Subgroups — #10617166

Order 640.

388 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	C₄×#711	C₂	1	1	yes	maximal, characteristic
2	#65773	C₂	1	2	yes	maximal, complemented
3	#65758	C₂	1	1	yes	maximal, characteristic, complemented
4	#65757	C₂	1	1	yes	maximal, characteristic, complemented
5	#65754	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#4004	C₂	1	1	yes	maximal, characteristic, complemented
7	#65707	C₂	1	2	yes	maximal, complemented
8	#65695	C₂	1	2	yes	maximal, complemented
9	#65693	C₂	1	2	yes	maximal, complemented
10	#65684	C₂	1	1	yes	maximal, characteristic, complemented
11	#65682	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#711	C₄	1	1	yes	characteristic
13	C₂×#704	C₂²	1	2	yes	—
14	C₂²×#170	C₂²	1	2	yes	—
15	C₂×#700	C₂²	1	2	yes	—
16	C₄×C₅×D₄	C₄	1	1	yes	characteristic
17	D₄×Dic₅	C₂²	1	1	yes	characteristic
18	C₄×C₂²⋊D₅	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊D₅	C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
22	#4038	C₄	1	1	yes	characteristic
23	#4033	C₂²	1	1	yes	characteristic
24	#4027	C₂²	1	1	yes	characteristic
25	#4027	C₂²	1	1	yes	characteristic
26	#4023	C₄	1	1	yes	characteristic
27	#4023	C₄	1	1	yes	characteristic
28	C₄²×D₅	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₀	C₄	1	1	yes	characteristic
30	C₂²⋊#170	C₂²	1	2	yes	—
31	C₂²⋊#170	C₂²	1	2	yes	—
32	#4004	C₂²	1	2	yes	—
33	#4004	C₂²	1	1	yes	characteristic
34	#4004	C₂²	1	1	yes	characteristic
35	#4004	C₂²	1	1	yes	characteristic
36	#4004	C₂²	1	1	yes	characteristic
37	#4004	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	#2989	—	5	1	no	maximal, Sylow
44	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
48	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
49	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
50	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
51	C₂²×Dic₅	C₂³	1	1	yes	characteristic
52	C₂²×Dic₅	C₂³	1	1	yes	characteristic
53	#711	—	2	4	no	—
54	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
57	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
58	#704	—	4	2	no	—
59	C₂×#170	—	4	2	no	—
60	C₂×#170	C₂³	1	2	yes	—
61	C₂×#170	C₂³	1	2	yes	—
62	C₂×#170	C₂³	1	2	yes	—
63	C₂×#170	C₂³	1	2	yes	—
64	#700	—	4	2	no	—
65	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
66	C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
67	C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
68	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
69	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
70	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
71	#685	C₂×C₄	1	1	yes	characteristic
72	#685	C₂×C₄	1	1	yes	characteristic
73	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
74	#683	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	C₄×Dic₅	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	#436	—	5	2	no	—
90	#435	—	5	2	no	—
91	#435	—	5	2	no	—
92	#432	—	5	1	no	—
93	#432	—	5	1	no	—
94	#431	—	5	1	no	—
95	#423	—	5	1	no	—
96	#423	—	5	1	no	—
97	C₂×C₄⋊C₈	—	5	1	no	—
98	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
99	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
100	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
101	C₅×D₄	—	2	2	no	—
102	C₂×C₄×C₅	—	4	1	no	—
103	C₂×C₄×C₅	—	4	1	no	—
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	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
109	C₂²⋊D₅	—	2	2	no	—
110	C₂²⋊D₅	—	2	2	no	—
111	C₂×Dic₅	—	2	2	no	—
112	C₂×Dic₅	—	2	1	no	—
113	C₂×Dic₅	—	2	1	no	—
114	C₂×Dic₅	—	2	1	no	—
115	C₂×Dic₅	—	2	1	no	—
116	C₂×Dic₅	M₁₆	1	4	yes	complemented
117	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₅	C₂⁴	1	1	yes	characteristic
124	C₄×D₅	—	2	2	no	—
125	C₄×D₅	—	2	1	no	—
126	C₄×D₅	—	2	1	no	—
127	C₄×D₅	—	4	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	C₂×M₁₆	—	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	AGL(1,𝔽₅)	1	1	yes	characteristic
145	#117	—	5	1	no	—
146	#117	—	5	1	no	—
147	#117	—	5	1	no	—
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	2	no	—
155	C₄⋊C₈	—	5	1	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	—	2	1	no	—
167	C₂²×C₅	—	2	1	no	—
168	C₂²×C₅	—	2	1	no	—
169	C₂²×C₅	C₂×M₁₆	1	2	yes	—
170	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
171	C₂×D₅	—	2	1	no	—
172	C₂×D₅	—	2	1	no	—
173	C₂×D₅	—	2	1	no	—
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	Dic₅	—	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	C₂×M₁₆	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	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
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	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
200	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
201	M₁₆	—	20	2	no	—
202	M₁₆	—	20	2	no	—
203	C₂×C₈	—	5	2	no	—
204	C₂×C₈	—	5	2	no	—
205	C₂×C₈	—	5	2	no	—
206	C₂×C₈	—	5	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	—	5	1	no	—
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	—	5	1	no	—
229	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
230	C₂×C₅	—	2	2	no	—
231	C₂×C₅	—	4	1	no	—
232	C₂×C₅	#520	1	1	yes	characteristic
233	C₂×C₅	C₂²×M₁₆	1	1	yes	characteristic
234	C₂×C₅	#568	1	1	yes	characteristic
235	D₅	—	4	1	no	—
236	C₂³	—	5	1	no	—
237	C₂³	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
238	C₂³	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
239	Q₈	—	10	2	no	—
240	D₄	—	2	2	no	—
241	D₄	—	10	2	no	—
242	D₄	—	10	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	5	4	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	—	5	1	no	—
251	C₂×C₄	—	5	1	no	—
252	C₂×C₄	—	5	1	no	—
253	C₂×C₄	—	10	1	no	—
254	C₂×C₄	—	10	2	no	—
255	C₂×C₄	—	10	1	no	—
256	C₂×C₄	—	10	1	no	—
257	C₂×C₄	—	10	1	no	—
258	C₂×C₄	—	10	2	no	—
259	C₂×C₄	—	10	1	no	—
260	C₂×C₄	—	10	1	no	—
261	C₂×C₄	—	20	1	no	—
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	#2989	1	1	yes	characteristic, complemented, Sylow
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	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	—	5	2	no	—
288	C₄	—	5	2	no	—
289	C₄	—	5	1	no	—
290	C₄	—	5	1	no	—
291	C₄	—	10	1	no	—
292	C₄	—	10	1	no	—
293	C₄	—	10	1	no	—
294	C₄	—	10	1	no	—
295	C₂	—	2	2	no	—
296	C₂	—	4	1	no	—
297	C₂	—	20	1	no	—
298	C₂	C₂×D₅⋊M₁₆	1	1	yes	characteristic
299	C₂	#66256	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	C₄×C₅×D₄	C₄
prime radical	C₂×#711	C₄
cosocle	C₂×Dic₅	C₂⁴
hypercenter	C₄×D₄	AGL(1,𝔽₅)
hypocenter	C₅	#2989
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #65773	C₂	semidirect
3. #65758	C₂	semidirect
4. #65757	C₂	semidirect
5. #65754	C₂	semidirect
6. C₂×#4004	C₂	semidirect
7. #65707	C₂	semidirect
8. #65695	C₂	semidirect
9. #65693	C₂	semidirect
10. #65684	C₂	semidirect
11. #65682	C₂	semidirect
116. C₂×Dic₅	M₁₆	semidirect
271. C₅	#2989	semidirect

Maximal subgroups

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

subgroup	index
C₄×#711	2
#65773	2
#65758	2
#65757	2
#65754	2
C₂×#4004	2
#65707	2
#65695	2
#65693	2
#65684	2
#65682	2
#2989	5