Subgroups — C₂²×#65193 #13005787

Order 1280.

2118 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#4050	C₂	1	2	yes	maximal, complemented
2	C₂³×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×Dic₄₀	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#65193	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#3932	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈₀	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#709	—	2	2	no	—
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂²×Dic₂₀	C₂²	1	2	yes	—
10	C₂×#4050	—	2	12	no	—
11	C₂×#4050	C₂²	1	24	yes	complemented
12	C₂²×D₄₀	C₂²	1	2	yes	—
13	C₂²×#677	—	2	2	no	—
14	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₅×C₁₆	C₂²	1	12	yes	complemented
17	C₂×Dic₄₀	C₂²	1	12	yes	complemented
18	#65193	C₂²	1	64	yes	direct product factor
19	C₂×#3932	C₂²	1	24	yes	complemented
20	C₂×D₈₀	C₂²	1	12	yes	complemented
21	C₂²×#508	—	5	1	no	maximal, Sylow
22	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂²×C₂²⋊D₅	—	4	2	no	—
24	C₂×#709	—	2	24	no	—
25	C₂×#709	—	4	12	no	—
26	C₂²×D₂₀	—	2	2	no	—
27	C₂²×C₄×D₅	—	4	2	no	—
28	C₂²×Dic₁₀	—	2	2	no	—
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	D₄	1	6	yes	—
31	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂³	1	6	yes	—
33	C₂×Dic₂₀	—	2	6	no	—
34	C₂×Dic₂₀	C₂³	1	12	yes	—
35	#4050	—	2	48	no	—
36	#4050	C₂³	1	32	yes	complemented
37	C₂×D₄₀	—	2	6	no	—
38	C₂×D₄₀	C₂³	1	12	yes	—
39	C₂×#677	—	2	24	no	—
40	C₂×C₅×C₁₆	C₂³	1	6	yes	—
41	C₂×C₅×C₁₆	C₂³	1	6	yes	—
42	C₂×C₅×C₁₆	C₂³	1	16	yes	complemented
43	Dic₄₀	C₂³	1	16	yes	complemented
44	#3932	C₂³	1	32	yes	complemented
45	D₈₀	C₂³	1	16	yes	complemented
46	C₂²×#135	—	5	2	no	—
47	C₂×#508	—	5	24	no	—
48	C₂²×Q₃₂	—	5	1	no	—
49	C₂²×SD₃₂	—	5	2	no	—
50	C₂²×D₁₆	—	5	1	no	—
51	C₂³×C₁₆	D₅	1	1	yes	characteristic, complemented
52	C₂⁴×C₅	D₈	1	1	yes	characteristic
53	C₂³×D₅	—	4	2	no	—
54	C₂²×C₄×C₅	D₈	1	6	yes	—
55	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
58	C₂×C₂²⋊D₅	—	4	24	no	—
59	C₂²×Dic₅	—	4	2	no	—
60	#709	—	2	32	no	—
61	#709	—	4	48	no	—
62	C₂×D₂₀	—	2	12	no	—
63	C₂×D₂₀	—	4	6	no	—
64	C₂×C₄×D₅	—	4	24	no	—
65	C₂×Dic₁₀	—	2	12	no	—
66	C₂×Dic₁₀	—	4	6	no	—
67	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
68	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
69	C₂×C₅×C₈	C₂×D₄	1	12	yes	—
70	C₂×C₅×C₈	C₂⁴	1	3	yes	—
71	C₂×C₅×C₈	C₂⁴	1	4	yes	—
72	Dic₂₀	—	2	12	no	—
73	Dic₂₀	C₂⁴	1	8	yes	—
74	D₄₀	—	2	12	no	—
75	D₄₀	C₂⁴	1	8	yes	—
76	#677	—	2	32	no	—
77	C₅×C₁₆	C₂⁴	1	4	yes	—
78	C₅×C₁₆	C₂⁴	1	4	yes	—
79	C₂²×#41	—	10	2	no	—
80	C₂×#135	—	5	24	no	—
81	C₂×#135	—	10	12	no	—
82	C₂²×Q₁₆	—	5	2	no	—
83	C₂²×SD₁₆	—	10	2	no	—
84	C₂²×D₈	—	5	2	no	—
85	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
86	#508	—	5	64	no	—
87	C₂×Q₃₂	—	5	12	no	—
88	C₂×SD₃₂	—	5	24	no	—
89	C₂×D₁₆	—	5	12	no	—
90	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
91	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
92	C₂²×C₁₆	C₂×D₅	1	12	yes	complemented
93	C₂³×C₅	—	2	4	no	—
94	C₂³×C₅	C₂×D₈	1	6	yes	—
95	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
96	C₂²×D₅	—	4	12	no	—
97	C₂²×D₅	—	8	8	no	—
98	C₂×C₄×C₅	C₂×D₈	1	3	yes	—
99	C₂×C₄×C₅	C₂×D₈	1	6	yes	—
100	C₂×C₄×C₅	C₂×D₈	1	12	yes	—
101	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
102	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
103	C₂²⋊D₅	—	4	32	no	—
104	C₂×Dic₅	—	4	12	no	—
105	D₂₀	—	2	8	no	—
106	D₂₀	—	4	12	no	—
107	C₄×D₅	—	4	32	no	—
108	Dic₁₀	—	2	8	no	—
109	Dic₁₀	—	4	12	no	—
110	C₅×C₈	C₂²×D₄	1	4	yes	—
111	C₅×C₈	C₂²×D₄	1	3	yes	—
112	C₅×C₈	C₂⁵	1	1	yes	characteristic
113	C₂×#41	—	10	24	no	—
114	C₂×#41	—	20	12	no	—
115	C₂²×Q₈	—	10	2	no	—
116	C₂²×D₄	—	10	2	no	—
117	C₂²×D₄	—	20	2	no	—
118	C₂³×C₄	—	20	2	no	—
119	C₂³×C₄	D₂₀	1	1	yes	characteristic
120	#135	—	5	32	no	—
121	#135	—	10	48	no	—
122	C₂×Q₁₆	—	5	12	no	—
123	C₂×Q₁₆	—	10	6	no	—
124	C₂×SD₁₆	—	10	24	no	—
125	C₂×D₈	—	5	12	no	—
126	C₂×D₈	—	10	6	no	—
127	C₂²×C₈	D₂₀	1	1	yes	characteristic
128	C₂²×C₈	D₂₀	1	6	yes	—
129	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₅	1	6	yes	—
131	Q₃₂	—	5	16	no	—
132	SD₃₂	—	5	32	no	—
133	D₁₆	—	5	16	no	—
134	C₂×C₁₆	C₂²×D₅	1	6	yes	—
135	C₂×C₁₆	C₂²×D₅	1	6	yes	—
136	C₂×C₁₆	C₂²×D₅	1	16	yes	complemented
137	C₂²×C₅	—	2	12	no	—
138	C₂²×C₅	#508	1	4	yes	complemented
139	C₂²×C₅	C₂²×D₈	1	4	yes	—
140	C₂²×C₅	C₂²×D₈	1	3	yes	—
141	C₂×D₅	—	4	8	no	—
142	C₂×D₅	—	8	24	no	—
143	C₄×C₅	C₂²×D₈	1	3	yes	—
144	C₄×C₅	C₂²×D₈	1	4	yes	—
145	C₄×C₅	C₂³×D₄	1	1	yes	characteristic
146	Dic₅	—	4	8	no	—
147	C₂⁴	—	20	2	no	—
148	C₂⁴	D₄₀	1	1	yes	characteristic
149	#41	—	10	32	no	—
150	#41	—	20	48	no	—
151	C₂×Q₈	—	10	12	no	—
152	C₂×Q₈	—	20	6	no	—
153	C₂×D₄	—	10	12	no	—
154	C₂×D₄	—	20	24	no	—
155	C₂×D₄	—	20	6	no	—
156	C₂²×C₄	—	20	2	no	—
157	C₂²×C₄	—	20	24	no	—
158	C₂²×C₄	D₄₀	1	6	yes	—
159	C₂²×C₄	D₄₀	1	1	yes	characteristic
160	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₂₀	1	6	yes	—
162	Q₁₆	—	5	8	no	—
163	Q₁₆	—	10	12	no	—
164	SD₁₆	—	10	32	no	—
165	D₈	—	5	8	no	—
166	D₈	—	10	12	no	—
167	C₂×C₈	C₂×D₂₀	1	3	yes	—
168	C₂×C₈	C₂×D₂₀	1	6	yes	—
169	C₂×C₈	C₂×D₂₀	1	12	yes	—
170	C₂×C₈	C₂³×D₅	1	3	yes	—
171	C₂×C₈	C₂³×D₅	1	4	yes	—
172	C₁₆	C₂³×D₅	1	4	yes	—
173	C₁₆	C₂³×D₅	1	4	yes	—
174	C₂×C₅	—	2	4	no	—
175	C₂×C₅	C₂×#508	1	6	yes	complemented
176	C₂×C₅	C₂³×D₈	1	1	yes	characteristic
177	D₅	—	8	8	no	—
178	C₂³	—	2	4	no	—
179	C₂³	—	20	12	no	—
180	C₂³	—	40	8	no	—
181	C₂³	C₂×D₄₀	1	6	yes	—
182	C₂³	C₂×D₄₀	1	1	yes	characteristic
183	Q₈	—	10	8	no	—
184	Q₈	—	20	12	no	—
185	D₄	—	10	8	no	—
186	D₄	—	20	32	no	—
187	D₄	—	20	12	no	—
188	C₂×C₄	—	20	12	no	—
189	C₂×C₄	—	20	32	no	—
190	C₂×C₄	C₂×D₄₀	1	6	yes	—
191	C₂×C₄	C₂×D₄₀	1	3	yes	—
192	C₂×C₄	C₂×D₄₀	1	12	yes	—
193	C₂×C₄	C₂²×D₂₀	1	3	yes	—
194	C₂×C₄	C₂²×D₂₀	1	4	yes	—
195	C₈	C₂²×D₂₀	1	3	yes	—
196	C₈	C₂²×D₂₀	1	4	yes	—
197	C₈	C₂⁴×D₅	1	1	yes	characteristic
198	C₅	C₂²×#508	1	1	yes	characteristic, complemented, Sylow
199	C₂²	—	2	12	no	—
200	C₂²	—	20	8	no	—
201	C₂²	—	40	24	no	—
202	C₂²	#65193	1	4	yes	direct product factor
203	C₂²	C₂²×D₄₀	1	4	yes	—
204	C₂²	C₂²×D₄₀	1	3	yes	—
205	C₄	—	20	8	no	—
206	C₄	C₂²×D₄₀	1	4	yes	—
207	C₄	C₂²×D₄₀	1	3	yes	—
208	C₄	C₂³×D₂₀	1	1	yes	characteristic
209	C₂	—	2	4	no	—
210	C₂	—	40	8	no	—
211	C₂	C₂×#65193	1	6	yes	direct product factor
212	C₂	C₂³×D₄₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄₀
commutator, cosocle	C₅×C₈	C₂⁵
Frattini	C₈	C₂⁴×D₅
Fitting	C₂³×C₅×C₁₆	C₂
socle	C₂³×C₅	C₂×D₈
hypercenter	C₂³×C₁₆	D₅
hypocenter	C₅	C₂²×#508
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#4050	C₂	semidirect
2. C₂³×C₅×C₁₆	C₂	semidirect
3. C₂²×Dic₄₀	C₂	semidirect
4. C₂×#65193	C₂	direct (N × H)
5. C₂²×#3932	C₂	semidirect
6. C₂²×D₈₀	C₂	semidirect
11. C₂×#4050	C₂²	semidirect
16. C₂²×C₅×C₁₆	C₂²	semidirect
17. C₂×Dic₄₀	C₂²	semidirect
18. #65193	C₂²	direct (N × H)
19. C₂×#3932	C₂²	semidirect
20. C₂×D₈₀	C₂²	semidirect
36. #4050	C₂³	semidirect
42. C₂×C₅×C₁₆	C₂³	semidirect
43. Dic₄₀	C₂³	semidirect
44. #3932	C₂³	semidirect
45. D₈₀	C₂³	semidirect
51. C₂³×C₁₆	D₅	semidirect
92. C₂²×C₁₆	C₂×D₅	semidirect
136. C₂×C₁₆	C₂²×D₅	semidirect
138. C₂²×C₅	#508	semidirect
175. C₂×C₅	C₂×#508	semidirect
198. C₅	C₂²×#508	semidirect
202. C₂²	#65193	direct (N × H)
211. C₂	C₂×#65193	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×#4050	2
C₂³×C₅×C₁₆	2
C₂²×Dic₄₀	2
C₂×#65193	2
C₂²×#3932	2
C₂²×D₈₀	2
C₂²×#508	5

Subgroups — C22×#65193 #13005787

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#65193 #13005787