Subgroups — #12014142

Order 1280.

614 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	C₂³×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#65390	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3944	C₂	1	1	yes	maximal, characteristic
4	#10598295	C₂	1	4	yes	maximal, complemented
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂×#4072	C₄	1	1	yes	characteristic
7	C₂²×#681	C₄	1	1	yes	characteristic
8	C₂²×C₅×C₁₆	—	2	2	no	—
9	C₂²×C₅×C₁₆	—	2	2	no	—
10	C₂²×C₅×C₁₆	C₄	1	4	yes	complemented
11	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
12	#65390	—	2	2	no	—
13	#65390	—	2	2	no	—
14	C₂×#3947	C₄	1	2	yes	—
15	C₂×C₈×Dic₅	C₂²	1	2	yes	—
16	C₂×#3944	—	2	2	no	—
17	C₂×#3944	—	2	2	no	—
18	C₂×#3944	C₄	1	4	yes	complemented
19	C₂²×#672	C₂²	1	2	yes	—
20	#12545	—	5	1	no	maximal, Sylow
21	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₅×C₈	D₄	1	2	yes	—
26	C₂²×C₅×C₈	D₄	1	2	yes	—
27	C₂²×C₅×C₈	D₄	1	2	yes	—
28	C₂²×C₅×C₈	Q₈	1	2	yes	—
29	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₅	C₈	1	2	yes	—
31	#4072	—	2	2	no	—
32	#4072	—	2	2	no	—
33	C₂×C₄⋊Dic₅	C₈	1	2	yes	—
34	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
35	C₂×#681	—	2	2	no	—
36	C₂×#681	—	2	2	no	—
37	C₂×#681	C₈	1	4	yes	—
38	C₂²×#168	C₂×C₄	1	2	yes	—
39	C₂×C₅×C₁₆	—	2	8	no	—
40	C₂×C₅×C₁₆	—	2	2	no	—
41	C₂×C₅×C₁₆	—	2	2	no	—
42	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
43	#3947	—	2	4	no	—
44	C₈×Dic₅	—	2	2	no	—
45	C₈×Dic₅	—	2	2	no	—
46	#3944	—	2	8	no	—
47	C₂×#672	—	2	2	no	—
48	C₂×#672	—	2	2	no	—
49	C₂×#672	C₂×C₄	1	4	yes	—
50	C₂²×M₃₂	—	5	1	no	—
51	C₂³×C₁₆	D₅	1	1	yes	characteristic
52	C₂×#405	—	5	1	no	—
53	#1374	—	5	4	no	—
54	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₄²	1	2	yes	—
56	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₅	M₁₆	1	2	yes	—
61	C₂²×C₄×C₅	M₁₆	1	2	yes	—
62	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₅	C₂×C₈	1	4	yes	—
64	C₂×C₅×C₈	C₄²	1	2	yes	—
65	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
66	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
67	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
68	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
69	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
70	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
71	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
77	C₂²⋊Dic₅	—	2	4	no	—
78	C₄⋊Dic₅	—	2	4	no	—
79	C₄×Dic₅	—	2	2	no	—
80	C₄×Dic₅	—	2	2	no	—
81	#681	—	2	8	no	—
82	C₂×#168	—	2	2	no	—
83	C₂×#168	—	2	2	no	—
84	C₂×#168	C₂×C₈	1	4	yes	—
85	C₅×C₁₆	—	2	4	no	—
86	#672	—	2	4	no	—
87	C₂²×M₁₆	—	5	1	no	—
88	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
89	C₂×#117	—	5	1	no	—
90	C₂×M₃₂	—	5	4	no	—
91	C₂×M₃₂	—	10	2	no	—
92	C₂×M₃₂	—	10	2	no	—
93	C₂²×C₁₆	—	2	2	no	—
94	C₂²×C₁₆	—	2	2	no	—
95	C₂²×C₁₆	—	5	2	no	—
96	C₂²×C₁₆	Dic₅	1	4	yes	complemented
97	C₂²×C₁₆	C₂×D₅	1	2	yes	—
98	#405	—	10	2	no	—
99	#405	—	10	2	no	—
100	C₂×#97	—	5	2	no	—
101	C₂×C₄×C₈	—	5	2	no	—
102	C₂³×C₅	—	2	4	no	—
103	C₂³×C₅	C₄×C₈	1	2	yes	—
104	C₂³×C₅	C₂²⋊C₈	1	2	yes	—
105	C₂³×C₅	C₄⋊C₈	1	2	yes	—
106	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₅	#95	1	4	yes	—
108	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
109	C₂×C₄×C₅	#97	1	4	yes	—
110	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
111	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
112	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
113	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
114	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
120	C₂×Dic₅	—	2	4	no	—
121	C₂×Dic₅	—	2	8	no	—
122	C₅×C₈	#95	1	4	yes	—
123	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
127	#168	—	2	4	no	—
128	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
129	C₂×M₁₆	—	5	4	no	—
130	C₂×M₁₆	—	10	2	no	—
131	C₂×M₁₆	—	10	2	no	—
132	C₂²×C₈	—	5	2	no	—
133	C₂²×C₈	Dic₁₀	1	2	yes	—
134	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
135	C₂²×C₈	C₄×D₅	1	2	yes	—
136	C₂²×C₈	D₂₀	1	2	yes	—
137	C₂²×C₈	C₂×Dic₅	1	2	yes	—
138	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
139	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
140	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
141	#117	—	10	2	no	—
142	#117	—	10	2	no	—
143	C₂×C₄⋊C₄	—	5	2	no	—
144	C₂×C₂²⋊C₄	—	5	2	no	—
145	C₂×C₄²	—	5	2	no	—
146	M₃₂	—	10	8	no	—
147	C₂×C₁₆	—	2	8	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	5	4	no	—
151	C₂×C₁₆	—	10	2	no	—
152	C₂×C₁₆	—	10	2	no	—
153	C₂×C₁₆	C₂×Dic₅	1	4	yes	—
154	#97	—	10	4	no	—
155	C₄×C₈	—	10	2	no	—
156	C₄×C₈	—	10	2	no	—
157	C₂²×C₅	—	2	4	no	—
158	C₂²×C₅	—	2	4	no	—
159	C₂²×C₅	—	2	4	no	—
160	C₂²×C₅	#336	1	4	yes	—
161	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
162	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
163	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
164	C₂²×C₅	#504	1	4	yes	—
165	C₄×C₅	#336	1	4	yes	—
166	C₄×C₅	C₂×#95	1	1	yes	characteristic
167	C₄×C₅	C₂×#97	1	1	yes	characteristic
168	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
169	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
170	Dic₅	—	4	4	no	—
171	C₂⁴	C₈×D₅	1	1	yes	characteristic
172	C₂²×C₄	—	5	4	no	—
173	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
174	C₂²×C₄	C₈×D₅	1	2	yes	—
175	C₂²×C₄	#676	1	2	yes	—
176	C₂²×C₄	#676	1	2	yes	—
177	C₂²×C₄	C₄×Dic₅	1	2	yes	—
178	C₂²×C₄	#683	1	2	yes	—
179	C₂²×C₄	#685	1	2	yes	—
180	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
181	M₁₆	—	10	8	no	—
182	C₂×C₈	—	5	4	no	—
183	C₂×C₈	—	10	2	no	—
184	C₂×C₈	—	10	2	no	—
185	C₂×C₈	C₄×Dic₅	1	2	yes	—
186	C₂×C₈	#683	1	2	yes	—
187	C₂×C₈	#683	1	4	yes	—
188	C₂×C₈	C₄⋊Dic₅	1	4	yes	—
189	C₂×C₈	#685	1	2	yes	—
190	C₂×C₈	#685	1	4	yes	—
191	C₂×C₈	C₂²⋊Dic₅	1	4	yes	—
192	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
193	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
194	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
195	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
196	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
197	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
198	C₄⋊C₄	—	10	4	no	—
199	C₂²⋊C₄	—	10	4	no	—
200	C₄²	—	10	2	no	—
201	C₄²	—	10	2	no	—
202	C₁₆	—	2	4	no	—
203	C₁₆	—	10	4	no	—
204	C₂×C₅	—	2	4	no	—
205	C₂×C₅	C₂×#336	1	1	yes	characteristic
206	C₂×C₅	#2108	1	2	yes	—
207	C₂×C₅	#2113	1	2	yes	—
208	C₂×C₅	#2151	1	2	yes	—
209	C₂³	—	2	4	no	—
210	C₂³	C₈×Dic₅	1	2	yes	—
211	C₂³	#3946	1	2	yes	—
212	C₂³	#3952	1	2	yes	—
213	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
214	C₂×C₄	—	10	4	no	—
215	C₂×C₄	—	10	8	no	—
216	C₂×C₄	C₈×Dic₅	1	2	yes	—
217	C₂×C₄	#3946	1	2	yes	—
218	C₂×C₄	#3946	1	4	yes	—
219	C₂×C₄	#3947	1	4	yes	—
220	C₂×C₄	#3952	1	2	yes	—
221	C₂×C₄	#3952	1	4	yes	—
222	C₂×C₄	#3963	1	4	yes	—
223	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
224	C₂×C₄	C₂×#676	1	1	yes	characteristic
225	C₂×C₄	C₂×#676	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
227	C₂×C₄	C₂×#683	1	1	yes	characteristic
228	C₂×C₄	C₂×#685	1	1	yes	characteristic
229	C₈	—	10	4	no	—
230	C₈	#3963	1	4	yes	—
231	C₈	C₂×#683	1	1	yes	characteristic
232	C₈	C₂×C₄⋊Dic₅	1	1	yes	characteristic
233	C₈	C₂×#685	1	1	yes	characteristic
234	C₈	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
235	C₅	#12545	1	1	yes	characteristic, complemented, Sylow
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	#64770	1	4	yes	—
240	C₂²	#65190	1	4	yes	—
241	C₂²	C₂×C₈×Dic₅	1	1	yes	characteristic
242	C₂²	C₂×#3946	1	1	yes	characteristic
243	C₂²	C₂×#3952	1	1	yes	characteristic
244	C₄	—	20	4	no	—
245	C₄	#64770	1	4	yes	—
246	C₄	C₂×#3946	1	1	yes	characteristic
247	C₄	C₂×#3947	1	1	yes	characteristic
248	C₄	C₂×#3952	1	1	yes	characteristic
249	C₄	C₂×#3963	1	1	yes	characteristic
250	C₂	—	2	4	no	—
251	C₂	#10602192	1	2	yes	—
252	C₂	#10602194	1	2	yes	—
253	C₂	#10602202	1	2	yes	—
254	C₂	C₂×#64770	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂×C₄×C₈
Fitting	C₂³×C₅×C₁₆	C₂
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂³×C₁₆	D₅
hypocenter	C₅	#12545
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
4. #10598295	C₂	semidirect
10. C₂²×C₅×C₁₆	C₄	semidirect
18. C₂×#3944	C₄	semidirect
96. C₂²×C₁₆	Dic₅	semidirect
235. C₅	#12545	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₅×C₁₆	2
C₂×#65390	2
C₂²×#3944	2
#10598295	2
#12545	5