Subgroups — C₂×#10598876 #12069071

Order 1280.

806 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄₀	C₂	1	1	yes	maximal, characteristic
2	C₂×#64977	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#64679	C₂	1	2	yes	maximal, complemented
5	C₂×#64677	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598876	C₂	1	8	yes	maximal, direct product factor
7	C₂²×D₄₀	C₄	1	1	yes	characteristic
8	C₂²×#676	—	2	2	no	—
9	C₂×C₄×D₂₀	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×#3953	C₄	1	2	yes	—
14	C₂×#3952	—	2	2	no	—
15	C₂×#3950	C₄	1	1	yes	characteristic
16	C₂×C₄⋊#168	C₂²	1	2	yes	—
17	C₄×D₄₀	C₄	1	4	yes	—
18	C₄×D₄₀	C₂²	1	4	yes	—
19	#64977	—	2	4	no	—
20	#64977	C₂²	1	8	yes	complemented
21	C₅×#322	C₂²	1	4	yes	complemented
22	#64679	C₂²	1	8	yes	complemented
23	#64677	C₂²	1	4	yes	complemented
24	C₂×#1594	—	5	1	no	maximal, Sylow
25	C₂²×D₂₀	C₂×C₄	1	2	yes	—
26	C₂²×C₄×D₅	—	2	2	no	—
27	C₂²×C₅×C₈	—	2	1	no	—
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	D₄	1	2	yes	—
31	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
32	C₂×#685	—	2	2	no	—
33	C₂×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
34	C₂²×#168	—	2	2	no	—
35	C₂×D₄₀	—	2	4	no	—
36	C₂×D₄₀	C₂×C₄	1	4	yes	—
37	C₂×#676	—	2	8	no	—
38	C₂×#676	—	4	8	no	—
39	C₄×D₂₀	—	2	2	no	—
40	C₄×D₂₀	—	2	2	no	—
41	C₄×D₂₀	C₂×C₄	1	4	yes	—
42	C₄×D₂₀	C₂³	1	4	yes	—
43	C₄×C₅×C₈	—	2	1	no	—
44	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
45	C₄×C₅×C₈	D₄	1	2	yes	—
46	C₄×C₅×C₈	C₂³	1	2	yes	—
47	C₄×C₅×C₈	C₂³	1	2	yes	—
48	C₄×C₅×C₈	C₂³	1	2	yes	—
49	#3953	C₂×C₄	1	8	yes	—
50	#3952	—	2	8	no	—
51	#3950	C₂×C₄	1	4	yes	—
52	C₄⋊#168	—	2	2	no	—
53	C₄⋊#168	C₂³	1	4	yes	—
54	C₂×C₄×D₈	—	5	1	no	—
55	C₂×#436	—	5	2	no	—
56	#1594	—	5	8	no	—
57	C₂×#334	—	5	1	no	—
58	C₂×D₄⋊C₈	—	5	2	no	—
59	C₂×#322	D₅	1	1	yes	characteristic, complemented
60	C₂³×D₅	—	2	2	no	—
61	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	#41	1	1	yes	characteristic
64	C₂²×Dic₅	—	2	2	no	—
65	C₂×D₂₀	—	2	4	no	—
66	C₂×D₂₀	—	2	4	no	—
67	C₂×D₂₀	—	2	2	no	—
68	C₂×D₂₀	C₂²×C₄	1	4	yes	—
69	C₂×C₄×D₅	—	2	4	no	—
70	C₂×C₄×D₅	—	2	4	no	—
71	C₂×C₄×D₅	—	4	4	no	—
72	C₂×C₄×D₅	—	4	4	no	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	2	4	no	—
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	4	2	no	—
77	C₂×C₅×C₈	M₁₆	1	4	yes	complemented
78	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
79	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
81	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅	#41	1	1	yes	characteristic
84	C₄²×C₅	C₂⁴	1	1	yes	characteristic
85	#685	—	2	4	no	—
86	#685	—	2	4	no	—
87	C₄⋊Dic₅	—	2	2	no	—
88	C₄⋊Dic₅	C₂²×C₄	1	4	yes	—
89	C₂×#168	—	2	4	no	—
90	C₂×#168	—	4	4	no	—
91	D₄₀	—	2	8	no	—
92	#676	—	4	16	no	—
93	C₂²×D₈	—	5	1	no	—
94	C₂²×M₁₆	—	10	2	no	—
95	C₂×C₄×D₄	—	5	2	no	—
96	C₄×D₈	—	5	4	no	—
97	C₄×D₈	—	5	4	no	—
98	#436	—	5	8	no	—
99	#436	—	10	4	no	—
100	C₂×#107	—	5	1	no	—
101	C₂×C₄⋊C₈	—	5	2	no	—
102	C₂×#102	—	5	2	no	—
103	C₂×C₂²⋊C₈	—	10	2	no	—
104	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
107	#334	—	5	4	no	—
108	D₄⋊C₈	—	5	8	no	—
109	#322	C₂×D₅	1	4	yes	complemented
110	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
111	C₂²×D₅	—	2	4	no	—
112	C₂²×D₅	—	4	4	no	—
113	C₂²×D₅	—	4	4	no	—
114	C₂²×D₅	—	4	4	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
120	C₂×C₄×C₅	#131	1	2	yes	—
121	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
122	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₅	—	2	4	no	—
126	C₂×Dic₅	—	4	4	no	—
127	D₂₀	—	2	4	no	—
128	D₂₀	—	2	4	no	—
129	D₂₀	—	4	4	no	—
130	C₄×D₅	—	4	4	no	—
131	C₄×D₅	—	4	4	no	—
132	C₄×D₅	—	4	8	no	—
133	C₅×C₈	—	2	2	no	—
134	C₅×C₈	—	2	4	no	—
135	C₅×C₈	—	4	2	no	—
136	C₅×C₈	C₂×M₁₆	1	4	yes	complemented
137	#168	—	4	4	no	—
138	C₂²×D₄	—	5	2	no	—
139	C₂³×C₄	—	10	2	no	—
140	C₂×D₈	—	5	4	no	—
141	C₂×D₈	—	10	4	no	—
142	C₂×M₁₆	—	10	8	no	—
143	C₂×M₁₆	—	20	8	no	—
144	C₂²×C₈	—	2	1	no	—
145	C₂²×C₈	—	10	2	no	—
146	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
147	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
148	C₂²×C₈	D₂₀	1	2	yes	—
149	C₄×D₄	—	5	4	no	—
150	C₄×D₄	—	5	4	no	—
151	C₄×D₄	—	10	2	no	—
152	C₄×D₄	—	10	2	no	—
153	C₂×C₄⋊C₄	—	5	2	no	—
154	C₂×C₂²⋊C₄	—	10	2	no	—
155	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
156	#107	—	5	4	no	—
157	C₄⋊C₈	—	5	4	no	—
158	C₄⋊C₈	—	10	2	no	—
159	#102	—	5	8	no	—
160	C₂²⋊C₈	—	10	8	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	C₄×D₅	1	2	yes	—
163	C₄×C₈	D₂₀	1	2	yes	—
164	C₄×C₈	C₂²×D₅	1	2	yes	—
165	C₄×C₈	C₂²×D₅	1	2	yes	—
166	C₄×C₈	C₂²×D₅	1	2	yes	—
167	C₂²×C₅	#436	1	2	yes	—
168	C₂²×C₅	#442	1	2	yes	—
169	C₂²×C₅	#444	1	2	yes	—
170	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×D₅	—	4	4	no	—
172	C₂×D₅	—	4	4	no	—
173	C₂×D₅	—	4	4	no	—
174	C₂×D₅	—	8	8	no	—
175	C₄×C₅	—	2	1	no	—
176	C₄×C₅	—	2	1	no	—
177	C₄×C₅	#436	1	2	yes	—
178	C₄×C₅	#442	1	2	yes	—
179	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
180	C₄×C₅	C₂×#131	1	1	yes	characteristic
181	C₄×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
182	Dic₅	—	4	4	no	—
183	C₂⁴	—	10	2	no	—
184	C₂×D₄	—	5	4	no	—
185	C₂×D₄	—	10	2	no	—
186	C₂×D₄	—	10	4	no	—
187	C₂×D₄	—	10	4	no	—
188	C₂²×C₄	—	10	2	no	—
189	C₂²×C₄	—	10	4	no	—
190	C₂²×C₄	—	10	4	no	—
191	C₂²×C₄	—	20	4	no	—
192	C₂²×C₄	—	20	4	no	—
193	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
195	C₂²×C₄	#709	1	1	yes	characteristic
196	D₈	—	10	8	no	—
197	M₁₆	—	20	16	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	10	4	no	—
203	C₂×C₈	—	20	4	no	—
204	C₂×C₈	#676	1	4	yes	complemented
205	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
206	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
207	C₂×C₈	C₂×D₂₀	1	4	yes	—
208	C₄⋊C₄	—	5	4	no	—
209	C₄⋊C₄	—	10	2	no	—
210	C₂²⋊C₄	—	10	4	no	—
211	C₂²⋊C₄	—	10	4	no	—
212	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
213	C₄²	C₂×D₂₀	1	1	yes	characteristic
214	C₄²	#709	1	1	yes	characteristic
215	C₄²	C₂³×D₅	1	1	yes	characteristic
216	C₂×C₅	#1594	1	4	yes	complemented
217	C₂×C₅	C₂×#436	1	1	yes	characteristic
218	C₂×C₅	C₂×#442	1	1	yes	characteristic
219	C₂×C₅	C₂×#444	1	1	yes	characteristic
220	D₅	—	8	4	no	—
221	C₂³	—	10	4	no	—
222	C₂³	—	20	4	no	—
223	C₂³	—	20	4	no	—
224	C₂³	—	20	4	no	—
225	C₂³	C₄×D₂₀	1	1	yes	characteristic
226	D₄	—	10	4	no	—
227	D₄	—	10	4	no	—
228	D₄	—	20	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	10	4	no	—
231	C₂×C₄	—	20	4	no	—
232	C₂×C₄	—	20	4	no	—
233	C₂×C₄	—	20	4	no	—
234	C₂×C₄	—	20	8	no	—
235	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
236	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
237	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
238	C₂×C₄	C₂×#676	1	2	yes	—
239	C₂×C₄	#4047	1	2	yes	—
240	C₂×C₄	#4054	1	4	yes	—
241	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
242	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
243	C₂×C₄	C₂×#709	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	4	no	—
246	C₈	—	4	2	no	—
247	C₈	—	20	4	no	—
248	C₈	C₂×#676	1	4	yes	complemented
249	C₅	C₂×#1594	1	1	yes	characteristic, complemented, Sylow
250	C₂²	—	20	4	no	—
251	C₂²	—	20	4	no	—
252	C₂²	—	20	4	no	—
253	C₂²	—	40	8	no	—
254	C₂²	#64977	1	2	yes	—
255	C₂²	#65000	1	2	yes	—
256	C₂²	#65006	1	2	yes	—
257	C₂²	C₂×C₄×D₂₀	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	20	4	no	—
261	C₄	#64977	1	2	yes	—
262	C₄	#65000	1	2	yes	—
263	C₄	C₂²×#676	1	1	yes	characteristic
264	C₄	C₂×#4047	1	1	yes	characteristic
265	C₄	C₂×#4054	1	2	yes	—
266	C₂	—	40	4	no	—
267	C₂	#10598876	1	4	yes	direct product factor
268	C₂	C₂×#64977	1	1	yes	characteristic
269	C₂	C₂×#65000	1	1	yes	characteristic
270	C₂	C₂×#65006	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₅×#322	C₂
socle	C₂³×C₅	C₄×D₄
prime radical	C₂²×D₄₀	C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₂×#322	D₅
hypocenter	C₅	C₂×#1594
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. C₂×#64977	C₂	semidirect
3. C₂×C₅×#322	C₂	semidirect
4. C₂×#64679	C₂	semidirect
5. C₂×#64677	C₂	semidirect
6. #10598876	C₂	direct (N × H)
20. #64977	C₂²	semidirect
21. C₅×#322	C₂²	semidirect
22. #64679	C₂²	semidirect
23. #64677	C₂²	semidirect
59. C₂×#322	D₅	semidirect
77. C₂×C₅×C₈	M₁₆	semidirect
109. #322	C₂×D₅	semidirect
136. C₅×C₈	C₂×M₁₆	semidirect
204. C₂×C₈	#676	semidirect
216. C₂×C₅	#1594	semidirect
248. C₈	C₂×#676	semidirect
249. C₅	C₂×#1594	semidirect
267. C₂	#10598876	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×C₄×D₄₀	2
C₂×#64977	2
C₂×C₅×#322	2
C₂×#64679	2
C₂×#64677	2
#10598876	2
C₂×#1594	5

Subgroups — C2×#10598876 #12069071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10598876 #12069071