Subgroups — #11997877

Order 1280.

404 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#64699	C₂	1	2	yes	maximal
3	#10600187	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
5	#10598194	C₂	1	2	yes	maximal
6	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	—	2	1	no	—
9	C₄³×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#168	—	2	1	no	—
11	C₂×C₄⋊#168	—	2	1	no	—
12	C₂×C₄⋊#168	C₄	1	1	yes	characteristic
13	C₂×C₄⋊#168	C₄	1	1	yes	characteristic
14	C₂×C₄⋊#168	C₂²	1	2	yes	—
15	C₂×C₄×#168	C₂²	1	2	yes	—
16	C₅×#336	—	2	1	no	—
17	#64744	C₄	1	2	yes	—
18	#64699	—	2	4	no	—
19	#9601	—	5	1	no	maximal, Sylow
20	C₂²×C₅×C₈	—	2	1	no	—
21	C₂²×C₅×C₈	—	2	1	no	—
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
28	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₂²×#168	—	2	2	no	—
30	C₂²×#168	C₂×C₄	1	2	yes	—
31	C₂²×#168	C₂×C₄	1	2	yes	—
32	C₅×C₄⋊C₈	—	2	1	no	—
33	C₅×C₄⋊C₈	—	2	1	no	—
34	C₅×C₄⋊C₈	D₄	1	2	yes	—
35	C₅×C₄⋊C₈	Q₈	1	2	yes	—
36	C₄×C₅×C₈	—	4	2	no	—
37	C₄⋊#168	—	2	2	no	—
38	C₄⋊#168	—	2	2	no	—
39	C₄⋊#168	—	2	2	no	—
40	C₄⋊#168	—	4	2	no	—
41	C₄⋊#168	—	4	2	no	—
42	C₄⋊#168	C₈	1	4	yes	complemented
43	C₄×#168	—	2	2	no	—
44	C₄×#168	—	2	2	no	—
45	#1842	—	5	1	no	—
46	C₄×C₄⋊C₈	—	5	1	no	—
47	C₄×C₄⋊C₈	D₅	1	1	yes	characteristic
48	C₂×#335	—	5	2	no	—
49	#1277	—	5	2	no	—
50	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	4	1	no	—
60	C₂×C₅×C₈	—	4	1	no	—
61	C₂×C₅×C₈	—	4	1	no	—
62	C₂×C₅×C₈	—	4	1	no	—
63	C₄²×C₅	—	2	2	no	—
64	C₄²×C₅	—	2	1	no	—
65	C₄²×C₅	—	2	1	no	—
66	C₄²×C₅	—	2	1	no	—
67	C₄²×C₅	—	2	1	no	—
68	C₄²×C₅	—	4	1	no	—
69	C₄²×C₅	C₂×C₈	1	2	yes	—
70	C₄²×C₅	M₁₆	1	2	yes	—
71	C₄²×C₅	D₈	1	2	yes	—
72	C₄²×C₅	Q₁₆	1	2	yes	—
73	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
75	C₄²×C₅	#41	1	1	yes	characteristic
76	C₄²×C₅	#41	1	1	yes	characteristic
77	C₂×#168	—	2	2	no	—
78	C₂×#168	—	2	2	no	—
79	C₂×#168	—	2	2	no	—
80	C₂×#168	—	2	2	no	—
81	C₂×#168	—	2	2	no	—
82	C₂×#168	—	4	2	no	—
83	C₂×#168	—	4	2	no	—
84	C₂×#168	—	4	2	no	—
85	C₂×#168	C₂×C₈	1	4	yes	—
86	C₂×C₄⋊C₈	—	5	1	no	—
87	C₂×C₄⋊C₈	—	5	1	no	—
88	C₂×C₄⋊C₈	—	5	2	no	—
89	C₂×C₄⋊C₈	—	10	1	no	—
90	C₂×C₄⋊C₈	—	10	1	no	—
91	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
92	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
93	C₂×C₄×C₈	—	2	1	no	—
94	C₂×C₄×C₈	—	5	2	no	—
95	C₄³	C₂×D₅	1	1	yes	characteristic
96	#336	—	2	1	no	—
97	#336	—	5	2	no	—
98	#335	—	10	4	no	—
99	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅	—	2	1	no	—
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	C₄⋊C₈	1	2	yes	—
110	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
111	C₂×C₄×C₅	#107	1	2	yes	—
112	C₂×C₄×C₅	#107	1	2	yes	—
113	C₂×C₄×C₅	#108	1	2	yes	—
114	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅	#121	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₅	#125	1	1	yes	characteristic
119	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₅	#131	1	1	yes	characteristic
123	C₂×C₄×C₅	#131	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₅	#137	1	1	yes	characteristic
128	C₅×C₈	—	4	2	no	—
129	C₅×C₈	—	4	2	no	—
130	#168	—	2	4	no	—
131	#168	—	4	2	no	—
132	#168	—	8	2	no	—
133	C₂²×C₈	—	2	1	no	—
134	C₂²×C₈	—	2	1	no	—
135	C₂²×C₈	—	5	2	no	—
136	C₂²×C₈	—	5	2	no	—
137	C₂²×C₈	—	10	2	no	—
138	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
139	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
140	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
141	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
142	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
143	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
144	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
145	C₄⋊C₈	—	2	1	no	—
146	C₄⋊C₈	—	2	1	no	—
147	C₄⋊C₈	—	5	4	no	—
148	C₄⋊C₈	—	10	2	no	—
149	C₄⋊C₈	—	10	2	no	—
150	C₄⋊C₈	—	10	2	no	—
151	C₄⋊C₈	—	20	2	no	—
152	C₄⋊C₈	—	20	2	no	—
153	C₄⋊C₈	Dic₁₀	1	2	yes	—
154	C₄⋊C₈	D₂₀	1	2	yes	—
155	C₄×C₈	—	4	2	no	—
156	C₄×C₈	—	10	2	no	—
157	C₄×C₈	—	10	2	no	—
158	C₂²×C₅	#389	1	1	yes	characteristic
159	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
160	C₂²×C₅	#424	1	1	yes	characteristic
161	C₂²×C₅	C₂×#107	1	1	yes	characteristic
162	C₂²×C₅	#428	1	1	yes	characteristic
163	C₂²×C₅	C₂×#108	1	1	yes	characteristic
164	C₂²×C₅	#430	1	1	yes	characteristic
165	C₄×C₅	—	2	2	no	—
166	C₄×C₅	—	2	2	no	—
167	C₄×C₅	—	2	1	no	—
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	—	4	1	no	—
170	C₄×C₅	#335	1	4	yes	complemented
171	C₄×C₅	C₈×D₄	1	1	yes	characteristic
172	C₄×C₅	#436	1	1	yes	characteristic
173	C₄×C₅	C₈×Q₈	1	1	yes	characteristic
174	C₄×C₅	#446	1	1	yes	characteristic
175	C₄×C₅	#459	1	1	yes	characteristic
176	C₄×C₅	#462	1	1	yes	characteristic
177	C₄×C₅	#474	1	1	yes	characteristic
178	C₄×C₅	#477	1	1	yes	characteristic
179	C₂²×C₄	#683	1	1	yes	characteristic
180	C₂²×C₄	#683	1	1	yes	characteristic
181	C₂²×C₄	#683	1	1	yes	characteristic
182	C₂²×C₄	#683	1	1	yes	characteristic
183	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	4	1	no	—
189	C₂×C₈	—	4	1	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	—	4	1	no	—
192	C₂×C₈	—	5	4	no	—
193	C₂×C₈	—	10	2	no	—
194	C₂×C₈	—	10	2	no	—
195	C₂×C₈	—	10	2	no	—
196	C₂×C₈	—	10	2	no	—
197	C₂×C₈	—	10	2	no	—
198	C₂×C₈	—	20	2	no	—
199	C₂×C₈	—	20	2	no	—
200	C₂×C₈	—	20	2	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	4	1	no	—
207	C₄²	C₈×D₅	1	2	yes	—
208	C₄²	#676	1	2	yes	—
209	C₄²	#686	1	2	yes	—
210	C₄²	#689	1	2	yes	—
211	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
212	C₄²	C₂×D₂₀	1	1	yes	characteristic
213	C₄²	#709	1	1	yes	characteristic
214	C₄²	#709	1	1	yes	characteristic
215	C₂×C₅	C₂×#335	1	1	yes	characteristic
216	C₂×C₅	#1565	1	1	yes	characteristic
217	C₂×C₅	#1571	1	1	yes	characteristic
218	C₂×C₅	#1574	1	1	yes	characteristic
219	C₂×C₅	#1939	1	1	yes	characteristic
220	C₂×C₅	#1947	1	1	yes	characteristic
221	C₂×C₅	#1950	1	1	yes	characteristic
222	C₂³	C₂×#683	1	1	yes	characteristic
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	#3939	1	2	yes	—
233	C₂×C₄	#3939	1	2	yes	—
234	C₂×C₄	#3946	1	2	yes	—
235	C₂×C₄	#3946	1	2	yes	—
236	C₂×C₄	#3954	1	2	yes	—
237	C₂×C₄	C₄×Dic₁₀	1	1	yes	characteristic
238	C₂×C₄	#4015	1	1	yes	characteristic
239	C₂×C₄	#4016	1	1	yes	characteristic
240	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
241	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
242	C₂×C₄	C₂×#676	1	1	yes	characteristic
243	C₂×C₄	#4047	1	1	yes	characteristic
244	C₂×C₄	#4047	1	1	yes	characteristic
245	C₂×C₄	#4070	1	1	yes	characteristic
246	C₂×C₄	#4076	1	1	yes	characteristic
247	C₂×C₄	C₂×#686	1	1	yes	characteristic
248	C₂×C₄	C₂×#689	1	1	yes	characteristic
249	C₂×C₄	#4095	1	1	yes	characteristic
250	C₂×C₄	#4097	1	1	yes	characteristic
251	C₈	—	4	2	no	—
252	C₈	—	4	2	no	—
253	C₈	—	10	4	no	—
254	C₈	—	20	2	no	—
255	C₈	—	40	2	no	—
256	C₅	#9601	1	1	yes	characteristic, complemented, Sylow
257	C₂²	#65217	1	1	yes	characteristic
258	C₂²	C₂×#3939	1	1	yes	characteristic
259	C₂²	#65284	1	1	yes	characteristic
260	C₂²	C₂×#3946	1	1	yes	characteristic
261	C₂²	#65391	1	1	yes	characteristic
262	C₂²	C₂×#3954	1	1	yes	characteristic
263	C₂²	#65413	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	1	no	—
269	C₄	#64699	1	4	yes	complemented
270	C₄	C₈×Dic₁₀	1	1	yes	characteristic
271	C₄	#64968	1	1	yes	characteristic
272	C₄	C₈×D₂₀	1	1	yes	characteristic
273	C₄	#64977	1	1	yes	characteristic
274	C₄	#65296	1	1	yes	characteristic
275	C₄	#65304	1	1	yes	characteristic
276	C₄	#65310	1	1	yes	characteristic
277	C₄	#65320	1	1	yes	characteristic
278	C₂	#10600259	1	1	yes	characteristic
279	C₂	#10600579	1	1	yes	characteristic
280	C₂	#10600668	1	1	yes	characteristic
281	C₂	C₂×#64699	1	1	yes	characteristic
282	C₂	#10601215	1	1	yes	characteristic
283	C₂	#10601539	1	1	yes	characteristic
284	C₂	#10601541	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₄×C₅×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₄×C₄⋊C₈	D₅
hypocenter	C₅	#9601
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
42. C₄⋊#168	C₈	semidirect
170. C₄×C₅	#335	semidirect
256. C₅	#9601	semidirect
269. C₄	#64699	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×C₄⋊C₈	2
C₂×#64699	2
#10600187	2
C₄×C₄⋊#168	2
#10598194	2
#9601	5