Subgroups — #10611451

Order 640.

456 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#65906	C₂	1	2	yes	maximal
3	C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	#65319	C₂	1	2	yes	maximal, complemented
6	#65169	C₂	1	1	yes	maximal, characteristic
7	#65162	C₂	1	1	yes	maximal, characteristic, complemented
8	#65091	C₂	1	2	yes	maximal
9	#65078	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₂₀	C₂	1	1	yes	maximal, characteristic, complemented
11	#64978	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₈×D₅	C₂²	1	2	yes	—
13	C₂×C₅×Q₁₆	C₂²	1	1	yes	characteristic
14	C₄×C₅×Q₈	C₂²	1	2	yes	—
15	Q₈×Dic₅	C₂²	1	2	yes	—
16	C₂×#689	C₂²	1	2	yes	—
17	D₅×Q₁₆	C₄	1	8	yes	complemented
18	C₂×Dic₂₀	C₂²	1	1	yes	characteristic
19	C₂×#676	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
21	#4038	—	2	2	no	—
22	#4038	C₂²	1	2	yes	—
23	#4033	—	2	2	no	—
24	#4033	C₂²	1	2	yes	—
25	#4018	—	2	2	no	—
26	#4018	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₀	—	2	2	no	—
28	C₄×Dic₁₀	C₂²	1	2	yes	—
29	C₅×#107	C₂²	1	1	yes	characteristic
30	C₅×#103	C₂²	1	2	yes	—
31	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
32	#3967	C₂²	1	2	yes	—
33	#3950	C₂²	1	1	yes	characteristic
34	#3948	C₂²	1	2	yes	—
35	#3947	C₂²	1	1	yes	characteristic
36	C₈×Dic₅	C₂²	1	1	yes	characteristic
37	#3942	C₂²	1	2	yes	—
38	#3940	C₂²	1	2	yes	—
39	#3935	C₂²	1	1	yes	characteristic
40	#2950	—	5	1	no	maximal, Sylow
41	C₂×C₅×Q₈	C₂³	1	2	yes	—
42	Q₈×D₅	—	2	4	no	—
43	Q₈×D₅	C₂×C₄	1	8	yes	—
44	C₂×C₄×D₅	—	2	2	no	—
45	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₀	—	2	2	no	—
47	C₂×Dic₁₀	C₂³	1	2	yes	—
48	C₅×Q₁₆	C₂×C₄	1	4	yes	—
49	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₅×C₄⋊C₄	—	2	2	no	—
52	C₅×C₄⋊C₄	C₂³	1	2	yes	—
53	C₄²×C₅	—	2	2	no	—
54	C₄²×C₅	C₂³	1	1	yes	characteristic
55	#689	C₂×C₄	1	8	yes	—
56	#685	—	2	2	no	—
57	#685	D₄	1	1	yes	characteristic
58	#685	D₄	1	1	yes	characteristic
59	C₄⋊Dic₅	—	2	2	no	—
60	C₄⋊Dic₅	C₂³	1	2	yes	—
61	#683	—	2	2	no	—
62	#683	—	2	2	no	—
63	#683	D₄	1	1	yes	characteristic
64	#683	D₄	1	1	yes	characteristic
65	C₄×Dic₅	—	2	2	no	—
66	C₄×Dic₅	—	2	2	no	—
67	C₄×Dic₅	—	2	2	no	—
68	C₄×Dic₅	—	2	2	no	—
69	C₄×Dic₅	C₂³	1	1	yes	characteristic
70	C₂×#168	C₂³	1	1	yes	characteristic
71	C₂×#168	C₂³	1	1	yes	characteristic
72	Dic₂₀	C₂×C₄	1	4	yes	—
73	#676	—	2	2	no	—
74	C₈×D₅	C₂×C₄	1	4	yes	—
75	C₂²×Q₁₆	—	5	1	no	—
76	#519	—	5	2	no	—
77	#441	—	5	2	no	—
78	#441	—	5	2	no	—
79	C₄×Q₁₆	—	5	1	no	—
80	C₄×Q₁₆	—	5	1	no	—
81	C₄×Q₁₆	—	5	1	no	—
82	C₄×Q₁₆	D₅	1	1	yes	characteristic, complemented
83	#427	—	5	1	no	—
84	#419	—	5	2	no	—
85	#405	—	5	1	no	—
86	C₂²×D₅	C₂×D₄	1	1	yes	characteristic
87	C₅×Q₈	—	2	2	no	—
88	C₅×Q₈	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₅	—	2	2	no	—
90	C₂×C₄×C₅	—	2	2	no	—
91	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
94	C₂×Dic₅	—	2	2	no	—
95	C₂×Dic₅	—	2	2	no	—
96	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
97	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
98	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
99	C₄×D₅	—	2	8	no	—
100	C₄×D₅	C₂²×C₄	1	2	yes	—
101	C₄×D₅	#41	1	2	yes	—
102	Dic₁₀	—	2	2	no	—
103	Dic₁₀	—	2	8	no	—
104	Dic₁₀	C₂²×C₄	1	4	yes	—
105	C₅×C₈	—	2	1	no	—
106	C₅×C₈	C₂²×C₄	1	2	yes	—
107	#168	—	2	1	no	—
108	#168	C₂²×C₄	1	2	yes	—
109	C₂²×Q₈	—	5	2	no	—
110	C₂×Q₁₆	—	5	2	no	—
111	C₂×Q₁₆	—	5	8	no	—
112	C₂×Q₁₆	—	5	1	no	—
113	C₂×Q₁₆	C₂×D₅	1	1	yes	characteristic
114	C₂×M₁₆	—	5	1	no	—
115	C₂²×C₈	—	5	1	no	—
116	C₄×Q₈	—	5	2	no	—
117	C₄×Q₈	—	5	2	no	—
118	C₄×Q₈	—	5	2	no	—
119	C₄×Q₈	—	10	2	no	—
120	C₄×Q₈	—	10	2	no	—
121	C₄×Q₈	C₂×D₅	1	2	yes	—
122	#117	—	5	1	no	—
123	#117	—	5	2	no	—
124	#117	—	10	2	no	—
125	#117	—	10	2	no	—
126	#107	—	5	1	no	—
127	#107	C₂×D₅	1	1	yes	characteristic
128	#106	—	5	2	no	—
129	#103	—	5	2	no	—
130	#103	—	5	2	no	—
131	#103	—	5	2	no	—
132	#103	C₂×D₅	1	2	yes	—
133	#97	—	5	1	no	—
134	#97	—	5	1	no	—
135	C₄×C₈	—	5	1	no	—
136	C₄×C₈	C₂×D₅	1	1	yes	characteristic
137	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
138	C₂×D₅	—	2	1	no	—
139	C₂×D₅	—	2	1	no	—
140	C₂×D₅	C₄×D₄	1	2	yes	—
141	C₄×C₅	—	2	4	no	—
142	C₄×C₅	—	2	1	no	—
143	C₄×C₅	—	2	1	no	—
144	C₄×C₅	—	4	2	no	—
145	C₄×C₅	C₂³×C₄	1	1	yes	characteristic
146	C₄×C₅	C₂×#41	1	1	yes	characteristic
147	Dic₅	—	2	4	no	—
148	Dic₅	—	2	1	no	—
149	Dic₅	—	2	1	no	—
150	Dic₅	—	4	2	no	—
151	Dic₅	C₄×D₄	1	2	yes	—
152	C₂×Q₈	—	5	8	no	—
153	C₂×Q₈	—	5	2	no	—
154	C₂×Q₈	—	10	2	no	—
155	C₂×Q₈	—	10	4	no	—
156	C₂×Q₈	C₂²×D₅	1	2	yes	—
157	C₂²×C₄	—	5	1	no	—
158	C₂²×C₄	—	10	2	no	—
159	Q₁₆	—	5	8	no	—
160	Q₁₆	—	5	4	no	—
161	Q₁₆	C₄×D₅	1	4	yes	—
162	M₁₆	—	10	2	no	—
163	C₂×C₈	—	5	1	no	—
164	C₂×C₈	—	5	4	no	—
165	C₂×C₈	—	5	1	no	—
166	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
167	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	5	1	no	—
170	C₄⋊C₄	—	5	1	no	—
171	C₄⋊C₄	—	5	2	no	—
172	C₄⋊C₄	—	10	2	no	—
173	C₄⋊C₄	—	10	2	no	—
174	C₄⋊C₄	—	10	2	no	—
175	C₄⋊C₄	C₂²×D₅	1	2	yes	—
176	C₂²⋊C₄	—	5	1	no	—
177	C₂²⋊C₄	—	5	1	no	—
178	C₂²⋊C₄	—	10	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	5	1	no	—
181	C₄²	—	10	2	no	—
182	C₄²	—	10	2	no	—
183	C₄²	—	10	2	no	—
184	C₄²	—	10	2	no	—
185	C₄²	C₂²×D₅	1	1	yes	characteristic
186	C₂×C₅	C₂×C₄×D₄	1	1	yes	characteristic
187	C₂×C₅	#578	1	1	yes	characteristic
188	C₂×C₅	#578	1	1	yes	characteristic
189	D₅	—	2	2	no	—
190	C₂³	—	5	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	5	4	no	—
193	Q₈	—	10	8	no	—
194	Q₈	—	10	2	no	—
195	Q₈	C₂×C₄×D₅	1	4	yes	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	5	1	no	—
199	C₂×C₄	—	5	2	no	—
200	C₂×C₄	—	5	2	no	—
201	C₂×C₄	—	5	1	no	—
202	C₂×C₄	—	5	1	no	—
203	C₂×C₄	—	10	8	no	—
204	C₂×C₄	—	10	2	no	—
205	C₂×C₄	—	10	2	no	—
206	C₂×C₄	D₄×D₅	1	1	yes	characteristic
207	C₂×C₄	D₄×D₅	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	5	2	no	—
211	C₈	—	10	1	no	—
212	C₈	C₂×C₄×D₅	1	2	yes	—
213	C₅	#2950	1	1	yes	characteristic, complemented, Sylow
214	C₂²	—	5	2	no	—
215	C₂²	—	10	1	no	—
216	C₂²	—	10	1	no	—
217	C₂²	C₂×D₄×D₅	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	4	no	—
221	C₄	—	4	2	no	—
222	C₄	—	5	2	no	—
223	C₄	—	10	1	no	—
224	C₄	—	10	1	no	—
225	C₄	—	10	4	no	—
226	C₄	—	20	2	no	—
227	C₄	C₂²×C₄×D₅	1	1	yes	characteristic
228	C₄	D₅×#41	1	1	yes	characteristic
229	C₂	—	10	2	no	—
230	C₂	C₄×D₄×D₅	1	1	yes	characteristic
231	C₂	#66100	1	1	yes	characteristic
232	C₂	#66105	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅
commutator	C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅
Fitting	C₄×C₅×Q₁₆	C₂
socle	C₂²×C₅	C₂²×D₄
prime radical	C₂²×D₅	C₂×D₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	C₄×Q₁₆	D₅
hypocenter	C₅	#2950
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×C₅×Q₁₆	C₂	semidirect
4. Q₁₆×Dic₅	C₂	semidirect
5. #65319	C₂	semidirect
7. #65162	C₂	semidirect
9. #65078	C₂	semidirect
10. C₄×Dic₂₀	C₂	semidirect
17. D₅×Q₁₆	C₄	semidirect
82. C₄×Q₁₆	D₅	semidirect
213. C₅	#2950	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅×Q₁₆	2
#65906	2
C₄×C₅×Q₁₆	2
Q₁₆×Dic₅	2
#65319	2
#65169	2
#65162	2
#65091	2
#65078	2
C₄×Dic₂₀	2
#64978	2
#2950	5