Subgroups — #10610396

Order 640.

422 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66162	C₂	1	1	yes	maximal, characteristic
2	C₅×#408	C₂	1	1	yes	maximal, characteristic, complemented
3	#65523	C₂	1	1	yes	maximal, characteristic, complemented
4	#65416	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
6	#65017	C₂	1	2	yes	maximal, complemented
7	#65014	C₂	1	2	yes	maximal, complemented
8	D₅×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
9	#65009	C₂	1	2	yes	maximal, complemented
10	C₂×C₅×#41	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₅	C₂²	1	1	yes	characteristic
12	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
14	#4092	C₄	1	2	yes	—
15	#4084	C₄	1	2	yes	—
16	#4083	C₄	1	1	yes	characteristic
17	#4083	C₄	1	1	yes	characteristic
18	#4081	C₄	1	1	yes	characteristic
19	#4081	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊D₅	C₂²	1	2	yes	—
21	C₄×C₂²⋊D₅	C₂²	1	2	yes	—
22	#4072	C₂²	1	1	yes	characteristic
23	C₂×#681	C₂²	1	1	yes	characteristic, complemented
24	C₂²×#168	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₈×D₅	C₂²	1	2	yes	—
26	C₂×C₈×D₅	C₂²	1	2	yes	—
27	C₅×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₅×C₂²⋊C₈	C₂²	1	2	yes	complemented
29	C₂²⋊#168	C₂²	1	2	yes	—
30	C₂²⋊#168	C₂²	1	2	yes	complemented
31	#3952	C₂²	1	2	yes	—
32	#3952	C₂²	1	2	yes	—
33	#3947	C₂²	1	2	yes	complemented
34	#3946	C₂²	1	2	yes	complemented
35	#3946	C₂²	1	2	yes	complemented
36	C₈×Dic₅	C₂²	1	2	yes	complemented
37	#2977	—	5	1	no	maximal, Sylow
38	C₂³×D₅	C₂×C₄	1	1	yes	characteristic
39	C₅×#41	—	4	1	no	—
40	C₅×#41	—	4	1	no	—
41	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
44	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
50	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₅	—	4	1	no	—
52	C₂×C₄×D₅	—	4	1	no	—
53	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
54	C₂×C₄×D₅	C₂³	1	2	yes	—
55	C₅×M₁₆	—	4	1	no	—
56	C₂×C₅×C₈	—	4	1	no	—
57	C₂×C₅×C₈	C₂³	1	2	yes	—
58	C₂×C₅×C₈	C₂³	1	2	yes	—
59	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
60	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
61	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
62	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
63	#685	C₂×C₄	1	2	yes	—
64	#685	C₂×C₄	1	2	yes	—
65	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
66	#683	C₂×C₄	1	2	yes	—
67	#683	C₂×C₄	1	2	yes	—
68	C₄×Dic₅	C₂³	1	2	yes	—
69	#681	—	4	1	no	—
70	C₂×#168	—	4	1	no	—
71	C₂×#168	C₂³	1	2	yes	—
72	C₂×#168	C₂³	1	2	yes	—
73	C₈×D₅	—	4	2	no	—
74	C₈×D₅	—	4	2	no	—
75	#525	—	5	1	no	—
76	#435	—	5	2	no	—
77	#435	—	5	2	no	—
78	C₈×D₄	—	5	2	no	—
79	C₈×D₄	—	5	2	no	—
80	#433	—	5	2	no	—
81	#408	—	5	1	no	—
82	#408	D₅	1	1	yes	characteristic, complemented
83	C₂×C₂²⋊C₈	—	5	2	no	—
84	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₅	—	2	4	no	—
88	C₂²×D₅	—	4	1	no	—
89	C₂²×D₅	C₂²×C₄	1	2	yes	—
90	C₅×Q₈	—	4	1	no	—
91	C₅×D₄	—	4	1	no	—
92	C₅×D₄	—	4	1	no	—
93	C₅×D₄	—	4	1	no	—
94	C₂×C₄×C₅	—	2	1	no	—
95	C₂×C₄×C₅	—	2	1	no	—
96	C₂×C₄×C₅	—	2	1	no	—
97	C₂×C₄×C₅	—	2	1	no	—
98	C₂×C₄×C₅	—	2	1	no	—
99	C₂×C₄×C₅	—	2	1	no	—
100	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₅	—	2	4	no	—
105	C₂²⋊D₅	—	2	4	no	—
106	C₂×Dic₅	—	4	1	no	—
107	C₂×Dic₅	C₂²×C₄	1	2	yes	—
108	C₂×Dic₅	C₂²×C₄	1	2	yes	—
109	C₂×Dic₅	C₂²×C₄	1	2	yes	—
110	C₄×D₅	—	2	2	no	—
111	C₄×D₅	—	2	2	no	—
112	C₄×D₅	—	4	2	no	—
113	C₅×C₈	—	2	2	no	—
114	C₅×C₈	—	2	2	no	—
115	#168	—	2	2	no	—
116	#168	—	2	2	no	—
117	C₂×#41	C₂×D₅	1	1	yes	characteristic
118	C₂³×C₄	—	5	1	no	—
119	C₂×M₁₆	—	5	1	no	—
120	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic, complemented
121	C₂²×C₈	—	5	1	no	—
122	C₂²×C₈	—	5	2	no	—
123	C₂²×C₈	—	5	2	no	—
124	C₂²×C₈	C₂×D₅	1	1	yes	characteristic, complemented
125	#123	—	5	1	no	—
126	#123	—	5	1	no	—
127	C₂²⋊Q₈	—	5	2	no	—
128	#121	—	5	2	no	—
129	#120	—	5	1	no	—
130	#120	—	5	1	no	—
131	C₄×D₄	—	5	2	no	—
132	C₄×D₄	—	5	2	no	—
133	#117	—	5	1	no	—
134	C₄⋊C₈	—	5	2	no	—
135	C₄⋊C₈	—	5	2	no	—
136	C₂²⋊C₈	—	5	2	no	—
137	C₂²⋊C₈	—	5	2	no	—
138	C₂²⋊C₈	—	5	2	no	—
139	C₂²⋊C₈	—	5	2	no	—
140	C₂²⋊C₈	C₂×D₅	1	2	yes	—
141	C₂²⋊C₈	C₂×D₅	1	2	yes	complemented
142	#97	—	5	2	no	—
143	C₄×C₈	—	5	2	no	—
144	C₂²×C₅	—	2	1	no	—
145	C₂²×C₅	—	2	1	no	—
146	C₂²×C₅	—	2	1	no	—
147	C₂²×C₅	—	2	2	no	—
148	C₂²×C₅	—	2	2	no	—
149	C₂²×C₅	—	2	2	no	—
150	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
151	C₂×D₅	—	2	2	no	—
152	C₂×D₅	—	2	2	no	—
153	C₂×D₅	—	4	4	no	—
154	C₂×D₅	#131	1	4	yes	—
155	C₄×C₅	—	2	1	no	—
156	C₄×C₅	—	2	1	no	—
157	C₄×C₅	—	2	1	no	—
158	C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
159	C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
160	Dic₅	—	2	2	no	—
161	Dic₅	—	2	2	no	—
162	Dic₅	—	2	2	no	—
163	C₂⁴	—	5	1	no	—
164	#41	—	4	1	no	—
165	#41	—	4	1	no	—
166	C₂×Q₈	C₄×D₅	1	1	yes	characteristic
167	C₂×D₄	—	5	2	no	—
168	C₂×D₄	—	5	2	no	—
169	C₂×D₄	C₄×D₅	1	1	yes	characteristic
170	C₂×D₄	C₄×D₅	1	1	yes	characteristic
171	C₂×D₄	C₄×D₅	1	1	yes	characteristic
172	C₂²×C₄	—	5	1	no	—
173	C₂²×C₄	—	5	2	no	—
174	C₂²×C₄	—	5	2	no	—
175	C₂²×C₄	—	20	1	no	—
176	C₂²×C₄	—	20	1	no	—
177	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
180	M₁₆	—	4	1	no	—
181	M₁₆	—	20	1	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	5	2	no	—
184	C₂×C₈	—	5	2	no	—
185	C₂×C₈	—	20	2	no	—
186	C₂×C₈	—	20	2	no	—
187	C₂×C₈	—	20	1	no	—
188	C₂×C₈	C₂²×D₅	1	2	yes	—
189	C₂×C₈	C₂²×D₅	1	2	yes	—
190	C₄⋊C₄	—	5	2	no	—
191	C₄⋊C₄	—	5	2	no	—
192	C₄⋊C₄	—	5	2	no	—
193	C₂²⋊C₄	—	5	1	no	—
194	C₂²⋊C₄	—	5	1	no	—
195	C₂²⋊C₄	—	5	2	no	—
196	C₂²⋊C₄	—	5	2	no	—
197	C₂²⋊C₄	—	5	2	no	—
198	C₂²⋊C₄	—	5	2	no	—
199	C₄²	—	5	2	no	—
200	C₂×C₅	—	4	1	no	—
201	C₂×C₅	—	4	1	no	—
202	C₂×C₅	—	4	1	no	—
203	C₂×C₅	#518	1	1	yes	characteristic
204	C₂×C₅	C₂×#131	1	2	yes	—
205	D₅	—	2	4	no	—
206	C₂³	—	5	2	no	—
207	C₂³	—	10	4	no	—
208	C₂³	—	20	1	no	—
209	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
210	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
211	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
212	Q₈	—	4	1	no	—
213	D₄	—	4	1	no	—
214	D₄	—	4	1	no	—
215	D₄	—	4	1	no	—
216	D₄	—	10	4	no	—
217	D₄	—	10	4	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	5	2	no	—
225	C₂×C₄	—	5	2	no	—
226	C₂×C₄	—	5	2	no	—
227	C₂×C₄	—	10	2	no	—
228	C₂×C₄	—	10	2	no	—
229	C₂×C₄	—	20	2	no	—
230	C₂×C₄	—	20	1	no	—
231	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
234	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	—	2	2	no	—
237	C₈	—	10	2	no	—
238	C₈	—	10	2	no	—
239	C₅	#2977	1	1	yes	characteristic, complemented, Sylow
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	5	4	no	—
247	C₂²	—	10	2	no	—
248	C₂²	—	10	2	no	—
249	C₂²	—	20	4	no	—
250	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	10	2	no	—
255	C₄	—	10	2	no	—
256	C₄	—	10	2	no	—
257	C₄	#4144	1	1	yes	characteristic
258	C₄	#4144	1	1	yes	characteristic
259	C₂	—	4	1	no	—
260	C₂	—	4	1	no	—
261	C₂	—	4	1	no	—
262	C₂	—	10	4	no	—
263	C₂	#65820	1	1	yes	characteristic
264	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	#66162	C₂
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#408	D₅
hypocenter	C₅	#2977
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. C₅×#408	C₂	semidirect
3. #65523	C₂	semidirect
4. #65416	C₂	semidirect
5. C₈×C₂²⋊D₅	C₂	semidirect
6. #65017	C₂	semidirect
7. #65014	C₂	semidirect
8. D₅×C₂²⋊C₈	C₂	semidirect
9. #65009	C₂	semidirect
12. C₂×C₅×M₁₆	C₂²	semidirect
13. C₂²×C₅×C₈	C₂²	semidirect
23. C₂×#681	C₂²	semidirect
24. C₂²×#168	C₂²	semidirect
28. C₅×C₂²⋊C₈	C₂²	semidirect
30. C₂²⋊#168	C₂²	semidirect
33. #3947	C₂²	semidirect
34. #3946	C₂²	semidirect
35. #3946	C₂²	semidirect
36. C₈×Dic₅	C₂²	semidirect
82. #408	D₅	semidirect
120. C₂×M₁₆	C₂×D₅	semidirect
124. C₂²×C₈	C₂×D₅	semidirect
141. C₂²⋊C₈	C₂×D₅	semidirect
239. C₅	#2977	semidirect

Maximal subgroups

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

subgroup	index
#66162	2
C₅×#408	2
#65523	2
#65416	2
C₈×C₂²⋊D₅	2
#65017	2
#65014	2
D₅×C₂²⋊C₈	2
#65009	2
#2977	5