Subgroups — #12252387

Order 1280.

590 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
2	#10602756	C₂	1	2	yes	maximal
3	#10600676	C₂	1	8	yes	maximal, complemented
4	#10600536	C₂	1	2	yes	maximal, complemented
5	C₄×#3944	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×M₁₆	C₄	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×#117	C₄	1	1	yes	characteristic
9	C₅×#405	C₄	1	4	yes	—
10	C₅×#405	C₂²	1	4	yes	—
11	C₂×C₅×#97	C₄	1	1	yes	characteristic
12	C₂×C₅×#97	C₄	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×#3944	C₂²	1	4	yes	—
16	C₂²⋊#672	C₂²	1	8	yes	—
17	C₄⋊#672	C₂²	1	8	yes	complemented
18	#64681	C₂²	1	4	yes	complemented
19	C₄×#672	C₂²	1	4	yes	complemented
20	#33419	—	5	1	no	maximal, Sylow
21	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₅×M₁₆	—	2	4	no	—
23	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
24	C₂²×C₅×C₈	—	2	4	no	—
25	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂³	1	2	yes	—
30	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₅×#117	C₈	1	4	yes	—
32	C₅×#117	C₂×C₄	1	4	yes	—
33	C₂×C₅×C₄⋊C₄	C₈	1	2	yes	—
34	C₂×C₅×C₂²⋊C₄	C₈	1	2	yes	—
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₅×#97	C₂×C₄	1	4	yes	—
38	C₅×#97	C₂×C₄	1	4	yes	—
39	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
40	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
41	C₄×C₅×C₈	C₂³	1	2	yes	—
42	C₄×C₅×C₈	C₂³	1	2	yes	—
43	#3944	—	2	8	no	—
44	C₂×#672	C₂³	1	8	yes	—
45	C₂×#405	D₅	1	1	yes	characteristic
46	#2164	—	5	8	no	—
47	#2150	—	5	2	no	—
48	#2112	—	5	2	no	—
49	C₄×M₃₂	—	5	2	no	—
50	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₅	—	2	2	no	—
52	C₂²×C₄×C₅	—	2	2	no	—
53	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₅×M₁₆	—	2	8	no	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	—	2	2	no	—
62	C₂×C₅×C₈	—	2	4	no	—
63	C₂×C₅×C₈	—	2	4	no	—
64	C₂×C₅×C₈	M₁₆	1	4	yes	—
65	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
66	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
67	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
69	C₂×C₅×C₈	#41	1	4	yes	—
70	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
71	C₅×C₄⋊C₄	C₂×C₈	1	8	yes	—
72	C₅×C₂²⋊C₄	C₂×C₈	1	8	yes	—
73	C₄²×C₅	C₂×C₈	1	2	yes	—
74	C₄²×C₅	C₂×C₈	1	2	yes	—
75	C₄²×C₅	C₂²×C₄	1	2	yes	—
76	C₄²×C₅	C₂²×C₄	1	2	yes	—
77	#672	—	2	8	no	—
78	C₂²×M₁₆	Dic₅	1	1	yes	characteristic
79	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
80	C₂×#117	Dic₅	1	1	yes	characteristic
81	C₂×M₃₂	—	5	4	no	—
82	#405	Dic₅	1	4	yes	—
83	#405	C₂×D₅	1	4	yes	—
84	C₂×#97	Dic₅	1	1	yes	characteristic
85	C₂×#97	Dic₅	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
88	C₄⋊C₁₆	—	5	8	no	—
89	C₂²⋊C₁₆	—	5	8	no	—
90	#346	—	5	4	no	—
91	C₄×C₁₆	—	5	4	no	—
92	C₂³×C₅	—	2	2	no	—
93	C₂³×C₅	—	4	2	no	—
94	C₂³×C₅	C₂²×C₈	1	2	yes	—
95	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₅	—	2	2	no	—
97	C₂×C₄×C₅	—	2	2	no	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	2	4	no	—
100	C₂×C₄×C₅	—	4	2	no	—
101	C₂×C₄×C₅	—	4	2	no	—
102	C₂×C₄×C₅	#117	1	2	yes	—
103	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
104	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
105	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
109	C₅×C₈	—	2	2	no	—
110	C₅×C₈	—	2	2	no	—
111	C₅×C₈	#117	1	2	yes	—
112	C₅×C₈	C₂×M₁₆	1	4	yes	—
113	C₅×C₈	C₂×#41	1	2	yes	—
114	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
115	C₂×M₁₆	—	2	4	no	—
116	C₂×M₁₆	C₂×Dic₅	1	4	yes	—
117	C₂²×C₈	—	2	4	no	—
118	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
119	C₂²×C₈	C₂×Dic₅	1	2	yes	—
120	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
121	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₅	1	2	yes	—
123	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
124	#117	#168	1	4	yes	—
125	#117	C₂×Dic₅	1	4	yes	—
126	C₂×C₄⋊C₄	#168	1	2	yes	—
127	C₂×C₂²⋊C₄	#168	1	2	yes	—
128	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
129	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
130	M₃₂	—	10	8	no	—
131	C₂×C₁₆	—	5	8	no	—
132	#97	C₂×Dic₅	1	4	yes	—
133	#97	C₂×Dic₅	1	4	yes	—
134	C₄×C₈	C₂×Dic₅	1	2	yes	—
135	C₄×C₈	C₂×Dic₅	1	2	yes	—
136	C₄×C₈	C₂²×D₅	1	2	yes	—
137	C₄×C₈	C₂²×D₅	1	2	yes	—
138	C₂²×C₅	—	2	2	no	—
139	C₂²×C₅	—	2	4	no	—
140	C₂²×C₅	—	2	2	no	—
141	C₂²×C₅	—	4	4	no	—
142	C₂²×C₅	#431	1	2	yes	—
143	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
144	C₄×C₅	—	2	4	no	—
145	C₄×C₅	—	2	4	no	—
146	C₄×C₅	—	2	2	no	—
147	C₄×C₅	#431	1	2	yes	—
148	C₄×C₅	C₂×#117	1	1	yes	characteristic
149	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
150	C₂⁴	C₂×#168	1	1	yes	characteristic
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	C₂×#168	1	1	yes	characteristic
154	C₂²×C₄	C₂×#168	1	2	yes	—
155	C₂²×C₄	C₂×#168	1	2	yes	—
156	C₂²×C₄	C₂×#168	1	2	yes	—
157	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
158	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
159	M₁₆	—	2	8	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	#681	1	4	yes	—
165	C₂×C₈	#709	1	4	yes	—
166	C₂×C₈	C₂²×Dic₅	1	2	yes	—
167	C₂×C₈	C₂²×Dic₅	1	2	yes	—
168	C₂×C₈	C₂²×Dic₅	1	2	yes	—
169	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
170	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
171	C₄⋊C₄	C₂×#168	1	8	yes	—
172	C₂²⋊C₄	C₂×#168	1	8	yes	—
173	C₄²	C₂×#168	1	2	yes	—
174	C₄²	C₂×#168	1	2	yes	—
175	C₄²	C₂²×Dic₅	1	2	yes	—
176	C₄²	C₂²×Dic₅	1	2	yes	—
177	C₁₆	—	10	8	no	—
178	C₂×C₅	—	2	2	no	—
179	C₂×C₅	—	4	2	no	—
180	C₂×C₅	C₂×#431	1	1	yes	characteristic
181	C₂×C₅	#3407	1	2	yes	—
182	C₂³	—	2	2	no	—
183	C₂³	—	4	2	no	—
184	C₂³	C₂²×#168	1	2	yes	—
185	C₂³	C₂²×#168	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	C₂²×#168	1	4	yes	—
193	C₂×C₄	C₂²×#168	1	4	yes	—
194	C₂×C₄	C₂²×#168	1	2	yes	—
195	C₂×C₄	C₂²×#168	1	1	yes	characteristic
196	C₂×C₄	C₂×#681	1	2	yes	—
197	C₂×C₄	#4072	1	2	yes	—
198	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	2	2	no	—
201	C₈	C₂×#681	1	4	yes	—
202	C₈	#4072	1	2	yes	—
203	C₈	C₂×#709	1	2	yes	—
204	C₅	#33419	1	1	yes	characteristic, complemented, Sylow
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	4	4	no	—
209	C₂²	#65214	1	2	yes	—
210	C₂²	C₂³×#168	1	1	yes	characteristic
211	C₄	—	2	2	no	—
212	C₄	—	2	4	no	—
213	C₄	—	2	4	no	—
214	C₄	#65214	1	2	yes	—
215	C₄	C₂²×#681	1	1	yes	characteristic
216	C₄	C₂×#4072	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	—	4	2	no	—
219	C₂	C₂×#65214	1	1	yes	characteristic
220	C₂	#10615511	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₂×C₅×#405	C₂
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×#405	D₅
hypocenter	C₅	#33419
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #10600676	C₂	semidirect
4. #10600536	C₂	semidirect
5. C₄×#3944	C₂	semidirect
17. C₄⋊#672	C₂²	semidirect
18. #64681	C₂²	semidirect
19. C₄×#672	C₂²	semidirect
204. C₅	#33419	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#405	2
#10602756	2
#10600676	2
#10600536	2
C₄×#3944	2
#33419	5