Subgroups — #11997895

Order 1280.

388 conjugacy classes of subgroups.

The subgroup lattice has 272 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×#64678	C₂	1	2	yes	maximal
3	#10600187	C₂	1	1	yes	maximal, characteristic
4	#10600184	C₂	1	1	yes	maximal, characteristic
5	#10598145	C₂	1	2	yes	maximal
6	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
8	C₄³×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#168	—	2	1	no	—
10	C₂×C₄⋊#168	—	2	1	no	—
11	C₂×C₄⋊#168	C₂²	1	2	yes	—
12	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
14	C₂×#3935	—	2	1	no	—
15	C₅×#336	C₄	1	1	yes	characteristic
16	C₅×#336	C₄	1	1	yes	characteristic
17	#64744	—	2	1	no	—
18	#64678	—	2	4	no	—
19	#9604	—	5	1	no	maximal, Sylow
20	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
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	1	no	—
30	C₂²×#168	—	2	1	no	—
31	C₂²×#168	—	2	2	no	—
32	C₅×C₄⋊C₈	—	2	4	no	—
33	C₄×C₅×C₈	—	2	2	no	—
34	C₄×C₅×C₈	—	2	2	no	—
35	C₄⋊#168	—	2	1	no	—
36	C₄⋊#168	—	2	1	no	—
37	C₄⋊#168	—	2	2	no	—
38	C₄⋊#168	—	2	2	no	—
39	C₄⋊#168	—	4	2	no	—
40	C₄⋊#168	—	4	2	no	—
41	C₄⋊#168	D₄	1	2	yes	—
42	C₄⋊#168	Q₈	1	2	yes	—
43	#3935	—	4	2	no	—
44	#1842	—	5	1	no	—
45	#1767	—	5	1	no	—
46	C₄×C₄⋊C₈	D₅	1	1	yes	characteristic
47	C₂×#335	—	5	2	no	—
48	#1277	—	5	2	no	—
49	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₅×C₈	—	2	2	no	—
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	2	2	no	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	M₁₆	1	4	yes	—
62	C₄²×C₅	—	2	2	no	—
63	C₄²×C₅	—	2	2	no	—
64	C₄²×C₅	—	2	2	no	—
65	C₄²×C₅	—	2	1	no	—
66	C₄²×C₅	—	2	1	no	—
67	C₄²×C₅	—	4	1	no	—
68	C₄²×C₅	D₈	1	2	yes	—
69	C₄²×C₅	Q₁₆	1	2	yes	—
70	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₅	#41	1	1	yes	characteristic
73	C₄²×C₅	#41	1	1	yes	characteristic
74	C₂×#168	—	2	2	no	—
75	C₂×#168	—	2	2	no	—
76	C₂×#168	—	4	1	no	—
77	C₂×#168	—	4	1	no	—
78	C₂×#168	—	4	1	no	—
79	C₂×#168	—	4	1	no	—
80	C₂×#168	—	4	2	no	—
81	C₂×#168	—	4	2	no	—
82	C₂×#168	—	4	2	no	—
83	C₂×C₄⋊C₈	—	5	1	no	—
84	C₂×C₄⋊C₈	—	5	1	no	—
85	C₂×C₄⋊C₈	—	5	2	no	—
86	C₂×C₄⋊C₈	—	10	1	no	—
87	C₂×C₄⋊C₈	—	10	1	no	—
88	C₂×C₄⋊C₈	Dic₅	1	2	yes	—
89	C₂×#97	—	10	1	no	—
90	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
91	C₄³	C₂×D₅	1	1	yes	characteristic
92	#336	—	10	1	no	—
93	#336	Dic₅	1	1	yes	characteristic
94	#336	Dic₅	1	1	yes	characteristic
95	#335	—	10	4	no	—
96	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅	—	2	1	no	—
98	C₂×C₄×C₅	—	2	1	no	—
99	C₂×C₄×C₅	—	2	1	no	—
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	#107	1	2	yes	—
109	C₂×C₄×C₅	#107	1	2	yes	—
110	C₂×C₄×C₅	#108	1	2	yes	—
111	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₅	#121	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₅	#125	1	1	yes	characteristic
116	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
118	C₂×C₄×C₅	#131	1	2	yes	—
119	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₅	#137	1	1	yes	characteristic
123	C₅×C₈	—	2	4	no	—
124	C₅×C₈	—	4	2	no	—
125	#168	—	4	2	no	—
126	#168	—	4	2	no	—
127	#168	—	8	2	no	—
128	C₂²×C₈	—	10	1	no	—
129	C₂²×C₈	—	10	1	no	—
130	C₂²×C₈	—	10	2	no	—
131	C₂²×C₈	C₂×Dic₅	1	2	yes	—
132	C₂²×C₈	C₂×Dic₅	1	2	yes	—
133	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
134	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
135	C₂×C₄²	D₂₀	1	1	yes	characteristic
136	C₂×C₄²	D₂₀	1	1	yes	characteristic
137	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
138	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
139	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
140	C₄⋊C₈	—	2	4	no	—
141	C₄⋊C₈	—	5	2	no	—
142	C₄⋊C₈	—	5	2	no	—
143	C₄⋊C₈	—	10	2	no	—
144	C₄⋊C₈	—	10	1	no	—
145	C₄⋊C₈	—	10	1	no	—
146	C₄⋊C₈	—	10	2	no	—
147	C₄⋊C₈	—	20	2	no	—
148	C₄⋊C₈	—	20	2	no	—
149	#97	—	20	2	no	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	—	2	2	no	—
152	C₂²×C₅	#389	1	1	yes	characteristic
153	C₂²×C₅	#423	1	1	yes	characteristic
154	C₂²×C₅	#424	1	1	yes	characteristic
155	C₂²×C₅	C₂×#107	1	1	yes	characteristic
156	C₂²×C₅	#428	1	1	yes	characteristic
157	C₂²×C₅	C₂×#108	1	1	yes	characteristic
158	C₂²×C₅	#430	1	1	yes	characteristic
159	C₄×C₅	—	2	2	no	—
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	2	2	no	—
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	—	4	1	no	—
165	C₄×C₅	#435	1	2	yes	—
166	C₄×C₅	#446	1	2	yes	—
167	C₄×C₅	#459	1	1	yes	characteristic
168	C₄×C₅	#462	1	1	yes	characteristic
169	C₄×C₅	#474	1	1	yes	characteristic
170	C₄×C₅	#477	1	1	yes	characteristic
171	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
172	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
173	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
174	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
175	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
177	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	10	2	no	—
184	C₂×C₈	—	10	2	no	—
185	C₂×C₈	—	20	1	no	—
186	C₂×C₈	—	20	1	no	—
187	C₂×C₈	—	20	1	no	—
188	C₂×C₈	—	20	1	no	—
189	C₂×C₈	—	20	2	no	—
190	C₂×C₈	—	20	2	no	—
191	C₂×C₈	—	20	2	no	—
192	C₂×C₈	#681	1	4	yes	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	4	1	no	—
199	C₄²	D₄₀	1	2	yes	—
200	C₄²	Dic₂₀	1	2	yes	—
201	C₄²	D₄×D₅	1	1	yes	characteristic
202	C₄²	#711	1	1	yes	characteristic
203	C₄²	Q₈×D₅	1	1	yes	characteristic
204	C₄²	#713	1	1	yes	characteristic
205	C₂×C₅	#1567	1	1	yes	characteristic
206	C₂×C₅	#1569	1	1	yes	characteristic
207	C₂×C₅	#1571	1	1	yes	characteristic
208	C₂×C₅	#1573	1	1	yes	characteristic
209	C₂×C₅	#1940	1	1	yes	characteristic
210	C₂×C₅	#1947	1	1	yes	characteristic
211	C₂×C₅	#1950	1	1	yes	characteristic
212	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	#3950	1	2	yes	—
225	C₂×C₄	#3950	1	2	yes	—
226	C₂×C₄	#3951	1	2	yes	—
227	C₂×C₄	#4034	1	1	yes	characteristic
228	C₂×C₄	#4036	1	1	yes	characteristic
229	C₂×C₄	#4041	1	1	yes	characteristic
230	C₂×C₄	#4043	1	1	yes	characteristic
231	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
232	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
233	C₂×C₄	#4054	1	1	yes	characteristic
234	C₂×C₄	#4055	1	1	yes	characteristic
235	C₂×C₄	C₂×#681	1	2	yes	—
236	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
237	C₂×C₄	Q₈×Dic₅	1	1	yes	characteristic
238	C₂×C₄	#4094	1	2	yes	—
239	C₈	—	2	4	no	—
240	C₈	—	4	2	no	—
241	C₈	—	20	2	no	—
242	C₈	—	20	2	no	—
243	C₈	—	40	2	no	—
244	C₅	#9604	1	1	yes	characteristic, complemented, Sylow
245	C₂²	#65213	1	1	yes	characteristic
246	C₂²	#65274	1	1	yes	characteristic
247	C₂²	#65288	1	1	yes	characteristic
248	C₂²	C₂×#3950	1	1	yes	characteristic
249	C₂²	C₂×#3951	1	1	yes	characteristic
250	C₂²	#65410	1	1	yes	characteristic
251	C₂²	#65414	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	1	no	—
258	C₄	#65132	1	1	yes	characteristic
259	C₄	#65135	1	1	yes	characteristic
260	C₄	#65138	1	1	yes	characteristic
261	C₄	#65139	1	1	yes	characteristic
262	C₄	#65300	1	2	yes	—
263	C₄	#65313	1	2	yes	—
264	C₂	#10600278	1	1	yes	characteristic
265	C₂	#10600524	1	1	yes	characteristic
266	C₂	#10600644	1	1	yes	characteristic
267	C₂	#10601246	1	1	yes	characteristic
268	C₂	#10601267	1	1	yes	characteristic
269	C₂	#10601543	1	1	yes	characteristic
270	C₂	#10601545	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂²×C₄×C₅	C₂²×C₄
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	#9604
LCS	G ▷ C₂²×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
244. C₅	#9604	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×#64678	2
#10600187	2
#10600184	2
#10598145	2
#9604	5