Subgroups — C₂×#10600185 #12099443

Order 1280.

1246 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#168	C₂	1	2	yes	maximal
3	C₂×#64744	C₂	1	4	yes	maximal
4	#10600185	C₂	1	8	yes	maximal, direct product factor
5	C₂²×C₄²×C₅	C₄	1	2	yes	—
6	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
8	C₂²×C₄²×C₅	C₂²	1	2	yes	—
9	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂³×#168	C₂²	1	4	yes	—
11	C₄³×C₅	C₄	1	4	yes	—
12	C₄³×C₅	C₂²	1	4	yes	—
13	C₂×C₄⋊#168	—	2	8	no	—
14	C₂×C₄⋊#168	C₂²	1	8	yes	—
15	#64744	C₂²	1	16	yes	—
16	C₂×#1831	—	5	1	no	maximal, Sylow
17	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₅	—	2	8	no	—
23	C₂×C₄²×C₅	—	2	4	no	—
24	C₂×C₄²×C₅	—	2	8	no	—
25	C₂×C₄²×C₅	C₈	1	8	yes	complemented
26	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₅	C₂×C₄	1	8	yes	—
28	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅	D₄	1	4	yes	—
32	C₂×C₄²×C₅	Q₈	1	4	yes	—
33	C₂×C₄²×C₅	C₂³	1	4	yes	—
34	C₂×C₄²×C₅	C₂³	1	2	yes	—
35	C₂²×#168	—	2	8	no	—
36	C₂²×#168	—	2	16	no	—
37	C₂²×#168	C₂³	1	8	yes	—
38	C₄⋊#168	—	2	16	no	—
39	C₂²×C₄⋊C₈	—	5	2	no	—
40	C₂×C₄³	D₅	1	1	yes	characteristic
41	#1831	—	5	8	no	—
42	C₂×#336	—	5	4	no	—
43	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	—	2	4	no	—
45	C₂²×C₄×C₅	—	2	4	no	—
46	C₂²×C₄×C₅	—	2	4	no	—
47	C₂²×C₄×C₅	—	2	4	no	—
48	C₂²×C₄×C₅	—	2	4	no	—
49	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₅	C₂×C₈	1	8	yes	—
52	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₅	M₁₆	1	8	yes	—
54	C₂²×C₄×C₅	M₁₆	1	4	yes	—
55	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₅	#41	1	8	yes	—
63	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
64	C₄²×C₅	—	2	8	no	—
65	C₄²×C₅	—	2	16	no	—
66	C₄²×C₅	—	2	8	no	—
67	C₄²×C₅	—	2	8	no	—
68	C₄²×C₅	C₄⋊C₄	1	8	yes	—
69	C₄²×C₅	C₂×C₈	1	16	yes	complemented
70	C₄²×C₅	C₂×D₄	1	4	yes	—
71	C₄²×C₅	C₂×Q₈	1	4	yes	—
72	C₂×#168	—	2	8	no	—
73	C₂×#168	—	2	16	no	—
74	C₂×#168	—	4	16	no	—
75	C₂³×C₈	—	5	4	no	—
76	C₂²×C₄²	Dic₅	1	2	yes	—
77	C₂²×C₄²	Dic₅	1	1	yes	characteristic
78	C₂²×C₄²	Dic₅	1	1	yes	characteristic
79	C₂²×C₄²	C₂×D₅	1	2	yes	—
80	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
81	C₂×C₄⋊C₈	—	5	8	no	—
82	C₂×C₄⋊C₈	—	10	8	no	—
83	C₄³	Dic₅	1	4	yes	—
84	C₄³	C₂×D₅	1	4	yes	—
85	#336	—	5	16	no	—
86	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
87	C₂³×C₅	#117	1	4	yes	—
88	C₂³×C₅	C₂²×C₈	1	2	yes	—
89	C₂³×C₅	C₂×M₁₆	1	2	yes	—
90	C₂³×C₅	C₂×M₁₆	1	4	yes	—
91	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₅	—	2	4	no	—
93	C₂×C₄×C₅	—	2	4	no	—
94	C₂×C₄×C₅	—	2	4	no	—
95	C₂×C₄×C₅	—	2	4	no	—
96	C₂×C₄×C₅	—	2	4	no	—
97	C₂×C₄×C₅	—	2	8	no	—
98	C₂×C₄×C₅	—	2	16	no	—
99	C₂×C₄×C₅	—	2	16	no	—
100	C₂×C₄×C₅	C₄⋊C₈	1	16	yes	—
101	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₅	#117	1	4	yes	—
105	C₂×C₄×C₅	#124	1	8	yes	—
106	C₂×C₄×C₅	#125	1	8	yes	—
107	C₂×C₄×C₅	C₂²×C₈	1	8	yes	—
108	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
109	C₂×C₄×C₅	C₂×M₁₆	1	8	yes	—
110	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
111	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×#41	1	4	yes	—
114	#168	—	4	8	no	—
115	C₂³×C₄	C₂×Dic₅	1	2	yes	—
116	C₂³×C₄	C₂×Dic₅	1	2	yes	—
117	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
118	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
119	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
120	C₂²×C₈	—	5	8	no	—
121	C₂²×C₈	—	10	8	no	—
122	C₂²×C₈	—	10	16	no	—
123	C₂×C₄²	—	2	8	no	—
124	C₂×C₄²	—	2	4	no	—
125	C₂×C₄²	—	2	8	no	—
126	C₂×C₄²	#168	1	8	yes	complemented
127	C₂×C₄²	Dic₁₀	1	4	yes	—
128	C₂×C₄²	D₂₀	1	4	yes	—
129	C₂×C₄²	C₂×Dic₅	1	4	yes	—
130	C₂×C₄²	C₂×Dic₅	1	8	yes	—
131	C₂×C₄²	C₂×Dic₅	1	4	yes	—
132	C₂×C₄²	C₂×Dic₅	1	4	yes	—
133	C₂×C₄²	C₂×Dic₅	1	2	yes	—
134	C₂×C₄²	C₂²×D₅	1	4	yes	—
135	C₂×C₄²	C₂²×D₅	1	2	yes	—
136	C₄⋊C₈	—	10	16	no	—
137	C₂²×C₅	#382	1	4	yes	—
138	C₂²×C₅	C₂×C₄⋊C₈	1	4	yes	—
139	C₂²×C₅	#423	1	4	yes	—
140	C₂²×C₅	#431	1	8	yes	—
141	C₂²×C₅	#432	1	8	yes	—
142	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₅	C₂×#117	1	2	yes	—
144	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
145	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
146	C₂²×C₅	C₂²×M₁₆	1	2	yes	—
147	C₄×C₅	—	2	4	no	—
148	C₄×C₅	—	2	8	no	—
149	C₄×C₅	—	2	8	no	—
150	C₄×C₅	#382	1	4	yes	—
151	C₄×C₅	C₂×C₄⋊C₈	1	8	yes	—
152	C₄×C₅	C₂×#124	1	2	yes	—
153	C₄×C₅	C₂×#125	1	2	yes	—
154	C₂⁴	C₂²×Dic₅	1	1	yes	characteristic
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	C₂×#168	1	8	yes	—
161	C₂²×C₄	C₂×#168	1	4	yes	—
162	C₂²×C₄	#681	1	8	yes	—
163	C₂²×C₄	#681	1	4	yes	—
164	C₂²×C₄	C₄⋊Dic₅	1	4	yes	—
165	C₂²×C₄	C₄⋊Dic₅	1	4	yes	—
166	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
167	C₂²×C₄	C₂×D₂₀	1	2	yes	—
168	C₂²×C₄	#709	1	8	yes	—
169	C₂²×C₄	C₂²×Dic₅	1	4	yes	—
170	C₂²×C₄	C₂²×Dic₅	1	4	yes	—
171	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
172	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
173	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
174	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
175	C₂×C₈	—	10	8	no	—
176	C₂×C₈	—	10	16	no	—
177	C₂×C₈	—	20	16	no	—
178	C₄²	—	2	8	no	—
179	C₄²	—	2	16	no	—
180	C₄²	—	2	8	no	—
181	C₄²	—	2	8	no	—
182	C₄²	C₂×#168	1	16	yes	complemented
183	C₄²	C₄⋊Dic₅	1	8	yes	—
184	C₄²	C₂×Dic₁₀	1	4	yes	—
185	C₄²	C₂×D₂₀	1	4	yes	—
186	C₂×C₅	#1831	1	8	yes	complemented
187	C₂×C₅	C₂×#382	1	1	yes	characteristic
188	C₂×C₅	C₂²×C₄⋊C₈	1	1	yes	characteristic
189	C₂×C₅	C₂×#423	1	1	yes	characteristic
190	C₂×C₅	C₂×#431	1	2	yes	—
191	C₂×C₅	C₂×#432	1	2	yes	—
192	C₂³	C₂²×#168	1	2	yes	—
193	C₂³	C₂×#681	1	2	yes	—
194	C₂³	C₂×#681	1	4	yes	—
195	C₂³	C₂×C₄⋊Dic₅	1	2	yes	—
196	C₂³	#4072	1	4	yes	—
197	C₂³	C₂³×Dic₅	1	1	yes	characteristic
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	16	no	—
205	C₂×C₄	—	2	16	no	—
206	C₂×C₄	C₄⋊#168	1	16	yes	—
207	C₂×C₄	#4016	1	8	yes	—
208	C₂×C₄	#4021	1	8	yes	—
209	C₂×C₄	C₂²×#168	1	8	yes	—
210	C₂×C₄	C₂²×#168	1	4	yes	—
211	C₂×C₄	C₂×#681	1	8	yes	—
212	C₂×C₄	C₂×#681	1	4	yes	—
213	C₂×C₄	C₂×C₄⋊Dic₅	1	4	yes	—
214	C₂×C₄	C₂×C₄⋊Dic₅	1	4	yes	—
215	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
216	C₂×C₄	#4072	1	4	yes	—
217	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
218	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
219	C₂×C₄	C₂×#709	1	4	yes	—
220	C₈	—	20	8	no	—
221	C₅	C₂×#1831	1	1	yes	characteristic, complemented, Sylow
222	C₂²	C₂×C₄⋊#168	1	4	yes	—
223	C₂²	#65213	1	4	yes	—
224	C₂²	#65214	1	8	yes	—
225	C₂²	#65215	1	8	yes	—
226	C₂²	#65225	1	4	yes	—
227	C₂²	C₂³×#168	1	1	yes	characteristic
228	C₂²	C₂²×#681	1	1	yes	characteristic
229	C₂²	C₂²×#681	1	2	yes	—
230	C₂²	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
231	C₂²	C₂×#4072	1	2	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	2	8	no	—
234	C₄	—	2	8	no	—
235	C₄	C₂×C₄⋊#168	1	8	yes	—
236	C₄	#65225	1	4	yes	—
237	C₄	C₂×#4016	1	2	yes	—
238	C₄	C₂×#4021	1	2	yes	—
239	C₂	#10600185	1	8	yes	direct product factor
240	C₂	C₂²×C₄⋊#168	1	1	yes	characteristic
241	C₂	C₂×#65213	1	1	yes	characteristic
242	C₂	C₂×#65214	1	2	yes	—
243	C₂	C₂×#65215	1	2	yes	—
244	C₂	C₂×#65225	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅
commutator	C₂²×C₅	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂×C₄³×C₅	C₂
socle, prime radical	C₂⁴×C₅	C₂²×C₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂×C₄³	D₅
hypocenter	C₅	C₂×#1831
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
4. #10600185	C₂	direct (N × H)
25. C₂×C₄²×C₅	C₈	semidirect
69. C₄²×C₅	C₂×C₈	semidirect
126. C₂×C₄²	#168	semidirect
182. C₄²	C₂×#168	semidirect
186. C₂×C₅	#1831	semidirect
221. C₅	C₂×#1831	semidirect
239. C₂	#10600185	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×C₄³×C₅	2
C₂²×C₄⋊#168	2
C₂×#64744	2
#10600185	2
C₂×#1831	5

Subgroups — C2×#10600185 #12099443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10600185 #12099443