Subgroups — #12009020

Order 1280.

394 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	#10610189	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598006	C₂	1	4	yes	maximal, complemented
5	C₂×#4047	—	2	1	no	—
6	#65811	C₄	1	1	yes	characteristic
7	C₂×C₅×M₃₂	—	2	1	no	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	#65399	—	2	1	no	—
10	C₂²×#672	—	2	1	no	—
11	#65213	C₄	1	1	yes	characteristic
12	#64977	—	2	1	no	—
13	#64977	C₂²	1	2	yes	—
14	C₈×D₂₀	—	2	1	no	—
15	C₈×D₂₀	C₄	1	2	yes	—
16	C₅×#346	C₂²	1	2	yes	—
17	C₄×C₅×C₁₆	—	2	1	no	—
18	C₄⋊#672	—	2	1	no	—
19	C₄⋊#672	C₂²	1	2	yes	—
20	#11586	—	5	1	no	maximal, Sylow
21	C₂×#709	—	2	1	no	—
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	#4076	—	2	1	no	—
26	#4076	C₈	1	2	yes	—
27	C₂×#681	—	2	1	no	—
28	#4047	—	4	2	no	—
29	#4047	—	4	2	no	—
30	C₂×#676	—	2	2	no	—
31	C₂×C₈×D₅	—	2	2	no	—
32	#4021	—	2	1	no	—
33	#4020	C₈	1	1	yes	characteristic
34	C₄×D₂₀	—	2	1	no	—
35	C₄×D₂₀	C₂×C₄	1	2	yes	—
36	#4015	C₈	1	1	yes	characteristic
37	C₅×M₃₂	—	2	4	no	—
38	C₂×C₅×C₁₆	—	2	2	no	—
39	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×C₅×C₈	D₄	1	1	yes	characteristic
41	C₄×C₅×C₈	D₄	1	1	yes	characteristic
42	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
43	#3952	—	2	2	no	—
44	#3952	—	2	2	no	—
45	C₂×#672	—	2	2	no	—
46	C₂×#672	—	4	2	no	—
47	C₄⋊#168	—	2	1	no	—
48	C₄⋊#168	C₂×C₄	1	2	yes	—
49	#2931	—	5	1	no	—
50	C₂×C₄⋊C₁₆	—	5	1	no	—
51	C₄×M₃₂	D₅	1	1	yes	characteristic, complemented
52	#1330	—	5	4	no	—
53	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
56	C₂×C₂²⋊D₅	—	2	2	no	—
57	#709	—	4	2	no	—
58	#709	—	4	2	no	—
59	C₂×D₂₀	—	2	1	no	—
60	C₂×D₂₀	—	2	1	no	—
61	C₂×D₂₀	C₂×C₈	1	2	yes	—
62	C₂×C₄×D₅	—	2	2	no	—
63	C₂×Dic₁₀	—	2	1	no	—
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	D₈	1	2	yes	—
68	C₂×C₅×C₈	SD₁₆	1	2	yes	—
69	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₅	M₁₆	1	1	yes	characteristic
74	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
75	#685	—	2	2	no	—
76	#685	—	2	2	no	—
77	C₄⋊Dic₅	—	2	1	no	—
78	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
79	#681	—	4	2	no	—
80	C₂×#168	—	2	2	no	—
81	#676	—	4	2	no	—
82	#676	—	4	2	no	—
83	C₈×D₅	—	4	4	no	—
84	C₅×C₁₆	—	2	4	no	—
85	#672	—	4	2	no	—
86	C₂×#131	—	10	1	no	—
87	#532	—	5	1	no	—
88	C₂×M₃₂	—	2	1	no	—
89	C₂²×C₁₆	—	10	1	no	—
90	#436	—	5	2	no	—
91	#436	—	10	1	no	—
92	C₈×D₄	—	5	2	no	—
93	C₈×D₄	—	10	1	no	—
94	#423	—	5	1	no	—
95	#408	—	10	1	no	—
96	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
97	C₄⋊C₁₆	—	5	2	no	—
98	C₄⋊C₁₆	—	10	1	no	—
99	#346	C₂×D₅	1	2	yes	—
100	C₄×C₁₆	—	2	1	no	—
101	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
102	C₂²×D₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	4	1	no	—
105	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
108	C₂×C₄×C₅	#102	1	2	yes	—
109	C₂×C₄×C₅	#104	1	2	yes	—
110	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
113	C₂²⋊D₅	—	4	4	no	—
114	C₂×Dic₅	—	2	2	no	—
115	D₂₀	—	4	2	no	—
116	D₂₀	—	4	2	no	—
117	D₂₀	—	4	2	no	—
118	C₄×D₅	—	4	2	no	—
119	C₄×D₅	—	4	2	no	—
120	Dic₁₀	—	4	2	no	—
121	C₅×C₈	—	2	1	no	—
122	C₅×C₈	—	2	1	no	—
123	C₅×C₈	#102	1	2	yes	—
124	C₅×C₈	C₂×D₈	1	1	yes	characteristic
125	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
126	#168	—	4	2	no	—
127	C₂×#41	—	10	1	no	—
128	#131	—	20	2	no	—
129	#131	—	20	2	no	—
130	C₂×M₁₆	—	10	1	no	—
131	C₂×M₁₆	—	10	2	no	—
132	C₂²×C₈	—	10	2	no	—
133	C₂²×C₈	D₂₀	1	1	yes	characteristic
134	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
135	C₄⋊Q₈	—	5	1	no	—
136	#127	—	5	1	no	—
137	#124	—	10	1	no	—
138	#121	—	5	2	no	—
139	#121	—	10	1	no	—
140	C₄×D₄	—	5	2	no	—
141	C₄×D₄	—	10	1	no	—
142	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
143	M₃₂	—	2	4	no	—
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	10	2	no	—
146	C₂×C₁₆	—	20	2	no	—
147	C₄⋊C₈	—	5	2	no	—
148	C₄⋊C₈	—	10	1	no	—
149	C₂²⋊C₈	—	10	2	no	—
150	C₂²⋊C₈	—	10	2	no	—
151	C₄×C₈	C₄×D₅	1	1	yes	characteristic
152	C₄×C₈	D₂₀	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
154	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
155	C₂²×C₅	—	2	1	no	—
156	C₂²×C₅	—	2	1	no	—
157	C₂²×C₅	D₄⋊C₈	1	2	yes	—
158	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
159	C₂×D₅	—	4	2	no	—
160	C₂×D₅	—	4	2	no	—
161	C₂×D₅	—	4	2	no	—
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	—	2	1	no	—
165	C₄×C₅	D₄⋊C₈	1	2	yes	—
166	C₄×C₅	#350	1	2	yes	—
167	C₄×C₅	C₂×#102	1	1	yes	characteristic
168	C₄×C₅	C₂×#104	1	1	yes	characteristic
169	Dic₅	—	4	2	no	—
170	#41	—	20	2	no	—
171	#41	—	20	2	no	—
172	C₂×Q₈	—	10	1	no	—
173	C₂×D₄	—	5	2	no	—
174	C₂×D₄	—	10	2	no	—
175	C₂×D₄	—	10	1	no	—
176	C₂×D₄	—	10	1	no	—
177	C₂²×C₄	—	10	2	no	—
178	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
179	C₂²×C₄	#676	1	1	yes	characteristic
180	C₂²×C₄	#685	1	1	yes	characteristic
181	M₁₆	—	20	2	no	—
182	M₁₆	—	20	2	no	—
183	M₁₆	—	20	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	10	2	no	—
186	C₂×C₈	—	20	4	no	—
187	C₂×C₈	#677	1	2	yes	—
188	C₂×C₈	D₄₀	1	2	yes	—
189	C₂×C₈	#685	1	1	yes	characteristic
190	C₂×C₈	#685	1	1	yes	characteristic
191	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
192	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
193	C₄⋊C₄	—	5	2	no	—
194	C₄⋊C₄	—	10	1	no	—
195	C₂²⋊C₄	—	10	2	no	—
196	C₂²⋊C₄	—	10	2	no	—
197	C₄²	C₈×D₅	1	1	yes	characteristic
198	C₄²	#676	1	1	yes	characteristic
199	C₄²	#685	1	1	yes	characteristic
200	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
201	C₁₆	—	2	4	no	—
202	C₁₆	—	20	2	no	—
203	C₂×C₅	—	2	2	no	—
204	C₂×C₅	C₂×D₄⋊C₈	1	1	yes	characteristic
205	C₂×C₅	#2115	1	1	yes	characteristic
206	C₂×C₅	C₂×#350	1	1	yes	characteristic
207	D₅	—	8	2	no	—
208	C₂³	—	10	2	no	—
209	C₂³	#3952	1	1	yes	characteristic
210	Q₈	—	20	2	no	—
211	D₄	—	20	2	no	—
212	D₄	—	20	4	no	—
213	D₄	—	20	2	no	—
214	D₄	—	20	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	10	2	no	—
218	C₂×C₄	—	20	2	no	—
219	C₂×C₄	—	20	2	no	—
220	C₂×C₄	#3937	1	2	yes	—
221	C₂×C₄	#3952	1	1	yes	characteristic
222	C₂×C₄	#3952	1	1	yes	characteristic
223	C₂×C₄	#3952	1	1	yes	characteristic
224	C₂×C₄	#3953	1	2	yes	—
225	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
226	C₂×C₄	C₂×#676	1	1	yes	characteristic
227	C₂×C₄	C₂×#685	1	1	yes	characteristic
228	C₈	—	2	1	no	—
229	C₈	—	2	1	no	—
230	C₈	—	20	2	no	—
231	C₈	#3953	1	2	yes	—
232	C₈	C₂×#677	1	1	yes	characteristic
233	C₈	C₂×D₄₀	1	1	yes	characteristic
234	C₅	#11586	1	1	yes	characteristic, complemented, Sylow
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	20	2	no	—
238	C₂²	—	20	2	no	—
239	C₂²	—	20	2	no	—
240	C₂²	#64679	1	2	yes	—
241	C₂²	C₂×#3952	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	20	2	no	—
246	C₄	#64679	1	2	yes	—
247	C₄	#64735	1	2	yes	—
248	C₄	C₂×#3937	1	1	yes	characteristic
249	C₄	C₂×#3953	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	—	40	2	no	—
252	C₂	C₂×#64679	1	1	yes	characteristic
253	C₂	#10602228	1	1	yes	characteristic
254	C₂	C₂×#64735	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#685
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₄×C₅×M₃₂	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	#65811	C₄
cosocle	C₄×C₅×C₈	C₂³
hypercenter	C₄×M₃₂	D₅
hypocenter	C₅	#11586
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₄×C₅×M₃₂	C₂	semidirect
3. C₂×C₄⋊#672	C₂	semidirect
4. #10598006	C₂	semidirect
51. C₄×M₃₂	D₅	semidirect
234. C₅	#11586	semidirect

Maximal subgroups

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

subgroup	index
#10610189	2
C₄×C₅×M₃₂	2
C₂×C₄⋊#672	2
#10598006	2
#11586	5