Subgroups — #13025479

Order 1280.

606 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65721	C₂	1	2	yes	maximal
2	C₂²×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#10598664	C₂	1	4	yes	maximal, complemented
4	C₂×#4130	—	2	2	no	—
5	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
6	#65721	—	2	4	no	—
7	#65721	C₄	1	8	yes	complemented
8	C₂×#3994	C₂²	1	4	yes	—
9	C₂×#3993	—	2	2	no	—
10	C₂²×C₅×C₁₆	C₄	1	1	yes	characteristic, complemented
11	C₂²×#672	C₄	1	1	yes	characteristic, complemented
12	C₂×D₅×C₁₆	—	2	4	no	—
13	C₂×D₅×C₁₆	C₄	1	2	yes	complemented
14	C₂×D₅×C₁₆	C₂²	1	2	yes	—
15	#12556	—	5	1	no	maximal, Sylow
16	C₂²×C₄×D₅	D₄	1	1	yes	characteristic
17	C₂×C₂²⋊AGL(1,𝔽₅)	—	4	2	no	—
18	#4130	—	2	8	no	—
19	#4130	—	4	4	no	—
20	C₂×C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
21	C₂×C₄×AGL(1,𝔽₅)	—	4	2	no	—
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×#168	Q₈	1	1	yes	characteristic
24	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₅	C₂×C₄	1	4	yes	—
26	C₂×C₈×D₅	D₄	1	1	yes	characteristic
27	C₂×C₈×D₅	D₄	1	4	yes	—
28	C₂×C₈×D₅	Q₈	1	1	yes	characteristic
29	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
30	#3994	—	2	4	no	—
31	#3994	C₂×C₄	1	8	yes	—
32	#3993	—	2	8	no	—
33	C₂×C₅×C₁₆	—	2	2	no	—
34	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
35	C₂×#672	—	2	2	no	—
36	C₂×#672	C₂×C₄	1	2	yes	—
37	D₅×C₁₆	—	2	4	no	—
38	D₅×C₁₆	—	2	4	no	—
39	C₂³×C₁₆	—	5	1	no	—
40	C₂×#427	—	5	2	no	—
41	#1380	—	5	4	no	—
42	C₂³×D₅	D₈	1	1	yes	characteristic
43	C₂²×AGL(1,𝔽₅)	—	4	4	no	—
44	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×Dic₅	Q₁₆	1	1	yes	characteristic
46	C₂×C₄×D₅	C₂²⋊C₄	1	4	yes	—
47	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×D₅	D₈	1	1	yes	characteristic
49	C₂×C₄×D₅	SD₁₆	1	4	yes	—
50	C₂×C₄×D₅	Q₁₆	1	1	yes	characteristic
51	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
52	C₂²⋊AGL(1,𝔽₅)	—	4	8	no	—
53	C₄⋊AGL(1,𝔽₅)	—	2	8	no	—
54	C₄⋊AGL(1,𝔽₅)	—	4	4	no	—
55	C₄×AGL(1,𝔽₅)	—	4	8	no	—
56	C₂×C₅×C₈	C₄²	1	2	yes	—
57	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
58	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
61	C₂×#168	C₄⋊C₄	1	4	yes	—
62	C₂×#168	C₂×Q₈	1	1	yes	characteristic
63	C₈×D₅	C₄²	1	2	yes	—
64	C₈×D₅	C₂²⋊C₄	1	4	yes	—
65	C₈×D₅	C₂²⋊C₄	1	2	yes	—
66	C₈×D₅	C₄⋊C₄	1	4	yes	—
67	C₈×D₅	C₂²×C₄	1	2	yes	—
68	C₈×D₅	C₂×D₄	1	2	yes	—
69	C₅×C₁₆	—	2	2	no	—
70	#672	—	2	2	no	—
71	C₂³×C₈	—	5	1	no	—
72	C₂×#117	—	10	2	no	—
73	C₂²×C₁₆	—	5	1	no	—
74	C₂²×C₁₆	—	5	2	no	—
75	C₂²×C₁₆	—	5	2	no	—
76	C₂²×C₁₆	—	10	4	no	—
77	C₂²×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
78	#427	—	5	8	no	—
79	#427	—	10	4	no	—
80	C₂×#107	—	5	4	no	—
81	C₂×#106	—	10	2	no	—
82	C₂³×C₅	#107	1	1	yes	characteristic
83	C₂²×D₅	—	2	4	no	—
84	C₂²×D₅	#102	1	4	yes	—
85	C₂²×D₅	#107	1	1	yes	characteristic
86	C₂²×D₅	C₂×D₈	1	1	yes	characteristic
87	C₂×AGL(1,𝔽₅)	—	4	8	no	—
88	C₂×AGL(1,𝔽₅)	—	8	8	no	—
89	C₂×C₄×C₅	#95	1	2	yes	—
90	C₂×C₄×C₅	#106	1	2	yes	—
91	C₂×C₄×C₅	#107	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×Dic₅	#103	1	4	yes	—
94	C₂×Dic₅	#107	1	1	yes	characteristic
95	C₂×Dic₅	C₂×Q₁₆	1	1	yes	characteristic
96	C₄×D₅	#95	1	2	yes	—
97	C₄×D₅	#102	1	4	yes	—
98	C₄×D₅	#103	1	4	yes	—
99	C₄×D₅	#106	1	2	yes	—
100	C₄×D₅	C₂×C₂²⋊C₄	1	2	yes	—
101	C₄×D₅	C₂×SD₁₆	1	2	yes	—
102	C₅×C₈	#95	1	2	yes	—
103	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
104	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	#168	#95	1	2	yes	—
106	#168	C₂×C₄⋊C₄	1	2	yes	—
107	C₂³×C₄	—	5	1	no	—
108	C₂²×C₈	—	5	1	no	—
109	C₂²×C₈	—	5	4	no	—
110	C₂²×C₈	—	5	1	no	—
111	C₂²×C₈	—	5	1	no	—
112	C₂²×C₈	—	5	4	no	—
113	C₂²×C₈	—	5	1	no	—
114	C₂²×C₈	—	5	1	no	—
115	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
116	#117	—	10	8	no	—
117	#117	—	20	4	no	—
118	C₂×C₄⋊C₄	—	10	4	no	—
119	C₂×C₂²⋊C₄	—	20	2	no	—
120	C₂×C₄²	—	20	2	no	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	5	2	no	—
123	C₂×C₁₆	—	10	4	no	—
124	C₂×C₁₆	—	10	4	no	—
125	C₂×C₁₆	—	10	2	no	—
126	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
127	#107	—	5	8	no	—
128	#107	—	10	4	no	—
129	#106	—	10	8	no	—
130	C₂²×C₅	—	2	2	no	—
131	C₂²×C₅	#340	1	2	yes	—
132	C₂²×C₅	C₂×#107	1	1	yes	characteristic
133	C₂×D₅	—	2	2	no	—
134	C₂×D₅	—	2	8	no	—
135	C₂×D₅	#340	1	2	yes	—
136	C₂×D₅	C₂×#102	1	2	yes	—
137	C₂×D₅	#508	1	4	yes	—
138	AGL(1,𝔽₅)	—	8	8	no	—
139	C₄×C₅	#340	1	2	yes	—
140	C₄×C₅	C₂×#95	1	1	yes	characteristic
141	C₄×C₅	C₂×#106	1	1	yes	characteristic
142	Dic₅	#340	1	2	yes	—
143	Dic₅	C₂×#103	1	2	yes	—
144	C₂⁴	—	5	1	no	—
145	C₂²×C₄	—	5	1	no	—
146	C₂²×C₄	—	5	1	no	—
147	C₂²×C₄	—	5	1	no	—
148	C₂²×C₄	—	5	4	no	—
149	C₂²×C₄	—	5	1	no	—
150	C₂²×C₄	—	5	1	no	—
151	C₂²×C₄	—	5	4	no	—
152	C₂²×C₄	—	20	4	no	—
153	C₂²×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
154	C₂×C₈	—	5	1	no	—
155	C₂×C₈	—	5	1	no	—
156	C₂×C₈	—	5	4	no	—
157	C₂×C₈	—	5	2	no	—
158	C₂×C₈	—	5	2	no	—
159	C₂×C₈	—	5	2	no	—
160	C₂×C₈	—	5	2	no	—
161	C₂×C₈	—	5	4	no	—
162	C₂×C₈	—	5	4	no	—
163	C₂×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
164	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
165	C₂×C₈	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
166	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
167	C₄⋊C₄	—	10	8	no	—
168	C₄⋊C₄	—	20	4	no	—
169	C₂²⋊C₄	—	20	8	no	—
170	C₄²	—	20	8	no	—
171	C₁₆	—	2	2	no	—
172	C₁₆	—	10	2	no	—
173	C₂×C₅	—	2	2	no	—
174	C₂×C₅	C₂×#340	1	1	yes	characteristic
175	C₂×C₅	#2158	1	2	yes	—
176	D₅	—	2	2	no	—
177	D₅	#2138	1	4	yes	—
178	C₂³	—	5	1	no	—
179	C₂³	—	5	4	no	—
180	C₂³	—	5	1	no	—
181	C₂³	—	10	4	no	—
182	C₂³	#3994	1	1	yes	characteristic
183	C₂×C₄	—	5	4	no	—
184	C₂×C₄	—	5	1	no	—
185	C₂×C₄	—	5	1	no	—
186	C₂×C₄	—	5	4	no	—
187	C₂×C₄	—	5	4	no	—
188	C₂×C₄	—	5	2	no	—
189	C₂×C₄	—	5	2	no	—
190	C₂×C₄	—	5	2	no	—
191	C₂×C₄	—	5	2	no	—
192	C₂×C₄	—	20	8	no	—
193	C₂×C₄	—	40	8	no	—
194	C₂×C₄	#3993	1	2	yes	—
195	C₂×C₄	#3994	1	1	yes	characteristic
196	C₂×C₄	(C₂×AGL(1,𝔽₅))⋊C₄	1	2	yes	—
197	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
198	C₈	—	5	2	no	—
199	C₈	—	5	2	no	—
200	C₈	(C₂×AGL(1,𝔽₅))⋊C₄	1	2	yes	—
201	C₈	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
202	C₈	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
203	C₅	#12556	1	1	yes	characteristic, complemented, Sylow
204	C₂²	—	2	2	no	—
205	C₂²	—	5	4	no	—
206	C₂²	—	5	2	no	—
207	C₂²	—	5	2	no	—
208	C₂²	—	10	8	no	—
209	C₂²	—	10	2	no	—
210	C₂²	#64893	1	2	yes	—
211	C₂²	C₂×#3994	1	1	yes	characteristic
212	C₄	—	5	2	no	—
213	C₄	—	5	2	no	—
214	C₄	—	40	8	no	—
215	C₄	#64893	1	2	yes	—
216	C₄	C₂×#3993	1	1	yes	characteristic
217	C₄	C₂×(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
218	C₂	—	2	2	no	—
219	C₂	—	5	4	no	—
220	C₂	—	10	2	no	—
221	C₂	#10604107	1	2	yes	—
222	C₂	C₂×#64893	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3994
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₂²×C₅×C₁₆	C₄
socle	C₂²×C₅	C₂×#107
prime radical	C₂³×D₅	D₈
cosocle	C₂×C₈×D₅	C₂³
hypercenter	C₂²×C₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#12556
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #10598664	C₂	semidirect
7. #65721	C₄	semidirect
10. C₂²×C₅×C₁₆	C₄	semidirect
11. C₂²×#672	C₄	semidirect
13. C₂×D₅×C₁₆	C₄	semidirect
77. C₂²×C₁₆	AGL(1,𝔽₅)	semidirect
203. C₅	#12556	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65721	2
C₂²×D₅×C₁₆	2
#10598664	2
#12556	5