Subgroups — #10611277

Order 640.

410 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65882	C₂	1	1	yes	maximal, characteristic
2	C₅×#436	C₂	1	1	yes	maximal, characteristic, complemented
3	#65418	C₂	1	2	yes	maximal, complemented
4	#65417	C₂	1	2	yes	maximal, complemented
5	D₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	#65127	C₂	1	1	yes	maximal, characteristic, complemented
7	#65122	C₂	1	1	yes	maximal, characteristic, complemented
8	#65016	C₂	1	2	yes	maximal, complemented
9	#65014	C₂	1	2	yes	maximal, complemented
10	C₈×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
11	#64974	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#709	C₂²	1	2	yes	—
13	C₂×C₅×M₁₆	C₂²	1	2	yes	—
14	C₄×C₅×D₄	C₂²	1	1	yes	characteristic
15	#4086	C₄	1	1	yes	characteristic
16	#4084	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₅	C₂²	1	2	yes	—
18	C₂²×#168	C₂²	1	2	yes	—
19	C₂×#676	C₂²	1	2	yes	—
20	C₂×C₈×D₅	C₂²	1	2	yes	—
21	#4041	C₄	1	1	yes	characteristic
22	#4034	C₄	1	1	yes	characteristic
23	#4031	C₄	1	2	yes	—
24	#4030	C₄	1	2	yes	—
25	C₄×D₂₀	C₂²	1	1	yes	characteristic
26	C₄²×D₅	C₂²	1	1	yes	characteristic
27	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₅×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
30	C₂²⋊#168	C₂²	1	2	yes	—
31	#3952	C₂²	1	2	yes	—
32	#3952	C₂²	1	2	yes	—
33	#3952	C₂²	1	2	yes	—
34	#3952	C₂²	1	1	yes	characteristic
35	#3952	C₂²	1	1	yes	characteristic
36	#3946	C₂²	1	1	yes	characteristic
37	#3946	C₂²	1	1	yes	characteristic
38	C₈×Dic₅	C₂²	1	2	yes	—
39	C₄⋊#168	C₂²	1	1	yes	characteristic
40	C₄×#168	C₂²	1	1	yes	characteristic
41	#2979	—	5	1	no	maximal, Sylow
42	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂³	1	2	yes	—
44	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
47	#709	—	4	2	no	—
48	#709	—	4	2	no	—
49	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
50	C₂×D₂₀	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₅	C₂³	1	2	yes	—
55	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
56	C₅×M₁₆	—	4	2	no	—
57	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
58	C₂×C₅×C₈	C₂³	1	2	yes	—
59	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
60	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
61	C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
62	C₄²×C₅	C₂³	1	1	yes	characteristic
63	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
64	#685	C₂×C₄	1	2	yes	—
65	#685	C₂×C₄	1	2	yes	—
66	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
67	#683	C₂×C₄	1	2	yes	—
68	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₅	C₂³	1	1	yes	characteristic
71	C₂×#168	—	4	2	no	—
72	C₂×#168	C₂³	1	1	yes	characteristic
73	C₂×#168	C₂³	1	1	yes	characteristic
74	C₂×#168	C₂³	1	2	yes	—
75	#676	—	4	2	no	—
76	C₈×D₅	—	4	2	no	—
77	#532	—	5	1	no	—
78	#436	—	5	2	no	—
79	#436	—	5	1	no	—
80	#436	D₅	1	1	yes	characteristic, complemented
81	C₈×D₄	—	5	1	no	—
82	C₈×D₄	—	5	2	no	—
83	C₈×D₄	—	5	1	no	—
84	#431	—	5	1	no	—
85	#431	—	5	1	no	—
86	#408	—	5	2	no	—
87	#408	—	5	2	no	—
88	C₂³×C₅	C₂²×C₄	1	2	yes	—
89	C₂²×D₅	C₂²×C₄	1	2	yes	—
90	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
91	C₅×D₄	—	2	2	no	—
92	C₂×C₄×C₅	—	2	2	no	—
93	C₂×C₄×C₅	—	2	2	no	—
94	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
98	C₂²⋊D₅	—	2	4	no	—
99	C₂²⋊D₅	—	4	2	no	—
100	C₂²⋊D₅	—	4	2	no	—
101	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₅	C₂²×C₄	1	2	yes	—
104	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
105	D₂₀	—	2	2	no	—
106	D₂₀	—	4	2	no	—
107	C₄×D₅	—	2	2	no	—
108	C₄×D₅	—	2	2	no	—
109	C₄×D₅	—	2	2	no	—
110	C₄×D₅	—	2	1	no	—
111	C₄×D₅	—	2	1	no	—
112	C₄×D₅	—	2	2	no	—
113	Dic₁₀	—	4	2	no	—
114	C₅×C₈	—	2	1	no	—
115	C₅×C₈	—	2	2	no	—
116	C₅×C₈	—	2	1	no	—
117	#168	—	2	2	no	—
118	#168	—	2	1	no	—
119	#168	—	2	1	no	—
120	C₂×#41	—	5	2	no	—
121	C₂×M₁₆	—	5	2	no	—
122	C₂×M₁₆	C₂×D₅	1	2	yes	—
123	C₂²×C₈	—	5	2	no	—
124	C₂²×C₈	—	5	2	no	—
125	C₄⋊Q₈	—	5	1	no	—
126	#127	—	5	1	no	—
127	#124	—	5	2	no	—
128	#121	—	5	1	no	—
129	#121	—	5	1	no	—
130	#121	—	5	2	no	—
131	C₄×D₄	—	5	2	no	—
132	C₄×D₄	—	5	1	no	—
133	C₄×D₄	C₂×D₅	1	1	yes	characteristic
134	C₂×C₄²	—	5	1	no	—
135	C₄⋊C₈	—	5	1	no	—
136	C₄⋊C₈	—	5	1	no	—
137	C₄⋊C₈	—	5	1	no	—
138	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
139	C₂²⋊C₈	—	5	2	no	—
140	C₂²⋊C₈	—	5	2	no	—
141	C₂²⋊C₈	—	5	1	no	—
142	C₂²⋊C₈	—	5	1	no	—
143	C₂²⋊C₈	—	5	2	no	—
144	C₂²⋊C₈	—	5	2	no	—
145	C₂²⋊C₈	C₂×D₅	1	2	yes	—
146	C₄×C₈	—	5	2	no	—
147	C₄×C₈	—	5	1	no	—
148	C₄×C₈	C₂×D₅	1	1	yes	characteristic
149	C₂²×C₅	—	2	2	no	—
150	C₂²×C₅	—	2	2	no	—
151	C₂²×C₅	—	2	2	no	—
152	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
153	C₂×D₅	—	2	1	no	—
154	C₂×D₅	—	2	2	no	—
155	C₂×D₅	—	2	1	no	—
156	C₂×D₅	—	2	1	no	—
157	C₂×D₅	—	2	2	no	—
158	C₂×D₅	—	2	2	no	—
159	C₄×C₅	—	2	2	no	—
160	C₄×C₅	—	2	1	no	—
161	C₄×C₅	#131	1	2	yes	—
162	C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
163	C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
164	Dic₅	—	2	1	no	—
165	Dic₅	—	2	2	no	—
166	Dic₅	—	2	1	no	—
167	Dic₅	#131	1	2	yes	—
168	#41	—	20	2	no	—
169	#41	—	20	2	no	—
170	C₂×Q₈	—	5	2	no	—
171	C₂×D₄	—	5	2	no	—
172	C₂×D₄	—	5	2	no	—
173	C₂×D₄	—	5	2	no	—
174	C₂×D₄	—	5	1	no	—
175	C₂×D₄	—	5	2	no	—
176	C₂×D₄	C₄×D₅	1	1	yes	characteristic
177	C₂²×C₄	—	5	1	no	—
178	C₂²×C₄	—	5	1	no	—
179	C₂²×C₄	—	5	1	no	—
180	C₂²×C₄	—	5	2	no	—
181	C₂²×C₄	C₂²×D₅	1	2	yes	—
182	M₁₆	—	4	2	no	—
183	M₁₆	—	20	2	no	—
184	C₂×C₈	—	5	2	no	—
185	C₂×C₈	—	5	1	no	—
186	C₂×C₈	—	5	1	no	—
187	C₂×C₈	—	20	2	no	—
188	C₂×C₈	—	20	2	no	—
189	C₂×C₈	C₂²×D₅	1	2	yes	—
190	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
191	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
192	C₄⋊C₄	—	5	2	no	—
193	C₄⋊C₄	—	5	1	no	—
194	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
195	C₂²⋊C₄	—	5	2	no	—
196	C₂²⋊C₄	—	5	2	no	—
197	C₂²⋊C₄	—	5	2	no	—
198	C₂²⋊C₄	C₄×D₅	1	2	yes	—
199	C₄²	—	5	1	no	—
200	C₄²	—	5	1	no	—
201	C₄²	—	5	1	no	—
202	C₄²	C₂²×D₅	1	1	yes	characteristic
203	C₂×C₅	—	4	2	no	—
204	C₂×C₅	#518	1	1	yes	characteristic
205	C₂×C₅	C₂×#131	1	1	yes	characteristic
206	C₂×C₅	C₂×#131	1	1	yes	characteristic
207	D₅	—	4	1	no	—
208	D₅	—	4	2	no	—
209	C₂³	—	5	2	no	—
210	C₂³	—	5	1	no	—
211	C₂³	C₂×C₄×D₅	1	2	yes	—
212	Q₈	—	20	2	no	—
213	D₄	—	2	2	no	—
214	D₄	—	10	4	no	—
215	D₄	—	10	2	no	—
216	D₄	—	20	2	no	—
217	D₄	—	20	2	no	—
218	D₄	—	20	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	5	1	no	—
222	C₂×C₄	—	5	1	no	—
223	C₂×C₄	—	5	1	no	—
224	C₂×C₄	—	5	2	no	—
225	C₂×C₄	—	10	2	no	—
226	C₂×C₄	—	10	2	no	—
227	C₂×C₄	—	10	1	no	—
228	C₂×C₄	—	10	2	no	—
229	C₂×C₄	—	10	1	no	—
230	C₂×C₄	—	10	2	no	—
231	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
233	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
234	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	2	no	—
237	C₈	—	2	1	no	—
238	C₈	—	10	1	no	—
239	C₈	—	10	2	no	—
240	C₈	—	10	1	no	—
241	C₅	#2979	1	1	yes	characteristic, complemented, Sylow
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	10	1	no	—
246	C₂²	—	10	2	no	—
247	C₂²	—	10	1	no	—
248	C₂²	—	10	2	no	—
249	C₂²	—	10	1	no	—
250	C₂²	—	10	2	no	—
251	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	2	no	—
254	C₄	—	5	2	no	—
255	C₄	—	10	1	no	—
256	C₄	—	10	1	no	—
257	C₄	—	10	2	no	—
258	C₄	#4053	1	2	yes	—
259	C₄	#4144	1	1	yes	characteristic
260	C₄	#4149	1	1	yes	characteristic
261	C₂	—	4	2	no	—
262	C₂	—	20	1	no	—
263	C₂	—	20	2	no	—
264	C₂	#65879	1	1	yes	characteristic
265	C₂	C₂×#4053	1	1	yes	characteristic
266	C₂	D₅×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#436	C₂
prime radical	#65882	C₂
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#436	D₅
hypocenter	C₅	#2979
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. C₅×#436	C₂	semidirect
3. #65418	C₂	semidirect
4. #65417	C₂	semidirect
5. D₄×#168	C₂	semidirect
6. #65127	C₂	semidirect
7. #65122	C₂	semidirect
8. #65016	C₂	semidirect
9. #65014	C₂	semidirect
10. C₈×D₂₀	C₂	semidirect
11. #64974	C₂	semidirect
80. #436	D₅	semidirect
241. C₅	#2979	semidirect

Maximal subgroups

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

subgroup	index
#65882	2
C₅×#436	2
#65418	2
#65417	2
D₄×#168	2
#65127	2
#65122	2
#65016	2
#65014	2
C₈×D₂₀	2
#64974	2
#2979	5