Subgroups — #12014346

Order 1280.

490 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65411	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#349	C₂	1	1	yes	maximal, characteristic
3	C₂×#64773	C₂	1	1	yes	maximal, characteristic
4	Dic₅×M₃₂	C₂	1	2	yes	maximal, complemented
5	#10598303	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
7	C₂×#4072	C₄	1	1	yes	characteristic
8	C₂³×#168	C₄	1	1	yes	characteristic
9	C₂×C₅×M₃₂	C₂²	1	2	yes	—
10	#65411	—	2	2	no	—
11	#65411	—	2	2	no	—
12	C₂×#3947	C₄	1	1	yes	characteristic
13	C₂×#3947	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
16	C₂×#3944	C₂²	1	2	yes	—
17	C₅×#349	C₄	1	4	yes	complemented
18	#64773	C₄	1	4	yes	complemented
19	#64722	—	2	2	no	—
20	C₁₆×Dic₅	—	2	2	no	—
21	#12602	—	5	1	no	maximal, Sylow
22	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₂×C₂²⋊Dic₅	C₈	1	2	yes	—
30	#4072	—	2	2	no	—
31	#4072	—	2	2	no	—
32	C₂×C₄⋊Dic₅	C₈	1	2	yes	—
33	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₂²×#168	—	2	1	no	—
36	C₂²×#168	—	2	1	no	—
37	C₂²×#168	—	2	2	no	—
38	C₂²×#168	C₈	1	4	yes	—
39	C₂²×#168	C₂×C₄	1	1	yes	characteristic
40	C₂²×#168	C₂×C₄	1	1	yes	characteristic
41	C₅×M₃₂	—	2	2	no	—
42	C₅×M₃₂	C₂×C₄	1	4	yes	—
43	C₂×C₅×C₁₆	—	2	2	no	—
44	#3947	—	2	2	no	—
45	#3947	—	2	2	no	—
46	C₈×Dic₅	—	2	1	no	—
47	C₈×Dic₅	—	2	1	no	—
48	C₈×Dic₅	D₄	1	2	yes	—
49	C₈×Dic₅	D₄	1	2	yes	—
50	#3944	—	2	2	no	—
51	#3944	C₂×C₄	1	4	yes	—
52	C₂×#672	—	2	2	no	—
53	C₂×#405	—	5	1	no	—
54	C₂×#349	—	5	1	no	—
55	C₂×#349	D₅	1	1	yes	characteristic
56	C₄×M₃₂	—	5	2	no	—
57	#1377	—	5	2	no	—
58	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₅	—	2	1	no	—
60	C₂²×C₄×C₅	—	2	1	no	—
61	C₂²×C₄×C₅	—	2	1	no	—
62	C₂²×C₄×C₅	—	2	1	no	—
63	C₂²×C₄×C₅	C₄²	1	2	yes	—
64	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₅	C₂×C₈	1	2	yes	—
68	C₂²×Dic₅	C₂×C₈	1	2	yes	—
69	C₅×M₁₆	—	2	4	no	—
70	C₅×M₁₆	—	2	4	no	—
71	C₂×C₅×C₈	—	2	1	no	—
72	C₂×C₅×C₈	—	2	1	no	—
73	C₂×C₅×C₈	C₄²	1	2	yes	—
74	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
77	C₂×C₅×C₈	#41	1	2	yes	—
78	C₂²⋊Dic₅	—	2	4	no	—
79	C₄⋊Dic₅	—	2	4	no	—
80	C₄×Dic₅	—	2	1	no	—
81	C₄×Dic₅	—	2	1	no	—
82	C₄×Dic₅	C₂²⋊C₄	1	2	yes	—
83	C₄×Dic₅	M₁₆	1	2	yes	—
84	C₂×#168	—	2	1	no	—
85	C₂×#168	—	2	1	no	—
86	C₂×#168	—	2	4	no	—
87	C₂×#168	—	2	4	no	—
88	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×#168	C₂×C₈	1	2	yes	—
91	C₂×#168	C₂×C₈	1	2	yes	—
92	C₂×#168	M₁₆	1	2	yes	—
93	C₅×C₁₆	—	2	4	no	—
94	#672	—	2	4	no	—
95	C₂²×M₁₆	C₂×D₅	1	1	yes	characteristic
96	C₂³×C₈	—	5	1	no	—
97	C₂×#117	—	5	1	no	—
98	C₂×M₃₂	—	5	2	no	—
99	C₂×M₃₂	C₂×D₅	1	2	yes	—
100	#405	—	10	2	no	—
101	#405	—	10	2	no	—
102	C₂×#97	—	5	1	no	—
103	C₂×#97	—	5	1	no	—
104	C₂×C₄×C₈	—	5	1	no	—
105	C₂×C₄×C₈	—	5	1	no	—
106	#349	—	5	4	no	—
107	#349	Dic₅	1	4	yes	complemented
108	#346	—	10	2	no	—
109	C₄×C₁₆	—	10	2	no	—
110	C₂³×C₅	—	2	1	no	—
111	C₂³×C₅	—	2	1	no	—
112	C₂³×C₅	—	4	2	no	—
113	C₂³×C₅	C₄×C₈	1	2	yes	—
114	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	—	2	1	no	—
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
122	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅	#117	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
128	C₂×Dic₅	—	2	2	no	—
129	C₂×Dic₅	—	4	2	no	—
130	C₂×Dic₅	—	4	2	no	—
131	C₂×Dic₅	C₂²⋊C₈	1	4	yes	—
132	C₅×C₈	—	2	2	no	—
133	C₅×C₈	C₄×D₄	1	2	yes	—
134	C₅×C₈	C₄×D₄	1	2	yes	—
135	#168	—	2	2	no	—
136	#168	C₂²⋊C₈	1	2	yes	—
137	#168	C₂²⋊C₈	1	2	yes	—
138	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	C₄×D₅	1	2	yes	—
142	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
143	C₂²×C₈	—	5	1	no	—
144	C₂²×C₈	—	5	4	no	—
145	C₂²×C₈	—	5	1	no	—
146	C₂²×C₈	—	10	1	no	—
147	C₂²×C₈	—	10	1	no	—
148	C₂²×C₈	—	10	2	no	—
149	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
151	#117	—	10	2	no	—
152	#117	—	10	2	no	—
153	C₂×C₄⋊C₄	—	5	2	no	—
154	C₂×C₂²⋊C₄	—	5	2	no	—
155	C₂×C₄²	—	5	1	no	—
156	C₂×C₄²	—	5	1	no	—
157	M₃₂	—	2	2	no	—
158	M₃₂	—	5	4	no	—
159	M₃₂	—	10	2	no	—
160	M₃₂	C₂×Dic₅	1	4	yes	—
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	—	10	2	no	—
163	#97	—	10	2	no	—
164	#97	—	10	2	no	—
165	C₄×C₈	—	5	2	no	—
166	C₄×C₈	—	5	2	no	—
167	C₄×C₈	—	10	1	no	—
168	C₄×C₈	—	10	1	no	—
169	C₂²×C₅	—	2	2	no	—
170	C₂²×C₅	—	2	2	no	—
171	C₂²×C₅	—	4	2	no	—
172	C₂²×C₅	—	4	2	no	—
173	C₂²×C₅	—	4	2	no	—
174	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
175	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
176	C₂²×C₅	#431	1	1	yes	characteristic
177	C₄×C₅	—	2	2	no	—
178	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
179	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
180	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
181	C₄×C₅	#431	1	1	yes	characteristic
182	Dic₅	—	2	4	no	—
183	Dic₅	—	4	2	no	—
184	C₂⁴	C₈×D₅	1	1	yes	characteristic
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	5	2	no	—
190	C₂²×C₄	—	5	2	no	—
191	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
192	C₂²×C₄	C₈×D₅	1	2	yes	—
193	C₂²×C₄	C₄×Dic₅	1	2	yes	—
194	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
195	M₁₆	—	2	4	no	—
196	M₁₆	—	2	4	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	5	1	no	—
200	C₂×C₈	—	5	1	no	—
201	C₂×C₈	—	5	2	no	—
202	C₂×C₈	—	5	2	no	—
203	C₂×C₈	—	5	2	no	—
204	C₂×C₈	—	10	4	no	—
205	C₂×C₈	—	10	4	no	—
206	C₂×C₈	—	10	1	no	—
207	C₂×C₈	—	10	1	no	—
208	C₂×C₈	C₄×Dic₅	1	2	yes	—
209	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
210	C₂×C₈	D₄×D₅	1	2	yes	—
211	C₂×C₈	#711	1	2	yes	—
212	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
213	C₄⋊C₄	—	10	4	no	—
214	C₂²⋊C₄	—	10	4	no	—
215	C₄²	—	5	2	no	—
216	C₄²	—	5	2	no	—
217	C₄²	—	10	1	no	—
218	C₄²	—	10	1	no	—
219	C₁₆	—	2	4	no	—
220	C₁₆	—	10	4	no	—
221	C₂×C₅	—	2	1	no	—
222	C₂×C₅	—	2	1	no	—
223	C₂×C₅	—	4	2	no	—
224	C₂×C₅	C₄×C₂²⋊C₈	1	1	yes	characteristic
225	C₂×C₅	#2115	1	2	yes	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	4	2	no	—
229	C₂³	C₈×Dic₅	1	2	yes	—
230	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	5	4	no	—
238	C₂×C₄	—	10	2	no	—
239	C₂×C₄	—	20	2	no	—
240	C₂×C₄	—	20	2	no	—
241	C₂×C₄	C₈×Dic₅	1	2	yes	—
242	C₂×C₄	#4023	1	1	yes	characteristic
243	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
245	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
246	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
247	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	5	2	no	—
250	C₈	—	5	2	no	—
251	C₈	—	10	2	no	—
252	C₈	#4027	1	2	yes	—
253	C₈	D₄×Dic₅	1	2	yes	—
254	C₅	#12602	1	1	yes	characteristic, complemented, Sylow
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	#65007	1	1	yes	characteristic
261	C₂²	D₅×C₂²⋊C₈	1	1	yes	characteristic
262	C₂²	C₂×C₈×Dic₅	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	10	4	no	—
265	C₄	—	20	2	no	—
266	C₄	#65007	1	1	yes	characteristic
267	C₄	D₅×C₂²⋊C₈	1	1	yes	characteristic
268	C₄	C₂²⋊C₄×Dic₅	1	1	yes	characteristic
269	C₄	M₁₆×Dic₅	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	4	2	no	—
273	C₂	#10599912	1	2	yes	—
274	C₂	Dic₅×C₂²⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₅×C₂²⋊C₄
commutator	C₂²×C₅	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#349	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#349	D₅
hypocenter	C₅	#12602
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
4. Dic₅×M₃₂	C₂	semidirect
5. #10598303	C₂	semidirect
17. C₅×#349	C₄	semidirect
18. #64773	C₄	semidirect
107. #349	Dic₅	semidirect
254. C₅	#12602	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65411	2
C₂×C₅×#349	2
C₂×#64773	2
Dic₅×M₃₂	2
#10598303	2
#12602	5