Subgroups — C₂×#10599382 #12081663

Order 1280.

638 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65139	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₀	C₂	1	1	yes	maximal, characteristic
3	C₂×#64971	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic
5	C₂×#64701	C₂	1	2	yes	maximal
6	#10599382	C₂	1	8	yes	maximal, direct product factor
7	C₂×#4015	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₀	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#3950	—	2	2	no	—
12	C₂×#3950	—	2	1	no	—
13	C₂×#3950	—	2	1	no	—
14	C₂×#3948	—	2	2	no	—
15	C₂×#3946	—	2	1	no	—
16	C₂×C₈×Dic₅	—	2	1	no	—
17	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
18	#65139	—	2	4	no	—
19	#65139	C₂²	1	8	yes	—
20	#64971	—	2	2	no	—
21	#64971	C₂²	1	4	yes	—
22	C₈×Dic₁₀	—	2	2	no	—
23	C₈×Dic₁₀	C₂²	1	4	yes	—
24	C₅×#335	C₂²	1	4	yes	—
25	#64701	C₂²	1	8	yes	—
26	C₂×#1676	—	5	1	no	maximal, Sylow
27	C₂²×Dic₁₀	—	2	2	no	—
28	C₂²×Dic₁₀	D₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊Dic₅	—	2	2	no	—
34	C₂×C₄⋊Dic₅	—	2	2	no	—
35	C₂×C₄⋊Dic₅	D₄	1	1	yes	characteristic
36	C₂×#683	—	2	1	no	—
37	C₂×C₄×Dic₅	—	2	1	no	—
38	C₂²×#168	—	2	1	no	—
39	#4015	—	2	2	no	—
40	#4015	—	2	2	no	—
41	#4015	—	2	2	no	—
42	#4015	C₂³	1	4	yes	—
43	C₄×Dic₁₀	—	2	1	no	—
44	C₄×Dic₁₀	—	2	1	no	—
45	C₄×Dic₁₀	D₄	1	2	yes	—
46	C₄×Dic₁₀	C₂³	1	2	yes	—
47	C₅×C₄⋊C₈	—	2	2	no	—
48	C₅×C₄⋊C₈	C₂³	1	4	yes	—
49	C₄×C₅×C₈	D₄	1	2	yes	—
50	C₄×C₅×C₈	C₂³	1	2	yes	—
51	#3950	—	2	4	no	—
52	#3950	—	2	4	no	—
53	#3950	—	2	8	no	—
54	#3948	—	2	8	no	—
55	#3946	—	2	4	no	—
56	C₈×Dic₅	—	2	4	no	—
57	C₄⋊#168	—	2	1	no	—
58	C₄⋊#168	C₂³	1	2	yes	—
59	C₂×#500	—	5	1	no	—
60	C₂×#477	—	5	2	no	—
61	C₂×C₈×Q₈	—	5	1	no	—
62	#1676	—	5	8	no	—
63	C₂×#335	D₅	1	1	yes	characteristic
64	C₂×#333	—	5	2	no	—
65	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	#41	1	1	yes	characteristic
68	C₂²×Dic₅	—	2	1	no	—
69	C₂²×Dic₅	—	2	1	no	—
70	C₂²×Dic₅	—	4	2	no	—
71	C₂×Dic₁₀	—	2	1	no	—
72	C₂×Dic₁₀	—	2	2	no	—
73	C₂×Dic₁₀	—	2	4	no	—
74	C₂×Dic₁₀	—	4	2	no	—
75	C₂×Dic₁₀	—	8	4	no	—
76	C₂×Dic₁₀	Q₁₆	1	4	yes	—
77	C₂×Dic₁₀	C₂×D₄	1	2	yes	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	4	4	no	—
81	C₂×C₅×C₈	Q₁₆	1	4	yes	—
82	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
83	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
84	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅	#41	1	1	yes	characteristic
87	C₄²×C₅	C₂⁴	1	1	yes	characteristic
88	C₄⋊Dic₅	—	2	1	no	—
89	C₄⋊Dic₅	—	2	4	no	—
90	C₄⋊Dic₅	—	2	4	no	—
91	C₄⋊Dic₅	—	4	2	no	—
92	C₄⋊Dic₅	—	4	2	no	—
93	C₄⋊Dic₅	C₂×D₄	1	2	yes	—
94	#683	—	2	2	no	—
95	#683	—	2	2	no	—
96	C₄×Dic₅	—	2	2	no	—
97	C₄×Dic₅	—	2	2	no	—
98	C₂×#168	—	2	2	no	—
99	C₂×#168	—	4	2	no	—
100	C₂×C₄⋊Q₈	—	5	2	no	—
101	C₂×C₄×Q₈	—	5	1	no	—
102	#500	—	5	4	no	—
103	#500	—	10	2	no	—
104	#477	—	5	8	no	—
105	#477	—	10	4	no	—
106	C₈×Q₈	—	5	4	no	—
107	C₈×Q₈	—	10	2	no	—
108	C₂×#107	—	10	2	no	—
109	C₂×#107	—	10	1	no	—
110	C₂×#107	—	10	1	no	—
111	C₂×C₄⋊C₈	—	5	1	no	—
112	C₂×C₄⋊C₈	—	10	1	no	—
113	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
114	C₂×#103	—	10	2	no	—
115	C₂×C₄×C₈	—	10	1	no	—
116	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
117	#335	C₂×D₅	1	4	yes	—
118	#333	—	5	8	no	—
119	C₂³×C₅	#121	1	1	yes	characteristic
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	#121	1	1	yes	characteristic
122	C₂×C₄×C₅	#121	1	1	yes	characteristic
123	C₂×C₄×C₅	#121	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
125	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
126	C₂×C₄×C₅	#135	1	2	yes	—
127	C₂×C₄×C₅	#137	1	2	yes	—
128	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
131	C₂×Dic₅	—	2	2	no	—
132	C₂×Dic₅	—	2	4	no	—
133	C₂×Dic₅	—	2	2	no	—
134	C₂×Dic₅	—	4	2	no	—
135	C₂×Dic₅	—	4	4	no	—
136	C₂×Dic₅	—	8	4	no	—
137	Dic₁₀	—	2	2	no	—
138	Dic₁₀	—	2	4	no	—
139	Dic₁₀	—	8	4	no	—
140	Dic₁₀	—	8	4	no	—
141	Dic₁₀	C₂×Q₁₆	1	4	yes	—
142	C₅×C₈	—	2	2	no	—
143	C₅×C₈	—	4	4	no	—
144	C₅×C₈	C₂×Q₁₆	1	4	yes	—
145	#168	—	4	2	no	—
146	C₂²×Q₈	—	5	1	no	—
147	C₂²×Q₈	—	10	2	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	10	1	no	—
150	C₂²×C₈	D₂₀	1	1	yes	characteristic
151	C₂²×C₈	D₂₀	1	1	yes	characteristic
152	C₄⋊Q₈	—	5	4	no	—
153	C₄⋊Q₈	—	10	2	no	—
154	C₄⋊Q₈	—	10	2	no	—
155	C₄⋊Q₈	—	10	2	no	—
156	C₄×Q₈	—	5	2	no	—
157	C₄×Q₈	—	5	2	no	—
158	C₄×Q₈	—	10	1	no	—
159	C₄×Q₈	—	10	1	no	—
160	C₂×C₄⋊C₄	—	5	1	no	—
161	C₂×C₄⋊C₄	—	10	1	no	—
162	C₂×C₄⋊C₄	—	10	2	no	—
163	C₂×C₄⋊C₄	—	10	2	no	—
164	C₂×C₄²	—	10	1	no	—
165	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
166	#107	—	10	8	no	—
167	#107	—	10	4	no	—
168	#107	—	10	4	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	5	2	no	—
171	C₄⋊C₈	—	10	4	no	—
172	C₄⋊C₈	—	10	1	no	—
173	C₄⋊C₈	C₂²×D₅	1	4	yes	—
174	#103	—	10	8	no	—
175	C₄×C₈	—	10	4	no	—
176	C₄×C₈	D₂₀	1	2	yes	—
177	C₄×C₈	C₂²×D₅	1	2	yes	—
178	C₂²×C₅	#462	1	2	yes	—
179	C₂²×C₅	#467	1	2	yes	—
180	C₂²×C₅	#472	1	2	yes	—
181	C₂²×C₅	C₂×#121	1	1	yes	characteristic
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	—	2	1	no	—
184	C₄×C₅	#462	1	2	yes	—
185	C₄×C₅	#467	1	2	yes	—
186	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
187	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
188	C₄×C₅	C₂×#135	1	1	yes	characteristic
189	C₄×C₅	C₂×#137	1	1	yes	characteristic
190	Dic₅	—	2	4	no	—
191	Dic₅	—	4	2	no	—
192	Dic₅	—	8	4	no	—
193	C₂×Q₈	—	5	2	no	—
194	C₂×Q₈	—	5	4	no	—
195	C₂×Q₈	—	10	1	no	—
196	C₂×Q₈	—	10	2	no	—
197	C₂×Q₈	—	10	4	no	—
198	C₂×Q₈	—	20	2	no	—
199	C₂×Q₈	—	40	4	no	—
200	C₂²×C₄	—	10	1	no	—
201	C₂²×C₄	—	10	1	no	—
202	C₂²×C₄	—	20	2	no	—
203	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
204	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
205	C₂²×C₄	#713	1	1	yes	characteristic
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	4	4	no	—
209	C₂×C₈	—	10	2	no	—
210	C₂×C₈	—	20	2	no	—
211	C₂×C₈	Dic₂₀	1	4	yes	—
212	C₂×C₈	C₂×D₂₀	1	2	yes	—
213	C₂×C₈	C₂×D₂₀	1	2	yes	—
214	C₄⋊C₄	—	5	2	no	—
215	C₄⋊C₄	—	10	2	no	—
216	C₄⋊C₄	—	10	1	no	—
217	C₄⋊C₄	—	10	2	no	—
218	C₄⋊C₄	—	10	4	no	—
219	C₄⋊C₄	—	10	4	no	—
220	C₄⋊C₄	—	20	2	no	—
221	C₄⋊C₄	—	20	2	no	—
222	C₄²	—	10	2	no	—
223	C₄²	—	10	2	no	—
224	C₄²	C₂×D₂₀	1	1	yes	characteristic
225	C₄²	D₄×D₅	1	1	yes	characteristic
226	C₄²	#713	1	1	yes	characteristic
227	C₄²	C₂³×D₅	1	1	yes	characteristic
228	C₂×C₅	#1676	1	4	yes	complemented
229	C₂×C₅	C₂×#462	1	1	yes	characteristic
230	C₂×C₅	C₂×#467	1	1	yes	characteristic
231	C₂×C₅	C₂×#472	1	1	yes	characteristic
232	C₂³	#4041	1	1	yes	characteristic
233	Q₈	—	5	4	no	—
234	Q₈	—	10	2	no	—
235	Q₈	—	10	4	no	—
236	Q₈	—	40	4	no	—
237	Q₈	—	40	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	10	2	no	—
240	C₂×C₄	—	10	4	no	—
241	C₂×C₄	—	10	2	no	—
242	C₂×C₄	—	20	2	no	—
243	C₂×C₄	—	20	4	no	—
244	C₂×C₄	—	40	4	no	—
245	C₂×C₄	#4041	1	1	yes	characteristic
246	C₂×C₄	#4041	1	1	yes	characteristic
247	C₂×C₄	#4041	1	1	yes	characteristic
248	C₂×C₄	C₂×Dic₂₀	1	2	yes	—
249	C₂×C₄	#4055	1	2	yes	—
250	C₂×C₄	#4058	1	2	yes	—
251	C₂×C₄	D₅×Q₁₆	1	2	yes	—
252	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
253	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
254	C₂×C₄	C₂×#713	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	4	4	no	—
257	C₈	—	20	2	no	—
258	C₈	C₂×Dic₂₀	1	4	yes	—
259	C₅	C₂×#1676	1	1	yes	characteristic, complemented, Sylow
260	C₂²	#65138	1	2	yes	—
261	C₂²	#65176	1	2	yes	—
262	C₂²	#65186	1	2	yes	—
263	C₂²	C₂×#4041	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	10	4	no	—
267	C₄	—	20	2	no	—
268	C₄	—	40	4	no	—
269	C₄	#65138	1	2	yes	—
270	C₄	#65176	1	2	yes	—
271	C₄	C₂²×Dic₂₀	1	1	yes	characteristic
272	C₄	C₂×#4055	1	1	yes	characteristic
273	C₄	C₂×#4058	1	1	yes	characteristic
274	C₄	C₂×D₅×Q₁₆	1	1	yes	characteristic
275	C₂	#10599382	1	4	yes	direct product factor
276	C₂	C₂×#65138	1	1	yes	characteristic
277	C₂	C₂×#65176	1	1	yes	characteristic
278	C₂	C₂×#65186	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4041
commutator, cosocle	C₄²×C₅	C₂⁴
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₅×#335	C₂
socle, prime radical	C₂³×C₅	#121
hypercenter	C₂×#335	D₅
hypocenter	C₅	C₂×#1676
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
6. #10599382	C₂	direct (N × H)
228. C₂×C₅	#1676	semidirect
259. C₅	C₂×#1676	semidirect
275. C₂	#10599382	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#65139	2
C₂×C₈×Dic₁₀	2
C₂×#64971	2
C₂×C₅×#335	2
C₂×#64701	2
#10599382	2
C₂×#1676	5

Subgroups — C2×#10599382 #12081663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10599382 #12081663