Subgroups — C₂²×#65266 #12257574

Order 1280.

4286 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂²×#3963	C₂	1	4	yes	maximal
5	C₂×#65266	C₂	1	24	yes	maximal, direct product factor
6	C₂⁴×C₄×C₅	C₂²	1	2	yes	—
7	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₅	C₂²	1	2	yes	—
9	C₂⁴×Dic₅	C₂²	1	2	yes	—
10	C₂²×C₅×C₄⋊C₄	—	2	8	no	—
11	C₂²×C₅×C₄⋊C₄	C₂²	1	12	yes	—
12	C₂²×C₄⋊Dic₅	—	2	8	no	—
13	C₂²×C₄⋊Dic₅	C₂²	1	12	yes	—
14	C₂²×C₄×Dic₅	C₄	1	16	yes	complemented
15	C₂²×C₄×Dic₅	C₂²	1	12	yes	—
16	C₂×#3963	C₂²	1	48	yes	—
17	#65266	C₂²	1	64	yes	direct product factor
18	C₂²×#382	—	5	1	no	maximal, Sylow
19	C₂⁵×C₅	C₂³	1	1	yes	characteristic
20	C₂³×C₄×C₅	—	2	8	no	—
21	C₂³×C₄×C₅	—	2	16	no	—
22	C₂³×C₄×C₅	C₂×C₄	1	8	yes	—
23	C₂³×C₄×C₅	D₄	1	8	yes	—
24	C₂³×C₄×C₅	Q₈	1	8	yes	—
25	C₂³×C₄×C₅	C₂³	1	12	yes	—
26	C₂³×C₄×C₅	C₂³	1	6	yes	—
27	C₂³×Dic₅	—	2	8	no	—
28	C₂³×Dic₅	—	2	8	no	—
29	C₂³×Dic₅	—	2	8	no	—
30	C₂³×Dic₅	—	2	16	no	—
31	C₂³×Dic₅	C₂×C₄	1	16	yes	—
32	C₂³×Dic₅	C₂³	1	12	yes	—
33	C₂³×Dic₅	C₂³	1	12	yes	—
34	C₂×C₅×C₄⋊C₄	—	2	48	no	—
35	C₂×C₅×C₄⋊C₄	C₂³	1	16	yes	—
36	C₂×C₄⋊Dic₅	—	2	48	no	—
37	C₂×C₄⋊Dic₅	C₂³	1	16	yes	—
38	C₂×C₄×Dic₅	C₂×C₄	1	96	yes	complemented
39	C₂×C₄×Dic₅	C₂³	1	16	yes	—
40	#3963	C₂³	1	64	yes	—
41	C₂³×C₄⋊C₄	—	5	1	no	—
42	C₂³×C₄⋊C₄	D₅	1	1	yes	characteristic
43	C₂³×C₄²	—	5	1	no	—
44	C₂×#382	—	5	24	no	—
45	C₂²×#95	—	5	4	no	—
46	C₂⁴×C₅	C₂²×C₄	1	4	yes	—
47	C₂⁴×C₅	C₂×D₄	1	4	yes	—
48	C₂⁴×C₅	C₂×Q₈	1	4	yes	—
49	C₂⁴×C₅	#41	1	8	yes	—
50	C₂⁴×C₅	#41	1	8	yes	—
51	C₂⁴×C₅	C₂⁴	1	3	yes	—
52	C₂²×C₄×C₅	—	2	24	no	—
53	C₂²×C₄×C₅	—	2	48	no	—
54	C₂²×C₄×C₅	—	4	32	no	—
55	C₂²×C₄×C₅	C₄⋊C₄	1	64	yes	—
56	C₂²×C₄×C₅	C₂²×C₄	1	24	yes	—
57	C₂²×C₄×C₅	C₂×D₄	1	24	yes	—
58	C₂²×C₄×C₅	C₂×Q₈	1	24	yes	—
59	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
60	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
61	C₂²×Dic₅	—	2	64	no	—
62	C₂²×Dic₅	—	2	24	no	—
63	C₂²×Dic₅	—	2	24	no	—
64	C₂²×Dic₅	—	2	24	no	—
65	C₂²×Dic₅	—	2	48	no	—
66	C₂²×Dic₅	—	4	32	no	—
67	C₂²×Dic₅	C₂²×C₄	1	48	yes	—
68	C₂²×Dic₅	C₂⁴	1	8	yes	—
69	C₂²×Dic₅	C₂⁴	1	8	yes	—
70	C₅×C₄⋊C₄	—	2	32	no	—
71	C₄⋊Dic₅	—	2	32	no	—
72	C₄×Dic₅	C₂²×C₄	1	64	yes	complemented
73	C₂⁴×C₄	—	5	2	no	—
74	C₂⁴×C₄	—	5	2	no	—
75	C₂⁴×C₄	C₂×D₅	1	2	yes	—
76	C₂⁴×C₄	C₂×D₅	1	1	yes	characteristic
77	C₂²×C₄⋊C₄	—	2	8	no	—
78	C₂²×C₄⋊C₄	—	5	12	no	—
79	C₂²×C₄⋊C₄	—	10	8	no	—
80	C₂²×C₄⋊C₄	C₂×D₅	1	12	yes	—
81	C₂²×C₄²	—	5	16	no	—
82	C₂²×C₄²	—	5	12	no	—
83	#382	—	5	64	no	—
84	C₂×#95	—	5	48	no	—
85	C₂³×C₅	C₂×C₄⋊C₄	1	16	yes	—
86	C₂³×C₅	#117	1	32	yes	—
87	C₂³×C₅	#124	1	16	yes	—
88	C₂³×C₅	#124	1	16	yes	—
89	C₂³×C₅	#125	1	32	yes	—
90	C₂³×C₅	C₂³×C₄	1	6	yes	—
91	C₂³×C₅	C₂²×D₄	1	6	yes	—
92	C₂³×C₅	C₂²×Q₈	1	6	yes	—
93	C₂³×C₅	C₂×#41	1	12	yes	—
94	C₂³×C₅	C₂×#41	1	12	yes	—
95	C₂³×C₅	C₂⁵	1	1	yes	characteristic
96	C₂×C₄×C₅	—	2	8	no	—
97	C₂×C₄×C₅	—	2	16	no	—
98	C₂×C₄×C₅	—	4	48	no	—
99	C₂×C₄×C₅	C₂×C₄⋊C₄	1	96	yes	—
100	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
101	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
102	C₂×C₄×C₅	C₂²×Q₈	1	8	yes	—
103	C₂×Dic₅	—	2	96	no	—
104	C₂×Dic₅	—	2	8	no	—
105	C₂×Dic₅	—	2	8	no	—
106	C₂×Dic₅	—	2	8	no	—
107	C₂×Dic₅	—	2	16	no	—
108	C₂×Dic₅	—	4	48	no	—
109	C₂×Dic₅	C₂³×C₄	1	16	yes	—
110	C₂⁵	C₂²×D₅	1	1	yes	characteristic
111	C₂³×C₄	—	2	16	no	—
112	C₂³×C₄	—	2	8	no	—
113	C₂³×C₄	—	5	16	no	—
114	C₂³×C₄	—	5	12	no	—
115	C₂³×C₄	—	5	12	no	—
116	C₂³×C₄	—	10	8	no	—
117	C₂³×C₄	—	10	8	no	—
118	C₂³×C₄	—	10	16	no	—
119	C₂³×C₄	—	10	8	no	—
120	C₂³×C₄	Dic₁₀	1	8	yes	—
121	C₂³×C₄	C₄×D₅	1	8	yes	—
122	C₂³×C₄	C₂²⋊D₅	1	8	yes	—
123	C₂³×C₄	C₂²×D₅	1	12	yes	—
124	C₂³×C₄	C₂²×D₅	1	6	yes	—
125	C₂×C₄⋊C₄	—	2	48	no	—
126	C₂×C₄⋊C₄	—	5	16	no	—
127	C₂×C₄⋊C₄	—	10	48	no	—
128	C₂×C₄⋊C₄	C₂²×D₅	1	16	yes	—
129	C₂×C₄²	—	5	96	no	—
130	C₂×C₄²	—	5	16	no	—
131	#95	—	5	64	no	—
132	C₂²×C₅	#382	1	64	yes	complemented
133	C₂²×C₅	C₂²×C₄⋊C₄	1	12	yes	—
134	C₂²×C₅	C₂×#117	1	24	yes	—
135	C₂²×C₅	C₂×#124	1	12	yes	—
136	C₂²×C₅	C₂×#124	1	12	yes	—
137	C₂²×C₅	C₂×#125	1	24	yes	—
138	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
139	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
140	C₂²×C₅	C₂³×Q₈	1	1	yes	characteristic
141	C₂²×C₅	C₂²×#41	1	2	yes	—
142	C₂²×C₅	C₂²×#41	1	2	yes	—
143	C₄×C₅	—	4	8	no	—
144	C₄×C₅	C₂²×C₄⋊C₄	1	16	yes	—
145	Dic₅	—	2	16	no	—
146	Dic₅	—	4	8	no	—
147	C₂⁴	C₂×Dic₁₀	1	4	yes	—
148	C₂⁴	C₂×C₄×D₅	1	4	yes	—
149	C₂⁴	#711	1	8	yes	—
150	C₂⁴	#713	1	8	yes	—
151	C₂⁴	C₂×C₂²⋊D₅	1	4	yes	—
152	C₂⁴	C₂³×D₅	1	3	yes	—
153	C₂²×C₄	—	2	48	no	—
154	C₂²×C₄	—	2	24	no	—
155	C₂²×C₄	—	4	32	no	—
156	C₂²×C₄	—	5	48	no	—
157	C₂²×C₄	—	5	8	no	—
158	C₂²×C₄	—	5	8	no	—
159	C₂²×C₄	—	10	24	no	—
160	C₂²×C₄	—	10	24	no	—
161	C₂²×C₄	—	10	64	no	—
162	C₂²×C₄	—	10	48	no	—
163	C₂²×C₄	—	10	24	no	—
164	C₂²×C₄	—	20	32	no	—
165	C₂²×C₄	#683	1	64	yes	—
166	C₂²×C₄	C₂×Dic₁₀	1	24	yes	—
167	C₂²×C₄	C₂×C₄×D₅	1	24	yes	—
168	C₂²×C₄	C₂×C₂²⋊D₅	1	24	yes	—
169	C₂²×C₄	C₂³×D₅	1	8	yes	—
170	C₂²×C₄	C₂³×D₅	1	4	yes	—
171	C₄⋊C₄	—	2	32	no	—
172	C₄⋊C₄	—	10	32	no	—
173	C₄²	—	5	64	no	—
174	C₂×C₅	C₂×#382	1	24	yes	complemented
175	C₂×C₅	C₂³×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₅	C₂²×#117	1	2	yes	—
177	C₂×C₅	C₂²×#124	1	1	yes	characteristic
178	C₂×C₅	C₂²×#124	1	1	yes	characteristic
179	C₂×C₅	C₂²×#125	1	2	yes	—
180	C₂³	#4036	1	32	yes	—
181	C₂³	#4038	1	32	yes	—
182	C₂³	C₂×#683	1	16	yes	—
183	C₂³	#4082	1	16	yes	—
184	C₂³	#4093	1	16	yes	—
185	C₂³	C₂²×Dic₁₀	1	6	yes	—
186	C₂³	C₂²×C₄×D₅	1	6	yes	—
187	C₂³	C₂×#711	1	12	yes	—
188	C₂³	C₂×#713	1	12	yes	—
189	C₂³	C₂²×C₂²⋊D₅	1	6	yes	—
190	C₂³	C₂⁴×D₅	1	1	yes	characteristic
191	C₂×C₄	—	2	16	no	—
192	C₂×C₄	—	2	8	no	—
193	C₂×C₄	—	4	48	no	—
194	C₂×C₄	—	5	16	no	—
195	C₂×C₄	—	10	96	no	—
196	C₂×C₄	—	10	8	no	—
197	C₂×C₄	—	10	8	no	—
198	C₂×C₄	—	10	16	no	—
199	C₂×C₄	—	10	8	no	—
200	C₂×C₄	—	20	48	no	—
201	C₂×C₄	C₂×#683	1	96	yes	—
202	C₂×C₄	C₂²×Dic₁₀	1	8	yes	—
203	C₂×C₄	C₂²×C₄×D₅	1	8	yes	—
204	C₂×C₄	C₂²×C₂²⋊D₅	1	8	yes	—
205	C₅	C₂²×#382	1	1	yes	characteristic, complemented, Sylow
206	C₂²	#65266	1	64	yes	direct product factor
207	C₂²	C₂×#4036	1	24	yes	—
208	C₂²	C₂×#4038	1	24	yes	—
209	C₂²	C₂²×#683	1	12	yes	—
210	C₂²	C₂×#4082	1	12	yes	—
211	C₂²	C₂×#4093	1	12	yes	—
212	C₂²	C₂³×Dic₁₀	1	1	yes	characteristic
213	C₂²	C₂³×C₄×D₅	1	1	yes	characteristic
214	C₂²	C₂²×#711	1	2	yes	—
215	C₂²	C₂²×#713	1	2	yes	—
216	C₂²	C₂³×C₂²⋊D₅	1	1	yes	characteristic
217	C₄	—	4	8	no	—
218	C₄	—	10	16	no	—
219	C₄	—	20	8	no	—
220	C₄	C₂²×#683	1	16	yes	—
221	C₂	C₂×#65266	1	24	yes	direct product factor
222	C₂	C₂²×#4038	1	2	yes	—
223	C₂	C₂²×#4036	1	2	yes	—
224	C₂	C₂³×#683	1	1	yes	characteristic
225	C₂	C₂²×#4082	1	1	yes	characteristic
226	C₂	C₂²×#4093	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₅
commutator	C₂²×C₅	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅
Fitting	C₂³×C₅×C₄⋊C₄	C₂
socle, prime radical	C₂⁵×C₅	C₂³
cosocle	C₂³×C₅	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₅
hypocenter	C₅	C₂²×#382
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
5. C₂×#65266	C₂	direct (N × H)
14. C₂²×C₄×Dic₅	C₄	semidirect
17. #65266	C₂²	direct (N × H)
38. C₂×C₄×Dic₅	C₂×C₄	semidirect
72. C₄×Dic₅	C₂²×C₄	semidirect
132. C₂²×C₅	#382	semidirect
174. C₂×C₅	C₂×#382	semidirect
205. C₅	C₂²×#382	semidirect
206. C₂²	#65266	direct (N × H)
221. C₂	C₂×#65266	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×C₅×C₄⋊C₄	2
C₂³×C₄⋊Dic₅	2
C₂³×C₄×Dic₅	2
C₂²×#3963	2
C₂×#65266	2
C₂²×#382	5

Subgroups — C22×#65266 #12257574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#65266 #12257574