Subgroups — #12014178

Order 1280.

606 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#65169	C₂	1	2	yes	maximal
3	C₅×#1380	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598297	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598213	C₂	1	2	yes	maximal, complemented
6	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
7	C₂×#4038	—	2	2	no	—
8	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₅×#107	C₂²	1	2	yes	—
10	C₂×#3950	C₂²	1	2	yes	—
11	C₂²×#672	C₂²	1	1	yes	characteristic
12	C₂×#3940	—	2	2	no	—
13	C₂×D₅×C₁₆	—	2	4	no	—
14	C₂×D₅×C₁₆	C₄	1	4	yes	complemented
15	#65169	—	2	4	no	—
16	#65169	C₄	1	8	yes	complemented
17	#12556	—	5	1	no	maximal, Sylow
18	C₂²×C₄×D₅	D₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
20	C₂×C₅×C₄⋊C₄	—	2	2	no	—
21	C₂×#685	—	4	2	no	—
22	C₂×C₄⋊Dic₅	—	2	2	no	—
23	C₂×C₄×Dic₅	—	4	2	no	—
24	C₂²×#168	D₄	1	1	yes	characteristic
25	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
26	C₂×C₈×D₅	C₂×C₄	1	4	yes	—
27	C₂×C₈×D₅	D₄	1	4	yes	—
28	C₂×C₈×D₅	Q₈	1	2	yes	—
29	#4038	—	2	8	no	—
30	#4038	—	4	4	no	—
31	C₂×C₅×C₁₆	—	2	2	no	—
32	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
33	C₅×#107	—	2	2	no	—
34	C₅×#107	C₂×C₄	1	4	yes	—
35	#3950	—	2	2	no	—
36	#3950	C₂×C₄	1	4	yes	—
37	C₂×#672	—	2	2	no	—
38	C₂×#672	C₂×C₄	1	2	yes	—
39	#3940	—	2	8	no	—
40	D₅×C₁₆	—	2	8	no	—
41	C₂³×C₁₆	—	5	1	no	—
42	C₂×#427	—	5	2	no	—
43	#1380	—	5	1	no	—
44	#1380	—	5	2	no	—
45	#1380	D₅	1	1	yes	characteristic, complemented
46	C₂³×D₅	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₅	—	4	2	no	—
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₅	—	4	2	no	—
50	C₂²×Dic₅	D₈	1	1	yes	characteristic
51	C₂×C₄×D₅	C₂²⋊C₄	1	4	yes	—
52	C₂×C₄×D₅	C₄⋊C₄	1	2	yes	—
53	C₂×C₄×D₅	SD₁₆	1	4	yes	—
54	C₂×C₄×D₅	Q₁₆	1	2	yes	—
55	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
57	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
58	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
59	C₅×C₄⋊C₄	—	2	4	no	—
60	C₅×C₄⋊C₄	—	4	2	no	—
61	#685	—	4	8	no	—
62	C₄⋊Dic₅	—	2	4	no	—
63	C₄⋊Dic₅	—	4	2	no	—
64	C₄×Dic₅	—	4	8	no	—
65	C₂×#168	C₂²⋊C₄	1	4	yes	—
66	C₂×#168	C₄⋊C₄	1	2	yes	—
67	C₈×D₅	C₄²	1	4	yes	—
68	C₈×D₅	C₂²⋊C₄	1	4	yes	—
69	C₈×D₅	C₄⋊C₄	1	8	yes	—
70	C₅×C₁₆	—	2	2	no	—
71	#672	—	2	2	no	—
72	C₂³×C₈	—	5	1	no	—
73	C₂×#117	—	10	2	no	—
74	C₂²×C₁₆	—	5	1	no	—
75	C₂²×C₁₆	—	5	4	no	—
76	C₂²×C₁₆	—	10	4	no	—
77	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
78	#427	—	5	8	no	—
79	#427	—	10	4	no	—
80	C₂×#107	—	5	2	no	—
81	C₂×#107	C₂×D₅	1	2	yes	—
82	C₂×#106	—	10	2	no	—
83	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
84	C₂²×D₅	—	2	1	no	—
85	C₂²×D₅	—	2	2	no	—
86	C₂²×D₅	—	2	1	no	—
87	C₂²×D₅	#102	1	4	yes	—
88	C₂²×D₅	#107	1	2	yes	—
89	C₂×C₄×C₅	—	4	4	no	—
90	C₂×C₄×C₅	—	8	2	no	—
91	C₂×C₄×C₅	—	8	2	no	—
92	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
95	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×Dic₅	—	4	4	no	—
97	C₂×Dic₅	—	8	2	no	—
98	C₂×Dic₅	—	8	2	no	—
99	C₂×Dic₅	#102	1	4	yes	—
100	C₂×Dic₅	#107	1	2	yes	—
101	C₄×D₅	#95	1	4	yes	—
102	C₄×D₅	#103	1	8	yes	—
103	C₄×D₅	#106	1	4	yes	—
104	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
105	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
107	#168	#95	1	4	yes	—
108	C₂³×C₄	—	5	1	no	—
109	C₂²×C₈	—	5	1	no	—
110	C₂²×C₈	—	5	4	no	—
111	C₂²×C₈	—	5	2	no	—
112	C₂²×C₈	—	5	4	no	—
113	C₂²×C₈	—	5	2	no	—
114	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
115	#117	—	10	8	no	—
116	#117	—	20	4	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	10	2	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	8	no	—
124	C₂×C₁₆	—	10	2	no	—
125	C₂×C₁₆	C₄×D₅	1	2	yes	—
126	#107	—	2	2	no	—
127	#107	—	5	4	no	—
128	#107	—	10	2	no	—
129	#107	C₄×D₅	1	4	yes	—
130	#106	—	10	8	no	—
131	C₂²×C₅	C₂×#102	1	2	yes	—
132	C₂²×C₅	C₂×#107	1	1	yes	characteristic
133	C₂²×C₅	#508	1	1	yes	characteristic
134	C₂²×C₅	#508	1	2	yes	—
135	C₂²×C₅	#508	1	1	yes	characteristic
136	C₂×D₅	—	2	4	no	—
137	C₂×D₅	—	2	4	no	—
138	C₂×D₅	—	2	2	no	—
139	C₂×D₅	—	2	2	no	—
140	C₂×D₅	#340	1	4	yes	—
141	C₄×C₅	—	8	4	no	—
142	C₄×C₅	C₂×#95	1	1	yes	characteristic
143	C₄×C₅	C₂×#103	1	2	yes	—
144	C₄×C₅	C₂×#106	1	1	yes	characteristic
145	Dic₅	—	8	4	no	—
146	Dic₅	#340	1	4	yes	—
147	C₂⁴	—	5	1	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	—	5	1	no	—
150	C₂²×C₄	—	5	4	no	—
151	C₂²×C₄	—	5	2	no	—
152	C₂²×C₄	—	5	4	no	—
153	C₂²×C₄	—	5	2	no	—
154	C₂²×C₄	—	20	2	no	—
155	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
156	C₂×C₈	—	5	4	no	—
157	C₂×C₈	—	5	2	no	—
158	C₂×C₈	—	5	4	no	—
159	C₂×C₈	—	5	4	no	—
160	C₂×C₈	—	5	8	no	—
161	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
162	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
163	C₂×C₈	D₄×D₅	1	2	yes	—
164	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	10	4	no	—
168	C₄⋊C₄	—	20	2	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	C₂×#340	1	1	yes	characteristic
174	C₂×C₅	#2138	1	2	yes	—
175	C₂×C₅	#2138	1	2	yes	—
176	C₂×C₅	#2158	1	1	yes	characteristic
177	C₂×C₅	#2158	1	1	yes	characteristic
178	D₅	—	2	4	no	—
179	C₂³	—	5	4	no	—
180	C₂³	—	5	2	no	—
181	C₂³	—	10	1	no	—
182	C₂³	—	10	2	no	—
183	C₂³	—	10	1	no	—
184	C₂³	D₅×D₈	1	1	yes	characteristic
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	5	4	no	—
187	C₂×C₄	—	5	2	no	—
188	C₂×C₄	—	5	8	no	—
189	C₂×C₄	—	5	4	no	—
190	C₂×C₄	—	5	4	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	20	4	no	—
194	C₂×C₄	—	40	2	no	—
195	C₂×C₄	—	40	2	no	—
196	C₂×C₄	D₅×C₂²⋊C₄	1	2	yes	—
197	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₄	D₅×SD₁₆	1	2	yes	—
199	C₂×C₄	D₅×Q₁₆	1	1	yes	characteristic
200	C₈	—	5	4	no	—
201	C₈	C₄²×D₅	1	1	yes	characteristic
202	C₈	D₅×C₂²⋊C₄	1	1	yes	characteristic
203	C₈	D₅×C₄⋊C₄	1	2	yes	—
204	C₅	#12556	1	1	yes	characteristic, complemented, Sylow
205	C₂²	—	5	4	no	—
206	C₂²	—	10	4	no	—
207	C₂²	—	10	4	no	—
208	C₂²	—	10	2	no	—
209	C₂²	—	10	2	no	—
210	C₂²	D₅×#102	1	2	yes	—
211	C₂²	D₅×#107	1	1	yes	characteristic
212	C₂²	#65201	1	1	yes	characteristic
213	C₂²	#65205	1	2	yes	—
214	C₂²	#65208	1	1	yes	characteristic
215	C₄	—	5	4	no	—
216	C₄	—	8	4	no	—
217	C₄	—	40	4	no	—
218	C₄	D₅×#95	1	1	yes	characteristic
219	C₄	D₅×#103	1	2	yes	—
220	C₄	D₅×#106	1	1	yes	characteristic
221	C₂	—	10	4	no	—
222	C₂	D₅×#340	1	1	yes	characteristic
223	C₂	#10599981	1	2	yes	—
224	C₂	#10600013	1	2	yes	—
225	C₂	#10600118	1	1	yes	characteristic
226	C₂	#10600138	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₅×D₈
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1380	C₂
socle	C₂³×C₅	C₂×D₈
prime radical	C₂³×D₅	D₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1380	D₅
hypocenter	C₅	#12556
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

Complements

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

N	H ≅ G/N	type
3. C₅×#1380	C₂	semidirect
4. #10598297	C₂	semidirect
5. #10598213	C₂	semidirect
14. C₂×D₅×C₁₆	C₄	semidirect
16. #65169	C₄	semidirect
45. #1380	D₅	semidirect
204. C₅	#12556	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₅×C₁₆	2
C₂×#65169	2
C₅×#1380	2
#10598297	2
#10598213	2
#12556	5