Subgroups — #10615526

Order 640.

492 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	C₂×C₅×#131	C₂	1	1	yes	maximal, characteristic, complemented
2	#66162	C₂	1	1	yes	maximal, characteristic
3	C₂²×#676	C₂	1	1	yes	maximal, characteristic, complemented
4	#65523	C₂	1	1	yes	maximal, characteristic, complemented
5	#65417	C₂	1	4	yes	maximal, complemented
6	#65415	C₂	1	1	yes	maximal, characteristic, complemented
7	#65408	C₂	1	1	yes	maximal, characteristic, complemented
8	#65400	C₂	1	4	yes	maximal, complemented
9	#65390	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#41	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₅	C₂²	1	1	yes	characteristic
12	C₅×#131	—	2	4	no	—
13	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₅×M₁₆	C₂²	1	2	yes	complemented
15	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₅×C₈	C₂²	1	2	yes	complemented
17	#4092	C₄	1	2	yes	—
18	#4084	C₄	1	2	yes	—
19	#4083	C₄	1	2	yes	—
20	#4081	C₄	1	2	yes	—
21	C₄×C₂²⋊D₅	C₂²	1	4	yes	—
22	#4072	C₂²	1	1	yes	characteristic
23	C₂×#681	C₂²	1	1	yes	characteristic
24	C₂²×#168	C₂²	1	1	yes	characteristic
25	C₂×#676	—	2	4	no	—
26	C₂×#676	C₂²	1	2	yes	—
27	C₂×#676	C₂²	1	2	yes	—
28	C₂²⋊#168	C₂²	1	4	yes	complemented
29	#3952	C₂²	1	4	yes	—
30	#3947	C₂²	1	2	yes	complemented
31	#3946	C₂²	1	4	yes	complemented
32	C₈×Dic₅	C₂²	1	2	yes	complemented
33	#2930	—	5	1	no	maximal, Sylow
34	C₂³×D₅	C₂×C₄	1	1	yes	characteristic
35	C₅×#41	—	2	2	no	—
36	C₅×#41	—	2	2	no	—
37	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅×D₄	C₂×C₄	1	2	yes	—
39	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂³	1	2	yes	—
41	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₂²⋊D₅	C₂×C₄	1	4	yes	—
43	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₅	—	2	2	no	—
45	C₂×C₄×D₅	—	2	2	no	—
46	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₅	C₂³	1	2	yes	—
48	C₅×M₁₆	—	2	2	no	—
49	C₅×M₁₆	—	2	4	no	—
50	C₂×C₅×C₈	—	2	4	no	—
51	C₂×C₅×C₈	D₄	1	4	yes	complemented
52	C₂×C₅×C₈	C₂³	1	2	yes	—
53	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
54	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
55	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
56	C₂²⋊Dic₅	C₂×C₄	1	4	yes	—
57	#685	C₂×C₄	1	4	yes	—
58	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
59	#683	C₂×C₄	1	4	yes	—
60	C₄×Dic₅	C₂³	1	2	yes	—
61	#681	—	2	2	no	—
62	C₂×#168	—	2	2	no	—
63	C₂×#168	C₂³	1	2	yes	—
64	C₂×#168	C₂³	1	2	yes	—
65	#676	—	2	4	no	—
66	#676	—	2	4	no	—
67	C₂×#131	D₅	1	1	yes	characteristic, complemented
68	C₂²×M₁₆	—	5	1	no	—
69	#525	—	5	1	no	—
70	#436	—	5	4	no	—
71	#435	—	5	4	no	—
72	#424	—	5	1	no	—
73	#408	—	5	1	no	—
74	#407	—	5	1	no	—
75	#405	—	5	1	no	—
76	C₂³×C₅	C₂²×C₄	1	2	yes	—
77	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂²×D₅	—	2	2	no	—
79	C₂²×D₅	—	4	2	no	—
80	C₂²×D₅	C₂²×C₄	1	2	yes	—
81	C₅×Q₈	—	2	2	no	—
82	C₅×D₄	—	2	4	no	—
83	C₅×D₄	—	2	2	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₅	#41	1	2	yes	—
90	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
91	C₂²⋊D₅	—	4	4	no	—
92	C₂×Dic₅	—	2	2	no	—
93	C₂×Dic₅	C₂²×C₄	1	4	yes	—
94	C₂×Dic₅	C₂²×C₄	1	2	yes	—
95	C₄×D₅	—	2	4	no	—
96	C₄×D₅	—	2	2	no	—
97	C₄×D₅	—	2	2	no	—
98	C₅×C₈	—	2	2	no	—
99	C₅×C₈	C₂×D₄	1	2	yes	—
100	C₅×C₈	C₂×D₄	1	2	yes	—
101	#168	—	2	2	no	—
102	#168	—	2	2	no	—
103	C₂×#41	C₂×D₅	1	1	yes	characteristic
104	C₂³×C₄	—	5	1	no	—
105	#131	—	2	4	no	—
106	C₂×M₁₆	—	5	1	no	—
107	C₂×M₁₆	—	5	2	no	—
108	C₂×M₁₆	—	5	2	no	—
109	C₂×M₁₆	—	10	4	no	—
110	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
111	C₂×M₁₆	C₂×D₅	1	2	yes	complemented
112	C₂²×C₈	—	5	1	no	—
113	C₂²×C₈	C₂×D₅	1	1	yes	characteristic
114	C₂²×C₈	C₂×D₅	1	2	yes	complemented
115	#123	—	5	2	no	—
116	C₂²⋊Q₈	—	5	2	no	—
117	#121	—	5	2	no	—
118	#120	—	5	2	no	—
119	C₄×D₄	—	5	4	no	—
120	#117	—	5	1	no	—
121	C₄⋊C₈	—	5	4	no	—
122	C₂²⋊C₈	—	5	4	no	—
123	C₂²⋊C₈	—	5	4	no	—
124	#97	—	5	2	no	—
125	C₄×C₈	—	5	2	no	—
126	C₂²×C₅	—	2	2	no	—
127	C₂²×C₅	—	2	2	no	—
128	C₂²×C₅	—	2	4	no	—
129	C₂²×C₅	C₄×D₄	1	2	yes	—
130	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
131	C₂×D₅	—	2	2	no	—
132	C₂×D₅	—	2	2	no	—
133	C₂×D₅	—	2	2	no	—
134	C₂×D₅	—	4	4	no	—
135	C₄×C₅	—	2	2	no	—
136	C₄×C₅	C₄×D₄	1	2	yes	—
137	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
138	C₄×C₅	C₂×#41	1	1	yes	characteristic
139	Dic₅	—	2	2	no	—
140	Dic₅	—	2	4	no	—
141	C₂⁴	—	5	1	no	—
142	#41	—	2	2	no	—
143	#41	—	2	2	no	—
144	C₂×Q₈	C₄×D₅	1	1	yes	characteristic
145	C₂×D₄	—	5	4	no	—
146	C₂×D₄	C₄×D₅	1	2	yes	—
147	C₂×D₄	C₄×D₅	1	1	yes	characteristic
148	C₂²×C₄	—	5	1	no	—
149	C₂²×C₄	—	5	2	no	—
150	C₂²×C₄	—	5	2	no	—
151	C₂²×C₄	—	10	2	no	—
152	C₂²×C₄	—	10	2	no	—
153	C₂²×C₄	C₂²×D₅	1	2	yes	—
154	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
155	M₁₆	—	2	2	no	—
156	M₁₆	—	2	4	no	—
157	M₁₆	—	10	2	no	—
158	M₁₆	—	10	4	no	—
159	M₁₆	—	10	4	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	5	2	no	—
162	C₂×C₈	—	5	2	no	—
163	C₂×C₈	—	10	2	no	—
164	C₂×C₈	C₂²⋊D₅	1	4	yes	complemented
165	C₂×C₈	C₂²×D₅	1	2	yes	—
166	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
167	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
168	C₄⋊C₄	—	5	2	no	—
169	C₄⋊C₄	—	5	4	no	—
170	C₂²⋊C₄	—	5	2	no	—
171	C₂²⋊C₄	—	5	4	no	—
172	C₂²⋊C₄	—	5	4	no	—
173	C₄²	—	5	2	no	—
174	C₂×C₅	—	2	2	no	—
175	C₂×C₅	—	4	2	no	—
176	C₂×C₅	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂×C₅	#568	1	2	yes	—
178	D₅	—	4	2	no	—
179	C₂³	—	5	2	no	—
180	C₂³	—	10	2	no	—
181	C₂³	—	20	2	no	—
182	C₂³	C₂×C₄×D₅	1	2	yes	—
183	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
184	Q₈	—	2	2	no	—
185	D₄	—	2	4	no	—
186	D₄	—	2	2	no	—
187	D₄	—	20	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	5	4	no	—
191	C₂×C₄	—	5	2	no	—
192	C₂×C₄	—	10	2	no	—
193	C₂×C₄	—	10	2	no	—
194	C₂×C₄	—	10	2	no	—
195	C₂×C₄	—	10	4	no	—
196	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
197	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
198	C₂×C₄	#709	1	2	yes	—
199	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
200	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
201	C₈	—	2	2	no	—
202	C₈	—	10	2	no	—
203	C₈	—	10	2	no	—
204	C₈	C₂×C₂²⋊D₅	1	2	yes	—
205	C₈	C₂×C₂²⋊D₅	1	2	yes	—
206	C₅	#2930	1	1	yes	characteristic, complemented, Sylow
207	C₂²	—	2	2	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	10	2	no	—
211	C₂²	—	10	2	no	—
212	C₂²	—	10	2	no	—
213	C₂²	—	20	4	no	—
214	C₂²	C₄×C₂²⋊D₅	1	2	yes	—
215	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	10	4	no	—
218	C₄	—	10	2	no	—
219	C₄	C₄×C₂²⋊D₅	1	2	yes	—
220	C₄	C₂×#709	1	1	yes	characteristic
221	C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	20	2	no	—
225	C₂	#66084	1	2	yes	—
226	C₂	C₂×C₄×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₂×C₅×#131	C₂
prime radical	#66162	C₂
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	C₂×#131	D₅
hypocenter	C₅	#2930
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#131	C₂	semidirect
3. C₂²×#676	C₂	semidirect
4. #65523	C₂	semidirect
5. #65417	C₂	semidirect
6. #65415	C₂	semidirect
7. #65408	C₂	semidirect
8. #65400	C₂	semidirect
9. #65390	C₂	semidirect
14. C₂×C₅×M₁₆	C₂²	semidirect
16. C₂²×C₅×C₈	C₂²	semidirect
28. C₂²⋊#168	C₂²	semidirect
30. #3947	C₂²	semidirect
31. #3946	C₂²	semidirect
32. C₈×Dic₅	C₂²	semidirect
51. C₂×C₅×C₈	D₄	semidirect
67. C₂×#131	D₅	semidirect
111. C₂×M₁₆	C₂×D₅	semidirect
114. C₂²×C₈	C₂×D₅	semidirect
164. C₂×C₈	C₂²⋊D₅	semidirect
206. C₅	#2930	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#131	2
#66162	2
C₂²×#676	2
#65523	2
#65417	2
#65415	2
#65408	2
#65400	2
#65390	2
#2930	5