Subgroups — #12253797

Order 1280.

714 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
2	#10615530	C₂	1	2	yes	maximal, complemented
3	C₂²×D₈₀	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602236	C₂	1	4	yes	maximal, complemented
5	#10602233	C₂	1	2	yes	maximal, complemented
6	#10602206	C₂	1	4	yes	maximal, complemented
7	#10602197	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#131	C₂²	1	1	yes	characteristic
9	#66164	—	2	2	no	—
10	C₂²×D₄₀	C₂²	1	2	yes	—
11	C₅×#504	—	2	4	no	—
12	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₅×M₃₂	C₂²	1	2	yes	complemented
14	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₁₆	C₂²	1	2	yes	complemented
16	#65424	—	2	4	no	—
17	#65420	—	2	2	no	—
18	#65404	C₂²	1	8	yes	complemented
19	#65395	C₂²	1	2	yes	—
20	C₂×D₈₀	—	2	4	no	—
21	C₂×D₈₀	C₂²	1	2	yes	—
22	C₂×D₈₀	C₂²	1	2	yes	—
23	#64729	C₂²	1	8	yes	complemented
24	#64725	C₂²	1	2	yes	complemented
25	#64724	C₂²	1	2	yes	complemented
26	#33607	—	5	1	no	maximal, Sylow
27	C₂×C₅×#41	D₄	1	1	yes	characteristic
28	C₂²×D₂₀	—	2	2	no	—
29	C₅×#131	—	2	2	no	—
30	C₅×#131	—	2	2	no	—
31	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
32	C₂×C₅×M₁₆	D₄	1	2	yes	—
33	C₂²×C₅×C₈	C₂³	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	#4093	—	4	2	no	—
36	#4086	—	4	2	no	—
37	#4083	—	4	4	no	—
38	#4076	—	2	8	no	—
39	#4072	—	2	2	no	—
40	C₂×D₄₀	—	2	4	no	—
41	C₂×D₄₀	—	2	4	no	—
42	C₂×D₄₀	—	2	2	no	—
43	C₂×D₄₀	C₂³	1	4	yes	—
44	C₅×M₃₂	—	2	2	no	—
45	C₅×M₃₂	—	2	4	no	—
46	C₂×C₅×C₁₆	—	2	4	no	—
47	C₂×C₅×C₁₆	D₄	1	4	yes	complemented
48	C₂×C₅×C₁₆	C₂³	1	2	yes	—
49	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
50	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
51	#3953	—	2	8	no	—
52	#3950	C₂³	1	4	yes	—
53	#3949	—	2	2	no	—
54	D₈₀	—	2	4	no	—
55	D₈₀	—	2	4	no	—
56	C₂²×D₁₆	—	5	1	no	—
57	C₂×#504	D₅	1	1	yes	characteristic, complemented
58	#3058	—	5	2	no	—
59	#2218	—	5	4	no	—
60	#2215	—	5	4	no	—
61	#2158	—	5	1	no	—
62	#2140	—	5	2	no	—
63	C₂³×D₅	—	4	2	no	—
64	C₅×#41	—	2	2	no	—
65	C₅×#41	—	2	2	no	—
66	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
67	C₂×C₅×D₄	D₈	1	2	yes	—
68	C₂×C₅×D₄	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂×C₂²⋊D₅	—	4	8	no	—
72	C₂×D₂₀	—	2	4	no	—
73	C₂×D₂₀	—	4	4	no	—
74	C₂×D₂₀	—	4	4	no	—
75	C₂×D₂₀	—	4	2	no	—
76	C₅×M₁₆	—	2	2	no	—
77	C₅×M₁₆	—	2	4	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
81	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
83	C₂×C₅×C₈	#41	1	2	yes	—
84	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
85	C₂²⋊Dic₅	—	4	2	no	—
86	#685	—	4	8	no	—
87	C₄⋊Dic₅	—	2	4	no	—
88	C₄×Dic₅	—	4	2	no	—
89	D₄₀	—	2	4	no	—
90	D₄₀	—	2	4	no	—
91	D₄₀	—	4	4	no	—
92	C₅×C₁₆	—	2	2	no	—
93	C₅×C₁₆	C₂×D₄	1	2	yes	—
94	C₅×C₁₆	C₂×D₄	1	2	yes	—
95	C₂²×D₈	—	5	2	no	—
96	C₂×#131	C₂×D₅	1	1	yes	characteristic
97	#535	—	10	2	no	—
98	C₂×D₁₆	—	5	2	no	—
99	C₂×D₁₆	—	5	2	no	—
100	C₂×D₁₆	—	10	4	no	—
101	#504	—	2	4	no	—
102	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
103	C₂×M₃₂	C₂×D₅	1	2	yes	complemented
104	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
105	C₂²×C₁₆	C₂×D₅	1	2	yes	complemented
106	#469	—	10	4	no	—
107	#466	—	5	8	no	—
108	#427	—	5	2	no	—
109	#418	—	10	2	no	—
110	#367	—	5	2	no	—
111	#366	—	5	2	no	—
112	#357	—	5	8	no	—
113	C₂³×C₅	C₂×D₈	1	2	yes	—
114	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₅	—	4	4	no	—
116	C₂²×D₅	—	8	4	no	—
117	C₂²×D₅	—	16	4	no	—
118	C₅×Q₈	—	2	2	no	—
119	C₅×D₄	—	2	4	no	—
120	C₅×D₄	—	2	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	#121	1	2	yes	—
124	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅	#135	1	2	yes	—
127	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
128	C₂²⋊D₅	—	16	8	no	—
129	C₂×Dic₅	—	4	4	no	—
130	D₂₀	—	4	4	no	—
131	D₂₀	—	4	4	no	—
132	D₂₀	—	8	4	no	—
133	C₅×C₈	—	2	2	no	—
134	C₅×C₈	#121	1	2	yes	—
135	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
136	C₅×C₈	C₂×#41	1	1	yes	characteristic
137	C₂×#41	D₂₀	1	1	yes	characteristic
138	C₂²×D₄	—	10	2	no	—
139	C₂×D₈	—	5	4	no	—
140	C₂×D₈	—	10	2	no	—
141	C₂×D₈	—	10	4	no	—
142	C₂×D₈	—	10	4	no	—
143	#131	—	2	2	no	—
144	#131	—	2	2	no	—
145	C₂×M₁₆	D₂₀	1	1	yes	characteristic
146	C₂×M₁₆	D₂₀	1	2	yes	—
147	C₂²×C₈	C₂²×D₅	1	2	yes	—
148	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
149	#127	—	20	2	no	—
150	#124	—	20	2	no	—
151	#121	—	10	8	no	—
152	#120	—	20	4	no	—
153	#117	—	10	2	no	—
154	D₁₆	—	10	4	no	—
155	D₁₆	—	10	4	no	—
156	M₃₂	—	2	2	no	—
157	M₃₂	—	2	4	no	—
158	C₂×C₁₆	—	2	4	no	—
159	C₂×C₁₆	C₂²⋊D₅	1	4	yes	complemented
160	C₂×C₁₆	C₂²×D₅	1	2	yes	—
161	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
162	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
163	#107	—	5	4	no	—
164	#106	—	10	2	no	—
165	#102	—	10	8	no	—
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	—	2	4	no	—
168	C₂²×C₅	—	2	2	no	—
169	C₂²×C₅	#466	1	2	yes	—
170	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
171	C₂×D₅	—	8	8	no	—
172	C₂×D₅	—	8	4	no	—
173	C₂×D₅	—	16	8	no	—
174	C₄×C₅	—	2	2	no	—
175	C₄×C₅	#466	1	2	yes	—
176	C₄×C₅	C₂×#121	1	1	yes	characteristic
177	C₄×C₅	C₂×#135	1	1	yes	characteristic
178	Dic₅	—	8	4	no	—
179	C₂⁴	—	20	2	no	—
180	#41	—	2	2	no	—
181	#41	—	2	2	no	—
182	C₂×Q₈	D₄₀	1	1	yes	characteristic
183	C₂×D₄	—	10	4	no	—
184	C₂×D₄	—	20	8	no	—
185	C₂×D₄	—	20	2	no	—
186	C₂×D₄	—	20	4	no	—
187	C₂×D₄	—	20	4	no	—
188	C₂×D₄	D₄₀	1	2	yes	—
189	C₂×D₄	D₄₀	1	1	yes	characteristic
190	C₂²×C₄	C₂×D₂₀	1	2	yes	—
191	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
192	D₈	—	10	4	no	—
193	D₈	—	10	4	no	—
194	D₈	—	20	4	no	—
195	M₁₆	—	2	2	no	—
196	M₁₆	—	2	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×D₂₀	1	2	yes	—
200	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
201	C₂×C₈	#709	1	2	yes	—
202	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
203	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
204	C₄⋊C₄	—	10	4	no	—
205	C₂²⋊C₄	—	20	2	no	—
206	C₂²⋊C₄	—	20	8	no	—
207	C₄²	—	20	2	no	—
208	C₁₆	—	2	2	no	—
209	C₁₆	C₂×C₂²⋊D₅	1	2	yes	—
210	C₁₆	C₂×C₂²⋊D₅	1	2	yes	—
211	C₂×C₅	—	2	2	no	—
212	C₂×C₅	—	4	2	no	—
213	C₂×C₅	C₂×#466	1	1	yes	characteristic
214	C₂×C₅	#3415	1	2	yes	—
215	D₅	—	16	4	no	—
216	C₂³	—	20	4	no	—
217	C₂³	—	40	4	no	—
218	C₂³	—	80	4	no	—
219	C₂³	C₂×D₄₀	1	2	yes	—
220	C₂³	C₂×D₄₀	1	1	yes	characteristic
221	Q₈	—	2	2	no	—
222	D₄	—	2	4	no	—
223	D₄	—	2	2	no	—
224	D₄	—	20	4	no	—
225	D₄	—	20	4	no	—
226	D₄	—	40	4	no	—
227	D₄	—	80	8	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	20	4	no	—
231	C₂×C₄	C₂×D₄₀	1	2	yes	—
232	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
233	C₂×C₄	#4050	1	2	yes	—
234	C₂×C₄	#4076	1	2	yes	—
235	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	#4076	1	2	yes	—
238	C₈	C₂×#709	1	1	yes	characteristic
239	C₈	C₂²×C₂²⋊D₅	1	1	yes	characteristic
240	C₅	#33607	1	1	yes	characteristic, complemented, Sylow
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	40	4	no	—
245	C₂²	—	40	8	no	—
246	C₂²	—	80	8	no	—
247	C₂²	#65404	1	2	yes	—
248	C₂²	C₂²×D₄₀	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	40	4	no	—
251	C₄	#65404	1	2	yes	—
252	C₄	C₂×#4050	1	1	yes	characteristic
253	C₄	C₂×#4076	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	4	2	no	—
256	C₂	—	80	4	no	—
257	C₂	#10613744	1	2	yes	—
258	C₂	C₂×#65404	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₀
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#504	C₂
socle	C₂²×C₅	C₂²×D₈
hypercenter	C₂×#504	D₅
hypocenter	C₅	#33607
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#504	C₂	semidirect
2. #10615530	C₂	semidirect
3. C₂²×D₈₀	C₂	semidirect
4. #10602236	C₂	semidirect
5. #10602233	C₂	semidirect
6. #10602206	C₂	semidirect
7. #10602197	C₂	semidirect
13. C₂×C₅×M₃₂	C₂²	semidirect
15. C₂²×C₅×C₁₆	C₂²	semidirect
18. #65404	C₂²	semidirect
23. #64729	C₂²	semidirect
24. #64725	C₂²	semidirect
25. #64724	C₂²	semidirect
47. C₂×C₅×C₁₆	D₄	semidirect
57. C₂×#504	D₅	semidirect
103. C₂×M₃₂	C₂×D₅	semidirect
105. C₂²×C₁₆	C₂×D₅	semidirect
159. C₂×C₁₆	C₂²⋊D₅	semidirect
240. C₅	#33607	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#504	2
#10615530	2
C₂²×D₈₀	2
#10602236	2
#10602233	2
#10602206	2
#10602197	2
#33607	5