Subgroups — C₂²×#16811258 #28291002

Order 13568.

20038 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	C₂³×D₄×D₅₃	C₂	1	4	yes	maximal, complemented
2	C₂×#16811258	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133188	C₂	1	4	yes	maximal, complemented
4	C₂²×#13133185	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×D₂₁₂	C₂	1	2	yes	maximal, complemented
6	C₂²×#127×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₅₃	C₂²	1	4	yes	complemented
8	C₂³×D₂₁₂	C₂²	1	2	yes	—
9	C₂³×C₂²⋊D₅₃	C₂²	1	8	yes	complemented
10	C₂²×D₄×D₅₃	—	2	64	no	—
11	C₂²×D₄×D₅₃	C₂²	1	96	yes	complemented
12	C₂³×C₄×D₅₃	C₂²	1	4	yes	complemented
13	#16811258	C₂²	1	256	yes	direct product factor
14	C₂×#13133188	C₂²	1	96	yes	complemented
15	C₂×#13133185	C₂²	1	96	yes	complemented
16	C₂×C₄×D₂₁₂	C₂²	1	48	yes	complemented
17	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	complemented
18	C₂²×C₂²⋊Dic₅₃	C₂²	1	4	yes	complemented
19	C₂²×#11719921	C₂²	1	4	yes	complemented
20	C₂³×D₄×C₅₃	C₂²	1	4	yes	complemented
21	C₂³×D₄×C₅₃	C₂²	1	2	yes	complemented
22	C₂×#127×C₅₃	C₂²	1	24	yes	complemented
23	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
24	C₂⁴×D₅₃	—	2	64	no	—
25	C₂⁴×D₅₃	—	4	16	no	—
26	C₂⁴×D₅₃	C₂³	1	4	yes	—
27	C₂⁴×D₅₃	C₂³	1	48	yes	complemented
28	C₂²×D₂₁₂	D₄	1	32	yes	complemented
29	C₂²×D₂₁₂	C₂³	1	24	yes	—
30	C₂²×C₂²⋊D₅₃	—	2	64	no	—
31	C₂²×C₂²⋊D₅₃	C₂³	1	96	yes	complemented
32	C₂×D₄×D₅₃	—	2	768	no	—
33	C₂×D₄×D₅₃	C₂³	1	256	yes	complemented
34	C₂³×Dic₅₃	C₂³	1	4	yes	—
35	C₂²×C₄×D₅₃	—	2	32	no	—
36	C₂²×C₄×D₅₃	C₂³	1	48	yes	complemented
37	#13133188	C₂³	1	256	yes	complemented
38	#13133185	C₂³	1	256	yes	complemented
39	C₄×D₂₁₂	C₂³	1	128	yes	complemented
40	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	complemented
41	C₂×C₂²⋊Dic₅₃	C₂³	1	48	yes	complemented
42	C₂×#11719921	C₂³	1	48	yes	complemented
43	C₂⁵×C₅₃	C₂³	1	4	yes	—
44	C₂²×D₄×C₅₃	—	2	32	no	—
45	C₂²×D₄×C₅₃	—	4	8	no	—
46	C₂²×D₄×C₅₃	C₂³	1	48	yes	complemented
47	C₂²×D₄×C₅₃	C₂³	1	24	yes	complemented
48	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
50	#127×C₅₃	C₂³	1	64	yes	complemented
51	C₂×C₄²×C₅₃	C₂³	1	12	yes	complemented
52	C₂⁴×C₅₃	—	2	32	no	—
53	C₂⁴×C₅₃	—	2	16	no	—
54	C₂⁴×C₅₃	C₂⁴	1	24	yes	—
55	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₂³×D₅₃	—	2	16	no	—
57	C₂³×D₅₃	—	2	16	no	—
58	C₂³×D₅₃	—	2	384	no	—
59	C₂³×D₅₃	—	4	96	no	—
60	C₂³×D₅₃	—	4	256	no	—
61	C₂³×D₅₃	C₂×D₄	1	32	yes	complemented
62	C₂³×D₅₃	C₂⁴	1	24	yes	—
63	C₂³×D₅₃	C₂⁴	1	64	yes	complemented
64	C₂×D₄×C₅₃	—	2	192	no	—
65	C₂×D₄×C₅₃	—	4	48	no	—
66	C₂×D₄×C₅₃	C₂⁴	1	64	yes	complemented
67	C₂×D₄×C₅₃	C₂⁴	1	32	yes	complemented
68	C₂²×C₄×C₅₃	—	2	4	no	—
69	C₂²×C₄×C₅₃	C₂×D₄	1	16	yes	complemented
70	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
71	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
72	C₂×C₂²⋊D₅₃	—	2	384	no	—
73	C₂×C₂²⋊D₅₃	C₂⁴	1	128	yes	complemented
74	C₂²×Dic₅₃	—	2	16	no	—
75	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
76	D₄×D₅₃	—	2	1024	no	—
77	C₂×D₂₁₂	C₂×D₄	1	192	yes	complemented
78	C₂×D₂₁₂	C₂⁴	1	32	yes	—
79	C₂×C₄×D₅₃	—	2	192	no	—
80	C₂×C₄×D₅₃	C₂⁴	1	64	yes	complemented
81	C₄²×C₅₃	C₂⁴	1	16	yes	complemented
82	C₂²⋊Dic₅₃	C₂⁴	1	64	yes	complemented
83	#11719921	C₂⁴	1	64	yes	complemented
84	C₄⋊Dic₅₃	C₂⁴	1	32	yes	complemented
85	C₂³×C₅₃	—	2	96	no	—
86	C₂³×C₅₃	—	2	48	no	—
87	C₂³×C₅₃	—	4	64	no	—
88	C₂³×C₅₃	C₂²×D₄	1	8	yes	—
89	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
90	C₂³×C₅₃	C₂⁵	1	16	yes	—
91	C₂³×C₅₃	C₂⁵	1	3	yes	—
92	C₂²×D₅₃	—	2	128	no	—
93	C₂²×D₅₃	—	2	48	no	—
94	C₂²×D₅₃	—	2	48	no	—
95	C₂²×D₅₃	—	2	256	no	—
96	C₂²×D₅₃	—	4	64	no	—
97	C₂²×D₅₃	—	4	768	no	—
98	C₂²×D₅₃	C₂²×D₄	1	96	yes	complemented
99	C₂²×D₅₃	C₂⁵	1	16	yes	—
100	D₄×C₅₃	—	2	128	no	—
101	D₄×C₅₃	—	4	32	no	—
102	C₂×C₄×C₅₃	—	2	12	no	—
103	C₂×C₄×C₅₃	C₂²×D₄	1	48	yes	complemented
104	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
105	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
106	C₂²⋊D₅₃	—	2	256	no	—
107	C₂×Dic₅₃	—	2	48	no	—
108	C₂×Dic₅₃	C₂⁵	1	16	yes	—
109	D₂₁₂	C₂²×D₄	1	128	yes	complemented
110	C₄×D₅₃	—	2	128	no	—
111	C₂²×D₄²	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	32	no	—
113	C₂²×C₅₃	—	2	16	no	—
114	C₂²×C₅₃	—	4	96	no	—
115	C₂²×C₅₃	D₄²	1	16	yes	complemented
116	C₂²×C₅₃	C₂³×D₄	1	12	yes	—
117	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
118	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
119	C₂×D₅₃	—	2	16	no	—
120	C₂×D₅₃	—	2	16	no	—
121	C₂×D₅₃	—	2	192	no	—
122	C₂×D₅₃	—	4	256	no	—
123	C₂×D₅₃	C₂³×D₄	1	32	yes	complemented
124	C₄×C₅₃	—	2	4	no	—
125	C₄×C₅₃	C₂³×D₄	1	16	yes	complemented
126	Dic₅₃	—	2	16	no	—
127	C₂⁴×D₄	—	53	4	no	—
128	C₂×D₄²	—	53	48	no	—
129	C₂²×#127	D₅₃	1	1	yes	characteristic, complemented
130	C₂²×#121	—	53	4	no	—
131	C₂²×#120	—	53	4	no	—
132	C₂²×C₄×D₄	—	53	2	no	—
133	C₂×C₅₃	—	4	16	no	—
134	C₂×C₅₃	C₂×D₄²	1	12	yes	complemented
135	C₂×C₅₃	C₂⁴×D₄	1	2	yes	—
136	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
137	D₅₃	—	2	32	no	—
138	C₂⁶	—	53	4	no	—
139	C₂³×D₄	—	53	8	no	—
140	C₂³×D₄	—	53	96	no	—
141	C₂³×D₄	—	53	2	no	—
142	C₂³×D₄	—	106	64	no	—
143	C₂³×D₄	C₂×D₅₃	1	4	yes	complemented
144	C₂³×D₄	C₂×D₅₃	1	2	yes	complemented
145	C₂⁴×C₄	—	53	4	no	—
146	D₄²	—	53	256	no	—
147	C₂×#127	C₂×D₅₃	1	24	yes	complemented
148	C₂×#121	—	53	96	no	—
149	C₂×#120	—	53	96	no	—
150	C₂×C₄×D₄	—	53	48	no	—
151	C₂²×C₄⋊C₄	—	53	2	no	—
152	C₂²×C₂²⋊C₄	—	53	4	no	—
153	C₂²×C₂²⋊C₄	—	53	4	no	—
154	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
155	C₅₃	C₂²×D₄²	1	1	yes	characteristic, complemented, Sylow
156	C₂⁵	—	53	4	no	—
157	C₂⁵	—	53	48	no	—
158	C₂⁵	—	106	64	no	—
159	C₂⁵	—	212	16	no	—
160	C₂⁵	C₂²×D₅₃	1	4	yes	—
161	C₂²×D₄	—	2	32	no	—
162	C₂²×D₄	—	4	8	no	—
163	C₂²×D₄	—	53	96	no	—
164	C₂²×D₄	—	53	256	no	—
165	C₂²×D₄	—	53	32	no	—
166	C₂²×D₄	—	53	24	no	—
167	C₂²×D₄	—	106	768	no	—
168	C₂²×D₄	—	106	64	no	—
169	C₂²×D₄	C₂²×D₅₃	1	48	yes	complemented
170	C₂²×D₄	C₂²×D₅₃	1	24	yes	complemented
171	C₂³×C₄	—	53	4	no	—
172	C₂³×C₄	—	53	48	no	—
173	C₂³×C₄	—	106	32	no	—
174	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
175	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
176	#127	C₂²×D₅₃	1	64	yes	complemented
177	#121	—	53	256	no	—
178	#120	—	53	256	no	—
179	C₄×D₄	—	53	128	no	—
180	C₂×C₄⋊C₄	—	53	24	no	—
181	C₂×C₂²⋊C₄	—	53	48	no	—
182	C₂×C₂²⋊C₄	—	53	48	no	—
183	C₂×C₄²	C₂²×D₅₃	1	12	yes	complemented
184	C₂⁴	—	2	32	no	—
185	C₂⁴	—	2	16	no	—
186	C₂⁴	—	53	32	no	—
187	C₂⁴	—	53	24	no	—
188	C₂⁴	—	53	64	no	—
189	C₂⁴	—	106	384	no	—
190	C₂⁴	—	106	16	no	—
191	C₂⁴	—	106	16	no	—
192	C₂⁴	—	212	96	no	—
193	C₂⁴	—	212	256	no	—
194	C₂⁴	C₂³×D₅₃	1	24	yes	—
195	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
196	C₂×D₄	—	2	192	no	—
197	C₂×D₄	—	4	48	no	—
198	C₂×D₄	—	53	128	no	—
199	C₂×D₄	—	53	192	no	—
200	C₂×D₄	—	53	32	no	—
201	C₂×D₄	—	106	1024	no	—
202	C₂×D₄	—	106	384	no	—
203	C₂×D₄	C₂³×D₅₃	1	64	yes	complemented
204	C₂×D₄	C₂³×D₅₃	1	32	yes	complemented
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	53	24	no	—
207	C₂²×C₄	—	53	64	no	—
208	C₂²×C₄	—	106	16	no	—
209	C₂²×C₄	—	106	192	no	—
210	C₂²×C₄	D₄×D₅₃	1	16	yes	complemented
211	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
212	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
213	C₄⋊C₄	—	53	32	no	—
214	C₂²⋊C₄	—	53	64	no	—
215	C₂²⋊C₄	—	53	64	no	—
216	C₄²	C₂³×D₅₃	1	16	yes	complemented
217	C₂³	—	2	96	no	—
218	C₂³	—	2	48	no	—
219	C₂³	—	4	64	no	—
220	C₂³	—	53	96	no	—
221	C₂³	—	53	16	no	—
222	C₂³	—	106	256	no	—
223	C₂³	—	106	48	no	—
224	C₂³	—	106	48	no	—
225	C₂³	—	106	128	no	—
226	C₂³	—	212	64	no	—
227	C₂³	—	212	768	no	—
228	C₂³	C₂×D₄×D₅₃	1	8	yes	—
229	C₂³	#13133262	1	4	yes	—
230	C₂³	C₂⁴×D₅₃	1	16	yes	—
231	C₂³	C₂⁴×D₅₃	1	3	yes	—
232	D₄	—	2	128	no	—
233	D₄	—	4	32	no	—
234	D₄	—	53	128	no	—
235	D₄	—	106	256	no	—
236	C₂×C₄	—	2	12	no	—
237	C₂×C₄	—	53	16	no	—
238	C₂×C₄	—	106	48	no	—
239	C₂×C₄	—	106	128	no	—
240	C₂×C₄	C₂×D₄×D₅₃	1	48	yes	complemented
241	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
242	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
243	C₂²	—	2	32	no	—
244	C₂²	—	2	16	no	—
245	C₂²	—	4	96	no	—
246	C₂²	—	53	32	no	—
247	C₂²	—	106	192	no	—
248	C₂²	—	106	16	no	—
249	C₂²	—	106	16	no	—
250	C₂²	—	212	256	no	—
251	C₂²	#16811258	1	16	yes	direct product factor
252	C₂²	C₂²×D₄×D₅₃	1	12	yes	—
253	C₂²	C₂×#13133262	1	6	yes	—
254	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
255	C₄	—	2	4	no	—
256	C₄	—	106	16	no	—
257	C₄	C₂²×D₄×D₅₃	1	16	yes	complemented
258	C₂	—	4	16	no	—
259	C₂	—	106	32	no	—
260	C₂	C₂×#16811258	1	12	yes	direct product factor
261	C₂	C₂³×D₄×D₅₃	1	2	yes	—
262	C₂	C₂²×#13133262	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×#127×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂²×#127	D₅₃
hypocenter	C₅₃	C₂²×D₄²
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×D₅₃	C₂	semidirect
2. C₂×#16811258	C₂	direct (N × H)
3. C₂²×#13133188	C₂	semidirect
4. C₂²×#13133185	C₂	semidirect
5. C₂²×C₄×D₂₁₂	C₂	semidirect
6. C₂²×#127×C₅₃	C₂	semidirect
7. C₂⁵×D₅₃	C₂²	semidirect
9. C₂³×C₂²⋊D₅₃	C₂²	semidirect
11. C₂²×D₄×D₅₃	C₂²	semidirect
12. C₂³×C₄×D₅₃	C₂²	semidirect
13. #16811258	C₂²	direct (N × H)
14. C₂×#13133188	C₂²	semidirect
15. C₂×#13133185	C₂²	semidirect
16. C₂×C₄×D₂₁₂	C₂²	semidirect
17. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
18. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
19. C₂²×#11719921	C₂²	semidirect
20. C₂³×D₄×C₅₃	C₂²	semidirect
21. C₂³×D₄×C₅₃	C₂²	semidirect
22. C₂×#127×C₅₃	C₂²	semidirect
23. C₂²×C₄²×C₅₃	C₂²	semidirect
27. C₂⁴×D₅₃	C₂³	semidirect
28. C₂²×D₂₁₂	D₄	semidirect
31. C₂²×C₂²⋊D₅₃	C₂³	semidirect
33. C₂×D₄×D₅₃	C₂³	semidirect
36. C₂²×C₄×D₅₃	C₂³	semidirect
37. #13133188	C₂³	semidirect
38. #13133185	C₂³	semidirect
39. C₄×D₂₁₂	C₂³	semidirect
40. C₂×C₄⋊Dic₅₃	C₂³	semidirect
41. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
42. C₂×#11719921	C₂³	semidirect
46. C₂²×D₄×C₅₃	C₂³	semidirect
47. C₂²×D₄×C₅₃	C₂³	semidirect
50. #127×C₅₃	C₂³	semidirect
51. C₂×C₄²×C₅₃	C₂³	semidirect
61. C₂³×D₅₃	C₂×D₄	semidirect
63. C₂³×D₅₃	C₂⁴	semidirect
66. C₂×D₄×C₅₃	C₂⁴	semidirect
67. C₂×D₄×C₅₃	C₂⁴	semidirect
69. C₂²×C₄×C₅₃	C₂×D₄	semidirect
73. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
77. C₂×D₂₁₂	C₂×D₄	semidirect
80. C₂×C₄×D₅₃	C₂⁴	semidirect
81. C₄²×C₅₃	C₂⁴	semidirect
82. C₂²⋊Dic₅₃	C₂⁴	semidirect
83. #11719921	C₂⁴	semidirect
84. C₄⋊Dic₅₃	C₂⁴	semidirect
98. C₂²×D₅₃	C₂²×D₄	semidirect
103. C₂×C₄×C₅₃	C₂²×D₄	semidirect
109. D₂₁₂	C₂²×D₄	semidirect
115. C₂²×C₅₃	D₄²	semidirect
123. C₂×D₅₃	C₂³×D₄	semidirect
125. C₄×C₅₃	C₂³×D₄	semidirect
129. C₂²×#127	D₅₃	semidirect
134. C₂×C₅₃	C₂×D₄²	semidirect
143. C₂³×D₄	C₂×D₅₃	semidirect
144. C₂³×D₄	C₂×D₅₃	semidirect
147. C₂×#127	C₂×D₅₃	semidirect
154. C₂²×C₄²	C₂×D₅₃	semidirect
155. C₅₃	C₂²×D₄²	semidirect
169. C₂²×D₄	C₂²×D₅₃	semidirect
170. C₂²×D₄	C₂²×D₅₃	semidirect
176. #127	C₂²×D₅₃	semidirect
183. C₂×C₄²	C₂²×D₅₃	semidirect
203. C₂×D₄	C₂³×D₅₃	semidirect
204. C₂×D₄	C₂³×D₅₃	semidirect
210. C₂²×C₄	D₄×D₅₃	semidirect
216. C₄²	C₂³×D₅₃	semidirect
240. C₂×C₄	C₂×D₄×D₅₃	semidirect
251. C₂²	#16811258	direct (N × H)
257. C₄	C₂²×D₄×D₅₃	semidirect
260. C₂	C₂×#16811258	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₄×D₅₃	2
C₂×#16811258	2
C₂²×#13133188	2
C₂²×#13133185	2
C₂²×C₄×D₂₁₂	2
C₂²×#127×C₅₃	2
C₂²×D₄²	53

Subgroups — C22×#16811258 #28291002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811258 #28291002