Subgroups — C₂×Q₈×D₄₀ #12979715

Order 1280.

2286 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₂₀	C₂	1	2	yes	maximal, complemented
2	C₂×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×D₄₀	C₂	1	16	yes	maximal, direct product factor
4	C₂×#65135	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×D₄₀	C₂	1	3	yes	maximal
6	C₂×#64971	C₂	1	3	yes	maximal, complemented
7	C₂²×Q₈×D₅	—	2	2	no	—
8	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×D₄₀	C₂²	1	1	yes	characteristic
10	Q₈×D₂₀	—	2	8	no	—
11	Q₈×D₂₀	C₂²	1	16	yes	complemented
12	C₂×#4043	—	2	6	no	—
13	C₂×C₄×D₂₀	C₂²	1	6	yes	—
14	C₂×#4015	C₂²	1	6	yes	—
15	C₅×C₈×Q₈	C₂²	1	8	yes	complemented
16	C₂×C₅×C₄⋊C₈	C₂²	1	3	yes	—
17	C₂×C₄×C₅×C₈	C₂²	1	3	yes	—
18	C₂×#3953	C₂²	1	6	yes	—
19	C₂×#3950	C₂²	1	3	yes	—
20	C₂×#3950	C₂²	1	6	yes	—
21	#65135	C₂²	1	48	yes	complemented
22	C₄×D₄₀	C₂²	1	24	yes	—
23	#64971	C₂²	1	24	yes	complemented
24	C₂×Q₈×D₈	—	5	1	no	maximal, Sylow
25	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
26	C₂×Q₈×D₅	—	2	16	no	—
27	C₂×Q₈×D₅	—	4	16	no	—
28	C₂²×D₂₀	C₂³	1	2	yes	—
29	C₂²×C₄×D₅	—	2	6	no	—
30	C₂²×Dic₁₀	—	2	6	no	—
31	C₂²×C₅×C₈	C₂³	1	3	yes	—
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₄×C₅×Q₈	D₄	1	4	yes	—
34	C₄×C₅×Q₈	C₂³	1	4	yes	—
35	C₂×C₅×C₄⋊C₄	D₄	1	3	yes	—
36	C₂×C₄²×C₅	C₂³	1	3	yes	—
37	C₂×#685	—	2	6	no	—
38	C₂×C₄⋊Dic₅	—	2	6	no	—
39	C₂×C₄⋊Dic₅	C₂³	1	6	yes	—
40	C₂×D₄₀	Q₈	1	8	yes	direct product factor
41	C₂×D₄₀	C₂³	1	4	yes	—
42	#4043	—	2	48	no	—
43	C₄×D₂₀	—	2	12	no	—
44	C₄×D₂₀	C₂³	1	24	yes	—
45	#4015	—	2	12	no	—
46	#4015	C₂³	1	24	yes	—
47	C₅×C₄⋊C₈	C₂³	1	12	yes	—
48	C₄×C₅×C₈	C₂³	1	12	yes	—
49	#3953	C₂³	1	24	yes	—
50	#3950	C₂³	1	24	yes	—
51	#3950	C₂³	1	12	yes	—
52	C₂×D₄×Q₈	—	5	2	no	—
53	Q₈×D₈	—	5	16	no	—
54	C₂×#500	—	5	3	no	—
55	C₂×#474	—	5	6	no	—
56	C₂×C₈×Q₈	D₅	1	1	yes	characteristic, complemented
57	C₂×C₄×D₈	—	5	3	no	—
58	C₂³×D₅	—	2	2	no	—
59	C₂×C₅×Q₈	D₈	1	8	yes	complemented
60	C₂×C₅×Q₈	C₂×D₄	1	4	yes	—
61	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
62	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
63	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₅	—	2	6	no	—
65	Q₈×D₅	—	4	64	no	—
66	C₂×D₂₀	—	2	2	no	—
67	C₂×D₂₀	—	2	4	no	—
68	C₂×D₂₀	C₂×Q₈	1	8	yes	complemented
69	C₂×D₂₀	C₂⁴	1	4	yes	—
70	C₂×C₄×D₅	—	2	24	no	—
71	C₂×C₄×D₅	—	4	24	no	—
72	C₂×Dic₁₀	—	2	24	no	—
73	C₂×Dic₁₀	—	4	24	no	—
74	C₂×C₅×C₈	—	2	6	no	—
75	C₂×C₅×C₈	C₂×Q₈	1	4	yes	complemented
76	C₂×C₅×C₈	C₂⁴	1	6	yes	—
77	C₂×C₅×C₈	C₂⁴	1	2	yes	—
78	C₅×C₄⋊C₄	C₂×D₄	1	12	yes	—
79	C₄²×C₅	C₂×D₄	1	6	yes	—
80	C₄²×C₅	C₂⁴	1	6	yes	—
81	#685	—	2	24	no	—
82	C₄⋊Dic₅	—	2	6	no	—
83	C₄⋊Dic₅	—	2	24	no	—
84	C₄⋊Dic₅	C₂⁴	1	12	yes	—
85	D₄₀	C₂×Q₈	1	16	yes	direct product factor
86	C₂³×Q₈	—	10	2	no	—
87	C₂²×D₈	—	5	1	no	—
88	D₄×Q₈	—	5	16	no	—
89	D₄×Q₈	—	10	8	no	—
90	C₂×C₄⋊Q₈	—	5	6	no	—
91	C₂×C₂²⋊Q₈	—	10	6	no	—
92	C₂×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
93	C₂×C₄×D₄	—	5	6	no	—
94	#500	—	5	24	no	—
95	#474	—	5	48	no	—
96	C₈×Q₈	C₂×D₅	1	8	yes	complemented
97	C₄×D₈	—	5	24	no	—
98	C₂×#107	—	5	3	no	—
99	C₂×#107	—	5	6	no	—
100	C₂×C₄⋊C₈	C₂×D₅	1	3	yes	—
101	C₂×#102	—	5	6	no	—
102	C₂×C₄×C₈	C₂×D₅	1	3	yes	—
103	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
104	C₂²×D₅	—	2	8	no	—
105	C₂²×D₅	—	2	4	no	—
106	C₂²×D₅	—	4	4	no	—
107	C₂²×D₅	—	4	4	no	—
108	C₅×Q₈	C₂×D₈	1	16	yes	complemented
109	C₂×C₄×C₅	—	2	6	no	—
110	C₂×C₄×C₅	C₂×D₈	1	12	yes	—
111	C₂×C₄×C₅	C₂²×D₄	1	6	yes	—
112	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂²×D₄	1	6	yes	—
114	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
115	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
116	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
117	C₂×Dic₅	—	2	12	no	—
118	C₂×Dic₅	—	4	12	no	—
119	D₂₀	—	2	8	no	—
120	D₂₀	—	2	4	no	—
121	D₂₀	C₂²×Q₈	1	8	yes	complemented
122	C₄×D₅	—	4	48	no	—
123	Dic₁₀	—	4	48	no	—
124	C₅×C₈	—	2	6	no	—
125	C₅×C₈	C₂²×Q₈	1	4	yes	complemented
126	C₂²×Q₈	—	10	16	no	—
127	C₂²×Q₈	—	10	6	no	—
128	C₂²×Q₈	—	20	16	no	—
129	C₂²×Q₈	D₂₀	1	1	yes	characteristic
130	C₂²×D₄	—	5	2	no	—
131	C₂³×C₄	—	10	6	no	—
132	C₂×D₈	—	5	8	no	—
133	C₂×D₈	—	5	4	no	—
134	C₂²×C₈	C₂²×D₅	1	3	yes	—
135	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
136	C₄⋊Q₈	—	5	24	no	—
137	C₄⋊Q₈	—	10	12	no	—
138	C₂²⋊Q₈	—	10	48	no	—
139	C₄×Q₈	D₂₀	1	4	yes	—
140	C₄×Q₈	C₂²×D₅	1	4	yes	—
141	C₄×D₄	—	5	24	no	—
142	C₄×D₄	—	10	12	no	—
143	C₂×C₄⋊C₄	—	5	6	no	—
144	C₂×C₄⋊C₄	—	10	6	no	—
145	C₂×C₄⋊C₄	D₂₀	1	3	yes	—
146	C₂×C₂²⋊C₄	—	10	6	no	—
147	C₂×C₄²	C₂²×D₅	1	3	yes	—
148	#107	—	5	12	no	—
149	#107	—	5	24	no	—
150	C₄⋊C₈	C₂²×D₅	1	12	yes	—
151	#102	—	5	24	no	—
152	C₄×C₈	C₂²×D₅	1	12	yes	—
153	C₂²×C₅	D₄×Q₈	1	2	yes	—
154	C₂²×C₅	C₂²×D₈	1	2	yes	—
155	C₂²×C₅	#578	1	2	yes	—
156	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
157	C₂×D₅	—	2	8	no	—
158	C₂×D₅	—	4	16	no	—
159	C₂×D₅	—	4	4	no	—
160	C₂×D₅	—	4	4	no	—
161	C₄×C₅	—	2	6	no	—
162	C₄×C₅	D₄×Q₈	1	2	yes	—
163	C₄×C₅	C₂²×D₈	1	12	yes	—
164	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
165	C₄×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
166	Dic₅	—	4	12	no	—
167	C₂⁴	—	10	2	no	—
168	C₂×Q₈	—	10	24	no	—
169	C₂×Q₈	—	20	64	no	—
170	C₂×Q₈	—	20	24	no	—
171	C₂×Q₈	D₄₀	1	8	yes	direct product factor
172	C₂×Q₈	C₂×D₂₀	1	4	yes	—
173	C₂×D₄	—	5	8	no	—
174	C₂×D₄	—	5	4	no	—
175	C₂×D₄	—	10	2	no	—
176	C₂×D₄	—	10	4	no	—
177	C₂²×C₄	—	10	6	no	—
178	C₂²×C₄	—	10	24	no	—
179	C₂²×C₄	—	20	24	no	—
180	C₂²×C₄	C₂×D₂₀	1	3	yes	—
181	C₂²×C₄	C₂×D₂₀	1	3	yes	—
182	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
183	D₈	—	5	16	no	—
184	C₂×C₈	—	2	6	no	—
185	C₂×C₈	Q₈×D₅	1	4	yes	complemented
186	C₂×C₈	C₂³×D₅	1	6	yes	—
187	C₂×C₈	C₂³×D₅	1	2	yes	—
188	C₄⋊C₄	—	5	12	no	—
189	C₄⋊C₄	—	10	6	no	—
190	C₄⋊C₄	—	10	24	no	—
191	C₄⋊C₄	C₂×D₂₀	1	12	yes	—
192	C₂²⋊C₄	—	10	24	no	—
193	C₄²	C₂×D₂₀	1	6	yes	—
194	C₄²	C₂³×D₅	1	6	yes	—
195	C₂×C₅	Q₈×D₈	1	4	yes	complemented
196	C₂×C₅	C₂×D₄×Q₈	1	1	yes	characteristic
197	C₂×C₅	C₂³×D₈	1	1	yes	characteristic
198	C₂×C₅	C₂×#578	1	1	yes	characteristic
199	D₅	—	4	8	no	—
200	C₂³	—	10	8	no	—
201	C₂³	—	10	4	no	—
202	C₂³	—	20	4	no	—
203	C₂³	—	20	4	no	—
204	C₂³	C₂²×D₂₀	1	1	yes	characteristic
205	Q₈	—	20	48	no	—
206	Q₈	C₂×D₄₀	1	16	yes	direct product factor
207	D₄	—	5	8	no	—
208	D₄	—	10	8	no	—
209	D₄	—	10	4	no	—
210	C₂×C₄	—	2	6	no	—
211	C₂×C₄	—	10	12	no	—
212	C₂×C₄	—	20	12	no	—
213	C₂×C₄	—	20	48	no	—
214	C₂×C₄	C₂×D₄₀	1	12	yes	—
215	C₂×C₄	C₂²×D₂₀	1	6	yes	—
216	C₂×C₄	C₂²×D₂₀	1	6	yes	—
217	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
218	C₂×C₄	C₂×Q₈×D₅	1	2	yes	—
219	C₂×C₄	#4150	1	2	yes	—
220	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
221	C₈	—	2	6	no	—
222	C₈	C₂×Q₈×D₅	1	4	yes	complemented
223	C₅	C₂×Q₈×D₈	1	1	yes	characteristic, complemented, Sylow
224	C₂²	—	10	8	no	—
225	C₂²	—	20	4	no	—
226	C₂²	—	20	4	no	—
227	C₂²	—	20	16	no	—
228	C₂²	Q₈×D₂₀	1	2	yes	—
229	C₂²	C₂²×D₄₀	1	2	yes	—
230	C₂²	#66087	1	2	yes	—
231	C₂²	C₂³×D₂₀	1	1	yes	characteristic
232	C₄	—	2	6	no	—
233	C₄	—	20	12	no	—
234	C₄	Q₈×D₂₀	1	2	yes	—
235	C₄	C₂²×D₄₀	1	12	yes	—
236	C₄	C₂²×Q₈×D₅	1	1	yes	characteristic
237	C₄	C₂×#4150	1	1	yes	characteristic
238	C₂	—	20	8	no	—
239	C₂	Q₈×D₄₀	1	4	yes	direct product factor
240	C₂	C₂×Q₈×D₂₀	1	1	yes	characteristic
241	C₂	C₂³×D₄₀	1	1	yes	characteristic
242	C₂	C₂×#66087	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂×C₅×C₈×Q₈	C₂
socle	C₂³×C₅	C₂²×D₄
prime radical	C₂²×D₄₀	C₂²
hypercenter	C₂×C₈×Q₈	D₅
hypocenter	C₅	C₂×Q₈×D₈
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₂₀	C₂	semidirect
2. C₂×C₅×C₈×Q₈	C₂	semidirect
3. Q₈×D₄₀	C₂	direct (N × H)
4. C₂×#65135	C₂	semidirect
6. C₂×#64971	C₂	semidirect
11. Q₈×D₂₀	C₂²	semidirect
15. C₅×C₈×Q₈	C₂²	semidirect
21. #65135	C₂²	semidirect
23. #64971	C₂²	semidirect
40. C₂×D₄₀	Q₈	direct (N × H)
56. C₂×C₈×Q₈	D₅	semidirect
59. C₂×C₅×Q₈	D₈	semidirect
68. C₂×D₂₀	C₂×Q₈	semidirect
75. C₂×C₅×C₈	C₂×Q₈	semidirect
85. D₄₀	C₂×Q₈	direct (N × H)
96. C₈×Q₈	C₂×D₅	semidirect
108. C₅×Q₈	C₂×D₈	semidirect
121. D₂₀	C₂²×Q₈	semidirect
125. C₅×C₈	C₂²×Q₈	semidirect
171. C₂×Q₈	D₄₀	direct (N × H)
185. C₂×C₈	Q₈×D₅	semidirect
195. C₂×C₅	Q₈×D₈	semidirect
206. Q₈	C₂×D₄₀	direct (N × H)
222. C₈	C₂×Q₈×D₅	semidirect
223. C₅	C₂×Q₈×D₈	semidirect
239. C₂	Q₈×D₄₀	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×Q₈×D₂₀	2
C₂×C₅×C₈×Q₈	2
Q₈×D₄₀	2
C₂×#65135	2
C₂×C₄×D₄₀	2
C₂×#64971	2
C₂×Q₈×D₈	5

Subgroups — C2×Q8×D40 #12979715

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×D₄₀ #12979715