Subgroups — #10610388

Order 640.

500 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66166	C₂	1	1	yes	maximal, characteristic
2	C₂×#4094	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#4053	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#408	C₂	1	1	yes	maximal, characteristic, complemented
5	#65407	C₂	1	1	yes	maximal, characteristic, complemented
6	#65399	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	#65018	C₂	1	4	yes	maximal, complemented
9	#65014	C₂	1	4	yes	maximal, complemented
10	C₂×C₅×#41	C₂²	1	1	yes	characteristic
11	C₂×#709	C₂²	1	1	yes	characteristic
12	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
14	#4094	—	2	4	no	—
15	#4093	C₄	1	1	yes	characteristic
16	#4090	C₄	1	1	yes	characteristic
17	#4086	C₄	1	1	yes	characteristic
18	#4085	C₄	1	4	yes	—
19	#4082	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊D₅	C₂²	1	4	yes	—
21	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
22	C₂×#681	C₂²	1	1	yes	characteristic
23	C₂×#681	C₂²	1	2	yes	—
24	C₂²×#168	C₂²	1	2	yes	—
25	C₂²×#168	C₂²	1	1	yes	characteristic
26	#4053	—	2	4	no	—
27	C₂×#676	C₂²	1	2	yes	—
28	C₂×C₈×D₅	C₂²	1	2	yes	—
29	C₅×C₂²⋊C₈	C₂²	1	4	yes	—
30	#3952	C₂²	1	4	yes	—
31	#3946	C₂²	1	4	yes	—
32	C₈×Dic₅	C₂²	1	2	yes	—
33	C₈×Dic₅	C₂²	1	2	yes	—
34	#2931	—	5	1	no	maximal, Sylow
35	C₅×#41	—	2	2	no	—
36	C₅×#41	—	2	2	no	—
37	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₅×D₄	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₅	C₂³	1	2	yes	—
41	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊D₅	C₂×C₄	1	4	yes	—
44	C₂²×Dic₅	C₂×C₄	1	2	yes	—
45	#709	—	2	2	no	—
46	#709	—	2	2	no	—
47	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₅	C₂³	1	2	yes	—
49	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
50	C₅×M₁₆	—	2	2	no	—
51	C₂×C₅×C₈	—	2	2	no	—
52	C₂×C₅×C₈	C₂³	1	2	yes	—
53	C₂×C₅×C₈	C₂³	1	2	yes	—
54	C₂²⋊Dic₅	C₂×C₄	1	4	yes	—
55	#685	C₂×C₄	1	4	yes	—
56	#683	C₂×C₄	1	4	yes	—
57	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
58	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₅	C₂³	1	2	yes	—
60	#681	—	2	2	no	—
61	#681	—	2	4	no	—
62	C₂×#168	—	2	4	no	—
63	C₂×#168	D₄	1	4	yes	complemented
64	C₂×#168	C₂³	1	2	yes	—
65	C₂×#168	C₂³	1	1	yes	characteristic
66	C₂×#168	C₂³	1	1	yes	characteristic
67	#676	—	2	4	no	—
68	C₈×D₅	—	2	4	no	—
69	C₂×#131	—	5	1	no	—
70	C₂×#131	—	5	1	no	—
71	#532	—	5	1	no	—
72	#436	—	5	4	no	—
73	C₈×D₄	—	5	4	no	—
74	#423	—	5	1	no	—
75	#408	—	5	1	no	—
76	#408	D₅	1	1	yes	characteristic, complemented
77	C₂×C₄×C₈	—	5	1	no	—
78	C₂³×C₅	C₂²×C₄	1	2	yes	—
79	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂²×D₅	C₂²×C₄	1	2	yes	—
81	C₅×Q₈	—	2	2	no	—
82	C₅×D₄	—	2	4	no	—
83	C₅×D₄	—	2	2	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₅	#41	1	2	yes	—
90	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
91	C₂²⋊D₅	—	2	4	no	—
92	C₂²⋊D₅	—	2	8	no	—
93	C₂×Dic₅	—	2	4	no	—
94	C₂×Dic₅	C₂²×C₄	1	2	yes	—
95	C₂×Dic₅	C₂²×C₄	1	2	yes	—
96	C₂×Dic₅	C₂²×C₄	1	2	yes	—
97	D₂₀	—	2	2	no	—
98	C₄×D₅	—	2	2	no	—
99	C₄×D₅	—	2	2	no	—
100	Dic₁₀	—	2	2	no	—
101	C₅×C₈	—	2	2	no	—
102	C₅×C₈	—	2	2	no	—
103	#168	—	2	2	no	—
104	#168	C₂×D₄	1	2	yes	—
105	#168	C₂×D₄	1	2	yes	—
106	C₂×#41	—	5	1	no	—
107	C₂×#41	C₂×D₅	1	1	yes	characteristic
108	#131	—	10	4	no	—
109	#131	—	10	4	no	—
110	C₂×M₁₆	—	5	1	no	—
111	C₂×M₁₆	—	5	2	no	—
112	C₂×M₁₆	—	5	2	no	—
113	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
114	C₂²×C₈	—	5	1	no	—
115	C₂²×C₈	—	5	2	no	—
116	C₂²×C₈	—	5	2	no	—
117	C₂²×C₈	C₂×D₅	1	1	yes	characteristic
118	C₄⋊Q₈	—	5	1	no	—
119	#127	—	5	1	no	—
120	#124	—	5	1	no	—
121	#124	—	5	1	no	—
122	#121	—	5	4	no	—
123	C₄×D₄	—	5	4	no	—
124	C₂×C₄²	—	5	1	no	—
125	C₄⋊C₈	—	5	4	no	—
126	C₂²⋊C₈	—	5	4	no	—
127	C₂²⋊C₈	C₂×D₅	1	4	yes	—
128	C₄×C₈	—	5	2	no	—
129	C₄×C₈	—	5	2	no	—
130	C₂²×C₅	—	2	2	no	—
131	C₂²×C₅	—	2	2	no	—
132	C₂²×C₅	—	2	4	no	—
133	C₂²×C₅	C₄×D₄	1	2	yes	—
134	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
135	C₂×D₅	—	2	2	no	—
136	C₂×D₅	—	2	4	no	—
137	C₄×C₅	—	2	2	no	—
138	C₄×C₅	C₄×D₄	1	2	yes	—
139	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₅	C₂×#41	1	1	yes	characteristic
141	Dic₅	—	2	2	no	—
142	Dic₅	—	2	2	no	—
143	Dic₅	#131	1	4	yes	complemented
144	#41	—	2	2	no	—
145	#41	—	2	2	no	—
146	#41	—	10	2	no	—
147	#41	—	10	2	no	—
148	C₂×Q₈	—	5	1	no	—
149	C₂×Q₈	C₄×D₅	1	1	yes	characteristic
150	C₂×D₄	—	5	4	no	—
151	C₂×D₄	—	5	2	no	—
152	C₂×D₄	—	5	1	no	—
153	C₂×D₄	C₄×D₅	1	2	yes	—
154	C₂×D₄	C₄×D₅	1	1	yes	characteristic
155	C₂²×C₄	—	5	2	no	—
156	C₂²×C₄	—	5	2	no	—
157	C₂²×C₄	C₂²×D₅	1	2	yes	—
158	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
159	M₁₆	—	2	2	no	—
160	M₁₆	—	10	4	no	—
161	M₁₆	—	10	4	no	—
162	M₁₆	—	10	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	5	4	no	—
165	C₂×C₈	—	5	1	no	—
166	C₂×C₈	—	5	1	no	—
167	C₂×C₈	—	5	2	no	—
168	C₂×C₈	—	10	4	no	—
169	C₂×C₈	—	10	4	no	—
170	C₂×C₈	C₂²×D₅	1	2	yes	—
171	C₂×C₈	C₂²×D₅	1	2	yes	—
172	C₄⋊C₄	—	5	4	no	—
173	C₂²⋊C₄	—	5	4	no	—
174	C₂²⋊C₄	—	5	4	no	—
175	C₄²	—	5	1	no	—
176	C₄²	—	5	1	no	—
177	C₄²	—	5	2	no	—
178	C₂×C₅	—	2	2	no	—
179	C₂×C₅	—	4	2	no	—
180	C₂×C₅	C₂×C₄×D₄	1	1	yes	characteristic
181	C₂×C₅	C₂×#131	1	2	yes	—
182	D₅	—	4	2	no	—
183	C₂³	—	5	2	no	—
184	C₂³	C₂×C₄×D₅	1	2	yes	—
185	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
186	Q₈	—	2	2	no	—
187	Q₈	—	10	2	no	—
188	D₄	—	2	4	no	—
189	D₄	—	2	2	no	—
190	D₄	—	10	8	no	—
191	D₄	—	10	4	no	—
192	D₄	—	10	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	5	2	no	—
196	C₂×C₄	—	5	2	no	—
197	C₂×C₄	—	5	2	no	—
198	C₂×C₄	—	10	4	no	—
199	C₂×C₄	—	10	2	no	—
200	C₂×C₄	—	10	2	no	—
201	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
202	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
203	C₂×C₄	D₄×D₅	1	2	yes	—
204	C₂×C₄	#711	1	2	yes	—
205	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	5	2	no	—
209	C₈	—	5	2	no	—
210	C₈	—	10	2	no	—
211	C₅	#2931	1	1	yes	characteristic, complemented, Sylow
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	10	2	no	—
216	C₂²	—	10	4	no	—
217	C₂²	#4027	1	2	yes	—
218	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	5	4	no	—
221	C₄	—	10	2	no	—
222	C₄	—	10	2	no	—
223	C₄	#4027	1	2	yes	—
224	C₄	C₂×D₄×D₅	1	1	yes	characteristic
225	C₄	C₂×#711	1	1	yes	characteristic
226	C₂	—	2	2	no	—
227	C₂	—	4	2	no	—
228	C₂	—	20	2	no	—
229	C₂	C₂×#4027	1	1	yes	characteristic
230	C₂	D₅×#131	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#408	C₂
prime radical	#66166	C₂
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#408	D₅
hypocenter	C₅	#2931
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
2. C₂×#4094	C₂	semidirect
3. C₂×#4053	C₂	semidirect
4. C₅×#408	C₂	semidirect
5. #65407	C₂	semidirect
6. #65399	C₂	semidirect
7. C₂×C₈×Dic₅	C₂	semidirect
8. #65018	C₂	semidirect
9. #65014	C₂	semidirect
63. C₂×#168	D₄	semidirect
76. #408	D₅	semidirect
143. Dic₅	#131	semidirect
211. C₅	#2931	semidirect

Maximal subgroups

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

subgroup	index
#66166	2
C₂×#4094	2
C₂×#4053	2
C₅×#408	2
#65407	2
#65399	2
C₂×C₈×Dic₅	2
#65018	2
#65014	2
#2931	5