Subgroups — #12252384

Order 1280.

590 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65390	C₂	1	1	yes	maximal, characteristic
2	C₅×#2112	C₂	1	1	yes	maximal, characteristic
3	#10602756	C₂	1	1	yes	maximal, characteristic
4	#10602213	C₂	1	2	yes	maximal, complemented
5	Dic₅×M₃₂	C₂	1	2	yes	maximal, complemented
6	#10599887	C₂	1	8	yes	maximal, complemented
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#4072	C₄	1	1	yes	characteristic
9	C₂²×#681	C₄	1	1	yes	characteristic
10	C₂×C₅×M₃₂	C₂²	1	2	yes	—
11	#65390	C₄	1	4	yes	—
12	#65390	C₂²	1	4	yes	—
13	C₂×#3947	C₄	1	1	yes	characteristic
14	C₂×#3947	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
17	C₂×#3944	C₂²	1	2	yes	—
18	C₅×C₂²⋊C₁₆	C₂²	1	4	yes	—
19	C₂²⋊#672	C₂²	1	4	yes	—
20	#64722	C₂²	1	4	yes	complemented
21	#64721	C₂²	1	8	yes	complemented
22	C₁₆×Dic₅	C₂²	1	4	yes	complemented
23	#33419	—	5	1	no	maximal, Sylow
24	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	—	2	4	no	—
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₅×C₈	C₂³	1	2	yes	—
29	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₅	C₈	1	2	yes	—
31	#4072	C₈	1	4	yes	—
32	#4072	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊Dic₅	C₈	1	2	yes	—
34	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
36	C₂×#681	—	2	4	no	—
37	C₂×#681	C₂×C₄	1	4	yes	—
38	C₂²×#168	C₂×C₄	1	1	yes	characteristic
39	C₂²×#168	C₂×C₄	1	1	yes	characteristic
40	C₅×M₃₂	—	2	4	no	—
41	C₂×C₅×C₁₆	C₂³	1	4	yes	—
42	#3947	C₂×C₄	1	4	yes	—
43	#3947	C₂×C₄	1	4	yes	—
44	C₈×Dic₅	C₂×C₄	1	2	yes	—
45	C₈×Dic₅	C₂×C₄	1	2	yes	—
46	C₈×Dic₅	C₂³	1	2	yes	—
47	C₈×Dic₅	C₂³	1	2	yes	—
48	#3944	—	2	4	no	—
49	C₂×#672	C₂³	1	4	yes	—
50	C₂×#405	—	5	1	no	—
51	#2164	—	5	8	no	—
52	#2150	—	5	2	no	—
53	#2112	—	5	1	no	—
54	#2112	D₅	1	1	yes	characteristic
55	C₄×M₃₂	—	5	2	no	—
56	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅	—	2	2	no	—
58	C₂²×C₄×C₅	—	2	2	no	—
59	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₅	C₂×C₈	1	2	yes	—
64	C₂²×Dic₅	C₂×C₈	1	2	yes	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	—	2	4	no	—
67	C₂×C₅×C₈	—	2	4	no	—
68	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
69	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₅×C₈	#41	1	4	yes	—
71	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
72	C₂²⋊Dic₅	C₂×C₈	1	8	yes	—
73	C₄⋊Dic₅	C₂×C₈	1	8	yes	—
74	C₄×Dic₅	C₂×C₈	1	2	yes	—
75	C₄×Dic₅	C₂×C₈	1	2	yes	—
76	C₄×Dic₅	C₂²×C₄	1	2	yes	—
77	C₄×Dic₅	C₂²×C₄	1	2	yes	—
78	#681	—	2	8	no	—
79	C₂×#168	—	2	2	no	—
80	C₂×#168	M₁₆	1	4	yes	—
81	C₂×#168	C₂²×C₄	1	2	yes	—
82	C₂×#168	C₂²×C₄	1	2	yes	—
83	C₅×C₁₆	—	2	4	no	—
84	#672	—	2	4	no	—
85	C₂²×M₁₆	—	5	1	no	—
86	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
87	C₂×#117	—	5	1	no	—
88	C₂×M₃₂	—	5	2	no	—
89	C₂×M₃₂	C₂×D₅	1	2	yes	—
90	#405	—	5	4	no	—
91	#405	—	5	4	no	—
92	C₂×#97	—	5	1	no	—
93	C₂×#97	—	5	1	no	—
94	C₂×C₄×C₈	—	5	1	no	—
95	C₂×C₄×C₈	—	5	1	no	—
96	C₄⋊C₁₆	—	5	8	no	—
97	C₂²⋊C₁₆	—	5	4	no	—
98	C₂²⋊C₁₆	C₂×D₅	1	4	yes	—
99	#346	—	5	4	no	—
100	C₄×C₁₆	—	5	4	no	—
101	C₂³×C₅	—	2	2	no	—
102	C₂³×C₅	—	4	2	no	—
103	C₂³×C₅	C₂²×C₈	1	2	yes	—
104	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	4	no	—
109	C₂×C₄×C₅	#117	1	2	yes	—
110	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
113	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
114	C₂×Dic₅	—	4	2	no	—
115	C₂×Dic₅	—	4	2	no	—
116	C₂×Dic₅	C₂²×C₈	1	4	yes	—
117	C₂×Dic₅	C₂²×C₈	1	4	yes	—
118	C₅×C₈	—	2	2	no	—
119	C₅×C₈	#117	1	2	yes	—
120	C₅×C₈	C₂×#41	1	2	yes	—
121	#168	—	2	2	no	—
122	#168	C₂×M₁₆	1	4	yes	—
123	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
124	C₂×M₁₆	—	5	4	no	—
125	C₂×M₁₆	—	10	4	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	5	1	no	—
128	C₂²×C₈	—	5	1	no	—
129	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
130	C₂²×C₈	C₄×D₅	1	2	yes	—
131	C₂²×C₈	C₂²×D₅	1	2	yes	—
132	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
133	#117	—	5	4	no	—
134	#117	—	5	4	no	—
135	C₂×C₄⋊C₄	—	5	2	no	—
136	C₂×C₂²⋊C₄	—	5	2	no	—
137	C₂×C₄²	—	5	1	no	—
138	C₂×C₄²	—	5	1	no	—
139	M₃₂	—	2	4	no	—
140	M₃₂	—	10	4	no	—
141	C₂×C₁₆	—	5	4	no	—
142	C₂×C₁₆	C₂²×D₅	1	4	yes	—
143	#97	—	5	4	no	—
144	#97	—	5	4	no	—
145	C₄×C₈	—	5	2	no	—
146	C₄×C₈	—	5	2	no	—
147	C₄×C₈	—	5	2	no	—
148	C₄×C₈	—	5	2	no	—
149	C₂²×C₅	—	2	2	no	—
150	C₂²×C₅	—	2	4	no	—
151	C₂²×C₅	—	2	2	no	—
152	C₂²×C₅	—	4	4	no	—
153	C₂²×C₅	#431	1	2	yes	—
154	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
155	C₄×C₅	—	2	2	no	—
156	C₄×C₅	#431	1	2	yes	—
157	C₄×C₅	C₂×#117	1	1	yes	characteristic
158	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
159	Dic₅	—	2	4	no	—
160	Dic₅	—	2	4	no	—
161	C₂⁴	C₈×D₅	1	1	yes	characteristic
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	5	2	no	—
165	C₂²×C₄	—	5	2	no	—
166	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
167	C₂²×C₄	C₈×D₅	1	2	yes	—
168	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
169	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
170	M₁₆	—	10	8	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	5	2	no	—
175	C₂×C₈	—	5	4	no	—
176	C₂×C₈	—	5	2	no	—
177	C₂×C₈	—	10	2	no	—
178	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
179	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
180	C₂×C₈	#711	1	4	yes	—
181	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
182	C₄⋊C₄	—	5	8	no	—
183	C₂²⋊C₄	—	5	8	no	—
184	C₄²	—	5	2	no	—
185	C₄²	—	5	2	no	—
186	C₄²	—	5	2	no	—
187	C₄²	—	5	2	no	—
188	C₁₆	—	2	4	no	—
189	C₁₆	—	10	4	no	—
190	C₂×C₅	—	2	2	no	—
191	C₂×C₅	—	4	2	no	—
192	C₂×C₅	C₂×#431	1	1	yes	characteristic
193	C₂×C₅	#3407	1	2	yes	—
194	C₂³	—	2	2	no	—
195	C₂³	—	4	2	no	—
196	C₂³	C₂×C₈×D₅	1	2	yes	—
197	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	5	4	no	—
203	C₂×C₄	—	5	4	no	—
204	C₂×C₄	—	20	2	no	—
205	C₂×C₄	—	20	2	no	—
206	C₂×C₄	#4023	1	2	yes	—
207	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
208	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
209	C₂×C₄	D₅×M₁₆	1	2	yes	—
210	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	—	5	4	no	—
213	C₈	—	10	2	no	—
214	C₈	#4023	1	2	yes	—
215	C₈	C₂×#711	1	2	yes	—
216	C₅	#33419	1	1	yes	characteristic, complemented, Sylow
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	4	4	no	—
221	C₂²	#65007	1	2	yes	—
222	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	10	4	no	—
225	C₄	—	10	4	no	—
226	C₄	#65007	1	2	yes	—
227	C₄	C₂×#4023	1	1	yes	characteristic
228	C₄	C₂×D₅×M₁₆	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	—	4	2	no	—
231	C₂	C₂×#65007	1	1	yes	characteristic
232	C₂	#10613738	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₅×#2112	C₂
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#2112	D₅
hypocenter	C₅	#33419
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ #2112

Complements

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

N	H ≅ G/N	type
4. #10602213	C₂	semidirect
5. Dic₅×M₃₂	C₂	semidirect
6. #10599887	C₂	semidirect
20. #64722	C₂²	semidirect
21. #64721	C₂²	semidirect
22. C₁₆×Dic₅	C₂²	semidirect
216. 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×#65390	2
C₅×#2112	2
#10602756	2
#10602213	2
Dic₅×M₃₂	2
#10599887	2
#33419	5