Subgroups — #12015451

Order 1280.

396 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#10602770	C₂	1	1	yes	maximal, characteristic
3	#10602763	C₂	1	1	yes	maximal, characteristic
4	C₂×#64733	C₂	1	2	yes	maximal, complemented
5	#10598307	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×M₃₂	—	2	4	no	—
8	C₂×C₅×M₃₂	C₄	1	2	yes	complemented
9	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₁₆	C₄	1	2	yes	complemented
12	#65501	—	2	1	no	—
13	#65496	—	2	1	no	—
14	C₂×#3951	C₂²	1	2	yes	—
15	C₂×#3949	C₂²	1	2	yes	—
16	#64772	—	2	1	no	—
17	#64771	—	2	1	no	—
18	#64733	—	2	4	no	—
19	#12831	—	5	1	no	maximal, Sylow
20	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
27	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₂×C₂²⋊Dic₅	—	4	1	no	—
30	C₂×C₄⋊Dic₅	—	2	2	no	—
31	C₂×#681	—	2	2	no	—
32	C₅×M₃₂	—	2	2	no	—
33	C₅×M₃₂	—	2	4	no	—
34	C₅×M₃₂	D₄	1	4	yes	—
35	C₂×C₅×C₁₆	—	2	4	no	—
36	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
38	#3963	—	4	1	no	—
39	C₂²⋊#168	—	4	1	no	—
40	C₂²⋊#168	—	4	1	no	—
41	#3951	—	2	2	no	—
42	#3951	—	2	2	no	—
43	#3949	—	2	2	no	—
44	#3949	—	2	2	no	—
45	C₂²×M₃₂	D₅	1	1	yes	characteristic
46	C₂×#365	—	5	2	no	—
47	#1810	—	5	1	no	—
48	#1807	—	5	1	no	—
49	#1383	—	5	2	no	—
50	C₂⁴×C₅	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₅	—	2	2	no	—
52	C₂²×C₄×C₅	—	2	2	no	—
53	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₅	—	4	2	no	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	—	2	1	no	—
62	C₂×C₅×C₈	—	2	1	no	—
63	C₂×C₅×C₈	—	2	1	no	—
64	C₂×C₅×C₈	—	2	1	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	—	2	2	no	—
67	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
71	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₅×C₈	#41	1	2	yes	—
73	C₂²⋊Dic₅	—	8	2	no	—
74	C₄⋊Dic₅	—	4	2	no	—
75	C₄⋊Dic₅	—	4	2	no	—
76	#681	—	4	2	no	—
77	#681	—	4	2	no	—
78	C₂×#168	—	4	2	no	—
79	C₅×C₁₆	—	2	2	no	—
80	C₅×C₁₆	C₂²⋊C₄	1	4	yes	complemented
81	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
82	C₂×M₃₂	—	2	4	no	—
83	C₂×M₃₂	Dic₅	1	2	yes	complemented
84	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
85	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
86	C₂²×C₁₆	Dic₅	1	2	yes	complemented
87	C₂×#108	—	5	2	no	—
88	C₂×#106	—	5	2	no	—
89	#407	—	10	1	no	—
90	#380	—	10	1	no	—
91	#365	—	10	4	no	—
92	#341	—	10	1	no	—
93	#340	—	10	1	no	—
94	C₂³×C₅	—	2	2	no	—
95	C₂³×C₅	—	2	2	no	—
96	C₂³×C₅	—	4	1	no	—
97	C₂³×C₅	#106	1	1	yes	characteristic
98	C₂³×C₅	#106	1	1	yes	characteristic
99	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	#106	1	1	yes	characteristic
108	C₂×C₄×C₅	#106	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅	#121	1	2	yes	—
111	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₅	#135	1	2	yes	—
113	C₂×Dic₅	—	8	2	no	—
114	C₂×Dic₅	—	8	2	no	—
115	C₂×Dic₅	—	8	2	no	—
116	C₅×C₈	—	2	1	no	—
117	C₅×C₈	—	2	1	no	—
118	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₅×C₈	#117	1	1	yes	characteristic
120	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
121	#168	—	8	2	no	—
122	C₂³×C₄	D₂₀	1	1	yes	characteristic
123	C₂×M₁₆	—	10	2	no	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
127	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
128	C₂²×C₈	D₂₀	1	1	yes	characteristic
129	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
130	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
132	C₂×C₄⋊C₄	—	10	2	no	—
133	C₂×C₂²⋊C₄	—	20	1	no	—
134	M₃₂	—	2	2	no	—
135	M₃₂	—	2	4	no	—
136	M₃₂	C₂²⋊D₅	1	4	yes	—
137	C₂×C₁₆	—	2	4	no	—
138	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
139	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
140	#108	—	10	2	no	—
141	#108	—	10	2	no	—
142	#106	—	10	2	no	—
143	#106	—	10	2	no	—
144	C₂²⋊C₈	—	20	1	no	—
145	C₂²⋊C₈	—	20	1	no	—
146	#95	—	20	1	no	—
147	C₂²×C₅	—	2	1	no	—
148	C₂²×C₅	—	2	1	no	—
149	C₂²×C₅	—	2	1	no	—
150	C₂²×C₅	—	2	1	no	—
151	C₂²×C₅	—	2	1	no	—
152	C₂²×C₅	—	2	1	no	—
153	C₂²×C₅	—	2	1	no	—
154	C₂²×C₅	—	4	4	no	—
155	C₂²×C₅	#365	1	2	yes	complemented
156	C₂²×C₅	C₂×#106	1	1	yes	characteristic
157	C₂²×C₅	#465	1	2	yes	—
158	C₄×C₅	—	2	1	no	—
159	C₄×C₅	—	2	1	no	—
160	C₄×C₅	#380	1	1	yes	characteristic
161	C₄×C₅	#427	1	1	yes	characteristic
162	C₄×C₅	#465	1	2	yes	—
163	Dic₅	—	16	2	no	—
164	C₂⁴	#677	1	1	yes	characteristic
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	20	2	no	—
168	C₂²×C₄	#677	1	1	yes	characteristic
169	C₂²×C₄	#677	1	1	yes	characteristic
170	C₂²×C₄	#677	1	1	yes	characteristic
171	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
172	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
174	M₁₆	—	20	2	no	—
175	M₁₆	—	20	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	20	2	no	—
184	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
185	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
186	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
187	C₂×C₈	#709	1	2	yes	—
188	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
189	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
190	C₄⋊C₄	—	20	2	no	—
191	C₄⋊C₄	—	20	2	no	—
192	C₂²⋊C₄	—	40	2	no	—
193	C₁₆	—	2	2	no	—
194	C₁₆	C₂²⋊Dic₅	1	4	yes	complemented
195	C₂×C₅	—	2	2	no	—
196	C₂×C₅	—	2	2	no	—
197	C₂×C₅	—	4	1	no	—
198	C₂×C₅	#1807	1	1	yes	characteristic
199	C₂×C₅	C₂×#365	1	1	yes	characteristic
200	C₂×C₅	#2155	1	1	yes	characteristic
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	4	1	no	—
204	C₂³	#3949	1	1	yes	characteristic
205	C₂³	#3949	1	1	yes	characteristic
206	C₂³	C₂×#677	1	1	yes	characteristic
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	40	2	no	—
215	C₂×C₄	—	40	2	no	—
216	C₂×C₄	—	40	2	no	—
217	C₂×C₄	#3949	1	1	yes	characteristic
218	C₂×C₄	#3949	1	1	yes	characteristic
219	C₂×C₄	C₂×#677	1	1	yes	characteristic
220	C₂×C₄	#4050	1	2	yes	—
221	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
222	C₂×C₄	#4076	1	2	yes	—
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	40	2	no	—
226	C₈	#4070	1	2	yes	—
227	C₈	#4072	1	1	yes	characteristic
228	C₈	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
229	C₅	#12831	1	1	yes	characteristic, complemented, Sylow
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	4	4	no	—
238	C₂²	#64733	1	2	yes	complemented
239	C₂²	C₂×#3949	1	1	yes	characteristic
240	C₂²	#65403	1	2	yes	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	80	2	no	—
244	C₄	#65395	1	1	yes	characteristic
245	C₄	#65403	1	2	yes	—
246	C₄	#65501	1	1	yes	characteristic
247	C₂	—	2	2	no	—
248	C₂	—	2	2	no	—
249	C₂	—	4	1	no	—
250	C₂	C₂×#64733	1	1	yes	characteristic
251	C₂	#10602220	1	1	yes	characteristic
252	C₂	#10602763	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#3949
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂²×C₅×M₃₂	C₂
socle	C₂²×C₅	C₂×#106
prime radical	C₂⁴×C₅	SD₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂²×M₃₂	D₅
hypocenter	C₅	#12831
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
4. C₂×#64733	C₂	semidirect
5. #10598307	C₂	semidirect
8. C₂×C₅×M₃₂	C₄	semidirect
11. C₂²×C₅×C₁₆	C₄	semidirect
80. C₅×C₁₆	C₂²⋊C₄	semidirect
83. C₂×M₃₂	Dic₅	semidirect
86. C₂²×C₁₆	Dic₅	semidirect
155. C₂²×C₅	#365	semidirect
194. C₁₆	C₂²⋊Dic₅	semidirect
229. C₅	#12831	semidirect
238. C₂²	#64733	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×M₃₂	2
#10602770	2
#10602763	2
C₂×#64733	2
#10598307	2
#12831	5