Subgroups — C₂²×#65163 #12968897

Order 1280.

2398 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	C₂²×#4034	C₂	1	2	yes	maximal
2	C₂²×C₅×#107	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3950	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₂²×#3939	C₂	1	2	yes	maximal
6	C₂×#65163	C₂	1	24	yes	maximal, direct product factor
7	C₂³×Dic₁₀	—	2	2	no	—
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂²×C₄⋊Dic₅	C₂²	1	2	yes	—
11	C₂²×#683	—	2	2	no	—
12	C₂²×C₄×Dic₅	C₂²	1	1	yes	characteristic
13	C₂³×#168	C₂²	1	1	yes	characteristic
14	C₂×#4034	—	2	12	no	—
15	C₂×#4034	C₂²	1	24	yes	—
16	C₂×C₅×#107	C₂²	1	12	yes	—
17	C₂×#3950	C₂²	1	12	yes	—
18	C₂×C₈×Dic₅	C₂²	1	12	yes	—
19	C₂×#3939	C₂²	1	24	yes	—
20	#65163	C₂²	1	64	yes	direct product factor
21	C₂²×#500	—	5	1	no	maximal, Sylow
22	C₂³×C₄×C₅	—	2	2	no	—
23	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂³×Dic₅	—	2	2	no	—
25	C₂³×Dic₅	D₄	1	1	yes	characteristic
26	C₂³×Dic₅	D₄	1	1	yes	characteristic
27	C₂²×Dic₁₀	—	2	24	no	—
28	C₂²×Dic₁₀	—	4	16	no	—
29	C₂²×C₅×C₈	Q₈	1	8	yes	complemented
30	C₂²×C₅×C₈	C₂³	1	6	yes	—
31	C₂×C₅×C₄⋊C₄	—	2	6	no	—
32	C₂×C₅×C₄⋊C₄	C₂³	1	12	yes	—
33	C₂×C₄⋊Dic₅	—	2	6	no	—
34	C₂×C₄⋊Dic₅	C₂³	1	12	yes	—
35	C₂×#683	—	2	24	no	—
36	C₂×C₄×Dic₅	D₄	1	6	yes	—
37	C₂×C₄×Dic₅	C₂³	1	6	yes	—
38	C₂²×#168	Q₈	1	8	yes	complemented
39	C₂²×#168	C₂³	1	6	yes	—
40	#4034	—	2	48	no	—
41	#4034	C₂³	1	32	yes	—
42	C₅×#107	C₂³	1	16	yes	—
43	#3950	C₂³	1	16	yes	—
44	C₈×Dic₅	C₂³	1	16	yes	—
45	#3939	C₂³	1	32	yes	—
46	C₂²×C₄⋊Q₈	—	5	2	no	—
47	C₂×#500	—	5	24	no	—
48	C₂²×#107	—	5	1	no	—
49	C₂²×#107	—	5	2	no	—
50	C₂²×#107	D₅	1	1	yes	characteristic
51	C₂²×C₄×C₈	—	5	1	no	—
52	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	—	2	12	no	—
54	C₂²×C₄×C₅	—	4	8	no	—
55	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
58	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
59	C₂²×Dic₅	—	2	12	no	—
60	C₂²×Dic₅	—	4	8	no	—
61	C₂²×Dic₅	D₈	1	8	yes	—
62	C₂²×Dic₅	Q₁₆	1	8	yes	—
63	C₂²×Dic₅	C₂×D₄	1	6	yes	—
64	C₂²×Dic₅	C₂×D₄	1	6	yes	—
65	C₂×Dic₁₀	—	2	32	no	—
66	C₂×Dic₁₀	—	4	96	no	—
67	C₂×C₅×C₈	C₂×Q₈	1	24	yes	complemented
68	C₂×C₅×C₈	C₂⁴	1	4	yes	—
69	C₅×C₄⋊C₄	—	2	12	no	—
70	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
71	C₄⋊Dic₅	—	2	12	no	—
72	C₄⋊Dic₅	C₂⁴	1	8	yes	—
73	#683	—	2	32	no	—
74	C₄×Dic₅	C₂×D₄	1	12	yes	—
75	C₄×Dic₅	C₂⁴	1	4	yes	—
76	C₂×#168	C₂×Q₈	1	24	yes	complemented
77	C₂×#168	C₂⁴	1	4	yes	—
78	C₂³×Q₈	—	10	2	no	—
79	C₂³×C₈	—	5	1	no	—
80	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
81	C₂×C₄⋊Q₈	—	5	24	no	—
82	C₂×C₄⋊Q₈	—	10	12	no	—
83	C₂²×C₄⋊C₄	—	5	2	no	—
84	C₂²×C₄⋊C₄	—	10	2	no	—
85	C₂²×C₄⋊C₄	C₂×D₅	1	2	yes	—
86	C₂²×C₄²	—	5	1	no	—
87	#500	—	5	64	no	—
88	C₂×#107	—	5	12	no	—
89	C₂×#107	—	5	24	no	—
90	C₂×#107	C₂×D₅	1	12	yes	—
91	C₂×C₄×C₈	—	5	12	no	—
92	C₂³×C₅	C₄⋊Q₈	1	4	yes	—
93	C₂³×C₅	C₂×D₈	1	4	yes	—
94	C₂³×C₅	C₂×Q₁₆	1	4	yes	—
95	C₂³×C₅	C₂²×D₄	1	3	yes	—
96	C₂×C₄×C₅	—	2	8	no	—
97	C₂×C₄×C₅	—	4	24	no	—
98	C₂×C₄×C₅	C₄⋊Q₈	1	12	yes	—
99	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
100	C₂×C₄×C₅	C₂²×Q₈	1	6	yes	—
101	C₂×C₄×C₅	C₂²×Q₈	1	6	yes	—
102	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
103	C₂×Dic₅	—	2	8	no	—
104	C₂×Dic₅	—	4	24	no	—
105	C₂×Dic₅	C₂×D₈	1	24	yes	—
106	C₂×Dic₅	C₂×Q₁₆	1	24	yes	—
107	C₂×Dic₅	C₂²×D₄	1	4	yes	—
108	C₂×Dic₅	C₂²×D₄	1	4	yes	—
109	Dic₁₀	—	4	64	no	—
110	C₅×C₈	C₂²×Q₈	1	8	yes	complemented
111	#168	C₂²×Q₈	1	8	yes	complemented
112	C₂²×Q₈	—	10	24	no	—
113	C₂²×Q₈	—	20	16	no	—
114	C₂³×C₄	—	2	2	no	—
115	C₂³×C₄	—	5	1	no	—
116	C₂³×C₄	—	5	1	no	—
117	C₂³×C₄	—	10	2	no	—
118	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
119	C₂²×C₈	—	5	8	no	—
120	C₂²×C₈	—	5	6	no	—
121	C₂²×C₈	Dic₁₀	1	8	yes	complemented
122	C₂²×C₈	C₂²×D₅	1	6	yes	—
123	C₄⋊Q₈	—	5	32	no	—
124	C₄⋊Q₈	—	10	48	no	—
125	C₂×C₄⋊C₄	—	2	6	no	—
126	C₂×C₄⋊C₄	—	5	12	no	—
127	C₂×C₄⋊C₄	—	10	24	no	—
128	C₂×C₄⋊C₄	—	10	6	no	—
129	C₂×C₄⋊C₄	C₂²×D₅	1	12	yes	—
130	C₂×C₄²	—	5	6	no	—
131	C₂×C₄²	—	5	6	no	—
132	#107	—	5	16	no	—
133	#107	—	5	32	no	—
134	#107	C₂²×D₅	1	16	yes	—
135	C₄×C₈	—	5	16	no	—
136	C₂²×C₅	#500	1	16	yes	complemented
137	C₂²×C₅	C₂×C₄⋊Q₈	1	6	yes	—
138	C₂²×C₅	C₂²×D₈	1	6	yes	—
139	C₂²×C₅	C₂²×Q₁₆	1	6	yes	—
140	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
141	C₄×C₅	—	4	8	no	—
142	C₄×C₅	C₂×C₄⋊Q₈	1	6	yes	—
143	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
144	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
145	Dic₅	—	4	8	no	—
146	Dic₅	C₂²×D₈	1	8	yes	—
147	Dic₅	C₂²×Q₁₆	1	8	yes	—
148	C₂⁴	D₄×D₅	1	1	yes	characteristic
149	C₂×Q₈	—	10	32	no	—
150	C₂×Q₈	—	20	96	no	—
151	C₂²×C₄	—	2	12	no	—
152	C₂²×C₄	—	4	8	no	—
153	C₂²×C₄	—	5	6	no	—
154	C₂²×C₄	—	5	6	no	—
155	C₂²×C₄	—	5	8	no	—
156	C₂²×C₄	—	5	8	no	—
157	C₂²×C₄	—	10	12	no	—
158	C₂²×C₄	—	20	8	no	—
159	C₂²×C₄	C₂×Dic₁₀	1	4	yes	—
160	C₂²×C₄	D₄×D₅	1	3	yes	—
161	C₂²×C₄	Q₈×D₅	1	4	yes	—
162	C₂²×C₄	C₂³×D₅	1	3	yes	—
163	C₂×C₈	—	5	24	no	—
164	C₂×C₈	—	5	4	no	—
165	C₂×C₈	C₂×Dic₁₀	1	24	yes	complemented
166	C₂×C₈	C₂³×D₅	1	4	yes	—
167	C₄⋊C₄	—	2	12	no	—
168	C₄⋊C₄	—	5	8	no	—
169	C₄⋊C₄	—	10	32	no	—
170	C₄⋊C₄	—	10	12	no	—
171	C₄⋊C₄	C₂³×D₅	1	8	yes	—
172	C₄²	—	5	12	no	—
173	C₄²	—	5	4	no	—
174	C₂×C₅	C₂×#500	1	12	yes	complemented
175	C₂×C₅	C₂²×C₄⋊Q₈	1	1	yes	characteristic
176	C₂×C₅	C₂³×D₈	1	1	yes	characteristic
177	C₂×C₅	C₂³×Q₁₆	1	1	yes	characteristic
178	C₂³	#4034	1	4	yes	—
179	C₂³	D₅×D₈	1	4	yes	—
180	C₂³	D₅×Q₁₆	1	4	yes	—
181	C₂³	C₂×D₄×D₅	1	3	yes	—
182	Q₈	—	20	64	no	—
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	4	24	no	—
185	C₂×C₄	—	5	24	no	—
186	C₂×C₄	—	5	24	no	—
187	C₂×C₄	—	5	4	no	—
188	C₂×C₄	—	5	4	no	—
189	C₂×C₄	—	10	8	no	—
190	C₂×C₄	—	20	24	no	—
191	C₂×C₄	#4034	1	12	yes	—
192	C₂×C₄	C₂²×Dic₁₀	1	6	yes	—
193	C₂×C₄	C₂×D₄×D₅	1	3	yes	—
194	C₂×C₄	C₂×Q₈×D₅	1	6	yes	—
195	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
196	C₈	—	5	8	no	—
197	C₈	C₂²×Dic₁₀	1	8	yes	complemented
198	C₅	C₂²×#500	1	1	yes	characteristic, complemented, Sylow
199	C₂²	#65163	1	16	yes	direct product factor
200	C₂²	C₂×#4034	1	6	yes	—
201	C₂²	C₂×D₅×D₈	1	6	yes	—
202	C₂²	C₂×D₅×Q₁₆	1	6	yes	—
203	C₂²	C₂²×D₄×D₅	1	1	yes	characteristic
204	C₄	—	4	8	no	—
205	C₄	—	5	8	no	—
206	C₄	—	5	8	no	—
207	C₄	—	20	8	no	—
208	C₄	C₂×#4034	1	6	yes	—
209	C₄	C₂³×Dic₁₀	1	1	yes	characteristic
210	C₄	C₂²×Q₈×D₅	1	1	yes	characteristic
211	C₂	C₂×#65163	1	12	yes	direct product factor
212	C₂	C₂²×#4034	1	1	yes	characteristic
213	C₂	C₂²×D₅×D₈	1	1	yes	characteristic
214	C₂	C₂²×D₅×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂²×C₅×#107	C₂
socle, prime radical	C₂⁴×C₅	C₂×D₄
hypercenter	C₂²×#107	D₅
hypocenter	C₅	C₂²×#500
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
6. C₂×#65163	C₂	direct (N × H)
20. #65163	C₂²	direct (N × H)
29. C₂²×C₅×C₈	Q₈	semidirect
38. C₂²×#168	Q₈	semidirect
67. C₂×C₅×C₈	C₂×Q₈	semidirect
76. C₂×#168	C₂×Q₈	semidirect
110. C₅×C₈	C₂²×Q₈	semidirect
111. #168	C₂²×Q₈	semidirect
121. C₂²×C₈	Dic₁₀	semidirect
136. C₂²×C₅	#500	semidirect
165. C₂×C₈	C₂×Dic₁₀	semidirect
174. C₂×C₅	C₂×#500	semidirect
197. C₈	C₂²×Dic₁₀	semidirect
198. C₅	C₂²×#500	semidirect
199. C₂²	#65163	direct (N × H)
211. C₂	C₂×#65163	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×#4034	2
C₂²×C₅×#107	2
C₂²×#3950	2
C₂²×C₈×Dic₅	2
C₂²×#3939	2
C₂×#65163	2
C₂²×#500	5

Subgroups — C22×#65163 #12968897

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#65163 #12968897