Subgroups — #10611071

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	C₄×#709	C₂	1	1	yes	maximal, characteristic
2	C₅×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#65126	C₂	1	1	yes	maximal, characteristic, complemented
5	#65124	C₂	1	1	yes	maximal, characteristic, complemented
6	#65122	C₂	1	1	yes	maximal, characteristic, complemented
7	D₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#65019	C₂	1	2	yes	maximal, complemented
9	#65014	C₂	1	2	yes	maximal, complemented
10	#64995	C₂	1	2	yes	maximal, complemented
11	#64990	C₂	1	2	yes	maximal, complemented
12	C₂×#709	C₄	1	1	yes	characteristic
13	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₄×C₂²⋊D₅	C₄	1	1	yes	characteristic
15	C₄×C₂²⋊D₅	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊D₅	C₂²	1	1	yes	characteristic
17	C₄×C₂²⋊D₅	C₂²	1	1	yes	characteristic
18	#4072	C₄	1	1	yes	characteristic
19	C₂²×#168	C₂²	1	2	yes	—
20	C₂×#676	C₂²	1	2	yes	—
21	C₂×C₈×D₅	C₂²	1	2	yes	—
22	C₄×D₂₀	C₄	1	1	yes	characteristic
23	C₄×D₂₀	C₂²	1	1	yes	characteristic
24	#4018	C₄	1	1	yes	characteristic
25	#4018	C₄	1	1	yes	characteristic
26	C₄²×D₅	C₂²	1	1	yes	characteristic
27	C₄²×D₅	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₀	C₄	1	1	yes	characteristic
29	C₄×Dic₁₀	C₂²	1	1	yes	characteristic
30	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₅×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₅×#97	C₂²	1	2	yes	—
34	#3952	C₂²	1	2	yes	—
35	#3952	C₂²	1	2	yes	—
36	#3947	C₂²	1	2	yes	—
37	#3946	C₂²	1	2	yes	—
38	#3946	C₂²	1	2	yes	—
39	C₈×Dic₅	C₂²	1	2	yes	—
40	C₄⋊#168	C₂²	1	1	yes	characteristic
41	C₄⋊#168	C₂²	1	1	yes	characteristic
42	C₄⋊#168	C₂²	1	2	yes	—
43	#2990	—	5	1	no	maximal, Sylow
44	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂³	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	#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×C₅×C₈	C₂³	1	2	yes	—
59	C₂×C₅×C₈	C₂³	1	2	yes	—
60	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
61	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
62	C₄²×C₅	C₂³	1	1	yes	characteristic
63	C₄²×C₅	C₂³	1	1	yes	characteristic
64	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
65	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
66	#685	C₂×C₄	1	1	yes	characteristic
67	#685	C₂×C₄	1	1	yes	characteristic
68	#685	C₂×C₄	1	1	yes	characteristic
69	#685	C₂×C₄	1	1	yes	characteristic
70	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
71	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
72	#683	C₂×C₄	1	1	yes	characteristic
73	#683	C₂×C₄	1	1	yes	characteristic
74	#683	C₂×C₄	1	1	yes	characteristic
75	#683	C₂×C₄	1	1	yes	characteristic
76	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₅	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	C₂³	1	1	yes	characteristic
81	C₄×Dic₅	C₂³	1	1	yes	characteristic
82	C₂×#168	—	4	2	no	—
83	C₂×#168	C₂³	1	2	yes	—
84	C₂×#168	C₂³	1	2	yes	—
85	#676	—	4	2	no	—
86	C₈×D₅	—	4	2	no	—
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	D₅	1	1	yes	characteristic, complemented
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	—	2	2	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	1	no	—
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	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
111	C₂²⋊D₅	—	2	2	no	—
112	C₂²⋊D₅	M₁₆	1	4	yes	complemented
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	C₂²×C₄	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	1	no	—
124	C₄×D₅	—	2	1	no	—
125	C₄×D₅	—	2	1	no	—
126	C₄×D₅	—	2	1	no	—
127	C₄×D₅	—	2	2	no	—
128	Dic₁₀	—	2	2	no	—
129	C₅×C₈	—	2	2	no	—
130	C₅×C₈	—	2	2	no	—
131	#168	—	2	2	no	—
132	#168	—	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	—	5	1	no	—
148	C₂×C₄²	—	5	1	no	—
149	C₂×C₄²	—	5	1	no	—
150	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
151	C₄⋊C₈	—	5	2	no	—
152	C₄⋊C₈	—	5	2	no	—
153	C₄⋊C₈	—	5	1	no	—
154	C₄⋊C₈	—	5	1	no	—
155	C₄⋊C₈	—	5	2	no	—
156	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
157	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
158	C₂²⋊C₈	—	5	2	no	—
159	C₂²⋊C₈	—	5	2	no	—
160	C₂²⋊C₈	C₂×D₅	1	2	yes	—
161	#97	—	5	2	no	—
162	#97	C₂×D₅	1	2	yes	—
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	#131	1	2	yes	—
179	C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
180	C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
181	Dic₅	—	2	1	no	—
182	Dic₅	—	2	1	no	—
183	Dic₅	—	2	1	no	—
184	Dic₅	—	2	1	no	—
185	Dic₅	—	2	1	no	—
186	Dic₅	C₂×M₁₆	1	2	yes	—
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	C₄×D₅	1	1	yes	characteristic
199	C₂²×C₄	C₄×D₅	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
201	M₁₆	—	20	2	no	—
202	C₂×C₈	—	5	2	no	—
203	C₂×C₈	—	5	2	no	—
204	C₂×C₈	—	20	2	no	—
205	C₂×C₈	—	20	2	no	—
206	C₂×C₈	C₂²×D₅	1	2	yes	—
207	C₂×C₈	C₂²×D₅	1	2	yes	—
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	—	5	1	no	—
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	—	5	1	no	—
219	C₂²⋊C₄	—	5	1	no	—
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	C₄×D₅	1	1	yes	characteristic
227	C₄²	C₄×D₅	1	1	yes	characteristic
228	C₄²	C₂²×D₅	1	1	yes	characteristic
229	C₄²	C₂²×D₅	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	C₂×C₄×D₅	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	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	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	2	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	1	no	—
259	C₂×C₄	—	10	2	no	—
260	C₂×C₄	—	10	2	no	—
261	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
266	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	2	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	C₂²×C₄×D₅	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	—	5	2	no	—
287	C₄	—	10	1	no	—
288	C₄	—	10	1	no	—
289	C₄	—	10	1	no	—
290	C₄	—	10	1	no	—
291	C₄	—	10	1	no	—
292	C₄	#4053	1	2	yes	—
293	C₄	#4149	1	1	yes	characteristic
294	C₄	#4150	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	10	2	no	—
297	C₂	—	20	1	no	—
298	C₂	#65857	1	1	yes	characteristic
299	C₂	C₂×D₅×M₁₆	1	1	yes	characteristic
300	C₂	C₂×#4053	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#432	C₂
prime radical	C₂×#709	C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#432	D₅
hypocenter	C₅	#2990
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. C₅×#432	C₂	semidirect
3. C₂×C₄⋊#168	C₂	semidirect
4. #65126	C₂	semidirect
5. #65124	C₂	semidirect
6. #65122	C₂	semidirect
7. D₅×C₄⋊C₈	C₂	semidirect
8. #65019	C₂	semidirect
9. #65014	C₂	semidirect
10. #64995	C₂	semidirect
11. #64990	C₂	semidirect
93. #432	D₅	semidirect
112. 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
C₄×#709	2
C₅×#432	2
C₂×C₄⋊#168	2
#65126	2
#65124	2
#65122	2
D₅×C₄⋊C₈	2
#65019	2
#65014	2
#64995	2
#64990	2
#2990	5