Subgroups — #10610381

Order 640.

392 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₅×#408	C₂	1	1	yes	maximal, characteristic
3	#65523	C₂	1	1	yes	maximal, characteristic
4	#65411	C₂	1	1	yes	maximal, characteristic, complemented
5	#65407	C₂	1	1	yes	maximal, characteristic, complemented
6	#65390	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#3946	C₂	1	1	yes	maximal, characteristic, complemented
8	#65009	C₂	1	2	yes	maximal, complemented
9	#65009	C₂	1	2	yes	maximal, complemented
10	#65008	C₂	1	2	yes	maximal, complemented
11	#65007	C₂	1	2	yes	maximal, complemented
12	C₂×C₅×#41	C₂²	1	1	yes	characteristic
13	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
15	Q₈×Dic₅	C₄	1	1	yes	characteristic
16	Q₈×Dic₅	C₄	1	1	yes	characteristic
17	D₄×Dic₅	C₄	1	2	yes	—
18	D₄×Dic₅	C₄	1	1	yes	characteristic
19	D₄×Dic₅	C₄	1	2	yes	—
20	D₄×Dic₅	C₄	1	1	yes	characteristic
21	#4072	C₂²	1	1	yes	characteristic
22	#4072	C₂²	1	2	yes	—
23	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₅	C₂²	1	2	yes	—
25	C₂×#681	C₂²	1	1	yes	characteristic
26	C₂²×#168	C₂²	1	1	yes	characteristic
27	C₅×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₅×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₂²⋊#168	C₂²	1	2	yes	—
30	C₂²⋊#168	C₂²	1	2	yes	—
31	#3947	C₂²	1	2	yes	—
32	#3947	C₂²	1	2	yes	—
33	#3946	C₂²	1	1	yes	characteristic
34	#3946	C₂²	1	1	yes	characteristic
35	#3946	C₂²	1	1	yes	characteristic
36	#3946	C₂²	1	1	yes	characteristic
37	#3946	C₂²	1	2	yes	—
38	#3946	C₂²	1	2	yes	—
39	C₈×Dic₅	C₂²	1	2	yes	—
40	C₈×Dic₅	C₂²	1	2	yes	—
41	#2925	—	5	1	no	maximal, Sylow
42	C₅×#41	—	2	2	no	—
43	C₅×#41	—	2	2	no	—
44	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₅×D₄	C₂×C₄	1	2	yes	—
46	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂³	1	2	yes	—
48	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
49	C₂²×Dic₅	C₂×C₄	1	2	yes	—
50	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
51	C₂²×Dic₅	C₂×C₄	1	2	yes	—
52	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
53	C₅×M₁₆	—	2	2	no	—
54	C₂×C₅×C₈	—	2	2	no	—
55	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
56	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂³	1	2	yes	—
58	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
59	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
60	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
62	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
63	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
64	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₅	C₂×C₄	1	2	yes	—
66	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
67	C₄×Dic₅	C₂×C₄	1	2	yes	—
68	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₅	C₂³	1	2	yes	—
70	C₄×Dic₅	C₂³	1	1	yes	characteristic
71	C₄×Dic₅	C₂³	1	1	yes	characteristic
72	#681	—	2	2	no	—
73	C₂×#168	—	2	2	no	—
74	C₂×#168	C₂³	1	1	yes	characteristic
75	C₂×#168	C₂³	1	2	yes	—
76	C₂×#168	C₂³	1	1	yes	characteristic
77	C₄×#41	—	5	1	no	—
78	#433	—	5	2	no	—
79	#433	—	5	2	no	—
80	#432	—	5	2	no	—
81	#431	—	5	2	no	—
82	#423	—	5	1	no	—
83	C₂×C₄⋊C₈	—	5	1	no	—
84	#408	—	5	1	no	—
85	#408	D₅	1	1	yes	characteristic
86	#405	—	5	1	no	—
87	#405	—	5	1	no	—
88	C₂³×C₅	C₂²×C₄	1	2	yes	—
89	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₅×Q₈	—	2	2	no	—
91	C₅×D₄	—	2	4	no	—
92	C₅×D₄	—	2	2	no	—
93	C₂×C₄×C₅	—	2	2	no	—
94	C₂×C₄×C₅	—	2	2	no	—
95	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₅	#41	1	2	yes	—
98	C₂×C₄×C₅	#41	1	2	yes	—
99	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
100	C₂×Dic₅	—	2	2	no	—
101	C₂×Dic₅	—	2	4	no	—
102	C₂×Dic₅	—	4	1	no	—
103	C₂×Dic₅	—	4	2	no	—
104	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₅	C₂²×C₄	1	2	yes	—
106	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
107	C₂×Dic₅	C₂²×C₄	1	2	yes	—
108	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
109	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
110	C₅×C₈	—	2	1	no	—
111	C₅×C₈	—	2	1	no	—
112	C₅×C₈	—	2	2	no	—
113	#168	—	2	1	no	—
114	#168	—	2	1	no	—
115	#168	—	2	2	no	—
116	C₂×#41	C₂×D₅	1	1	yes	characteristic
117	C₂×M₁₆	—	5	1	no	—
118	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
119	C₂²×C₈	—	5	1	no	—
120	C₂²×C₈	C₂×D₅	1	1	yes	characteristic
121	C₄×Q₈	—	5	1	no	—
122	C₄×Q₈	—	5	1	no	—
123	C₄×D₄	—	5	2	no	—
124	C₄×D₄	—	5	1	no	—
125	C₄×D₄	—	5	2	no	—
126	C₄×D₄	—	5	1	no	—
127	#117	—	5	1	no	—
128	#117	—	5	2	no	—
129	C₂×C₄²	—	5	1	no	—
130	C₂×C₄²	—	5	2	no	—
131	C₄⋊C₈	—	5	1	no	—
132	C₄⋊C₈	—	5	1	no	—
133	C₄⋊C₈	—	5	1	no	—
134	C₄⋊C₈	—	5	1	no	—
135	C₄⋊C₈	—	5	2	no	—
136	C₄⋊C₈	—	5	2	no	—
137	C₂²⋊C₈	—	5	2	no	—
138	C₂²⋊C₈	—	5	2	no	—
139	C₂²⋊C₈	C₂×D₅	1	2	yes	—
140	C₂²⋊C₈	C₂×D₅	1	2	yes	—
141	#97	—	5	2	no	—
142	#97	—	5	2	no	—
143	C₄×C₈	—	5	2	no	—
144	C₄×C₈	—	5	2	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	#117	1	2	yes	—
151	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
152	C₄×C₅	—	2	2	no	—
153	C₄×C₅	#117	1	2	yes	—
154	C₄×C₅	C₂×#41	1	1	yes	characteristic
155	C₄×C₅	C₂×#41	1	1	yes	characteristic
156	Dic₅	—	2	1	no	—
157	Dic₅	—	2	2	no	—
158	Dic₅	—	2	1	no	—
159	Dic₅	—	2	1	no	—
160	Dic₅	—	2	2	no	—
161	Dic₅	#131	1	2	yes	—
162	#41	—	2	2	no	—
163	#41	—	2	2	no	—
164	C₂×Q₈	C₄×D₅	1	1	yes	characteristic
165	C₂×D₄	C₄×D₅	1	2	yes	—
166	C₂×D₄	C₄×D₅	1	1	yes	characteristic
167	C₂²×C₄	—	5	2	no	—
168	C₂²×C₄	—	5	1	no	—
169	C₂²×C₄	—	5	2	no	—
170	C₂²×C₄	—	5	1	no	—
171	C₂²×C₄	C₂²×D₅	1	2	yes	—
172	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
173	M₁₆	—	2	2	no	—
174	M₁₆	—	10	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	5	1	no	—
177	C₂×C₈	—	5	2	no	—
178	C₂×C₈	—	5	1	no	—
179	C₂×C₈	—	10	2	no	—
180	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
181	C₂×C₈	C₂²×D₅	1	2	yes	—
182	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
183	C₄⋊C₄	—	5	2	no	—
184	C₄⋊C₄	—	5	2	no	—
185	C₄⋊C₄	—	5	1	no	—
186	C₄⋊C₄	—	5	1	no	—
187	C₂²⋊C₄	—	5	2	no	—
188	C₂²⋊C₄	—	5	2	no	—
189	C₂²⋊C₄	—	5	2	no	—
190	C₄²	—	5	2	no	—
191	C₄²	—	5	1	no	—
192	C₄²	—	5	2	no	—
193	C₄²	—	5	1	no	—
194	C₄²	—	5	2	no	—
195	C₄²	—	5	1	no	—
196	C₄²	—	5	1	no	—
197	C₂×C₅	—	2	2	no	—
198	C₂×C₅	—	4	2	no	—
199	C₂×C₅	C₂×#117	1	1	yes	characteristic
200	C₂×C₅	C₂×#131	1	1	yes	characteristic
201	C₂×C₅	#568	1	1	yes	characteristic
202	C₂³	C₂×C₄×D₅	1	2	yes	—
203	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
204	Q₈	—	2	2	no	—
205	D₄	—	2	4	no	—
206	D₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	5	1	no	—
210	C₂×C₄	—	5	1	no	—
211	C₂×C₄	—	5	2	no	—
212	C₂×C₄	—	5	1	no	—
213	C₂×C₄	—	5	2	no	—
214	C₂×C₄	—	5	1	no	—
215	C₂×C₄	—	10	2	no	—
216	C₂×C₄	—	10	4	no	—
217	C₂×C₄	—	20	1	no	—
218	C₂×C₄	—	20	2	no	—
219	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
220	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
221	C₂×C₄	#711	1	2	yes	—
222	C₂×C₄	#711	1	2	yes	—
223	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	2	2	no	—
227	C₈	—	10	1	no	—
228	C₈	—	10	1	no	—
229	C₈	—	10	2	no	—
230	C₅	#2925	1	1	yes	characteristic, complemented, Sylow
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	#4023	1	2	yes	—
237	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	5	2	no	—
240	C₄	—	10	2	no	—
241	C₄	—	10	1	no	—
242	C₄	—	10	1	no	—
243	C₄	—	10	2	no	—
244	C₄	—	10	1	no	—
245	C₄	#4023	1	2	yes	—
246	C₄	C₂×#711	1	1	yes	characteristic
247	C₄	C₂×#711	1	1	yes	characteristic
248	C₂	—	2	2	no	—
249	C₂	—	4	2	no	—
250	C₂	C₂×#4023	1	1	yes	characteristic
251	C₂	D₅×#131	1	1	yes	characteristic
252	C₂	#66084	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#408	C₂
prime radical	C₂×C₅×#41	C₂²
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#408	D₅
hypocenter	C₅	#2925
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
4. #65411	C₂	semidirect
5. #65407	C₂	semidirect
6. #65390	C₂	semidirect
7. C₂×#3946	C₂	semidirect
8. #65009	C₂	semidirect
9. #65009	C₂	semidirect
10. #65008	C₂	semidirect
11. #65007	C₂	semidirect
230. C₅	#2925	semidirect

Maximal subgroups

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

subgroup	index
#41×Dic₅	2
C₅×#408	2
#65523	2
#65411	2
#65407	2
#65390	2
C₂×#3946	2
#65009	2
#65008	2
#65007	2
#2925	5