Subgroups — #12015239

Order 1280.

538 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602224	C₂	1	2	yes	maximal
4	C₂×#64683	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598301	C₂	1	2	yes	maximal, complemented
6	C₂²×C₈×D₅	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₅	C₄	1	1	yes	characteristic
9	C₂×C₅×M₃₂	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
13	C₂×#3944	C₂²	1	2	yes	—
14	C₂×C₄×#168	C₄	1	1	yes	characteristic
15	C₄×C₈×D₅	—	2	2	no	—
16	C₄×C₈×D₅	—	2	2	no	—
17	C₅×#364	—	2	2	no	—
18	#64727	—	2	4	no	—
19	#64683	—	2	2	no	—
20	#12788	—	5	1	no	maximal, Sylow
21	C₂²×C₄×D₅	—	2	1	no	—
22	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₅	—	2	1	no	—
27	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×#168	C₂×C₄	1	1	yes	characteristic
29	C₂²×#168	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₅	—	2	2	no	—
31	C₂×C₈×D₅	—	2	2	no	—
32	C₂×C₈×D₅	—	2	1	no	—
33	C₂×C₈×D₅	—	2	1	no	—
34	C₂×C₈×D₅	—	2	1	no	—
35	C₂×C₈×D₅	—	2	1	no	—
36	C₂×C₈×D₅	C₈	1	4	yes	—
37	C₂×C₈×D₅	D₄	1	2	yes	—
38	C₂×C₈×D₅	Q₈	1	2	yes	—
39	C₄²×D₅	—	2	2	no	—
40	C₄²×D₅	—	2	2	no	—
41	C₅×M₃₂	—	2	2	no	—
42	C₅×M₃₂	—	2	2	no	—
43	C₂×C₅×C₁₆	—	2	2	no	—
44	C₄×C₅×C₈	D₄	1	2	yes	—
45	C₄×C₅×C₈	D₄	1	2	yes	—
46	C₈×Dic₅	—	2	2	no	—
47	C₈×Dic₅	—	2	1	no	—
48	C₈×Dic₅	—	2	1	no	—
49	#3944	—	2	2	no	—
50	#3944	—	2	2	no	—
51	C₂×#672	—	2	2	no	—
52	C₄×#168	—	2	2	no	—
53	C₂²×C₄×C₈	—	5	1	no	—
54	C₂×#364	—	5	1	no	—
55	C₂×#364	D₅	1	1	yes	characteristic, complemented
56	#2112	—	5	2	no	—
57	#1376	—	5	2	no	—
58	C₂³×D₅	M₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₅	—	2	1	no	—
60	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂²×Dic₅	—	2	1	no	—
62	C₂²×Dic₅	M₁₆	1	1	yes	characteristic
63	C₂×C₄×D₅	—	2	8	no	—
64	C₂×C₄×D₅	—	2	4	no	—
65	C₂×C₄×D₅	—	2	1	no	—
66	C₂×C₄×D₅	—	2	1	no	—
67	C₂×C₄×D₅	—	2	1	no	—
68	C₂×C₄×D₅	—	2	1	no	—
69	C₂×C₄×D₅	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
71	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
72	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
73	C₂×C₅×C₈	M₁₆	1	2	yes	—
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×D₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₅×C₈	#41	1	1	yes	characteristic
79	C₂×C₅×C₈	#41	1	1	yes	characteristic
80	C₄²×C₅	—	2	1	no	—
81	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
82	C₄×Dic₅	—	2	4	no	—
83	C₄×Dic₅	—	2	1	no	—
84	C₄×Dic₅	M₁₆	1	2	yes	—
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	C₄⋊C₄	1	2	yes	—
90	C₂×#168	C₂×C₈	1	2	yes	—
91	C₈×D₅	—	2	8	no	—
92	C₈×D₅	—	2	2	no	—
93	C₈×D₅	—	2	2	no	—
94	C₈×D₅	—	2	2	no	—
95	C₈×D₅	—	2	2	no	—
96	C₅×C₁₆	—	4	2	no	—
97	#672	—	4	2	no	—
98	C₂³×C₈	—	5	1	no	—
99	C₂³×C₈	—	5	1	no	—
100	C₂²×C₄²	—	5	1	no	—
101	C₂×M₃₂	—	5	2	no	—
102	C₂×M₃₂	C₂×D₅	1	2	yes	—
103	C₂×C₄×C₈	—	5	1	no	—
104	C₂×C₄×C₈	—	5	1	no	—
105	C₂×C₄×C₈	—	5	1	no	—
106	C₂×C₄×C₈	—	10	2	no	—
107	C₂×C₄×C₈	—	10	2	no	—
108	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
109	#364	—	2	2	no	—
110	#364	—	10	2	no	—
111	C₂²⋊C₁₆	—	10	4	no	—
112	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
113	C₂²×D₅	—	2	1	no	—
114	C₂²×D₅	—	2	1	no	—
115	C₂²×D₅	—	2	4	no	—
116	C₂²×D₅	C₄⋊C₈	1	2	yes	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅	#117	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
125	C₂×Dic₅	—	2	4	no	—
126	C₂×Dic₅	—	2	1	no	—
127	C₂×Dic₅	—	2	2	no	—
128	C₂×Dic₅	—	2	1	no	—
129	C₂×Dic₅	C₄⋊C₈	1	2	yes	—
130	C₄×D₅	—	2	16	no	—
131	C₄×D₅	—	2	2	no	—
132	C₄×D₅	—	2	2	no	—
133	C₄×D₅	—	2	2	no	—
134	C₄×D₅	—	2	2	no	—
135	C₅×C₈	C₂²⋊C₈	1	4	yes	—
136	C₅×C₈	#121	1	1	yes	characteristic
137	C₅×C₈	#121	1	1	yes	characteristic
138	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
139	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
140	#168	—	2	2	no	—
141	#168	—	2	2	no	—
142	C₂³×C₄	—	5	1	no	—
143	C₂³×C₄	—	10	1	no	—
144	C₂²×C₈	—	5	1	no	—
145	C₂²×C₈	—	5	1	no	—
146	C₂²×C₈	—	5	4	no	—
147	C₂²×C₈	—	5	2	no	—
148	C₂²×C₈	—	5	2	no	—
149	C₂²×C₈	—	10	2	no	—
150	C₂²×C₈	—	10	2	no	—
151	C₂²×C₈	—	10	1	no	—
152	C₂²×C₈	—	10	1	no	—
153	C₂²×C₈	—	10	1	no	—
154	C₂²×C₈	—	10	1	no	—
155	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
156	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
157	C₂×C₄²	—	5	1	no	—
158	C₂×C₄²	—	10	2	no	—
159	C₂×C₄²	—	10	1	no	—
160	C₂×C₄²	—	10	2	no	—
161	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
162	M₃₂	—	2	2	no	—
163	M₃₂	—	2	2	no	—
164	M₃₂	—	10	2	no	—
165	M₃₂	—	10	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	10	2	no	—
168	C₄×C₈	—	10	2	no	—
169	C₄×C₈	—	10	1	no	—
170	C₄×C₈	—	10	1	no	—
171	C₄×C₈	—	10	2	no	—
172	C₄×C₈	D₂₀	1	2	yes	—
173	C₄×C₈	C₂²⋊D₅	1	2	yes	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	#407	1	1	yes	characteristic
176	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
177	C₂²×C₅	#432	1	1	yes	characteristic
178	C₂×D₅	—	2	4	no	—
179	C₂×D₅	—	2	4	no	—
180	C₂×D₅	—	2	2	no	—
181	C₂×D₅	—	2	2	no	—
182	C₂×D₅	#364	1	4	yes	complemented
183	C₄×C₅	—	2	4	no	—
184	C₄×C₅	#380	1	1	yes	characteristic
185	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
186	C₄×C₅	#423	1	1	yes	characteristic
187	C₄×C₅	#431	1	1	yes	characteristic
188	Dic₅	—	2	4	no	—
189	Dic₅	—	2	2	no	—
190	C₂⁴	—	5	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	5	1	no	—
193	C₂²×C₄	—	5	2	no	—
194	C₂²×C₄	—	5	2	no	—
195	C₂²×C₄	—	10	4	no	—
196	C₂²×C₄	—	10	8	no	—
197	C₂²×C₄	—	10	1	no	—
198	C₂²×C₄	—	10	1	no	—
199	C₂²×C₄	—	10	1	no	—
200	C₂²×C₄	—	10	1	no	—
201	C₂²×C₄	—	10	1	no	—
202	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
203	C₂×C₈	—	5	2	no	—
204	C₂×C₈	—	5	2	no	—
205	C₂×C₈	—	10	1	no	—
206	C₂×C₈	—	10	1	no	—
207	C₂×C₈	—	10	1	no	—
208	C₂×C₈	—	10	1	no	—
209	C₂×C₈	—	10	8	no	—
210	C₂×C₈	—	10	2	no	—
211	C₂×C₈	—	10	2	no	—
212	C₂×C₈	—	10	2	no	—
213	C₂×C₈	—	10	2	no	—
214	C₂×C₈	C₈×D₅	1	2	yes	—
215	C₂×C₈	#676	1	2	yes	—
216	C₂×C₈	#685	1	2	yes	—
217	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
218	C₂×C₈	D₄×D₅	1	1	yes	characteristic
219	C₂×C₈	#711	1	1	yes	characteristic
220	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
221	C₂×C₈	#713	1	1	yes	characteristic
222	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
223	C₄²	—	2	1	no	—
224	C₄²	—	5	2	no	—
225	C₄²	—	10	4	no	—
226	C₄²	—	10	1	no	—
227	C₄²	#685	1	2	yes	—
228	C₁₆	—	4	2	no	—
229	C₁₆	—	20	2	no	—
230	C₂×C₅	—	2	1	no	—
231	C₂×C₅	—	2	1	no	—
232	C₂×C₅	#1797	1	1	yes	characteristic
233	C₂×C₅	C₂×#364	1	2	yes	—
234	D₅	—	2	4	no	—
235	C₂³	—	5	2	no	—
236	C₂³	—	10	4	no	—
237	C₂³	—	10	1	no	—
238	C₂³	—	10	1	no	—
239	C₂³	D₅×M₁₆	1	1	yes	characteristic
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	5	2	no	—
243	C₂×C₄	—	10	16	no	—
244	C₂×C₄	—	10	4	no	—
245	C₂×C₄	—	10	1	no	—
246	C₂×C₄	—	10	2	no	—
247	C₂×C₄	—	10	1	no	—
248	C₂×C₄	—	10	2	no	—
249	C₂×C₄	—	10	2	no	—
250	C₂×C₄	—	10	2	no	—
251	C₂×C₄	—	10	2	no	—
252	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
253	C₂×C₄	#4038	1	1	yes	characteristic
254	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
255	C₂×C₄	C₂×#676	1	1	yes	characteristic
256	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
257	C₂×C₄	C₂×#685	1	1	yes	characteristic
258	C₈	—	10	2	no	—
259	C₈	—	10	2	no	—
260	C₈	#3952	1	4	yes	—
261	C₈	#4041	1	1	yes	characteristic
262	C₈	#4043	1	1	yes	characteristic
263	C₈	#4084	1	1	yes	characteristic
264	C₈	#4092	1	1	yes	characteristic
265	C₅	#12788	1	1	yes	characteristic, complemented, Sylow
266	C₂²	—	2	2	no	—
267	C₂²	—	5	4	no	—
268	C₂²	—	10	4	no	—
269	C₂²	—	10	4	no	—
270	C₂²	—	10	2	no	—
271	C₂²	—	10	2	no	—
272	C₂²	D₅×C₄⋊C₈	1	1	yes	characteristic
273	C₂²	#65124	1	1	yes	characteristic
274	C₂²	#65415	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	10	4	no	—
277	C₄	—	10	2	no	—
278	C₄	#65122	1	1	yes	characteristic
279	C₄	#65126	1	1	yes	characteristic
280	C₄	#65276	1	1	yes	characteristic
281	C₄	C₂×#3952	1	1	yes	characteristic
282	C₂	—	2	1	no	—
283	C₂	—	2	1	no	—
284	C₂	—	10	4	no	—
285	C₂	D₅×#364	1	2	yes	—
286	C₂	#10601268	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#364	C₂
socle	C₂²×C₅	C₂×C₄⋊C₈
prime radical	C₂³×D₅	M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#364	D₅
hypocenter	C₅	#12788
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#364	C₂	semidirect
4. C₂×#64683	C₂	semidirect
5. #10598301	C₂	semidirect
55. C₂×#364	D₅	semidirect
182. C₂×D₅	#364	semidirect
265. C₅	#12788	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₅	2
C₂×C₅×#364	2
#10602224	2
C₂×#64683	2
#10598301	2
#12788	5