Subgroups — C₂²×C₄×#681 #12962688

Order 1280.

3950 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	C₂³×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×#681	C₂	1	2	yes	maximal
3	C₂×C₄×#681	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#3935	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×#168	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₅	C₄	1	2	yes	—
7	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₅	C₄	1	12	yes	—
9	C₂²×C₄²×C₅	C₄	1	2	yes	—
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₅	C₂²	1	12	yes	—
12	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂²×#681	C₄	1	32	yes	direct product factor
14	C₂²×#681	C₂²	1	24	yes	—
15	C₂³×#168	C₂²	1	4	yes	—
16	C₄×#681	C₂²	1	64	yes	direct product factor
17	C₂×#3935	C₂²	1	24	yes	complemented
18	C₂×C₄×#168	C₂²	1	24	yes	complemented
19	C₂²×C₄×M₁₆	—	5	1	no	maximal, Sylow
20	C₂⁵×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₅	—	2	16	no	—
22	C₂³×C₄×C₅	C₂×C₄	1	16	yes	—
23	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₅	C₂×C₄	1	24	yes	—
25	C₂³×C₄×C₅	C₂×C₄	1	6	yes	—
26	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₅	C₂³	1	6	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	6	yes	—
31	C₂×C₄²×C₅	C₂×C₄	1	48	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	24	yes	—
33	C₂×C₄²×C₅	C₂×C₄	1	6	yes	—
34	C₂×C₄²×C₅	C₂³	1	6	yes	—
35	C₂×C₄²×C₅	C₂³	1	6	yes	—
36	C₂×C₄²×C₅	C₂³	1	16	yes	—
37	C₂×#681	C₂×C₄	1	192	yes	direct product factor
38	C₂×#681	C₂³	1	32	yes	—
39	C₂²×#168	C₂×C₄	1	32	yes	complemented
40	C₂²×#168	C₂³	1	24	yes	—
41	#3935	C₂³	1	32	yes	complemented
42	C₄×#168	C₂³	1	32	yes	complemented
43	C₂³×M₁₆	—	5	2	no	—
44	C₂³×C₄²	D₅	1	1	yes	characteristic
45	C₂×C₄×M₁₆	—	5	24	no	—
46	C₂²×#97	—	5	2	no	—
47	C₂²×C₄×C₈	—	5	2	no	—
48	C₂⁴×C₅	—	2	8	no	—
49	C₂⁴×C₅	C₄²	1	8	yes	—
50	C₂⁴×C₅	C₂²×C₄	1	6	yes	—
51	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	—	2	8	no	—
53	C₂²×C₄×C₅	—	2	96	no	—
54	C₂²×C₄×C₅	C₄²	1	48	yes	—
55	C₂²×C₄×C₅	C₄²	1	8	yes	—
56	C₂²×C₄×C₅	M₁₆	1	16	yes	complemented
57	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
58	C₂²×C₄×C₅	C₂²×C₄	1	48	yes	—
59	C₂²×C₄×C₅	C₂²×C₄	1	12	yes	—
60	C₂²×C₄×C₅	C₂²×C₄	1	32	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	12	yes	—
62	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
63	C₂²×C₄×C₅	C₂²×C₄	1	6	yes	—
64	C₂²×C₄×C₅	C₂²×C₄	1	12	yes	—
65	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
66	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
67	C₄²×C₅	C₂²×C₄	1	12	yes	—
68	C₄²×C₅	C₂²×C₄	1	32	yes	—
69	C₄²×C₅	C₂²×C₄	1	12	yes	—
70	C₄²×C₅	C₂⁴	1	4	yes	—
71	C₄²×C₅	C₂⁴	1	4	yes	—
72	#681	C₂²×C₄	1	128	yes	direct product factor
73	C₂×#168	C₂²×C₄	1	96	yes	complemented
74	C₂×#168	C₂⁴	1	16	yes	—
75	C₂⁴×C₄	Dic₅	1	2	yes	—
76	C₂⁴×C₄	C₂×D₅	1	1	yes	characteristic
77	C₂²×M₁₆	—	5	32	no	—
78	C₂²×M₁₆	—	5	24	no	—
79	C₂³×C₈	—	5	4	no	—
80	C₂²×C₄²	Dic₅	1	2	yes	—
81	C₂²×C₄²	Dic₅	1	12	yes	—
82	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
83	C₂²×C₄²	C₂×D₅	1	12	yes	—
84	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
85	C₄×M₁₆	—	5	64	no	—
86	C₂×#97	—	5	24	no	—
87	C₂×C₄×C₈	—	5	24	no	—
88	C₂³×C₅	—	2	24	no	—
89	C₂³×C₅	—	2	32	no	—
90	C₂³×C₅	C₂×C₄²	1	24	yes	—
91	C₂³×C₅	C₂×C₄²	1	4	yes	—
92	C₂³×C₅	C₂×M₁₆	1	8	yes	—
93	C₂³×C₅	C₂³×C₄	1	4	yes	—
94	C₂³×C₅	C₂³×C₄	1	3	yes	—
95	C₂×C₄×C₅	—	2	24	no	—
96	C₂×C₄×C₅	—	2	64	no	—
97	C₂×C₄×C₅	C₂×C₄²	1	48	yes	—
98	C₂×C₄×C₅	C₂×C₄²	1	12	yes	—
99	C₂×C₄×C₅	C₂×C₄²	1	24	yes	—
100	C₂×C₄×C₅	C₂×M₁₆	1	48	yes	complemented
101	C₂×C₄×C₅	C₂³×C₄	1	12	yes	—
102	C₂×C₄×C₅	C₂³×C₄	1	16	yes	—
103	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
104	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
105	C₂×C₄×C₅	C₂³×C₄	1	3	yes	—
106	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
107	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
108	#168	C₂³×C₄	1	32	yes	complemented
109	C₂⁵	C₂×Dic₅	1	1	yes	characteristic
110	C₂³×C₄	—	2	16	no	—
111	C₂³×C₄	C₄×D₅	1	16	yes	—
112	C₂³×C₄	C₂×Dic₅	1	2	yes	—
113	C₂³×C₄	C₂×Dic₅	1	24	yes	—
114	C₂³×C₄	C₂×Dic₅	1	2	yes	—
115	C₂³×C₄	C₂×Dic₅	1	6	yes	—
116	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
117	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
118	C₂³×C₄	C₂²×D₅	1	6	yes	—
119	C₂×M₁₆	—	5	192	no	—
120	C₂×M₁₆	—	5	32	no	—
121	C₂²×C₈	—	5	32	no	—
122	C₂²×C₈	—	5	24	no	—
123	C₂×C₄²	C₂×Dic₅	1	6	yes	—
124	C₂×C₄²	C₂×Dic₅	1	48	yes	—
125	C₂×C₄²	C₂×Dic₅	1	24	yes	—
126	C₂×C₄²	C₂×Dic₅	1	6	yes	—
127	C₂×C₄²	C₂²×D₅	1	6	yes	—
128	C₂×C₄²	C₂²×D₅	1	6	yes	—
129	C₂×C₄²	C₂²×D₅	1	16	yes	—
130	#97	—	5	32	no	—
131	C₄×C₈	—	5	32	no	—
132	C₂²×C₅	—	2	48	no	—
133	C₂²×C₅	—	2	8	no	—
134	C₂²×C₅	C₄×M₁₆	1	16	yes	complemented
135	C₂²×C₅	C₂²×C₄²	1	8	yes	—
136	C₂²×C₅	C₂²×C₄²	1	6	yes	—
137	C₂²×C₅	C₂²×M₁₆	1	12	yes	—
138	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
139	C₄×C₅	—	2	8	no	—
140	C₄×C₅	C₂²×C₄²	1	6	yes	—
141	C₄×C₅	C₂²×C₄²	1	8	yes	—
142	C₄×C₅	C₂²×M₁₆	1	16	yes	complemented
143	C₄×C₅	C₂⁴×C₄	1	2	yes	—
144	C₂⁴	—	2	8	no	—
145	C₂⁴	C₄×Dic₅	1	8	yes	—
146	C₂⁴	C₂²×Dic₅	1	6	yes	—
147	C₂⁴	C₂²×Dic₅	1	1	yes	characteristic
148	C₂²×C₄	—	2	8	no	—
149	C₂²×C₄	—	2	96	no	—
150	C₂²×C₄	#681	1	16	yes	direct product factor
151	C₂²×C₄	C₄×Dic₅	1	48	yes	—
152	C₂²×C₄	C₄×Dic₅	1	8	yes	—
153	C₂²×C₄	C₂×C₄×D₅	1	8	yes	—
154	C₂²×C₄	C₂×C₄×D₅	1	48	yes	—
155	C₂²×C₄	C₂²×Dic₅	1	12	yes	—
156	C₂²×C₄	C₂²×Dic₅	1	32	yes	—
157	C₂²×C₄	C₂²×Dic₅	1	12	yes	—
158	C₂²×C₄	C₂²×Dic₅	1	12	yes	—
159	C₂²×C₄	C₂²×Dic₅	1	6	yes	—
160	C₂²×C₄	C₂²×Dic₅	1	3	yes	—
161	C₂²×C₄	C₂³×D₅	1	3	yes	—
162	C₂²×C₄	C₂³×D₅	1	4	yes	—
163	M₁₆	—	5	128	no	—
164	C₂×C₈	—	5	96	no	—
165	C₂×C₈	—	5	16	no	—
166	C₄²	C₂²×Dic₅	1	12	yes	—
167	C₄²	C₂²×Dic₅	1	32	yes	—
168	C₄²	C₂²×Dic₅	1	12	yes	—
169	C₄²	C₂³×D₅	1	4	yes	—
170	C₄²	C₂³×D₅	1	4	yes	—
171	C₂×C₅	—	2	8	no	—
172	C₂×C₅	C₂×C₄×M₁₆	1	12	yes	complemented
173	C₂×C₅	C₂³×C₄²	1	1	yes	characteristic
174	C₂×C₅	C₂³×M₁₆	1	2	yes	—
175	C₂³	—	2	24	no	—
176	C₂³	—	2	32	no	—
177	C₂³	C₂×#681	1	8	yes	—
178	C₂³	C₂×C₄×Dic₅	1	24	yes	—
179	C₂³	C₂×C₄×Dic₅	1	4	yes	—
180	C₂³	C₂³×Dic₅	1	4	yes	—
181	C₂³	C₂³×Dic₅	1	3	yes	—
182	C₂×C₄	—	2	24	no	—
183	C₂×C₄	—	2	64	no	—
184	C₂×C₄	C₂×#681	1	48	yes	direct product factor
185	C₂×C₄	C₂×C₄×Dic₅	1	48	yes	—
186	C₂×C₄	C₂×C₄×Dic₅	1	12	yes	—
187	C₂×C₄	C₂×C₄×Dic₅	1	24	yes	—
188	C₂×C₄	C₂²×C₄×D₅	1	12	yes	—
189	C₂×C₄	C₂²×C₄×D₅	1	16	yes	—
190	C₂×C₄	C₂³×Dic₅	1	8	yes	—
191	C₂×C₄	C₂³×Dic₅	1	8	yes	—
192	C₂×C₄	C₂³×Dic₅	1	3	yes	—
193	C₂×C₄	C₂³×Dic₅	1	4	yes	—
194	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
195	C₈	—	5	32	no	—
196	C₅	C₂²×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
197	C₂²	—	2	48	no	—
198	C₂²	—	2	8	no	—
199	C₂²	C₄×#681	1	16	yes	direct product factor
200	C₂²	C₂²×#681	1	12	yes	—
201	C₂²	C₂²×C₄×Dic₅	1	8	yes	—
202	C₂²	C₂²×C₄×Dic₅	1	6	yes	—
203	C₂²	C₂⁴×Dic₅	1	1	yes	characteristic
204	C₄	—	2	8	no	—
205	C₄	C₂²×#681	1	16	yes	direct product factor
206	C₄	C₂²×C₄×Dic₅	1	6	yes	—
207	C₄	C₂²×C₄×Dic₅	1	8	yes	—
208	C₄	C₂³×C₄×D₅	1	2	yes	—
209	C₂	—	2	8	no	—
210	C₂	C₂×C₄×#681	1	12	yes	direct product factor
211	C₂	C₂³×#681	1	2	yes	—
212	C₂	C₂³×C₄×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄²	C₂×D₅
commutator	C₂×C₅	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂³×C₄²×C₅	C₂
socle	C₂⁴×C₅	C₂²×C₄
prime radical	C₂⁵×C₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂⁵
hypercenter	C₂³×C₄²	D₅
hypocenter	C₅	C₂²×C₄×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄² ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×#681	C₂	direct (N × H)
4. C₂²×#3935	C₂	semidirect
5. C₂²×C₄×#168	C₂	semidirect
13. C₂²×#681	C₄	direct (N × H)
16. C₄×#681	C₂²	direct (N × H)
17. C₂×#3935	C₂²	semidirect
18. C₂×C₄×#168	C₂²	semidirect
37. C₂×#681	C₂×C₄	direct (N × H)
39. C₂²×#168	C₂×C₄	semidirect
41. #3935	C₂³	semidirect
42. C₄×#168	C₂³	semidirect
56. C₂²×C₄×C₅	M₁₆	semidirect
72. #681	C₂²×C₄	direct (N × H)
73. C₂×#168	C₂²×C₄	semidirect
100. C₂×C₄×C₅	C₂×M₁₆	semidirect
108. #168	C₂³×C₄	semidirect
134. C₂²×C₅	C₄×M₁₆	semidirect
142. C₄×C₅	C₂²×M₁₆	semidirect
150. C₂²×C₄	#681	direct (N × H)
172. C₂×C₅	C₂×C₄×M₁₆	semidirect
184. C₂×C₄	C₂×#681	direct (N × H)
196. C₅	C₂²×C₄×M₁₆	semidirect
199. C₂²	C₄×#681	direct (N × H)
205. C₄	C₂²×#681	direct (N × H)
210. C₂	C₂×C₄×#681	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×C₄²×C₅	2
C₂³×#681	2
C₂×C₄×#681	2
C₂²×#3935	2
C₂²×C₄×#168	2
C₂²×C₄×M₁₆	5

Subgroups — C22×C4×#681 #12962688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#681 #12962688