Subgroups — #12015230

Order 1280.

454 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64978	C₂	1	1	yes	maximal, characteristic
2	C₅×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#64734	C₂	1	2	yes	maximal
4	#10600536	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598303	C₂	1	2	yes	maximal, complemented
6	C₂²×#676	C₂²	1	1	yes	characteristic
7	C₂²×C₈×D₅	C₄	1	1	yes	characteristic
8	C₂×#4018	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×#3947	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
13	C₂×#3944	C₂²	1	2	yes	—
14	C₂×#3935	C₄	1	1	yes	characteristic
15	#64978	—	2	2	no	—
16	#64978	—	2	2	no	—
17	C₅×C₄⋊C₁₆	—	2	2	no	—
18	#64734	—	2	4	no	—
19	C₄⋊#672	—	2	2	no	—
20	#12786	—	5	1	no	maximal, Sylow
21	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
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×#685	C₈	1	2	yes	—
26	C₂×#683	C₈	1	2	yes	—
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×#676	—	2	1	no	—
31	C₂×#676	—	2	1	no	—
32	C₂×#676	—	2	1	no	—
33	C₂×#676	—	2	1	no	—
34	C₂×#676	D₄	1	2	yes	—
35	C₂×#676	Q₈	1	2	yes	—
36	C₂×C₈×D₅	—	2	2	no	—
37	C₂×C₈×D₅	—	2	2	no	—
38	C₂×C₈×D₅	—	2	2	no	—
39	#4018	—	2	2	no	—
40	#4018	—	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	#3947	—	2	1	no	—
47	#3947	—	2	1	no	—
48	C₈×Dic₅	—	2	2	no	—
49	#3944	—	2	2	no	—
50	#3944	—	2	2	no	—
51	C₂×#672	—	2	2	no	—
52	#3935	—	2	2	no	—
53	C₂×#405	—	5	1	no	—
54	#2150	—	5	1	no	—
55	#2150	D₅	1	1	yes	characteristic, complemented
56	C₂×#349	—	5	2	no	—
57	#1377	—	5	2	no	—
58	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
62	C₂²×Dic₅	C₂×C₈	1	2	yes	—
63	C₂×C₄×D₅	—	2	1	no	—
64	C₂×C₄×D₅	—	2	1	no	—
65	C₂×C₄×D₅	—	2	1	no	—
66	C₂×C₄×D₅	—	2	1	no	—
67	C₂×C₄×D₅	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×D₅	M₁₆	1	2	yes	—
69	C₂×C₅×C₈	—	2	1	no	—
70	C₂×C₅×C₈	—	2	1	no	—
71	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
72	C₂×C₅×C₈	C₂×D₄	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×Q₈	1	1	yes	characteristic
76	C₂×C₅×C₈	#41	1	1	yes	characteristic
77	C₂×C₅×C₈	#41	1	1	yes	characteristic
78	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₅	M₁₆	1	2	yes	—
80	#685	—	2	4	no	—
81	#683	—	2	4	no	—
82	C₄×Dic₅	—	2	1	no	—
83	C₄×Dic₅	—	2	1	no	—
84	C₂×#168	—	2	1	no	—
85	C₂×#168	—	2	1	no	—
86	C₂×#168	—	2	1	no	—
87	C₂×#168	—	2	1	no	—
88	C₂×#168	C₄⋊C₄	1	2	yes	—
89	C₂×#168	M₁₆	1	2	yes	—
90	#676	—	2	2	no	—
91	#676	—	2	2	no	—
92	#676	—	2	2	no	—
93	#676	—	2	2	no	—
94	C₈×D₅	—	2	8	no	—
95	C₅×C₁₆	—	4	2	no	—
96	#672	—	4	2	no	—
97	C₂²×M₁₆	—	5	1	no	—
98	C₂³×C₈	—	5	1	no	—
99	C₂×#117	—	5	1	no	—
100	C₂×M₃₂	—	5	2	no	—
101	C₂×M₃₂	C₂×D₅	1	2	yes	—
102	#405	—	10	2	no	—
103	#405	—	10	2	no	—
104	C₂×#97	—	5	1	no	—
105	C₂×#97	—	5	1	no	—
106	C₂×C₄×C₈	—	5	1	no	—
107	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
108	C₄⋊C₁₆	—	2	2	no	—
109	C₄⋊C₁₆	—	10	2	no	—
110	#349	—	10	4	no	—
111	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
112	C₂²×D₅	—	2	1	no	—
113	C₂²×D₅	—	2	1	no	—
114	C₂²×D₅	—	4	2	no	—
115	C₂²×D₅	C₄⋊C₈	1	2	yes	—
116	C₂×C₄×C₅	—	4	2	no	—
117	C₂×C₄×C₅	C₂²⋊C₈	1	4	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	1	no	—
125	C₂×Dic₅	—	2	1	no	—
126	C₂×Dic₅	—	2	2	no	—
127	C₂×Dic₅	—	4	2	no	—
128	C₂×Dic₅	C₄⋊C₈	1	2	yes	—
129	C₄×D₅	—	2	2	no	—
130	C₄×D₅	—	2	2	no	—
131	C₄×D₅	—	2	2	no	—
132	C₄×D₅	—	2	2	no	—
133	C₅×C₈	—	2	2	no	—
134	C₅×C₈	#121	1	1	yes	characteristic
135	C₅×C₈	#121	1	1	yes	characteristic
136	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
137	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
138	#168	—	2	2	no	—
139	#168	—	2	2	no	—
140	C₂³×C₄	—	5	1	no	—
141	C₂×M₁₆	—	5	2	no	—
142	C₂×M₁₆	—	5	2	no	—
143	C₂×M₁₆	—	10	1	no	—
144	C₂×M₁₆	—	10	1	no	—
145	C₂×M₁₆	—	10	1	no	—
146	C₂×M₁₆	—	10	1	no	—
147	C₂²×C₈	—	5	1	no	—
148	C₂²×C₈	—	5	1	no	—
149	C₂²×C₈	—	10	2	no	—
150	C₂²×C₈	—	10	2	no	—
151	C₂²×C₈	—	10	2	no	—
152	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
153	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
154	#117	—	10	2	no	—
155	#117	—	10	2	no	—
156	C₂×C₄⋊C₄	—	5	2	no	—
157	C₂×C₂²⋊C₄	—	5	2	no	—
158	C₂×C₄²	—	5	1	no	—
159	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
160	M₃₂	—	2	2	no	—
161	M₃₂	—	2	2	no	—
162	M₃₂	—	10	2	no	—
163	M₃₂	—	10	2	no	—
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	—	10	2	no	—
166	#97	—	10	1	no	—
167	#97	—	10	1	no	—
168	#97	—	10	2	no	—
169	C₄×C₈	—	10	2	no	—
170	C₄×C₈	D₂₀	1	2	yes	—
171	C₄×C₈	C₂²⋊D₅	1	2	yes	—
172	C₂²×C₅	—	2	2	no	—
173	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
174	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
175	C₂²×C₅	#431	1	1	yes	characteristic
176	C₂×D₅	—	2	2	no	—
177	C₂×D₅	—	4	2	no	—
178	C₂×D₅	—	4	2	no	—
179	C₂×D₅	—	4	2	no	—
180	C₄×C₅	—	2	4	no	—
181	C₄×C₅	#380	1	1	yes	characteristic
182	C₄×C₅	#407	1	1	yes	characteristic
183	C₄×C₅	#423	1	1	yes	characteristic
184	C₄×C₅	#432	1	1	yes	characteristic
185	Dic₅	—	2	2	no	—
186	Dic₅	—	4	2	no	—
187	C₂⁴	—	5	1	no	—
188	C₂²×C₄	—	5	1	no	—
189	C₂²×C₄	—	5	2	no	—
190	C₂²×C₄	—	5	2	no	—
191	C₂²×C₄	—	5	2	no	—
192	C₂²×C₄	—	10	1	no	—
193	C₂²×C₄	—	10	1	no	—
194	C₂²×C₄	—	10	1	no	—
195	C₂²×C₄	—	10	1	no	—
196	C₂²×C₄	C₈×D₅	1	2	yes	—
197	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
198	M₁₆	—	10	2	no	—
199	M₁₆	—	10	2	no	—
200	M₁₆	—	10	2	no	—
201	M₁₆	—	10	2	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	5	2	no	—
205	C₂×C₈	—	5	2	no	—
206	C₂×C₈	—	10	8	no	—
207	C₂×C₈	—	10	1	no	—
208	C₂×C₈	—	10	1	no	—
209	C₂×C₈	—	10	1	no	—
210	C₂×C₈	—	10	1	no	—
211	C₂×C₈	#685	1	2	yes	—
212	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
213	C₂×C₈	D₄×D₅	1	1	yes	characteristic
214	C₂×C₈	#711	1	1	yes	characteristic
215	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
216	C₂×C₈	#713	1	1	yes	characteristic
217	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
218	C₄⋊C₄	—	10	4	no	—
219	C₂²⋊C₄	—	10	4	no	—
220	C₄²	—	10	1	no	—
221	C₄²	—	10	1	no	—
222	C₄²	#676	1	2	yes	—
223	C₄²	#685	1	2	yes	—
224	C₁₆	—	4	2	no	—
225	C₁₆	—	20	2	no	—
226	C₂×C₅	—	2	1	no	—
227	C₂×C₅	—	2	1	no	—
228	C₂×C₅	#1797	1	1	yes	characteristic
229	C₂×C₅	#2115	1	2	yes	—
230	D₅	—	4	2	no	—
231	C₂³	—	5	2	no	—
232	C₂³	—	10	1	no	—
233	C₂³	—	10	1	no	—
234	C₂³	—	20	2	no	—
235	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	5	2	no	—
238	C₂×C₄	—	10	1	no	—
239	C₂×C₄	—	10	1	no	—
240	C₂×C₄	—	10	2	no	—
241	C₂×C₄	—	10	2	no	—
242	C₂×C₄	—	10	2	no	—
243	C₂×C₄	—	10	2	no	—
244	C₂×C₄	—	10	2	no	—
245	C₂×C₄	—	20	2	no	—
246	C₂×C₄	#3952	1	4	yes	—
247	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
248	C₂×C₄	#4038	1	1	yes	characteristic
249	C₂×C₄	C₂×#676	1	1	yes	characteristic
250	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
251	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
252	C₂×C₄	C₂×#685	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	10	2	no	—
255	C₈	—	10	2	no	—
256	C₈	#4041	1	1	yes	characteristic
257	C₈	#4043	1	1	yes	characteristic
258	C₈	#4084	1	1	yes	characteristic
259	C₈	#4092	1	1	yes	characteristic
260	C₅	#12786	1	1	yes	characteristic, complemented, Sylow
261	C₂²	—	2	2	no	—
262	C₂²	—	10	2	no	—
263	C₂²	—	20	2	no	—
264	C₂²	—	20	2	no	—
265	C₂²	—	20	2	no	—
266	C₂²	D₅×C₄⋊C₈	1	1	yes	characteristic
267	C₂²	#65122	1	1	yes	characteristic
268	C₂²	C₂×#3952	1	1	yes	characteristic
269	C₄	—	2	4	no	—
270	C₄	—	10	2	no	—
271	C₄	—	20	2	no	—
272	C₄	#65124	1	1	yes	characteristic
273	C₄	#65126	1	1	yes	characteristic
274	C₄	#65276	1	1	yes	characteristic
275	C₄	#65415	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	20	2	no	—
279	C₂	#10601268	1	1	yes	characteristic
280	C₂	#10602228	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#685
commutator	C₂³×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#2150	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#2150	D₅
hypocenter	C₅	#12786
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. C₅×#2150	C₂	semidirect
4. #10600536	C₂	semidirect
5. #10598303	C₂	semidirect
55. #2150	D₅	semidirect
260. C₅	#12786	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64978	2
C₅×#2150	2
C₂×#64734	2
#10600536	2
#10598303	2
#12786	5