Subgroups — #10610366

Order 640.

412 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66125	C₂	1	1	yes	maximal, characteristic
2	C₅×#407	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#65417	C₂	1	2	yes	maximal, complemented
5	#65416	C₂	1	2	yes	maximal, complemented
6	#65019	C₂	1	2	yes	maximal, complemented
7	#65016	C₂	1	2	yes	maximal, complemented
8	#65014	C₂	1	2	yes	maximal, complemented
9	#65009	C₂	1	2	yes	maximal, complemented
10	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂×#709	C₂²	1	1	yes	characteristic
12	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
14	#4099	C₄	1	2	yes	—
15	#4076	C₄	1	1	yes	characteristic
16	#4076	C₄	1	1	yes	characteristic
17	#4075	C₄	1	2	yes	—
18	C₄×C₂²⋊D₅	C₂²	1	2	yes	—
19	C₄×C₂²⋊D₅	C₂²	1	2	yes	—
20	#4072	C₂²	1	1	yes	characteristic
21	#4070	C₄	1	1	yes	characteristic
22	#4070	C₄	1	1	yes	characteristic
23	C₂²×#168	C₂²	1	1	yes	characteristic
24	C₂²×#168	C₂²	1	1	yes	characteristic
25	C₂×#676	C₂²	1	2	yes	complemented
26	C₂×C₈×D₅	C₂²	1	2	yes	complemented
27	C₅×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₅×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₂²⋊#168	C₂²	1	2	yes	—
30	C₂²⋊#168	C₂²	1	2	yes	—
31	#3952	C₂²	1	2	yes	complemented
32	#3952	C₂²	1	2	yes	complemented
33	#3947	C₂²	1	2	yes	complemented
34	#3946	C₂²	1	2	yes	complemented
35	#3946	C₂²	1	2	yes	complemented
36	C₈×Dic₅	C₂²	1	2	yes	complemented
37	#2976	—	5	1	no	maximal, Sylow
38	C₂⁴×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅	—	4	1	no	—
40	C₂²×C₄×C₅	—	4	1	no	—
41	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
50	#709	—	4	1	no	—
51	#709	—	4	1	no	—
52	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₅	C₂³	1	2	yes	—
54	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
55	C₅×M₁₆	—	4	1	no	—
56	C₅×M₁₆	—	4	1	no	—
57	C₂×C₅×C₈	C₂³	1	2	yes	—
58	C₂×C₅×C₈	C₂³	1	2	yes	—
59	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
60	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
62	#685	C₂×C₄	1	2	yes	—
63	#685	C₂×C₄	1	2	yes	—
64	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
65	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
66	#683	C₂×C₄	1	2	yes	—
67	#683	C₂×C₄	1	2	yes	—
68	C₄×Dic₅	C₂³	1	2	yes	—
69	C₂×#168	—	4	1	no	—
70	C₂×#168	—	4	1	no	—
71	C₂×#168	C₂³	1	2	yes	—
72	C₂×#168	C₂³	1	2	yes	—
73	#676	—	4	2	no	—
74	C₈×D₅	—	4	2	no	—
75	#525	—	5	1	no	—
76	#436	—	5	2	no	—
77	#435	—	5	2	no	—
78	#435	—	5	2	no	—
79	C₈×D₄	—	5	2	no	—
80	#433	—	5	2	no	—
81	#408	—	5	2	no	—
82	#407	D₅	1	1	yes	characteristic, complemented
83	C₂×C₂²⋊C₈	—	5	1	no	—
84	C₂³×C₅	—	2	2	no	—
85	C₂³×C₅	—	4	1	no	—
86	C₂³×C₅	—	4	1	no	—
87	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₂²×D₅	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₅	—	2	1	no	—
92	C₂×C₄×C₅	—	2	1	no	—
93	C₂×C₄×C₅	—	2	1	no	—
94	C₂×C₄×C₅	—	2	1	no	—
95	C₂×C₄×C₅	—	2	1	no	—
96	C₂×C₄×C₅	—	2	1	no	—
97	C₂×C₄×C₅	—	4	1	no	—
98	C₂×C₄×C₅	—	4	1	no	—
99	C₂×C₄×C₅	—	4	1	no	—
100	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₅	—	2	4	no	—
105	C₂²⋊D₅	—	4	2	no	—
106	C₂²⋊D₅	—	4	2	no	—
107	C₂×Dic₅	C₂²×C₄	1	2	yes	—
108	C₂×Dic₅	C₂²×C₄	1	2	yes	—
109	C₂×Dic₅	C₂²×C₄	1	2	yes	—
110	D₂₀	—	4	1	no	—
111	C₄×D₅	—	2	2	no	—
112	C₄×D₅	—	2	2	no	—
113	Dic₁₀	—	4	1	no	—
114	C₅×C₈	—	2	2	no	—
115	C₅×C₈	—	2	2	no	—
116	#168	—	2	2	no	—
117	#168	—	2	2	no	—
118	C₂×#41	—	5	1	no	—
119	C₂³×C₄	C₂×D₅	1	1	yes	characteristic
120	C₂×M₁₆	—	5	2	no	—
121	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
122	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
123	C₂²×C₈	—	5	1	no	—
124	C₂²×C₈	—	5	1	no	—
125	C₂²×C₈	—	5	2	no	—
126	#123	—	5	2	no	—
127	C₂²⋊Q₈	—	5	1	no	—
128	C₂²⋊Q₈	—	5	1	no	—
129	#121	—	5	1	no	—
130	#121	—	5	1	no	—
131	#120	—	5	2	no	—
132	C₄×D₄	—	5	2	no	—
133	C₄×D₄	—	5	2	no	—
134	#117	—	5	1	no	—
135	C₄⋊C₈	—	5	2	no	—
136	C₄⋊C₈	—	5	2	no	—
137	C₂²⋊C₈	—	5	2	no	—
138	C₂²⋊C₈	—	5	2	no	—
139	C₂²⋊C₈	—	5	2	no	—
140	C₂²⋊C₈	—	5	2	no	—
141	C₂²⋊C₈	C₂×D₅	1	2	yes	—
142	C₂²⋊C₈	C₂×D₅	1	2	yes	—
143	#97	—	5	2	no	—
144	C₄×C₈	—	5	2	no	—
145	C₂²×C₅	—	2	1	no	—
146	C₂²×C₅	—	2	1	no	—
147	C₂²×C₅	—	2	1	no	—
148	C₂²×C₅	—	2	1	no	—
149	C₂²×C₅	—	2	1	no	—
150	C₂²×C₅	—	2	1	no	—
151	C₂²×C₅	—	2	1	no	—
152	C₂²×C₅	—	2	1	no	—
153	C₂²×C₅	—	4	4	no	—
154	C₂²×C₅	#131	1	2	yes	—
155	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
156	C₂×D₅	—	2	2	no	—
157	C₂×D₅	—	2	2	no	—
158	C₂×D₅	—	2	2	no	—
159	C₄×C₅	—	2	1	no	—
160	C₄×C₅	—	2	1	no	—
161	C₄×C₅	—	2	1	no	—
162	C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
163	C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
164	Dic₅	—	2	2	no	—
165	Dic₅	—	2	2	no	—
166	Dic₅	—	2	2	no	—
167	C₂⁴	C₄×D₅	1	1	yes	characteristic
168	#41	—	20	1	no	—
169	#41	—	20	1	no	—
170	C₂×Q₈	—	5	1	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	C₂²×C₄	—	4	1	no	—
176	C₂²×C₄	—	4	1	no	—
177	C₂²×C₄	—	5	2	no	—
178	C₂²×C₄	C₄×D₅	1	1	yes	characteristic
179	C₂²×C₄	C₄×D₅	1	1	yes	characteristic
180	C₂²×C₄	C₄×D₅	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
182	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
183	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
184	M₁₆	—	4	1	no	—
185	M₁₆	—	4	1	no	—
186	M₁₆	—	20	2	no	—
187	C₂×C₈	—	5	2	no	—
188	C₂×C₈	—	5	2	no	—
189	C₂×C₈	—	20	2	no	—
190	C₂×C₈	—	20	1	no	—
191	C₂×C₈	—	20	1	no	—
192	C₂×C₈	C₂²×D₅	1	2	yes	—
193	C₂×C₈	C₂²×D₅	1	2	yes	—
194	C₄⋊C₄	—	5	1	no	—
195	C₄⋊C₄	—	5	1	no	—
196	C₄⋊C₄	—	5	2	no	—
197	C₄⋊C₄	—	5	2	no	—
198	C₂²⋊C₄	—	5	2	no	—
199	C₂²⋊C₄	—	5	2	no	—
200	C₂²⋊C₄	—	5	2	no	—
201	C₂²⋊C₄	—	5	2	no	—
202	C₂²⋊C₄	—	5	2	no	—
203	C₄²	—	5	2	no	—
204	C₂×C₅	—	2	2	no	—
205	C₂×C₅	—	4	1	no	—
206	C₂×C₅	—	4	1	no	—
207	C₂×C₅	#518	1	1	yes	characteristic
208	C₂×C₅	C₂×#131	1	1	yes	characteristic
209	C₂×C₅	#568	1	1	yes	characteristic
210	D₅	—	4	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	4	1	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	5	2	no	—
215	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
216	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
217	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
218	Q₈	—	20	1	no	—
219	D₄	—	10	4	no	—
220	D₄	—	20	1	no	—
221	D₄	—	20	2	no	—
222	D₄	—	20	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	5	2	no	—
233	C₂×C₄	—	5	2	no	—
234	C₂×C₄	—	5	2	no	—
235	C₂×C₄	—	10	2	no	—
236	C₂×C₄	—	10	2	no	—
237	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
240	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	10	2	no	—
244	C₈	—	10	2	no	—
245	C₅	#2976	1	1	yes	characteristic, complemented, Sylow
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	10	2	no	—
256	C₂²	—	10	2	no	—
257	C₂²	—	10	2	no	—
258	C₂²	#4053	1	2	yes	—
259	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	10	2	no	—
264	C₄	—	10	2	no	—
265	C₄	—	10	2	no	—
266	C₄	#4144	1	1	yes	characteristic
267	C₄	#4144	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	4	1	no	—
270	C₂	—	4	1	no	—
271	C₂	—	20	2	no	—
272	C₂	#65820	1	1	yes	characteristic
273	C₂	C₂×#4053	1	1	yes	characteristic
274	C₂	#66079	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#407	C₂
prime radical	#66125	C₂
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#407	D₅
hypocenter	C₅	#2976
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
2. C₅×#407	C₂	semidirect
3. C₂×C₂²⋊#168	C₂	semidirect
4. #65417	C₂	semidirect
5. #65416	C₂	semidirect
6. #65019	C₂	semidirect
7. #65016	C₂	semidirect
8. #65014	C₂	semidirect
9. #65009	C₂	semidirect
25. C₂×#676	C₂²	semidirect
26. C₂×C₈×D₅	C₂²	semidirect
31. #3952	C₂²	semidirect
32. #3952	C₂²	semidirect
33. #3947	C₂²	semidirect
34. #3946	C₂²	semidirect
35. #3946	C₂²	semidirect
36. C₈×Dic₅	C₂²	semidirect
82. #407	D₅	semidirect
245. C₅	#2976	semidirect

Maximal subgroups

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

subgroup	index
#66125	2
C₅×#407	2
C₂×C₂²⋊#168	2
#65417	2
#65416	2
#65019	2
#65016	2
#65014	2
#65009	2
#2976	5