Subgroups — #11986251

Order 1280.

386 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#336	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#672	C₂	1	2	yes	maximal
3	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
4	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
5	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
6	C₂²×#672	—	2	2	no	—
7	C₅×#336	—	2	2	no	—
8	C₂²⋊#672	—	2	2	no	—
9	C₂²⋊#672	C₄	1	4	yes	complemented
10	#7183	—	5	1	no	maximal, Sylow
11	C₂³×C₄×C₅	C₈	1	2	yes	—
12	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₅×C₈	—	2	1	no	—
14	C₂²×C₅×C₈	—	2	1	no	—
15	C₂²×C₅×C₈	—	4	1	no	—
16	C₂²×C₅×C₈	—	4	1	no	—
17	C₂²×C₅×C₈	—	4	1	no	—
18	C₂²×C₅×C₈	—	4	1	no	—
19	C₂²×C₅×C₈	C₈	1	2	yes	—
20	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	D₄	1	2	yes	—
23	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₅	—	2	1	no	—
25	C₂×C₄²×C₅	—	2	1	no	—
26	C₂×C₄²×C₅	—	4	1	no	—
27	C₂×C₄²×C₅	—	4	1	no	—
28	C₂×#672	—	2	4	no	—
29	C₂×#672	—	4	2	no	—
30	C₂×#672	—	4	2	no	—
31	C₂×C₂²⋊C₁₆	—	5	2	no	—
32	C₂×#336	D₅	1	1	yes	characteristic
33	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₅	—	2	1	no	—
35	C₂²×C₄×C₅	—	2	1	no	—
36	C₂²×C₄×C₅	—	2	1	no	—
37	C₂²×C₄×C₅	—	2	1	no	—
38	C₂²×C₄×C₅	—	2	2	no	—
39	C₂²×C₄×C₅	—	2	2	no	—
40	C₂²×C₄×C₅	—	4	2	no	—
41	C₂²×C₄×C₅	—	4	2	no	—
42	C₂²×C₄×C₅	C₁₆	1	4	yes	complemented
43	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
49	C₂×C₅×C₈	—	2	2	no	—
50	C₂×C₅×C₈	—	2	2	no	—
51	C₂×C₅×C₈	—	2	2	no	—
52	C₂×C₅×C₈	—	4	2	no	—
53	C₂×C₅×C₈	—	4	1	no	—
54	C₂×C₅×C₈	—	4	1	no	—
55	C₂×C₅×C₈	—	4	2	no	—
56	C₂×C₅×C₈	—	4	1	no	—
57	C₂×C₅×C₈	—	4	1	no	—
58	C₂×C₅×C₈	—	4	2	no	—
59	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
60	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
61	C₄²×C₅	—	4	2	no	—
62	C₄²×C₅	—	4	1	no	—
63	C₄²×C₅	—	4	1	no	—
64	#672	—	4	4	no	—
65	C₂³×C₈	Dic₅	1	1	yes	characteristic
66	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
67	C₂²×C₄²	Dic₅	1	1	yes	characteristic
68	C₂²×C₁₆	—	10	2	no	—
69	C₂²⋊C₁₆	—	5	4	no	—
70	C₂²⋊C₁₆	—	10	2	no	—
71	#336	—	2	2	no	—
72	C₂³×C₅	—	2	1	no	—
73	C₂³×C₅	—	2	1	no	—
74	C₂³×C₅	—	2	2	no	—
75	C₂³×C₅	C₄×C₈	1	1	yes	characteristic
76	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
77	C₂³×C₅	C₄⋊C₈	1	1	yes	characteristic
78	C₂³×C₅	C₂×C₁₆	1	2	yes	—
79	C₂³×C₅	M₃₂	1	2	yes	—
80	C₂×C₄×C₅	—	2	1	no	—
81	C₂×C₄×C₅	—	2	1	no	—
82	C₂×C₄×C₅	—	2	1	no	—
83	C₂×C₄×C₅	—	2	1	no	—
84	C₂×C₄×C₅	—	2	1	no	—
85	C₂×C₄×C₅	—	2	1	no	—
86	C₂×C₄×C₅	—	2	1	no	—
87	C₂×C₄×C₅	—	2	1	no	—
88	C₂×C₄×C₅	—	2	1	no	—
89	C₂×C₄×C₅	—	2	1	no	—
90	C₂×C₄×C₅	—	2	4	no	—
91	C₂×C₄×C₅	—	4	4	no	—
92	C₂×C₄×C₅	—	4	2	no	—
93	C₂×C₄×C₅	—	4	2	no	—
94	C₂×C₄×C₅	—	4	4	no	—
95	C₂×C₄×C₅	#95	1	1	yes	characteristic
96	C₂×C₄×C₅	#97	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₅	#99	1	1	yes	characteristic
99	C₂×C₄×C₅	#99	1	1	yes	characteristic
100	C₂×C₄×C₅	#100	1	1	yes	characteristic
101	C₂×C₄×C₅	#101	1	1	yes	characteristic
102	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
103	C₅×C₈	—	2	2	no	—
104	C₅×C₈	—	2	2	no	—
105	C₅×C₈	—	4	2	no	—
106	C₂³×C₄	#168	1	2	yes	—
107	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
108	C₂²×C₈	—	2	1	no	—
109	C₂²×C₈	—	2	1	no	—
110	C₂²×C₈	—	4	1	no	—
111	C₂²×C₈	—	4	1	no	—
112	C₂²×C₈	—	4	1	no	—
113	C₂²×C₈	—	4	1	no	—
114	C₂²×C₈	#168	1	2	yes	—
115	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
116	C₂²×C₈	C₄×D₅	1	2	yes	—
117	C₂²×C₈	D₂₀	1	1	yes	characteristic
118	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
119	C₂×C₄²	—	2	1	no	—
120	C₂×C₄²	—	2	1	no	—
121	C₂×C₄²	—	4	1	no	—
122	C₂×C₄²	—	4	1	no	—
123	C₂×C₁₆	—	10	4	no	—
124	C₂×C₁₆	—	20	2	no	—
125	C₂×C₁₆	—	20	2	no	—
126	C₂²×C₅	—	2	1	no	—
127	C₂²×C₅	—	2	1	no	—
128	C₂²×C₅	—	2	2	no	—
129	C₂²×C₅	—	2	2	no	—
130	C₂²×C₅	—	2	2	no	—
131	C₂²×C₅	—	2	2	no	—
132	C₂²×C₅	—	2	1	no	—
133	C₂²×C₅	—	2	1	no	—
134	C₂²×C₅	#323	1	1	yes	characteristic
135	C₂²×C₅	#324	1	1	yes	characteristic
136	C₂²×C₅	#336	1	1	yes	characteristic
137	C₂²×C₅	C₄×C₁₆	1	1	yes	characteristic
138	C₂²×C₅	#346	1	1	yes	characteristic
139	C₂²×C₅	C₂²⋊C₁₆	1	2	yes	—
140	C₂²×C₅	#349	1	2	yes	—
141	C₂²×C₅	C₄⋊C₁₆	1	2	yes	—
142	C₄×C₅	—	2	1	no	—
143	C₄×C₅	—	2	1	no	—
144	C₄×C₅	—	4	4	no	—
145	C₄×C₅	#323	1	1	yes	characteristic
146	C₄×C₅	#324	1	1	yes	characteristic
147	C₄×C₅	#342	1	1	yes	characteristic
148	C₄×C₅	#343	1	1	yes	characteristic
149	C₂⁴	C₂×#168	1	1	yes	characteristic
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	#672	1	4	yes	complemented
159	C₂²×C₄	C₂×#168	1	1	yes	characteristic
160	C₂²×C₄	#681	1	1	yes	characteristic
161	C₂²×C₄	#681	1	1	yes	characteristic
162	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
163	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
164	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	4	1	no	—
173	C₂×C₈	—	4	1	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	#683	1	2	yes	—
176	C₂×C₈	#685	1	2	yes	—
177	C₄²	—	4	2	no	—
178	C₄²	—	4	1	no	—
179	C₄²	—	4	1	no	—
180	C₁₆	—	20	4	no	—
181	C₂×C₅	—	2	1	no	—
182	C₂×C₅	—	2	1	no	—
183	C₂×C₅	—	2	2	no	—
184	C₂×C₅	#1280	1	1	yes	characteristic
185	C₂×C₅	#1314	1	2	yes	—
186	C₂×C₅	#1322	1	2	yes	—
187	C₂×C₅	#1374	1	1	yes	characteristic
188	C₂×C₅	#1377	1	1	yes	characteristic
189	C₂³	—	2	2	no	—
190	C₂³	—	2	1	no	—
191	C₂³	—	2	1	no	—
192	C₂³	C₄×#168	1	1	yes	characteristic
193	C₂³	C₄⋊#168	1	1	yes	characteristic
194	C₂³	C₂×#672	1	2	yes	—
195	C₂³	#3944	1	2	yes	—
196	C₂³	C₂²⋊#168	1	1	yes	characteristic
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	#3935	1	1	yes	characteristic
213	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
214	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
215	C₂×C₄	#3963	1	1	yes	characteristic
216	C₂×C₄	#3965	1	1	yes	characteristic
217	C₂×C₄	#3966	1	1	yes	characteristic
218	C₂×C₄	#3966	1	1	yes	characteristic
219	C₂×C₄	#3968	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	4	2	no	—
223	C₅	#7183	1	1	yes	characteristic, complemented, Sylow
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	C₄×#672	1	1	yes	characteristic
233	C₂²	#64681	1	1	yes	characteristic
234	C₂²	C₄⋊#672	1	2	yes	—
235	C₂²	#64744	1	1	yes	characteristic
236	C₂²	#64745	1	1	yes	characteristic
237	C₂²	#64748	1	1	yes	characteristic
238	C₂²	C₂²⋊#672	1	2	yes	—
239	C₂²	#64773	1	2	yes	—
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	4	4	no	—
243	C₄	#64745	1	1	yes	characteristic
244	C₄	#64746	1	1	yes	characteristic
245	C₄	#64748	1	1	yes	characteristic
246	C₄	#64750	1	1	yes	characteristic
247	C₂	—	2	1	no	—
248	C₂	—	2	1	no	—
249	C₂	—	2	2	no	—
250	C₂	#10598104	1	1	yes	characteristic
251	C₂	#10598148	1	1	yes	characteristic
252	C₂	#10598155	1	1	yes	characteristic
253	C₂	#10598164	1	2	yes	—
254	C₂	#10598200	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₅
commutator	C₂²×C₅	C₄×C₁₆
Frattini	C₂³×C₈	C₂×D₅
Fitting	C₂×C₅×#336	C₂
socle	C₂³×C₅	C₂²⋊C₈
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂³×C₅×C₈	C₂²
hypercenter	C₂×#336	D₅
hypocenter	C₅	#7183
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
9. C₂²⋊#672	C₄	semidirect
42. C₂²×C₄×C₅	C₁₆	semidirect
158. C₂²×C₄	#672	semidirect
223. C₅	#7183	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#336	2
C₂×C₂²⋊#672	2
#7183	5