Subgroups — #11998596

Order 1280.

506 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64988	C₂	1	1	yes	maximal, characteristic
2	C₅×#1770	C₂	1	1	yes	maximal, characteristic
3	C₂×#64702	C₂	1	2	yes	maximal
4	C₂×#64677	C₂	1	1	yes	maximal, characteristic
5	#10600283	C₂	1	2	yes	maximal
6	C₂²×Dic₂₀	—	2	1	no	—
7	C₂²×Dic₂₀	—	2	1	no	—
8	C₂×#4015	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×#97	—	2	1	no	—
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#168	C₂²	1	2	yes	—
14	#64988	—	2	2	no	—
15	#64988	C₄	1	4	yes	complemented
16	C₅×#336	—	2	1	no	—
17	#64749	—	2	4	no	—
18	#64702	—	2	4	no	—
19	#64677	—	2	2	no	—
20	#9742	—	5	1	no	maximal, Sylow
21	C₂²×Dic₁₀	—	2	2	no	—
22	C₂²×C₅×C₈	—	2	1	no	—
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
26	C₂×C₄²×C₅	—	2	1	no	—
27	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₅	—	2	2	no	—
29	C₂×C₄⋊Dic₅	—	2	2	no	—
30	C₂²×#168	—	2	2	no	—
31	C₂×Dic₂₀	—	2	4	no	—
32	C₂×Dic₂₀	—	2	4	no	—
33	C₂×Dic₂₀	—	4	2	no	—
34	C₂×Dic₂₀	—	4	2	no	—
35	C₂×Dic₂₀	—	4	2	no	—
36	C₂×Dic₂₀	—	4	2	no	—
37	#4015	—	2	2	no	—
38	#4015	—	2	2	no	—
39	#4015	—	2	2	no	—
40	#4015	C₂×C₄	1	4	yes	—
41	C₅×C₄⋊C₈	—	2	1	no	—
42	C₅×C₄⋊C₈	D₄	1	2	yes	—
43	C₅×#97	—	4	2	no	—
44	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
45	C₄×C₅×C₈	D₄	1	2	yes	—
46	C₅×#95	—	2	1	no	—
47	C₄⋊#168	—	2	2	no	—
48	C₄⋊#168	—	2	2	no	—
49	C₂×#494	—	5	1	no	—
50	#1988	—	5	2	no	—
51	#1770	D₅	1	1	yes	characteristic
52	C₂×#334	—	5	1	no	—
53	C₂×#332	—	5	2	no	—
54	C₂²×C₄×C₅	—	2	2	no	—
55	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	#41	1	1	yes	characteristic
58	C₂²×Dic₅	—	4	2	no	—
59	C₂×Dic₁₀	—	2	4	no	—
60	C₂×Dic₁₀	—	4	2	no	—
61	C₂×Dic₁₀	—	4	2	no	—
62	C₂×Dic₁₀	—	4	2	no	—
63	C₂×Dic₁₀	—	4	2	no	—
64	C₂×Dic₁₀	—	4	2	no	—
65	C₂×C₅×C₈	—	2	1	no	—
66	C₂×C₅×C₈	—	2	1	no	—
67	C₂×C₅×C₈	—	2	2	no	—
68	C₂×C₅×C₈	—	4	1	no	—
69	C₂×C₅×C₈	—	4	1	no	—
70	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
71	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
72	C₂×C₅×C₈	D₈	1	2	yes	—
73	C₂×C₅×C₈	SD₁₆	1	2	yes	—
74	C₅×C₄⋊C₄	—	2	4	no	—
75	C₄²×C₅	—	4	1	no	—
76	C₄²×C₅	—	4	1	no	—
77	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅	#41	1	1	yes	characteristic
81	C₄⋊Dic₅	—	2	4	no	—
82	C₄⋊Dic₅	—	2	4	no	—
83	C₄⋊Dic₅	—	4	2	no	—
84	C₄⋊Dic₅	—	4	2	no	—
85	C₂×#168	—	4	2	no	—
86	C₂×#168	—	4	4	no	—
87	Dic₂₀	—	4	8	no	—
88	Dic₂₀	—	4	8	no	—
89	C₂²×Q₁₆	—	10	1	no	—
90	C₂²×Q₁₆	—	10	1	no	—
91	C₂×C₄⋊Q₈	—	5	2	no	—
92	#494	—	5	4	no	—
93	#494	—	10	2	no	—
94	C₂×C₄⋊C₈	—	5	2	no	—
95	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
96	C₂×#97	—	2	1	no	—
97	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
98	C₄×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
99	#340	—	10	4	no	—
100	#336	—	2	1	no	—
101	#334	—	10	2	no	—
102	#332	—	10	4	no	—
103	C₂³×C₅	#121	1	1	yes	characteristic
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	4	2	no	—
106	C₂×C₄×C₅	—	4	2	no	—
107	C₂×C₄×C₅	—	4	2	no	—
108	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅	#121	1	1	yes	characteristic
112	C₂×C₄×C₅	#124	1	1	yes	characteristic
113	C₂×C₄×C₅	#127	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₅	#137	1	4	yes	—
119	C₂×Dic₅	—	4	4	no	—
120	C₂×Dic₅	—	8	2	no	—
121	C₂×Dic₅	—	8	2	no	—
122	Dic₁₀	—	4	4	no	—
123	Dic₁₀	—	4	4	no	—
124	Dic₁₀	—	8	2	no	—
125	Dic₁₀	—	8	2	no	—
126	C₅×C₈	—	2	2	no	—
127	C₅×C₈	—	4	2	no	—
128	C₅×C₈	#102	1	4	yes	—
129	#168	—	8	2	no	—
130	C₂²×Q₈	—	10	2	no	—
131	C₂×Q₁₆	—	10	4	no	—
132	C₂×Q₁₆	—	10	4	no	—
133	C₂×Q₁₆	—	20	2	no	—
134	C₂×Q₁₆	—	20	2	no	—
135	C₂×Q₁₆	—	20	2	no	—
136	C₂×Q₁₆	—	20	2	no	—
137	C₂²×C₈	—	2	1	no	—
138	C₂²×C₈	—	10	2	no	—
139	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
140	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
141	C₄⋊Q₈	—	5	4	no	—
142	C₄⋊Q₈	—	10	2	no	—
143	C₄⋊Q₈	—	10	2	no	—
144	C₄⋊Q₈	—	10	2	no	—
145	C₂×C₄⋊C₄	—	10	2	no	—
146	C₂×C₄⋊C₄	—	10	2	no	—
147	C₂×C₄⋊C₄	D₂₀	1	2	yes	—
148	C₂×C₄²	—	2	1	no	—
149	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
150	C₄⋊C₈	—	2	1	no	—
151	C₄⋊C₈	—	10	2	no	—
152	C₄⋊C₈	—	10	2	no	—
153	C₄⋊C₈	D₂₀	1	2	yes	—
154	#97	—	4	2	no	—
155	C₄×C₈	C₄×D₅	1	2	yes	—
156	C₄×C₈	D₂₀	1	2	yes	—
157	#95	—	2	1	no	—
158	C₂²×C₅	#390	1	1	yes	characteristic
159	C₂²×C₅	#459	1	1	yes	characteristic
160	C₂²×C₅	#460	1	1	yes	characteristic
161	C₂²×C₅	#470	1	2	yes	—
162	C₂²×C₅	#473	1	2	yes	—
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	—	2	1	no	—
165	C₄×C₅	—	8	2	no	—
166	C₄×C₅	C₂×#102	1	1	yes	characteristic
167	C₄×C₅	#418	1	1	yes	characteristic
168	C₄×C₅	#441	1	2	yes	—
169	C₄×C₅	#459	1	1	yes	characteristic
170	C₄×C₅	#460	1	1	yes	characteristic
171	C₄×C₅	#497	1	2	yes	—
172	Dic₅	—	8	4	no	—
173	C₂×Q₈	—	10	4	no	—
174	C₂×Q₈	—	20	2	no	—
175	C₂×Q₈	—	20	2	no	—
176	C₂×Q₈	—	20	2	no	—
177	C₂×Q₈	—	20	2	no	—
178	C₂×Q₈	—	20	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	20	2	no	—
181	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
182	C₂²×C₄	#709	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
184	Q₁₆	—	20	8	no	—
185	Q₁₆	—	20	8	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	4	1	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	—	20	2	no	—
192	C₂×C₈	—	20	4	no	—
193	C₂×C₈	#685	1	2	yes	—
194	C₂×C₈	#685	1	2	yes	—
195	C₂×C₈	#686	1	2	yes	—
196	C₂×C₈	#688	1	2	yes	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	10	4	no	—
199	C₄⋊C₄	—	10	4	no	—
200	C₄⋊C₄	—	20	2	no	—
201	C₄⋊C₄	—	20	2	no	—
202	C₄²	—	4	1	no	—
203	C₄²	—	4	1	no	—
204	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
205	C₄²	C₂×D₂₀	1	1	yes	characteristic
206	C₄²	C₂×D₂₀	1	1	yes	characteristic
207	C₄²	#709	1	1	yes	characteristic
208	C₂×C₅	#1689	1	2	yes	—
209	C₂×C₅	#1690	1	2	yes	—
210	C₂×C₅	#1971	1	1	yes	characteristic
211	C₂×C₅	#1979	1	1	yes	characteristic
212	C₂×C₅	#1982	1	1	yes	characteristic
213	C₂³	#4076	1	1	yes	characteristic
214	Q₈	—	20	4	no	—
215	Q₈	—	20	4	no	—
216	Q₈	—	40	2	no	—
217	Q₈	—	40	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	20	4	no	—
223	C₂×C₄	—	40	2	no	—
224	C₂×C₄	—	40	2	no	—
225	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
226	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
227	C₂×C₄	#4020	1	1	yes	characteristic
228	C₂×C₄	#4021	1	1	yes	characteristic
229	C₂×C₄	#4055	1	4	yes	—
230	C₂×C₄	C₂×#685	1	1	yes	characteristic
231	C₂×C₄	#4076	1	1	yes	characteristic
232	C₂×C₄	C₂×#686	1	1	yes	characteristic
233	C₂×C₄	C₂×#688	1	1	yes	characteristic
234	C₂×C₄	#4095	1	1	yes	characteristic
235	C₂×C₄	#4095	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	4	2	no	—
238	C₈	—	40	2	no	—
239	C₈	#3941	1	4	yes	—
240	C₅	#9742	1	1	yes	characteristic, complemented, Sylow
241	C₂²	#65228	1	1	yes	characteristic
242	C₂²	#65304	1	1	yes	characteristic
243	C₂²	#65316	1	1	yes	characteristic
244	C₂²	#65426	1	2	yes	—
245	C₂²	#65432	1	2	yes	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	8	2	no	—
249	C₄	—	40	4	no	—
250	C₄	#65004	1	2	yes	—
251	C₄	#65005	1	2	yes	—
252	C₄	C₂×#3941	1	1	yes	characteristic
253	C₄	#65290	1	1	yes	characteristic
254	C₄	#65304	1	1	yes	characteristic
255	C₄	#65316	1	1	yes	characteristic
256	C₂	#10600280	1	1	yes	characteristic
257	C₂	#10600617	1	1	yes	characteristic
258	C₂	#10600697	1	1	yes	characteristic
259	C₂	#10601664	1	2	yes	—
260	C₂	#10601755	1	2	yes	—

	subgroup	G/H
center	C₂³	#4076
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1770	C₂
socle, prime radical	C₂³×C₅	#121
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1770	D₅
hypocenter	C₅	#9742
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1770

Complements

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

N	H ≅ G/N	type
15. #64988	C₄	semidirect
240. C₅	#9742	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64988	2
C₅×#1770	2
C₂×#64702	2
C₂×#64677	2
#10600283	2
#9742	5