Subgroups — #11998542

Order 1280.

518 conjugacy classes of subgroups.

The subgroup lattice has 268 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
2	#10602573	C₂	1	2	yes	maximal
3	C₂×#64767	C₂	1	2	yes	maximal
4	C₈×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₂×#64678	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×Q₁₆	—	2	1	no	—
7	C₂²×C₅×Q₁₆	—	2	1	no	—
8	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₅×#494	—	2	2	no	—
10	C₅×#494	C₄	1	4	yes	complemented
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₅	—	2	1	no	—
13	C₄×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#168	C₂²	1	2	yes	—
15	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
16	#64770	—	2	1	no	—
17	#64767	—	2	4	no	—
18	#64749	—	2	4	no	—
19	#64678	—	2	2	no	—
20	#9733	—	5	1	no	maximal, Sylow
21	C₂²×C₅×Q₈	—	2	2	no	—
22	C₂×C₅×Q₁₆	—	2	4	no	—
23	C₂×C₅×Q₁₆	—	2	4	no	—
24	C₂×C₅×Q₁₆	—	4	2	no	—
25	C₂×C₅×Q₁₆	—	4	2	no	—
26	C₂×C₅×Q₁₆	—	4	2	no	—
27	C₂×C₅×Q₁₆	—	4	2	no	—
28	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₅×C₄⋊Q₈	—	2	2	no	—
31	C₅×C₄⋊Q₈	—	2	2	no	—
32	C₅×C₄⋊Q₈	—	2	2	no	—
33	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
34	C₂×C₅×C₄⋊C₄	—	2	2	no	—
35	C₂×C₅×C₄⋊C₄	—	2	2	no	—
36	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊Dic₅	D₄	1	2	yes	—
38	C₂×C₄×Dic₅	—	2	1	no	—
39	C₂²×#168	—	2	1	no	—
40	C₂²×#168	—	2	2	no	—
41	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₅×C₈	D₄	1	2	yes	—
43	#3963	—	2	1	no	—
44	C₈×Dic₅	—	4	2	no	—
45	C₄⋊#168	—	2	1	no	—
46	C₄⋊#168	—	2	2	no	—
47	C₄⋊#168	—	2	2	no	—
48	C₄⋊#168	D₄	1	2	yes	—
49	C₂×#494	D₅	1	1	yes	characteristic
50	#1988	—	5	2	no	—
51	C₈×C₄⋊C₄	—	5	1	no	—
52	C₂×#335	—	5	1	no	—
53	C₂×#332	—	5	2	no	—
54	C₂×C₅×Q₈	—	2	4	no	—
55	C₂×C₅×Q₈	—	4	2	no	—
56	C₂×C₅×Q₈	—	4	2	no	—
57	C₂×C₅×Q₈	—	4	2	no	—
58	C₂×C₅×Q₈	—	4	2	no	—
59	C₂×C₅×Q₈	—	4	2	no	—
60	C₂²×C₄×C₅	—	4	2	no	—
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	#41	1	1	yes	characteristic
64	C₂²×Dic₅	—	2	2	no	—
65	C₅×Q₁₆	—	4	8	no	—
66	C₅×Q₁₆	—	4	8	no	—
67	C₂×C₅×C₈	—	2	1	no	—
68	C₂×C₅×C₈	—	2	1	no	—
69	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
70	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
71	C₂×C₅×C₈	D₈	1	2	yes	—
72	C₂×C₅×C₈	SD₁₆	1	2	yes	—
73	C₅×C₄⋊C₄	—	2	4	no	—
74	C₅×C₄⋊C₄	—	2	4	no	—
75	C₅×C₄⋊C₄	—	4	2	no	—
76	C₅×C₄⋊C₄	—	4	2	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	2	no	—
82	C₄⋊Dic₅	Q₁₆	1	4	yes	complemented
83	C₄×Dic₅	—	4	1	no	—
84	C₄×Dic₅	—	4	1	no	—
85	C₂×#168	—	2	2	no	—
86	C₂×#168	—	4	1	no	—
87	C₂×#168	—	4	1	no	—
88	C₂×#168	—	4	2	no	—
89	C₂×#168	—	4	4	no	—
90	C₂²×Q₁₆	—	2	1	no	—
91	C₂²×Q₁₆	—	2	1	no	—
92	C₂×C₄⋊Q₈	C₂×D₅	1	2	yes	—
93	#494	—	2	2	no	—
94	#494	Dic₅	1	4	yes	complemented
95	C₂×C₄⋊C₈	—	5	1	no	—
96	C₂×C₄⋊C₈	—	5	2	no	—
97	C₂×C₄×C₈	—	10	1	no	—
98	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
99	C₄×C₄⋊C₄	—	5	1	no	—
100	#340	—	10	4	no	—
101	#336	—	10	1	no	—
102	#335	—	10	2	no	—
103	#332	—	10	4	no	—
104	C₂³×C₅	#121	1	1	yes	characteristic
105	C₅×Q₈	—	4	4	no	—
106	C₅×Q₈	—	4	4	no	—
107	C₅×Q₈	—	8	2	no	—
108	C₅×Q₈	—	8	2	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	4	4	no	—
111	C₂×C₄×C₅	—	8	2	no	—
112	C₂×C₄×C₅	—	8	2	no	—
113	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅	#121	1	1	yes	characteristic
117	C₂×C₄×C₅	#124	1	1	yes	characteristic
118	C₂×C₄×C₅	#127	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
122	C₂×C₄×C₅	#135	1	2	yes	—
123	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×Dic₅	—	2	4	no	—
126	C₂×Dic₅	—	4	2	no	—
127	C₂×Dic₅	—	4	2	no	—
128	C₅×C₈	—	2	2	no	—
129	C₅×C₈	#102	1	4	yes	—
130	#168	—	4	2	no	—
131	#168	—	8	2	no	—
132	C₂²×Q₈	—	2	2	no	—
133	C₂×Q₁₆	—	2	4	no	—
134	C₂×Q₁₆	—	2	4	no	—
135	C₂×Q₁₆	—	4	2	no	—
136	C₂×Q₁₆	—	4	2	no	—
137	C₂×Q₁₆	—	4	2	no	—
138	C₂×Q₁₆	—	4	2	no	—
139	C₂²×C₈	—	10	1	no	—
140	C₂²×C₈	—	10	2	no	—
141	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
142	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	—	2	2	no	—
146	C₄⋊Q₈	C₂×Dic₅	1	4	yes	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	5	2	no	—
150	C₂×C₄²	—	10	1	no	—
151	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
152	C₄⋊C₈	—	5	2	no	—
153	C₄⋊C₈	—	10	2	no	—
154	C₄⋊C₈	—	10	1	no	—
155	C₄⋊C₈	—	10	2	no	—
156	C₄×C₈	—	20	2	no	—
157	C₄×C₈	C₂×Dic₅	1	2	yes	—
158	C₄×C₈	C₂²⋊D₅	1	2	yes	—
159	#95	—	10	1	no	—
160	C₂²×C₅	#390	1	1	yes	characteristic
161	C₂²×C₅	#459	1	1	yes	characteristic
162	C₂²×C₅	#460	1	1	yes	characteristic
163	C₂²×C₅	#467	1	2	yes	—
164	C₂²×C₅	#473	1	2	yes	—
165	C₄×C₅	—	2	1	no	—
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	—	8	4	no	—
168	C₄×C₅	C₂×#102	1	1	yes	characteristic
169	C₄×C₅	#418	1	1	yes	characteristic
170	C₄×C₅	C₄×Q₁₆	1	2	yes	—
171	C₄×C₅	#459	1	1	yes	characteristic
172	C₄×C₅	#460	1	1	yes	characteristic
173	C₄×C₅	#494	1	1	yes	characteristic
174	C₄×C₅	#495	1	1	yes	characteristic
175	Dic₅	—	4	4	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	10	2	no	—
184	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
185	C₂²×C₄	#711	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
187	Q₁₆	—	4	8	no	—
188	Q₁₆	—	4	8	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	10	2	no	—
192	C₂×C₈	—	20	1	no	—
193	C₂×C₈	—	20	1	no	—
194	C₂×C₈	—	20	4	no	—
195	C₂×C₈	—	20	2	no	—
196	C₂×C₈	#686	1	2	yes	—
197	C₂×C₈	#687	1	2	yes	—
198	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
199	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	5	4	no	—
205	C₄⋊C₄	—	10	2	no	—
206	C₄²	—	20	1	no	—
207	C₄²	—	20	1	no	—
208	C₄²	D₄×D₅	1	1	yes	characteristic
209	C₄²	#711	1	1	yes	characteristic
210	C₄²	C₂²×Dic₅	1	1	yes	characteristic
211	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
212	C₂×C₅	#1669	1	2	yes	—
213	C₂×C₅	#1670	1	2	yes	—
214	C₂×C₅	#1969	1	1	yes	characteristic
215	C₂×C₅	#1979	1	1	yes	characteristic
216	C₂×C₅	#1982	1	1	yes	characteristic
217	C₂³	#4084	1	1	yes	characteristic
218	Q₈	—	4	4	no	—
219	Q₈	—	4	4	no	—
220	Q₈	—	8	2	no	—
221	Q₈	—	8	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	10	4	no	—
227	C₂×C₄	—	20	2	no	—
228	C₂×C₄	—	20	2	no	—
229	C₂×C₄	D₅×Q₁₆	1	2	yes	—
230	C₂×C₄	#4065	1	2	yes	—
231	C₂×C₄	C₂×#686	1	1	yes	characteristic
232	C₂×C₄	#4078	1	1	yes	characteristic
233	C₂×C₄	#4078	1	1	yes	characteristic
234	C₂×C₄	C₂×#687	1	1	yes	characteristic
235	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
236	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
237	C₂×C₄	#4082	1	1	yes	characteristic
238	C₂×C₄	#4084	1	1	yes	characteristic
239	C₂×C₄	#4086	1	1	yes	characteristic
240	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	20	2	no	—
243	C₈	—	40	2	no	—
244	C₈	#3964	1	4	yes	—
245	C₅	#9733	1	1	yes	characteristic, complemented, Sylow
246	C₂²	#65361	1	1	yes	characteristic
247	C₂²	#65364	1	1	yes	characteristic
248	C₂²	#65477	1	2	yes	—
249	C₂²	#65481	1	2	yes	—
250	C₂²	#65510	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	8	4	no	—
254	C₄	—	20	4	no	—
255	C₄	#65361	1	1	yes	characteristic
256	C₄	#65364	1	1	yes	characteristic
257	C₄	#65470	1	1	yes	characteristic
258	C₄	Q₁₆×Dic₅	1	2	yes	—
259	C₄	#65480	1	1	yes	characteristic
260	C₄	C₂×#3964	1	1	yes	characteristic
261	C₄	#65504	1	1	yes	characteristic
262	C₂	#10602098	1	2	yes	—
263	C₂	#10602101	1	2	yes	—
264	C₂	#10602554	1	1	yes	characteristic
265	C₂	#10602859	1	1	yes	characteristic
266	C₂	#10602862	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4084
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#494	C₂
socle, prime radical	C₂³×C₅	#121
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#494	D₅
hypocenter	C₅	#9733
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
10. C₅×#494	C₄	semidirect
82. C₄⋊Dic₅	Q₁₆	semidirect
94. #494	Dic₅	semidirect
245. C₅	#9733	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
#10602573	2
C₂×#64767	2
C₈×C₄⋊Dic₅	2
C₂×#64678	2
#9733	5