Subgroups — #12005209

Order 1280.

806 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3951	C₂	1	2	yes	maximal
3	C₂×#64772	C₂	1	2	yes	maximal
4	#10602320	C₂	1	2	yes	maximal
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂²×#681	C₂²	1	4	yes	—
9	C₂×C₄×C₅×C₈	—	2	4	no	—
10	C₂×C₄×C₅×C₈	C₄	1	4	yes	—
11	C₂×#3951	—	2	4	no	—
12	C₂×#3951	—	2	4	no	—
13	C₂×#3951	C₄	1	8	yes	complemented
14	#64772	—	2	4	no	—
15	#64744	—	2	4	no	—
16	#10864	—	5	1	no	maximal, Sylow
17	C₂³×C₄×C₅	Q₈	1	2	yes	—
18	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
19	C₂²×C₅×C₈	—	2	2	no	—
20	C₂²×C₅×C₈	—	2	2	no	—
21	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
22	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₅×C₈	D₄	1	2	yes	—
25	C₂²×C₅×C₈	D₄	1	2	yes	—
26	C₂²×C₅×C₈	D₄	1	4	yes	—
27	C₂²×C₅×C₈	Q₈	1	2	yes	—
28	C₂²×C₅×C₈	Q₈	1	2	yes	—
29	C₂×C₄²×C₅	—	2	2	no	—
30	C₂×C₄²×C₅	—	2	2	no	—
31	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅	D₄	1	2	yes	—
33	C₂×#681	—	2	4	no	—
34	C₂×#681	—	2	4	no	—
35	C₂×#681	—	2	4	no	—
36	C₂×#681	—	2	4	no	—
37	C₂×#681	—	2	4	no	—
38	C₂×#681	C₂×C₄	1	8	yes	—
39	C₂²×#168	—	2	4	no	—
40	C₄×C₅×C₈	—	2	8	no	—
41	#3951	—	2	16	no	—
42	C₂²×#108	—	5	2	no	—
43	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
44	C₂×#341	—	5	2	no	—
45	#1728	—	5	2	no	—
46	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₅	—	2	4	no	—
48	C₂²×C₄×C₅	—	2	8	no	—
49	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₅	#41	1	2	yes	—
59	C₂²×C₄×C₅	#41	1	2	yes	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	—	2	2	no	—
62	C₂×C₅×C₈	—	2	4	no	—
63	C₂×C₅×C₈	—	2	4	no	—
64	C₂×C₅×C₈	C₄²	1	4	yes	—
65	C₂×C₅×C₈	C₂²⋊C₄	1	8	yes	—
66	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
67	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
68	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
69	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
70	C₄²×C₅	—	2	2	no	—
71	C₄²×C₅	—	2	2	no	—
72	C₄²×C₅	—	2	4	no	—
73	#681	—	2	8	no	—
74	#681	—	2	8	no	—
75	#681	—	4	4	no	—
76	#681	—	4	4	no	—
77	C₂×#168	—	2	8	no	—
78	C₂×#168	—	4	4	no	—
79	C₂×#168	—	4	4	no	—
80	C₂²×M₁₆	—	5	4	no	—
81	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
82	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
83	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
84	C₂×#108	—	5	8	no	—
85	C₂×#108	—	10	4	no	—
86	C₂×#108	—	10	4	no	—
87	C₂×C₄×C₈	—	2	4	no	—
88	C₂×C₄×C₈	Dic₅	1	4	yes	—
89	#341	—	10	4	no	—
90	#336	—	10	4	no	—
91	C₂³×C₅	—	2	4	no	—
92	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₅	C₄×Q₈	1	2	yes	—
94	C₂³×C₅	C₂²⋊Q₈	1	2	yes	—
95	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
96	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₅	—	2	16	no	—
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	4	no	—
100	C₂×C₄×C₅	#108	1	8	yes	complemented
101	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
104	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅	#117	1	1	yes	characteristic
108	C₂×C₄×C₅	#117	1	1	yes	characteristic
109	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₅	#121	1	2	yes	—
113	C₂×C₄×C₅	#121	1	2	yes	—
114	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₅	#124	1	2	yes	—
116	C₂×C₄×C₅	#125	1	2	yes	—
117	C₂×C₄×C₅	#127	1	1	yes	characteristic
118	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
119	C₅×C₈	—	2	4	no	—
120	C₅×C₈	#95	1	8	yes	—
121	#168	—	4	8	no	—
122	C₂³×C₄	Dic₁₀	1	2	yes	—
123	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
124	C₂×M₁₆	—	5	8	no	—
125	C₂×M₁₆	—	10	4	no	—
126	C₂×M₁₆	—	10	4	no	—
127	C₂×M₁₆	—	10	4	no	—
128	C₂×M₁₆	—	10	4	no	—
129	C₂×M₁₆	—	10	4	no	—
130	C₂²×C₈	—	2	2	no	—
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	—	10	4	no	—
133	C₂²×C₈	Dic₁₀	1	2	yes	—
134	C₂²×C₈	Dic₁₀	1	2	yes	—
135	C₂²×C₈	C₄×D₅	1	4	yes	—
136	C₂²×C₈	D₂₀	1	2	yes	—
137	C₂²×C₈	D₂₀	1	2	yes	—
138	C₂²×C₈	C₂×Dic₅	1	2	yes	—
139	C₂²×C₈	C₂×Dic₅	1	2	yes	—
140	C₂²×C₈	C₂²⋊D₅	1	4	yes	—
141	C₂×C₄²	—	2	2	no	—
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	D₂₀	1	2	yes	—
144	C₂×C₄²	C₂×Dic₅	1	2	yes	—
145	#108	—	10	16	no	—
146	C₄×C₈	—	2	8	no	—
147	C₂²×C₅	—	2	4	no	—
148	C₂²×C₅	—	2	2	no	—
149	C₂²×C₅	—	2	2	no	—
150	C₂²×C₅	—	2	4	no	—
151	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₅	#380	1	1	yes	characteristic
153	C₂²×C₅	#384	1	1	yes	characteristic
154	C₂²×C₅	#389	1	2	yes	—
155	C₂²×C₅	#391	1	2	yes	—
156	C₂²×C₅	C₂×#108	1	4	yes	—
157	C₄×C₅	—	2	8	no	—
158	C₄×C₅	C₂×#95	1	1	yes	characteristic
159	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
160	C₄×C₅	#380	1	1	yes	characteristic
161	C₄×C₅	#382	1	1	yes	characteristic
162	C₄×C₅	#389	1	2	yes	—
163	C₄×C₅	#390	1	2	yes	—
164	C₂⁴	C₂×Dic₁₀	1	1	yes	characteristic
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	C₄⋊Dic₅	1	4	yes	—
168	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
169	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
171	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
174	C₂²×C₄	#709	1	2	yes	—
175	C₂²×C₄	#709	1	2	yes	—
176	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
178	M₁₆	—	10	8	no	—
179	M₁₆	—	10	8	no	—
180	M₁₆	—	20	4	no	—
181	M₁₆	—	20	4	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	10	8	no	—
187	C₂×C₈	—	20	4	no	—
188	C₂×C₈	—	20	4	no	—
189	C₂×C₈	C₄×Dic₅	1	4	yes	—
190	C₂×C₈	#683	1	8	yes	—
191	C₂×C₈	C₄⋊Dic₅	1	4	yes	—
192	C₂×C₈	C₄⋊Dic₅	1	4	yes	—
193	C₂×C₈	#685	1	8	yes	—
194	C₂×C₈	C₂²⋊Dic₅	1	4	yes	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	4	no	—
198	C₂×C₅	—	2	2	no	—
199	C₂×C₅	—	2	2	no	—
200	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
201	C₂×C₅	C₄×#108	1	2	yes	—
202	C₂×C₅	#1810	1	2	yes	—
203	C₂×C₅	#1848	1	2	yes	—
204	C₂³	—	2	4	no	—
205	C₂³	C₄×Dic₁₀	1	2	yes	—
206	C₂³	#4015	1	1	yes	characteristic
207	C₂³	#4015	1	1	yes	characteristic
208	C₂³	#4070	1	2	yes	—
209	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
210	C₂×C₄	—	2	16	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	#3951	1	8	yes	complemented
214	C₂×C₄	C₄×Dic₁₀	1	2	yes	—
215	C₂×C₄	#4015	1	1	yes	characteristic
216	C₂×C₄	#4016	1	2	yes	—
217	C₂×C₄	C₄×D₂₀	1	2	yes	—
218	C₂×C₄	C₄×D₂₀	1	2	yes	—
219	C₂×C₄	#4020	1	1	yes	characteristic
220	C₂×C₄	#4021	1	2	yes	—
221	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
222	C₂×C₄	C₂×#683	1	2	yes	—
223	C₂×C₄	#4070	1	2	yes	—
224	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
226	C₂×C₄	#4072	1	1	yes	characteristic
227	C₂×C₄	#4072	1	1	yes	characteristic
228	C₂×C₄	C₂×#685	1	2	yes	—
229	C₂×C₄	#4076	1	2	yes	—
230	C₂×C₄	#4076	1	2	yes	—
231	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
232	C₈	—	2	4	no	—
233	C₈	—	20	8	no	—
234	C₈	#3963	1	8	yes	—
235	C₅	#10864	1	1	yes	characteristic, complemented, Sylow
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	#65217	1	2	yes	—
241	C₂²	#65218	1	2	yes	—
242	C₂²	C₄×C₄⋊Dic₅	1	1	yes	characteristic
243	C₂²	#65224	1	1	yes	characteristic
244	C₂²	C₂×#3951	1	4	yes	—
245	C₂²	#65501	1	1	yes	characteristic
246	C₄	—	2	8	no	—
247	C₄	#65217	1	2	yes	—
248	C₄	C₄×C₄⋊Dic₅	1	1	yes	characteristic
249	C₄	#65225	1	1	yes	characteristic
250	C₄	#65228	1	2	yes	—
251	C₄	C₂×#3963	1	1	yes	characteristic
252	C₄	#65501	1	1	yes	characteristic
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	C₄×#3951	1	2	yes	—
256	C₂	#10600585	1	2	yes	—
257	C₂	#10602328	1	1	yes	characteristic
258	C₂	#10602770	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₂²×C₄×C₅×C₈	C₂
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	#10864
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
13. C₂×#3951	C₄	semidirect
100. C₂×C₄×C₅	#108	semidirect
213. C₂×C₄	#3951	semidirect
235. C₅	#10864	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×C₈	2
C₂²×#3951	2
C₂×#64772	2
#10602320	2
#10864	5