Subgroups — C₂²×#65928 #13006173

Order 1280.

10246 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#120	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#4099	C₂	1	2	yes	maximal, complemented
3	C₂²×#4083	C₂	1	3	yes	maximal, complemented
4	C₂²×#4082	C₂	1	3	yes	maximal, complemented
5	C₂×#65928	C₂	1	48	yes	maximal, direct product factor
6	C₂²×#4031	C₂	1	6	yes	maximal, complemented
7	C₂⁶×C₅	C₂²	1	1	yes	characteristic, complemented
8	C₂⁵×D₅	C₂²	1	1	yes	characteristic, complemented
9	C₂³×C₅×D₄	C₂²	1	3	yes	—
10	C₂³×C₂²⋊D₅	C₂²	1	6	yes	—
11	C₂³×Dic₁₀	C₂²	1	3	yes	complemented
12	C₂×C₅×#120	C₂²	1	24	yes	complemented
13	C₂²×C₅×C₂²⋊C₄	C₂²	1	3	yes	complemented
14	C₂×#4099	C₂²	1	48	yes	complemented
15	C₂²×C₂²⋊Dic₅	C₂²	1	6	yes	complemented
16	C₂²×C₂²⋊Dic₅	C₂²	1	3	yes	complemented
17	C₂×#4083	C₂²	1	72	yes	complemented
18	C₂×#4082	C₂²	1	72	yes	complemented
19	C₂²×#685	C₂²	1	6	yes	complemented
20	C₂²×C₄×Dic₅	C₂²	1	3	yes	complemented
21	#65928	C₂²	1	256	yes	direct product factor
22	C₂×#4031	C₂²	1	144	yes	complemented
23	C₂²×H(3,𝔽₄)	—	5	1	no	maximal, Sylow
24	C₂⁵×C₅	—	4	12	no	—
25	C₂⁵×C₅	C₂³	1	1	yes	characteristic
26	C₂⁵×C₅	C₂³	1	3	yes	—
27	C₂⁵×C₅	C₂³	1	12	yes	complemented
28	C₂⁴×D₅	—	4	12	no	—
29	C₂⁴×D₅	C₂³	1	2	yes	—
30	C₂⁴×D₅	C₂³	1	12	yes	complemented
31	C₂²×C₅×D₄	—	4	12	no	—
32	C₂²×C₅×D₄	C₂³	1	36	yes	—
33	C₂³×C₄×C₅	C₂³	1	3	yes	—
34	C₂²×C₂²⋊D₅	—	4	24	no	—
35	C₂²×C₂²⋊D₅	C₂³	1	72	yes	—
36	C₂³×Dic₅	C₂³	1	6	yes	—
37	C₂²×Dic₁₀	—	4	12	no	—
38	C₂²×Dic₁₀	C₂³	1	36	yes	complemented
39	C₅×#120	C₂³	1	64	yes	complemented
40	C₂×C₅×C₂²⋊C₄	C₂³	1	36	yes	complemented
41	#4099	C₂³	1	128	yes	complemented
42	C₂×C₂²⋊Dic₅	C₂³	1	72	yes	complemented
43	C₂×C₂²⋊Dic₅	C₂³	1	36	yes	complemented
44	#4083	C₂³	1	192	yes	complemented
45	#4082	C₂³	1	192	yes	complemented
46	C₂×#685	C₂³	1	72	yes	complemented
47	C₂×C₄×Dic₅	C₂³	1	36	yes	complemented
48	#4031	C₂³	1	384	yes	complemented
49	C₂×H(3,𝔽₄)	—	5	48	no	—
50	C₂²×#124	—	5	3	no	—
51	C₂²×#124	—	5	6	no	—
52	C₂²×#120	—	5	2	no	—
53	C₂²×#120	—	5	3	no	—
54	C₂²×#120	D₅	1	1	yes	characteristic, complemented
55	C₂⁴×C₅	—	2	24	no	—
56	C₂⁴×C₅	—	2	12	no	—
57	C₂⁴×C₅	—	2	12	no	—
58	C₂⁴×C₅	—	4	72	no	—
59	C₂⁴×C₅	—	4	64	no	—
60	C₂⁴×C₅	C₂⁴	1	6	yes	—
61	C₂⁴×C₅	C₂⁴	1	18	yes	—
62	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
63	C₂⁴×C₅	C₂⁴	1	16	yes	complemented
64	C₂³×D₅	—	2	24	no	—
65	C₂³×D₅	—	4	72	no	—
66	C₂³×D₅	—	4	64	no	—
67	C₂³×D₅	C₂⁴	1	12	yes	—
68	C₂³×D₅	C₂⁴	1	16	yes	complemented
69	C₂×C₅×D₄	—	4	72	no	—
70	C₂×C₅×D₄	C₂⁴	1	48	yes	—
71	C₂²×C₄×C₅	—	2	12	no	—
72	C₂²×C₄×C₅	C₂⁴	1	18	yes	—
73	C₂×C₂²⋊D₅	—	4	144	no	—
74	C₂×C₂²⋊D₅	C₂⁴	1	96	yes	—
75	C₂²×Dic₅	—	2	24	no	—
76	C₂²×Dic₅	C₂⁴	1	36	yes	—
77	C₂×Dic₁₀	—	4	72	no	—
78	C₂×Dic₁₀	C₂⁴	1	48	yes	complemented
79	C₅×C₂²⋊C₄	C₂⁴	1	48	yes	complemented
80	C₂²⋊Dic₅	C₂⁴	1	96	yes	complemented
81	C₂²⋊Dic₅	C₂⁴	1	48	yes	complemented
82	#685	C₂⁴	1	96	yes	complemented
83	C₄×Dic₅	C₂⁴	1	48	yes	complemented
84	C₂⁶	—	5	1	no	—
85	C₂⁶	C₂×D₅	1	1	yes	characteristic, complemented
86	C₂³×Q₈	—	5	3	no	—
87	C₂³×D₄	—	5	6	no	—
88	C₂³×D₄	C₂×D₅	1	3	yes	—
89	H(3,𝔽₄)	—	5	256	no	—
90	C₂×#124	—	5	72	no	—
91	C₂×#124	—	5	144	no	—
92	C₂×#120	—	5	48	no	—
93	C₂×#120	—	5	72	no	—
94	C₂×#120	C₂×D₅	1	24	yes	complemented
95	C₂²×C₂²⋊C₄	—	5	3	no	—
96	C₂²×C₂²⋊C₄	—	5	6	no	—
97	C₂²×C₂²⋊C₄	—	5	6	no	—
98	C₂²×C₂²⋊C₄	C₂×D₅	1	3	yes	complemented
99	C₂²×C₄²	—	5	3	no	—
100	C₂³×C₅	—	2	72	no	—
101	C₂³×C₅	—	2	36	no	—
102	C₂³×C₅	—	2	36	no	—
103	C₂³×C₅	—	4	48	no	—
104	C₂³×C₅	—	4	192	no	—
105	C₂³×C₅	—	4	16	no	—
106	C₂³×C₅	—	4	48	no	—
107	C₂³×C₅	H(5,𝔽₂)	1	12	yes	—
108	C₂³×C₅	C₂⁵	1	4	yes	—
109	C₂³×C₅	C₂⁵	1	12	yes	—
110	C₂³×C₅	C₂⁵	1	3	yes	—
111	C₂²×D₅	—	2	72	no	—
112	C₂²×D₅	—	4	48	no	—
113	C₂²×D₅	—	4	192	no	—
114	C₂²×D₅	—	4	32	no	—
115	C₂²×D₅	C₂⁵	1	8	yes	—
116	C₅×D₄	—	4	48	no	—
117	C₂×C₄×C₅	—	2	36	no	—
118	C₂×C₄×C₅	C₂⁵	1	12	yes	—
119	C₂²⋊D₅	—	4	96	no	—
120	C₂×Dic₅	—	2	72	no	—
121	C₂×Dic₅	C₂⁵	1	24	yes	—
122	Dic₁₀	—	4	48	no	—
123	C₂⁵	—	4	12	no	—
124	C₂⁵	—	5	2	no	—
125	C₂⁵	—	5	12	no	—
126	C₂⁵	—	20	12	no	—
127	C₂⁵	C₂²×D₅	1	1	yes	characteristic
128	C₂⁵	C₂²×D₅	1	3	yes	—
129	C₂⁵	C₂²×D₅	1	12	yes	complemented
130	C₂²×Q₈	—	5	36	no	—
131	C₂²×Q₈	—	20	12	no	—
132	C₂²×D₄	—	4	12	no	—
133	C₂²×D₄	—	5	72	no	—
134	C₂²×D₄	—	20	24	no	—
135	C₂²×D₄	C₂²×D₅	1	36	yes	—
136	C₂³×C₄	—	5	6	no	—
137	C₂³×C₄	C₂²×D₅	1	3	yes	—
138	#124	—	5	192	no	—
139	#124	—	5	384	no	—
140	#120	—	5	128	no	—
141	#120	—	5	192	no	—
142	#120	C₂²×D₅	1	64	yes	complemented
143	C₂×C₂²⋊C₄	—	5	36	no	—
144	C₂×C₂²⋊C₄	—	5	72	no	—
145	C₂×C₂²⋊C₄	—	5	72	no	—
146	C₂×C₂²⋊C₄	C₂²×D₅	1	36	yes	complemented
147	C₂×C₄²	—	5	36	no	—
148	C₂²×C₅	—	2	24	no	—
149	C₂²×C₅	—	2	12	no	—
150	C₂²×C₅	—	2	12	no	—
151	C₂²×C₅	—	4	64	no	—
152	C₂²×C₅	—	4	24	no	—
153	C₂²×C₅	—	4	72	no	—
154	C₂²×C₅	H(3,𝔽₄)	1	16	yes	complemented
155	C₂²×C₅	C₂×H(5,𝔽₂)	1	18	yes	—
156	C₂²×C₅	C₂⁶	1	1	yes	characteristic
157	C₂×D₅	—	2	24	no	—
158	C₂×D₅	—	4	64	no	—
159	C₂×D₅	—	4	48	no	—
160	C₄×C₅	—	2	12	no	—
161	Dic₅	—	2	24	no	—
162	C₂⁴	—	2	12	no	—
163	C₂⁴	—	2	12	no	—
164	C₂⁴	—	2	24	no	—
165	C₂⁴	—	4	72	no	—
166	C₂⁴	—	4	64	no	—
167	C₂⁴	—	5	12	no	—
168	C₂⁴	—	5	16	no	—
169	C₂⁴	—	10	24	no	—
170	C₂⁴	—	20	72	no	—
171	C₂⁴	—	20	64	no	—
172	C₂⁴	C₂³×D₅	1	6	yes	—
173	C₂⁴	C₂³×D₅	1	18	yes	—
174	C₂⁴	C₂³×D₅	1	1	yes	characteristic
175	C₂⁴	C₂³×D₅	1	16	yes	complemented
176	C₂×Q₈	—	5	48	no	—
177	C₂×Q₈	—	20	72	no	—
178	C₂×D₄	—	4	72	no	—
179	C₂×D₄	—	5	96	no	—
180	C₂×D₄	—	20	144	no	—
181	C₂×D₄	C₂³×D₅	1	48	yes	—
182	C₂²×C₄	—	2	12	no	—
183	C₂²×C₄	—	5	36	no	—
184	C₂²×C₄	—	10	24	no	—
185	C₂²×C₄	C₂³×D₅	1	18	yes	—
186	C₂²⋊C₄	—	5	48	no	—
187	C₂²⋊C₄	—	5	96	no	—
188	C₂²⋊C₄	—	5	96	no	—
189	C₂²⋊C₄	C₂³×D₅	1	48	yes	complemented
190	C₄²	—	5	48	no	—
191	C₂×C₅	—	4	4	no	—
192	C₂×C₅	—	4	12	no	—
193	C₂×C₅	C₂×H(3,𝔽₄)	1	12	yes	complemented
194	C₂×C₅	C₂²×H(5,𝔽₂)	1	3	yes	—
195	D₅	—	4	8	no	—
196	C₂³	—	2	36	no	—
197	C₂³	—	2	36	no	—
198	C₂³	—	2	72	no	—
199	C₂³	—	4	48	no	—
200	C₂³	—	4	192	no	—
201	C₂³	—	4	48	no	—
202	C₂³	—	4	16	no	—
203	C₂³	—	5	8	no	—
204	C₂³	—	10	72	no	—
205	C₂³	—	20	48	no	—
206	C₂³	—	20	192	no	—
207	C₂³	—	20	32	no	—
208	C₂³	#4144	1	12	yes	—
209	C₂³	C₂⁴×D₅	1	4	yes	—
210	C₂³	C₂⁴×D₅	1	12	yes	—
211	C₂³	C₂⁴×D₅	1	3	yes	—
212	Q₈	—	20	48	no	—
213	D₄	—	4	48	no	—
214	D₄	—	20	96	no	—
215	C₂×C₄	—	2	36	no	—
216	C₂×C₄	—	5	24	no	—
217	C₂×C₄	—	10	72	no	—
218	C₂×C₄	C₂⁴×D₅	1	12	yes	—
219	C₅	C₂²×H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
220	C₂²	—	2	12	no	—
221	C₂²	—	2	12	no	—
222	C₂²	—	2	24	no	—
223	C₂²	—	4	64	no	—
224	C₂²	—	4	72	no	—
225	C₂²	—	4	24	no	—
226	C₂²	—	10	24	no	—
227	C₂²	—	20	64	no	—
228	C₂²	—	20	48	no	—
229	C₂²	#65928	1	16	yes	direct product factor
230	C₂²	C₂×#4144	1	18	yes	—
231	C₂²	C₂⁵×D₅	1	1	yes	characteristic
232	C₄	—	2	12	no	—
233	C₄	—	10	24	no	—
234	C₂	—	4	12	no	—
235	C₂	—	4	4	no	—
236	C₂	—	20	8	no	—
237	C₂	C₂×#65928	1	12	yes	direct product factor
238	C₂	C₂²×#4144	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁶
Frattini	C₂²	C₂⁵×D₅
Fitting	C₂²×C₅×#120	C₂
socle	C₂⁴×C₅	C₂⁴
hypercenter	C₂²×#120	D₅
hypocenter	C₅	C₂²×H(3,𝔽₄)
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#120

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×#120	C₂	semidirect
2. C₂²×#4099	C₂	semidirect
3. C₂²×#4083	C₂	semidirect
4. C₂²×#4082	C₂	semidirect
5. C₂×#65928	C₂	direct (N × H)
6. C₂²×#4031	C₂	semidirect
7. C₂⁶×C₅	C₂²	semidirect
8. C₂⁵×D₅	C₂²	semidirect
11. C₂³×Dic₁₀	C₂²	semidirect
12. C₂×C₅×#120	C₂²	semidirect
13. C₂²×C₅×C₂²⋊C₄	C₂²	semidirect
14. C₂×#4099	C₂²	semidirect
15. C₂²×C₂²⋊Dic₅	C₂²	semidirect
16. C₂²×C₂²⋊Dic₅	C₂²	semidirect
17. C₂×#4083	C₂²	semidirect
18. C₂×#4082	C₂²	semidirect
19. C₂²×#685	C₂²	semidirect
20. C₂²×C₄×Dic₅	C₂²	semidirect
21. #65928	C₂²	direct (N × H)
22. C₂×#4031	C₂²	semidirect
27. C₂⁵×C₅	C₂³	semidirect
30. C₂⁴×D₅	C₂³	semidirect
38. C₂²×Dic₁₀	C₂³	semidirect
39. C₅×#120	C₂³	semidirect
40. C₂×C₅×C₂²⋊C₄	C₂³	semidirect
41. #4099	C₂³	semidirect
42. C₂×C₂²⋊Dic₅	C₂³	semidirect
43. C₂×C₂²⋊Dic₅	C₂³	semidirect
44. #4083	C₂³	semidirect
45. #4082	C₂³	semidirect
46. C₂×#685	C₂³	semidirect
47. C₂×C₄×Dic₅	C₂³	semidirect
48. #4031	C₂³	semidirect
54. C₂²×#120	D₅	semidirect
63. C₂⁴×C₅	C₂⁴	semidirect
68. C₂³×D₅	C₂⁴	semidirect
78. C₂×Dic₁₀	C₂⁴	semidirect
79. C₅×C₂²⋊C₄	C₂⁴	semidirect
80. C₂²⋊Dic₅	C₂⁴	semidirect
81. C₂²⋊Dic₅	C₂⁴	semidirect
82. #685	C₂⁴	semidirect
83. C₄×Dic₅	C₂⁴	semidirect
85. C₂⁶	C₂×D₅	semidirect
94. C₂×#120	C₂×D₅	semidirect
98. C₂²×C₂²⋊C₄	C₂×D₅	semidirect
129. C₂⁵	C₂²×D₅	semidirect
142. #120	C₂²×D₅	semidirect
146. C₂×C₂²⋊C₄	C₂²×D₅	semidirect
154. C₂²×C₅	H(3,𝔽₄)	semidirect
175. C₂⁴	C₂³×D₅	semidirect
189. C₂²⋊C₄	C₂³×D₅	semidirect
193. C₂×C₅	C₂×H(3,𝔽₄)	semidirect
219. C₅	C₂²×H(3,𝔽₄)	semidirect
229. C₂²	#65928	direct (N × H)
237. C₂	C₂×#65928	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×C₅×#120	2
C₂²×#4099	2
C₂²×#4083	2
C₂²×#4082	2
C₂×#65928	2
C₂²×#4031	2
C₂²×H(3,𝔽₄)	5

Subgroups — C22×#65928 #13006173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#65928 #13006173