Subgroups — #10611253

Order 640.

402 conjugacy classes of subgroups.

The subgroup lattice has 252 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	C₅×#436	C₂	1	1	yes	maximal, characteristic, complemented
3	#65417	C₂	1	2	yes	maximal, complemented
4	M₁₆×Dic₅	C₂	1	2	yes	maximal, complemented
5	#65301	C₂	1	1	yes	maximal, characteristic, complemented
6	#65124	C₂	1	1	yes	maximal, characteristic, complemented
7	#65119	C₂	1	1	yes	maximal, characteristic, complemented
8	#65018	C₂	1	2	yes	maximal, complemented
9	#65008	C₂	1	2	yes	maximal, complemented
10	C₄×#676	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#711	C₄	1	1	yes	characteristic
13	C₂×C₅×M₁₆	C₂²	1	2	yes	—
14	C₄×C₅×D₄	C₂²	1	1	yes	characteristic
15	D₄×Dic₅	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊D₅	C₂²	1	2	yes	—
17	C₂×C₄×Dic₅	C₂²	1	2	yes	—
18	C₂×#681	C₂²	1	2	yes	—
19	C₂×#676	C₂²	1	1	yes	characteristic
20	C₂×#676	C₂²	1	1	yes	characteristic
21	#4038	C₄	1	1	yes	characteristic
22	#4033	C₄	1	1	yes	characteristic
23	#4027	C₄	1	2	yes	—
24	#4023	C₄	1	2	yes	—
25	C₄²×D₅	C₂²	1	1	yes	characteristic
26	C₄×Dic₁₀	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	#3947	C₂²	1	2	yes	—
33	#3947	C₂²	1	1	yes	characteristic
34	#3947	C₂²	1	1	yes	characteristic
35	#3946	C₂²	1	2	yes	—
36	#3946	C₂²	1	2	yes	—
37	C₈×Dic₅	C₂²	1	2	yes	—
38	C₈×Dic₅	C₂²	1	2	yes	—
39	C₄⋊#168	C₂²	1	1	yes	characteristic
40	C₄×#168	C₂²	1	1	yes	characteristic
41	#2971	—	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	C₂²×Dic₅	C₂×C₄	1	2	yes	—
46	C₂²×Dic₅	C₂×C₄	1	2	yes	—
47	#711	—	2	4	no	—
48	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
51	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
52	C₅×M₁₆	—	2	4	no	—
53	C₂×C₅×C₈	C₂³	1	2	yes	—
54	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
55	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
56	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
57	C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
58	C₄²×C₅	C₂³	1	1	yes	characteristic
59	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
60	#685	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
62	#683	C₂×C₄	1	2	yes	—
63	#683	C₂×C₄	1	2	yes	—
64	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₅	C₂×C₄	1	2	yes	—
67	C₄×Dic₅	C₂×C₄	1	2	yes	—
68	C₄×Dic₅	C₂³	1	2	yes	—
69	C₄×Dic₅	C₂³	1	1	yes	characteristic
70	#681	—	2	4	no	—
71	C₂×#168	C₂³	1	1	yes	characteristic
72	C₂×#168	C₂³	1	2	yes	—
73	C₂×#168	C₂³	1	1	yes	characteristic
74	#676	—	2	2	no	—
75	#676	—	2	2	no	—
76	C₄×#41	—	5	1	no	—
77	C₈×Q₈	—	5	1	no	—
78	C₈×Q₈	—	5	1	no	—
79	#436	—	5	2	no	—
80	#436	—	5	1	no	—
81	#436	—	5	2	no	—
82	#436	D₅	1	1	yes	characteristic, complemented
83	#432	—	5	1	no	—
84	#432	—	5	2	no	—
85	C₄×M₁₆	—	5	2	no	—
86	C₄×M₁₆	—	5	1	no	—
87	C₂³×C₅	C₂²×C₄	1	2	yes	—
88	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
89	C₅×D₄	—	2	2	no	—
90	C₂×C₄×C₅	—	2	2	no	—
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₅	—	2	4	no	—
97	C₂×Dic₅	—	2	4	no	—
98	C₂×Dic₅	—	4	2	no	—
99	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
100	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₅	C₂²×C₄	1	2	yes	—
103	C₂×Dic₅	C₂²×C₄	1	2	yes	—
104	C₄×D₅	—	2	2	no	—
105	C₄×D₅	—	2	1	no	—
106	C₄×D₅	—	2	1	no	—
107	C₄×D₅	—	4	1	no	—
108	Dic₁₀	M₁₆	1	4	yes	complemented
109	C₅×C₈	—	2	1	no	—
110	C₅×C₈	—	2	2	no	—
111	C₅×C₈	#41	1	2	yes	complemented
112	#168	—	2	1	no	—
113	#168	—	2	2	no	—
114	#168	#41	1	2	yes	complemented
115	C₂×#41	—	5	1	no	—
116	C₂×M₁₆	—	5	2	no	—
117	C₂×M₁₆	—	5	1	no	—
118	C₂×M₁₆	—	5	1	no	—
119	C₂×M₁₆	C₂×D₅	1	2	yes	—
120	C₄×Q₈	—	5	1	no	—
121	C₄×Q₈	—	5	1	no	—
122	C₄×D₄	—	5	1	no	—
123	C₄×D₄	—	5	2	no	—
124	C₄×D₄	—	5	2	no	—
125	C₄×D₄	C₂×D₅	1	1	yes	characteristic
126	#117	—	5	1	no	—
127	#117	—	5	2	no	—
128	C₂×C₄²	—	5	2	no	—
129	C₂×C₄²	—	5	1	no	—
130	C₄⋊C₈	—	5	2	no	—
131	C₄⋊C₈	—	5	2	no	—
132	C₄⋊C₈	—	5	1	no	—
133	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
134	C₂²⋊C₈	—	5	2	no	—
135	C₂²⋊C₈	—	5	2	no	—
136	C₂²⋊C₈	C₂×D₅	1	2	yes	—
137	#97	—	5	1	no	—
138	#97	—	5	2	no	—
139	#97	—	5	1	no	—
140	C₄×C₈	—	5	2	no	—
141	C₄×C₈	—	5	2	no	—
142	C₄×C₈	—	5	1	no	—
143	C₄×C₈	C₂×D₅	1	1	yes	characteristic
144	C₂²×C₅	—	2	2	no	—
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	—	2	2	no	—
147	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
148	C₂×D₅	—	2	1	no	—
149	C₂×D₅	—	2	1	no	—
150	C₂×D₅	—	2	1	no	—
151	C₄×C₅	—	2	2	no	—
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	C₂×M₁₆	1	2	yes	—
154	C₄×C₅	C₂×#41	1	1	yes	characteristic
155	C₄×C₅	C₂×#41	1	1	yes	characteristic
156	Dic₅	—	2	1	no	—
157	Dic₅	—	2	1	no	—
158	Dic₅	—	2	1	no	—
159	Dic₅	—	2	2	no	—
160	Dic₅	C₂×M₁₆	1	4	yes	—
161	#41	—	10	4	no	—
162	C₂×Q₈	—	5	1	no	—
163	C₂×D₄	—	5	2	no	—
164	C₂×D₄	C₄×D₅	1	1	yes	characteristic
165	C₂²×C₄	—	5	2	no	—
166	C₂²×C₄	—	5	2	no	—
167	C₂²×C₄	—	5	1	no	—
168	C₂²×C₄	—	5	1	no	—
169	C₂²×C₄	—	5	1	no	—
170	C₂²×C₄	C₂²×D₅	1	2	yes	—
171	M₁₆	—	2	4	no	—
172	M₁₆	—	10	2	no	—
173	M₁₆	—	10	4	no	—
174	M₁₆	—	10	2	no	—
175	C₂×C₈	—	5	1	no	—
176	C₂×C₈	—	5	1	no	—
177	C₂×C₈	—	5	2	no	—
178	C₂×C₈	C₂²×D₅	1	2	yes	—
179	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
180	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
181	C₄⋊C₄	—	5	2	no	—
182	C₄⋊C₄	—	5	2	no	—
183	C₄⋊C₄	—	5	1	no	—
184	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
185	C₂²⋊C₄	—	5	2	no	—
186	C₂²⋊C₄	—	5	2	no	—
187	C₂²⋊C₄	C₄×D₅	1	2	yes	—
188	C₄²	—	5	2	no	—
189	C₄²	—	5	1	no	—
190	C₄²	—	5	2	no	—
191	C₄²	—	5	1	no	—
192	C₄²	—	5	2	no	—
193	C₄²	—	5	1	no	—
194	C₄²	C₂²×D₅	1	1	yes	characteristic
195	C₂×C₅	—	4	2	no	—
196	C₂×C₅	C₄×#41	1	1	yes	characteristic
197	C₂×C₅	C₂²×M₁₆	1	1	yes	characteristic
198	C₂×C₅	#568	1	1	yes	characteristic
199	D₅	—	4	1	no	—
200	C₂³	—	5	1	no	—
201	C₂³	C₂×C₄×D₅	1	2	yes	—
202	Q₈	—	5	4	no	—
203	D₄	—	2	2	no	—
204	D₄	—	10	4	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	5	2	no	—
208	C₂×C₄	—	5	1	no	—
209	C₂×C₄	—	5	1	no	—
210	C₂×C₄	—	5	2	no	—
211	C₂×C₄	—	5	1	no	—
212	C₂×C₄	—	10	4	no	—
213	C₂×C₄	—	10	1	no	—
214	C₂×C₄	—	10	1	no	—
215	C₂×C₄	—	10	2	no	—
216	C₂×C₄	—	20	1	no	—
217	C₂×C₄	—	20	2	no	—
218	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
219	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
220	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
221	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	—	5	2	no	—
225	C₈	—	10	1	no	—
226	C₈	—	10	2	no	—
227	C₈	#711	1	2	yes	complemented
228	C₅	#2971	1	1	yes	characteristic, complemented, Sylow
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	10	1	no	—
233	C₂²	—	10	1	no	—
234	C₂²	—	10	1	no	—
235	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	5	4	no	—
239	C₄	—	10	1	no	—
240	C₄	—	10	1	no	—
241	C₄	—	10	2	no	—
242	C₄	—	10	1	no	—
243	C₄	D₅×M₁₆	1	2	yes	—
244	C₄	C₂×#711	1	1	yes	characteristic
245	C₄	D₅×#41	1	1	yes	characteristic
246	C₂	—	4	2	no	—
247	C₂	—	20	1	no	—
248	C₂	C₄×#711	1	1	yes	characteristic
249	C₂	C₂×D₅×M₁₆	1	1	yes	characteristic
250	C₂	#66084	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	C₂×#711	C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#436	D₅
hypocenter	C₅	#2971
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. #65417	C₂	semidirect
4. M₁₆×Dic₅	C₂	semidirect
5. #65301	C₂	semidirect
6. #65124	C₂	semidirect
7. #65119	C₂	semidirect
8. #65018	C₂	semidirect
9. #65008	C₂	semidirect
10. C₄×#676	C₂	semidirect
11. C₈×Dic₁₀	C₂	semidirect
82. #436	D₅	semidirect
108. Dic₁₀	M₁₆	semidirect
111. C₅×C₈	#41	semidirect
114. #168	#41	semidirect
227. C₈	#711	semidirect
228. C₅	#2971	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
C₅×#436	2
#65417	2
M₁₆×Dic₅	2
#65301	2
#65124	2
#65119	2
#65018	2
#65008	2
C₄×#676	2
C₈×Dic₁₀	2
#2971	5