Subgroups — #10617123

Order 640.

398 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65856	C₂	1	1	yes	maximal, characteristic
2	#65757	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#4004	C₂	1	1	yes	maximal, characteristic, complemented
4	#65707	C₂	1	2	yes	maximal, complemented
5	#65704	C₂	1	2	yes	maximal, complemented
6	#65693	C₂	1	2	yes	maximal, complemented
7	#65692	C₂	1	2	yes	maximal, complemented
8	#65684	C₂	1	1	yes	maximal, characteristic, complemented
9	#65682	C₂	1	1	yes	maximal, characteristic, complemented
10	D₅⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	#65678	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#709	C₄	1	1	yes	characteristic
13	C₂²×#170	C₂²	1	2	yes	—
14	C₂×#700	C₂²	1	2	yes	—
15	C₂×D₅⋊C₈	C₂²	1	2	yes	—
16	C₅×#117	C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
18	#4039	C₄	1	1	yes	characteristic
19	#4039	C₂²	1	1	yes	characteristic
20	#4033	C₄	1	1	yes	characteristic
21	#4033	C₂²	1	1	yes	characteristic
22	#4027	C₄	1	1	yes	characteristic
23	#4027	C₄	1	1	yes	characteristic
24	#4027	C₂²	1	1	yes	characteristic
25	#4027	C₂²	1	1	yes	characteristic
26	#4018	C₄	1	1	yes	characteristic
27	#4018	C₄	1	1	yes	characteristic
28	C₄²×D₅	C₂²	1	1	yes	characteristic
29	C₄²×D₅	C₂²	1	1	yes	characteristic
30	C₂²⋊#170	C₂²	1	2	yes	—
31	#4004	C₂²	1	1	yes	characteristic
32	#4004	C₂²	1	1	yes	characteristic
33	#4004	C₂²	1	1	yes	characteristic
34	#4004	C₂²	1	1	yes	characteristic
35	#4004	C₂²	1	2	yes	—
36	#4004	C₂²	1	2	yes	—
37	#4003	C₂²	1	2	yes	—
38	#4003	C₂²	1	2	yes	—
39	#4002	C₂²	1	2	yes	—
40	#4002	C₂²	1	2	yes	—
41	C₄⋊#170	C₂²	1	2	yes	—
42	C₄×#170	C₂²	1	2	yes	—
43	#2990	—	5	1	no	maximal, Sylow
44	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
45	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
47	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
48	C₂²×Dic₅	C₂³	1	1	yes	characteristic
49	#709	—	2	4	no	—
50	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
57	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
58	C₂×#170	—	4	2	no	—
59	C₂×#170	C₂³	1	2	yes	—
60	C₂×#170	C₂³	1	2	yes	—
61	C₂×#170	C₂³	1	2	yes	—
62	C₂×#170	C₂³	1	2	yes	—
63	#700	—	4	2	no	—
64	D₅⋊C₈	—	4	2	no	—
65	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
66	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
67	C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
68	C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
69	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
71	#685	C₂×C₄	1	1	yes	characteristic
72	#685	C₂×C₄	1	1	yes	characteristic
73	#685	C₂×C₄	1	1	yes	characteristic
74	#685	C₂×C₄	1	1	yes	characteristic
75	#683	C₂×C₄	1	1	yes	characteristic
76	#683	C₂×C₄	1	1	yes	characteristic
77	#683	C₂×C₄	1	1	yes	characteristic
78	#683	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
82	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
83	C₄×Dic₅	C₂³	1	1	yes	characteristic
84	C₄×Dic₅	C₂³	1	1	yes	characteristic
85	C₄×Dic₅	C₂³	1	1	yes	characteristic
86	C₄×Dic₅	C₂³	1	1	yes	characteristic
87	C₄×#41	—	5	1	no	—
88	#446	—	5	2	no	—
89	#435	—	5	2	no	—
90	#435	—	5	2	no	—
91	C₈×D₄	—	5	2	no	—
92	#432	—	5	1	no	—
93	#432	—	5	1	no	—
94	#431	—	5	1	no	—
95	#423	—	5	1	no	—
96	C₂×C₄⋊C₈	—	5	1	no	—
97	C₂×C₄⋊C₈	—	5	1	no	—
98	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
99	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
100	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
108	C₂²⋊D₅	—	2	2	no	—
109	C₂²⋊D₅	M₁₆	1	4	yes	complemented
110	C₂×Dic₅	—	2	2	no	—
111	C₂×Dic₅	—	2	1	no	—
112	C₂×Dic₅	—	2	1	no	—
113	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
114	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
115	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
117	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₅	C₂⁴	1	1	yes	characteristic
119	D₂₀	—	2	2	no	—
120	C₄×D₅	—	2	2	no	—
121	C₄×D₅	—	2	2	no	—
122	C₄×D₅	—	2	2	no	—
123	C₄×D₅	—	2	2	no	—
124	C₄×D₅	—	2	1	no	—
125	C₄×D₅	—	2	1	no	—
126	C₄×D₅	—	2	1	no	—
127	C₄×D₅	—	2	1	no	—
128	Dic₁₀	—	2	2	no	—
129	#170	—	2	2	no	—
130	#170	—	2	2	no	—
131	#170	—	2	2	no	—
132	#170	—	2	2	no	—
133	C₂×#41	—	5	1	no	—
134	C₂×M₁₆	—	5	2	no	—
135	C₂²×C₈	—	5	2	no	—
136	C₂²×C₈	—	5	2	no	—
137	C₄×Q₈	—	5	1	no	—
138	C₄×Q₈	—	5	1	no	—
139	C₄×D₄	—	5	1	no	—
140	C₄×D₄	—	5	1	no	—
141	C₄×D₄	—	5	1	no	—
142	C₄×D₄	—	5	1	no	—
143	C₄×D₄	—	5	1	no	—
144	C₄×D₄	—	5	1	no	—
145	#117	—	5	1	no	—
146	#117	—	5	1	no	—
147	#117	AGL(1,𝔽₅)	1	1	yes	characteristic
148	C₂×C₄²	—	5	1	no	—
149	C₂×C₄²	—	5	1	no	—
150	C₂×C₄²	—	5	1	no	—
151	C₄⋊C₈	—	5	2	no	—
152	C₄⋊C₈	—	5	1	no	—
153	C₄⋊C₈	—	5	1	no	—
154	C₄⋊C₈	—	5	1	no	—
155	C₄⋊C₈	—	5	2	no	—
156	C₄⋊C₈	—	5	2	no	—
157	C₄⋊C₈	—	5	1	no	—
158	C₂²⋊C₈	—	5	2	no	—
159	C₂²⋊C₈	—	5	2	no	—
160	C₂²⋊C₈	—	5	2	no	—
161	#97	—	5	2	no	—
162	#97	—	5	2	no	—
163	C₄×C₈	—	5	2	no	—
164	C₂²×C₅	—	2	1	no	—
165	C₂²×C₅	—	2	1	no	—
166	C₂²×C₅	C₂×M₁₆	1	2	yes	—
167	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
168	C₂×D₅	—	2	1	no	—
169	C₂×D₅	—	2	1	no	—
170	C₂×D₅	—	2	1	no	—
171	C₂×D₅	—	2	1	no	—
172	C₂×D₅	—	2	1	no	—
173	C₂×D₅	C₂×M₁₆	1	2	yes	—
174	C₄×C₅	—	2	1	no	—
175	C₄×C₅	—	2	1	no	—
176	C₄×C₅	—	2	1	no	—
177	C₄×C₅	—	2	1	no	—
178	C₄×C₅	—	2	1	no	—
179	C₄×C₅	—	2	1	no	—
180	Dic₅	—	2	1	no	—
181	Dic₅	—	2	1	no	—
182	Dic₅	—	2	1	no	—
183	Dic₅	C₂×M₁₆	1	2	yes	—
184	Dic₅	#131	1	2	yes	—
185	Dic₅	H(5,𝔽₂)	1	1	yes	characteristic
186	Dic₅	2₋¹⁺⁴	1	1	yes	characteristic
187	#41	—	10	4	no	—
188	C₂×Q₈	—	5	1	no	—
189	C₂×D₄	—	5	1	no	—
190	C₂×D₄	—	5	1	no	—
191	C₂×D₄	—	5	1	no	—
192	C₂²×C₄	—	5	1	no	—
193	C₂²×C₄	—	5	1	no	—
194	C₂²×C₄	—	5	1	no	—
195	C₂²×C₄	—	5	1	no	—
196	C₂²×C₄	—	5	1	no	—
197	C₂²×C₄	—	5	1	no	—
198	C₂²×C₄	—	5	1	no	—
199	C₂²×C₄	—	5	1	no	—
200	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
201	M₁₆	—	20	2	no	—
202	C₂×C₈	—	5	2	no	—
203	C₂×C₈	—	5	2	no	—
204	C₂×C₈	—	5	2	no	—
205	C₂×C₈	—	5	2	no	—
206	C₂×C₈	—	20	2	no	—
207	C₂×C₈	—	20	2	no	—
208	C₄⋊C₄	—	5	1	no	—
209	C₄⋊C₄	—	5	1	no	—
210	C₄⋊C₄	—	5	1	no	—
211	C₄⋊C₄	—	5	1	no	—
212	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
213	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
214	C₂²⋊C₄	—	5	1	no	—
215	C₂²⋊C₄	—	5	1	no	—
216	C₂²⋊C₄	—	5	1	no	—
217	C₂²⋊C₄	—	5	1	no	—
218	C₂²⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
219	C₂²⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
220	C₄²	—	5	1	no	—
221	C₄²	—	5	1	no	—
222	C₄²	—	5	1	no	—
223	C₄²	—	5	1	no	—
224	C₄²	—	5	1	no	—
225	C₄²	—	5	1	no	—
226	C₄²	—	5	1	no	—
227	C₄²	—	5	1	no	—
228	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
229	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
230	C₂×C₅	—	2	2	no	—
231	C₂×C₅	#520	1	1	yes	characteristic
232	C₂×C₅	C₂²×M₁₆	1	1	yes	characteristic
233	C₂×C₅	C₂×#131	1	1	yes	characteristic
234	D₅	—	2	2	no	—
235	D₅	—	4	1	no	—
236	C₂³	—	5	1	no	—
237	C₂³	—	5	1	no	—
238	C₂³	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
239	Q₈	—	10	2	no	—
240	D₄	—	5	4	no	—
241	D₄	—	10	2	no	—
242	D₄	—	10	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	5	1	no	—
245	C₂×C₄	—	5	1	no	—
246	C₂×C₄	—	5	1	no	—
247	C₂×C₄	—	5	1	no	—
248	C₂×C₄	—	5	1	no	—
249	C₂×C₄	—	5	1	no	—
250	C₂×C₄	—	10	1	no	—
251	C₂×C₄	—	10	1	no	—
252	C₂×C₄	—	10	2	no	—
253	C₂×C₄	—	10	2	no	—
254	C₂×C₄	—	10	2	no	—
255	C₂×C₄	—	10	1	no	—
256	C₂×C₄	—	10	1	no	—
257	C₂×C₄	—	10	2	no	—
258	C₂×C₄	—	10	1	no	—
259	C₂×C₄	—	10	2	no	—
260	C₂×C₄	—	10	1	no	—
261	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
262	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
263	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
264	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
265	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
266	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
267	C₈	—	10	2	no	—
268	C₈	—	10	2	no	—
269	C₈	—	10	2	no	—
270	C₈	—	10	2	no	—
271	C₅	#2990	1	1	yes	characteristic, complemented, Sylow
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	5	2	no	—
275	C₂²	—	10	1	no	—
276	C₂²	—	10	1	no	—
277	C₂²	—	10	1	no	—
278	C₂²	—	10	1	no	—
279	C₂²	—	10	1	no	—
280	C₂²	D₅⋊M₁₆	1	2	yes	—
281	C₂²	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	5	2	no	—
289	C₄	—	5	2	no	—
290	C₄	—	5	1	no	—
291	C₄	—	5	1	no	—
292	C₄	—	10	1	no	—
293	C₄	—	10	1	no	—
294	C₄	—	10	1	no	—
295	C₂	—	2	2	no	—
296	C₂	—	10	2	no	—
297	C₂	—	20	1	no	—
298	C₂	C₂×D₅⋊M₁₆	1	1	yes	characteristic
299	C₂	D₅⋊#131	1	1	yes	characteristic
300	C₂	#66266	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#117	C₄
prime radical	C₂×#709	C₄
cosocle	C₂×Dic₅	C₂⁴
hypercenter	#117	AGL(1,𝔽₅)
hypocenter	C₅	#2990
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. #65757	C₂	semidirect
3. C₂×#4004	C₂	semidirect
4. #65707	C₂	semidirect
5. #65704	C₂	semidirect
6. #65693	C₂	semidirect
7. #65692	C₂	semidirect
8. #65684	C₂	semidirect
9. #65682	C₂	semidirect
10. D₅⋊(C₄⋊C₈)	C₂	semidirect
11. #65678	C₂	semidirect
109. C₂²⋊D₅	M₁₆	semidirect
271. C₅	#2990	semidirect

Maximal subgroups

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

subgroup	index
#65856	2
#65757	2
C₂×#4004	2
#65707	2
#65704	2
#65693	2
#65692	2
#65684	2
#65682	2
D₅⋊(C₄⋊C₈)	2
#65678	2
#2990	5