Subgroups — #10611698

Order 640.

330 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65912	C₂	1	1	yes	maximal, characteristic
2	C₅×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#65129	C₂	1	2	yes	maximal
5	#65122	C₂	1	1	yes	maximal, characteristic
6	#65120	C₂	1	1	yes	maximal, characteristic, complemented
7	#65119	C₂	1	2	yes	maximal, complemented
8	#64998	C₂	1	2	yes	maximal
9	#64990	C₂	1	2	yes	maximal, complemented
10	#64974	C₂	1	1	yes	maximal, characteristic
11	C₈×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
13	#4092	C₄	1	1	yes	characteristic
14	#4090	C₄	1	1	yes	characteristic
15	#4043	C₄	1	1	yes	characteristic
16	#4042	C₄	1	2	yes	—
17	#4035	C₄	1	2	yes	—
18	#4034	C₄	1	1	yes	characteristic
19	#4018	C₂²	1	2	yes	—
20	C₄²×D₅	C₂²	1	1	yes	characteristic
21	C₄×Dic₁₀	C₂²	1	2	yes	—
22	C₄×Dic₁₀	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊C₈	C₂²	1	2	yes	—
24	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₅×#97	C₂²	1	2	yes	—
26	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
27	#3952	C₂²	1	2	yes	—
28	#3952	C₂²	1	1	yes	characteristic
29	#3952	C₂²	1	1	yes	characteristic
30	#3947	C₂²	1	2	yes	—
31	#3946	C₂²	1	1	yes	characteristic
32	#3946	C₂²	1	2	yes	—
33	#3946	C₂²	1	2	yes	—
34	#3946	C₂²	1	1	yes	characteristic
35	C₈×Dic₅	C₂²	1	2	yes	—
36	C₈×Dic₅	C₂²	1	2	yes	—
37	C₄⋊#168	C₂²	1	1	yes	characteristic
38	C₄⋊#168	C₂²	1	2	yes	—
39	C₄×#168	C₂²	1	2	yes	—
40	C₄×#168	C₂²	1	1	yes	characteristic
41	#2986	—	5	1	no	maximal, Sylow
42	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
47	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
48	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂³	1	2	yes	—
50	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
52	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
53	C₄²×C₅	C₂³	1	1	yes	characteristic
54	C₄²×C₅	C₂³	1	2	yes	—
55	#685	C₂×C₄	1	2	yes	—
56	#685	C₂×C₄	1	2	yes	—
57	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
58	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
59	#683	C₂×C₄	1	2	yes	—
60	#683	C₂×C₄	1	2	yes	—
61	#683	C₂×C₄	1	2	yes	—
62	#683	C₂×C₄	1	2	yes	—
63	#683	C₂×C₄	1	2	yes	—
64	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₅	C₂³	1	1	yes	characteristic
67	C₄×Dic₅	C₂³	1	2	yes	—
68	C₂×#168	C₂³	1	1	yes	characteristic
69	C₂×#168	C₂³	1	2	yes	—
70	C₂×#168	C₂³	1	1	yes	characteristic
71	#533	—	5	1	no	—
72	#446	—	5	1	no	—
73	#446	—	5	2	no	—
74	#446	D₅	1	1	yes	characteristic, complemented
75	C₈×Q₈	—	5	1	no	—
76	C₈×Q₈	—	5	2	no	—
77	C₈×Q₈	—	5	1	no	—
78	#433	—	5	2	no	—
79	#433	—	5	2	no	—
80	#431	—	5	1	no	—
81	#431	—	5	1	no	—
82	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
83	C₅×Q₈	—	2	2	no	—
84	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
89	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₅	C₂²×C₄	1	2	yes	—
92	C₂×Dic₅	C₂²×C₄	1	2	yes	—
93	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
94	C₄×D₅	—	2	2	no	—
95	C₄×D₅	—	2	1	no	—
96	C₄×D₅	—	2	1	no	—
97	C₄×D₅	—	2	2	no	—
98	Dic₁₀	—	2	2	no	—
99	Dic₁₀	—	2	4	no	—
100	C₅×C₈	—	2	1	no	—
101	C₅×C₈	—	2	1	no	—
102	C₅×C₈	—	2	2	no	—
103	#168	—	2	2	no	—
104	#168	—	2	1	no	—
105	#168	—	2	1	no	—
106	C₄⋊Q₈	—	5	1	no	—
107	C₄⋊Q₈	—	5	1	no	—
108	#125	—	5	2	no	—
109	C₂²⋊Q₈	—	5	1	no	—
110	C₂²⋊Q₈	—	5	2	no	—
111	C₂²⋊Q₈	—	5	1	no	—
112	C₄×Q₈	—	5	2	no	—
113	C₄×Q₈	—	5	1	no	—
114	C₄×Q₈	C₂×D₅	1	1	yes	characteristic
115	#117	—	5	2	no	—
116	C₂×C₄²	—	5	1	no	—
117	C₄⋊C₈	—	5	1	no	—
118	C₄⋊C₈	—	5	1	no	—
119	C₄⋊C₈	—	5	2	no	—
120	C₄⋊C₈	—	5	2	no	—
121	C₄⋊C₈	—	5	1	no	—
122	C₄⋊C₈	—	5	2	no	—
123	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
124	C₄⋊C₈	C₂×D₅	1	2	yes	—
125	C₂²⋊C₈	—	5	2	no	—
126	C₂²⋊C₈	—	5	1	no	—
127	C₂²⋊C₈	—	5	1	no	—
128	#97	—	5	2	no	—
129	#97	C₂×D₅	1	2	yes	—
130	C₄×C₈	—	5	2	no	—
131	C₄×C₈	—	5	2	no	—
132	C₄×C₈	—	5	1	no	—
133	C₄×C₈	—	5	2	no	—
134	C₄×C₈	C₂×D₅	1	1	yes	characteristic
135	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
136	C₂×D₅	—	2	1	no	—
137	C₂×D₅	—	2	1	no	—
138	C₂×D₅	—	2	1	no	—
139	C₄×C₅	—	2	2	no	—
140	C₄×C₅	—	2	1	no	—
141	C₄×C₅	—	2	2	no	—
142	C₄×C₅	#131	1	2	yes	—
143	C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
144	C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
145	Dic₅	—	2	1	no	—
146	Dic₅	—	2	2	no	—
147	Dic₅	—	2	2	no	—
148	Dic₅	—	2	1	no	—
149	Dic₅	#131	1	2	yes	—
150	C₂×Q₈	—	5	2	no	—
151	C₂×Q₈	—	5	1	no	—
152	C₂×Q₈	C₄×D₅	1	1	yes	characteristic
153	C₂²×C₄	—	5	1	no	—
154	C₂²×C₄	—	5	1	no	—
155	C₂²×C₄	—	5	1	no	—
156	C₂×C₈	—	5	2	no	—
157	C₂×C₈	—	5	1	no	—
158	C₂×C₈	—	5	1	no	—
159	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
160	C₂×C₈	C₂²×D₅	1	2	yes	—
161	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
162	C₄⋊C₄	—	5	2	no	—
163	C₄⋊C₄	—	5	2	no	—
164	C₄⋊C₄	—	5	2	no	—
165	C₄⋊C₄	—	5	2	no	—
166	C₄⋊C₄	—	5	2	no	—
167	C₄⋊C₄	—	5	2	no	—
168	C₄⋊C₄	—	5	1	no	—
169	C₄⋊C₄	C₄×D₅	1	2	yes	—
170	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
171	C₂²⋊C₄	—	5	2	no	—
172	C₂²⋊C₄	—	5	2	no	—
173	C₄²	—	5	1	no	—
174	C₄²	—	5	1	no	—
175	C₄²	—	5	1	no	—
176	C₄²	—	5	2	no	—
177	C₄²	C₂²×D₅	1	1	yes	characteristic
178	C₄²	C₂²×D₅	1	2	yes	—
179	C₂×C₅	#519	1	1	yes	characteristic
180	C₂×C₅	C₂×#131	1	1	yes	characteristic
181	C₂×C₅	C₂×#131	1	1	yes	characteristic
182	D₅	—	4	1	no	—
183	C₂³	—	5	1	no	—
184	Q₈	—	2	2	no	—
185	Q₈	—	10	4	no	—
186	Q₈	—	10	2	no	—
187	C₂×C₄	—	5	1	no	—
188	C₂×C₄	—	5	1	no	—
189	C₂×C₄	—	5	1	no	—
190	C₂×C₄	—	5	2	no	—
191	C₂×C₄	—	5	2	no	—
192	C₂×C₄	—	10	2	no	—
193	C₂×C₄	—	10	1	no	—
194	C₂×C₄	—	10	1	no	—
195	C₂×C₄	—	10	2	no	—
196	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
197	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
200	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	2	2	no	—
204	C₈	—	10	2	no	—
205	C₈	—	10	1	no	—
206	C₈	—	10	1	no	—
207	C₅	#2986	1	1	yes	characteristic, complemented, Sylow
208	C₂²	—	10	1	no	—
209	C₂²	—	10	1	no	—
210	C₂²	—	10	1	no	—
211	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	2	no	—
215	C₄	—	5	2	no	—
216	C₄	—	10	1	no	—
217	C₄	—	10	2	no	—
218	C₄	—	10	2	no	—
219	C₄	—	10	1	no	—
220	C₄	#4053	1	2	yes	—
221	C₄	#4147	1	1	yes	characteristic
222	C₄	#4150	1	1	yes	characteristic
223	C₂	—	20	1	no	—
224	C₂	#65906	1	1	yes	characteristic
225	C₂	C₂×#4053	1	1	yes	characteristic
226	C₂	D₅×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#446	C₂
prime radical	C₂²×D₅	C₂²×C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#446	D₅
hypocenter	C₅	#2986
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. C₅×#446	C₂	semidirect
3. Q₈×#168	C₂	semidirect
6. #65120	C₂	semidirect
7. #65119	C₂	semidirect
9. #64990	C₂	semidirect
11. C₈×Dic₁₀	C₂	semidirect
74. #446	D₅	semidirect
207. C₅	#2986	semidirect

Maximal subgroups

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

subgroup	index
#65912	2
C₅×#446	2
Q₈×#168	2
#65129	2
#65122	2
#65120	2
#65119	2
#64998	2
#64990	2
#64974	2
C₈×Dic₁₀	2
#2986	5