Subgroups — #12065021

Order 1280.

814 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	#10614248	C₂	1	2	yes	maximal
3	#10600642	C₂	1	2	yes	maximal, complemented
4	#10600641	C₂	1	2	yes	maximal, complemented
5	#10600640	C₂	1	2	yes	maximal, complemented
6	#10600517	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#681	C₂	1	4	yes	maximal, complemented
8	C₂×C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#131	C₄	1	2	yes	—
10	C₄×C₅×#41	C₂²	1	1	yes	characteristic
11	C₅×#405	C₄	1	4	yes	—
12	C₅×#405	C₄	1	2	yes	—
13	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
14	C₄×C₅×M₁₆	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
17	#65313	C₄	1	4	yes	—
18	#65301	C₄	1	4	yes	—
19	#65300	C₄	1	8	yes	—
20	#65214	C₂²	1	4	yes	—
21	#65214	C₂²	1	2	yes	—
22	C₄×#681	C₂²	1	4	yes	—
23	C₂×C₄×#168	C₂²	1	2	yes	—
24	#64675	C₂²	1	2	yes	—
25	#64674	C₂²	1	4	yes	—
26	#64674	C₂²	1	4	yes	—
27	C₈×#168	C₂²	1	2	yes	—
28	C₈×#168	C₂²	1	1	yes	characteristic
29	C₈×#168	C₂²	1	1	yes	characteristic
30	C₈×#168	C₂²	1	2	yes	—
31	#17335	—	5	1	no	maximal, Sylow
32	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
33	C₅×#131	—	2	8	no	—
34	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
36	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
37	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
38	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
39	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
40	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
41	C₅×#117	C₂×C₄	1	2	yes	—
42	C₅×#117	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂³	1	2	yes	—
45	C₂×#681	C₂×C₄	1	8	yes	—
46	C₂²×#168	C₂×C₄	1	4	yes	—
47	C₅×#97	C₂×C₄	1	2	yes	—
48	C₅×#97	C₂×C₄	1	4	yes	—
49	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
50	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
51	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
52	C₄×C₅×C₈	C₂³	1	2	yes	—
53	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
54	C₂²⋊#168	C₂×C₄	1	4	yes	—
55	C₂²⋊#168	C₂×C₄	1	8	yes	—
56	C₄⋊#168	C₂×C₄	1	8	yes	—
57	C₄⋊#168	C₂×C₄	1	4	yes	—
58	#3935	C₂×C₄	1	8	yes	—
59	C₄×#168	C₂×C₄	1	4	yes	—
60	C₄×#168	C₂³	1	2	yes	—
61	C₄×#168	C₂³	1	2	yes	—
62	C₄×#168	C₂³	1	4	yes	—
63	#2965	—	5	2	no	—
64	C₄×#131	D₅	1	1	yes	characteristic
65	#1455	—	5	2	no	—
66	#1453	—	5	2	no	—
67	#1451	—	5	2	no	—
68	#1450	—	5	1	no	—
69	C₈×M₁₆	—	5	4	no	—
70	C₂×C₈²	—	5	1	no	—
71	C₅×#41	—	2	4	no	—
72	C₂×C₅×Q₈	C₄²	1	1	yes	characteristic
73	C₂×C₅×D₄	C₄²	1	2	yes	—
74	C₂×C₅×D₄	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₅×M₁₆	—	2	4	no	—
80	C₅×M₁₆	—	2	8	no	—
81	C₂×C₅×C₈	—	2	8	no	—
82	C₂×C₅×C₈	M₁₆	1	8	yes	complemented
83	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
84	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
86	C₅×C₄⋊C₄	C₄²	1	4	yes	—
87	C₅×C₄⋊C₄	C₄²	1	2	yes	—
88	C₅×C₂²⋊C₄	C₄²	1	4	yes	—
89	C₅×C₂²⋊C₄	C₄²	1	2	yes	—
90	C₄²×C₅	C₂²×C₄	1	2	yes	—
91	C₄²×C₅	C₂²×C₄	1	4	yes	—
92	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅	C₂²×C₄	1	2	yes	—
94	C₄²×C₅	C₂⁴	1	1	yes	characteristic
95	#681	—	2	16	no	—
96	C₂×#168	—	2	8	no	—
97	C₂×#168	C₂²×C₄	1	4	yes	—
98	C₂×#168	C₂²×C₄	1	8	yes	—
99	C₂×#168	C₂²×C₄	1	4	yes	—
100	C₂×#131	Dic₅	1	2	yes	—
101	C₄×#41	C₂×D₅	1	1	yes	characteristic
102	#446	—	5	4	no	—
103	#436	—	5	4	no	—
104	#435	—	5	8	no	—
105	#431	—	5	4	no	—
106	#431	—	5	2	no	—
107	#405	Dic₅	1	4	yes	—
108	#405	Dic₅	1	2	yes	—
109	C₄×M₁₆	—	5	4	no	—
110	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
111	C₄×M₁₆	C₂×D₅	1	2	yes	—
112	C₂×C₄×C₈	—	5	2	no	—
113	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
115	#322	—	5	4	no	—
116	#322	—	5	4	no	—
117	#322	—	5	2	no	—
118	C₈²	—	5	1	no	—
119	C₈²	—	5	1	no	—
120	C₈²	—	5	2	no	—
121	C₈²	—	5	2	no	—
122	C₂³×C₅	C₂×C₄²	1	2	yes	—
123	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
124	C₅×Q₈	—	2	2	no	—
125	C₅×D₄	—	2	4	no	—
126	C₅×D₄	—	2	2	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	—	2	8	no	—
130	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
132	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
135	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
136	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
137	C₅×C₈	—	2	4	no	—
138	C₅×C₈	C₂×M₁₆	1	4	yes	—
139	C₅×C₈	C₂×M₁₆	1	4	yes	—
140	#168	—	2	4	no	—
141	#168	—	2	8	no	—
142	#168	#131	1	8	yes	complemented
143	C₂×#41	C₂×Dic₅	1	1	yes	characteristic
144	#131	—	2	8	no	—
145	C₂×M₁₆	—	5	8	no	—
146	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
147	C₂×M₁₆	C₂×Dic₅	1	4	yes	—
148	C₂²×C₈	—	5	4	no	—
149	C₂²×C₈	C₂×Dic₅	1	4	yes	—
150	C₂²×C₈	C₂×Dic₅	1	2	yes	—
151	C₄×Q₈	C₄×D₅	1	2	yes	—
152	C₄×D₄	C₄×D₅	1	4	yes	—
153	C₄×D₄	C₄×D₅	1	2	yes	—
154	#117	C₂×Dic₅	1	2	yes	—
155	#117	C₂×Dic₅	1	1	yes	characteristic
156	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
157	C₂×C₄²	C₂²×D₅	1	2	yes	—
158	C₄⋊C₈	—	5	8	no	—
159	C₄⋊C₈	—	5	4	no	—
160	C₂²⋊C₈	—	5	4	no	—
161	C₂²⋊C₈	—	5	8	no	—
162	#97	—	5	8	no	—
163	#97	C₂×Dic₅	1	2	yes	—
164	#97	C₂×Dic₅	1	4	yes	—
165	C₄×C₈	—	5	4	no	—
166	C₄×C₈	—	5	2	no	—
167	C₄×C₈	—	5	2	no	—
168	C₄×C₈	—	5	4	no	—
169	C₄×C₈	C₂×Dic₅	1	4	yes	—
170	C₄×C₈	C₂×Dic₅	1	2	yes	—
171	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
172	C₄×C₈	C₂²×D₅	1	2	yes	—
173	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	—	2	4	no	—
177	C₂²×C₅	C₄×M₁₆	1	2	yes	—
178	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
179	C₄×C₅	—	2	2	no	—
180	C₄×C₅	—	2	2	no	—
181	C₄×C₅	—	2	4	no	—
182	C₄×C₅	C₄×M₁₆	1	2	yes	—
183	C₄×C₅	C₂²×M₁₆	1	2	yes	—
184	C₄×C₅	C₂×#131	1	4	yes	—
185	#41	—	2	4	no	—
186	C₂×Q₈	C₄×Dic₅	1	1	yes	characteristic
187	C₂×D₄	C₄×Dic₅	1	2	yes	—
188	C₂×D₄	C₄×Dic₅	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
190	C₂²×C₄	C₂×C₄×D₅	1	4	yes	—
191	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
192	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
193	M₁₆	—	2	4	no	—
194	M₁₆	—	2	8	no	—
195	M₁₆	—	10	16	no	—
196	C₂×C₈	—	2	8	no	—
197	C₂×C₈	—	5	4	no	—
198	C₂×C₈	—	5	4	no	—
199	C₂×C₈	—	5	8	no	—
200	C₂×C₈	—	10	8	no	—
201	C₂×C₈	#681	1	8	yes	complemented
202	C₂×C₈	C₂²×Dic₅	1	4	yes	—
203	C₂×C₈	C₂²×Dic₅	1	2	yes	—
204	C₂×C₈	C₂²×Dic₅	1	2	yes	—
205	C₄⋊C₄	C₄×Dic₅	1	4	yes	—
206	C₄⋊C₄	C₄×Dic₅	1	2	yes	—
207	C₂²⋊C₄	C₄×Dic₅	1	4	yes	—
208	C₂²⋊C₄	C₄×Dic₅	1	2	yes	—
209	C₄²	C₂×C₄×D₅	1	2	yes	—
210	C₄²	C₂×C₄×D₅	1	4	yes	—
211	C₄²	C₂²×Dic₅	1	1	yes	characteristic
212	C₄²	C₂²×Dic₅	1	2	yes	—
213	C₄²	C₂³×D₅	1	1	yes	characteristic
214	C₂×C₅	—	2	2	no	—
215	C₂×C₅	—	4	2	no	—
216	C₂×C₅	C₂×C₄×M₁₆	1	1	yes	characteristic
217	C₂×C₅	C₄×#131	1	2	yes	—
218	C₂³	C₂×C₄×Dic₅	1	2	yes	—
219	C₂³	C₂×C₄×Dic₅	1	1	yes	characteristic
220	Q₈	—	2	2	no	—
221	D₄	—	2	4	no	—
222	D₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	C₂×#681	1	4	yes	—
227	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
228	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
229	C₂×C₄	C₂×C₄×Dic₅	1	4	yes	—
230	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
231	C₂×C₄	C₂²×C₄×D₅	1	2	yes	—
232	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
233	C₈	—	2	4	no	—
234	C₈	—	5	8	no	—
235	C₈	—	10	4	no	—
236	C₈	—	10	8	no	—
237	C₈	C₂×#681	1	4	yes	—
238	C₈	C₂×#681	1	4	yes	—
239	C₅	#17335	1	1	yes	characteristic, complemented, Sylow
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	C₄×#681	1	2	yes	—
244	C₂²	C₂²×C₄×Dic₅	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	4	no	—
248	C₄	C₄×#681	1	2	yes	—
249	C₄	D₅×#131	1	4	yes	—
250	C₄	C₂²×#681	1	2	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	4	2	no	—
253	C₂	C₂×C₄×#681	1	1	yes	characteristic
254	C₂	Dic₅×#131	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₅
commutator, socle	C₂²×C₅	C₂²×C₄²
Frattini	C₄²	C₂³×D₅
Fitting	C₄×C₅×#131	C₂
prime radical	C₂×C₅×#41	C₂×C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₄×#131	D₅
hypocenter	C₅	#17335
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
3. #10600642	C₂	semidirect
4. #10600641	C₂	semidirect
5. #10600640	C₂	semidirect
6. #10600517	C₂	semidirect
7. C₈×#681	C₂	semidirect
8. C₂×C₈×#168	C₂	semidirect
82. C₂×C₅×C₈	M₁₆	semidirect
142. #168	#131	semidirect
201. C₂×C₈	#681	semidirect
239. C₅	#17335	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#131	2
#10614248	2
#10600642	2
#10600641	2
#10600640	2
#10600517	2
C₈×#681	2
C₂×C₈×#168	2
#17335	5