Subgroups — #12254735

Order 1280.

814 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#65468	C₂	1	4	yes	maximal, complemented
3	#10610198	C₂	1	1	yes	maximal, characteristic
4	#10602107	C₂	1	4	yes	maximal, complemented
5	#10602106	C₂	1	2	yes	maximal, complemented
6	#10602105	C₂	1	2	yes	maximal, complemented
7	C₄×#3944	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×Q₁₆	—	2	1	no	—
9	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
11	C₂×#4050	—	2	1	no	—
12	C₂×#4050	C₂²	1	2	yes	—
13	#65811	—	2	1	no	—
14	C₅×#494	—	2	1	no	—
15	C₅×#494	—	2	1	no	—
16	C₅×#494	C₂²	1	2	yes	—
17	C₅×#494	C₂²	1	2	yes	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₂×#3961	C₂²	1	8	yes	—
20	#65468	—	2	16	no	—
21	C₂×#3960	C₂²	1	8	yes	—
22	C₂×#3944	C₂²	1	2	yes	—
23	#64988	C₂²	1	1	yes	characteristic
24	#64985	—	2	1	no	—
25	#64985	C₂²	1	2	yes	—
26	#64984	C₂²	1	1	yes	characteristic
27	#64982	—	2	1	no	—
28	#64681	C₂²	1	2	yes	—
29	C₄×#672	C₂²	1	2	yes	—
30	#33718	—	5	1	no	maximal, Sylow
31	C₂²×C₅×Q₈	—	2	2	no	—
32	C₂×#709	—	2	2	no	—
33	C₂×C₅×Q₁₆	—	2	8	no	—
34	C₂×C₅×Q₁₆	—	2	2	no	—
35	C₂×C₅×Q₁₆	—	2	2	no	—
36	C₂×C₅×Q₁₆	—	2	2	no	—
37	C₂×C₅×Q₁₆	—	4	4	no	—
38	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
39	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₅×C₄⋊Q₈	—	2	2	no	—
42	C₅×C₄⋊Q₈	—	2	2	no	—
43	C₅×C₄⋊Q₈	—	4	2	no	—
44	C₂×C₅×C₄⋊C₄	—	4	2	no	—
45	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
46	#4076	—	4	2	no	—
47	C₂×Dic₂₀	—	2	1	no	—
48	C₂×Dic₂₀	C₂³	1	2	yes	—
49	#4050	—	2	8	no	—
50	#4050	—	4	4	no	—
51	C₂×D₄₀	—	2	1	no	—
52	C₂×D₄₀	C₂³	1	2	yes	—
53	C₂×#677	—	2	2	no	—
54	C₂×#677	—	2	2	no	—
55	#4021	—	2	2	no	—
56	#4020	—	2	1	no	—
57	C₄×D₂₀	—	4	2	no	—
58	#4015	—	2	1	no	—
59	C₄×C₅×C₈	D₄	1	1	yes	characteristic
60	C₄×C₅×C₈	D₄	1	1	yes	characteristic
61	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
62	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
63	#3961	—	2	16	no	—
64	#3960	—	2	16	no	—
65	#3944	D₄	1	8	yes	complemented
66	C₂×#672	C₂³	1	4	yes	—
67	C₂×#510	—	5	4	no	—
68	#3147	—	5	1	no	—
69	C₂×#494	D₅	1	1	yes	characteristic, complemented
70	#2251	—	5	4	no	—
71	#2248	—	5	2	no	—
72	#2247	—	5	2	no	—
73	C₄×M₃₂	—	5	1	no	—
74	C₂×C₅×Q₈	—	2	4	no	—
75	C₂×C₅×Q₈	—	4	2	no	—
76	C₂×C₅×Q₈	—	4	4	no	—
77	C₂×C₅×Q₈	—	4	4	no	—
78	C₂²×C₄×C₅	—	4	2	no	—
79	C₂²×C₄×C₅	D₈	1	2	yes	—
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₅	—	4	2	no	—
82	#709	—	4	4	no	—
83	#709	—	4	4	no	—
84	C₂×D₂₀	—	2	2	no	—
85	C₂×D₂₀	—	4	2	no	—
86	C₂×C₄×D₅	—	4	2	no	—
87	C₂×Dic₁₀	—	2	2	no	—
88	C₅×Q₁₆	—	2	8	no	—
89	C₅×Q₁₆	—	4	4	no	—
90	C₅×Q₁₆	—	4	4	no	—
91	C₅×Q₁₆	—	4	4	no	—
92	C₂×C₅×C₈	D₈	1	4	yes	—
93	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
94	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
98	C₅×C₄⋊C₄	—	4	4	no	—
99	C₅×C₄⋊C₄	—	4	4	no	—
100	C₄²×C₅	D₈	1	2	yes	—
101	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
103	#685	—	4	4	no	—
104	C₄⋊Dic₅	—	4	2	no	—
105	Dic₂₀	—	2	4	no	—
106	Dic₂₀	—	4	2	no	—
107	D₄₀	—	2	4	no	—
108	D₄₀	—	4	2	no	—
109	#677	—	4	4	no	—
110	#677	—	4	4	no	—
111	#672	C₂×D₄	1	8	yes	—
112	C₂×#135	—	5	2	no	—
113	C₂×#135	—	10	1	no	—
114	C₂²×Q₁₆	—	2	1	no	—
115	C₂²×Q₁₆	C₂×D₅	1	2	yes	—
116	#532	—	10	1	no	—
117	C₂×C₄⋊Q₈	—	2	1	no	—
118	#510	—	10	16	no	—
119	C₂×Q₃₂	—	5	8	no	—
120	C₂×SD₃₂	—	5	8	no	—
121	C₂×M₃₂	—	5	2	no	—
122	#495	—	5	2	no	—
123	#495	—	10	1	no	—
124	#494	—	2	1	no	—
125	#494	—	2	1	no	—
126	#494	—	5	1	no	—
127	#494	C₂×D₅	1	2	yes	—
128	#494	C₂×D₅	1	2	yes	—
129	#493	—	5	1	no	—
130	#492	—	10	1	no	—
131	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
132	#346	—	5	2	no	—
133	C₄×C₁₆	—	5	2	no	—
134	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₅	—	4	2	no	—
136	C₅×Q₈	—	4	4	no	—
137	C₅×Q₈	—	4	4	no	—
138	C₅×Q₈	—	8	4	no	—
139	C₂×C₄×C₅	—	2	4	no	—
140	C₂×C₄×C₅	—	4	4	no	—
141	C₂×C₄×C₅	—	8	4	no	—
142	C₂×C₄×C₅	#127	1	2	yes	—
143	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
147	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
148	C₂²⋊D₅	—	8	4	no	—
149	C₂×Dic₅	—	4	2	no	—
150	D₂₀	—	4	2	no	—
151	D₂₀	—	4	2	no	—
152	D₂₀	—	8	4	no	—
153	C₄×D₅	—	8	4	no	—
154	Dic₁₀	—	4	2	no	—
155	Dic₁₀	—	4	2	no	—
156	C₅×C₈	#127	1	2	yes	—
157	C₅×C₈	C₂×D₈	1	2	yes	—
158	C₅×C₈	C₂×D₈	1	2	yes	—
159	C₅×C₈	C₂²×D₄	1	2	yes	—
160	C₂×#41	—	10	2	no	—
161	C₂²×Q₈	—	2	2	no	—
162	#135	—	10	8	no	—
163	#135	—	20	4	no	—
164	C₂×Q₁₆	—	2	8	no	—
165	C₂×Q₁₆	—	2	2	no	—
166	C₂×Q₁₆	—	2	2	no	—
167	C₂×Q₁₆	—	2	2	no	—
168	C₂×Q₁₆	—	4	4	no	—
169	C₂×Q₁₆	—	5	2	no	—
170	C₂×Q₁₆	—	10	1	no	—
171	C₂×Q₁₆	C₂²×D₅	1	4	yes	—
172	C₂×SD₁₆	—	10	2	no	—
173	C₂×SD₁₆	—	10	2	no	—
174	C₂×D₈	—	5	2	no	—
175	C₂×D₈	—	10	1	no	—
176	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	—	4	2	no	—
181	C₄⋊Q₈	—	10	1	no	—
182	#127	—	10	1	no	—
183	#124	—	10	2	no	—
184	#121	—	20	2	no	—
185	C₄×D₄	—	20	2	no	—
186	C₂×C₄⋊C₄	—	4	2	no	—
187	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
188	Q₃₂	—	10	16	no	—
189	SD₃₂	—	10	16	no	—
190	M₃₂	—	5	8	no	—
191	C₂×C₁₆	—	5	4	no	—
192	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
193	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
195	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	#493	1	2	yes	—
198	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
199	C₂×D₅	—	8	2	no	—
200	C₂×D₅	—	8	4	no	—
201	C₄×C₅	—	2	2	no	—
202	C₄×C₅	—	8	4	no	—
203	C₄×C₅	#493	1	2	yes	—
204	C₄×C₅	#510	1	4	yes	complemented
205	C₄×C₅	C₂×#127	1	1	yes	characteristic
206	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
207	Dic₅	—	8	2	no	—
208	#41	—	20	4	no	—
209	#41	—	20	4	no	—
210	C₂×Q₈	—	2	4	no	—
211	C₂×Q₈	—	4	2	no	—
212	C₂×Q₈	—	4	4	no	—
213	C₂×Q₈	—	4	4	no	—
214	C₂×Q₈	—	10	2	no	—
215	C₂×D₄	—	10	2	no	—
216	C₂×D₄	—	20	2	no	—
217	C₂×D₄	—	20	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	20	2	no	—
220	C₂²×C₄	#686	1	2	yes	—
221	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
222	Q₁₆	—	2	8	no	—
223	Q₁₆	—	4	4	no	—
224	Q₁₆	—	4	4	no	—
225	Q₁₆	—	4	4	no	—
226	Q₁₆	—	10	4	no	—
227	Q₁₆	—	20	2	no	—
228	SD₁₆	—	20	4	no	—
229	SD₁₆	—	20	4	no	—
230	D₈	—	10	4	no	—
231	D₈	—	20	2	no	—
232	C₂×C₈	#686	1	4	yes	—
233	C₂×C₈	D₄×D₅	1	4	yes	—
234	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
235	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
236	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
237	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	4	no	—
240	C₄⋊C₄	—	20	2	no	—
241	C₂²⋊C₄	—	20	4	no	—
242	C₄²	#686	1	2	yes	—
243	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
244	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
245	C₁₆	—	5	8	no	—
246	C₂×C₅	—	2	2	no	—
247	C₂×C₅	C₂×#493	1	1	yes	characteristic
248	C₂×C₅	C₂×#510	1	2	yes	—
249	D₅	—	16	2	no	—
250	C₂³	—	20	2	no	—
251	C₂³	C₂×#686	1	1	yes	characteristic
252	Q₈	—	4	4	no	—
253	Q₈	—	4	4	no	—
254	Q₈	—	8	4	no	—
255	Q₈	—	20	2	no	—
256	Q₈	—	20	2	no	—
257	D₄	—	20	2	no	—
258	D₄	—	20	2	no	—
259	D₄	—	40	4	no	—
260	D₄	—	40	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	8	4	no	—
264	C₂×C₄	—	20	2	no	—
265	C₂×C₄	—	40	4	no	—
266	C₂×C₄	C₂×#686	1	2	yes	—
267	C₂×C₄	C₂×#686	1	2	yes	—
268	C₂×C₄	C₂×#686	1	1	yes	characteristic
269	C₂×C₄	C₂×#686	1	2	yes	—
270	C₂×C₄	#4086	1	2	yes	—
271	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
272	C₈	C₂×#686	1	2	yes	—
273	C₈	C₂×#686	1	2	yes	—
274	C₈	#4086	1	2	yes	—
275	C₈	C₂×D₄×D₅	1	2	yes	—
276	C₅	#33718	1	1	yes	characteristic, complemented, Sylow
277	C₂²	—	2	2	no	—
278	C₂²	—	40	4	no	—
279	C₂²	—	40	2	no	—
280	C₂²	#65362	1	2	yes	—
281	C₂²	C₂²×#686	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	8	4	no	—
284	C₄	—	40	2	no	—
285	C₄	#65362	1	2	yes	—
286	C₄	#65468	1	4	yes	complemented
287	C₄	C₂²×#686	1	1	yes	characteristic
288	C₄	C₂×#4086	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	—	80	2	no	—
291	C₂	C₂×#65362	1	1	yes	characteristic
292	C₂	C₂×#65468	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#686
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#494	C₂
socle	C₂²×C₅	C₂²×D₈
prime radical	#10610198	C₂
hypercenter	C₂×#494	D₅
hypocenter	C₅	#33718
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#494	C₂	semidirect
2. C₂×#65468	C₂	semidirect
4. #10602107	C₂	semidirect
5. #10602106	C₂	semidirect
6. #10602105	C₂	semidirect
7. C₄×#3944	C₂	semidirect
65. #3944	D₄	semidirect
69. C₂×#494	D₅	semidirect
204. C₄×C₅	#510	semidirect
276. C₅	#33718	semidirect
286. C₄	#65468	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#494	2
C₂×#65468	2
#10610198	2
#10602107	2
#10602106	2
#10602105	2
C₄×#3944	2
#33718	5