Subgroups — #10610784

Order 640.

354 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	#65813	C₂	1	1	yes	maximal, characteristic
2	C₅×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	#65417	C₂	1	2	yes	maximal, complemented
4	#65416	C₂	1	2	yes	maximal, complemented
5	#65214	C₂	1	1	yes	maximal, characteristic, complemented
6	#65129	C₂	1	2	yes	maximal, complemented
7	#65128	C₂	1	2	yes	maximal, complemented
8	#65127	C₂	1	1	yes	maximal, characteristic, complemented
9	#65123	C₂	1	1	yes	maximal, characteristic, complemented
10	#65120	C₂	1	1	yes	maximal, characteristic, complemented
11	#65119	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅×M₁₆	C₂²	1	2	yes	—
13	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
14	#4076	C₄	1	1	yes	characteristic
15	#4075	C₄	1	2	yes	—
16	C₄×C₂²⋊D₅	C₂²	1	2	yes	—
17	#4070	C₄	1	1	yes	characteristic
18	C₂×#676	C₂²	1	2	yes	—
19	C₂×C₈×D₅	C₂²	1	2	yes	—
20	#4022	C₄	1	2	yes	—
21	#4021	C₄	1	1	yes	characteristic
22	C₄×D₂₀	C₂²	1	1	yes	characteristic
23	#4018	C₂²	1	2	yes	—
24	#4016	C₄	1	1	yes	characteristic
25	C₄×Dic₁₀	C₂²	1	1	yes	characteristic
26	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₅×C₄⋊C₈	C₂²	1	2	yes	—
29	C₂²⋊#168	C₂²	1	1	yes	characteristic
30	C₂²⋊#168	C₂²	1	1	yes	characteristic
31	#3952	C₂²	1	2	yes	—
32	#3952	C₂²	1	2	yes	—
33	#3947	C₂²	1	2	yes	—
34	#3946	C₂²	1	2	yes	—
35	#3946	C₂²	1	2	yes	—
36	C₈×Dic₅	C₂²	1	2	yes	—
37	C₄⋊#168	C₂²	1	1	yes	characteristic
38	C₄⋊#168	C₂²	1	1	yes	characteristic
39	C₄×#168	C₂²	1	1	yes	characteristic
40	C₄×#168	C₂²	1	1	yes	characteristic
41	#2993	—	5	1	no	maximal, Sylow
42	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
46	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₅	C₂³	1	2	yes	—
48	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
49	C₅×M₁₆	—	4	2	no	—
50	C₂×C₅×C₈	C₂³	1	2	yes	—
51	C₂×C₅×C₈	C₂³	1	2	yes	—
52	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₅	C₂³	1	1	yes	characteristic
55	C₄²×C₅	C₂³	1	1	yes	characteristic
56	C₂²⋊Dic₅	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	1	yes	characteristic
62	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
63	#683	C₂×C₄	1	2	yes	—
64	#683	C₂×C₄	1	2	yes	—
65	#683	C₂×C₄	1	2	yes	—
66	#683	C₂×C₄	1	2	yes	—
67	C₄×Dic₅	C₂³	1	2	yes	—
68	C₂×#168	C₂³	1	1	yes	characteristic
69	C₂×#168	C₂³	1	1	yes	characteristic
70	C₂×#168	C₂³	1	1	yes	characteristic
71	C₂×#168	C₂³	1	1	yes	characteristic
72	#676	—	4	2	no	—
73	C₈×D₅	—	4	2	no	—
74	#529	—	5	1	no	—
75	#446	—	5	1	no	—
76	C₈×Q₈	—	5	1	no	—
77	#436	—	5	2	no	—
78	#436	—	5	1	no	—
79	#435	—	5	2	no	—
80	C₈×D₄	—	5	1	no	—
81	#433	—	5	2	no	—
82	#431	—	5	1	no	—
83	#424	—	5	2	no	—
84	#423	D₅	1	1	yes	characteristic, complemented
85	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₅	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	—	2	1	no	—
89	C₂×C₄×C₅	—	2	1	no	—
90	C₂×C₄×C₅	—	4	1	no	—
91	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₅	—	4	2	no	—
98	C₂×Dic₅	C₂²×C₄	1	2	yes	—
99	C₂×Dic₅	C₂²×C₄	1	2	yes	—
100	C₂×Dic₅	C₂²×C₄	1	2	yes	—
101	D₂₀	—	2	2	no	—
102	C₄×D₅	—	2	2	no	—
103	C₄×D₅	—	2	2	no	—
104	Dic₁₀	—	2	2	no	—
105	C₅×C₈	—	2	2	no	—
106	C₅×C₈	—	2	2	no	—
107	#168	—	2	1	no	—
108	#168	—	2	1	no	—
109	#168	—	2	1	no	—
110	#168	—	2	1	no	—
111	C₂×M₁₆	—	5	2	no	—
112	C₂×M₁₆	C₂×D₅	1	2	yes	—
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	—	5	1	no	—
121	C₄×D₄	—	5	2	no	—
122	C₄×D₄	—	5	1	no	—
123	#117	—	5	2	no	—
124	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
125	C₄⋊C₈	—	5	2	no	—
126	C₄⋊C₈	—	5	2	no	—
127	C₄⋊C₈	—	5	1	no	—
128	C₄⋊C₈	—	5	1	no	—
129	C₄⋊C₈	C₂×D₅	1	2	yes	—
130	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
131	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
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	2	no	—
138	C₄×C₈	—	5	1	no	—
139	C₄×C₈	—	5	1	no	—
140	C₂²×C₅	—	2	1	no	—
141	C₂²×C₅	—	2	1	no	—
142	C₂²×C₅	—	2	1	no	—
143	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
144	C₂×D₅	—	2	2	no	—
145	C₂×D₅	—	2	2	no	—
146	C₂×D₅	—	2	2	no	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	—	2	1	no	—
149	C₄×C₅	—	2	1	no	—
150	C₄×C₅	—	2	1	no	—
151	C₄×C₅	#131	1	2	yes	—
152	C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
153	C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
154	Dic₅	—	2	2	no	—
155	Dic₅	—	2	2	no	—
156	Dic₅	—	2	2	no	—
157	C₂×Q₈	—	5	1	no	—
158	C₂×D₄	—	5	2	no	—
159	C₂×D₄	—	5	1	no	—
160	C₂²×C₄	—	5	2	no	—
161	C₂²×C₄	C₄×D₅	1	1	yes	characteristic
162	C₂²×C₄	C₄×D₅	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
164	M₁₆	—	4	2	no	—
165	M₁₆	—	20	2	no	—
166	C₂×C₈	—	5	1	no	—
167	C₂×C₈	—	5	1	no	—
168	C₂×C₈	—	5	1	no	—
169	C₂×C₈	—	5	1	no	—
170	C₂×C₈	—	20	2	no	—
171	C₂×C₈	C₂²×D₅	1	2	yes	—
172	C₂×C₈	C₂²×D₅	1	2	yes	—
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	—	5	1	no	—
178	C₄⋊C₄	—	5	2	no	—
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	2	no	—
185	C₄²	C₄×D₅	1	1	yes	characteristic
186	C₄²	C₄×D₅	1	1	yes	characteristic
187	C₄²	C₂²×D₅	1	1	yes	characteristic
188	C₄²	C₂²×D₅	1	1	yes	characteristic
189	C₂×C₅	—	4	1	no	—
190	C₂×C₅	#520	1	1	yes	characteristic
191	C₂×C₅	C₂×#131	1	1	yes	characteristic
192	C₂×C₅	#568	1	1	yes	characteristic
193	D₅	—	4	2	no	—
194	C₂³	—	5	2	no	—
195	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
196	Q₈	—	10	2	no	—
197	D₄	—	10	2	no	—
198	D₄	—	20	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	5	2	no	—
204	C₂×C₄	—	5	2	no	—
205	C₂×C₄	—	5	2	no	—
206	C₂×C₄	—	10	2	no	—
207	C₂×C₄	—	10	2	no	—
208	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
209	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
213	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	2	2	no	—
216	C₈	—	10	1	no	—
217	C₈	—	10	1	no	—
218	C₈	—	10	1	no	—
219	C₈	—	10	1	no	—
220	C₅	#2993	1	1	yes	characteristic, complemented, Sylow
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	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	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	10	2	no	—
233	C₄	—	10	2	no	—
234	C₄	—	10	2	no	—
235	C₄	#4053	1	2	yes	—
236	C₄	#4144	1	1	yes	characteristic
237	C₄	#4147	1	1	yes	characteristic
238	C₂	—	4	1	no	—
239	C₂	—	20	2	no	—
240	C₂	#65838	1	1	yes	characteristic
241	C₂	C₂×#4053	1	1	yes	characteristic
242	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	#4076	C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#423	D₅
hypocenter	C₅	#2993
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
2. C₅×#423	C₂	semidirect
3. #65417	C₂	semidirect
4. #65416	C₂	semidirect
5. #65214	C₂	semidirect
6. #65129	C₂	semidirect
7. #65128	C₂	semidirect
8. #65127	C₂	semidirect
9. #65123	C₂	semidirect
10. #65120	C₂	semidirect
11. #65119	C₂	semidirect
84. #423	D₅	semidirect
220. C₅	#2993	semidirect

Maximal subgroups

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

subgroup	index
#65813	2
C₅×#423	2
#65417	2
#65416	2
#65214	2
#65129	2
#65128	2
#65127	2
#65123	2
#65120	2
#65119	2
#2993	5