Subgroups — #10610777

Order 640.

388 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	C₅×#423	C₂	1	1	yes	maximal, characteristic
3	#65407	C₂	1	2	yes	maximal
4	M₁₆×Dic₅	C₂	1	2	yes	maximal
5	C₄×#681	C₂	1	1	yes	maximal, characteristic
6	#65120	C₂	1	2	yes	maximal, complemented
7	#65119	C₂	1	2	yes	maximal, complemented
8	#65117	C₂	1	4	yes	maximal, complemented
9	C₂²×Dic₁₀	C₄	1	1	yes	characteristic
10	C₂×C₅×M₁₆	C₂²	1	2	yes	—
11	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₅	C₄	1	1	yes	characteristic
13	C₂×#683	C₄	1	2	yes	—
14	C₂×C₄×Dic₅	C₂²	1	2	yes	—
15	C₂×#681	C₂²	1	1	yes	characteristic
16	C₂×#681	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₀	C₄	1	1	yes	characteristic
18	C₄×Dic₁₀	C₄	1	2	yes	—
19	C₄×Dic₁₀	C₄	1	1	yes	characteristic
20	C₄×Dic₁₀	C₂²	1	2	yes	—
21	C₄×Dic₁₀	C₂²	1	1	yes	characteristic
22	C₄×Dic₁₀	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊C₈	C₂²	1	2	yes	—
24	C₅×C₄⋊C₈	C₂²	1	2	yes	—
25	#3947	C₂²	1	4	yes	—
26	#3946	C₂²	1	4	yes	—
27	#3946	C₂²	1	4	yes	—
28	C₈×Dic₅	C₂²	1	4	yes	—
29	#3935	C₂²	1	2	yes	—
30	C₄×#168	C₂²	1	2	yes	—
31	Q₈×M₁₆	—	5	1	no	maximal, Sylow
32	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂²×Dic₅	C₂×C₄	1	2	yes	—
36	C₂²×Dic₅	C₂×C₄	1	2	yes	—
37	C₂×Dic₁₀	—	2	4	no	—
38	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
39	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
40	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
41	C₅×M₁₆	—	2	4	no	—
42	C₂×C₅×C₈	C₂³	1	4	yes	—
43	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
48	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
49	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
50	#683	C₂×C₄	1	2	yes	—
51	#683	C₂×C₄	1	2	yes	—
52	#683	C₂×C₄	1	2	yes	—
53	#683	C₂×C₄	1	2	yes	—
54	C₄×Dic₅	C₂×C₄	1	2	yes	—
55	C₄×Dic₅	C₂×C₄	1	2	yes	—
56	C₄×Dic₅	C₂³	1	2	yes	—
57	C₄×Dic₅	C₂³	1	2	yes	—
58	#681	—	2	2	no	—
59	#681	Q₈	1	4	yes	complemented
60	C₂×#168	C₂³	1	2	yes	—
61	C₂×#168	C₂³	1	2	yes	—
62	C₂×C₄×Q₈	—	5	1	no	—
63	#446	—	5	4	no	—
64	#446	—	5	2	no	—
65	C₈×Q₈	—	5	2	no	—
66	#423	—	5	2	no	—
67	#423	D₅	1	1	yes	characteristic
68	C₄×M₁₆	—	5	2	no	—
69	C₄×M₁₆	—	5	1	no	—
70	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₅	—	2	2	no	—
72	C₂×C₄×C₅	—	4	1	no	—
73	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
79	C₂×C₄×C₅	#41	1	2	yes	—
80	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₅	—	2	4	no	—
82	C₂×Dic₅	—	4	2	no	—
83	C₂×Dic₅	C₂²×C₄	1	2	yes	—
84	C₂×Dic₅	C₂²×C₄	1	2	yes	—
85	C₂×Dic₅	C₂²×C₄	1	2	yes	—
86	C₂×Dic₅	C₂²×C₄	1	2	yes	—
87	Dic₁₀	—	2	2	no	—
88	Dic₁₀	—	2	4	no	—
89	Dic₁₀	M₁₆	1	4	yes	complemented
90	C₅×C₈	—	2	4	no	—
91	#168	—	2	2	no	—
92	#168	C₂×Q₈	1	4	yes	complemented
93	C₂²×Q₈	—	5	1	no	—
94	C₂×M₁₆	—	5	1	no	—
95	C₂×M₁₆	—	5	1	no	—
96	C₂×M₁₆	C₂×D₅	1	2	yes	—
97	C₄×Q₈	—	5	1	no	—
98	C₄×Q₈	—	5	2	no	—
99	C₄×Q₈	—	5	1	no	—
100	C₄×Q₈	—	5	1	no	—
101	C₄×Q₈	—	5	2	no	—
102	C₄×Q₈	—	5	1	no	—
103	C₂×C₄⋊C₄	—	5	2	no	—
104	C₂×C₄⋊C₄	—	5	1	no	—
105	C₂×C₄²	—	5	2	no	—
106	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
107	C₄⋊C₈	—	5	4	no	—
108	C₄⋊C₈	—	5	4	no	—
109	C₄⋊C₈	C₂×D₅	1	2	yes	—
110	C₄⋊C₈	C₂×D₅	1	2	yes	—
111	#97	—	5	4	no	—
112	#97	—	5	2	no	—
113	C₄×C₈	—	5	4	no	—
114	C₄×C₈	—	5	2	no	—
115	C₂²×C₅	—	2	1	no	—
116	C₂²×C₅	—	2	1	no	—
117	C₂²×C₅	C₄×Q₈	1	2	yes	—
118	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
119	C₄×C₅	—	2	1	no	—
120	C₄×C₅	—	2	1	no	—
121	C₄×C₅	—	2	1	no	—
122	C₄×C₅	C₄×Q₈	1	2	yes	—
123	C₄×C₅	C₂×M₁₆	1	2	yes	—
124	C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
125	C₄×C₅	C₂×#41	1	1	yes	characteristic
126	Dic₅	—	2	2	no	—
127	Dic₅	—	2	2	no	—
128	Dic₅	—	2	2	no	—
129	Dic₅	C₂×M₁₆	1	4	yes	—
130	C₂×Q₈	—	5	1	no	—
131	C₂×Q₈	—	5	2	no	—
132	C₂×Q₈	—	5	1	no	—
133	C₂×Q₈	—	10	4	no	—
134	C₂²×C₄	—	5	2	no	—
135	C₂²×C₄	—	5	2	no	—
136	C₂²×C₄	C₄×D₅	1	1	yes	characteristic
137	C₂²×C₄	C₄×D₅	1	1	yes	characteristic
138	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
139	M₁₆	—	2	4	no	—
140	M₁₆	—	5	4	no	—
141	M₁₆	—	10	2	no	—
142	C₂×C₈	—	5	2	no	—
143	C₂×C₈	—	5	2	no	—
144	C₂×C₈	C₂²×D₅	1	4	yes	—
145	C₄⋊C₄	—	5	2	no	—
146	C₄⋊C₄	—	5	2	no	—
147	C₄⋊C₄	—	5	2	no	—
148	C₄⋊C₄	—	5	2	no	—
149	C₄⋊C₄	—	5	2	no	—
150	C₄⋊C₄	—	5	1	no	—
151	C₄⋊C₄	—	5	1	no	—
152	C₄²	—	5	2	no	—
153	C₄²	—	5	2	no	—
154	C₄²	—	5	2	no	—
155	C₄²	—	5	2	no	—
156	C₄²	C₄×D₅	1	1	yes	characteristic
157	C₄²	C₄×D₅	1	1	yes	characteristic
158	C₄²	C₂²×D₅	1	1	yes	characteristic
159	C₄²	C₂²×D₅	1	1	yes	characteristic
160	C₂×C₅	—	2	2	no	—
161	C₂×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
162	C₂×C₅	C₂²×M₁₆	1	1	yes	characteristic
163	C₂×C₅	#568	1	1	yes	characteristic
164	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
165	Q₈	—	5	4	no	—
166	Q₈	—	10	4	no	—
167	Q₈	—	10	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	5	2	no	—
171	C₂×C₄	—	5	2	no	—
172	C₂×C₄	—	5	2	no	—
173	C₂×C₄	—	5	2	no	—
174	C₂×C₄	—	10	4	no	—
175	C₂×C₄	—	20	2	no	—
176	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
181	C₂×C₄	#711	1	2	yes	—
182	C₂×C₄	Q₈×D₅	1	2	yes	—
183	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
184	C₈	—	2	4	no	—
185	C₈	—	5	4	no	—
186	C₈	—	10	2	no	—
187	C₅	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	#4033	1	2	yes	—
191	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	5	4	no	—
196	C₄	—	10	2	no	—
197	C₄	—	10	2	no	—
198	C₄	—	10	2	no	—
199	C₄	#4033	1	2	yes	—
200	C₄	D₅×M₁₆	1	2	yes	—
201	C₄	C₂×#711	1	1	yes	characteristic
202	C₄	C₂×Q₈×D₅	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	C₂×#4033	1	1	yes	characteristic
205	C₂	C₂×D₅×M₁₆	1	1	yes	characteristic
206	C₂	#66079	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#423	C₂
prime radical	C₂³×C₅	C₂²×C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#423	D₅
hypocenter	C₅	Q₈×M₁₆
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
6. #65120	C₂	semidirect
7. #65119	C₂	semidirect
8. #65117	C₂	semidirect
59. #681	Q₈	semidirect
89. Dic₁₀	M₁₆	semidirect
92. #168	C₂×Q₈	semidirect
187. C₅	Q₈×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₀	2
C₅×#423	2
#65407	2
M₁₆×Dic₅	2
C₄×#681	2
#65120	2
#65119	2
#65117	2
Q₈×M₁₆	5