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	C₈×Q₈	D₅
hypocenter	C₅	#2994
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. C₅×C₈×Q₈	C₂	semidirect
3. #65313	C₂	semidirect
4. #65128	C₂	semidirect
5. #65125	C₂	semidirect
6. #65123	C₂	semidirect
7. D₅×C₄⋊C₈	C₂	semidirect
8. #64979	C₂	semidirect
9. #64977	C₂	semidirect
10. C₈×D₂₀	C₂	semidirect
11. #64974	C₂	semidirect
76. C₈×Q₈	D₅	semidirect
220. C₅	#2994	semidirect

Maximal subgroups

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

subgroup	index
#65914	2
C₅×C₈×Q₈	2
#65313	2
#65128	2
#65125	2
#65123	2
D₅×C₄⋊C₈	2
#64979	2
#64977	2
C₈×D₂₀	2
#64974	2
#2994	5