Subgroups — #13018737

Order 1280.

422 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	#10604191	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	#10598771	C₂	1	2	yes	maximal
4	#10598771	C₂	1	2	yes	maximal
5	C₂×#4004	—	2	2	no	—
6	C₂×#4004	—	2	2	no	—
7	C₂×C₄⋊#170	C₂²	1	4	yes	—
8	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
13	C₄²×Dic₅	C₂²	1	1	yes	characteristic
14	C₂×C₄×#168	C₄	1	1	yes	characteristic
15	#8784	—	5	1	no	maximal, Sylow
16	C₂²×#170	—	2	4	no	—
17	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅	Q₈	1	1	yes	characteristic
24	C₂×C₄×Dic₅	Q₈	1	1	yes	characteristic
25	C₂×C₄×Dic₅	C₂³	1	1	yes	characteristic
26	C₂²×#168	C₂×C₄	1	1	yes	characteristic
27	C₂²×#168	C₂×C₄	1	1	yes	characteristic
28	#4004	—	4	4	no	—
29	#4004	—	4	4	no	—
30	C₄⋊#170	—	2	4	no	—
31	C₄⋊#170	—	2	4	no	—
32	C₄×C₅×C₈	—	2	2	no	—
33	C₈×Dic₅	—	2	2	no	—
34	C₈×Dic₅	—	2	2	no	—
35	C₈×Dic₅	—	2	2	no	—
36	C₈×Dic₅	C₈	1	4	yes	complemented
37	C₄×#168	—	2	2	no	—
38	#1842	—	5	2	no	—
39	C₄²×C₈	—	5	1	no	—
40	#1277	—	5	2	no	—
41	#1277	—	5	2	no	—
42	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₅	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×Dic₅	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×Dic₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₅	C₂×Q₈	1	1	yes	characteristic
49	C₂×#170	—	4	4	no	—
50	C₂×#170	—	4	4	no	—
51	C₂×#170	—	4	4	no	—
52	C₂×C₅×C₈	—	2	1	no	—
53	C₂×C₅×C₈	—	2	1	no	—
54	C₂×C₅×C₈	—	2	1	no	—
55	C₂×C₅×C₈	—	2	1	no	—
56	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
57	C₂×C₅×C₈	M₁₆	1	2	yes	—
58	C₄²×C₅	—	2	1	no	—
59	C₄²×C₅	—	2	1	no	—
60	C₄×Dic₅	—	2	2	no	—
61	C₄×Dic₅	—	2	2	no	—
62	C₄×Dic₅	C₂×C₈	1	2	yes	—
63	C₄×Dic₅	M₁₆	1	2	yes	—
64	C₄×Dic₅	D₈	1	2	yes	—
65	C₄×Dic₅	SD₁₆	1	4	yes	—
66	C₄×Dic₅	Q₁₆	1	2	yes	—
67	C₄×Dic₅	#41	1	2	yes	—
68	C₄×Dic₅	#41	1	2	yes	—
69	C₂×#168	—	2	1	no	—
70	C₂×#168	—	2	1	no	—
71	C₂×#168	—	2	1	no	—
72	C₂×#168	—	2	1	no	—
73	C₂×#168	C₂×C₈	1	2	yes	—
74	C₂×#168	M₁₆	1	2	yes	—
75	C₂×C₄⋊C₈	—	5	4	no	—
76	C₂×C₄⋊C₈	—	10	2	no	—
77	C₂×C₄⋊C₈	—	10	2	no	—
78	C₂×C₄×C₈	—	5	1	no	—
79	C₂×C₄×C₈	—	5	1	no	—
80	C₂×C₄×C₈	—	5	1	no	—
81	C₂×C₄×C₈	—	5	1	no	—
82	C₂×C₄×C₈	—	5	1	no	—
83	C₂×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
84	C₄³	—	5	1	no	—
85	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₅	—	2	1	no	—
87	C₂×C₄×C₅	—	2	1	no	—
88	C₂×C₄×C₅	—	2	1	no	—
89	C₂×C₄×C₅	—	2	1	no	—
90	C₂×C₄×C₅	#106	1	2	yes	—
91	C₂×C₄×C₅	#107	1	2	yes	—
92	C₂×C₄×C₅	#117	1	1	yes	characteristic
93	C₂×C₄×C₅	#117	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
98	C₂×Dic₅	—	2	1	no	—
99	C₂×Dic₅	—	2	1	no	—
100	C₂×Dic₅	C₄⋊C₈	1	2	yes	—
101	C₂×Dic₅	C₄⋊C₈	1	2	yes	—
102	C₂×Dic₅	#106	1	2	yes	—
103	C₂×Dic₅	#107	1	2	yes	—
104	C₂×Dic₅	#108	1	2	yes	—
105	C₂×Dic₅	#108	1	2	yes	—
106	C₂×Dic₅	#124	1	2	yes	—
107	C₂×Dic₅	#125	1	2	yes	—
108	C₂×Dic₅	C₂×D₈	1	1	yes	characteristic
109	C₂×Dic₅	C₂×SD₁₆	1	2	yes	—
110	C₂×Dic₅	C₂×Q₁₆	1	1	yes	characteristic
111	#170	—	8	4	no	—
112	C₅×C₈	—	2	2	no	—
113	C₅×C₈	—	2	2	no	—
114	#168	—	2	2	no	—
115	#168	—	2	2	no	—
116	C₂²×C₈	—	5	1	no	—
117	C₂²×C₈	—	5	1	no	—
118	C₂²×C₈	—	10	4	no	—
119	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
120	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
121	C₂×C₄²	—	5	1	no	—
122	C₂×C₄²	—	5	1	no	—
123	C₂×C₄²	—	5	1	no	—
124	C₂×C₄²	—	5	1	no	—
125	C₂×C₄²	—	5	1	no	—
126	C₂×C₄²	—	5	1	no	—
127	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
128	C₄⋊C₈	—	10	4	no	—
129	C₄⋊C₈	—	10	4	no	—
130	C₄⋊C₈	—	20	4	no	—
131	C₄⋊C₈	—	20	4	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	5	4	no	—
134	C₄×C₈	—	10	2	no	—
135	C₄×C₈	—	10	2	no	—
136	C₄×C₈	—	10	2	no	—
137	C₄×C₈	—	10	2	no	—
138	C₂²×C₅	#382	1	1	yes	characteristic
139	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
140	C₂²×C₅	#423	1	1	yes	characteristic
141	C₂²×C₅	C₂×#106	1	1	yes	characteristic
142	C₂²×C₅	C₂×#107	1	1	yes	characteristic
143	C₂²×C₅	C₂×#108	1	1	yes	characteristic
144	C₂²×C₅	C₂×#108	1	1	yes	characteristic
145	C₄×C₅	—	2	2	no	—
146	C₄×C₅	—	2	2	no	—
147	C₄×C₅	#431	1	1	yes	characteristic
148	C₄×C₅	#431	1	1	yes	characteristic
149	C₄×C₅	#432	1	1	yes	characteristic
150	C₄×C₅	#432	1	1	yes	characteristic
151	Dic₅	—	2	2	no	—
152	Dic₅	#334	1	4	yes	—
153	Dic₅	#335	1	4	yes	—
154	Dic₅	#486	1	2	yes	—
155	Dic₅	#487	1	2	yes	—
156	C₂²×C₄	—	5	1	no	—
157	C₂²×C₄	—	5	1	no	—
158	C₂²×C₄	—	5	1	no	—
159	C₂²×C₄	—	5	1	no	—
160	C₂²×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
161	C₂²×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
162	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	5	2	no	—
168	C₂×C₈	—	5	2	no	—
169	C₂×C₈	—	10	1	no	—
170	C₂×C₈	—	10	1	no	—
171	C₂×C₈	—	10	1	no	—
172	C₂×C₈	—	10	1	no	—
173	C₂×C₈	—	20	4	no	—
174	C₂×C₈	—	20	4	no	—
175	C₂×C₈	—	20	4	no	—
176	C₂×C₈	D₅⋊C₈	1	2	yes	—
177	C₂×C₈	#704	1	2	yes	—
178	C₄²	—	2	1	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	5	2	no	—
181	C₄²	—	5	2	no	—
182	C₄²	—	5	2	no	—
183	C₄²	—	5	2	no	—
184	C₄²	—	5	4	no	—
185	C₄²	—	5	2	no	—
186	C₄²	—	5	2	no	—
187	C₄²	—	10	2	no	—
188	C₄²	—	10	2	no	—
189	C₂×C₅	C₂×#334	1	1	yes	characteristic
190	C₂×C₅	C₂×#335	1	1	yes	characteristic
191	C₂×C₅	#1566	1	1	yes	characteristic
192	C₂×C₅	#1567	1	1	yes	characteristic
193	C₂×C₅	#1831	1	1	yes	characteristic
194	C₂×C₅	#1834	1	1	yes	characteristic
195	C₂×C₅	#1837	1	1	yes	characteristic
196	C₂³	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	5	2	no	—
202	C₂×C₄	—	5	2	no	—
203	C₂×C₄	—	5	2	no	—
204	C₂×C₄	—	5	2	no	—
205	C₂×C₄	—	5	2	no	—
206	C₂×C₄	—	5	1	no	—
207	C₂×C₄	—	5	2	no	—
208	C₂×C₄	—	5	1	no	—
209	C₂×C₄	—	5	2	no	—
210	C₂×C₄	—	5	2	no	—
211	C₂×C₄	—	5	2	no	—
212	C₂×C₄	—	10	1	no	—
213	C₂×C₄	—	10	1	no	—
214	C₂×C₄	#3993	1	2	yes	—
215	C₂×C₄	#3994	1	2	yes	—
216	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
217	C₂×C₄	D₅⋊M₁₆	1	1	yes	characteristic
218	C₂×C₄	D₅⋊M₁₆	1	1	yes	characteristic
219	C₂×C₄	#4130	1	1	yes	characteristic
220	C₂×C₄	#4130	1	1	yes	characteristic
221	C₂×C₄	C₂×#704	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	2	2	no	—
224	C₈	—	10	2	no	—
225	C₈	—	10	2	no	—
226	C₈	—	40	4	no	—
227	C₅	#8784	1	1	yes	characteristic, complemented, Sylow
228	C₂²	D₅⋊(C₄⋊C₈)	1	1	yes	characteristic
229	C₂²	#65689	1	1	yes	characteristic
230	C₂²	C₂×#3993	1	1	yes	characteristic
231	C₂²	C₂×#3994	1	1	yes	characteristic
232	C₂²	D₅⋊#108	1	1	yes	characteristic
233	C₂²	D₅⋊#108	1	1	yes	characteristic
234	C₂²	#65758	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	5	4	no	—
238	C₄	—	5	4	no	—
239	C₄	—	5	2	no	—
240	C₄	—	5	2	no	—
241	C₄	—	10	2	no	—
242	C₄	#65678	1	1	yes	characteristic
243	C₄	#65684	1	1	yes	characteristic
244	C₄	#65754	1	1	yes	characteristic
245	C₄	#65757	1	1	yes	characteristic
246	C₂	D₅⋊#334	1	1	yes	characteristic
247	C₂	D₅⋊#335	1	1	yes	characteristic
248	C₂	#10603845	1	1	yes	characteristic
249	C₂	#10603857	1	1	yes	characteristic
250	C₂	#10604186	1	1	yes	characteristic
251	C₂	#10604696	1	1	yes	characteristic
252	C₂	#10604697	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅×C₈	C₄
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₅	C₂³
hypercenter	C₂×C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#8784
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
36. C₈×Dic₅	C₈	semidirect
227. C₅	#8784	semidirect

Maximal subgroups

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

subgroup	index
#10604191	2
C₄×C₈×Dic₅	2
#10598771	2
#8784	5