Subgroups — #12015378

Order 1280.

410 conjugacy classes of subgroups.

The subgroup lattice has 290 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	C₅×#1848	C₂	1	1	yes	maximal, characteristic
3	C₂×#64716	C₂	1	2	yes	maximal
4	#10600536	C₂	1	1	yes	maximal, characteristic
5	#10598223	C₂	1	2	yes	maximal
6	C₂²×Dic₂₀	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀	—	2	1	no	—
8	C₂×C₅×#108	—	2	1	no	—
9	C₂×C₅×#108	C₂²	1	2	yes	—
10	C₅×#423	—	2	1	no	—
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×#3950	C₂²	1	1	yes	characteristic
13	C₂×#3948	—	2	1	no	—
14	C₂×#3944	C₂²	1	2	yes	—
15	C₄×Dic₂₀	—	2	1	no	—
16	C₄×Dic₂₀	—	2	1	no	—
17	C₄×Dic₂₀	C₄	1	2	yes	—
18	C₄×Dic₂₀	C₄	1	2	yes	—
19	#64716	—	2	4	no	—
20	C₄⋊#672	—	2	1	no	—
21	C₄⋊#672	—	2	1	no	—
22	#12819	—	5	1	no	maximal, Sylow
23	C₂²×Dic₁₀	—	2	1	no	—
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
28	C₂×C₄⋊Dic₅	—	2	1	no	—
29	C₂×#683	—	4	1	no	—
30	C₂×C₄×Dic₅	—	4	1	no	—
31	C₂×Dic₂₀	—	2	1	no	—
32	C₂×Dic₂₀	—	2	1	no	—
33	C₂×Dic₂₀	—	2	1	no	—
34	C₂×Dic₂₀	C₂×C₄	1	2	yes	—
35	C₂×Dic₂₀	D₄	1	2	yes	—
36	C₂×Dic₂₀	Q₈	1	2	yes	—
37	C₄×Dic₁₀	—	2	2	no	—
38	C₄×Dic₁₀	—	2	2	no	—
39	C₄×Dic₁₀	—	4	1	no	—
40	C₄×Dic₁₀	—	4	1	no	—
41	C₅×#108	—	2	2	no	—
42	C₅×#108	—	2	2	no	—
43	C₅×#108	—	4	2	no	—
44	C₅×C₄⋊C₈	—	4	1	no	—
45	C₅×C₄⋊C₈	—	4	1	no	—
46	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₅×C₈	D₄	1	1	yes	characteristic
49	C₄×C₅×C₈	D₄	1	1	yes	characteristic
50	#3950	—	2	1	no	—
51	#3950	C₂×C₄	1	2	yes	—
52	#3948	—	2	2	no	—
53	#3948	—	2	2	no	—
54	#3944	—	2	2	no	—
55	#3944	—	2	2	no	—
56	C₂×#672	—	2	2	no	—
57	C₂×C₄×Q₁₆	—	5	1	no	—
58	#2150	—	5	1	no	—
59	C₂×#362	—	5	2	no	—
60	#1848	D₅	1	1	yes	characteristic
61	#1387	—	5	2	no	—
62	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₅	—	4	1	no	—
66	C₂²×Dic₅	—	4	1	no	—
67	C₂×Dic₁₀	—	2	2	no	—
68	C₂×Dic₁₀	—	2	2	no	—
69	C₂×Dic₁₀	—	2	2	no	—
70	C₂×Dic₁₀	—	4	1	no	—
71	C₂×Dic₁₀	—	4	1	no	—
72	C₂×Dic₁₀	—	4	1	no	—
73	C₅×M₁₆	—	4	2	no	—
74	C₅×M₁₆	—	4	2	no	—
75	C₂×C₅×C₈	—	2	1	no	—
76	C₂×C₅×C₈	—	2	1	no	—
77	C₂×C₅×C₈	—	4	2	no	—
78	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₅×C₈	#41	1	1	yes	characteristic
85	C₂×C₅×C₈	#41	1	1	yes	characteristic
86	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₅	D₈	1	1	yes	characteristic
89	C₄²×C₅	D₈	1	1	yes	characteristic
90	C₄⋊Dic₅	—	2	2	no	—
91	C₄⋊Dic₅	—	4	1	no	—
92	#683	—	4	2	no	—
93	#683	—	4	2	no	—
94	C₄×Dic₅	—	4	2	no	—
95	C₄×Dic₅	—	4	2	no	—
96	Dic₂₀	—	2	2	no	—
97	Dic₂₀	—	2	2	no	—
98	Dic₂₀	—	2	2	no	—
99	Dic₂₀	C₄⋊C₄	1	4	yes	—
100	#672	—	4	2	no	—
101	C₂²×Q₁₆	—	5	1	no	—
102	C₂×C₄×Q₈	—	10	1	no	—
103	C₂×M₃₂	—	5	2	no	—
104	C₄×Q₁₆	—	5	2	no	—
105	C₄×Q₁₆	—	5	2	no	—
106	C₄×Q₁₆	—	10	1	no	—
107	C₄×Q₁₆	—	10	1	no	—
108	C₂×#108	—	2	1	no	—
109	C₂×#108	C₂×D₅	1	2	yes	—
110	C₂×#107	—	5	1	no	—
111	#423	—	2	1	no	—
112	C₂×#103	—	10	1	no	—
113	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
114	C₄⋊C₁₆	—	10	1	no	—
115	C₄⋊C₁₆	—	10	1	no	—
116	#362	—	10	4	no	—
117	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	4	1	no	—
120	C₂×C₄×C₅	#102	1	1	yes	characteristic
121	C₂×C₄×C₅	#102	1	1	yes	characteristic
122	C₂×C₄×C₅	#102	1	1	yes	characteristic
123	C₂×C₄×C₅	#102	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	#121	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₅	#137	1	1	yes	characteristic
130	C₂×Dic₅	—	4	2	no	—
131	C₂×Dic₅	—	4	2	no	—
132	C₂×Dic₅	—	4	2	no	—
133	C₂×Dic₅	—	4	2	no	—
134	C₂×Dic₅	—	8	2	no	—
135	Dic₁₀	—	2	4	no	—
136	Dic₁₀	—	4	2	no	—
137	Dic₁₀	—	4	2	no	—
138	Dic₁₀	—	4	2	no	—
139	C₅×C₈	—	2	1	no	—
140	C₅×C₈	—	2	1	no	—
141	C₅×C₈	—	8	2	no	—
142	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₅×C₈	#117	1	1	yes	characteristic
144	C₅×C₈	#121	1	1	yes	characteristic
145	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
146	C₂²×Q₈	—	10	1	no	—
147	C₂×Q₁₆	—	5	2	no	—
148	C₂×Q₁₆	—	5	2	no	—
149	C₂×Q₁₆	—	5	2	no	—
150	C₂×Q₁₆	—	10	1	no	—
151	C₂×Q₁₆	—	10	1	no	—
152	C₂×Q₁₆	—	10	1	no	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
156	C₄×Q₈	—	10	2	no	—
157	C₄×Q₈	—	10	2	no	—
158	C₄×Q₈	—	20	1	no	—
159	C₄×Q₈	—	20	1	no	—
160	C₂×C₄⋊C₄	—	10	1	no	—
161	C₂×C₄⋊C₄	—	20	1	no	—
162	C₂×C₄²	—	20	1	no	—
163	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
164	M₃₂	—	10	2	no	—
165	M₃₂	—	10	2	no	—
166	C₂×C₁₆	—	10	2	no	—
167	#108	—	2	2	no	—
168	#108	—	2	2	no	—
169	#108	—	4	2	no	—
170	#107	—	5	2	no	—
171	#107	—	10	1	no	—
172	C₄⋊C₈	—	4	1	no	—
173	C₄⋊C₈	—	4	1	no	—
174	#103	—	10	2	no	—
175	#103	—	10	2	no	—
176	C₄×C₈	C₄×D₅	1	1	yes	characteristic
177	C₄×C₈	C₄×D₅	1	1	yes	characteristic
178	C₄×C₈	D₂₀	1	1	yes	characteristic
179	C₄×C₈	D₂₀	1	1	yes	characteristic
180	C₂²×C₅	—	2	1	no	—
181	C₂²×C₅	—	2	1	no	—
182	C₂²×C₅	C₂×#102	1	1	yes	characteristic
183	C₂²×C₅	#461	1	1	yes	characteristic
184	C₂²×C₅	#476	1	1	yes	characteristic
185	C₄×C₅	—	2	1	no	—
186	C₄×C₅	—	2	1	no	—
187	C₄×C₅	—	2	1	no	—
188	C₄×C₅	#362	1	2	yes	complemented
189	C₄×C₅	#380	1	1	yes	characteristic
190	C₄×C₅	#417	1	1	yes	characteristic
191	C₄×C₅	#459	1	1	yes	characteristic
192	C₄×C₅	#474	1	1	yes	characteristic
193	Dic₅	—	4	4	no	—
194	Dic₅	—	8	2	no	—
195	C₂×Q₈	—	10	2	no	—
196	C₂×Q₈	—	10	2	no	—
197	C₂×Q₈	—	10	2	no	—
198	C₂×Q₈	—	20	1	no	—
199	C₂×Q₈	—	20	1	no	—
200	C₂×Q₈	—	20	1	no	—
201	C₂²×C₄	—	20	1	no	—
202	C₂²×C₄	—	20	1	no	—
203	C₂²×C₄	#688	1	1	yes	characteristic
204	C₂²×C₄	#688	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
206	Q₁₆	—	5	4	no	—
207	Q₁₆	—	10	2	no	—
208	Q₁₆	—	10	2	no	—
209	Q₁₆	—	10	2	no	—
210	M₁₆	—	4	2	no	—
211	M₁₆	—	4	2	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	#685	1	1	yes	characteristic
216	C₂×C₈	#685	1	1	yes	characteristic
217	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
218	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
219	C₂×C₈	D₄×D₅	1	1	yes	characteristic
220	C₂×C₈	#711	1	1	yes	characteristic
221	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
222	C₂×C₈	#713	1	1	yes	characteristic
223	C₄⋊C₄	—	10	2	no	—
224	C₄⋊C₄	—	20	2	no	—
225	C₄⋊C₄	—	20	2	no	—
226	C₄⋊C₄	—	20	1	no	—
227	C₄²	—	20	2	no	—
228	C₄²	—	20	2	no	—
229	C₄²	#685	1	1	yes	characteristic
230	C₄²	#685	1	1	yes	characteristic
231	C₄²	#686	1	1	yes	characteristic
232	C₄²	#686	1	1	yes	characteristic
233	C₁₆	—	20	2	no	—
234	C₂×C₅	—	2	1	no	—
235	C₂×C₅	—	2	1	no	—
236	C₂×C₅	#1802	1	1	yes	characteristic
237	C₂×C₅	C₂×#362	1	1	yes	characteristic
238	C₂×C₅	#2148	1	1	yes	characteristic
239	C₂³	C₂×#688	1	1	yes	characteristic
240	Q₈	—	10	4	no	—
241	Q₈	—	20	2	no	—
242	Q₈	—	20	2	no	—
243	Q₈	—	20	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	20	2	no	—
247	C₂×C₄	—	20	2	no	—
248	C₂×C₄	—	20	2	no	—
249	C₂×C₄	—	20	2	no	—
250	C₂×C₄	—	40	2	no	—
251	C₂×C₄	#3941	1	1	yes	characteristic
252	C₂×C₄	#3941	1	1	yes	characteristic
253	C₂×C₄	#3941	1	1	yes	characteristic
254	C₂×C₄	#3941	1	1	yes	characteristic
255	C₂×C₄	C₂×#685	1	1	yes	characteristic
256	C₂×C₄	C₂×#686	1	1	yes	characteristic
257	C₂×C₄	#4078	1	1	yes	characteristic
258	C₂×C₄	#4084	1	1	yes	characteristic
259	C₂×C₄	#4088	1	1	yes	characteristic
260	C₂×C₄	#4092	1	1	yes	characteristic
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	8	2	no	—
264	C₈	D₅×C₄⋊C₄	1	1	yes	characteristic
265	C₈	#4038	1	1	yes	characteristic
266	C₈	#4041	1	1	yes	characteristic
267	C₈	#4043	1	1	yes	characteristic
268	C₅	#12819	1	1	yes	characteristic, complemented, Sylow
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×#3941	1	1	yes	characteristic
272	C₂²	#65372	1	1	yes	characteristic
273	C₂²	#65373	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	20	4	no	—
278	C₄	—	40	2	no	—
279	C₄	#64716	1	2	yes	complemented
280	C₄	#65255	1	1	yes	characteristic
281	C₄	#65276	1	1	yes	characteristic
282	C₄	#65361	1	1	yes	characteristic
283	C₄	#65376	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	#10600377	1	1	yes	characteristic
287	C₂	#10601480	1	1	yes	characteristic
288	C₂	C₂×#64716	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#3941
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1848	C₂
socle	C₂²×C₅	C₂×#102
prime radical	C₂³×C₅	C₂×SD₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1848	D₅
hypocenter	C₅	#12819
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
188. C₄×C₅	#362	semidirect
268. C₅	#12819	semidirect
279. C₄	#64716	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
C₅×#1848	2
C₂×#64716	2
#10600536	2
#10598223	2
#12819	5