Subgroups — C₂×D₅×H(3,𝔽₄) #13006171

Order 1280.

10246 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic, complemented
2	D₅×H(3,𝔽₄)	C₂	1	32	yes	maximal, direct product factor
3	C₂×#66017	C₂	1	9	yes	maximal, complemented
4	C₂×D₅×#124	C₂	1	9	yes	maximal, complemented
5	C₂×D₅×#120	C₂	1	6	yes	maximal, complemented
6	C₂×#65928	C₂	1	6	yes	maximal, complemented
7	C₂⁵×D₅	C₂²	1	2	yes	complemented
8	C₂²×Q₈×D₅	C₂²	1	3	yes	complemented
9	C₂²×D₄×D₅	C₂²	1	9	yes	—
10	C₅×H(3,𝔽₄)	C₂²	1	16	yes	complemented
11	C₂×C₅×#124	C₂²	1	9	yes	complemented
12	C₂×C₅×#120	C₂²	1	6	yes	complemented
13	C₂×#4099	C₂²	1	6	yes	complemented
14	C₂×#4093	C₂²	1	9	yes	complemented
15	C₂×#4083	C₂²	1	18	yes	complemented
16	C₂×#4082	C₂²	1	9	yes	complemented
17	#66017	C₂²	1	144	yes	complemented
18	D₅×#124	C₂²	1	144	yes	complemented
19	#65928	C₂²	1	96	yes	complemented
20	D₅×#120	C₂²	1	96	yes	complemented
21	C₂×#4031	C₂²	1	36	yes	complemented
22	C₂×#4028	C₂²	1	18	yes	complemented
23	C₂×D₅×C₂²⋊C₄	C₂²	1	18	yes	complemented
24	C₂×#4021	C₂²	1	9	yes	complemented
25	C₂×C₄²×D₅	C₂²	1	3	yes	complemented
26	C₂²×H(3,𝔽₄)	—	5	1	no	maximal, Sylow
27	C₂⁵×C₅	C₂³	1	2	yes	complemented
28	C₂⁴×D₅	—	4	24	no	—
29	C₂⁴×D₅	C₂³	1	6	yes	—
30	C₂⁴×D₅	C₂³	1	16	yes	complemented
31	C₂⁴×D₅	C₂³	1	6	yes	complemented
32	C₂²×C₅×Q₈	C₂³	1	3	yes	complemented
33	C₂²×C₅×D₄	C₂³	1	9	yes	—
34	C₂²×C₂²⋊D₅	C₂³	1	18	yes	—
35	C₂×Q₈×D₅	—	4	12	no	—
36	C₂×Q₈×D₅	C₂³	1	24	yes	complemented
37	C₂×D₄×D₅	—	4	36	no	—
38	C₂×D₄×D₅	C₂³	1	72	yes	—
39	C₂²×D₂₀	C₂³	1	9	yes	—
40	C₂²×C₄×D₅	C₂³	1	9	yes	—
41	C₂²×Dic₁₀	C₂³	1	9	yes	complemented
42	C₅×#124	C₂³	1	72	yes	complemented
43	C₅×#120	C₂³	1	48	yes	complemented
44	C₂×C₅×C₂²⋊C₄	C₂³	1	18	yes	complemented
45	C₂×C₄²×C₅	C₂³	1	3	yes	complemented
46	#4099	C₂³	1	48	yes	complemented
47	C₂×C₂²⋊Dic₅	C₂³	1	18	yes	complemented
48	#4093	C₂³	1	72	yes	complemented
49	#4083	C₂³	1	144	yes	complemented
50	#4082	C₂³	1	72	yes	complemented
51	C₂×#685	C₂³	1	36	yes	complemented
52	C₂×C₄×Dic₅	C₂³	1	9	yes	complemented
53	#4031	C₂³	1	288	yes	complemented
54	#4028	C₂³	1	144	yes	complemented
55	D₅×C₂²⋊C₄	C₂³	1	144	yes	complemented
56	#4021	C₂³	1	72	yes	complemented
57	C₄²×D₅	C₂³	1	24	yes	complemented
58	C₂×H(3,𝔽₄)	—	5	32	no	—
59	C₂×H(3,𝔽₄)	—	5	9	no	—
60	C₂×H(3,𝔽₄)	—	5	6	no	—
61	C₂×H(3,𝔽₄)	D₅	1	1	yes	characteristic, direct product factor
62	C₂²×#124	—	5	9	no	—
63	C₂²×#120	—	5	6	no	—
64	C₂⁴×C₅	—	4	12	no	—
65	C₂⁴×C₅	C₂⁴	1	6	yes	—
66	C₂⁴×C₅	C₂⁴	1	8	yes	complemented
67	C₂³×D₅	—	2	72	no	—
68	C₂³×D₅	—	4	96	no	—
69	C₂³×D₅	—	4	36	no	—
70	C₂³×D₅	—	4	128	no	—
71	C₂³×D₅	C₂⁴	1	6	yes	—
72	C₂³×D₅	C₂⁴	1	24	yes	—
73	C₂³×D₅	C₂⁴	1	1	yes	characteristic
74	C₂³×D₅	C₂⁴	1	24	yes	complemented
75	C₂×C₅×Q₈	—	4	6	no	—
76	C₂×C₅×Q₈	C₂⁴	1	12	yes	complemented
77	C₂×C₅×D₄	—	4	18	no	—
78	C₂×C₅×D₄	C₂⁴	1	36	yes	—
79	C₂²×C₄×C₅	C₂⁴	1	9	yes	—
80	C₂×C₂²⋊D₅	—	4	36	no	—
81	C₂×C₂²⋊D₅	C₂⁴	1	72	yes	—
82	C₂²×Dic₅	C₂⁴	1	9	yes	—
83	Q₈×D₅	—	4	48	no	—
84	D₄×D₅	—	4	144	no	—
85	C₂×D₂₀	—	4	18	no	—
86	C₂×D₂₀	C₂⁴	1	36	yes	—
87	C₂×C₄×D₅	—	2	36	no	—
88	C₂×C₄×D₅	C₂⁴	1	36	yes	—
89	C₂×Dic₁₀	—	4	18	no	—
90	C₂×Dic₁₀	C₂⁴	1	36	yes	complemented
91	C₅×C₂²⋊C₄	C₂⁴	1	72	yes	complemented
92	C₄²×C₅	C₂⁴	1	12	yes	complemented
93	C₂²⋊Dic₅	C₂⁴	1	72	yes	complemented
94	#685	C₂⁴	1	144	yes	complemented
95	C₄×Dic₅	C₂⁴	1	36	yes	complemented
96	C₂⁶	—	5	2	no	—
97	C₂³×Q₈	—	5	3	no	—
98	C₂³×D₄	—	5	9	no	—
99	H(3,𝔽₄)	—	5	144	no	—
100	H(3,𝔽₄)	—	5	96	no	—
101	H(3,𝔽₄)	C₂×D₅	1	16	yes	direct product factor
102	C₂×#124	—	5	9	no	—
103	C₂×#124	—	5	9	no	—
104	C₂×#124	—	5	144	no	—
105	C₂×#124	—	5	36	no	—
106	C₂×#124	—	5	9	no	—
107	C₂×#124	C₂×D₅	1	9	yes	complemented
108	C₂×#120	—	5	6	no	—
109	C₂×#120	—	5	18	no	—
110	C₂×#120	—	5	96	no	—
111	C₂×#120	—	5	18	no	—
112	C₂×#120	C₂×D₅	1	6	yes	complemented
113	C₂²×C₂²⋊C₄	—	5	18	no	—
114	C₂²×C₄²	—	5	3	no	—
115	C₂³×C₅	—	2	36	no	—
116	C₂³×C₅	—	4	24	no	—
117	C₂³×C₅	—	4	32	no	—
118	C₂³×C₅	C₂⁵	1	12	yes	—
119	C₂³×C₅	C₂⁵	1	1	yes	characteristic
120	C₂²×D₅	—	2	36	no	—
121	C₂²×D₅	—	2	144	no	—
122	C₂²×D₅	—	4	72	no	—
123	C₂²×D₅	—	4	96	no	—
124	C₂²×D₅	—	4	256	no	—
125	C₂²×D₅	—	4	96	no	—
126	C₂²×D₅	H(5,𝔽₂)	1	12	yes	—
127	C₂²×D₅	C₂⁵	1	12	yes	—
128	C₂²×D₅	C₂⁵	1	2	yes	—
129	C₅×Q₈	—	4	12	no	—
130	C₅×D₄	—	4	36	no	—
131	C₂×C₄×C₅	—	2	18	no	—
132	C₂×C₄×C₅	C₂⁵	1	18	yes	—
133	C₂²⋊D₅	—	4	72	no	—
134	C₂×Dic₅	—	2	18	no	—
135	C₂×Dic₅	C₂⁵	1	18	yes	—
136	D₂₀	—	4	36	no	—
137	C₄×D₅	—	2	72	no	—
138	Dic₁₀	—	4	36	no	—
139	C₂⁵	—	5	6	no	—
140	C₂⁵	—	5	6	no	—
141	C₂⁵	—	5	16	no	—
142	C₂⁵	—	20	24	no	—
143	C₂⁵	C₂²×D₅	1	2	yes	complemented
144	C₂²×Q₈	—	5	24	no	—
145	C₂²×Q₈	—	5	9	no	—
146	C₂²×Q₈	—	20	12	no	—
147	C₂²×Q₈	C₂²×D₅	1	3	yes	complemented
148	C₂²×D₄	—	5	18	no	—
149	C₂²×D₄	—	5	72	no	—
150	C₂²×D₄	—	5	9	no	—
151	C₂²×D₄	—	20	36	no	—
152	C₂²×D₄	C₂²×D₅	1	9	yes	—
153	C₂³×C₄	—	5	9	no	—
154	#124	—	5	72	no	—
155	#124	—	5	72	no	—
156	#124	—	5	288	no	—
157	#124	—	5	72	no	—
158	#124	C₂²×D₅	1	72	yes	complemented
159	#120	—	5	48	no	—
160	#120	—	5	144	no	—
161	#120	—	5	144	no	—
162	#120	C₂²×D₅	1	48	yes	complemented
163	C₂×C₂²⋊C₄	—	5	18	no	—
164	C₂×C₂²⋊C₄	—	5	36	no	—
165	C₂×C₂²⋊C₄	—	5	144	no	—
166	C₂×C₂²⋊C₄	C₂²×D₅	1	18	yes	complemented
167	C₂×C₄²	—	5	9	no	—
168	C₂×C₄²	—	5	24	no	—
169	C₂×C₄²	C₂²×D₅	1	3	yes	complemented
170	C₂²×C₅	—	2	36	no	—
171	C₂²×C₅	—	4	32	no	—
172	C₂²×C₅	—	4	24	no	—
173	C₂²×C₅	C₂×H(5,𝔽₂)	1	6	yes	—
174	C₂²×C₅	C₂⁶	1	1	yes	characteristic
175	C₂×D₅	—	2	36	no	—
176	C₂×D₅	—	4	96	no	—
177	C₂×D₅	—	4	24	no	—
178	C₂×D₅	—	4	96	no	—
179	C₂×D₅	H(3,𝔽₄)	1	16	yes	direct product factor
180	C₂×D₅	C₂×H(5,𝔽₂)	1	12	yes	—
181	C₄×C₅	—	2	18	no	—
182	Dic₅	—	2	18	no	—
183	C₂⁴	—	4	12	no	—
184	C₂⁴	—	5	1	no	—
185	C₂⁴	—	5	24	no	—
186	C₂⁴	—	5	6	no	—
187	C₂⁴	—	5	24	no	—
188	C₂⁴	—	10	72	no	—
189	C₂⁴	—	20	36	no	—
190	C₂⁴	—	20	96	no	—
191	C₂⁴	—	20	128	no	—
192	C₂⁴	C₂³×D₅	1	6	yes	—
193	C₂⁴	C₂³×D₅	1	8	yes	complemented
194	C₂×Q₈	—	4	6	no	—
195	C₂×Q₈	—	5	36	no	—
196	C₂×Q₈	—	20	18	no	—
197	C₂×Q₈	—	20	48	no	—
198	C₂×Q₈	C₂³×D₅	1	12	yes	complemented
199	C₂×D₄	—	4	18	no	—
200	C₂×D₄	—	5	72	no	—
201	C₂×D₄	—	5	36	no	—
202	C₂×D₄	—	20	18	no	—
203	C₂×D₄	—	20	144	no	—
204	C₂×D₄	—	20	36	no	—
205	C₂×D₄	C₂³×D₅	1	36	yes	—
206	C₂²×C₄	—	5	9	no	—
207	C₂²×C₄	—	5	36	no	—
208	C₂²×C₄	—	10	36	no	—
209	C₂²×C₄	C₂³×D₅	1	9	yes	—
210	C₂²⋊C₄	—	5	72	no	—
211	C₂²⋊C₄	—	5	144	no	—
212	C₂²⋊C₄	C₂³×D₅	1	72	yes	complemented
213	C₄²	—	5	36	no	—
214	C₄²	C₂³×D₅	1	12	yes	complemented
215	C₂×C₅	—	4	12	no	—
216	C₂×C₅	C₂×H(3,𝔽₄)	1	4	yes	complemented
217	C₂×C₅	C₂²×H(5,𝔽₂)	1	3	yes	—
218	D₅	—	4	12	no	—
219	D₅	C₂×H(3,𝔽₄)	1	8	yes	direct product factor
220	C₂³	—	2	36	no	—
221	C₂³	—	4	24	no	—
222	C₂³	—	4	32	no	—
223	C₂³	—	5	12	no	—
224	C₂³	—	5	2	no	—
225	C₂³	—	5	12	no	—
226	C₂³	—	10	144	no	—
227	C₂³	—	10	36	no	—
228	C₂³	—	20	72	no	—
229	C₂³	—	20	96	no	—
230	C₂³	—	20	256	no	—
231	C₂³	—	20	96	no	—
232	C₂³	C₂⁴×D₅	1	12	yes	—
233	C₂³	C₂⁴×D₅	1	1	yes	characteristic
234	Q₈	—	4	12	no	—
235	Q₈	—	20	36	no	—
236	D₄	—	4	36	no	—
237	D₄	—	20	36	no	—
238	D₄	—	20	72	no	—
239	C₂×C₄	—	2	18	no	—
240	C₂×C₄	—	5	18	no	—
241	C₂×C₄	—	10	72	no	—
242	C₂×C₄	—	10	18	no	—
243	C₂×C₄	C₂⁴×D₅	1	18	yes	—
244	C₅	C₂²×H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
245	C₂²	—	2	36	no	—
246	C₂²	—	4	32	no	—
247	C₂²	—	4	24	no	—
248	C₂²	—	5	12	no	—
249	C₂²	—	5	16	no	—
250	C₂²	—	10	36	no	—
251	C₂²	—	20	96	no	—
252	C₂²	—	20	96	no	—
253	C₂²	—	20	24	no	—
254	C₂²	D₅×H(5,𝔽₂)	1	6	yes	—
255	C₂²	C₂⁵×D₅	1	1	yes	characteristic
256	C₄	—	2	18	no	—
257	C₄	—	10	18	no	—
258	C₂	—	4	12	no	—
259	C₂	—	5	8	no	—
260	C₂	—	20	12	no	—
261	C₂	D₅×H(3,𝔽₄)	1	4	yes	direct product factor
262	C₂	C₂×D₅×H(5,𝔽₂)	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×H(3,𝔽₄)	C₂	semidirect
2. D₅×H(3,𝔽₄)	C₂	direct (N × H)
3. C₂×#66017	C₂	semidirect
4. C₂×D₅×#124	C₂	semidirect
5. C₂×D₅×#120	C₂	semidirect
6. C₂×#65928	C₂	semidirect
7. C₂⁵×D₅	C₂²	semidirect
8. C₂²×Q₈×D₅	C₂²	semidirect
10. C₅×H(3,𝔽₄)	C₂²	semidirect
11. C₂×C₅×#124	C₂²	semidirect
12. C₂×C₅×#120	C₂²	semidirect
13. C₂×#4099	C₂²	semidirect
14. C₂×#4093	C₂²	semidirect
15. C₂×#4083	C₂²	semidirect
16. C₂×#4082	C₂²	semidirect
17. #66017	C₂²	semidirect
18. D₅×#124	C₂²	semidirect
19. #65928	C₂²	semidirect
20. D₅×#120	C₂²	semidirect
21. C₂×#4031	C₂²	semidirect
22. C₂×#4028	C₂²	semidirect
23. C₂×D₅×C₂²⋊C₄	C₂²	semidirect
24. C₂×#4021	C₂²	semidirect
25. C₂×C₄²×D₅	C₂²	semidirect
27. C₂⁵×C₅	C₂³	semidirect
30. C₂⁴×D₅	C₂³	semidirect
31. C₂⁴×D₅	C₂³	semidirect
32. C₂²×C₅×Q₈	C₂³	semidirect
36. C₂×Q₈×D₅	C₂³	semidirect
41. C₂²×Dic₁₀	C₂³	semidirect
42. C₅×#124	C₂³	semidirect
43. C₅×#120	C₂³	semidirect
44. C₂×C₅×C₂²⋊C₄	C₂³	semidirect
45. C₂×C₄²×C₅	C₂³	semidirect
46. #4099	C₂³	semidirect
47. C₂×C₂²⋊Dic₅	C₂³	semidirect
48. #4093	C₂³	semidirect
49. #4083	C₂³	semidirect
50. #4082	C₂³	semidirect
51. C₂×#685	C₂³	semidirect
52. C₂×C₄×Dic₅	C₂³	semidirect
53. #4031	C₂³	semidirect
54. #4028	C₂³	semidirect
55. D₅×C₂²⋊C₄	C₂³	semidirect
56. #4021	C₂³	semidirect
57. C₄²×D₅	C₂³	semidirect
61. C₂×H(3,𝔽₄)	D₅	direct (N × H)
66. C₂⁴×C₅	C₂⁴	semidirect
74. C₂³×D₅	C₂⁴	semidirect
76. C₂×C₅×Q₈	C₂⁴	semidirect
90. C₂×Dic₁₀	C₂⁴	semidirect
91. C₅×C₂²⋊C₄	C₂⁴	semidirect
92. C₄²×C₅	C₂⁴	semidirect
93. C₂²⋊Dic₅	C₂⁴	semidirect
94. #685	C₂⁴	semidirect
95. C₄×Dic₅	C₂⁴	semidirect
101. H(3,𝔽₄)	C₂×D₅	direct (N × H)
107. C₂×#124	C₂×D₅	semidirect
112. C₂×#120	C₂×D₅	semidirect
143. C₂⁵	C₂²×D₅	semidirect
147. C₂²×Q₈	C₂²×D₅	semidirect
158. #124	C₂²×D₅	semidirect
162. #120	C₂²×D₅	semidirect
166. C₂×C₂²⋊C₄	C₂²×D₅	semidirect
169. C₂×C₄²	C₂²×D₅	semidirect
179. C₂×D₅	H(3,𝔽₄)	direct (N × H)
193. C₂⁴	C₂³×D₅	semidirect
198. C₂×Q₈	C₂³×D₅	semidirect
212. C₂²⋊C₄	C₂³×D₅	semidirect
214. C₄²	C₂³×D₅	semidirect
216. C₂×C₅	C₂×H(3,𝔽₄)	semidirect
219. D₅	C₂×H(3,𝔽₄)	direct (N × H)
244. C₅	C₂²×H(3,𝔽₄)	semidirect
261. C₂	D₅×H(3,𝔽₄)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×H(3,𝔽₄)	2
D₅×H(3,𝔽₄)	2
C₂×#66017	2
C₂×D₅×#124	2
C₂×D₅×#120	2
C₂×#65928	2
C₂²×H(3,𝔽₄)	5

Subgroups — C2×D5×H(3,𝔽4) #13006171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₅×H(3,𝔽₄) #13006171