Subgroups — C₂²×#64982 #12968698

Order 1280.

3950 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#677	C₂	1	4	yes	maximal, complemented
2	C₂²×#4020	C₂	1	1	yes	maximal, characteristic
3	C₂²×#4015	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#64982	C₂	1	24	yes	maximal, direct product factor
6	C₂³×D₂₀	—	2	2	no	—
7	C₂³×D₂₀	C₂²	1	2	yes	—
8	C₂³×Dic₁₀	C₂²	1	2	yes	—
9	C₂³×C₅×C₈	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊Dic₅	—	2	2	no	—
12	C₂²×#677	—	2	32	no	—
13	C₂²×#677	C₂²	1	48	yes	complemented
14	C₂×#4020	—	2	6	no	—
15	C₂×#4020	C₂²	1	12	yes	—
16	C₂×#4015	—	2	6	no	—
17	C₂×#4015	C₂²	1	12	yes	complemented
18	C₂×C₄×C₅×C₈	C₂²	1	12	yes	complemented
19	#64982	C₂²	1	64	yes	direct product factor
20	C₂²×#492	—	5	1	no	maximal, Sylow
21	C₂⁴×D₅	—	2	2	no	—
22	C₂³×C₄×C₅	D₄	1	2	yes	—
23	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂³×Dic₅	—	2	2	no	—
25	C₂²×D₂₀	—	2	16	no	—
26	C₂²×D₂₀	—	2	6	no	—
27	C₂²×D₂₀	—	2	24	no	—
28	C₂²×D₂₀	—	4	16	no	—
29	C₂²×D₂₀	C₂³	1	12	yes	—
30	C₂²×Dic₁₀	—	2	16	no	—
31	C₂²×Dic₁₀	—	2	6	no	—
32	C₂²×Dic₁₀	C₂³	1	12	yes	—
33	C₂²×C₅×C₈	D₄	1	16	yes	complemented
34	C₂²×C₅×C₈	C₂³	1	12	yes	—
35	C₂×C₄²×C₅	D₄	1	6	yes	—
36	C₂×C₄²×C₅	C₂³	1	6	yes	—
37	C₂×C₄⋊Dic₅	—	2	24	no	—
38	C₂×#677	—	2	192	no	—
39	C₂×#677	C₂³	1	64	yes	complemented
40	#4020	—	2	24	no	—
41	#4020	C₂³	1	16	yes	—
42	#4015	—	2	24	no	—
43	#4015	C₂³	1	16	yes	complemented
44	C₄×C₅×C₈	C₂³	1	16	yes	complemented
45	C₂³×SD₁₆	—	5	4	no	—
46	C₂²×C₄⋊Q₈	—	5	1	no	—
47	C₂²×#127	—	5	1	no	—
48	C₂×#492	—	5	24	no	—
49	C₂²×C₄×C₈	D₅	1	1	yes	characteristic, complemented
50	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
51	C₂³×D₅	—	2	12	no	—
52	C₂³×D₅	—	4	8	no	—
53	C₂³×D₅	—	4	16	no	—
54	C₂²×C₄×C₅	SD₁₆	1	16	yes	complemented
55	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
57	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
58	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
59	C₂²×Dic₅	—	2	12	no	—
60	C₂²×Dic₅	—	4	8	no	—
61	C₂×D₂₀	—	2	48	no	—
62	C₂×D₂₀	—	2	32	no	—
63	C₂×D₂₀	—	2	12	no	—
64	C₂×D₂₀	—	4	24	no	—
65	C₂×D₂₀	—	4	96	no	—
66	C₂×D₂₀	C₂⁴	1	8	yes	—
67	C₂×Dic₁₀	—	2	48	no	—
68	C₂×Dic₁₀	—	2	12	no	—
69	C₂×Dic₁₀	—	4	24	no	—
70	C₂×Dic₁₀	C₂⁴	1	8	yes	—
71	C₂×C₅×C₈	C₂×D₄	1	48	yes	complemented
72	C₂×C₅×C₈	C₂⁴	1	8	yes	—
73	C₄²×C₅	C₂×D₄	1	12	yes	—
74	C₄²×C₅	C₂⁴	1	4	yes	—
75	C₄⋊Dic₅	—	2	32	no	—
76	#677	—	2	128	no	—
77	C₂³×Q₈	—	5	2	no	—
78	C₂³×D₄	—	5	2	no	—
79	C₂³×D₄	—	10	2	no	—
80	C₂²×SD₁₆	—	5	48	no	—
81	C₂²×SD₁₆	—	10	32	no	—
82	C₂³×C₈	C₂×D₅	1	2	yes	—
83	C₂×C₄⋊Q₈	—	5	12	no	—
84	C₂×C₄⋊Q₈	—	10	6	no	—
85	C₂×#127	—	5	12	no	—
86	C₂×#127	—	10	6	no	—
87	C₂²×C₄⋊C₄	—	10	2	no	—
88	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
89	#492	—	5	64	no	—
90	C₂×C₄×C₈	C₂×D₅	1	12	yes	complemented
91	C₂³×C₅	#127	1	4	yes	—
92	C₂³×C₅	C₂×SD₁₆	1	8	yes	—
93	C₂³×C₅	C₂²×D₄	1	3	yes	—
94	C₂²×D₅	—	2	8	no	—
95	C₂²×D₅	—	4	24	no	—
96	C₂²×D₅	—	4	48	no	—
97	C₂²×D₅	—	8	32	no	—
98	C₂×C₄×C₅	#127	1	12	yes	—
99	C₂×C₄×C₅	C₂×SD₁₆	1	48	yes	complemented
100	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
101	C₂×C₄×C₅	C₂²×D₄	1	12	yes	—
102	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
103	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
104	C₂×Dic₅	—	2	8	no	—
105	C₂×Dic₅	—	4	24	no	—
106	D₂₀	—	2	16	no	—
107	D₂₀	—	4	24	no	—
108	D₂₀	—	4	64	no	—
109	Dic₁₀	—	2	16	no	—
110	Dic₁₀	—	4	24	no	—
111	C₅×C₈	C₂²×D₄	1	16	yes	complemented
112	C₂⁵	—	10	2	no	—
113	C₂²×Q₈	—	5	12	no	—
114	C₂²×Q₈	—	10	6	no	—
115	C₂²×Q₈	—	10	16	no	—
116	C₂²×D₄	—	5	12	no	—
117	C₂²×D₄	—	10	16	no	—
118	C₂²×D₄	—	10	24	no	—
119	C₂²×D₄	—	10	6	no	—
120	C₂²×D₄	—	20	16	no	—
121	C₂³×C₄	—	10	2	no	—
122	C₂³×C₄	D₂₀	1	2	yes	—
123	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
124	C₂×SD₁₆	—	5	64	no	—
125	C₂×SD₁₆	—	10	192	no	—
126	C₂²×C₈	D₂₀	1	16	yes	complemented
127	C₂²×C₈	C₂²×D₅	1	12	yes	—
128	C₄⋊Q₈	—	5	16	no	—
129	C₄⋊Q₈	—	10	24	no	—
130	#127	—	5	16	no	—
131	#127	—	10	24	no	—
132	C₂×C₄⋊C₄	—	10	24	no	—
133	C₂×C₄²	D₂₀	1	6	yes	—
134	C₂×C₄²	C₂²×D₅	1	6	yes	—
135	C₄×C₈	C₂²×D₅	1	16	yes	complemented
136	C₂²×C₅	#492	1	16	yes	complemented
137	C₂²×C₅	C₂×#127	1	6	yes	—
138	C₂²×C₅	C₂²×SD₁₆	1	12	yes	—
139	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
140	C₂×D₅	—	4	8	no	—
141	C₂×D₅	—	4	16	no	—
142	C₂×D₅	—	8	48	no	—
143	C₄×C₅	C₂×#127	1	6	yes	—
144	C₄×C₅	C₂²×SD₁₆	1	16	yes	complemented
145	C₄×C₅	C₂³×D₄	1	2	yes	—
146	Dic₅	—	4	8	no	—
147	C₂⁴	—	10	12	no	—
148	C₂⁴	—	20	8	no	—
149	C₂⁴	—	20	16	no	—
150	C₂⁴	C₂×D₂₀	1	1	yes	characteristic
151	C₂×Q₈	—	5	8	no	—
152	C₂×Q₈	—	10	12	no	—
153	C₂×Q₈	—	10	48	no	—
154	C₂×Q₈	—	20	24	no	—
155	C₂×D₄	—	5	8	no	—
156	C₂×D₄	—	10	48	no	—
157	C₂×D₄	—	10	32	no	—
158	C₂×D₄	—	10	12	no	—
159	C₂×D₄	—	20	24	no	—
160	C₂×D₄	—	20	96	no	—
161	C₂²×C₄	—	10	12	no	—
162	C₂²×C₄	—	20	8	no	—
163	C₂²×C₄	#677	1	16	yes	complemented
164	C₂²×C₄	C₂×D₂₀	1	8	yes	—
165	C₂²×C₄	C₂×D₂₀	1	3	yes	—
166	C₂²×C₄	C₂×D₂₀	1	12	yes	—
167	C₂²×C₄	C₂³×D₅	1	3	yes	—
168	SD₁₆	—	10	128	no	—
169	C₂×C₈	C₂×D₂₀	1	48	yes	complemented
170	C₂×C₈	C₂³×D₅	1	8	yes	—
171	C₄⋊C₄	—	10	32	no	—
172	C₄²	C₂×D₂₀	1	12	yes	—
173	C₄²	C₂³×D₅	1	4	yes	—
174	C₂×C₅	C₂×#492	1	12	yes	complemented
175	C₂×C₅	C₂²×#127	1	1	yes	characteristic
176	C₂×C₅	C₂³×SD₁₆	1	2	yes	—
177	D₅	—	8	8	no	—
178	C₂³	—	10	8	no	—
179	C₂³	—	20	24	no	—
180	C₂³	—	20	48	no	—
181	C₂³	—	40	32	no	—
182	C₂³	#4020	1	4	yes	—
183	C₂³	C₂×#677	1	8	yes	—
184	C₂³	C₂²×D₂₀	1	3	yes	—
185	Q₈	—	10	16	no	—
186	Q₈	—	20	24	no	—
187	D₄	—	10	16	no	—
188	D₄	—	20	24	no	—
189	D₄	—	20	64	no	—
190	C₂×C₄	—	10	8	no	—
191	C₂×C₄	—	20	24	no	—
192	C₂×C₄	#4020	1	12	yes	—
193	C₂×C₄	C₂×#677	1	48	yes	complemented
194	C₂×C₄	C₂²×D₂₀	1	3	yes	—
195	C₂×C₄	C₂²×D₂₀	1	12	yes	—
196	C₂×C₄	C₂²×D₂₀	1	8	yes	—
197	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
198	C₈	C₂²×D₂₀	1	16	yes	complemented
199	C₅	C₂²×#492	1	1	yes	characteristic, complemented, Sylow
200	C₂²	—	20	16	no	—
201	C₂²	—	20	8	no	—
202	C₂²	—	40	48	no	—
203	C₂²	#64982	1	16	yes	direct product factor
204	C₂²	C₂×#4020	1	6	yes	—
205	C₂²	C₂²×#677	1	12	yes	—
206	C₂²	C₂³×D₂₀	1	1	yes	characteristic
207	C₄	—	20	8	no	—
208	C₄	C₂×#4020	1	6	yes	—
209	C₄	C₂²×#677	1	16	yes	complemented
210	C₄	C₂³×D₂₀	1	2	yes	—
211	C₂	—	40	8	no	—
212	C₂	C₂×#64982	1	12	yes	direct product factor
213	C₂	C₂²×#4020	1	1	yes	characteristic
214	C₂	C₂³×#677	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂²×C₄×C₅×C₈	C₂
socle	C₂⁴×C₅	C₂×D₄
prime radical	C₂²×#4020	C₂
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	C₂²×#492
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×#677	C₂	semidirect
3. C₂²×#4015	C₂	semidirect
4. C₂²×C₄×C₅×C₈	C₂	semidirect
5. C₂×#64982	C₂	direct (N × H)
13. C₂²×#677	C₂²	semidirect
17. C₂×#4015	C₂²	semidirect
18. C₂×C₄×C₅×C₈	C₂²	semidirect
19. #64982	C₂²	direct (N × H)
33. C₂²×C₅×C₈	D₄	semidirect
39. C₂×#677	C₂³	semidirect
43. #4015	C₂³	semidirect
44. C₄×C₅×C₈	C₂³	semidirect
49. C₂²×C₄×C₈	D₅	semidirect
54. C₂²×C₄×C₅	SD₁₆	semidirect
71. C₂×C₅×C₈	C₂×D₄	semidirect
90. C₂×C₄×C₈	C₂×D₅	semidirect
99. C₂×C₄×C₅	C₂×SD₁₆	semidirect
111. C₅×C₈	C₂²×D₄	semidirect
126. C₂²×C₈	D₂₀	semidirect
135. C₄×C₈	C₂²×D₅	semidirect
136. C₂²×C₅	#492	semidirect
144. C₄×C₅	C₂²×SD₁₆	semidirect
163. C₂²×C₄	#677	semidirect
169. C₂×C₈	C₂×D₂₀	semidirect
174. C₂×C₅	C₂×#492	semidirect
193. C₂×C₄	C₂×#677	semidirect
198. C₈	C₂²×D₂₀	semidirect
199. C₅	C₂²×#492	semidirect
203. C₂²	#64982	direct (N × H)
209. C₄	C₂²×#677	semidirect
212. C₂	C₂×#64982	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#677	2
C₂²×#4020	2
C₂²×#4015	2
C₂²×C₄×C₅×C₈	2
C₂×#64982	2
C₂²×#492	5

Subgroups — C22×#64982 #12968698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#64982 #12968698