Subgroups — C₂²×#65265 #12257589

Order 1280.

5502 conjugacy classes of subgroups.

The subgroup lattice has 234 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×#683	C₂	1	4	yes	maximal
3	C₂³×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₂×#65265	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×#683	—	2	32	no	—
15	C₂²×#683	C₂²	1	48	yes	—
16	C₂²×C₄×Dic₅	C₄	1	16	yes	complemented
17	C₂²×C₄×Dic₅	C₂²	1	12	yes	—
18	#65265	C₂²	1	64	yes	direct product factor
19	C₂²×#384	—	5	1	no	maximal, Sylow
20	C₂⁵×C₅	C₂³	1	1	yes	characteristic
21	C₂³×C₄×C₅	—	2	16	no	—
22	C₂³×C₄×C₅	—	2	8	no	—
23	C₂³×C₄×C₅	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₅	D₄	1	8	yes	—
25	C₂³×C₄×C₅	Q₈	1	8	yes	—
26	C₂³×C₄×C₅	C₂³	1	12	yes	—
27	C₂³×C₄×C₅	C₂³	1	6	yes	—
28	C₂³×Dic₅	—	2	16	no	—
29	C₂³×Dic₅	—	2	8	no	—
30	C₂³×Dic₅	C₂×C₄	1	16	yes	—
31	C₂³×Dic₅	D₄	1	16	yes	—
32	C₂³×Dic₅	Q₈	1	16	yes	—
33	C₂³×Dic₅	C₂³	1	12	yes	—
34	C₂³×Dic₅	C₂³	1	12	yes	—
35	C₂×C₅×C₄⋊C₄	—	2	48	no	—
36	C₂×C₅×C₄⋊C₄	C₂³	1	16	yes	—
37	C₂×C₄⋊Dic₅	—	2	48	no	—
38	C₂×C₄⋊Dic₅	C₂³	1	16	yes	—
39	C₂×#683	—	2	192	no	—
40	C₂×#683	C₂³	1	64	yes	—
41	C₂×C₄×Dic₅	C₂×C₄	1	96	yes	complemented
42	C₂×C₄×Dic₅	C₂³	1	16	yes	—
43	C₂³×C₄⋊C₄	—	5	1	no	—
44	C₂³×C₄⋊C₄	—	5	4	no	—
45	C₂³×C₄⋊C₄	D₅	1	1	yes	characteristic
46	C₂³×C₄²	—	5	1	no	—
47	C₂×#384	—	5	24	no	—
48	C₂⁴×C₅	C₂²×C₄	1	4	yes	—
49	C₂⁴×C₅	C₂×D₄	1	8	yes	—
50	C₂⁴×C₅	C₂×D₄	1	4	yes	—
51	C₂⁴×C₅	C₂×Q₈	1	4	yes	—
52	C₂⁴×C₅	C₂×Q₈	1	8	yes	—
53	C₂⁴×C₅	C₂⁴	1	3	yes	—
54	C₂²×C₄×C₅	—	2	24	no	—
55	C₂²×C₄×C₅	—	2	48	no	—
56	C₂²×C₄×C₅	—	4	32	no	—
57	C₂²×C₄×C₅	C₄⋊C₄	1	64	yes	complemented
58	C₂²×C₄×C₅	C₂²×C₄	1	24	yes	—
59	C₂²×C₄×C₅	C₂×D₄	1	24	yes	—
60	C₂²×C₄×C₅	C₂×Q₈	1	24	yes	—
61	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
62	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
63	C₂²×Dic₅	—	2	48	no	—
64	C₂²×Dic₅	—	2	24	no	—
65	C₂²×Dic₅	—	4	32	no	—
66	C₂²×Dic₅	C₄⋊C₄	1	128	yes	complemented
67	C₂²×Dic₅	C₂²×C₄	1	48	yes	—
68	C₂²×Dic₅	C₂×D₄	1	48	yes	—
69	C₂²×Dic₅	C₂×Q₈	1	48	yes	—
70	C₂²×Dic₅	C₂⁴	1	8	yes	—
71	C₂²×Dic₅	C₂⁴	1	8	yes	—
72	C₅×C₄⋊C₄	—	2	32	no	—
73	C₄⋊Dic₅	—	2	32	no	—
74	#683	—	2	128	no	—
75	C₄×Dic₅	C₂²×C₄	1	64	yes	complemented
76	C₂⁴×C₄	—	5	2	no	—
77	C₂⁴×C₄	—	5	2	no	—
78	C₂⁴×C₄	C₂×D₅	1	2	yes	—
79	C₂⁴×C₄	C₂×D₅	1	1	yes	characteristic
80	C₂²×C₄⋊C₄	—	2	8	no	—
81	C₂²×C₄⋊C₄	—	5	48	no	—
82	C₂²×C₄⋊C₄	—	5	12	no	—
83	C₂²×C₄⋊C₄	—	10	32	no	—
84	C₂²×C₄⋊C₄	—	10	8	no	—
85	C₂²×C₄⋊C₄	C₂×D₅	1	12	yes	—
86	C₂²×C₄²	—	5	16	no	—
87	C₂²×C₄²	—	5	12	no	—
88	#384	—	5	64	no	—
89	C₂³×C₅	C₂×C₄⋊C₄	1	32	yes	—
90	C₂³×C₅	C₂×C₄⋊C₄	1	16	yes	—
91	C₂³×C₅	#127	1	16	yes	—
92	C₂³×C₅	C₄⋊Q₈	1	32	yes	—
93	C₂³×C₅	C₄⋊Q₈	1	16	yes	—
94	C₂³×C₅	C₂³×C₄	1	6	yes	—
95	C₂³×C₅	C₂²×D₄	1	12	yes	—
96	C₂³×C₅	C₂²×D₄	1	6	yes	—
97	C₂³×C₅	C₂²×Q₈	1	6	yes	—
98	C₂³×C₅	C₂²×Q₈	1	12	yes	—
99	C₂³×C₅	C₂⁵	1	1	yes	characteristic
100	C₂×C₄×C₅	—	2	16	no	—
101	C₂×C₄×C₅	—	2	8	no	—
102	C₂×C₄×C₅	—	4	48	no	—
103	C₂×C₄×C₅	C₂×C₄⋊C₄	1	96	yes	complemented
104	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
105	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
106	C₂×C₄×C₅	C₂²×Q₈	1	8	yes	—
107	C₂×Dic₅	—	2	16	no	—
108	C₂×Dic₅	—	2	8	no	—
109	C₂×Dic₅	—	4	48	no	—
110	C₂×Dic₅	C₂×C₄⋊C₄	1	192	yes	complemented
111	C₂×Dic₅	C₂³×C₄	1	16	yes	—
112	C₂×Dic₅	C₂²×D₄	1	16	yes	—
113	C₂×Dic₅	C₂²×Q₈	1	16	yes	—
114	C₂⁵	C₂²×D₅	1	1	yes	characteristic
115	C₂³×C₄	—	2	16	no	—
116	C₂³×C₄	—	2	8	no	—
117	C₂³×C₄	—	5	16	no	—
118	C₂³×C₄	—	5	16	no	—
119	C₂³×C₄	—	5	16	no	—
120	C₂³×C₄	—	5	12	no	—
121	C₂³×C₄	—	5	12	no	—
122	C₂³×C₄	—	10	16	no	—
123	C₂³×C₄	—	10	8	no	—
124	C₂³×C₄	Dic₁₀	1	8	yes	—
125	C₂³×C₄	C₄×D₅	1	8	yes	—
126	C₂³×C₄	C₂²⋊D₅	1	8	yes	—
127	C₂³×C₄	C₂²×D₅	1	12	yes	—
128	C₂³×C₄	C₂²×D₅	1	6	yes	—
129	C₂×C₄⋊C₄	—	2	48	no	—
130	C₂×C₄⋊C₄	—	5	64	no	—
131	C₂×C₄⋊C₄	—	5	16	no	—
132	C₂×C₄⋊C₄	—	10	192	no	—
133	C₂×C₄⋊C₄	—	10	48	no	—
134	C₂×C₄⋊C₄	C₂²×D₅	1	16	yes	—
135	C₂×C₄²	—	5	96	no	—
136	C₂×C₄²	—	5	16	no	—
137	C₂²×C₅	#384	1	64	yes	complemented
138	C₂²×C₅	C₂²×C₄⋊C₄	1	24	yes	—
139	C₂²×C₅	C₂²×C₄⋊C₄	1	12	yes	—
140	C₂²×C₅	C₂×#127	1	12	yes	—
141	C₂²×C₅	C₂×C₄⋊Q₈	1	24	yes	—
142	C₂²×C₅	C₂×C₄⋊Q₈	1	12	yes	—
143	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
144	C₂²×C₅	C₂³×D₄	1	2	yes	—
145	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
146	C₂²×C₅	C₂³×Q₈	1	1	yes	characteristic
147	C₂²×C₅	C₂³×Q₈	1	2	yes	—
148	C₄×C₅	—	4	8	no	—
149	C₄×C₅	C₂²×C₄⋊C₄	1	16	yes	complemented
150	Dic₅	—	4	8	no	—
151	Dic₅	C₂²×C₄⋊C₄	1	32	yes	complemented
152	C₂⁴	C₂×Dic₁₀	1	4	yes	—
153	C₂⁴	C₂×C₄×D₅	1	4	yes	—
154	C₂⁴	D₄×D₅	1	8	yes	—
155	C₂⁴	Q₈×D₅	1	8	yes	—
156	C₂⁴	C₂×C₂²⋊D₅	1	4	yes	—
157	C₂⁴	C₂³×D₅	1	3	yes	—
158	C₂²×C₄	—	2	48	no	—
159	C₂²×C₄	—	2	24	no	—
160	C₂²×C₄	—	4	32	no	—
161	C₂²×C₄	—	5	48	no	—
162	C₂²×C₄	—	5	48	no	—
163	C₂²×C₄	—	5	48	no	—
164	C₂²×C₄	—	5	8	no	—
165	C₂²×C₄	—	5	128	no	—
166	C₂²×C₄	—	5	8	no	—
167	C₂²×C₄	—	10	48	no	—
168	C₂²×C₄	—	10	24	no	—
169	C₂²×C₄	—	20	32	no	—
170	C₂²×C₄	#683	1	64	yes	complemented
171	C₂²×C₄	C₂×Dic₁₀	1	24	yes	—
172	C₂²×C₄	C₂×C₄×D₅	1	24	yes	—
173	C₂²×C₄	C₂×C₂²⋊D₅	1	24	yes	—
174	C₂²×C₄	C₂³×D₅	1	8	yes	—
175	C₂²×C₄	C₂³×D₅	1	4	yes	—
176	C₄⋊C₄	—	2	32	no	—
177	C₄⋊C₄	—	10	128	no	—
178	C₄⋊C₄	—	10	32	no	—
179	C₄²	—	5	64	no	—
180	C₂×C₅	C₂×#384	1	24	yes	complemented
181	C₂×C₅	C₂³×C₄⋊C₄	1	2	yes	—
182	C₂×C₅	C₂³×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₅	C₂²×#127	1	1	yes	characteristic
184	C₂×C₅	C₂²×C₄⋊Q₈	1	2	yes	—
185	C₂×C₅	C₂²×C₄⋊Q₈	1	1	yes	characteristic
186	C₂³	#4034	1	32	yes	—
187	C₂³	D₅×C₄⋊C₄	1	32	yes	—
188	C₂³	C₂×#683	1	16	yes	—
189	C₂³	#4086	1	16	yes	—
190	C₂³	#4090	1	16	yes	—
191	C₂³	C₂²×Dic₁₀	1	6	yes	—
192	C₂³	C₂²×C₄×D₅	1	6	yes	—
193	C₂³	C₂×D₄×D₅	1	12	yes	—
194	C₂³	C₂×Q₈×D₅	1	12	yes	—
195	C₂³	C₂²×C₂²⋊D₅	1	6	yes	—
196	C₂³	C₂⁴×D₅	1	1	yes	characteristic
197	C₂×C₄	—	2	16	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	4	48	no	—
200	C₂×C₄	—	5	192	no	—
201	C₂×C₄	—	5	16	no	—
202	C₂×C₄	—	5	16	no	—
203	C₂×C₄	—	5	16	no	—
204	C₂×C₄	—	10	16	no	—
205	C₂×C₄	—	10	8	no	—
206	C₂×C₄	—	20	48	no	—
207	C₂×C₄	C₂×#683	1	96	yes	complemented
208	C₂×C₄	C₂²×Dic₁₀	1	8	yes	—
209	C₂×C₄	C₂²×C₄×D₅	1	8	yes	—
210	C₂×C₄	C₂²×C₂²⋊D₅	1	8	yes	—
211	C₅	C₂²×#384	1	1	yes	characteristic, complemented, Sylow
212	C₂²	#65265	1	64	yes	direct product factor
213	C₂²	C₂×#4034	1	24	yes	—
214	C₂²	C₂×D₅×C₄⋊C₄	1	24	yes	—
215	C₂²	C₂²×#683	1	12	yes	—
216	C₂²	C₂×#4086	1	12	yes	—
217	C₂²	C₂×#4090	1	12	yes	—
218	C₂²	C₂³×Dic₁₀	1	1	yes	characteristic
219	C₂²	C₂³×C₄×D₅	1	1	yes	characteristic
220	C₂²	C₂²×D₄×D₅	1	2	yes	—
221	C₂²	C₂²×Q₈×D₅	1	2	yes	—
222	C₂²	C₂³×C₂²⋊D₅	1	1	yes	characteristic
223	C₄	—	4	8	no	—
224	C₄	—	5	32	no	—
225	C₄	—	20	8	no	—
226	C₄	C₂²×#683	1	16	yes	complemented
227	C₂	C₂×#65265	1	24	yes	direct product factor
228	C₂	C₂²×#4034	1	2	yes	—
229	C₂	C₂²×D₅×C₄⋊C₄	1	2	yes	—
230	C₂	C₂³×#683	1	1	yes	characteristic
231	C₂	C₂²×#4086	1	1	yes	characteristic
232	C₂	C₂²×#4090	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×#384
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×#65265	C₂	direct (N × H)
16. C₂²×C₄×Dic₅	C₄	semidirect
18. #65265	C₂²	direct (N × H)
41. C₂×C₄×Dic₅	C₂×C₄	semidirect
57. C₂²×C₄×C₅	C₄⋊C₄	semidirect
66. C₂²×Dic₅	C₄⋊C₄	semidirect
75. C₄×Dic₅	C₂²×C₄	semidirect
103. C₂×C₄×C₅	C₂×C₄⋊C₄	semidirect
110. C₂×Dic₅	C₂×C₄⋊C₄	semidirect
137. C₂²×C₅	#384	semidirect
149. C₄×C₅	C₂²×C₄⋊C₄	semidirect
151. Dic₅	C₂²×C₄⋊C₄	semidirect
170. C₂²×C₄	#683	semidirect
180. C₂×C₅	C₂×#384	semidirect
207. C₂×C₄	C₂×#683	semidirect
211. C₅	C₂²×#384	semidirect
212. C₂²	#65265	direct (N × H)
226. C₄	C₂²×#683	semidirect
227. C₂	C₂×#65265	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×#683	2
C₂³×C₄⋊Dic₅	2
C₂³×C₄×Dic₅	2
C₂×#65265	2
C₂²×#384	5

Subgroups — C22×#65265 #12257589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#65265 #12257589