Subgroups — #10618162

Order 640.

1232 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	#66059	C₂	1	4	yes	maximal, complemented
3	#66032	C₂	1	4	yes	maximal, complemented
4	#65964	C₂	1	8	yes	maximal, complemented
5	#65919	C₂	1	4	yes	maximal, complemented
6	#65911	C₂	1	4	yes	maximal, complemented
7	#65859	C₂	1	2	yes	maximal, complemented
8	#65855	C₂	1	2	yes	maximal, complemented
9	C₂×#4020	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#709	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#713	C₂²	1	8	yes	complemented
12	C₂×#709	C₂²	1	1	yes	characteristic
13	C₂²×D₂₀	C₂²	1	2	yes	—
14	C₂²×D₂₀	C₂²	1	4	yes	—
15	C₅×C₄⋊Q₈	C₂²	1	2	yes	complemented
16	C₅×#125	C₂²	1	2	yes	complemented
17	C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
18	C₄×C₅×Q₈	C₂²	1	4	yes	complemented
19	C₅×#117	C₂²	1	2	yes	—
20	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
21	#4093	C₂²	1	8	yes	complemented
22	C₄×C₂²⋊D₅	C₂²	1	4	yes	—
23	#4072	C₂²	1	1	yes	characteristic
24	#4041	C₂²	1	8	yes	complemented
25	#4041	C₂²	1	8	yes	complemented
26	#4041	C₂²	1	16	yes	complemented
27	#4040	C₂²	1	16	yes	complemented
28	#4039	C₂²	1	8	yes	complemented
29	#4039	C₂²	1	8	yes	complemented
30	#4028	C₂²	1	8	yes	—
31	D₅×C₂²⋊C₄	C₂²	1	8	yes	—
32	#4021	C₂²	1	4	yes	complemented
33	#4020	C₂²	1	2	yes	—
34	#4020	C₂²	1	4	yes	—
35	#4020	C₂²	1	2	yes	—
36	#4020	C₂²	1	4	yes	complemented
37	C₄×D₂₀	C₂²	1	2	yes	complemented
38	C₄×D₂₀	C₂²	1	8	yes	complemented
39	#4018	C₂²	1	2	yes	complemented
40	C₄²×D₅	C₂²	1	2	yes	complemented
41	C₄×Dic₁₀	C₂²	1	2	yes	complemented
42	#3508	—	5	1	no	maximal, Sylow
43	C₂³×D₅	C₂³	1	4	yes	—
44	C₂×C₅×Q₈	C₂³	1	4	yes	—
45	C₂²×C₄×C₅	C₂³	1	2	yes	—
46	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₅	C₂³	1	2	yes	—
48	#713	—	4	16	no	—
49	#709	—	2	4	no	—
50	C₂×D₂₀	—	2	8	no	—
51	C₂×D₂₀	—	4	8	no	—
52	C₂×D₂₀	C₂³	1	8	yes	—
53	C₂×D₂₀	C₂³	1	4	yes	—
54	C₂×D₂₀	C₂³	1	4	yes	—
55	C₂×D₂₀	C₂³	1	1	yes	characteristic
56	C₂×D₂₀	C₂³	1	8	yes	—
57	C₂×D₂₀	C₂³	1	16	yes	—
58	C₂×C₄×D₅	C₂³	1	2	yes	—
59	C₂×C₄×D₅	C₂³	1	4	yes	—
60	C₂×C₄×D₅	C₂³	1	16	yes	—
61	C₂×Dic₁₀	C₂³	1	1	yes	characteristic
62	C₅×C₄⋊C₄	C₂³	1	4	yes	—
63	C₅×C₄⋊C₄	C₂³	1	8	yes	—
64	C₅×C₄⋊C₄	C₂³	1	4	yes	—
65	C₅×C₂²⋊C₄	C₂³	1	4	yes	—
66	C₄²×C₅	C₂³	1	2	yes	—
67	C₄²×C₅	C₂³	1	1	yes	characteristic
68	C₄²×C₅	C₂³	1	1	yes	characteristic
69	C₄²×C₅	C₂³	1	4	yes	—
70	C₂²⋊Dic₅	C₂³	1	2	yes	—
71	#685	C₂³	1	4	yes	—
72	#685	C₂³	1	16	yes	—
73	#685	C₂³	1	16	yes	—
74	C₄⋊Dic₅	C₂³	1	2	yes	—
75	#683	C₂³	1	4	yes	—
76	C₄×Dic₅	C₂³	1	4	yes	—
77	C₄×Dic₅	C₂³	1	2	yes	—
78	#559	—	5	4	no	—
79	#550	—	5	4	no	—
80	#546	—	5	8	no	—
81	#540	—	5	4	no	—
82	#535	—	5	2	no	—
83	#533	D₅	1	1	yes	characteristic, complemented
84	#532	—	5	4	no	—
85	C₂×#127	—	5	1	no	—
86	#518	—	5	2	no	—
87	C₄×#41	—	5	1	no	—
88	C₂³×C₅	C₂⁴	1	1	yes	characteristic
89	C₂²×D₅	—	2	8	no	—
90	C₂²×D₅	—	4	8	no	—
91	C₂²×D₅	C₂⁴	1	8	yes	—
92	C₂²×D₅	C₂⁴	1	2	yes	—
93	C₅×Q₈	—	2	8	no	—
94	C₂×C₄×C₅	—	2	4	no	—
95	C₂×C₄×C₅	#41	1	4	yes	complemented
96	C₂×C₄×C₅	C₂⁴	1	2	yes	—
97	C₂×C₄×C₅	C₂⁴	1	8	yes	—
98	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
100	C₂×C₄×C₅	C₂⁴	1	2	yes	—
101	C₂²⋊D₅	—	2	4	no	—
102	C₂×Dic₅	C₂⁴	1	2	yes	—
103	C₂×Dic₅	C₂⁴	1	4	yes	—
104	D₂₀	—	2	8	no	—
105	D₂₀	—	2	8	no	—
106	D₂₀	—	2	2	no	—
107	D₂₀	—	2	16	no	—
108	D₂₀	—	4	8	no	—
109	D₂₀	—	4	16	no	—
110	C₄×D₅	—	2	4	no	—
111	C₄×D₅	—	2	8	no	—
112	C₄×D₅	—	4	16	no	—
113	Dic₁₀	—	2	2	no	—
114	C₂×#41	—	5	8	no	—
115	C₂×#41	—	5	1	no	—
116	C₂²×D₄	—	5	2	no	—
117	C₂²×D₄	—	5	4	no	—
118	C₄⋊Q₈	C₂×D₅	1	2	yes	complemented
119	#127	—	5	2	no	—
120	#127	—	5	2	no	—
121	#127	—	5	4	no	—
122	#127	—	5	4	no	—
123	#125	C₂×D₅	1	2	yes	complemented
124	#124	—	5	8	no	—
125	#124	—	5	4	no	—
126	#123	—	5	16	no	—
127	C₂²⋊Q₈	C₂×D₅	1	4	yes	—
128	#121	—	5	8	no	—
129	#121	—	5	8	no	—
130	#121	—	5	16	no	—
131	#120	—	5	8	no	—
132	C₄×Q₈	—	5	2	no	—
133	C₄×Q₈	C₂×D₅	1	4	yes	complemented
134	C₄×D₄	—	5	4	no	—
135	C₄×D₄	—	5	8	no	—
136	C₄×D₄	—	5	8	no	—
137	C₄×D₄	—	5	2	no	—
138	C₄×D₄	—	5	8	no	—
139	#117	—	5	1	no	—
140	#117	—	5	2	no	—
141	#117	C₂×D₅	1	2	yes	—
142	C₂×C₂²⋊C₄	—	5	8	no	—
143	C₂×C₄²	—	5	2	no	—
144	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	C₂×#41	1	2	yes	—
147	C₂²×C₅	C₂⁵	1	1	yes	characteristic
148	C₂×D₅	—	2	2	no	—
149	C₂×D₅	—	2	8	no	—
150	C₂×D₅	—	2	4	no	—
151	C₂×D₅	—	2	16	no	—
152	C₂×D₅	—	8	8	no	—
153	C₄×C₅	—	2	8	no	—
154	C₄×C₅	—	2	2	no	—
155	C₄×C₅	C₂×#41	1	2	yes	—
156	C₄×C₅	C₂×#41	1	2	yes	—
157	C₄×C₅	H(5,𝔽₂)	1	4	yes	—
158	Dic₅	—	2	2	no	—
159	Dic₅	—	2	4	no	—
160	C₂⁴	—	5	4	no	—
161	#41	—	10	4	no	—
162	#41	—	20	16	no	—
163	C₂×Q₈	—	5	1	no	—
164	C₂×Q₈	C₂²×D₅	1	4	yes	—
165	C₂×D₄	—	5	2	no	—
166	C₂×D₄	—	5	4	no	—
167	C₂×D₄	—	5	4	no	—
168	C₂×D₄	—	5	1	no	—
169	C₂×D₄	—	5	8	no	—
170	C₂×D₄	—	5	16	no	—
171	C₂×D₄	—	5	8	no	—
172	C₂×D₄	—	10	8	no	—
173	C₂×D₄	—	20	8	no	—
174	C₂²×C₄	—	5	2	no	—
175	C₂²×C₄	—	5	4	no	—
176	C₂²×C₄	—	5	16	no	—
177	C₂²×C₄	C₂²×D₅	1	2	yes	—
178	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
179	C₄⋊C₄	—	5	2	no	—
180	C₄⋊C₄	—	5	4	no	—
181	C₄⋊C₄	C₂²×D₅	1	4	yes	—
182	C₄⋊C₄	C₂²×D₅	1	4	yes	—
183	C₄⋊C₄	C₂²×D₅	1	8	yes	—
184	C₂²⋊C₄	—	5	2	no	—
185	C₂²⋊C₄	—	5	16	no	—
186	C₂²⋊C₄	—	5	4	no	—
187	C₂²⋊C₄	—	5	16	no	—
188	C₂²⋊C₄	C₂²×D₅	1	4	yes	—
189	C₄²	—	5	2	no	—
190	C₄²	—	5	4	no	—
191	C₄²	C₂²×D₅	1	1	yes	characteristic
192	C₄²	C₂²×D₅	1	1	yes	characteristic
193	C₄²	C₂²×D₅	1	2	yes	—
194	C₄²	C₂²×D₅	1	4	yes	—
195	C₂×C₅	—	2	2	no	—
196	C₂×C₅	C₂²×#41	1	1	yes	characteristic
197	C₂×C₅	C₂×H(5,𝔽₂)	1	2	yes	—
198	D₅	—	4	2	no	—
199	D₅	—	4	8	no	—
200	C₂³	—	5	8	no	—
201	C₂³	—	5	2	no	—
202	C₂³	—	10	8	no	—
203	C₂³	—	20	8	no	—
204	C₂³	C₂³×D₅	1	1	yes	characteristic
205	Q₈	—	2	8	no	—
206	Q₈	—	10	2	no	—
207	D₄	—	10	4	no	—
208	D₄	—	10	8	no	—
209	D₄	—	10	8	no	—
210	D₄	—	10	2	no	—
211	D₄	—	10	16	no	—
212	D₄	—	20	8	no	—
213	D₄	—	20	16	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	5	2	no	—
216	C₂×C₄	—	5	4	no	—
217	C₂×C₄	—	10	4	no	—
218	C₂×C₄	—	10	8	no	—
219	C₂×C₄	—	20	16	no	—
220	C₂×C₄	#713	1	4	yes	complemented
221	C₂×C₄	C₂³×D₅	1	2	yes	—
222	C₂×C₄	C₂³×D₅	1	8	yes	—
223	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
224	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
225	C₂×C₄	C₂³×D₅	1	2	yes	—
226	C₅	#3508	1	1	yes	characteristic, complemented, Sylow
227	C₂²	—	2	2	no	—
228	C₂²	—	10	2	no	—
229	C₂²	—	10	8	no	—
230	C₂²	—	10	4	no	—
231	C₂²	—	10	16	no	—
232	C₂²	—	40	8	no	—
233	C₂²	C₂×#713	1	2	yes	—
234	C₂²	C₂⁴×D₅	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	8	no	—
237	C₄	—	10	2	no	—
238	C₄	—	10	4	no	—
239	C₄	C₂×#713	1	2	yes	—
240	C₄	C₂×#713	1	2	yes	—
241	C₄	#4149	1	4	yes	—
242	C₂	—	2	2	no	—
243	C₂	—	20	8	no	—
244	C₂	—	20	2	no	—
245	C₂	C₂²×#713	1	1	yes	characteristic
246	C₂	C₂×#4149	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁵
Fitting	C₅×#533	C₂
hypercenter	#533	D₅
hypocenter	C₅	#3508
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. C₅×#533	C₂	semidirect
2. #66059	C₂	semidirect
3. #66032	C₂	semidirect
4. #65964	C₂	semidirect
5. #65919	C₂	semidirect
6. #65911	C₂	semidirect
7. #65859	C₂	semidirect
8. #65855	C₂	semidirect
9. C₂×#4020	C₂	semidirect
10. C₄×#709	C₂	semidirect
11. C₂×#713	C₂²	semidirect
15. C₅×C₄⋊Q₈	C₂²	semidirect
16. C₅×#125	C₂²	semidirect
18. C₄×C₅×Q₈	C₂²	semidirect
21. #4093	C₂²	semidirect
24. #4041	C₂²	semidirect
25. #4041	C₂²	semidirect
26. #4041	C₂²	semidirect
27. #4040	C₂²	semidirect
28. #4039	C₂²	semidirect
29. #4039	C₂²	semidirect
32. #4021	C₂²	semidirect
36. #4020	C₂²	semidirect
37. C₄×D₂₀	C₂²	semidirect
38. C₄×D₂₀	C₂²	semidirect
39. #4018	C₂²	semidirect
40. C₄²×D₅	C₂²	semidirect
41. C₄×Dic₁₀	C₂²	semidirect
83. #533	D₅	semidirect
95. C₂×C₄×C₅	#41	semidirect
118. C₄⋊Q₈	C₂×D₅	semidirect
123. #125	C₂×D₅	semidirect
133. C₄×Q₈	C₂×D₅	semidirect
220. C₂×C₄	#713	semidirect
226. C₅	#3508	semidirect

Maximal subgroups

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

subgroup	index
C₅×#533	2
#66059	2
#66032	2
#65964	2
#65919	2
#65911	2
#65859	2
#65855	2
C₂×#4020	2
C₄×#709	2
#3508	5