Subgroups — C₂²×C₄×C₅×C₂²⋊C₄ #11956708

Order 1280.

9534 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₅×C₂²⋊C₄	C₂	1	2	yes	maximal
3	C₂³×C₄²×C₅	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₅×C₂²⋊C₄	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₅×#95	C₂	1	2	yes	maximal, complemented
6	C₂⁶×C₅	C₂²	1	1	yes	characteristic
7	C₂⁴×C₄×C₅	—	2	16	no	—
8	C₂⁴×C₄×C₅	C₄	1	16	yes	complemented
9	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₅	C₂²	1	12	yes	—
11	C₂⁴×C₄×C₅	C₂²	1	4	yes	—
12	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₂²⋊C₄	C₄	1	32	yes	direct product factor
14	C₂²×C₅×C₂²⋊C₄	C₂²	1	24	yes	—
15	C₂²×C₄²×C₅	—	2	16	no	—
16	C₂²×C₄²×C₅	C₂²	1	24	yes	complemented
17	C₄×C₅×C₂²⋊C₄	C₂²	1	64	yes	direct product factor
18	C₂×C₅×#95	C₂²	1	24	yes	complemented
19	C₂²×C₄×C₂²⋊C₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂⁵×C₅	—	2	8	no	—
21	C₂⁵×C₅	—	2	8	no	—
22	C₂⁵×C₅	C₂×C₄	1	16	yes	—
23	C₂⁵×C₅	C₂×C₄	1	8	yes	—
24	C₂⁵×C₅	C₂³	1	6	yes	—
25	C₂⁵×C₅	C₂³	1	1	yes	characteristic
26	C₂³×C₄×C₅	—	2	16	no	—
27	C₂³×C₄×C₅	—	2	8	no	—
28	C₂³×C₄×C₅	—	2	16	no	—
29	C₂³×C₄×C₅	—	2	128	no	—
30	C₂³×C₄×C₅	—	2	96	no	—
31	C₂³×C₄×C₅	C₂×C₄	1	8	yes	—
32	C₂³×C₄×C₅	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₅	C₂×C₄	1	96	yes	complemented
34	C₂³×C₄×C₅	C₂×C₄	1	32	yes	complemented
35	C₂³×C₄×C₅	D₄	1	16	yes	—
36	C₂³×C₄×C₅	C₂³	1	6	yes	—
37	C₂³×C₄×C₅	C₂³	1	24	yes	—
38	C₂³×C₄×C₅	C₂³	1	16	yes	—
39	C₂³×C₄×C₅	C₂³	1	6	yes	—
40	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	192	yes	direct product factor
41	C₂×C₅×C₂²⋊C₄	C₂³	1	32	yes	—
42	C₂×C₄²×C₅	—	2	96	no	—
43	C₂×C₄²×C₅	C₂³	1	32	yes	complemented
44	C₅×#95	C₂³	1	32	yes	complemented
45	C₂⁵×C₄	C₂×C₅	1	1	yes	characteristic
46	C₂³×C₂²⋊C₄	C₂×C₅	1	2	yes	—
47	C₂³×C₄²	C₂×C₅	1	2	yes	complemented
48	C₂×C₄×C₂²⋊C₄	C₂×C₅	1	24	yes	direct product factor
49	C₂²×#95	C₂×C₅	1	2	yes	complemented
50	C₂⁴×C₅	—	2	32	no	—
51	C₂⁴×C₅	—	2	32	no	—
52	C₂⁴×C₅	—	2	128	no	—
53	C₂⁴×C₅	—	2	24	no	—
54	C₂⁴×C₅	—	2	24	no	—
55	C₂⁴×C₅	C₄²	1	64	yes	complemented
56	C₂⁴×C₅	C₂²×C₄	1	48	yes	—
57	C₂⁴×C₅	C₂²×C₄	1	24	yes	—
58	C₂⁴×C₅	C₂²×C₄	1	4	yes	—
59	C₂⁴×C₅	C₂²×C₄	1	8	yes	—
60	C₂⁴×C₅	C₂×D₄	1	8	yes	—
61	C₂⁴×C₅	#41	1	8	yes	—
62	C₂⁴×C₅	C₂⁴	1	4	yes	—
63	C₂⁴×C₅	C₂⁴	1	3	yes	—
64	C₂²×C₄×C₅	—	2	32	no	—
65	C₂²×C₄×C₅	—	2	128	no	—
66	C₂²×C₄×C₅	—	2	48	no	—
67	C₂²×C₄×C₅	—	2	24	no	—
68	C₂²×C₄×C₅	—	2	48	no	—
69	C₂²×C₄×C₅	—	2	384	no	—
70	C₂²×C₄×C₅	—	2	64	no	—
71	C₂²×C₄×C₅	C₂²⋊C₄	1	64	yes	direct product factor
72	C₂²×C₄×C₅	C₂²×C₄	1	24	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	24	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	64	yes	complemented
75	C₂²×C₄×C₅	C₂²×C₄	1	96	yes	complemented
76	C₂²×C₄×C₅	C₂×D₄	1	48	yes	—
77	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
78	C₂²×C₄×C₅	C₂⁴	1	16	yes	—
79	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
80	C₅×C₂²⋊C₄	C₂²×C₄	1	128	yes	direct product factor
81	C₄²×C₅	—	2	64	no	—
82	C₂⁶	C₂²×C₅	1	1	yes	characteristic
83	C₂⁴×C₄	—	2	16	no	—
84	C₂⁴×C₄	C₄×C₅	1	16	yes	complemented
85	C₂⁴×C₄	C₂²×C₅	1	1	yes	characteristic
86	C₂⁴×C₄	C₂²×C₅	1	12	yes	—
87	C₂⁴×C₄	C₂²×C₅	1	4	yes	—
88	C₂⁴×C₄	C₂²×C₅	1	1	yes	characteristic
89	C₂²×C₂²⋊C₄	C₄×C₅	1	32	yes	direct product factor
90	C₂²×C₂²⋊C₄	C₂²×C₅	1	24	yes	—
91	C₂²×C₄²	—	2	16	no	—
92	C₂²×C₄²	C₂²×C₅	1	24	yes	complemented
93	C₄×C₂²⋊C₄	C₂²×C₅	1	64	yes	direct product factor
94	C₂×#95	C₂²×C₅	1	24	yes	complemented
95	C₂³×C₅	—	2	48	no	—
96	C₂³×C₅	—	2	48	no	—
97	C₂³×C₅	—	2	192	no	—
98	C₂³×C₅	—	2	8	no	—
99	C₂³×C₅	—	2	8	no	—
100	C₂³×C₅	—	2	256	no	—
101	C₂³×C₅	C₂×C₄²	1	16	yes	—
102	C₂³×C₅	C₂×C₄²	1	96	yes	complemented
103	C₂³×C₅	C₂×C₂²⋊C₄	1	16	yes	—
104	C₂³×C₅	#117	1	16	yes	—
105	C₂³×C₅	C₄×D₄	1	64	yes	—
106	C₂³×C₅	C₂³×C₄	1	16	yes	—
107	C₂³×C₅	C₂³×C₄	1	8	yes	—
108	C₂³×C₅	C₂³×C₄	1	6	yes	—
109	C₂³×C₅	C₂³×C₄	1	12	yes	—
110	C₂³×C₅	C₂²×D₄	1	12	yes	—
111	C₂³×C₅	C₂×#41	1	12	yes	—
112	C₂³×C₅	C₂⁵	1	1	yes	characteristic
113	C₂×C₄×C₅	—	2	48	no	—
114	C₂×C₄×C₅	—	2	192	no	—
115	C₂×C₄×C₅	—	2	16	no	—
116	C₂×C₄×C₅	—	2	8	no	—
117	C₂×C₄×C₅	—	2	16	no	—
118	C₂×C₄×C₅	—	2	128	no	—
119	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	96	yes	direct product factor
120	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
121	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
122	C₂×C₄×C₅	C₂³×C₄	1	32	yes	complemented
123	C₂×C₄×C₅	C₂²×D₄	1	16	yes	—
124	C₂⁵	—	2	8	no	—
125	C₂⁵	—	2	8	no	—
126	C₂⁵	C₂×C₄×C₅	1	16	yes	—
127	C₂⁵	C₂×C₄×C₅	1	8	yes	—
128	C₂⁵	C₂³×C₅	1	6	yes	—
129	C₂⁵	C₂³×C₅	1	1	yes	characteristic
130	C₂³×C₄	—	2	8	no	—
131	C₂³×C₄	—	2	16	no	—
132	C₂³×C₄	—	2	16	no	—
133	C₂³×C₄	—	2	128	no	—
134	C₂³×C₄	—	2	96	no	—
135	C₂³×C₄	C₂×C₄×C₅	1	8	yes	—
136	C₂³×C₄	C₂×C₄×C₅	1	8	yes	—
137	C₂³×C₄	C₂×C₄×C₅	1	96	yes	complemented
138	C₂³×C₄	C₂×C₄×C₅	1	32	yes	complemented
139	C₂³×C₄	C₅×D₄	1	16	yes	—
140	C₂³×C₄	C₂³×C₅	1	6	yes	—
141	C₂³×C₄	C₂³×C₅	1	24	yes	—
142	C₂³×C₄	C₂³×C₅	1	16	yes	—
143	C₂³×C₄	C₂³×C₅	1	6	yes	—
144	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	192	yes	direct product factor
145	C₂×C₂²⋊C₄	C₂³×C₅	1	32	yes	—
146	C₂×C₄²	—	2	96	no	—
147	C₂×C₄²	C₂³×C₅	1	32	yes	complemented
148	#95	C₂³×C₅	1	32	yes	complemented
149	C₂²×C₅	—	2	8	no	—
150	C₂²×C₅	—	2	8	no	—
151	C₂²×C₅	—	2	32	no	—
152	C₂²×C₅	—	2	192	no	—
153	C₂²×C₅	C₄×C₂²⋊C₄	1	64	yes	direct product factor
154	C₂²×C₅	C₂²×C₄²	1	12	yes	—
155	C₂²×C₅	C₂²×C₄²	1	16	yes	complemented
156	C₂²×C₅	C₂²×C₂²⋊C₄	1	12	yes	—
157	C₂²×C₅	C₂×#117	1	12	yes	—
158	C₂²×C₅	C₂×C₄×D₄	1	48	yes	—
159	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
160	C₂²×C₅	C₂⁴×C₄	1	2	yes	—
161	C₂²×C₅	C₂³×D₄	1	2	yes	—
162	C₂²×C₅	C₂²×#41	1	2	yes	—
163	C₄×C₅	—	2	8	no	—
164	C₄×C₅	—	2	32	no	—
165	C₄×C₅	C₂²×C₂²⋊C₄	1	16	yes	direct product factor
166	C₂⁴	—	2	32	no	—
167	C₂⁴	—	2	32	no	—
168	C₂⁴	—	2	128	no	—
169	C₂⁴	—	2	24	no	—
170	C₂⁴	—	2	24	no	—
171	C₂⁴	C₄²×C₅	1	64	yes	complemented
172	C₂⁴	C₂²×C₄×C₅	1	48	yes	—
173	C₂⁴	C₂²×C₄×C₅	1	24	yes	—
174	C₂⁴	C₂²×C₄×C₅	1	4	yes	—
175	C₂⁴	C₂²×C₄×C₅	1	8	yes	—
176	C₂⁴	C₂×C₅×D₄	1	8	yes	—
177	C₂⁴	C₅×#41	1	8	yes	—
178	C₂⁴	C₂⁴×C₅	1	4	yes	—
179	C₂⁴	C₂⁴×C₅	1	3	yes	—
180	C₂²×C₄	—	2	32	no	—
181	C₂²×C₄	—	2	128	no	—
182	C₂²×C₄	—	2	48	no	—
183	C₂²×C₄	—	2	24	no	—
184	C₂²×C₄	—	2	48	no	—
185	C₂²×C₄	—	2	384	no	—
186	C₂²×C₄	—	2	64	no	—
187	C₂²×C₄	C₅×C₂²⋊C₄	1	64	yes	direct product factor
188	C₂²×C₄	C₂²×C₄×C₅	1	24	yes	—
189	C₂²×C₄	C₂²×C₄×C₅	1	24	yes	—
190	C₂²×C₄	C₂²×C₄×C₅	1	64	yes	complemented
191	C₂²×C₄	C₂²×C₄×C₅	1	96	yes	complemented
192	C₂²×C₄	C₂×C₅×D₄	1	48	yes	—
193	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
194	C₂²×C₄	C₂⁴×C₅	1	16	yes	—
195	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
196	C₂²⋊C₄	C₂²×C₄×C₅	1	128	yes	direct product factor
197	C₄²	—	2	64	no	—
198	C₂×C₅	—	2	16	no	—
199	C₂×C₅	C₂×C₄×C₂²⋊C₄	1	24	yes	direct product factor
200	C₂×C₅	C₂³×C₄²	1	1	yes	characteristic
201	C₂×C₅	C₂³×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₅	C₂²×#117	1	1	yes	characteristic
203	C₂×C₅	C₂²×C₄×D₄	1	4	yes	—
204	C₂³	—	2	256	no	—
205	C₂³	—	2	48	no	—
206	C₂³	—	2	48	no	—
207	C₂³	—	2	192	no	—
208	C₂³	—	2	8	no	—
209	C₂³	—	2	8	no	—
210	C₂³	C₂×C₄²×C₅	1	16	yes	—
211	C₂³	C₂×C₄²×C₅	1	96	yes	complemented
212	C₂³	C₂×C₅×C₂²⋊C₄	1	16	yes	—
213	C₂³	C₅×#117	1	16	yes	—
214	C₂³	C₄×C₅×D₄	1	64	yes	—
215	C₂³	C₂³×C₄×C₅	1	16	yes	—
216	C₂³	C₂³×C₄×C₅	1	8	yes	—
217	C₂³	C₂³×C₄×C₅	1	6	yes	—
218	C₂³	C₂³×C₄×C₅	1	12	yes	—
219	C₂³	C₂²×C₅×D₄	1	12	yes	—
220	C₂³	C₂×C₅×#41	1	12	yes	—
221	C₂³	C₂⁵×C₅	1	1	yes	characteristic
222	C₂×C₄	—	2	48	no	—
223	C₂×C₄	—	2	192	no	—
224	C₂×C₄	—	2	16	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	16	no	—
227	C₂×C₄	—	2	128	no	—
228	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	96	yes	direct product factor
229	C₂×C₄	C₂³×C₄×C₅	1	8	yes	—
230	C₂×C₄	C₂³×C₄×C₅	1	8	yes	—
231	C₂×C₄	C₂³×C₄×C₅	1	32	yes	complemented
232	C₂×C₄	C₂²×C₅×D₄	1	16	yes	—
233	C₅	C₂²×C₄×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
234	C₂²	—	2	192	no	—
235	C₂²	—	2	8	no	—
236	C₂²	—	2	8	no	—
237	C₂²	—	2	32	no	—
238	C₂²	C₄×C₅×C₂²⋊C₄	1	64	yes	direct product factor
239	C₂²	C₂²×C₄²×C₅	1	12	yes	—
240	C₂²	C₂²×C₄²×C₅	1	16	yes	complemented
241	C₂²	C₂²×C₅×C₂²⋊C₄	1	12	yes	—
242	C₂²	C₂×C₅×#117	1	12	yes	—
243	C₂²	C₂×C₄×C₅×D₄	1	48	yes	—
244	C₂²	C₂⁴×C₄×C₅	1	1	yes	characteristic
245	C₂²	C₂⁴×C₄×C₅	1	2	yes	—
246	C₂²	C₂³×C₅×D₄	1	2	yes	—
247	C₂²	C₂²×C₅×#41	1	2	yes	—
248	C₄	—	2	8	no	—
249	C₄	—	2	32	no	—
250	C₄	C₂²×C₅×C₂²⋊C₄	1	16	yes	direct product factor
251	C₂	—	2	16	no	—
252	C₂	C₂×C₄×C₅×C₂²⋊C₄	1	24	yes	direct product factor
253	C₂	C₂³×C₄²×C₅	1	1	yes	characteristic
254	C₂	C₂³×C₅×C₂²⋊C₄	1	1	yes	characteristic
255	C₂	C₂²×C₅×#117	1	1	yes	characteristic
256	C₂	C₂²×C₄×C₅×D₄	1	4	yes	—

	subgroup	G/H
center	C₂⁴×C₄×C₅	C₂²
commutator	C₂	C₂³×C₄²×C₅
Frattini, cosocle	C₂³	C₂⁵×C₅
socle	C₂⁵×C₅	C₂³
prime radical	C₂⁶×C₅	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂³×C₄²×C₅	C₂	semidirect
4. C₂×C₄×C₅×C₂²⋊C₄	C₂	direct (N × H)
5. C₂²×C₅×#95	C₂	semidirect
8. C₂⁴×C₄×C₅	C₄	semidirect
13. C₂²×C₅×C₂²⋊C₄	C₄	direct (N × H)
16. C₂²×C₄²×C₅	C₂²	semidirect
17. C₄×C₅×C₂²⋊C₄	C₂²	direct (N × H)
18. C₂×C₅×#95	C₂²	semidirect
19. C₂²×C₄×C₂²⋊C₄	C₅	direct (N × H)
33. C₂³×C₄×C₅	C₂×C₄	semidirect
34. C₂³×C₄×C₅	C₂×C₄	semidirect
40. C₂×C₅×C₂²⋊C₄	C₂×C₄	direct (N × H)
43. C₂×C₄²×C₅	C₂³	semidirect
44. C₅×#95	C₂³	semidirect
47. C₂³×C₄²	C₂×C₅	semidirect
48. C₂×C₄×C₂²⋊C₄	C₂×C₅	direct (N × H)
49. C₂²×#95	C₂×C₅	semidirect
55. C₂⁴×C₅	C₄²	semidirect
71. C₂²×C₄×C₅	C₂²⋊C₄	direct (N × H)
74. C₂²×C₄×C₅	C₂²×C₄	semidirect
75. C₂²×C₄×C₅	C₂²×C₄	semidirect
80. C₅×C₂²⋊C₄	C₂²×C₄	direct (N × H)
84. C₂⁴×C₄	C₄×C₅	semidirect
89. C₂²×C₂²⋊C₄	C₄×C₅	direct (N × H)
92. C₂²×C₄²	C₂²×C₅	semidirect
93. C₄×C₂²⋊C₄	C₂²×C₅	direct (N × H)
94. C₂×#95	C₂²×C₅	semidirect
102. C₂³×C₅	C₂×C₄²	semidirect
119. C₂×C₄×C₅	C₂×C₂²⋊C₄	direct (N × H)
122. C₂×C₄×C₅	C₂³×C₄	semidirect
137. C₂³×C₄	C₂×C₄×C₅	semidirect
138. C₂³×C₄	C₂×C₄×C₅	semidirect
144. C₂×C₂²⋊C₄	C₂×C₄×C₅	direct (N × H)
147. C₂×C₄²	C₂³×C₅	semidirect
148. #95	C₂³×C₅	semidirect
153. C₂²×C₅	C₄×C₂²⋊C₄	direct (N × H)
155. C₂²×C₅	C₂²×C₄²	semidirect
165. C₄×C₅	C₂²×C₂²⋊C₄	direct (N × H)
171. C₂⁴	C₄²×C₅	semidirect
187. C₂²×C₄	C₅×C₂²⋊C₄	direct (N × H)
190. C₂²×C₄	C₂²×C₄×C₅	semidirect
191. C₂²×C₄	C₂²×C₄×C₅	semidirect
196. C₂²⋊C₄	C₂²×C₄×C₅	direct (N × H)
199. C₂×C₅	C₂×C₄×C₂²⋊C₄	direct (N × H)
211. C₂³	C₂×C₄²×C₅	semidirect
228. C₂×C₄	C₂×C₅×C₂²⋊C₄	direct (N × H)
231. C₂×C₄	C₂³×C₄×C₅	semidirect
233. C₅	C₂²×C₄×C₂²⋊C₄	direct (N × H)
238. C₂²	C₄×C₅×C₂²⋊C₄	direct (N × H)
240. C₂²	C₂²×C₄²×C₅	semidirect
250. C₄	C₂²×C₅×C₂²⋊C₄	direct (N × H)
252. C₂	C₂×C₄×C₅×C₂²⋊C₄	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁵×C₄×C₅	2
C₂³×C₅×C₂²⋊C₄	2
C₂³×C₄²×C₅	2
C₂×C₄×C₅×C₂²⋊C₄	2
C₂²×C₅×#95	2
C₂²×C₄×C₂²⋊C₄	5

Subgroups — C22×C4×C5×C22⋊C4 #11956708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₅×C₂²⋊C₄ #11956708