Subgroups — #11986250

Order 1280.

386 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#64770	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊#672	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
5	C₂²×C₄×Dic₅	C₄	1	1	yes	characteristic
6	C₂³×#168	C₄	1	1	yes	characteristic
7	C₂²×C₅×C₁₆	—	2	1	no	—
8	C₂²×#672	—	2	1	no	—
9	C₅×C₂²⋊C₁₆	—	2	1	no	—
10	C₅×C₂²⋊C₁₆	C₄	1	2	yes	complemented
11	#64770	—	2	1	no	—
12	#64770	—	2	1	no	—
13	C₂²⋊#672	—	2	1	no	—
14	C₂²⋊#672	C₄	1	2	yes	complemented
15	#7183	—	5	1	no	maximal, Sylow
16	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂³×Dic₅	C₈	1	2	yes	—
18	C₂²×C₅×C₈	—	4	1	no	—
19	C₂²×C₅×C₈	—	4	1	no	—
20	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
26	C₂×C₄×Dic₅	—	2	1	no	—
27	C₂×C₄×Dic₅	—	2	1	no	—
28	C₂×C₄×Dic₅	—	4	1	no	—
29	C₂×C₄×Dic₅	—	4	1	no	—
30	C₂²×#168	—	2	1	no	—
31	C₂²×#168	—	2	1	no	—
32	C₂²×#168	—	4	1	no	—
33	C₂²×#168	—	4	1	no	—
34	C₂²×#168	C₈	1	2	yes	—
35	C₂×C₅×C₁₆	—	2	2	no	—
36	C₂×C₅×C₁₆	—	4	1	no	—
37	C₂×C₅×C₁₆	—	4	1	no	—
38	C₂×#672	—	2	2	no	—
39	C₂×#672	—	4	1	no	—
40	C₂×#672	—	4	1	no	—
41	C₂×C₂²⋊C₁₆	—	5	1	no	—
42	C₂×C₂²⋊C₁₆	D₅	1	1	yes	characteristic
43	C₂×#336	—	5	1	no	—
44	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₅	—	2	1	no	—
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	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
55	C₂²×Dic₅	—	2	2	no	—
56	C₂²×Dic₅	—	2	2	no	—
57	C₂²×Dic₅	—	4	2	no	—
58	C₂²×Dic₅	—	4	2	no	—
59	C₂²×Dic₅	C₁₆	1	4	yes	complemented
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	1	no	—
64	C₂×C₅×C₈	—	4	1	no	—
65	C₂×C₅×C₈	—	4	1	no	—
66	C₂×C₅×C₈	—	4	1	no	—
67	C₂×C₅×C₈	—	4	1	no	—
68	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
72	C₄×Dic₅	—	4	2	no	—
73	C₄×Dic₅	—	4	1	no	—
74	C₄×Dic₅	—	4	1	no	—
75	C₂×#168	—	2	2	no	—
76	C₂×#168	—	4	2	no	—
77	C₂×#168	—	4	1	no	—
78	C₂×#168	—	4	1	no	—
79	C₂×#168	—	4	2	no	—
80	C₅×C₁₆	—	4	2	no	—
81	#672	—	4	2	no	—
82	C₂³×C₈	—	5	1	no	—
83	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
84	C₂²×C₄²	—	5	1	no	—
85	C₂²×C₁₆	—	2	1	no	—
86	C₂²×C₁₆	—	10	1	no	—
87	C₂²⋊C₁₆	—	2	1	no	—
88	C₂²⋊C₁₆	—	5	2	no	—
89	C₂²⋊C₁₆	—	10	1	no	—
90	C₂²⋊C₁₆	Dic₅	1	2	yes	complemented
91	#336	—	10	1	no	—
92	#336	—	10	1	no	—
93	C₂³×C₅	—	2	1	no	—
94	C₂³×C₅	—	2	1	no	—
95	C₂³×C₅	—	2	2	no	—
96	C₂³×C₅	C₄×C₈	1	1	yes	characteristic
97	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
98	C₂³×C₅	C₄⋊C₈	1	1	yes	characteristic
99	C₂³×C₅	C₂×C₁₆	1	2	yes	—
100	C₂³×C₅	M₃₂	1	2	yes	—
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	#95	1	1	yes	characteristic
112	C₂×C₄×C₅	#97	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
114	C₂×C₄×C₅	#99	1	1	yes	characteristic
115	C₂×C₄×C₅	#99	1	1	yes	characteristic
116	C₂×C₄×C₅	#100	1	1	yes	characteristic
117	C₂×C₄×C₅	#101	1	1	yes	characteristic
118	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
119	C₂×Dic₅	—	2	4	no	—
120	C₂×Dic₅	—	4	4	no	—
121	C₂×Dic₅	—	4	2	no	—
122	C₂×Dic₅	—	4	2	no	—
123	C₂×Dic₅	—	4	4	no	—
124	C₅×C₈	—	2	1	no	—
125	C₅×C₈	—	2	1	no	—
126	C₅×C₈	—	2	1	no	—
127	C₅×C₈	—	2	1	no	—
128	#168	—	4	2	no	—
129	C₂³×C₄	—	5	2	no	—
130	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
131	C₂²×C₈	—	4	1	no	—
132	C₂²×C₈	—	4	1	no	—
133	C₂²×C₈	—	5	2	no	—
134	C₂²×C₈	—	10	1	no	—
135	C₂²×C₈	—	10	1	no	—
136	C₂²×C₈	—	20	1	no	—
137	C₂²×C₈	—	20	1	no	—
138	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
139	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
140	C₂²×C₈	D₂₀	1	1	yes	characteristic
141	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
142	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
143	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
144	C₂×C₄²	—	10	1	no	—
145	C₂×C₄²	—	10	1	no	—
146	C₂×C₄²	—	20	1	no	—
147	C₂×C₄²	—	20	1	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	4	1	no	—
150	C₂×C₁₆	—	4	1	no	—
151	C₂×C₁₆	—	10	2	no	—
152	C₂×C₁₆	—	20	1	no	—
153	C₂×C₁₆	—	20	1	no	—
154	C₂²×C₅	—	2	1	no	—
155	C₂²×C₅	—	2	1	no	—
156	C₂²×C₅	—	2	2	no	—
157	C₂²×C₅	—	2	2	no	—
158	C₂²×C₅	—	2	2	no	—
159	C₂²×C₅	—	2	2	no	—
160	C₂²×C₅	—	2	1	no	—
161	C₂²×C₅	—	2	1	no	—
162	C₂²×C₅	#323	1	1	yes	characteristic
163	C₂²×C₅	#324	1	1	yes	characteristic
164	C₂²×C₅	#336	1	1	yes	characteristic
165	C₂²×C₅	C₄×C₁₆	1	1	yes	characteristic
166	C₂²×C₅	#346	1	1	yes	characteristic
167	C₂²×C₅	C₂²⋊C₁₆	1	2	yes	—
168	C₂²×C₅	#349	1	2	yes	—
169	C₂²×C₅	C₄⋊C₁₆	1	2	yes	—
170	C₄×C₅	—	2	1	no	—
171	C₄×C₅	—	2	1	no	—
172	C₄×C₅	#323	1	1	yes	characteristic
173	C₄×C₅	#324	1	1	yes	characteristic
174	C₄×C₅	#342	1	1	yes	characteristic
175	C₄×C₅	#343	1	1	yes	characteristic
176	Dic₅	—	4	4	no	—
177	C₂⁴	C₈×D₅	1	1	yes	characteristic
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	5	4	no	—
183	C₂²×C₄	—	10	2	no	—
184	C₂²×C₄	—	10	2	no	—
185	C₂²×C₄	—	20	2	no	—
186	C₂²×C₄	—	20	2	no	—
187	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
188	C₂²×C₄	#676	1	1	yes	characteristic
189	C₂²×C₄	#676	1	1	yes	characteristic
190	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
191	C₂²×C₄	#683	1	1	yes	characteristic
192	C₂²×C₄	#685	1	1	yes	characteristic
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	10	2	no	—
202	C₂×C₈	—	20	2	no	—
203	C₂×C₈	—	20	1	no	—
204	C₂×C₈	—	20	1	no	—
205	C₂×C₈	—	20	2	no	—
206	C₂×C₈	#683	1	1	yes	characteristic
207	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
208	C₂×C₈	#685	1	1	yes	characteristic
209	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
210	C₄²	—	20	2	no	—
211	C₄²	—	20	1	no	—
212	C₄²	—	20	1	no	—
213	C₁₆	—	4	2	no	—
214	C₁₆	—	20	2	no	—
215	C₂×C₅	—	2	1	no	—
216	C₂×C₅	—	2	1	no	—
217	C₂×C₅	—	2	2	no	—
218	C₂×C₅	#1280	1	1	yes	characteristic
219	C₂×C₅	#1314	1	2	yes	—
220	C₂×C₅	#1322	1	2	yes	—
221	C₂×C₅	#1374	1	1	yes	characteristic
222	C₂×C₅	#1377	1	1	yes	characteristic
223	C₂³	—	2	1	no	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	2	no	—
226	C₂³	D₅×C₁₆	1	2	yes	—
227	C₂³	#3930	1	2	yes	—
228	C₂³	C₈×Dic₅	1	1	yes	characteristic
229	C₂³	#3946	1	1	yes	characteristic
230	C₂³	#3952	1	1	yes	characteristic
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	10	4	no	—
242	C₂×C₄	—	20	4	no	—
243	C₂×C₄	—	20	2	no	—
244	C₂×C₄	—	20	2	no	—
245	C₂×C₄	—	20	4	no	—
246	C₂×C₄	#3938	1	1	yes	characteristic
247	C₂×C₄	#3938	1	1	yes	characteristic
248	C₂×C₄	#3946	1	1	yes	characteristic
249	C₂×C₄	#3947	1	1	yes	characteristic
250	C₂×C₄	#3952	1	1	yes	characteristic
251	C₂×C₄	#3955	1	1	yes	characteristic
252	C₂×C₄	#3956	1	1	yes	characteristic
253	C₂×C₄	#3963	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	20	2	no	—
259	C₅	#7183	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	#64688	1	1	yes	characteristic
269	C₂²	#64689	1	1	yes	characteristic
270	C₂²	C₁₆×Dic₅	1	1	yes	characteristic
271	C₂²	#64721	1	2	yes	—
272	C₂²	#64722	1	1	yes	characteristic
273	C₂²	#64727	1	2	yes	—
274	C₂²	#64734	1	2	yes	—
275	C₂²	#64770	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	20	4	no	—
279	C₄	#64688	1	1	yes	characteristic
280	C₄	#64689	1	1	yes	characteristic
281	C₄	#64747	1	1	yes	characteristic
282	C₄	#64774	1	1	yes	characteristic
283	C₂	—	2	1	no	—
284	C₂	—	2	2	no	—
285	C₂	—	2	1	no	—
286	C₂	#10598013	1	2	yes	—
287	C₂	#10598015	1	2	yes	—
288	C₂	#10598165	1	1	yes	characteristic
289	C₂	#10598295	1	1	yes	characteristic
290	C₂	#10598303	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#685
commutator	C₂²×C₅	C₄×C₁₆
Frattini	C₂³×C₈	C₂×D₅
Fitting	C₂×C₅×C₂²⋊C₁₆	C₂
socle	C₂³×C₅	C₂²⋊C₈
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂³×C₅×C₈	C₂²
hypercenter	C₂×C₂²⋊C₁₆	D₅
hypocenter	C₅	#7183
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
10. C₅×C₂²⋊C₁₆	C₄	semidirect
14. C₂²⋊#672	C₄	semidirect
59. C₂²×Dic₅	C₁₆	semidirect
90. C₂²⋊C₁₆	Dic₅	semidirect
259. 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×C₂²⋊C₁₆	2
C₂×#64770	2
C₂×C₂²⋊#672	2
#7183	5