Subgroups — C₂×#10601485 #12097403

Order 1280.

1046 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×Dic₅	C₂	1	2	yes	maximal
3	C₂²×#3951	C₂	1	1	yes	maximal, characteristic
4	C₂²×#3947	C₂	1	1	yes	maximal, characteristic
5	C₂×#64778	C₂	1	2	yes	maximal
6	#10601485	C₂	1	8	yes	maximal, direct product factor
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₅	C₂²	1	1	yes	characteristic
10	C₂²×#681	C₂²	1	2	yes	—
11	C₂³×#168	C₂²	1	1	yes	characteristic
12	C₂×C₅×#108	C₄	1	8	yes	complemented
13	C₂×C₅×#108	C₂²	1	4	yes	—
14	M₁₆×Dic₅	—	2	4	no	—
15	M₁₆×Dic₅	C₂²	1	8	yes	—
16	C₂×#3951	C₄	1	8	yes	complemented
17	C₂×#3951	C₂²	1	4	yes	—
18	C₂×#3947	—	2	2	no	—
19	C₂×#3947	C₄	1	4	yes	—
20	C₂×#3947	C₄	1	4	yes	—
21	C₂×#3947	C₂²	1	4	yes	—
22	C₂×C₈×Dic₅	—	2	2	no	—
23	#64778	C₂²	1	8	yes	—
24	C₂×#1778	—	5	1	no	maximal, Sylow
25	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂³×Dic₅	Q₈	1	2	yes	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	4	no	—
29	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
30	C₂×C₅×M₁₆	C₂³	1	4	yes	—
31	C₂²×C₅×C₈	—	2	2	no	—
32	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
35	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
36	C₂²×C₅×C₈	C₂³	1	2	yes	—
37	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
38	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₅	D₄	1	2	yes	—
40	C₂×C₄×Dic₅	D₄	1	2	yes	—
41	C₂×C₄×Dic₅	Q₈	1	2	yes	—
42	C₂×C₄×Dic₅	C₂³	1	2	yes	—
43	C₂×#681	—	2	2	no	—
44	C₂×#681	—	2	4	no	—
45	C₂×#681	C₂×C₄	1	8	yes	—
46	C₂×#681	C₂³	1	4	yes	—
47	C₂²×#168	—	2	2	no	—
48	C₂²×#168	—	2	2	no	—
49	C₂²×#168	—	2	2	no	—
50	C₂²×#168	C₂×C₄	1	2	yes	—
51	C₂²×#168	C₂×C₄	1	2	yes	—
52	C₂²×#168	C₂³	1	2	yes	—
53	C₅×#108	C₂×C₄	1	16	yes	complemented
54	#3951	C₂×C₄	1	16	yes	complemented
55	#3947	—	2	8	no	—
56	#3947	C₂×C₄	1	16	yes	—
57	C₈×Dic₅	—	2	8	no	—
58	C₂²×#108	—	5	1	no	—
59	C₂²×#108	D₅	1	1	yes	characteristic
60	C₂×C₄×M₁₆	—	5	2	no	—
61	C₂²×#97	—	5	1	no	—
62	#1778	—	5	8	no	—
63	C₂×#341	—	5	2	no	—
64	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	#41	1	2	yes	—
73	C₂²×C₄×C₅	#41	1	2	yes	—
74	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₅	—	4	4	no	—
76	C₂²×Dic₅	C₄⋊C₄	1	4	yes	—
77	C₂²×Dic₅	C₄⋊C₄	1	4	yes	—
78	C₂²×Dic₅	C₂×Q₈	1	4	yes	—
79	C₅×M₁₆	—	2	8	no	—
80	C₅×M₁₆	—	2	4	no	—
81	C₅×M₁₆	C₂²×C₄	1	8	yes	—
82	C₂×C₅×C₈	—	2	8	no	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	C₄²	1	8	yes	—
85	C₂×C₅×C₈	C₄²	1	8	yes	—
86	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
87	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
88	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
89	C₄×Dic₅	C₄⋊C₄	1	4	yes	—
90	C₄×Dic₅	C₄⋊C₄	1	4	yes	—
91	C₄×Dic₅	C₂²×C₄	1	4	yes	—
92	C₄×Dic₅	C₂×D₄	1	4	yes	—
93	#681	—	2	8	no	—
94	#681	—	2	4	no	—
95	#681	C₂²×C₄	1	8	yes	—
96	C₂×#168	—	2	4	no	—
97	C₂×#168	—	2	4	no	—
98	C₂×#168	—	2	8	no	—
99	C₂×#168	—	2	4	no	—
100	C₂×#168	—	2	2	no	—
101	C₂×#168	—	2	2	no	—
102	C₂×#168	C₂²×C₄	1	4	yes	—
103	C₂²×M₁₆	—	5	2	no	—
104	C₂²×M₁₆	C₂×D₅	1	2	yes	—
105	C₂³×C₈	—	5	1	no	—
106	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
107	C₂²×C₄²	—	5	1	no	—
108	C₂×#108	—	5	8	no	—
109	C₂×#108	—	5	4	no	—
110	C₂×#108	Dic₅	1	8	yes	complemented
111	C₂×#108	C₂×D₅	1	4	yes	—
112	C₄×M₁₆	—	5	8	no	—
113	C₄×M₁₆	—	10	4	no	—
114	C₂×#97	—	5	4	no	—
115	C₂×#97	—	5	4	no	—
116	C₂×#97	—	5	4	no	—
117	C₂×#97	—	10	2	no	—
118	C₂×C₄×C₈	—	10	2	no	—
119	#341	—	5	8	no	—
120	C₂³×C₅	—	2	4	no	—
121	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂³×C₅	C₄×Q₈	1	2	yes	—
124	C₂³×C₅	C₄×Q₈	1	2	yes	—
125	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	#117	1	2	yes	—
131	C₂×C₄×C₅	#117	1	2	yes	—
132	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
137	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
138	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
144	C₂×Dic₅	—	2	8	no	—
145	C₂×Dic₅	—	4	4	no	—
146	C₂×Dic₅	—	4	4	no	—
147	C₂×Dic₅	C₂×C₄⋊C₄	1	8	yes	—
148	C₅×C₈	—	2	8	no	—
149	C₅×C₈	C₂×C₄²	1	8	yes	—
150	#168	—	2	4	no	—
151	#168	—	2	8	no	—
152	C₂³×C₄	—	5	2	no	—
153	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	2	4	no	—
156	C₂×M₁₆	—	5	8	no	—
157	C₂×M₁₆	—	5	4	no	—
158	C₂×M₁₆	—	10	2	no	—
159	C₂×M₁₆	—	10	4	no	—
160	C₂×M₁₆	C₂×Dic₅	1	8	yes	—
161	C₂×M₁₆	C₂²×D₅	1	4	yes	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	5	2	no	—
164	C₂²×C₈	—	5	2	no	—
165	C₂²×C₈	—	5	2	no	—
166	C₂²×C₈	—	10	2	no	—
167	C₂²×C₈	—	10	2	no	—
168	C₂²×C₈	—	10	2	no	—
169	C₂²×C₈	C₄×D₅	1	2	yes	—
170	C₂²×C₈	C₄×D₅	1	2	yes	—
171	C₂²×C₈	C₄×D₅	1	4	yes	—
172	C₂²×C₈	C₂×Dic₅	1	4	yes	—
173	C₂²×C₈	C₂²×D₅	1	2	yes	—
174	C₂×C₄²	—	5	2	no	—
175	C₂×C₄²	—	5	2	no	—
176	C₂×C₄²	—	5	2	no	—
177	C₂×C₄²	—	5	2	no	—
178	C₂×C₄²	—	5	2	no	—
179	C₂×C₄²	—	5	2	no	—
180	#108	—	5	16	no	—
181	#108	C₂×Dic₅	1	16	yes	complemented
182	#97	—	5	16	no	—
183	#97	—	10	8	no	—
184	C₄×C₈	—	10	8	no	—
185	C₂²×C₅	—	2	2	no	—
186	C₂²×C₅	—	2	2	no	—
187	C₂²×C₅	—	2	4	no	—
188	C₂²×C₅	—	2	4	no	—
189	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
190	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
191	C₂²×C₅	#430	1	4	yes	—
192	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
193	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
194	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
195	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
196	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
197	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
198	C₄×C₅	C₂×#117	1	1	yes	characteristic
199	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
200	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
201	Dic₅	—	2	8	no	—
202	C₂⁴	Q₈×D₅	1	1	yes	characteristic
203	C₂²×C₄	—	5	4	no	—
204	C₂²×C₄	—	5	4	no	—
205	C₂²×C₄	—	5	4	no	—
206	C₂²×C₄	—	20	4	no	—
207	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
208	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
210	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
211	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
212	C₂²×C₄	#711	1	2	yes	—
213	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
214	C₂²×C₄	#713	1	2	yes	—
215	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
216	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	8	no	—
219	M₁₆	—	5	8	no	—
220	M₁₆	—	10	4	no	—
221	M₁₆	—	10	8	no	—
222	M₁₆	C₂²×Dic₅	1	8	yes	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	5	4	no	—
226	C₂×C₈	—	10	4	no	—
227	C₂×C₈	—	10	2	no	—
228	C₂×C₈	—	10	2	no	—
229	C₂×C₈	—	10	4	no	—
230	C₂×C₈	—	10	4	no	—
231	C₂×C₈	—	10	8	no	—
232	C₂×C₈	C₄×Dic₅	1	8	yes	—
233	C₂×C₈	C₄×Dic₅	1	8	yes	—
234	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
235	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
236	C₂×C₈	C₂²×Dic₅	1	4	yes	—
237	C₄²	—	5	4	no	—
238	C₄²	—	5	4	no	—
239	C₄²	—	5	4	no	—
240	C₄²	—	5	4	no	—
241	C₂×C₅	—	2	1	no	—
242	C₂×C₅	—	2	1	no	—
243	C₂×C₅	—	2	2	no	—
244	C₂×C₅	#1778	1	4	yes	complemented
245	C₂×C₅	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
246	C₂×C₅	C₂×#430	1	2	yes	—
247	C₂³	—	2	4	no	—
248	C₂³	#4033	1	2	yes	—
249	C₂³	D₅×C₄⋊C₄	1	1	yes	characteristic
250	C₂³	D₅×C₄⋊C₄	1	1	yes	characteristic
251	C₂³	Q₈×Dic₅	1	2	yes	—
252	C₂³	C₂×Q₈×D₅	1	1	yes	characteristic
253	C₂×C₄	—	5	8	no	—
254	C₂×C₄	—	10	8	no	—
255	C₂×C₄	—	20	4	no	—
256	C₂×C₄	—	20	4	no	—
257	C₂×C₄	#4033	1	2	yes	—
258	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
259	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
260	C₂×C₄	#4038	1	2	yes	—
261	C₂×C₄	#4038	1	2	yes	—
262	C₂×C₄	#4039	1	2	yes	—
263	C₂×C₄	#4039	1	2	yes	—
264	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
265	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
266	C₂×C₄	D₄×Dic₅	1	2	yes	—
267	C₂×C₄	D₄×Dic₅	1	2	yes	—
268	C₂×C₄	Q₈×Dic₅	1	2	yes	—
269	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
270	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
271	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
272	C₂×C₄	C₂×#711	1	1	yes	characteristic
273	C₂×C₄	C₂×#713	1	1	yes	characteristic
274	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
275	C₈	—	2	8	no	—
276	C₈	—	10	4	no	—
277	C₈	—	10	8	no	—
278	C₈	C₂×C₄×Dic₅	1	8	yes	—
279	C₅	C₂×#1778	1	1	yes	characteristic, complemented, Sylow
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	#65182	1	4	yes	—
285	C₂²	C₄⋊C₄×Dic₅	1	2	yes	—
286	C₂²	C₄⋊C₄×Dic₅	1	2	yes	—
287	C₂²	C₂×#4033	1	1	yes	characteristic
288	C₂²	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
289	C₂²	C₂×Q₈×Dic₅	1	1	yes	characteristic
290	C₄	—	10	8	no	—
291	C₄	C₄⋊C₄×Dic₅	1	2	yes	—
292	C₄	C₄⋊C₄×Dic₅	1	2	yes	—
293	C₄	C₂×#4038	1	1	yes	characteristic
294	C₄	C₂×#4039	1	1	yes	characteristic
295	C₄	C₂²×C₄×Dic₅	1	1	yes	characteristic
296	C₄	C₂×D₄×Dic₅	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	#10601485	1	4	yes	direct product factor
301	C₂	C₂×#65182	1	2	yes	—
302	C₂	C₂×C₄⋊C₄×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅
commutator	C₄×C₅	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂²×C₅×#108	C₂
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂²×#108	D₅
hypocenter	C₅	C₂×#1778
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
6. #10601485	C₂	direct (N × H)
12. C₂×C₅×#108	C₄	semidirect
16. C₂×#3951	C₄	semidirect
53. C₅×#108	C₂×C₄	semidirect
54. #3951	C₂×C₄	semidirect
110. C₂×#108	Dic₅	semidirect
181. #108	C₂×Dic₅	semidirect
244. C₂×C₅	#1778	semidirect
279. C₅	C₂×#1778	semidirect
300. C₂	#10601485	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#108	2
C₂×M₁₆×Dic₅	2
C₂²×#3951	2
C₂²×#3947	2
C₂×#64778	2
#10601485	2
C₂×#1778	5

Subgroups — C2×#10601485 #12097403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10601485 #12097403