Subgroups — #12016195

Order 1280.

508 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2929	C₂	1	1	yes	maximal, characteristic
2	C₂³×#672	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#64781	C₂	1	2	yes	maximal, complemented
4	#10602756	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598162	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#131	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×#525	C₄	1	1	yes	characteristic
9	C₅×#435	—	2	2	no	—
10	C₅×C₈×D₄	—	2	2	no	—
11	C₅×#424	C₄	1	1	yes	characteristic
12	C₅×#408	C₄	1	1	yes	characteristic
13	C₅×#407	C₄	1	1	yes	characteristic
14	C₅×#405	C₂²	1	1	yes	characteristic
15	C₂×#3944	C₂²	1	2	yes	—
16	C₂²×#672	—	2	4	no	—
17	C₂²×#672	—	2	2	no	—
18	C₂²×#672	C₂²	1	2	yes	—
19	#64781	—	2	4	no	—
20	C₂²⋊#672	—	2	2	no	—
21	#12983	—	5	1	no	maximal, Sylow
22	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₅×#131	—	2	2	no	—
25	C₅×#131	—	2	2	no	—
26	C₂×C₅×M₁₆	—	2	1	no	—
27	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	—	2	1	no	—
32	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	D₄	1	2	yes	—
34	C₂²×C₅×C₈	D₄	1	2	yes	—
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₅×#123	—	2	1	no	—
37	C₅×C₂²⋊Q₈	C₈	1	2	yes	—
38	C₅×#121	C₈	1	2	yes	—
39	C₅×#120	—	2	1	no	—
40	C₄×C₅×D₄	—	2	2	no	—
41	C₅×#117	C₂×C₄	1	1	yes	characteristic
42	C₅×C₄⋊C₈	—	2	2	no	—
43	C₅×C₂²⋊C₈	—	2	2	no	—
44	C₅×C₂²⋊C₈	—	2	2	no	—
45	C₅×#97	—	2	1	no	—
46	C₄×C₅×C₈	—	2	1	no	—
47	#3944	—	2	2	no	—
48	#3944	—	2	2	no	—
49	C₂×#672	—	2	8	no	—
50	C₂×#672	—	2	2	no	—
51	C₂×#672	—	2	2	no	—
52	C₂×#672	—	2	2	no	—
53	C₂×#672	D₄	1	4	yes	complemented
54	C₂³×C₁₆	—	5	1	no	—
55	#2929	D₅	1	1	yes	characteristic
56	C₂×#350	—	5	2	no	—
57	#2112	—	5	1	no	—
58	#1379	—	5	2	no	—
59	C₂⁴×C₅	M₁₆	1	1	yes	characteristic
60	C₅×#41	—	2	2	no	—
61	C₅×#41	—	2	2	no	—
62	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
63	C₂×C₅×D₄	—	2	2	no	—
64	C₂×C₅×D₄	—	2	1	no	—
65	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₅	—	2	2	no	—
67	C₂²×C₄×C₅	—	2	2	no	—
68	C₂²×C₄×C₅	—	2	1	no	—
69	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₅×M₁₆	—	2	2	no	—
74	C₅×M₁₆	—	2	2	no	—
75	C₅×M₁₆	—	4	1	no	—
76	C₅×M₁₆	—	4	1	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	1	no	—
82	C₂×C₅×C₈	—	2	1	no	—
83	C₂×C₅×C₈	—	4	2	no	—
84	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
85	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
86	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₅×C₈	#41	1	2	yes	—
89	C₅×C₄⋊C₄	—	2	2	no	—
90	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
91	C₅×C₂²⋊C₄	—	2	2	no	—
92	C₅×C₂²⋊C₄	—	2	1	no	—
93	C₅×C₂²⋊C₄	—	2	2	no	—
94	C₄²×C₅	—	2	1	no	—
95	#672	—	2	4	no	—
96	#672	—	4	2	no	—
97	C₂×#131	C₂×D₅	1	1	yes	characteristic
98	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
99	#525	Dic₅	1	1	yes	characteristic
100	C₂×M₃₂	—	5	2	no	—
101	C₂²×C₁₆	—	5	2	no	—
102	C₂²×C₁₆	—	10	4	no	—
103	C₂²×C₁₆	—	10	2	no	—
104	#435	—	2	2	no	—
105	C₈×D₄	—	2	2	no	—
106	#424	Dic₅	1	1	yes	characteristic
107	#408	Dic₅	1	1	yes	characteristic
108	#407	Dic₅	1	1	yes	characteristic
109	#405	C₂×D₅	1	1	yes	characteristic
110	#350	—	10	4	no	—
111	C₂²⋊C₁₆	—	10	2	no	—
112	C₂³×C₅	—	2	4	no	—
113	C₂³×C₅	—	2	2	no	—
114	C₂³×C₅	—	2	1	no	—
115	C₂³×C₅	C₂²⋊C₈	1	2	yes	—
116	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
117	C₅×Q₈	—	2	2	no	—
118	C₅×D₄	—	2	2	no	—
119	C₅×D₄	—	4	4	no	—
120	C₅×D₄	—	4	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	1	no	—
127	C₂×C₄×C₅	—	4	1	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
130	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₅	#131	1	2	yes	—
134	C₅×C₈	—	2	2	no	—
135	C₅×C₈	—	4	1	no	—
136	C₅×C₈	—	4	1	no	—
137	C₅×C₈	#120	1	1	yes	characteristic
138	C₅×C₈	#121	1	2	yes	—
139	C₅×C₈	#123	1	1	yes	characteristic
140	C₂×#41	C₂×Dic₅	1	1	yes	characteristic
141	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
142	#131	—	2	2	no	—
143	#131	—	2	2	no	—
144	C₂×M₁₆	—	2	1	no	—
145	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
146	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
151	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
152	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
153	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
154	#123	—	2	1	no	—
155	C₂²⋊Q₈	#168	1	2	yes	—
156	#121	#168	1	2	yes	—
157	#120	—	2	1	no	—
158	C₄×D₄	—	2	2	no	—
159	#117	C₂×Dic₅	1	1	yes	characteristic
160	M₃₂	—	10	2	no	—
161	M₃₂	—	10	2	no	—
162	C₂×C₁₆	—	5	4	no	—
163	C₂×C₁₆	—	10	8	no	—
164	C₂×C₁₆	—	10	2	no	—
165	C₂×C₁₆	—	10	2	no	—
166	C₂×C₁₆	—	10	2	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₂²⋊C₈	—	2	2	no	—
169	C₂²⋊C₈	—	2	2	no	—
170	#97	—	2	1	no	—
171	C₄×C₈	—	2	1	no	—
172	C₂²×C₅	—	2	8	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	—	4	2	no	—
177	C₂²×C₅	—	4	1	no	—
178	C₂²×C₅	#350	1	4	yes	complemented
179	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
180	C₂²×C₅	#435	1	2	yes	—
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	4	2	no	—
183	C₄×C₅	—	4	1	no	—
184	C₄×C₅	#386	1	1	yes	characteristic
185	C₄×C₅	#408	1	1	yes	characteristic
186	C₄×C₅	C₈×D₄	1	2	yes	—
187	C₂⁴	#681	1	1	yes	characteristic
188	#41	—	2	2	no	—
189	#41	—	2	2	no	—
190	C₂×Q₈	C₂×#168	1	1	yes	characteristic
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	C₂×#168	1	1	yes	characteristic
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	C₂×#168	1	2	yes	—
198	C₂²×C₄	#681	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
200	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	4	1	no	—
204	M₁₆	—	4	1	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
213	C₂×C₈	D₄×D₅	1	2	yes	—
214	C₂×C₈	#711	1	2	yes	—
215	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
216	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	C₂×#168	1	2	yes	—
219	C₂²⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	2	1	no	—
221	C₂²⋊C₄	—	2	2	no	—
222	C₄²	—	2	1	no	—
223	C₁₆	—	10	4	no	—
224	C₁₆	—	20	2	no	—
225	C₂×C₅	—	2	4	no	—
226	C₂×C₅	—	2	2	no	—
227	C₂×C₅	—	8	1	no	—
228	C₂×C₅	#1869	1	1	yes	characteristic
229	C₂×C₅	C₂×#350	1	2	yes	—
230	C₂³	—	2	4	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	1	no	—
233	C₂³	C₂²⋊#168	1	2	yes	—
234	C₂³	C₂×#681	1	1	yes	characteristic
235	Q₈	—	2	2	no	—
236	D₄	—	2	2	no	—
237	D₄	—	4	4	no	—
238	D₄	—	4	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	C₂²⋊#168	1	2	yes	—
248	C₂×C₄	C₂²×#168	1	1	yes	characteristic
249	C₂×C₄	D₄×Dic₅	1	2	yes	—
250	C₂×C₄	#4094	1	2	yes	—
251	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	4	1	no	—
254	C₈	—	4	1	no	—
255	C₈	#4081	1	1	yes	characteristic
256	C₈	#4083	1	1	yes	characteristic
257	C₈	#4085	1	2	yes	—
258	C₅	#12983	1	1	yes	characteristic, complemented, Sylow
259	C₂²	—	2	8	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	#64781	1	4	yes	complemented
266	C₂²	#65300	1	2	yes	—
267	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	1	no	—
271	C₄	D₄×#168	1	2	yes	—
272	C₄	#65509	1	1	yes	characteristic
273	C₄	#65523	1	1	yes	characteristic
274	C₂	—	2	4	no	—
275	C₂	—	2	2	no	—
276	C₂	—	8	1	no	—
277	C₂	#10602403	1	1	yes	characteristic
278	C₂	C₂×#64781	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#2929	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₅×#525	C₄
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#2929	D₅
hypocenter	C₅	#12983
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ #2929

Complements

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

N	H ≅ G/N	type
2. C₂³×#672	C₂	semidirect
3. C₂×#64781	C₂	semidirect
4. #10602756	C₂	semidirect
5. #10598162	C₂	semidirect
53. C₂×#672	D₄	semidirect
178. C₂²×C₅	#350	semidirect
258. C₅	#12983	semidirect
265. C₂²	#64781	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2929	2
C₂³×#672	2
C₂×#64781	2
#10602756	2
#10598162	2
#12983	5