Subgroups — #12015347

Order 1280.

446 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#64723	C₂	1	2	yes	maximal
4	#10600558	C₂	1	1	yes	maximal, characteristic
5	#10598297	C₂	1	2	yes	maximal
6	C₂²×Dic₂₀	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀	—	2	1	no	—
8	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×#3950	—	2	1	no	—
11	C₂×#3950	C₂²	1	2	yes	—
12	C₂×#3950	C₂²	1	1	yes	characteristic
13	C₂×#3948	—	2	1	no	—
14	#65224	—	2	1	no	—
15	C₄×Dic₂₀	—	2	2	no	—
16	C₄×Dic₂₀	C₄	1	4	yes	complemented
17	C₅×C₄⋊C₁₆	—	2	2	no	—
18	#64723	—	2	4	no	—
19	#12812	—	5	1	no	maximal, Sylow
20	C₂²×Dic₁₀	—	2	1	no	—
21	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
24	C₂×C₄⋊Dic₅	—	2	1	no	—
25	C₂×C₄⋊Dic₅	—	2	2	no	—
26	C₂×C₄⋊Dic₅	—	4	1	no	—
27	C₂×C₄⋊Dic₅	—	4	1	no	—
28	C₂×#683	—	4	1	no	—
29	C₂×C₄×Dic₅	—	4	1	no	—
30	C₂×Dic₂₀	—	2	1	no	—
31	C₂×Dic₂₀	—	2	1	no	—
32	C₂×Dic₂₀	—	2	1	no	—
33	C₂×Dic₂₀	C₂×C₄	1	2	yes	—
34	C₂×Dic₂₀	D₄	1	2	yes	—
35	C₂×Dic₂₀	Q₈	1	2	yes	—
36	C₄×Dic₁₀	—	2	4	no	—
37	C₄×Dic₁₀	—	4	2	no	—
38	C₂×C₅×C₁₆	—	2	2	no	—
39	C₂×C₅×C₁₆	—	2	2	no	—
40	C₂×C₅×C₁₆	—	2	2	no	—
41	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₅×C₈	D₄	1	2	yes	—
43	#3950	—	2	1	no	—
44	#3950	—	2	2	no	—
45	#3950	—	2	2	no	—
46	#3950	—	4	2	no	—
47	#3950	C₂×C₄	1	2	yes	—
48	#3948	—	2	4	no	—
49	C₂×C₄×Q₁₆	—	5	1	no	—
50	C₂×C₄⋊C₁₆	D₅	1	1	yes	characteristic
51	C₂×#358	—	5	2	no	—
52	#1845	—	5	1	no	—
53	#1380	—	5	2	no	—
54	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₅	—	4	1	no	—
58	C₂²×Dic₅	—	4	1	no	—
59	C₂²×Dic₅	—	4	2	no	—
60	C₂×Dic₁₀	—	2	2	no	—
61	C₂×Dic₁₀	—	2	2	no	—
62	C₂×Dic₁₀	—	2	2	no	—
63	C₂×Dic₁₀	—	4	1	no	—
64	C₂×Dic₁₀	—	4	1	no	—
65	C₂×Dic₁₀	—	4	1	no	—
66	C₂×C₅×C₈	—	2	1	no	—
67	C₂×C₅×C₈	—	2	1	no	—
68	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₅×C₈	#41	1	1	yes	characteristic
74	C₂×C₅×C₈	#41	1	1	yes	characteristic
75	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₅	SD₁₆	1	2	yes	—
77	C₄⋊Dic₅	—	2	2	no	—
78	C₄⋊Dic₅	—	4	1	no	—
79	C₄⋊Dic₅	—	4	2	no	—
80	C₄⋊Dic₅	—	4	2	no	—
81	C₄⋊Dic₅	—	8	2	no	—
82	C₄⋊Dic₅	—	8	2	no	—
83	#683	—	4	4	no	—
84	C₄×Dic₅	—	4	4	no	—
85	Dic₂₀	—	2	2	no	—
86	Dic₂₀	—	2	2	no	—
87	Dic₂₀	—	2	2	no	—
88	Dic₂₀	C₄⋊C₄	1	4	yes	complemented
89	C₅×C₁₆	—	4	2	no	—
90	C₂²×Q₁₆	—	5	1	no	—
91	C₂×C₄×Q₈	—	10	1	no	—
92	C₂²×C₁₆	C₂×D₅	1	2	yes	—
93	C₄×Q₁₆	—	5	4	no	—
94	C₄×Q₁₆	—	10	2	no	—
95	C₂×#107	—	5	1	no	—
96	C₂×#107	—	5	2	no	—
97	C₂×#107	—	10	1	no	—
98	C₂×#103	—	10	1	no	—
99	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
100	#384	—	10	1	no	—
101	C₄⋊C₁₆	—	2	2	no	—
102	#358	—	10	4	no	—
103	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	#102	1	2	yes	—
107	C₂×C₄×C₅	#102	1	2	yes	—
108	C₂×C₄×C₅	SD₃₂	1	2	yes	—
109	C₂×C₄×C₅	Q₃₂	1	2	yes	—
110	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅	#121	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₄×C₅	#137	1	1	yes	characteristic
116	C₂×Dic₅	—	4	2	no	—
117	C₂×Dic₅	—	4	2	no	—
118	C₂×Dic₅	—	4	2	no	—
119	C₂×Dic₅	—	4	2	no	—
120	C₂×Dic₅	—	8	1	no	—
121	C₂×Dic₅	—	8	1	no	—
122	C₂×Dic₅	—	8	2	no	—
123	C₂×Dic₅	—	8	2	no	—
124	C₂×Dic₅	—	8	2	no	—
125	Dic₁₀	—	2	4	no	—
126	Dic₁₀	—	4	2	no	—
127	Dic₁₀	—	4	2	no	—
128	Dic₁₀	—	4	2	no	—
129	C₅×C₈	—	2	2	no	—
130	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₅×C₈	#117	1	1	yes	characteristic
132	C₅×C₈	#121	1	1	yes	characteristic
133	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂²×Q₈	—	10	1	no	—
135	C₂×Q₁₆	—	5	2	no	—
136	C₂×Q₁₆	—	5	2	no	—
137	C₂×Q₁₆	—	5	2	no	—
138	C₂×Q₁₆	—	10	1	no	—
139	C₂×Q₁₆	—	10	1	no	—
140	C₂×Q₁₆	—	10	1	no	—
141	C₂²×C₈	D₂₀	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
143	C₄×Q₈	—	10	4	no	—
144	C₄×Q₈	—	20	2	no	—
145	C₂×C₄⋊C₄	—	10	1	no	—
146	C₂×C₄⋊C₄	—	10	2	no	—
147	C₂×C₄⋊C₄	—	20	1	no	—
148	C₂×C₄⋊C₄	—	20	1	no	—
149	C₂×C₄⋊C₄	—	20	1	no	—
150	C₂×C₄²	—	20	1	no	—
151	C₂×C₄²	D₂₀	1	1	yes	characteristic
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	2	2	no	—
155	#107	—	5	2	no	—
156	#107	—	10	1	no	—
157	#107	—	10	2	no	—
158	#107	—	10	2	no	—
159	#107	—	20	2	no	—
160	#103	—	10	4	no	—
161	C₄×C₈	C₄×D₅	1	2	yes	—
162	C₄×C₈	C₂²⋊D₅	1	2	yes	—
163	C₂²×C₅	C₂×#102	1	1	yes	characteristic
164	C₂²×C₅	#459	1	1	yes	characteristic
165	C₂²×C₅	#474	1	1	yes	characteristic
166	C₂²×C₅	C₂×SD₃₂	1	1	yes	characteristic
167	C₂²×C₅	C₂×Q₃₂	1	1	yes	characteristic
168	C₂²×C₅	#509	1	1	yes	characteristic
169	C₂²×C₅	#510	1	1	yes	characteristic
170	C₄×C₅	—	2	2	no	—
171	C₄×C₅	#358	1	4	yes	complemented
172	C₄×C₅	#380	1	1	yes	characteristic
173	C₄×C₅	#417	1	1	yes	characteristic
174	C₄×C₅	#461	1	1	yes	characteristic
175	C₄×C₅	#476	1	1	yes	characteristic
176	Dic₅	—	4	4	no	—
177	Dic₅	—	8	2	no	—
178	Dic₅	—	16	2	no	—
179	C₂×Q₈	—	10	2	no	—
180	C₂×Q₈	—	10	2	no	—
181	C₂×Q₈	—	10	2	no	—
182	C₂×Q₈	—	20	1	no	—
183	C₂×Q₈	—	20	1	no	—
184	C₂×Q₈	—	20	1	no	—
185	C₂²×C₄	—	20	1	no	—
186	C₂²×C₄	—	20	1	no	—
187	C₂²×C₄	—	20	2	no	—
188	C₂²×C₄	D₄₀	1	1	yes	characteristic
189	C₂²×C₄	D₄₀	1	1	yes	characteristic
190	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
191	Q₁₆	—	5	4	no	—
192	Q₁₆	—	10	2	no	—
193	Q₁₆	—	10	2	no	—
194	Q₁₆	—	10	2	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	#685	1	2	yes	—
198	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
199	C₂×C₈	D₄×D₅	1	1	yes	characteristic
200	C₂×C₈	#711	1	1	yes	characteristic
201	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
202	C₂×C₈	#713	1	1	yes	characteristic
203	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
204	C₄⋊C₄	—	10	2	no	—
205	C₄⋊C₄	—	20	4	no	—
206	C₄⋊C₄	—	20	1	no	—
207	C₄⋊C₄	—	20	2	no	—
208	C₄⋊C₄	—	20	2	no	—
209	C₄⋊C₄	—	40	2	no	—
210	C₄⋊C₄	—	40	2	no	—
211	C₄²	—	20	4	no	—
212	C₄²	#677	1	2	yes	—
213	C₄²	#685	1	2	yes	—
214	C₁₆	—	4	2	no	—
215	C₂×C₅	#1802	1	1	yes	characteristic
216	C₂×C₅	C₂×#358	1	1	yes	characteristic
217	C₂×C₅	#2139	1	1	yes	characteristic
218	C₂×C₅	#2207	1	1	yes	characteristic
219	C₂×C₅	#2209	1	1	yes	characteristic
220	C₂×C₅	#2225	1	1	yes	characteristic
221	C₂×C₅	#2227	1	1	yes	characteristic
222	C₂³	C₂×D₄₀	1	1	yes	characteristic
223	Q₈	—	10	4	no	—
224	Q₈	—	20	2	no	—
225	Q₈	—	20	2	no	—
226	Q₈	—	20	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	20	2	no	—
230	C₂×C₄	—	20	2	no	—
231	C₂×C₄	—	20	2	no	—
232	C₂×C₄	—	20	2	no	—
233	C₂×C₄	—	40	1	no	—
234	C₂×C₄	—	40	1	no	—
235	C₂×C₄	—	40	2	no	—
236	C₂×C₄	—	40	2	no	—
237	C₂×C₄	—	40	2	no	—
238	C₂×C₄	#3932	1	2	yes	—
239	C₂×C₄	Dic₄₀	1	2	yes	—
240	C₂×C₄	#3953	1	2	yes	—
241	C₂×C₄	#3953	1	2	yes	—
242	C₂×C₄	#4041	1	1	yes	characteristic
243	C₂×C₄	#4043	1	1	yes	characteristic
244	C₂×C₄	C₂×#677	1	1	yes	characteristic
245	C₂×C₄	#4054	1	1	yes	characteristic
246	C₂×C₄	#4055	1	1	yes	characteristic
247	C₂×C₄	C₂×#685	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	D₅×C₄⋊C₄	1	1	yes	characteristic
250	C₈	#4038	1	1	yes	characteristic
251	C₈	#4084	1	1	yes	characteristic
252	C₈	#4092	1	1	yes	characteristic
253	C₅	#12812	1	1	yes	characteristic, complemented, Sylow
254	C₂²	#65132	1	1	yes	characteristic
255	C₂²	#65135	1	1	yes	characteristic
256	C₂²	C₂×#3932	1	1	yes	characteristic
257	C₂²	C₂×Dic₄₀	1	1	yes	characteristic
258	C₂²	#65197	1	1	yes	characteristic
259	C₂²	#65198	1	1	yes	characteristic
260	C₂²	C₂×#3953	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	20	4	no	—
263	C₄	—	40	2	no	—
264	C₄	—	80	2	no	—
265	C₄	#64723	1	4	yes	complemented
266	C₄	#65130	1	1	yes	characteristic
267	C₄	#65131	1	1	yes	characteristic
268	C₄	#65276	1	1	yes	characteristic
269	C₄	#65422	1	1	yes	characteristic
270	C₂	#10600085	1	1	yes	characteristic
271	C₂	#10600086	1	1	yes	characteristic
272	C₂	#10600088	1	1	yes	characteristic
273	C₂	#10600089	1	1	yes	characteristic
274	C₂	#10601272	1	1	yes	characteristic
275	C₂	C₂×#64723	1	1	yes	characteristic
276	C₂	#10602232	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₀
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅	C₂×D₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅
hypocenter	C₅	#12812
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
16. C₄×Dic₂₀	C₄	semidirect
88. Dic₂₀	C₄⋊C₄	semidirect
171. C₄×C₅	#358	semidirect
253. C₅	#12812	semidirect
265. C₄	#64723	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₀	2
C₂×C₅×C₄⋊C₁₆	2
C₂×#64723	2
#10600558	2
#10598297	2
#12812	5