Subgroups — #12119056

Order 1280.

782 conjugacy classes of subgroups.

The subgroup lattice has 322 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	#10606056	C₂	1	2	yes	maximal
3	C₅×#1845	C₂	1	1	yes	maximal, characteristic
4	#10601420	C₂	1	2	yes	maximal
5	#10601417	C₂	1	4	yes	maximal
6	Dic₅×#107	C₂	1	2	yes	maximal
7	C₈×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
8	#10600381	C₂	1	2	yes	maximal
9	C₂²×Dic₂₀	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀	C₂²	1	2	yes	—
11	C₂×C₅×#107	C₂²	1	2	yes	—
12	C₂×C₅×#107	C₂²	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×#384	C₂²	1	2	yes	—
15	C₂×#3950	C₂²	1	1	yes	characteristic
16	C₂×#3948	C₂²	1	2	yes	—
17	C₂×C₈×Dic₅	C₂²	1	2	yes	—
18	#65267	—	2	4	no	—
19	#65266	—	2	2	no	—
20	C₄⋊C₄×Dic₅	—	2	2	no	—
21	C₄⋊C₄×Dic₅	C₂²	1	4	yes	—
22	#65263	C₂²	1	4	yes	—
23	C₂×#3942	C₂²	1	4	yes	—
24	C₄×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₀	C₄	1	8	yes	complemented
27	#64770	C₂²	1	2	yes	—
28	#64749	C₂²	1	4	yes	—
29	#22808	—	5	1	no	maximal, Sylow
30	C₂²×Dic₁₀	C₂³	1	2	yes	—
31	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₂×C₅×C₄⋊C₄	—	2	2	no	—
34	C₂×C₅×C₄⋊C₄	—	2	2	no	—
35	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
36	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊Dic₅	D₄	1	1	yes	characteristic
38	C₂×C₄⋊Dic₅	D₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₅	C₂³	1	2	yes	—
40	C₂×#683	—	2	2	no	—
41	C₂×C₄×Dic₅	—	2	2	no	—
42	C₂×C₄×Dic₅	—	2	4	no	—
43	C₂×C₄×Dic₅	C₂³	1	2	yes	—
44	C₂²×#168	C₂³	1	2	yes	—
45	C₂×Dic₂₀	—	2	2	no	—
46	C₂×Dic₂₀	—	2	2	no	—
47	C₂×Dic₂₀	C₂×C₄	1	4	yes	—
48	C₄×Dic₁₀	—	2	4	no	—
49	C₄×Dic₁₀	C₂×C₄	1	8	yes	—
50	C₅×#107	—	2	4	no	—
51	C₅×#107	—	4	2	no	—
52	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
53	#3963	—	2	4	no	—
54	#3963	—	2	4	no	—
55	#3963	—	2	4	no	—
56	#3963	D₄	1	2	yes	—
57	#3950	C₂×C₄	1	4	yes	—
58	#3948	C₂×C₄	1	8	yes	—
59	C₈×Dic₅	—	2	4	no	—
60	#3942	—	4	4	no	—
61	C₄⋊#168	—	2	2	no	—
62	C₂×C₄×Q₁₆	—	5	1	no	—
63	#2355	—	5	2	no	—
64	#1969	—	5	2	no	—
65	#1928	—	5	4	no	—
66	#1845	D₅	1	1	yes	characteristic
67	#1803	—	5	2	no	—
68	C₄×#107	—	5	2	no	—
69	C₈×C₄⋊C₄	—	5	1	no	—
70	C₂²×C₄×C₅	—	2	4	no	—
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₅	—	2	2	no	—
75	C₂²×Dic₅	—	2	2	no	—
76	C₂²×Dic₅	C₂×D₄	1	2	yes	—
77	C₂²×Dic₅	C₂×D₄	1	2	yes	—
78	C₂×Dic₁₀	—	2	2	no	—
79	C₂×Dic₁₀	—	2	2	no	—
80	C₂×Dic₁₀	—	2	2	no	—
81	C₂×Dic₁₀	—	2	2	no	—
82	C₂×Dic₁₀	—	2	2	no	—
83	C₂×Dic₁₀	C₂²×C₄	1	4	yes	—
84	C₂×C₅×C₈	—	2	1	no	—
85	C₂×C₅×C₈	—	2	1	no	—
86	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₅×C₈	#41	1	2	yes	—
89	C₂×C₅×C₈	#41	1	2	yes	—
90	C₅×C₄⋊C₄	—	2	4	no	—
91	C₅×C₄⋊C₄	—	2	4	no	—
92	C₅×C₄⋊C₄	—	4	4	no	—
93	C₅×C₄⋊C₄	—	8	2	no	—
94	C₄²×C₅	C₂²×C₄	1	2	yes	—
95	C₄²×C₅	#41	1	2	yes	—
96	C₄⋊Dic₅	—	2	2	no	—
97	C₄⋊Dic₅	—	2	2	no	—
98	C₄⋊Dic₅	Q₁₆	1	4	yes	complemented
99	C₄⋊Dic₅	C₂²×C₄	1	4	yes	—
100	#683	—	2	8	no	—
101	C₄×Dic₅	—	2	8	no	—
102	C₄×Dic₅	—	2	2	no	—
103	C₄×Dic₅	—	2	2	no	—
104	C₄×Dic₅	—	8	4	no	—
105	C₂×#168	—	2	2	no	—
106	C₂×#168	—	2	2	no	—
107	C₂×#168	—	2	2	no	—
108	Dic₂₀	—	2	8	no	—
109	C₂²×Q₁₆	—	5	1	no	—
110	C₂×C₄×Q₈	—	5	2	no	—
111	C₄×Q₁₆	—	5	8	no	—
112	C₂×#107	—	5	1	no	—
113	C₂×#107	C₂×D₅	1	2	yes	—
114	C₂×#107	C₂×D₅	1	2	yes	—
115	C₂×C₄⋊C₈	—	5	1	no	—
116	C₂×#103	—	5	2	no	—
117	C₂×#103	—	5	4	no	—
118	C₂×C₄×C₈	—	5	2	no	—
119	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
120	#391	—	5	4	no	—
121	#387	—	10	4	no	—
122	#384	C₂×D₅	1	2	yes	—
123	#382	—	10	2	no	—
124	C₄×C₄⋊C₄	—	5	1	no	—
125	C₄×C₄⋊C₄	—	5	4	no	—
126	C₄×C₄⋊C₄	—	10	2	no	—
127	#340	—	5	4	no	—
128	#336	—	5	2	no	—
129	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	4	4	no	—
133	C₂×C₄×C₅	—	4	4	no	—
134	C₂×C₄×C₅	—	4	4	no	—
135	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
137	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
138	C₂×C₄×C₅	#135	1	2	yes	—
139	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
145	C₂×Dic₅	—	2	2	no	—
146	C₂×Dic₅	—	2	2	no	—
147	C₂×Dic₅	—	2	4	no	—
148	C₂×Dic₅	—	2	4	no	—
149	C₂×Dic₅	—	2	2	no	—
150	C₂×Dic₅	—	2	2	no	—
151	C₂×Dic₅	—	2	2	no	—
152	C₂×Dic₅	—	4	2	no	—
153	C₂×Dic₅	—	4	2	no	—
154	C₂×Dic₅	—	4	2	no	—
155	C₂×Dic₅	—	4	2	no	—
156	C₂×Dic₅	C₂×Q₁₆	1	4	yes	—
157	Dic₁₀	—	2	4	no	—
158	Dic₁₀	—	2	4	no	—
159	Dic₁₀	—	4	2	no	—
160	Dic₁₀	—	4	2	no	—
161	C₅×C₈	—	2	2	no	—
162	C₅×C₈	#117	1	4	yes	—
163	#168	—	4	2	no	—
164	C₂²×Q₈	—	5	2	no	—
165	C₂×Q₁₆	—	5	4	no	—
166	C₂×Q₁₆	—	10	2	no	—
167	C₂×Q₁₆	—	10	2	no	—
168	C₂²×C₈	—	5	2	no	—
169	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
171	C₄×Q₈	—	5	8	no	—
172	C₄×Q₈	—	10	4	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	5	1	no	—
176	C₂×C₄⋊C₄	—	5	2	no	—
177	C₂×C₄⋊C₄	—	5	1	no	—
178	C₂×C₄⋊C₄	—	10	2	no	—
179	C₂×C₄⋊C₄	C₂²×D₅	1	4	yes	—
180	C₂×C₄²	—	5	2	no	—
181	C₂×C₄²	—	10	2	no	—
182	C₂×C₄²	—	10	4	no	—
183	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
184	#107	—	2	4	no	—
185	#107	—	4	2	no	—
186	#107	—	5	4	no	—
187	C₄⋊C₈	—	10	2	no	—
188	#103	—	5	8	no	—
189	#103	—	20	4	no	—
190	C₄×C₈	—	10	4	no	—
191	C₄×C₈	C₄×D₅	1	4	yes	—
192	#95	—	5	2	no	—
193	#95	—	10	4	no	—
194	#95	—	10	4	no	—
195	#95	—	10	4	no	—
196	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
197	C₂²×C₅	#547	1	1	yes	characteristic
198	C₂²×C₅	#550	1	1	yes	characteristic
199	C₂²×C₅	C₂²×Q₁₆	1	1	yes	characteristic
200	C₂²×C₅	C₂×#135	1	1	yes	characteristic
201	C₂²×C₅	#576	1	1	yes	characteristic
202	C₂²×C₅	#578	1	1	yes	characteristic
203	C₄×C₅	—	2	2	no	—
204	C₄×C₅	—	8	4	no	—
205	C₄×C₅	C₄×Q₁₆	1	4	yes	—
206	C₄×C₅	C₂×#117	1	1	yes	characteristic
207	C₄×C₅	#520	1	1	yes	characteristic
208	C₄×C₅	#555	1	1	yes	characteristic
209	C₄×C₅	#556	1	1	yes	characteristic
210	Dic₅	—	2	4	no	—
211	Dic₅	—	4	2	no	—
212	Dic₅	—	4	4	no	—
213	Dic₅	—	4	4	no	—
214	C₂×Q₈	—	5	4	no	—
215	C₂×Q₈	—	10	2	no	—
216	C₂×Q₈	—	10	2	no	—
217	C₂×Q₈	—	10	2	no	—
218	C₂×Q₈	—	10	2	no	—
219	C₂×Q₈	—	10	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	5	2	no	—
222	C₂²×C₄	—	5	2	no	—
223	C₂²×C₄	—	10	2	no	—
224	C₂²×C₄	—	10	2	no	—
225	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
226	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
227	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
228	Q₁₆	—	10	8	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	10	2	no	—
232	C₂×C₈	—	10	2	no	—
233	C₂×C₈	—	10	2	no	—
234	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
236	C₂×C₈	#711	1	2	yes	—
237	C₂×C₈	#713	1	2	yes	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₄⋊C₄	—	5	4	no	—
242	C₄⋊C₄	—	5	4	no	—
243	C₄⋊C₄	—	8	2	no	—
244	C₄⋊C₄	—	10	2	no	—
245	C₄⋊C₄	—	10	8	no	—
246	C₄⋊C₄	—	10	2	no	—
247	C₄²	—	10	2	no	—
248	C₄²	—	10	2	no	—
249	C₄²	—	10	8	no	—
250	C₄²	—	40	4	no	—
251	C₄²	C₂×C₄×D₅	1	2	yes	—
252	C₄²	#713	1	2	yes	—
253	C₂×C₅	#2354	1	1	yes	characteristic
254	C₂×C₅	C₂×C₄×Q₁₆	1	1	yes	characteristic
255	C₂×C₅	#2951	1	1	yes	characteristic
256	C₂×C₅	#3338	1	1	yes	characteristic
257	C₂×C₅	#3342	1	1	yes	characteristic
258	C₂×C₅	#3395	1	1	yes	characteristic
259	C₂×C₅	#3396	1	1	yes	characteristic
260	C₂³	C₂×D₄×D₅	1	1	yes	characteristic
261	Q₈	—	10	4	no	—
262	Q₈	—	10	4	no	—
263	Q₈	—	20	2	no	—
264	Q₈	—	20	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	5	4	no	—
271	C₂×C₄	—	10	2	no	—
272	C₂×C₄	—	10	2	no	—
273	C₂×C₄	—	10	2	no	—
274	C₂×C₄	—	10	2	no	—
275	C₂×C₄	—	10	4	no	—
276	C₂×C₄	—	10	4	no	—
277	C₂×C₄	—	10	2	no	—
278	C₂×C₄	—	20	2	no	—
279	C₂×C₄	—	20	2	no	—
280	C₂×C₄	—	20	2	no	—
281	C₂×C₄	—	20	2	no	—
282	C₂×C₄	#4039	1	2	yes	—
283	C₂×C₄	#4039	1	2	yes	—
284	C₂×C₄	#4058	1	2	yes	—
285	C₂×C₄	D₅×Q₁₆	1	2	yes	—
286	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
287	C₂×C₄	C₂×#711	1	1	yes	characteristic
288	C₂×C₄	#4144	1	1	yes	characteristic
289	C₂×C₄	C₂×#713	1	1	yes	characteristic
290	C₂×C₄	C₂×#713	1	1	yes	characteristic
291	C₂×C₄	#4147	1	1	yes	characteristic
292	C₈	—	2	2	no	—
293	C₈	—	20	2	no	—
294	C₈	#4038	1	4	yes	—
295	C₅	#22808	1	1	yes	characteristic, complemented, Sylow
296	C₂²	C₂×#4039	1	1	yes	characteristic
297	C₂²	#66052	1	1	yes	characteristic
298	C₂²	#66060	1	1	yes	characteristic
299	C₂²	C₂×#4058	1	1	yes	characteristic
300	C₂²	#66091	1	1	yes	characteristic
301	C₂²	C₂×D₅×Q₁₆	1	1	yes	characteristic
302	C₂²	#66100	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	8	4	no	—
305	C₄	—	10	4	no	—
306	C₄	—	20	2	no	—
307	C₄	—	20	4	no	—
308	C₄	—	20	4	no	—
309	C₄	#65162	1	4	yes	—
310	C₄	C₂×#4038	1	1	yes	characteristic
311	C₄	#65838	1	1	yes	characteristic
312	C₄	#66066	1	1	yes	characteristic
313	C₄	#66068	1	1	yes	characteristic
314	C₂	#10606055	1	1	yes	characteristic
315	C₂	C₂×#65162	1	1	yes	characteristic
316	C₂	#10610867	1	1	yes	characteristic
317	C₂	#10613510	1	1	yes	characteristic
318	C₂	#10613527	1	1	yes	characteristic
319	C₂	#10613678	1	1	yes	characteristic
320	C₂	#10613680	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₅×#1845	C₂
socle, prime radical	C₂³×C₅	C₂²×D₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	#1845	D₅
hypocenter	C₅	#22808
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
26. C₄×Dic₂₀	C₄	semidirect
98. C₄⋊Dic₅	Q₁₆	semidirect
295. C₅	#22808	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
#10606056	2
C₅×#1845	2
#10601420	2
#10601417	2
Dic₅×#107	2
C₈×C₄⋊Dic₅	2
#10600381	2
#22808	5