Subgroups — #12098091

Order 1280.

1090 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×#65214	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×#168	C₂	1	4	yes	maximal, complemented
5	C₂²⋊C₄×#168	C₂	1	4	yes	maximal
6	C₄×C₄⋊#168	C₂	1	4	yes	maximal, complemented
7	C₂×C₅×#117	C₄	1	2	yes	—
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂³×#168	C₂²	1	2	yes	—
10	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
11	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₄×C₅×C₂²⋊C₄	C₄	1	2	yes	—
13	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
14	C₅×#376	C₄	1	2	yes	—
15	C₄³×C₅	C₂²	1	2	yes	—
16	C₂×C₂²⋊#168	C₂²	1	2	yes	—
17	#65214	C₄	1	8	yes	complemented
18	C₂×C₄⋊#168	C₄	1	8	yes	complemented
19	C₂×C₄⋊#168	C₂²	1	2	yes	—
20	C₂×C₄⋊#168	C₂²	1	2	yes	—
21	C₂×C₄⋊#168	C₂²	1	2	yes	—
22	C₂×C₄×#168	C₂²	1	8	yes	—
23	C₂×C₄×#168	C₂²	1	2	yes	—
24	#64744	C₂²	1	8	yes	—
25	#20543	—	5	1	no	maximal, Sylow
26	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₅×#117	C₈	1	16	yes	complemented
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₅	C₂³	1	4	yes	—
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₂²×#168	—	4	4	no	—
42	C₂²×#168	C₂×C₄	1	8	yes	—
43	C₂²×#168	C₂³	1	4	yes	—
44	C₂²×#168	C₂³	1	4	yes	—
45	C₅×#95	C₂×C₄	1	4	yes	—
46	C₅×#95	C₂×C₄	1	4	yes	—
47	C₂²⋊#168	C₂×C₄	1	8	yes	—
48	C₄⋊#168	C₂×C₄	1	8	yes	—
49	C₄⋊#168	C₂×C₄	1	8	yes	—
50	C₄⋊#168	C₂×C₄	1	8	yes	—
51	C₄×#168	—	4	8	no	—
52	C₄×#168	C₂×C₄	1	8	yes	—
53	C₂×#431	—	5	1	no	—
54	C₂²×C₄⋊C₈	—	5	1	no	—
55	C₄×#117	D₅	1	1	yes	characteristic
56	C₈×C₄⋊C₄	—	5	4	no	—
57	C₄×C₄⋊C₈	—	5	4	no	—
58	C₈×C₂²⋊C₄	—	5	4	no	—
59	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	—	2	2	no	—
61	C₂²×C₄×C₅	—	2	2	no	—
62	C₂²×C₄×C₅	—	4	2	no	—
63	C₂²×C₄×C₅	C₄²	1	4	yes	—
64	C₂²×C₄×C₅	C₄²	1	4	yes	—
65	C₂²×C₄×C₅	C₂×C₈	1	8	yes	—
66	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
75	C₅×C₄⋊C₄	C₂×C₈	1	16	yes	complemented
76	C₅×C₂²⋊C₄	C₂×C₈	1	16	yes	—
77	C₄²×C₅	—	2	4	no	—
78	C₄²×C₅	—	2	4	no	—
79	C₄²×C₅	—	4	4	no	—
80	C₄²×C₅	C₄²	1	4	yes	—
81	C₄²×C₅	C₄²	1	4	yes	—
82	C₄²×C₅	C₂×C₈	1	16	yes	complemented
83	C₄²×C₅	C₂²×C₄	1	2	yes	—
84	C₄²×C₅	C₂²×C₄	1	2	yes	—
85	C₄²×C₅	C₂²×C₄	1	2	yes	—
86	C₄²×C₅	C₂²×C₄	1	2	yes	—
87	C₂×#168	—	2	4	no	—
88	C₂×#168	—	2	8	no	—
89	C₂×#168	—	2	4	no	—
90	C₂×#168	—	4	8	no	—
91	C₂×#168	C₂²×C₄	1	8	yes	—
92	C₂×#168	C₂²×C₄	1	8	yes	—
93	C₂³×C₈	—	5	2	no	—
94	C₂×#117	Dic₅	1	2	yes	—
95	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
96	#431	—	5	8	no	—
97	C₂×C₄⋊C₈	—	5	8	no	—
98	C₂×C₄⋊C₈	—	5	2	no	—
99	C₂×C₄⋊C₈	—	5	2	no	—
100	C₂×C₄⋊C₈	—	5	2	no	—
101	C₂×C₂²⋊C₈	—	5	2	no	—
102	C₂×C₄×C₈	—	5	2	no	—
103	C₂×C₄×C₈	—	5	8	no	—
104	C₄×C₄⋊C₄	Dic₅	1	2	yes	—
105	C₄×C₄⋊C₄	C₂×D₅	1	2	yes	—
106	C₄×C₂²⋊C₄	Dic₅	1	2	yes	—
107	C₄×C₂²⋊C₄	C₂×D₅	1	2	yes	—
108	#376	Dic₅	1	2	yes	—
109	C₄³	C₂×D₅	1	2	yes	—
110	#336	—	5	8	no	—
111	C₂³×C₅	—	2	2	no	—
112	C₂³×C₅	—	2	2	no	—
113	C₂³×C₅	C₂×C₄²	1	2	yes	—
114	C₂³×C₅	C₂²×C₈	1	4	yes	—
115	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₅	—	2	8	no	—
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	8	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	4	4	no	—
125	C₂×C₄×C₅	C₄×C₈	1	16	yes	complemented
126	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
130	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₅	C₂²×C₈	1	16	yes	—
132	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
133	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
134	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
137	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
138	#168	—	2	8	no	—
139	#168	—	2	8	no	—
140	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
141	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
142	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
143	C₂²×C₈	—	5	4	no	—
144	C₂²×C₈	—	5	4	no	—
145	C₂²×C₈	—	5	8	no	—
146	C₂²×C₈	—	20	4	no	—
147	#117	#168	1	16	yes	complemented
148	C₂×C₄⋊C₄	C₂×Dic₅	1	4	yes	—
149	C₂×C₂²⋊C₄	C₂×Dic₅	1	4	yes	—
150	C₂×C₄²	C₄×D₅	1	4	yes	—
151	C₂×C₄²	C₄×D₅	1	4	yes	—
152	C₂×C₄²	C₂×Dic₅	1	4	yes	—
153	C₂×C₄²	C₂×Dic₅	1	4	yes	—
154	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
155	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
156	C₂×C₄²	C₂²×D₅	1	4	yes	—
157	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
158	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
159	C₄⋊C₈	—	5	8	no	—
160	C₄⋊C₈	—	5	8	no	—
161	C₄⋊C₈	—	5	8	no	—
162	C₂²⋊C₈	—	5	8	no	—
163	C₄×C₈	—	5	8	no	—
164	C₄×C₈	—	20	8	no	—
165	#95	C₂×Dic₅	1	4	yes	—
166	#95	C₂×Dic₅	1	4	yes	—
167	C₂²×C₅	—	2	2	no	—
168	C₂²×C₅	—	2	8	no	—
169	C₂²×C₅	—	2	2	no	—
170	C₂²×C₅	C₂×C₄×C₈	1	4	yes	—
171	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
172	C₂²×C₅	#520	1	2	yes	—
173	C₂²×C₅	C₂³×C₈	1	2	yes	—
174	C₂²×C₅	#568	1	2	yes	—
175	C₄×C₅	—	2	2	no	—
176	C₄×C₅	—	2	2	no	—
177	C₄×C₅	—	2	16	no	—
178	C₄×C₅	—	2	2	no	—
179	C₄×C₅	C₂×C₄×C₈	1	4	yes	—
180	C₄×C₅	C₂×C₄×C₈	1	4	yes	—
181	C₄×C₅	#518	1	1	yes	characteristic
182	C₄×C₅	#519	1	1	yes	characteristic
183	C₄×C₅	#520	1	2	yes	—
184	C₂⁴	C₂²×Dic₅	1	1	yes	characteristic
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	C₂×#168	1	8	yes	—
189	C₂²×C₄	C₄×Dic₅	1	4	yes	—
190	C₂²×C₄	C₄×Dic₅	1	4	yes	—
191	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
192	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
193	C₂²×C₄	C₂²×Dic₅	1	8	yes	—
194	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
195	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
196	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
197	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
198	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
199	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
200	C₂×C₈	—	5	8	no	—
201	C₂×C₈	—	5	8	no	—
202	C₂×C₈	—	10	4	no	—
203	C₂×C₈	—	10	8	no	—
204	C₂×C₈	—	10	4	no	—
205	C₂×C₈	—	20	8	no	—
206	C₄⋊C₄	C₂×#168	1	16	yes	complemented
207	C₂²⋊C₄	C₂×#168	1	16	yes	—
208	C₄²	—	2	4	no	—
209	C₄²	—	2	4	no	—
210	C₄²	—	4	4	no	—
211	C₄²	C₂×#168	1	16	yes	complemented
212	C₄²	C₄×Dic₅	1	4	yes	—
213	C₄²	C₄×Dic₅	1	4	yes	—
214	C₄²	C₂×C₄×D₅	1	2	yes	—
215	C₄²	C₂×C₄×D₅	1	2	yes	—
216	C₄²	C₂×C₄×D₅	1	2	yes	—
217	C₄²	C₂×C₄×D₅	1	2	yes	—
218	C₂×C₅	—	2	4	no	—
219	C₂×C₅	#2274	1	1	yes	characteristic
220	C₂×C₅	C₂²×C₄×C₈	1	1	yes	characteristic
221	C₂×C₅	#2873	1	1	yes	characteristic
222	C₂×C₅	#2988	1	4	yes	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	C₂²×#168	1	4	yes	—
226	C₂³	C₂×C₄×Dic₅	1	2	yes	—
227	C₂³	C₂³×Dic₅	1	1	yes	characteristic
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	C₄×#168	1	16	yes	complemented
238	C₂×C₄	C₂²×#168	1	16	yes	—
239	C₂×C₄	C₂²×#168	1	4	yes	—
240	C₂×C₄	C₂²×#168	1	4	yes	—
241	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
242	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
243	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
244	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
245	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
246	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
248	C₂×C₄	#4149	1	2	yes	—
249	C₂×C₄	#4150	1	2	yes	—
250	C₈	—	10	8	no	—
251	C₈	—	10	8	no	—
252	C₅	#20543	1	1	yes	characteristic, complemented, Sylow
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	8	no	—
256	C₂²	C₂×C₄×#168	1	4	yes	—
257	C₂²	C₂³×#168	1	2	yes	—
258	C₂²	C₂²×C₄×Dic₅	1	1	yes	characteristic
259	C₂²	#66153	1	2	yes	—
260	C₂²	#66161	1	2	yes	—
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	16	no	—
265	C₄	C₂×C₄×#168	1	4	yes	—
266	C₄	C₂×C₄×#168	1	4	yes	—
267	C₄	#65850	1	1	yes	characteristic
268	C₄	#65855	1	1	yes	characteristic
269	C₄	#65857	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	C₂²×C₄×#168	1	1	yes	characteristic
272	C₂	#10613933	1	1	yes	characteristic
273	C₂	#10613954	1	1	yes	characteristic
274	C₂	#10614251	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅
commutator	C₂×C₅	C₂²×C₄×C₈
Fitting	C₄×C₅×#117	C₂
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂⁴×C₅	C₂²×C₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₄×#117	D₅
hypocenter	C₅	#20543
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
4. C₄⋊C₄×#168	C₂	semidirect
6. C₄×C₄⋊#168	C₂	semidirect
17. #65214	C₄	semidirect
18. C₂×C₄⋊#168	C₄	semidirect
29. C₅×#117	C₈	semidirect
75. C₅×C₄⋊C₄	C₂×C₈	semidirect
82. C₄²×C₅	C₂×C₈	semidirect
125. C₂×C₄×C₅	C₄×C₈	semidirect
147. #117	#168	semidirect
206. C₄⋊C₄	C₂×#168	semidirect
211. C₄²	C₂×#168	semidirect
237. C₂×C₄	C₄×#168	semidirect
252. C₅	#20543	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#117	2
C₂×#65214	2
C₂²×C₄⋊#168	2
C₄⋊C₄×#168	2
C₂²⋊C₄×#168	2
C₄×C₄⋊#168	2
#20543	5