Subgroups — #13025511

Order 1280.

554 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅⋊#108	C₂	1	2	yes	maximal
2	C₂×D₅×M₃₂	C₂	1	1	yes	maximal, characteristic
3	#10598668	C₂	1	4	yes	maximal, complemented
4	C₂×D₅⋊M₁₆	—	2	2	no	—
5	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
6	D₅⋊#108	—	2	2	no	—
7	D₅⋊#108	—	2	2	no	—
8	D₅⋊#108	C₄	1	4	yes	—
9	D₅⋊#108	C₂²	1	4	yes	—
10	C₂×#3996	—	2	2	no	—
11	C₂×#3995	C₄	1	4	yes	—
12	C₂×C₅×M₃₂	C₄	1	1	yes	characteristic
13	C₂×#3944	C₄	1	1	yes	characteristic
14	D₅×M₃₂	—	2	2	no	—
15	D₅×M₃₂	—	2	2	no	—
16	C₂×#3930	C₂²	1	2	yes	—
17	C₂×D₅×C₁₆	C₄	1	2	yes	—
18	C₂×#1385	—	5	1	no	maximal, Sylow
19	C₂²×C₄×D₅	D₄	1	1	yes	characteristic
20	C₂×#704	—	4	2	no	—
21	D₅⋊M₁₆	—	2	4	no	—
22	D₅⋊M₁₆	—	2	4	no	—
23	D₅⋊M₁₆	—	4	2	no	—
24	D₅⋊M₁₆	—	4	2	no	—
25	C₂×#700	—	2	4	no	—
26	C₂×D₅⋊C₈	—	4	2	no	—
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×#168	Q₈	1	1	yes	characteristic
29	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₅	D₄	1	1	yes	characteristic
33	C₂×C₈×D₅	D₄	1	2	yes	—
34	C₂×C₈×D₅	D₄	1	2	yes	—
35	C₂×C₈×D₅	Q₈	1	1	yes	characteristic
36	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
37	#3996	—	2	4	no	—
38	#3996	—	2	4	no	—
39	#3995	—	2	4	no	—
40	#3995	C₂×C₄	1	8	yes	—
41	C₅×M₃₂	—	2	2	no	—
42	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
43	#3944	—	2	2	no	—
44	C₂×#672	C₂×C₄	1	2	yes	—
45	#3930	—	2	4	no	—
46	D₅×C₁₆	—	2	2	no	—
47	D₅×C₁₆	—	2	2	no	—
48	C₂²×M₃₂	—	5	1	no	—
49	C₂²×#108	—	5	2	no	—
50	#1385	—	5	4	no	—
51	C₂³×D₅	Q₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×Dic₅	D₈	1	1	yes	characteristic
54	C₂×C₄×D₅	C₂²⋊C₄	1	2	yes	—
55	C₂×C₄×D₅	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×D₅	D₈	1	1	yes	characteristic
58	C₂×C₄×D₅	SD₁₆	1	2	yes	—
59	C₂×C₄×D₅	SD₁₆	1	2	yes	—
60	C₂×C₄×D₅	Q₁₆	1	1	yes	characteristic
61	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
62	#704	—	4	4	no	—
63	#704	—	4	4	no	—
64	C₂×#170	—	4	4	no	—
65	#700	—	2	8	no	—
66	#700	—	4	4	no	—
67	D₅⋊C₈	—	4	4	no	—
68	D₅⋊C₈	—	4	4	no	—
69	C₂×C₅×C₈	C₄²	1	2	yes	—
70	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
71	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
74	C₂×#168	C₄⋊C₄	1	2	yes	—
75	C₂×#168	C₄⋊C₄	1	2	yes	—
76	C₂×#168	C₂×Q₈	1	1	yes	characteristic
77	C₈×D₅	C₄²	1	1	yes	characteristic
78	C₈×D₅	C₄²	1	1	yes	characteristic
79	C₈×D₅	C₂²⋊C₄	1	2	yes	—
80	C₈×D₅	C₂²⋊C₄	1	2	yes	—
81	C₈×D₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₈×D₅	C₂²⋊C₄	1	1	yes	characteristic
83	C₈×D₅	C₄⋊C₄	1	2	yes	—
84	C₈×D₅	C₄⋊C₄	1	2	yes	—
85	C₈×D₅	C₂²×C₄	1	2	yes	—
86	C₈×D₅	C₂×D₄	1	2	yes	—
87	C₅×C₁₆	—	2	2	no	—
88	#672	—	2	2	no	—
89	C₂²×M₁₆	—	10	2	no	—
90	C₂³×C₈	—	5	1	no	—
91	C₂×M₃₂	—	5	1	no	—
92	C₂×M₃₂	—	5	2	no	—
93	C₂×M₃₂	—	10	2	no	—
94	C₂×M₃₂	—	10	2	no	—
95	C₂×M₃₂	AGL(1,𝔽₅)	1	1	yes	characteristic
96	C₂²×C₁₆	—	5	2	no	—
97	C₂×#108	—	5	4	no	—
98	C₂×#108	—	5	4	no	—
99	C₂×#108	—	5	4	no	—
100	C₂×#108	—	10	2	no	—
101	C₂×#108	—	10	2	no	—
102	C₂×#108	—	10	2	no	—
103	C₂³×C₅	#107	1	1	yes	characteristic
104	C₂²×D₅	—	4	2	no	—
105	C₂²×D₅	#103	1	2	yes	—
106	C₂²×D₅	#103	1	2	yes	—
107	C₂²×D₅	#107	1	1	yes	characteristic
108	C₂²×D₅	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₄×C₅	#95	1	2	yes	—
110	C₂×C₄×C₅	#106	1	2	yes	—
111	C₂×C₄×C₅	#107	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×Dic₅	#102	1	2	yes	—
114	C₂×Dic₅	#102	1	2	yes	—
115	C₂×Dic₅	#107	1	1	yes	characteristic
116	C₂×Dic₅	C₂×D₈	1	1	yes	characteristic
117	C₄×D₅	#95	1	1	yes	characteristic
118	C₄×D₅	#95	1	1	yes	characteristic
119	C₄×D₅	#102	1	2	yes	—
120	C₄×D₅	#102	1	2	yes	—
121	C₄×D₅	#103	1	2	yes	—
122	C₄×D₅	#103	1	2	yes	—
123	C₄×D₅	#106	1	1	yes	characteristic
124	C₄×D₅	#106	1	1	yes	characteristic
125	C₄×D₅	C₂×C₂²⋊C₄	1	2	yes	—
126	C₄×D₅	C₂×SD₁₆	1	2	yes	—
127	#170	—	4	8	no	—
128	C₅×C₈	#95	1	2	yes	—
129	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
130	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	#168	#95	1	1	yes	characteristic
132	#168	#95	1	1	yes	characteristic
133	#168	C₂×C₄⋊C₄	1	2	yes	—
134	C₂³×C₄	—	5	1	no	—
135	C₂×M₁₆	—	10	4	no	—
136	C₂×M₁₆	—	10	4	no	—
137	C₂×M₁₆	—	10	4	no	—
138	C₂×M₁₆	—	20	2	no	—
139	C₂×M₁₆	—	20	2	no	—
140	C₂×M₁₆	—	20	2	no	—
141	C₂²×C₈	—	5	1	no	—
142	C₂²×C₈	—	5	2	no	—
143	C₂²×C₈	—	5	2	no	—
144	C₂²×C₈	—	5	1	no	—
145	C₂²×C₈	—	5	1	no	—
146	C₂²×C₈	—	5	2	no	—
147	C₂²×C₈	—	5	2	no	—
148	C₂²×C₈	—	5	1	no	—
149	C₂²×C₈	—	5	1	no	—
150	C₂²×C₈	—	20	2	no	—
151	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
152	M₃₂	—	2	2	no	—
153	M₃₂	—	10	4	no	—
154	M₃₂	—	10	2	no	—
155	C₂×C₁₆	—	5	2	no	—
156	C₂×C₁₆	—	10	2	no	—
157	C₂×C₁₆	—	10	2	no	—
158	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
159	#108	—	5	8	no	—
160	#108	—	10	4	no	—
161	#108	—	10	4	no	—
162	#108	—	10	4	no	—
163	C₂²×C₅	—	4	1	no	—
164	C₂²×C₅	#340	1	2	yes	—
165	C₂²×C₅	C₂×#107	1	1	yes	characteristic
166	C₂×D₅	—	2	4	no	—
167	C₂×D₅	—	2	2	no	—
168	C₂×D₅	—	4	2	no	—
169	C₂×D₅	—	4	1	no	—
170	C₂×D₅	#340	1	1	yes	characteristic
171	C₂×D₅	#340	1	1	yes	characteristic
172	C₂×D₅	C₂×#103	1	2	yes	—
173	C₄×C₅	#340	1	2	yes	—
174	C₄×C₅	C₂×#95	1	1	yes	characteristic
175	C₄×C₅	C₂×#106	1	1	yes	characteristic
176	Dic₅	#340	1	1	yes	characteristic
177	Dic₅	#340	1	1	yes	characteristic
178	Dic₅	C₂×#102	1	2	yes	—
179	C₂⁴	—	5	1	no	—
180	C₂²×C₄	—	5	1	no	—
181	C₂²×C₄	—	5	1	no	—
182	C₂²×C₄	—	5	1	no	—
183	C₂²×C₄	—	5	2	no	—
184	C₂²×C₄	—	5	2	no	—
185	C₂²×C₄	—	5	1	no	—
186	C₂²×C₄	—	5	1	no	—
187	C₂²×C₄	—	5	2	no	—
188	C₂²×C₄	—	5	2	no	—
189	C₂²×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
190	M₁₆	—	10	8	no	—
191	M₁₆	—	20	4	no	—
192	M₁₆	—	20	4	no	—
193	M₁₆	—	20	4	no	—
194	C₂×C₈	—	5	1	no	—
195	C₂×C₈	—	5	1	no	—
196	C₂×C₈	—	5	2	no	—
197	C₂×C₈	—	5	2	no	—
198	C₂×C₈	—	5	2	no	—
199	C₂×C₈	—	5	1	no	—
200	C₂×C₈	—	5	1	no	—
201	C₂×C₈	—	5	2	no	—
202	C₂×C₈	—	5	1	no	—
203	C₂×C₈	—	5	1	no	—
204	C₂×C₈	—	5	2	no	—
205	C₂×C₈	—	5	2	no	—
206	C₂×C₈	—	5	2	no	—
207	C₂×C₈	—	5	2	no	—
208	C₂×C₈	—	20	4	no	—
209	C₂×C₈	—	20	4	no	—
210	C₂×C₈	—	20	4	no	—
211	C₂×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
212	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
213	C₂×C₈	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
214	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
215	C₁₆	—	2	2	no	—
216	C₁₆	—	10	2	no	—
217	C₂×C₅	—	2	2	no	—
218	C₂×C₅	—	2	1	no	—
219	C₂×C₅	C₂×#340	1	1	yes	characteristic
220	D₅	—	2	1	no	—
221	D₅	—	2	2	no	—
222	D₅	#1385	1	2	yes	complemented
223	C₂³	—	5	2	no	—
224	C₂³	—	5	1	no	—
225	C₂³	—	5	2	no	—
226	C₂³	—	5	1	no	—
227	C₂³	—	20	2	no	—
228	C₂³	#3994	1	1	yes	characteristic
229	C₂×C₄	—	5	2	no	—
230	C₂×C₄	—	5	2	no	—
231	C₂×C₄	—	5	1	no	—
232	C₂×C₄	—	5	1	no	—
233	C₂×C₄	—	5	2	no	—
234	C₂×C₄	—	5	2	no	—
235	C₂×C₄	—	5	2	no	—
236	C₂×C₄	—	5	2	no	—
237	C₂×C₄	—	5	2	no	—
238	C₂×C₄	—	5	1	no	—
239	C₂×C₄	—	5	1	no	—
240	C₂×C₄	—	5	2	no	—
241	C₂×C₄	—	5	1	no	—
242	C₂×C₄	—	5	1	no	—
243	C₂×C₄	#3993	1	2	yes	—
244	C₂×C₄	#3994	1	1	yes	characteristic
245	C₂×C₄	(C₂×AGL(1,𝔽₅))⋊C₄	1	2	yes	—
246	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
247	C₈	—	5	1	no	—
248	C₈	—	5	1	no	—
249	C₈	—	5	2	no	—
250	C₈	—	20	8	no	—
251	C₈	(C₂×AGL(1,𝔽₅))⋊C₄	1	2	yes	—
252	C₈	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
253	C₈	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
254	C₅	C₂×#1385	1	1	yes	characteristic, complemented, Sylow
255	C₂²	—	4	1	no	—
256	C₂²	—	5	1	no	—
257	C₂²	—	5	1	no	—
258	C₂²	—	5	2	no	—
259	C₂²	—	10	4	no	—
260	C₂²	—	10	2	no	—
261	C₂²	—	20	2	no	—
262	C₂²	—	20	1	no	—
263	C₂²	#64893	1	2	yes	—
264	C₂²	C₂×#3994	1	1	yes	characteristic
265	C₄	—	5	2	no	—
266	C₄	—	5	1	no	—
267	C₄	—	5	1	no	—
268	C₄	#64893	1	2	yes	—
269	C₄	C₂×#3993	1	1	yes	characteristic
270	C₄	C₂×(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	2	1	no	—
273	C₂	—	5	2	no	—
274	C₂	—	10	1	no	—
275	C₂	—	10	2	no	—
276	C₂	C₂×#64893	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#3993
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×M₃₂	C₄
socle	C₂×C₅	C₂×#340
prime radical	C₂³×D₅	Q₁₆
cosocle	C₂×C₈×D₅	C₂³
hypercenter	C₂×M₃₂	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#1385
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #10598668	C₂	semidirect
222. D₅	#1385	semidirect
254. C₅	C₂×#1385	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅⋊#108	2
C₂×D₅×M₃₂	2
#10598668	2
C₂×#1385	5