Subgroups — C₂×H(3,𝔽₄)×D₅₃ #28290963

Order 13568.

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×#16811366	C₂	1	6	yes	maximal, complemented
2	C₂×#16811365	C₂	1	9	yes	maximal, complemented
3	H(3,𝔽₄)×D₅₃	C₂	1	32	yes	maximal, direct product factor
4	C₂×#124×D₅₃	C₂	1	9	yes	maximal, complemented
5	C₂×#120×D₅₃	C₂	1	6	yes	maximal, complemented
6	C₂×C₅₃×H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₅₃	C₂²	1	2	yes	complemented
8	C₂²×Q₈×D₅₃	C₂²	1	3	yes	complemented
9	C₂²×D₄×D₅₃	C₂²	1	9	yes	—
10	#16811366	C₂²	1	96	yes	complemented
11	#16811365	C₂²	1	144	yes	complemented
12	C₂×#13133206	C₂²	1	9	yes	complemented
13	C₂×#13133205	C₂²	1	9	yes	complemented
14	C₂×#13133204	C₂²	1	9	yes	complemented
15	C₂×#13133203	C₂²	1	36	yes	complemented
16	#124×D₅₃	C₂²	1	144	yes	complemented
17	C₂×#13133187	C₂²	1	18	yes	complemented
18	C₂×#13133186	C₂²	1	6	yes	complemented
19	C₂×#13133185	C₂²	1	18	yes	complemented
20	#120×D₅₃	C₂²	1	96	yes	complemented
21	C₂×C₂²⋊C₄×D₅₃	C₂²	1	18	yes	complemented
22	C₂×C₄²×D₅₃	C₂²	1	3	yes	complemented
23	C₅₃×H(3,𝔽₄)	C₂²	1	16	yes	complemented
24	C₂×#124×C₅₃	C₂²	1	9	yes	complemented
25	C₂×#120×C₅₃	C₂²	1	6	yes	complemented
26	C₂⁴×D₅₃	—	4	24	no	—
27	C₂⁴×D₅₃	C₂³	1	6	yes	—
28	C₂⁴×D₅₃	C₂³	1	16	yes	complemented
29	C₂⁴×D₅₃	C₂³	1	6	yes	complemented
30	C₂²×Dic₁₀₆	C₂³	1	9	yes	complemented
31	C₂×Q₈×D₅₃	—	4	12	no	—
32	C₂×Q₈×D₅₃	C₂³	1	24	yes	complemented
33	C₂²×D₂₁₂	C₂³	1	9	yes	—
34	C₂²×C₂²⋊D₅₃	C₂³	1	18	yes	—
35	C₂×D₄×D₅₃	—	4	36	no	—
36	C₂×D₄×D₅₃	C₂³	1	72	yes	—
37	C₂²×C₄×D₅₃	C₂³	1	9	yes	—
38	#13133206	C₂³	1	72	yes	complemented
39	#13133205	C₂³	1	72	yes	complemented
40	#13133204	C₂³	1	72	yes	complemented
41	#13133203	C₂³	1	288	yes	complemented
42	#13133187	C₂³	1	144	yes	complemented
43	#13133186	C₂³	1	48	yes	complemented
44	#13133185	C₂³	1	144	yes	complemented
45	C₂×C₂²⋊Dic₅₃	C₂³	1	18	yes	complemented
46	C₂×#11719921	C₂³	1	36	yes	complemented
47	C₂²⋊C₄×D₅₃	C₂³	1	144	yes	complemented
48	C₂×C₄×Dic₅₃	C₂³	1	9	yes	complemented
49	C₄²×D₅₃	C₂³	1	24	yes	complemented
50	C₂⁵×C₅₃	C₂³	1	2	yes	complemented
51	C₂²×Q₈×C₅₃	C₂³	1	3	yes	complemented
52	C₂²×D₄×C₅₃	C₂³	1	9	yes	—
53	#124×C₅₃	C₂³	1	72	yes	complemented
54	#120×C₅₃	C₂³	1	48	yes	complemented
55	C₂×C₂²⋊C₄×C₅₃	C₂³	1	18	yes	complemented
56	C₂×C₄²×C₅₃	C₂³	1	3	yes	complemented
57	C₂⁴×C₅₃	—	4	12	no	—
58	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
59	C₂⁴×C₅₃	C₂⁴	1	8	yes	complemented
60	C₂³×D₅₃	—	2	72	no	—
61	C₂³×D₅₃	—	4	96	no	—
62	C₂³×D₅₃	—	4	36	no	—
63	C₂³×D₅₃	—	4	128	no	—
64	C₂³×D₅₃	C₂⁴	1	24	yes	—
65	C₂³×D₅₃	C₂⁴	1	6	yes	—
66	C₂³×D₅₃	C₂⁴	1	1	yes	characteristic
67	C₂³×D₅₃	C₂⁴	1	24	yes	complemented
68	C₂×Q₈×C₅₃	—	4	6	no	—
69	C₂×Q₈×C₅₃	C₂⁴	1	12	yes	complemented
70	C₂×D₄×C₅₃	—	4	18	no	—
71	C₂×D₄×C₅₃	C₂⁴	1	36	yes	—
72	C₂²×C₄×C₅₃	C₂⁴	1	9	yes	—
73	C₂×C₂²⋊D₅₃	—	4	36	no	—
74	C₂×C₂²⋊D₅₃	C₂⁴	1	72	yes	—
75	C₂²×Dic₅₃	C₂⁴	1	9	yes	—
76	Q₈×D₅₃	—	4	48	no	—
77	D₄×D₅₃	—	4	144	no	—
78	C₂×D₂₁₂	—	4	18	no	—
79	C₂×D₂₁₂	C₂⁴	1	36	yes	—
80	C₂×C₄×D₅₃	—	2	36	no	—
81	C₂×C₄×D₅₃	C₂⁴	1	36	yes	—
82	C₂×Dic₁₀₆	—	4	18	no	—
83	C₂×Dic₁₀₆	C₂⁴	1	36	yes	complemented
84	C₂²⋊C₄×C₅₃	C₂⁴	1	72	yes	complemented
85	C₄²×C₅₃	C₂⁴	1	12	yes	complemented
86	C₂²⋊Dic₅₃	C₂⁴	1	72	yes	complemented
87	#11719921	C₂⁴	1	144	yes	complemented
88	C₄×Dic₅₃	C₂⁴	1	36	yes	complemented
89	C₂³×C₅₃	—	2	36	no	—
90	C₂³×C₅₃	—	4	24	no	—
91	C₂³×C₅₃	—	4	32	no	—
92	C₂³×C₅₃	C₂⁵	1	12	yes	—
93	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
94	C₂²×D₅₃	—	2	36	no	—
95	C₂²×D₅₃	—	2	144	no	—
96	C₂²×D₅₃	—	4	72	no	—
97	C₂²×D₅₃	—	4	96	no	—
98	C₂²×D₅₃	—	4	256	no	—
99	C₂²×D₅₃	—	4	96	no	—
100	C₂²×D₅₃	H(5,𝔽₂)	1	12	yes	—
101	C₂²×D₅₃	C₂⁵	1	12	yes	—
102	C₂²×D₅₃	C₂⁵	1	2	yes	—
103	Q₈×C₅₃	—	4	12	no	—
104	D₄×C₅₃	—	4	36	no	—
105	C₂×C₄×C₅₃	—	2	18	no	—
106	C₂×C₄×C₅₃	C₂⁵	1	18	yes	—
107	C₂²⋊D₅₃	—	4	72	no	—
108	C₂×Dic₅₃	—	2	18	no	—
109	C₂×Dic₅₃	C₂⁵	1	18	yes	—
110	D₂₁₂	—	4	36	no	—
111	C₄×D₅₃	—	2	72	no	—
112	Dic₁₀₆	—	4	36	no	—
113	C₂²×H(3,𝔽₄)	—	53	1	no	maximal, Sylow
114	C₂²×C₅₃	—	2	36	no	—
115	C₂²×C₅₃	—	4	32	no	—
116	C₂²×C₅₃	—	4	24	no	—
117	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
118	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
119	C₂×D₅₃	—	2	36	no	—
120	C₂×D₅₃	—	4	96	no	—
121	C₂×D₅₃	—	4	24	no	—
122	C₂×D₅₃	—	4	96	no	—
123	C₂×D₅₃	H(3,𝔽₄)	1	16	yes	direct product factor
124	C₂×D₅₃	C₂×H(5,𝔽₂)	1	12	yes	—
125	C₄×C₅₃	—	2	18	no	—
126	Dic₅₃	—	2	18	no	—
127	C₂×H(3,𝔽₄)	—	53	32	no	—
128	C₂×H(3,𝔽₄)	—	53	9	no	—
129	C₂×H(3,𝔽₄)	—	53	6	no	—
130	C₂×H(3,𝔽₄)	D₅₃	1	1	yes	characteristic, direct product factor
131	C₂²×#124	—	53	9	no	—
132	C₂²×#120	—	53	6	no	—
133	C₂×C₅₃	—	4	12	no	—
134	C₂×C₅₃	C₂×H(3,𝔽₄)	1	4	yes	complemented
135	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	3	yes	—
136	D₅₃	—	4	12	no	—
137	D₅₃	C₂×H(3,𝔽₄)	1	8	yes	direct product factor
138	C₂⁶	—	53	2	no	—
139	C₂³×Q₈	—	53	3	no	—
140	C₂³×D₄	—	53	9	no	—
141	H(3,𝔽₄)	—	53	144	no	—
142	H(3,𝔽₄)	—	53	96	no	—
143	H(3,𝔽₄)	C₂×D₅₃	1	16	yes	direct product factor
144	C₂×#124	—	53	9	no	—
145	C₂×#124	—	53	9	no	—
146	C₂×#124	—	53	144	no	—
147	C₂×#124	—	53	36	no	—
148	C₂×#124	—	53	9	no	—
149	C₂×#124	C₂×D₅₃	1	9	yes	complemented
150	C₂×#120	—	53	6	no	—
151	C₂×#120	—	53	18	no	—
152	C₂×#120	—	53	96	no	—
153	C₂×#120	—	53	18	no	—
154	C₂×#120	C₂×D₅₃	1	6	yes	complemented
155	C₂²×C₂²⋊C₄	—	53	18	no	—
156	C₂²×C₄²	—	53	3	no	—
157	C₅₃	C₂²×H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
158	C₂⁵	—	53	6	no	—
159	C₂⁵	—	53	6	no	—
160	C₂⁵	—	53	16	no	—
161	C₂⁵	—	212	24	no	—
162	C₂⁵	C₂²×D₅₃	1	2	yes	complemented
163	C₂²×Q₈	—	53	24	no	—
164	C₂²×Q₈	—	53	9	no	—
165	C₂²×Q₈	—	212	12	no	—
166	C₂²×Q₈	C₂²×D₅₃	1	3	yes	complemented
167	C₂²×D₄	—	53	18	no	—
168	C₂²×D₄	—	53	72	no	—
169	C₂²×D₄	—	53	9	no	—
170	C₂²×D₄	—	212	36	no	—
171	C₂²×D₄	C₂²×D₅₃	1	9	yes	—
172	C₂³×C₄	—	53	9	no	—
173	#124	—	53	72	no	—
174	#124	—	53	72	no	—
175	#124	—	53	288	no	—
176	#124	—	53	72	no	—
177	#124	C₂²×D₅₃	1	72	yes	complemented
178	#120	—	53	48	no	—
179	#120	—	53	144	no	—
180	#120	—	53	144	no	—
181	#120	C₂²×D₅₃	1	48	yes	complemented
182	C₂×C₂²⋊C₄	—	53	18	no	—
183	C₂×C₂²⋊C₄	—	53	36	no	—
184	C₂×C₂²⋊C₄	—	53	144	no	—
185	C₂×C₂²⋊C₄	C₂²×D₅₃	1	18	yes	complemented
186	C₂×C₄²	—	53	9	no	—
187	C₂×C₄²	—	53	24	no	—
188	C₂×C₄²	C₂²×D₅₃	1	3	yes	complemented
189	C₂⁴	—	4	12	no	—
190	C₂⁴	—	53	1	no	—
191	C₂⁴	—	53	24	no	—
192	C₂⁴	—	53	6	no	—
193	C₂⁴	—	53	24	no	—
194	C₂⁴	—	106	72	no	—
195	C₂⁴	—	212	36	no	—
196	C₂⁴	—	212	96	no	—
197	C₂⁴	—	212	128	no	—
198	C₂⁴	C₂³×D₅₃	1	6	yes	—
199	C₂⁴	C₂³×D₅₃	1	8	yes	complemented
200	C₂×Q₈	—	4	6	no	—
201	C₂×Q₈	—	53	36	no	—
202	C₂×Q₈	—	212	18	no	—
203	C₂×Q₈	—	212	48	no	—
204	C₂×Q₈	C₂³×D₅₃	1	12	yes	complemented
205	C₂×D₄	—	4	18	no	—
206	C₂×D₄	—	53	72	no	—
207	C₂×D₄	—	53	36	no	—
208	C₂×D₄	—	212	18	no	—
209	C₂×D₄	—	212	144	no	—
210	C₂×D₄	—	212	36	no	—
211	C₂×D₄	C₂³×D₅₃	1	36	yes	—
212	C₂²×C₄	—	53	9	no	—
213	C₂²×C₄	—	53	36	no	—
214	C₂²×C₄	—	106	36	no	—
215	C₂²×C₄	C₂³×D₅₃	1	9	yes	—
216	C₂²⋊C₄	—	53	72	no	—
217	C₂²⋊C₄	—	53	144	no	—
218	C₂²⋊C₄	C₂³×D₅₃	1	72	yes	complemented
219	C₄²	—	53	36	no	—
220	C₄²	C₂³×D₅₃	1	12	yes	complemented
221	C₂³	—	2	36	no	—
222	C₂³	—	4	24	no	—
223	C₂³	—	4	32	no	—
224	C₂³	—	53	12	no	—
225	C₂³	—	53	2	no	—
226	C₂³	—	53	12	no	—
227	C₂³	—	106	144	no	—
228	C₂³	—	106	36	no	—
229	C₂³	—	212	72	no	—
230	C₂³	—	212	96	no	—
231	C₂³	—	212	256	no	—
232	C₂³	—	212	96	no	—
233	C₂³	C₂⁴×D₅₃	1	12	yes	—
234	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
235	Q₈	—	4	12	no	—
236	Q₈	—	212	36	no	—
237	D₄	—	4	36	no	—
238	D₄	—	212	36	no	—
239	D₄	—	212	72	no	—
240	C₂×C₄	—	2	18	no	—
241	C₂×C₄	—	53	18	no	—
242	C₂×C₄	—	106	72	no	—
243	C₂×C₄	—	106	18	no	—
244	C₂×C₄	C₂⁴×D₅₃	1	18	yes	—
245	C₂²	—	2	36	no	—
246	C₂²	—	4	32	no	—
247	C₂²	—	4	24	no	—
248	C₂²	—	53	16	no	—
249	C₂²	—	53	12	no	—
250	C₂²	—	106	36	no	—
251	C₂²	—	212	96	no	—
252	C₂²	—	212	96	no	—
253	C₂²	—	212	24	no	—
254	C₂²	H(5,𝔽₂)×D₅₃	1	6	yes	—
255	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
256	C₄	—	2	18	no	—
257	C₄	—	106	18	no	—
258	C₂	—	4	12	no	—
259	C₂	—	53	8	no	—
260	C₂	—	212	12	no	—
261	C₂	H(3,𝔽₄)×D₅₃	1	4	yes	direct product factor
262	C₂	C₂×H(5,𝔽₂)×D₅₃	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×C₅₃×H(3,𝔽₄)	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×H(3,𝔽₄)	D₅₃
hypocenter	C₅₃	C₂²×H(3,𝔽₄)
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
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×#16811366	C₂	semidirect
2. C₂×#16811365	C₂	semidirect
3. H(3,𝔽₄)×D₅₃	C₂	direct (N × H)
4. C₂×#124×D₅₃	C₂	semidirect
5. C₂×#120×D₅₃	C₂	semidirect
6. C₂×C₅₃×H(3,𝔽₄)	C₂	semidirect
7. C₂⁵×D₅₃	C₂²	semidirect
8. C₂²×Q₈×D₅₃	C₂²	semidirect
10. #16811366	C₂²	semidirect
11. #16811365	C₂²	semidirect
12. C₂×#13133206	C₂²	semidirect
13. C₂×#13133205	C₂²	semidirect
14. C₂×#13133204	C₂²	semidirect
15. C₂×#13133203	C₂²	semidirect
16. #124×D₅₃	C₂²	semidirect
17. C₂×#13133187	C₂²	semidirect
18. C₂×#13133186	C₂²	semidirect
19. C₂×#13133185	C₂²	semidirect
20. #120×D₅₃	C₂²	semidirect
21. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
22. C₂×C₄²×D₅₃	C₂²	semidirect
23. C₅₃×H(3,𝔽₄)	C₂²	semidirect
24. C₂×#124×C₅₃	C₂²	semidirect
25. C₂×#120×C₅₃	C₂²	semidirect
28. C₂⁴×D₅₃	C₂³	semidirect
29. C₂⁴×D₅₃	C₂³	semidirect
30. C₂²×Dic₁₀₆	C₂³	semidirect
32. C₂×Q₈×D₅₃	C₂³	semidirect
38. #13133206	C₂³	semidirect
39. #13133205	C₂³	semidirect
40. #13133204	C₂³	semidirect
41. #13133203	C₂³	semidirect
42. #13133187	C₂³	semidirect
43. #13133186	C₂³	semidirect
44. #13133185	C₂³	semidirect
45. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
46. C₂×#11719921	C₂³	semidirect
47. C₂²⋊C₄×D₅₃	C₂³	semidirect
48. C₂×C₄×Dic₅₃	C₂³	semidirect
49. C₄²×D₅₃	C₂³	semidirect
50. C₂⁵×C₅₃	C₂³	semidirect
51. C₂²×Q₈×C₅₃	C₂³	semidirect
53. #124×C₅₃	C₂³	semidirect
54. #120×C₅₃	C₂³	semidirect
55. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
56. C₂×C₄²×C₅₃	C₂³	semidirect
59. C₂⁴×C₅₃	C₂⁴	semidirect
67. C₂³×D₅₃	C₂⁴	semidirect
69. C₂×Q₈×C₅₃	C₂⁴	semidirect
83. C₂×Dic₁₀₆	C₂⁴	semidirect
84. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
85. C₄²×C₅₃	C₂⁴	semidirect
86. C₂²⋊Dic₅₃	C₂⁴	semidirect
87. #11719921	C₂⁴	semidirect
88. C₄×Dic₅₃	C₂⁴	semidirect
123. C₂×D₅₃	H(3,𝔽₄)	direct (N × H)
130. C₂×H(3,𝔽₄)	D₅₃	direct (N × H)
134. C₂×C₅₃	C₂×H(3,𝔽₄)	semidirect
137. D₅₃	C₂×H(3,𝔽₄)	direct (N × H)
143. H(3,𝔽₄)	C₂×D₅₃	direct (N × H)
149. C₂×#124	C₂×D₅₃	semidirect
154. C₂×#120	C₂×D₅₃	semidirect
157. C₅₃	C₂²×H(3,𝔽₄)	semidirect
162. C₂⁵	C₂²×D₅₃	semidirect
166. C₂²×Q₈	C₂²×D₅₃	semidirect
177. #124	C₂²×D₅₃	semidirect
181. #120	C₂²×D₅₃	semidirect
185. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
188. C₂×C₄²	C₂²×D₅₃	semidirect
199. C₂⁴	C₂³×D₅₃	semidirect
204. C₂×Q₈	C₂³×D₅₃	semidirect
218. C₂²⋊C₄	C₂³×D₅₃	semidirect
220. C₄²	C₂³×D₅₃	semidirect
261. C₂	H(3,𝔽₄)×D₅₃	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×#16811366	2
C₂×#16811365	2
H(3,𝔽₄)×D₅₃	2
C₂×#124×D₅₃	2
C₂×#120×D₅₃	2
C₂×C₅₃×H(3,𝔽₄)	2
C₂²×H(3,𝔽₄)	53

Subgroups — C2×H(3,𝔽4)×D53 #28290963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×H(3,𝔽₄)×D₅₃ #28290963