Subgroups — #12017485

Order 1280.

430 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	C₂×#64989	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#64728	C₂	1	2	yes	maximal, complemented
4	#10602202	C₂	1	2	yes	maximal, complemented
5	C₂×#64683	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#4050	C₄	1	1	yes	characteristic
7	C₂×#4047	C₂²	1	2	yes	—
8	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#3951	C₄	1	1	yes	characteristic
12	C₂×#3944	C₂²	1	2	yes	—
13	C₂×#3937	C₄	1	2	yes	—
14	#64989	—	2	2	no	—
15	#64989	—	2	2	no	—
16	C₅×#346	—	2	2	no	—
17	#64728	—	2	4	no	—
18	#64727	—	2	4	no	—
19	#64683	—	2	2	no	—
20	#13196	—	5	1	no	maximal, Sylow
21	C₂×#709	C₂×C₄	1	2	yes	—
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×#681	C₂×C₄	1	2	yes	—
26	C₂×Dic₂₀	C₈	1	1	yes	characteristic
27	#4050	—	2	2	no	—
28	#4050	—	2	2	no	—
29	C₂×D₄₀	C₈	1	1	yes	characteristic
30	C₂×#677	C₈	1	2	yes	—
31	#4047	—	2	2	no	—
32	#4047	—	2	2	no	—
33	#4047	—	2	2	no	—
34	#4047	—	2	2	no	—
35	C₂×#676	—	2	2	no	—
36	C₂×C₈×D₅	—	2	2	no	—
37	C₂×C₅×C₁₆	—	2	1	no	—
38	C₂×C₅×C₁₆	—	2	1	no	—
39	C₂×C₅×C₁₆	—	2	1	no	—
40	C₂×C₅×C₁₆	—	2	1	no	—
41	C₂×C₅×C₁₆	D₄	1	2	yes	—
42	C₂×C₅×C₁₆	D₄	1	2	yes	—
43	C₄×C₅×C₈	D₄	1	2	yes	—
44	C₄×C₅×C₈	D₄	1	2	yes	—
45	#3951	—	2	2	no	—
46	#3944	—	2	2	no	—
47	#3944	—	2	2	no	—
48	C₂×#672	—	2	2	no	—
49	#3937	—	2	4	no	—
50	C₂×#443	—	5	1	no	—
51	C₂×#364	—	5	1	no	—
52	C₂×#350	—	5	2	no	—
53	#2113	—	5	2	no	—
54	C₂×#346	D₅	1	1	yes	characteristic, complemented
55	C₂²×C₄×C₅	—	2	1	no	—
56	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊D₅	—	2	2	no	—
58	#709	—	2	2	no	—
59	#709	—	2	2	no	—
60	#709	—	2	2	no	—
61	#709	—	2	2	no	—
62	C₂×D₂₀	C₂×C₈	1	2	yes	—
63	C₂×C₄×D₅	—	2	2	no	—
64	C₂×Dic₁₀	C₂×C₈	1	2	yes	—
65	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
66	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
68	C₂×C₅×C₈	M₁₆	1	2	yes	—
69	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂×C₅×C₈	#41	1	1	yes	characteristic
74	C₂×C₅×C₈	#41	1	1	yes	characteristic
75	C₄²×C₅	—	2	1	no	—
76	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
77	#681	—	2	2	no	—
78	#681	—	2	2	no	—
79	C₂×#168	—	2	2	no	—
80	Dic₂₀	—	2	2	no	—
81	D₄₀	—	2	2	no	—
82	#677	—	2	4	no	—
83	#676	—	4	2	no	—
84	#676	—	4	2	no	—
85	C₈×D₅	—	4	2	no	—
86	C₈×D₅	—	4	2	no	—
87	C₅×C₁₆	—	2	2	no	—
88	C₅×C₁₆	—	2	2	no	—
89	#672	—	4	2	no	—
90	C₂×#135	—	5	1	no	—
91	C₂×#131	—	5	2	no	—
92	C₂×M₃₂	—	5	2	no	—
93	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
94	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
95	#443	—	10	2	no	—
96	#443	—	10	2	no	—
97	C₂×#108	—	5	1	no	—
98	C₂×#104	—	5	2	no	—
99	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
100	#364	—	10	2	no	—
101	#350	—	10	4	no	—
102	C₂²⋊C₁₆	—	10	4	no	—
103	#346	—	2	2	no	—
104	C₂³×C₅	#131	1	1	yes	characteristic
105	C₂²×D₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	4	2	no	—
108	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅	C₄×D₄	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	C₂×C₄×C₅	#131	1	1	yes	characteristic
114	C₂²⋊D₅	—	4	2	no	—
115	C₂²⋊D₅	—	4	2	no	—
116	C₂×Dic₅	—	2	2	no	—
117	D₂₀	—	2	2	no	—
118	D₂₀	—	2	2	no	—
119	C₄×D₅	—	4	2	no	—
120	C₄×D₅	—	4	2	no	—
121	Dic₁₀	—	2	2	no	—
122	Dic₁₀	—	2	2	no	—
123	C₅×C₈	C₂²⋊C₈	1	2	yes	—
124	C₅×C₈	C₂²⋊C₈	1	2	yes	—
125	C₅×C₈	#121	1	1	yes	characteristic
126	C₅×C₈	#121	1	1	yes	characteristic
127	C₅×C₈	#124	1	1	yes	characteristic
128	C₅×C₈	#127	1	1	yes	characteristic
129	#168	—	4	2	no	—
130	C₂×#41	—	5	2	no	—
131	#135	—	10	2	no	—
132	#135	—	10	2	no	—
133	C₂×Q₁₆	—	5	1	no	—
134	C₂×SD₁₆	—	5	2	no	—
135	C₂×D₈	—	5	1	no	—
136	#131	—	10	2	no	—
137	#131	—	10	2	no	—
138	#131	—	10	2	no	—
139	#131	—	10	2	no	—
140	C₂×M₁₆	—	5	2	no	—
141	C₂×M₁₆	—	10	2	no	—
142	C₂²×C₈	—	10	2	no	—
143	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
144	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
145	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
146	M₃₂	—	10	2	no	—
147	M₃₂	—	10	2	no	—
148	C₂×C₁₆	—	2	1	no	—
149	C₂×C₁₆	—	2	1	no	—
150	C₂×C₁₆	—	2	1	no	—
151	C₂×C₁₆	—	2	1	no	—
152	C₂×C₁₆	—	10	2	no	—
153	C₂×C₁₆	D₂₀	1	2	yes	—
154	C₂×C₁₆	D₂₀	1	2	yes	—
155	#108	—	10	2	no	—
156	#104	—	10	4	no	—
157	C₄×C₈	D₂₀	1	2	yes	—
158	C₄×C₈	C₂²⋊D₅	1	2	yes	—
159	C₂²×C₅	—	2	2	no	—
160	C₂²×C₅	#408	1	1	yes	characteristic
161	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
162	C₂²×C₅	#436	1	1	yes	characteristic
163	C₂×D₅	—	4	4	no	—
164	C₂×D₅	—	4	2	no	—
165	C₄×C₅	—	4	2	no	—
166	C₄×C₅	#390	1	1	yes	characteristic
167	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
168	C₄×C₅	C₈×D₄	1	1	yes	characteristic
169	C₄×C₅	#436	1	1	yes	characteristic
170	Dic₅	—	4	2	no	—
171	#41	—	10	2	no	—
172	#41	—	10	2	no	—
173	#41	—	10	2	no	—
174	#41	—	10	2	no	—
175	C₂×Q₈	—	5	2	no	—
176	C₂×D₄	—	5	2	no	—
177	C₂×D₄	—	10	2	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	10	2	no	—
180	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
181	Q₁₆	—	10	2	no	—
182	SD₁₆	—	10	4	no	—
183	D₈	—	10	2	no	—
184	M₁₆	—	10	2	no	—
185	M₁₆	—	10	2	no	—
186	M₁₆	—	20	2	no	—
187	M₁₆	—	20	2	no	—
188	C₂×C₈	—	10	2	no	—
189	C₂×C₈	—	20	2	no	—
190	C₂×C₈	—	20	2	no	—
191	C₂×C₈	C₈×D₅	1	1	yes	characteristic
192	C₂×C₈	C₈×D₅	1	1	yes	characteristic
193	C₂×C₈	#676	1	2	yes	—
194	C₂×C₈	#685	1	2	yes	—
195	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
196	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
197	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
198	C₂×C₈	#709	1	1	yes	characteristic
199	C₂×C₈	#709	1	1	yes	characteristic
200	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
201	C₄²	—	2	1	no	—
202	C₄²	#685	1	2	yes	—
203	C₁₆	—	2	2	no	—
204	C₁₆	—	2	2	no	—
205	C₁₆	—	20	2	no	—
206	C₂×C₅	—	2	1	no	—
207	C₂×C₅	—	2	1	no	—
208	C₂×C₅	#1954	1	1	yes	characteristic
209	C₂×C₅	#2171	1	2	yes	—
210	D₅	—	8	2	no	—
211	C₂³	—	10	2	no	—
212	C₂³	#4047	1	1	yes	characteristic
213	Q₈	—	10	2	no	—
214	Q₈	—	10	2	no	—
215	D₄	—	10	2	no	—
216	D₄	—	10	2	no	—
217	D₄	—	20	2	no	—
218	D₄	—	20	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	10	2	no	—
222	C₂×C₄	—	20	2	no	—
223	C₂×C₄	—	20	2	no	—
224	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
225	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
226	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
227	C₂×C₄	C₂×#676	1	1	yes	characteristic
228	C₂×C₄	#4047	1	1	yes	characteristic
229	C₂×C₄	C₂×#685	1	1	yes	characteristic
230	C₈	—	20	2	no	—
231	C₈	#3952	1	2	yes	—
232	C₈	#3952	1	2	yes	—
233	C₈	#4020	1	1	yes	characteristic
234	C₈	#4021	1	1	yes	characteristic
235	C₈	#4076	1	1	yes	characteristic
236	C₈	#4076	1	1	yes	characteristic
237	C₅	#13196	1	1	yes	characteristic, complemented, Sylow
238	C₂²	—	2	2	no	—
239	C₂²	—	20	2	no	—
240	C₂²	—	20	4	no	—
241	C₂²	C₈×D₂₀	1	1	yes	characteristic
242	C₂²	#64977	1	1	yes	characteristic
243	C₂²	#65399	1	1	yes	characteristic
244	C₄	—	4	2	no	—
245	C₄	—	20	2	no	—
246	C₄	C₈×D₂₀	1	1	yes	characteristic
247	C₄	#64977	1	1	yes	characteristic
248	C₄	#65228	1	1	yes	characteristic
249	C₄	C₂×#3952	1	1	yes	characteristic
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	40	2	no	—
253	C₂	#10599869	1	2	yes	—
254	C₂	#10600580	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#346	C₂
socle	C₂²×C₅	C₈×D₄
prime radical	C₂×#4050	C₄
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#346	D₅
hypocenter	C₅	#13196
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#346	C₂	semidirect
3. C₂×#64728	C₂	semidirect
4. #10602202	C₂	semidirect
5. C₂×#64683	C₂	semidirect
54. C₂×#346	D₅	semidirect
237. C₅	#13196	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64989	2
C₂×C₅×#346	2
C₂×#64728	2
#10602202	2
C₂×#64683	2
#13196	5