Subgroups — C₂×#10600584 #12099526

Order 1280.

870 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#65213	C₂	1	2	yes	maximal
3	C₂×#64772	C₂	1	4	yes	maximal
4	#10600584	C₂	1	8	yes	maximal, direct product factor
5	C₂³×C₅×C₈	C₂²	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂²×#681	C₂²	1	4	yes	—
8	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
9	C₂×C₄×C₅×C₈	C₄	1	4	yes	—
10	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
12	#65213	—	2	4	no	—
13	#65213	C₂²	1	8	yes	—
14	C₂×C₄⋊#168	—	2	4	no	—
15	#64772	C₂²	1	16	yes	—
16	C₂×#1837	—	5	1	no	maximal, Sylow
17	C₂³×C₄×C₅	Q₈	1	2	yes	—
18	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
19	C₂²×C₅×C₈	—	2	8	no	—
20	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
21	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
22	C₂²×C₅×C₈	C₂³	1	4	yes	—
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	D₄	1	2	yes	—
27	C₂×C₄²×C₅	D₄	1	2	yes	—
28	C₂×C₄²×C₅	Q₈	1	2	yes	—
29	C₂×C₄²×C₅	C₂³	1	2	yes	—
30	C₂×#681	—	2	4	no	—
31	C₂×#681	—	2	16	no	—
32	C₂×#681	C₂³	1	8	yes	—
33	C₂²×#168	—	2	4	no	—
34	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
35	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
36	C₄⋊#168	—	2	16	no	—
37	C₂×#423	—	5	2	no	—
38	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
39	#1837	—	5	8	no	—
40	C₂×#341	—	5	4	no	—
41	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅	—	2	8	no	—
43	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
51	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₅	#41	1	8	yes	—
53	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
54	C₂×C₅×C₈	—	2	8	no	—
55	C₂×C₅×C₈	—	2	8	no	—
56	C₂×C₅×C₈	—	2	8	no	—
57	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
58	C₄²×C₅	C₄⋊C₄	1	2	yes	—
59	C₄²×C₅	C₄⋊C₄	1	4	yes	—
60	C₄²×C₅	C₄⋊C₄	1	2	yes	—
61	C₄²×C₅	C₂²×C₄	1	2	yes	—
62	C₄²×C₅	C₂²×C₄	1	2	yes	—
63	C₄²×C₅	C₂×D₄	1	4	yes	—
64	#681	—	2	16	no	—
65	#681	—	4	8	no	—
66	C₂×#168	—	2	8	no	—
67	C₂×#168	—	4	8	no	—
68	C₂²×M₁₆	—	5	4	no	—
69	C₂³×C₈	C₂×D₅	1	2	yes	—
70	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
71	#423	—	5	8	no	—
72	#423	—	10	4	no	—
73	C₂×C₄⋊C₈	—	10	4	no	—
74	C₂×C₄×C₈	Dic₅	1	2	yes	—
75	C₂×C₄×C₈	Dic₅	1	4	yes	—
76	C₂×C₄×C₈	Dic₅	1	2	yes	—
77	C₂×C₄×C₈	C₂×D₅	1	4	yes	—
78	#341	—	5	16	no	—
79	C₂³×C₅	—	2	4	no	—
80	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₅	#125	1	4	yes	—
83	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₅	—	2	4	no	—
85	C₂×C₄×C₅	—	2	8	no	—
86	C₂×C₄×C₅	—	2	8	no	—
87	C₂×C₄×C₅	#108	1	8	yes	—
88	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
89	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₅	#117	1	4	yes	—
93	C₂×C₄×C₅	#117	1	4	yes	—
94	C₂×C₄×C₅	#124	1	4	yes	—
95	C₂×C₄×C₅	#124	1	4	yes	—
96	C₂×C₄×C₅	#125	1	4	yes	—
97	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂×#41	1	4	yes	—
100	C₅×C₈	—	2	8	no	—
101	#168	—	4	8	no	—
102	C₂³×C₄	Dic₁₀	1	2	yes	—
103	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
104	C₂×M₁₆	—	5	8	no	—
105	C₂×M₁₆	—	10	4	no	—
106	C₂×M₁₆	—	10	16	no	—
107	C₂²×C₈	—	2	8	no	—
108	C₂²×C₈	—	10	4	no	—
109	C₂²×C₈	C₂×Dic₅	1	4	yes	—
110	C₂²×C₈	C₂×Dic₅	1	4	yes	—
111	C₂²×C₈	C₂²×D₅	1	4	yes	—
112	C₂×C₄²	Dic₁₀	1	2	yes	—
113	C₂×C₄²	D₂₀	1	2	yes	—
114	C₂×C₄²	D₂₀	1	2	yes	—
115	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
116	C₂×C₄²	C₂×Dic₅	1	2	yes	—
117	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
118	C₂×C₄²	C₂²×D₅	1	2	yes	—
119	C₄⋊C₈	—	10	16	no	—
120	C₄×C₈	C₂×Dic₅	1	8	yes	—
121	C₄×C₈	C₂×Dic₅	1	8	yes	—
122	C₂²×C₅	—	2	8	no	—
123	C₂²×C₅	—	2	2	no	—
124	C₂²×C₅	—	2	2	no	—
125	C₂²×C₅	#382	1	2	yes	—
126	C₂²×C₅	#382	1	2	yes	—
127	C₂²×C₅	C₂×#108	1	4	yes	—
128	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₅	C₂×#125	1	2	yes	—
130	C₄×C₅	—	2	4	no	—
131	C₄×C₅	#382	1	2	yes	—
132	C₄×C₅	#382	1	2	yes	—
133	C₄×C₅	C₂×#108	1	8	yes	—
134	C₄×C₅	C₂×#117	1	2	yes	—
135	C₄×C₅	C₂×#124	1	2	yes	—
136	C₂⁴	C₂×Dic₁₀	1	1	yes	characteristic
137	C₂²×C₄	—	2	8	no	—
138	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
139	C₂²×C₄	C₄⋊Dic₅	1	4	yes	—
140	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
141	C₂²×C₄	C₂×Dic₁₀	1	4	yes	—
142	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
143	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
145	C₂²×C₄	#709	1	8	yes	—
146	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
147	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
148	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
149	M₁₆	—	10	16	no	—
150	M₁₆	—	20	8	no	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	—	2	8	no	—
154	C₂×C₈	—	10	8	no	—
155	C₂×C₈	—	20	8	no	—
156	C₂×C₈	C₂²×Dic₅	1	8	yes	—
157	C₄²	C₄⋊Dic₅	1	2	yes	—
158	C₄²	C₄⋊Dic₅	1	4	yes	—
159	C₄²	C₄⋊Dic₅	1	2	yes	—
160	C₄²	C₂×D₂₀	1	4	yes	—
161	C₄²	C₂²×Dic₅	1	2	yes	—
162	C₄²	C₂²×Dic₅	1	2	yes	—
163	C₂×C₅	—	2	2	no	—
164	C₂×C₅	—	2	2	no	—
165	C₂×C₅	#1837	1	4	yes	complemented
166	C₂×C₅	C₂×#382	1	1	yes	characteristic
167	C₂×C₅	C₂²×#108	1	2	yes	—
168	C₂³	—	2	4	no	—
169	C₂³	#4016	1	4	yes	—
170	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
171	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
172	C₂³	C₂²×Dic₁₀	1	1	yes	characteristic
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	8	no	—
175	C₂×C₄	—	2	8	no	—
176	C₂×C₄	#3951	1	8	yes	—
177	C₂×C₄	#4016	1	4	yes	—
178	C₂×C₄	#4021	1	4	yes	—
179	C₂×C₄	#4021	1	4	yes	—
180	C₂×C₄	C₂×C₄⋊Dic₅	1	4	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₅	1	4	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
184	C₂×C₄	#4072	1	4	yes	—
185	C₂×C₄	#4072	1	4	yes	—
186	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
187	C₂×C₄	C₂×#709	1	4	yes	—
188	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
189	C₈	—	2	8	no	—
190	C₈	—	20	8	no	—
191	C₅	C₂×#1837	1	1	yes	characteristic, complemented, Sylow
192	C₂²	—	2	8	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	#65225	1	2	yes	—
196	C₂²	#65225	1	2	yes	—
197	C₂²	C₂×#3951	1	4	yes	—
198	C₂²	C₂×#4016	1	2	yes	—
199	C₂²	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
200	C₄	—	2	4	no	—
201	C₄	#65225	1	2	yes	—
202	C₄	#65225	1	2	yes	—
203	C₄	C₂×#3951	1	8	yes	—
204	C₄	C₂×#4021	1	2	yes	—
205	C₄	C₂×#4072	1	2	yes	—
206	C₂	—	2	2	no	—
207	C₂	—	2	2	no	—
208	C₂	#10600584	1	4	yes	direct product factor
209	C₂	C₂×#65225	1	1	yes	characteristic
210	C₂	C₂²×#3951	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂²×C₄×C₅×C₈	C₂
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	C₂×#1837
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #10600584	C₂	direct (N × H)
165. C₂×C₅	#1837	semidirect
191. C₅	C₂×#1837	semidirect
208. C₂	#10600584	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×C₄×C₅×C₈	2
C₂×#65213	2
C₂×#64772	2
#10600584	2
C₂×#1837	5

Subgroups — C2×#10600584 #12099526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10600584 #12099526