Subgroups — #12015227

Order 1280.

486 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×Dic₅	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	#10602213	C₂	1	2	yes	maximal, complemented
5	#10598303	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₄×Dic₅	C₄	1	1	yes	characteristic
8	C₂²×#681	C₄	1	1	yes	characteristic
9	C₂×C₅×M₃₂	C₂²	1	2	yes	—
10	M₁₆×Dic₅	—	2	4	no	—
11	C₂×#3947	C₄	1	1	yes	characteristic
12	C₂×#3947	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
15	C₂×#3944	C₂²	1	2	yes	—
16	C₅×#349	—	2	2	no	—
17	#64773	—	2	2	no	—
18	#64721	—	2	4	no	—
19	#12785	—	5	1	no	maximal, Sylow
20	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂³×Dic₅	C₈	1	2	yes	—
22	C₂×C₅×M₁₆	—	2	2	no	—
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	D₄	1	2	yes	—
25	C₂×C₅×M₁₆	Q₈	1	2	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₂×C₄×Dic₅	—	2	2	no	—
29	C₂×C₄×Dic₅	—	2	1	no	—
30	C₂×C₄×Dic₅	—	2	1	no	—
31	C₂×C₄×Dic₅	C₈	1	2	yes	—
32	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
34	C₂×#681	—	2	2	no	—
35	C₂×#681	—	2	2	no	—
36	C₂×#681	—	2	2	no	—
37	C₂²×#168	C₂×C₄	1	1	yes	characteristic
38	C₂²×#168	C₂×C₄	1	1	yes	characteristic
39	C₅×M₃₂	—	2	2	no	—
40	C₅×M₃₂	—	2	2	no	—
41	C₂×C₅×C₁₆	—	2	2	no	—
42	#3947	—	2	2	no	—
43	#3947	—	2	2	no	—
44	C₈×Dic₅	—	2	2	no	—
45	C₈×Dic₅	D₄	1	4	yes	—
46	#3944	—	2	2	no	—
47	#3944	—	2	2	no	—
48	C₂×#672	—	2	2	no	—
49	C₂×C₄×M₁₆	—	5	1	no	—
50	#2150	—	5	2	no	—
51	C₂×#349	—	5	1	no	—
52	C₂×#349	D₅	1	1	yes	characteristic
53	#1377	—	5	2	no	—
54	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	—	2	1	no	—
56	C₂²×C₄×C₅	—	2	1	no	—
57	C₂²×C₄×C₅	—	2	1	no	—
58	C₂²×C₄×C₅	—	2	1	no	—
59	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	M₁₆	1	2	yes	—
62	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₅	—	2	2	no	—
64	C₂²×Dic₅	—	2	2	no	—
65	C₂²×Dic₅	—	4	4	no	—
66	C₂²×Dic₅	C₂×C₈	1	2	yes	—
67	C₂²×Dic₅	C₂×C₈	1	2	yes	—
68	C₅×M₁₆	—	2	4	no	—
69	C₅×M₁₆	—	2	4	no	—
70	C₂×C₅×C₈	—	2	1	no	—
71	C₂×C₅×C₈	—	2	1	no	—
72	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
73	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
74	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₅×C₈	#41	1	2	yes	—
77	C₄×Dic₅	—	2	1	no	—
78	C₄×Dic₅	—	2	1	no	—
79	C₄×Dic₅	—	2	2	no	—
80	C₄×Dic₅	—	2	2	no	—
81	C₄×Dic₅	C₂²⋊C₄	1	2	yes	—
82	C₄×Dic₅	M₁₆	1	1	yes	characteristic
83	C₄×Dic₅	M₁₆	1	1	yes	characteristic
84	#681	—	2	8	no	—
85	C₂×#168	—	2	1	no	—
86	C₂×#168	—	2	1	no	—
87	C₂×#168	—	2	1	no	—
88	C₂×#168	—	2	1	no	—
89	C₂×#168	C₂²⋊C₄	1	2	yes	—
90	C₂×#168	M₁₆	1	2	yes	—
91	C₅×C₁₆	—	4	2	no	—
92	#672	—	4	2	no	—
93	C₂²×M₁₆	—	5	1	no	—
94	C₂²×M₁₆	C₂×D₅	1	1	yes	characteristic
95	C₂²×C₄²	—	5	1	no	—
96	C₂×M₃₂	—	5	2	no	—
97	C₂×M₃₂	C₂×D₅	1	2	yes	—
98	C₄×M₁₆	—	10	4	no	—
99	C₂×#97	—	5	1	no	—
100	C₂×#97	—	5	1	no	—
101	C₂×C₄×C₈	—	5	1	no	—
102	C₂×C₄×C₈	—	5	1	no	—
103	C₄⋊C₁₆	—	10	4	no	—
104	#349	—	2	2	no	—
105	#349	—	10	2	no	—
106	C₂³×C₅	—	2	1	no	—
107	C₂³×C₅	—	2	1	no	—
108	C₂³×C₅	—	4	2	no	—
109	C₂³×C₅	C₄⋊C₈	1	2	yes	—
110	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
118	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	#117	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
124	C₂×Dic₅	—	2	4	no	—
125	C₂×Dic₅	—	2	2	no	—
126	C₂×Dic₅	—	4	4	no	—
127	C₂×Dic₅	—	4	4	no	—
128	C₂×Dic₅	—	4	2	no	—
129	C₂×Dic₅	C₂²⋊C₈	1	2	yes	—
130	C₂×Dic₅	C₂²⋊C₈	1	2	yes	—
131	C₅×C₈	—	2	2	no	—
132	C₅×C₈	#121	1	2	yes	—
133	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
134	#168	—	2	2	no	—
135	#168	—	2	2	no	—
136	C₂³×C₄	—	5	2	no	—
137	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
138	C₂×M₁₆	—	2	2	no	—
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	10	2	no	—
141	C₂×M₁₆	—	10	2	no	—
142	C₂×M₁₆	—	10	2	no	—
143	C₂×M₁₆	Dic₁₀	1	2	yes	—
144	C₂×M₁₆	C₂²⋊D₅	1	2	yes	—
145	C₂²×C₈	—	5	1	no	—
146	C₂²×C₈	—	5	1	no	—
147	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
149	C₂×C₄²	—	5	1	no	—
150	C₂×C₄²	—	5	2	no	—
151	C₂×C₄²	—	5	1	no	—
152	C₂×C₄²	—	10	2	no	—
153	C₂×C₄²	—	10	1	no	—
154	C₂×C₄²	—	10	1	no	—
155	M₃₂	—	2	2	no	—
156	M₃₂	—	2	2	no	—
157	M₃₂	—	10	2	no	—
158	M₃₂	—	10	2	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	10	2	no	—
161	#97	—	10	2	no	—
162	#97	—	10	2	no	—
163	C₄×C₈	—	5	4	no	—
164	C₄×C₈	—	10	2	no	—
165	C₂²×C₅	—	2	2	no	—
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	—	4	2	no	—
168	C₂²×C₅	—	4	2	no	—
169	C₂²×C₅	—	4	2	no	—
170	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
171	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
172	C₂²×C₅	#431	1	1	yes	characteristic
173	C₄×C₅	—	2	2	no	—
174	C₄×C₅	#380	1	1	yes	characteristic
175	C₄×C₅	#407	1	1	yes	characteristic
176	C₄×C₅	#423	1	1	yes	characteristic
177	C₄×C₅	#432	1	1	yes	characteristic
178	Dic₅	—	2	2	no	—
179	Dic₅	—	4	2	no	—
180	Dic₅	#349	1	4	yes	complemented
181	C₂⁴	C₈×D₅	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	5	2	no	—
187	C₂²×C₄	—	5	2	no	—
188	C₂²×C₄	—	10	2	no	—
189	C₂²×C₄	—	10	2	no	—
190	C₂²×C₄	—	20	4	no	—
191	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
192	C₂²×C₄	#676	1	2	yes	—
193	C₂²×C₄	#683	1	2	yes	—
194	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
195	M₁₆	—	2	4	no	—
196	M₁₆	—	2	4	no	—
197	M₁₆	—	10	8	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	5	2	no	—
201	C₂×C₈	—	5	2	no	—
202	C₂×C₈	—	10	1	no	—
203	C₂×C₈	—	10	1	no	—
204	C₂×C₈	—	10	1	no	—
205	C₂×C₈	—	10	1	no	—
206	C₂×C₈	#683	1	2	yes	—
207	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
208	C₂×C₈	D₄×D₅	1	2	yes	—
209	C₂×C₈	#711	1	2	yes	—
210	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
211	C₄²	—	5	2	no	—
212	C₄²	—	5	1	no	—
213	C₄²	—	5	1	no	—
214	C₄²	—	10	2	no	—
215	C₄²	—	10	2	no	—
216	C₄²	—	10	1	no	—
217	C₄²	—	10	1	no	—
218	C₁₆	—	4	2	no	—
219	C₁₆	—	20	2	no	—
220	C₂×C₅	—	2	1	no	—
221	C₂×C₅	—	2	1	no	—
222	C₂×C₅	—	4	2	no	—
223	C₂×C₅	#1797	1	1	yes	characteristic
224	C₂×C₅	C₂×#349	1	2	yes	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	4	2	no	—
228	C₂³	#3946	1	2	yes	—
229	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	5	2	no	—
237	C₂×C₄	—	5	2	no	—
238	C₂×C₄	—	10	4	no	—
239	C₂×C₄	—	10	2	no	—
240	C₂×C₄	—	20	4	no	—
241	C₂×C₄	—	20	4	no	—
242	C₂×C₄	—	20	2	no	—
243	C₂×C₄	#3946	1	2	yes	—
244	C₂×C₄	#4023	1	1	yes	characteristic
245	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
246	C₂×C₄	C₂×#676	1	1	yes	characteristic
247	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
248	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
249	C₂×C₄	C₂×#683	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	10	2	no	—
252	C₈	—	10	2	no	—
253	C₈	#4024	1	2	yes	—
254	C₈	#4085	1	2	yes	—
255	C₅	#12785	1	1	yes	characteristic, complemented, Sylow
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	#65007	1	1	yes	characteristic
262	C₂²	D₅×C₂²⋊C₈	1	1	yes	characteristic
263	C₂²	C₂×#3946	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	5	4	no	—
266	C₄	—	10	2	no	—
267	C₄	—	20	2	no	—
268	C₄	#65008	1	1	yes	characteristic
269	C₄	#65015	1	1	yes	characteristic
270	C₄	#65231	1	1	yes	characteristic
271	C₄	#65407	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	4	2	no	—
275	C₂	D₅×#349	1	2	yes	—
276	C₂	#10600701	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₅×C₂²⋊C₄
commutator	C₂³×C₅	C₂²×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	#12785
LCS	G ▷ C₂³×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. #10602213	C₂	semidirect
5. #10598303	C₂	semidirect
180. Dic₅	#349	semidirect
255. C₅	#12785	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×Dic₅	2
C₂×C₅×#349	2
C₂×#64773	2
#10602213	2
#10598303	2
#12785	5