Subgroups — C₂×H(5,𝔽₂)×Dic₅₃ #28290696

Order 13568.

12378 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(5,𝔽₂)×Dic₅₃	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16811106	C₂	1	6	yes	maximal, complemented
3	C₂×#16811098	C₂	1	9	yes	maximal, complemented
4	C₂×#41×Dic₅₃	C₂	1	6	yes	maximal, complemented
5	C₂²×D₄×Dic₅₃	C₂	1	9	yes	maximal, complemented
6	C₂²×H(5,𝔽₂)×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₅₃	C₂²	1	6	yes	complemented
8	#16811106	C₂²	1	96	yes	complemented
9	#16811098	C₂²	1	144	yes	complemented
10	#41×Dic₅₃	C₂²	1	96	yes	complemented
11	C₂×Q₈×Dic₅₃	C₂²	1	2	yes	complemented
12	C₂×Q₈×Dic₅₃	C₂²	1	6	yes	complemented
13	C₂×D₄×Dic₅₃	C₂²	1	18	yes	complemented
14	C₂×D₄×Dic₅₃	C₂²	1	36	yes	complemented
15	C₂×D₄×Dic₅₃	C₂²	1	18	yes	complemented
16	C₂×D₄×Dic₅₃	C₂²	1	144	yes	complemented
17	C₂×#13133176	C₂²	1	18	yes	complemented
18	C₂²×C₄⋊Dic₅₃	C₂²	1	9	yes	complemented
19	C₂²×C₂²⋊Dic₅₃	C₂²	1	18	yes	complemented
20	C₂²×C₄×Dic₅₃	C₂²	1	9	yes	complemented
21	C₂×H(5,𝔽₂)×C₅₃	C₄	1	32	yes	complemented
22	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	16	yes	—
23	C₂²×#41×C₅₃	C₂²	1	6	yes	—
24	C₂³×D₄×C₅₃	C₂²	1	9	yes	—
25	C₂³×Dic₅₃	—	4	24	no	—
26	C₂³×Dic₅₃	C₂³	1	9	yes	—
27	C₂³×Dic₅₃	C₂³	1	9	yes	—
28	C₂³×Dic₅₃	C₂³	1	18	yes	—
29	C₂³×Dic₅₃	C₂³	1	48	yes	complemented
30	Q₈×Dic₅₃	C₂³	1	16	yes	complemented
31	Q₈×Dic₅₃	C₂³	1	48	yes	complemented
32	D₄×Dic₅₃	C₂³	1	144	yes	complemented
33	D₄×Dic₅₃	C₂³	1	288	yes	complemented
34	D₄×Dic₅₃	C₂³	1	144	yes	complemented
35	#13133176	C₂³	1	144	yes	complemented
36	C₂×C₄⋊Dic₅₃	C₂³	1	18	yes	—
37	C₂×C₄⋊Dic₅₃	C₂³	1	6	yes	—
38	C₂×C₄⋊Dic₅₃	C₂³	1	72	yes	complemented
39	C₂×C₂²⋊Dic₅₃	C₂³	1	36	yes	—
40	C₂×C₂²⋊Dic₅₃	C₂³	1	144	yes	complemented
41	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
42	C₂×C₄×Dic₅₃	C₂³	1	18	yes	—
43	C₂×C₄×Dic₅₃	C₂³	1	72	yes	complemented
44	C₂⁵×C₅₃	C₂³	1	6	yes	—
45	H(5,𝔽₂)×C₅₃	C₂×C₄	1	256	yes	complemented
46	C₂×#41×C₅₃	C₂×C₄	1	96	yes	complemented
47	C₂×#41×C₅₃	C₂³	1	48	yes	—
48	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
49	C₂²×D₄×C₅₃	C₂×C₄	1	144	yes	complemented
50	C₂²×D₄×C₅₃	C₂³	1	18	yes	—
51	C₂²×D₄×C₅₃	C₂³	1	72	yes	—
52	C₂³×C₄×C₅₃	C₂³	1	9	yes	—
53	C₂⁴×C₅₃	—	4	12	no	—
54	C₂⁴×C₅₃	—	4	12	no	—
55	C₂⁴×C₅₃	C₂²×C₄	1	48	yes	complemented
56	C₂⁴×C₅₃	C₂⁴	1	9	yes	—
57	C₂⁴×C₅₃	C₂⁴	1	24	yes	—
58	#41×C₅₃	C₂²×C₄	1	384	yes	complemented
59	C₂×Q₈×C₅₃	C₂²×C₄	1	16	yes	complemented
60	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
61	C₂×D₄×C₅₃	C₂²×C₄	1	144	yes	complemented
62	C₂×D₄×C₅₃	C₂²×C₄	1	576	yes	complemented
63	C₂×D₄×C₅₃	C₂⁴	1	72	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	72	yes	complemented
65	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
66	C₂²×C₄×C₅₃	C₂⁴	1	36	yes	—
67	C₂²×Dic₅₃	—	2	36	no	—
68	C₂²×Dic₅₃	—	4	96	no	—
69	C₂²×Dic₅₃	—	4	36	no	—
70	C₂²×Dic₅₃	—	4	18	no	—
71	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
72	C₂²×Dic₅₃	C₂⁴	1	9	yes	—
73	C₂²×Dic₅₃	C₂⁴	1	36	yes	—
74	C₂²×Dic₅₃	C₂⁴	1	36	yes	—
75	C₂²×Dic₅₃	C₂⁴	1	72	yes	—
76	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
77	C₂²⋊Dic₅₃	C₂⁴	1	144	yes	—
78	C₄⋊Dic₅₃	C₂⁴	1	72	yes	—
79	C₄⋊Dic₅₃	C₂⁴	1	24	yes	—
80	C₄×Dic₅₃	C₂⁴	1	24	yes	—
81	C₄×Dic₅₃	C₂⁴	1	72	yes	—
82	C₂³×C₅₃	—	2	18	no	—
83	C₂³×C₅₃	—	2	18	no	—
84	C₂³×C₅₃	—	4	24	no	—
85	C₂³×C₅₃	—	4	96	no	—
86	C₂³×C₅₃	—	4	24	no	—
87	C₂³×C₅₃	C₂³×C₄	1	36	yes	—
88	C₂³×C₅₃	C₂³×C₄	1	96	yes	complemented
89	C₂³×C₅₃	C₂⁵	1	18	yes	—
90	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
91	Q₈×C₅₃	C₂³×C₄	1	32	yes	complemented
92	D₄×C₅₃	C₂³×C₄	1	288	yes	complemented
93	C₂×C₄×C₅₃	C₂³×C₄	1	24	yes	—
94	C₂×C₄×C₅₃	C₂³×C₄	1	144	yes	complemented
95	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
96	C₂×Dic₅₃	—	2	18	no	—
97	C₂×Dic₅₃	—	2	12	no	—
98	C₂×Dic₅₃	—	2	72	no	—
99	C₂×Dic₅₃	—	4	72	no	—
100	C₂×Dic₅₃	—	4	36	no	—
101	C₂×Dic₅₃	H(5,𝔽₂)	1	4	yes	direct product factor
102	C₂×Dic₅₃	C₂⁵	1	18	yes	—
103	C₂×Dic₅₃	C₂⁵	1	12	yes	—
104	C₂×Dic₅₃	C₂⁵	1	2	yes	—
105	C₂×C₄×H(5,𝔽₂)	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	—	2	18	no	—
107	C₂²×C₅₃	—	2	18	no	—
108	C₂²×C₅₃	—	2	72	no	—
109	C₂²×C₅₃	—	4	96	no	—
110	C₂²×C₅₃	C₂⁴×C₄	1	36	yes	—
111	C₂²×C₅₃	C₂⁴×C₄	1	2	yes	—
112	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
113	C₂²×C₅₃	#585	1	2	yes	—
114	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
115	C₄×C₅₃	C₂⁴×C₄	1	24	yes	—
116	Dic₅₃	—	2	18	no	—
117	Dic₅₃	—	2	12	no	—
118	Dic₅₃	C₂×H(5,𝔽₂)	1	4	yes	direct product factor
119	C₂²×H(5,𝔽₂)	D₅₃	1	1	yes	characteristic
120	C₄×H(5,𝔽₂)	—	53	32	no	—
121	C₂×#520	—	53	6	no	—
122	C₂×#518	—	53	9	no	—
123	C₂×C₄×#41	—	53	6	no	—
124	C₂²×C₄×D₄	—	53	9	no	—
125	C₂×C₅₃	—	2	36	no	—
126	C₂×C₅₃	C₄×H(5,𝔽₂)	1	4	yes	complemented
127	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
128	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×C₅₃	C₂×#585	1	1	yes	characteristic
130	C₂×H(5,𝔽₂)	Dic₅₃	1	32	yes	direct product factor
131	C₂×H(5,𝔽₂)	C₂×D₅₃	1	16	yes	—
132	C₂²×#41	C₂×D₅₃	1	6	yes	—
133	C₂³×D₄	C₂×D₅₃	1	9	yes	—
134	C₂⁴×C₄	—	53	6	no	—
135	#520	—	53	96	no	—
136	#518	—	53	144	no	—
137	C₄×#41	—	53	96	no	—
138	C₂×C₄×Q₈	—	53	2	no	—
139	C₂×C₄×Q₈	—	53	6	no	—
140	C₂×C₄×D₄	—	53	18	no	—
141	C₂×C₄×D₄	—	53	36	no	—
142	C₂×C₄×D₄	—	53	144	no	—
143	C₂×C₄×D₄	—	53	18	no	—
144	C₂×#117	—	53	18	no	—
145	C₂²×C₄⋊C₄	—	53	9	no	—
146	C₂²×C₂²⋊C₄	—	53	18	no	—
147	C₂²×C₄²	—	53	9	no	—
148	C₅₃	C₂×C₄×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
149	C₂⁵	C₂²×D₅₃	1	6	yes	—
150	H(5,𝔽₂)	C₂×Dic₅₃	1	256	yes	direct product factor
151	C₂×#41	C₂×Dic₅₃	1	96	yes	complemented
152	C₂×#41	C₂²×D₅₃	1	48	yes	—
153	C₂²×Q₈	C₂²×D₅₃	1	2	yes	—
154	C₂²×D₄	C₂×Dic₅₃	1	144	yes	complemented
155	C₂²×D₄	C₂²×D₅₃	1	18	yes	—
156	C₂²×D₄	C₂²×D₅₃	1	72	yes	—
157	C₂³×C₄	—	53	9	no	—
158	C₂³×C₄	—	53	9	no	—
159	C₂³×C₄	—	53	18	no	—
160	C₂³×C₄	—	53	48	no	—
161	C₂³×C₄	—	212	24	no	—
162	C₂³×C₄	C₂²×D₅₃	1	9	yes	—
163	C₄×Q₈	—	53	16	no	—
164	C₄×Q₈	—	53	48	no	—
165	C₄×D₄	—	53	144	no	—
166	C₄×D₄	—	53	288	no	—
167	C₄×D₄	—	53	144	no	—
168	#117	—	53	144	no	—
169	C₂×C₄⋊C₄	—	53	18	no	—
170	C₂×C₄⋊C₄	—	53	72	no	—
171	C₂×C₄⋊C₄	—	53	6	no	—
172	C₂×C₂²⋊C₄	—	53	36	no	—
173	C₂×C₂²⋊C₄	—	53	144	no	—
174	C₂×C₄²	—	53	18	no	—
175	C₂×C₄²	—	53	6	no	—
176	C₂×C₄²	—	53	72	no	—
177	C₂⁴	—	4	12	no	—
178	C₂⁴	—	4	12	no	—
179	C₂⁴	C₂²×Dic₅₃	1	48	yes	complemented
180	C₂⁴	C₂³×D₅₃	1	9	yes	—
181	C₂⁴	C₂³×D₅₃	1	24	yes	—
182	#41	C₂²×Dic₅₃	1	384	yes	complemented
183	C₂×Q₈	C₂²×Dic₅₃	1	16	yes	complemented
184	C₂×Q₈	C₂³×D₅₃	1	8	yes	—
185	C₂×D₄	C₂²×Dic₅₃	1	144	yes	complemented
186	C₂×D₄	C₂²×Dic₅₃	1	576	yes	complemented
187	C₂×D₄	C₂³×D₅₃	1	72	yes	—
188	C₂²×C₄	—	53	1	no	—
189	C₂²×C₄	—	53	36	no	—
190	C₂²×C₄	—	53	6	no	—
191	C₂²×C₄	—	53	9	no	—
192	C₂²×C₄	—	53	36	no	—
193	C₂²×C₄	—	53	72	no	—
194	C₂²×C₄	—	106	36	no	—
195	C₂²×C₄	—	212	96	no	—
196	C₂²×C₄	—	212	18	no	—
197	C₂²×C₄	—	212	36	no	—
198	C₂²×C₄	C₂²×Dic₅₃	1	72	yes	complemented
199	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
200	C₂²×C₄	C₂³×D₅₃	1	36	yes	—
201	C₄⋊C₄	—	53	72	no	—
202	C₄⋊C₄	—	53	24	no	—
203	C₂²⋊C₄	—	53	144	no	—
204	C₄²	—	53	24	no	—
205	C₄²	—	53	72	no	—
206	C₂³	—	2	18	no	—
207	C₂³	—	2	18	no	—
208	C₂³	—	4	24	no	—
209	C₂³	—	4	96	no	—
210	C₂³	—	4	24	no	—
211	C₂³	C₂³×Dic₅₃	1	36	yes	—
212	C₂³	C₂³×Dic₅₃	1	96	yes	complemented
213	C₂³	C₂⁴×D₅₃	1	18	yes	—
214	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
215	Q₈	C₂³×Dic₅₃	1	32	yes	complemented
216	D₄	C₂³×Dic₅₃	1	288	yes	complemented
217	C₂×C₄	—	53	4	no	—
218	C₂×C₄	—	53	2	no	—
219	C₂×C₄	—	53	12	no	—
220	C₂×C₄	—	53	18	no	—
221	C₂×C₄	—	106	72	no	—
222	C₂×C₄	—	106	18	no	—
223	C₂×C₄	—	106	12	no	—
224	C₂×C₄	—	212	36	no	—
225	C₂×C₄	—	212	72	no	—
226	C₂×C₄	C₂³×Dic₅₃	1	24	yes	—
227	C₂×C₄	C₂³×Dic₅₃	1	144	yes	complemented
228	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
229	C₂²	—	2	72	no	—
230	C₂²	—	2	18	no	—
231	C₂²	—	2	18	no	—
232	C₂²	—	4	96	no	—
233	C₂²	C₂⁴×Dic₅₃	1	36	yes	—
234	C₂²	C₂⁴×Dic₅₃	1	2	yes	—
235	C₂²	H(5,𝔽₂)×D₅₃	1	2	yes	—
236	C₂²	#16811458	1	2	yes	—
237	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
238	C₄	—	53	4	no	—
239	C₄	—	106	18	no	—
240	C₄	—	106	12	no	—
241	C₄	C₂⁴×Dic₅₃	1	24	yes	—
242	C₂	—	2	36	no	—
243	C₂	H(5,𝔽₂)×Dic₅₃	1	4	yes	direct product factor
244	C₂	C₂⁵×Dic₅₃	1	1	yes	characteristic
245	C₂	C₂×H(5,𝔽₂)×D₅₃	1	1	yes	characteristic
246	C₂	C₂×#16811458	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₃
Fitting, prime radical	C₂²×H(5,𝔽₂)×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
cosocle	C₂²×C₅₃	C₂⁶
hypercenter	C₂²×H(5,𝔽₂)	D₅₃
hypocenter	C₅₃	C₂×C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. H(5,𝔽₂)×Dic₅₃	C₂	direct (N × H)
2. C₂×#16811106	C₂	semidirect
3. C₂×#16811098	C₂	semidirect
4. C₂×#41×Dic₅₃	C₂	semidirect
5. C₂²×D₄×Dic₅₃	C₂	semidirect
7. C₂⁴×Dic₅₃	C₂²	semidirect
8. #16811106	C₂²	semidirect
9. #16811098	C₂²	semidirect
10. #41×Dic₅₃	C₂²	semidirect
11. C₂×Q₈×Dic₅₃	C₂²	semidirect
12. C₂×Q₈×Dic₅₃	C₂²	semidirect
13. C₂×D₄×Dic₅₃	C₂²	semidirect
14. C₂×D₄×Dic₅₃	C₂²	semidirect
15. C₂×D₄×Dic₅₃	C₂²	semidirect
16. C₂×D₄×Dic₅₃	C₂²	semidirect
17. C₂×#13133176	C₂²	semidirect
18. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
19. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
20. C₂²×C₄×Dic₅₃	C₂²	semidirect
21. C₂×H(5,𝔽₂)×C₅₃	C₄	semidirect
29. C₂³×Dic₅₃	C₂³	semidirect
30. Q₈×Dic₅₃	C₂³	semidirect
31. Q₈×Dic₅₃	C₂³	semidirect
32. D₄×Dic₅₃	C₂³	semidirect
33. D₄×Dic₅₃	C₂³	semidirect
34. D₄×Dic₅₃	C₂³	semidirect
35. #13133176	C₂³	semidirect
38. C₂×C₄⋊Dic₅₃	C₂³	semidirect
40. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
43. C₂×C₄×Dic₅₃	C₂³	semidirect
45. H(5,𝔽₂)×C₅₃	C₂×C₄	semidirect
46. C₂×#41×C₅₃	C₂×C₄	semidirect
49. C₂²×D₄×C₅₃	C₂×C₄	semidirect
55. C₂⁴×C₅₃	C₂²×C₄	semidirect
58. #41×C₅₃	C₂²×C₄	semidirect
59. C₂×Q₈×C₅₃	C₂²×C₄	semidirect
61. C₂×D₄×C₅₃	C₂²×C₄	semidirect
62. C₂×D₄×C₅₃	C₂²×C₄	semidirect
64. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
88. C₂³×C₅₃	C₂³×C₄	semidirect
91. Q₈×C₅₃	C₂³×C₄	semidirect
92. D₄×C₅₃	C₂³×C₄	semidirect
94. C₂×C₄×C₅₃	C₂³×C₄	semidirect
101. C₂×Dic₅₃	H(5,𝔽₂)	direct (N × H)
118. Dic₅₃	C₂×H(5,𝔽₂)	direct (N × H)
126. C₂×C₅₃	C₄×H(5,𝔽₂)	semidirect
130. C₂×H(5,𝔽₂)	Dic₅₃	direct (N × H)
148. C₅₃	C₂×C₄×H(5,𝔽₂)	semidirect
150. H(5,𝔽₂)	C₂×Dic₅₃	direct (N × H)
151. C₂×#41	C₂×Dic₅₃	semidirect
154. C₂²×D₄	C₂×Dic₅₃	semidirect
179. C₂⁴	C₂²×Dic₅₃	semidirect
182. #41	C₂²×Dic₅₃	semidirect
183. C₂×Q₈	C₂²×Dic₅₃	semidirect
185. C₂×D₄	C₂²×Dic₅₃	semidirect
186. C₂×D₄	C₂²×Dic₅₃	semidirect
198. C₂²×C₄	C₂²×Dic₅₃	semidirect
212. C₂³	C₂³×Dic₅₃	semidirect
215. Q₈	C₂³×Dic₅₃	semidirect
216. D₄	C₂³×Dic₅₃	semidirect
227. C₂×C₄	C₂³×Dic₅₃	semidirect
243. C₂	H(5,𝔽₂)×Dic₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
H(5,𝔽₂)×Dic₅₃	2
C₂×#16811106	2
C₂×#16811098	2
C₂×#41×Dic₅₃	2
C₂²×D₄×Dic₅₃	2
C₂²×H(5,𝔽₂)×C₅₃	2
C₂×C₄×H(5,𝔽₂)	53

Subgroups — C2×H(5,𝔽2)×Dic53 #28290696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×H(5,𝔽₂)×Dic₅₃ #28290696