Subgroups — #12014362

Order 1280.

466 conjugacy classes of subgroups.

The subgroup lattice has 238 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	C₂×#64686	C₂	1	2	yes	maximal, complemented
4	#10598154	C₂	1	2	yes	maximal, complemented
5	C₂²×C₅×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×#117	C₄	1	1	yes	characteristic
8	C₅×#405	—	2	4	no	—
9	C₂×C₅×#97	C₂²	1	1	yes	characteristic
10	C₂×C₅×#97	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
13	C₂×#3944	C₂²	1	4	yes	—
14	C₂²⋊#672	C₄	1	8	yes	complemented
15	#64686	—	2	4	no	—
16	#12606	—	5	1	no	maximal, Sylow
17	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₅×M₁₆	—	2	2	no	—
19	C₂×C₅×M₁₆	—	2	2	no	—
20	C₂×C₅×M₁₆	—	2	2	no	—
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₅×#117	—	2	2	no	—
29	C₅×#117	—	2	2	no	—
30	C₂×C₅×C₄⋊C₄	—	2	1	no	—
31	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₅×#97	—	2	2	no	—
35	C₅×#97	D₄	1	4	yes	—
36	C₄×C₅×C₈	—	2	2	no	—
37	C₄×C₅×C₈	—	2	2	no	—
38	#3944	—	2	4	no	—
39	#3944	—	2	4	no	—
40	C₂×#672	C₂×C₄	1	8	yes	—
41	C₂×#405	D₅	1	1	yes	characteristic
42	#2112	—	5	2	no	—
43	C₂×#347	—	5	2	no	—
44	#1375	—	5	2	no	—
45	C₂⁴×C₅	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₅	—	2	1	no	—
47	C₂²×C₄×C₅	—	2	1	no	—
48	C₂²×C₄×C₅	—	2	1	no	—
49	C₂²×C₄×C₅	—	2	1	no	—
50	C₂²×C₄×C₅	—	2	1	no	—
51	C₂²×C₄×C₅	—	2	1	no	—
52	C₂²×C₄×C₅	C₄²	1	2	yes	—
53	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₅	M₁₆	1	2	yes	—
55	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
56	C₅×M₁₆	—	2	8	no	—
57	C₂×C₅×C₈	—	2	1	no	—
58	C₂×C₅×C₈	—	2	1	no	—
59	C₂×C₅×C₈	—	2	1	no	—
60	C₂×C₅×C₈	—	2	1	no	—
61	C₂×C₅×C₈	—	2	1	no	—
62	C₂×C₅×C₈	—	2	1	no	—
63	C₂×C₅×C₈	—	2	8	no	—
64	C₂×C₅×C₈	C₄²	1	2	yes	—
65	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
66	C₂×C₅×C₈	M₁₆	1	2	yes	—
67	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
68	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
69	C₂×C₅×C₈	#41	1	2	yes	—
70	C₅×C₄⋊C₄	—	2	4	no	—
71	C₅×C₂²⋊C₄	—	2	4	no	—
72	C₄²×C₅	—	2	1	no	—
73	C₄²×C₅	—	2	1	no	—
74	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₅	M₁₆	1	2	yes	—
76	#672	—	2	8	no	—
77	C₂²×M₁₆	Dic₅	1	1	yes	characteristic
78	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
79	C₂×#117	Dic₅	1	1	yes	characteristic
80	C₂×M₃₂	—	5	4	no	—
81	#405	—	2	4	no	—
82	C₂×#97	C₂×D₅	1	1	yes	characteristic
83	C₂×#97	C₂×D₅	1	1	yes	characteristic
84	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
85	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
86	C₂²⋊C₁₆	—	5	8	no	—
87	#347	—	10	4	no	—
88	C₂³×C₅	—	2	1	no	—
89	C₂³×C₅	—	2	1	no	—
90	C₂³×C₅	—	4	2	no	—
91	C₂³×C₅	#97	1	2	yes	—
92	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅	—	2	1	no	—
94	C₂×C₄×C₅	—	2	1	no	—
95	C₂×C₄×C₅	—	2	2	no	—
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	2	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	4	2	no	—
102	C₂×C₄×C₅	—	4	2	no	—
103	C₂×C₄×C₅	#97	1	2	yes	—
104	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅	#117	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
110	C₅×C₈	—	2	2	no	—
111	C₅×C₈	—	2	2	no	—
112	C₅×C₈	—	2	2	no	—
113	C₅×C₈	C₄×D₄	1	4	yes	—
114	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
115	C₂×M₁₆	—	2	2	no	—
116	C₂×M₁₆	—	2	2	no	—
117	C₂×M₁₆	—	2	2	no	—
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	C₄×D₅	1	4	yes	—
121	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
122	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
123	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
125	#117	—	2	2	no	—
126	#117	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	1	no	—
128	C₂×C₂²⋊C₄	—	2	1	no	—
129	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
130	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
131	M₃₂	—	10	4	no	—
132	M₃₂	—	10	4	no	—
133	C₂×C₁₆	—	5	8	no	—
134	#97	—	2	2	no	—
135	#97	C₂²⋊D₅	1	4	yes	—
136	C₄×C₈	—	2	2	no	—
137	C₄×C₈	—	2	2	no	—
138	C₂²×C₅	—	2	2	no	—
139	C₂²×C₅	—	2	4	no	—
140	C₂²×C₅	—	2	2	no	—
141	C₂²×C₅	—	4	2	no	—
142	C₂²×C₅	—	4	2	no	—
143	C₂²×C₅	C₂×#97	1	1	yes	characteristic
144	C₂²×C₅	#407	1	1	yes	characteristic
145	C₂²×C₅	#432	1	1	yes	characteristic
146	C₄×C₅	—	2	2	no	—
147	C₄×C₅	—	4	2	no	—
148	C₄×C₅	—	4	2	no	—
149	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
151	C₄×C₅	#407	1	1	yes	characteristic
152	C₄×C₅	#432	1	1	yes	characteristic
153	C₂⁴	#681	1	1	yes	characteristic
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	#681	1	1	yes	characteristic
161	C₂²×C₄	#681	1	2	yes	—
162	C₂²×C₄	C₄×Dic₅	1	2	yes	—
163	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
164	M₁₆	—	2	8	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	8	no	—
172	C₂×C₈	#681	1	2	yes	—
173	C₂×C₈	C₄×Dic₅	1	2	yes	—
174	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
175	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
176	C₂×C₈	#709	1	2	yes	—
177	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
178	C₄⋊C₄	—	2	4	no	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₄²	—	2	1	no	—
181	C₄²	—	2	1	no	—
182	C₄²	#681	1	2	yes	—
183	C₄²	C₂²⋊Dic₅	1	2	yes	—
184	C₁₆	—	10	8	no	—
185	C₂×C₅	—	2	1	no	—
186	C₂×C₅	—	2	1	no	—
187	C₂×C₅	—	4	2	no	—
188	C₂×C₅	#1749	1	1	yes	characteristic
189	C₂×C₅	#2110	1	2	yes	—
190	C₂³	—	2	1	no	—
191	C₂³	—	2	1	no	—
192	C₂³	—	4	2	no	—
193	C₂³	#3935	1	2	yes	—
194	C₂³	C₂×#681	1	1	yes	characteristic
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	#3935	1	2	yes	—
206	C₂×C₄	C₂×#681	1	1	yes	characteristic
207	C₂×C₄	C₂×#681	1	1	yes	characteristic
208	C₂×C₄	C₂×#681	1	1	yes	characteristic
209	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
210	C₂×C₄	#4072	1	1	yes	characteristic
211	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	2	2	no	—
214	C₈	—	2	2	no	—
215	C₈	C₄×C₂²⋊D₅	1	4	yes	—
216	C₅	#12606	1	1	yes	characteristic, complemented, Sylow
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	C₂×#3935	1	1	yes	characteristic
223	C₂²	#65215	1	1	yes	characteristic
224	C₂²	#65496	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	2	no	—
228	C₄	C₄×#681	1	1	yes	characteristic
229	C₄	#65215	1	1	yes	characteristic
230	C₄	#65496	1	1	yes	characteristic
231	C₄	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	—	4	2	no	—
235	C₂	#10600677	1	2	yes	—
236	C₂	#10602322	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#405	C₂
socle	C₂²×C₅	C₂×#97
prime radical	C₂⁴×C₅	M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#405	D₅
hypocenter	C₅	#12606
LCS	G ▷ C₂×C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. C₂×#64686	C₂	semidirect
4. #10598154	C₂	semidirect
14. C₂²⋊#672	C₄	semidirect
216. C₅	#12606	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
C₂×#64686	2
#10598154	2
#12606	5