Subgroups — C₂²×#16811360 #28290958

Order 13568.

9606 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811360	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13133211	C₂	1	2	yes	maximal, complemented
3	C₂²×#13133198	C₂	1	3	yes	maximal, complemented
4	C₂²×#13133189	C₂	1	6	yes	maximal, complemented
5	C₂²×#13133185	C₂	1	3	yes	maximal, complemented
6	C₂²×#127×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
8	C₂³×C₂²⋊D₅₃	C₂²	1	6	yes	—
9	C₂⁴×Dic₅₃	C₂²	1	3	yes	complemented
10	#16811360	C₂²	1	256	yes	direct product factor
11	C₂×#13133211	C₂²	1	48	yes	complemented
12	C₂×#13133198	C₂²	1	72	yes	complemented
13	C₂×#13133189	C₂²	1	144	yes	complemented
14	C₂×#13133185	C₂²	1	72	yes	complemented
15	C₂²×#11719919	C₂²	1	6	yes	complemented
16	C₂²×C₂²⋊Dic₅₃	C₂²	1	3	yes	—
17	C₂²×C₂²⋊Dic₅₃	C₂²	1	3	yes	complemented
18	C₂²×#11719921	C₂²	1	6	yes	—
19	C₂³×D₄×C₅₃	C₂²	1	3	yes	—
20	C₂³×D₄×C₅₃	C₂²	1	3	yes	complemented
21	C₂×#127×C₅₃	C₂²	1	24	yes	complemented
22	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
23	C₂⁴×D₅₃	—	4	12	no	—
24	C₂⁴×D₅₃	C₂³	1	2	yes	—
25	C₂⁴×D₅₃	C₂³	1	12	yes	—
26	C₂²×C₂²⋊D₅₃	—	4	24	no	—
27	C₂²×C₂²⋊D₅₃	C₂³	1	72	yes	—
28	C₂³×Dic₅₃	—	4	12	no	—
29	C₂³×Dic₅₃	C₂³	1	6	yes	—
30	C₂³×Dic₅₃	C₂³	1	36	yes	complemented
31	#13133211	C₂³	1	128	yes	complemented
32	#13133198	C₂³	1	192	yes	complemented
33	#13133189	C₂³	1	384	yes	complemented
34	#13133185	C₂³	1	192	yes	complemented
35	C₂×#11719919	C₂³	1	72	yes	complemented
36	C₂×C₂²⋊Dic₅₃	C₂³	1	36	yes	—
37	C₂×C₂²⋊Dic₅₃	C₂³	1	36	yes	complemented
38	C₂×#11719921	C₂³	1	72	yes	—
39	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
40	C₂⁵×C₅₃	C₂³	1	3	yes	—
41	C₂²×D₄×C₅₃	—	4	12	no	—
42	C₂²×D₄×C₅₃	—	4	12	no	—
43	C₂²×D₄×C₅₃	C₂³	1	36	yes	—
44	C₂²×D₄×C₅₃	C₂³	1	36	yes	complemented
45	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
46	#127×C₅₃	C₂³	1	64	yes	complemented
47	C₂×C₄²×C₅₃	C₂³	1	12	yes	complemented
48	C₂⁴×C₅₃	—	2	12	no	—
49	C₂⁴×C₅₃	—	2	24	no	—
50	C₂⁴×C₅₃	—	2	12	no	—
51	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
52	C₂⁴×C₅₃	C₂⁴	1	18	yes	—
53	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
54	C₂³×D₅₃	—	2	24	no	—
55	C₂³×D₅₃	—	4	72	no	—
56	C₂³×D₅₃	—	4	64	no	—
57	C₂³×D₅₃	C₂⁴	1	12	yes	—
58	C₂³×D₅₃	C₂⁴	1	16	yes	—
59	C₂×D₄×C₅₃	—	4	72	no	—
60	C₂×D₄×C₅₃	—	4	72	no	—
61	C₂×D₄×C₅₃	C₂⁴	1	48	yes	—
62	C₂×D₄×C₅₃	C₂⁴	1	48	yes	complemented
63	C₂²×C₄×C₅₃	—	2	12	no	—
64	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
65	C₂×C₂²⋊D₅₃	—	4	144	no	—
66	C₂×C₂²⋊D₅₃	C₂⁴	1	96	yes	—
67	C₂²×Dic₅₃	—	2	24	no	—
68	C₂²×Dic₅₃	—	4	72	no	—
69	C₂²×Dic₅₃	C₂⁴	1	36	yes	—
70	C₂²×Dic₅₃	C₂⁴	1	48	yes	complemented
71	C₄²×C₅₃	C₂⁴	1	16	yes	complemented
72	C₂²⋊Dic₅₃	C₂⁴	1	48	yes	—
73	C₂²⋊Dic₅₃	C₂⁴	1	48	yes	complemented
74	#11719921	C₂⁴	1	96	yes	—
75	#11719919	C₂⁴	1	96	yes	complemented
76	C₂³×C₅₃	—	2	36	no	—
77	C₂³×C₅₃	—	2	72	no	—
78	C₂³×C₅₃	—	2	36	no	—
79	C₂³×C₅₃	—	4	16	no	—
80	C₂³×C₅₃	—	4	48	no	—
81	C₂³×C₅₃	H(5,𝔽₂)	1	12	yes	—
82	C₂³×C₅₃	C₂⁵	1	12	yes	—
83	C₂³×C₅₃	C₂⁵	1	4	yes	—
84	C₂³×C₅₃	C₂⁵	1	3	yes	—
85	C₂²×D₅₃	—	2	72	no	—
86	C₂²×D₅₃	—	4	48	no	—
87	C₂²×D₅₃	—	4	192	no	—
88	C₂²×D₅₃	—	4	32	no	—
89	C₂²×D₅₃	C₂⁵	1	8	yes	—
90	D₄×C₅₃	—	4	48	no	—
91	D₄×C₅₃	—	4	48	no	—
92	C₂×C₄×C₅₃	—	2	36	no	—
93	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
94	C₂²⋊D₅₃	—	4	96	no	—
95	C₂×Dic₅₃	—	2	72	no	—
96	C₂×Dic₅₃	—	4	48	no	—
97	C₂×Dic₅₃	C₂⁵	1	24	yes	—
98	C₂²×#560	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	—	2	12	no	—
100	C₂²×C₅₃	—	2	24	no	—
101	C₂²×C₅₃	—	2	12	no	—
102	C₂²×C₅₃	—	4	24	no	—
103	C₂²×C₅₃	—	4	72	no	—
104	C₂²×C₅₃	#560	1	16	yes	complemented
105	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	18	yes	—
106	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
107	C₂×D₅₃	—	2	24	no	—
108	C₂×D₅₃	—	4	64	no	—
109	C₂×D₅₃	—	4	48	no	—
110	C₄×C₅₃	—	2	12	no	—
111	Dic₅₃	—	2	24	no	—
112	C₂×#560	—	53	48	no	—
113	C₂²×#127	D₅₃	1	1	yes	characteristic, complemented
114	C₂²×#126	—	53	2	no	—
115	C₂²×#123	—	53	3	no	—
116	C₂²×#121	—	53	6	no	—
117	C₂²×#120	—	53	3	no	—
118	C₂×C₅₃	—	4	4	no	—
119	C₂×C₅₃	—	4	12	no	—
120	C₂×C₅₃	C₂×#560	1	12	yes	complemented
121	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	3	yes	—
122	D₅₃	—	4	8	no	—
123	C₂⁶	—	53	1	no	—
124	C₂³×D₄	—	53	6	no	—
125	C₂³×D₄	C₂×D₅₃	1	3	yes	—
126	C₂³×D₄	C₂×D₅₃	1	3	yes	complemented
127	C₂⁴×C₄	—	53	3	no	—
128	#560	—	53	256	no	—
129	C₂×#127	C₂×D₅₃	1	24	yes	complemented
130	C₂×#126	—	53	48	no	—
131	C₂×#123	—	53	72	no	—
132	C₂×#121	—	53	144	no	—
133	C₂×#120	—	53	72	no	—
134	C₂²×C₄⋊C₄	—	53	6	no	—
135	C₂²×C₂²⋊C₄	—	53	3	no	—
136	C₂²×C₂²⋊C₄	—	53	3	no	—
137	C₂²×C₂²⋊C₄	—	53	6	no	—
138	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
139	C₅₃	C₂²×#560	1	1	yes	characteristic, complemented, Sylow
140	C₂⁵	—	53	2	no	—
141	C₂⁵	—	53	12	no	—
142	C₂⁵	—	212	12	no	—
143	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
144	C₂⁵	C₂²×D₅₃	1	3	yes	—
145	C₂²×D₄	—	4	12	no	—
146	C₂²×D₄	—	4	12	no	—
147	C₂²×D₄	—	53	72	no	—
148	C₂²×D₄	—	212	24	no	—
149	C₂²×D₄	C₂²×D₅₃	1	36	yes	—
150	C₂²×D₄	C₂²×D₅₃	1	36	yes	complemented
151	C₂³×C₄	—	53	6	no	—
152	C₂³×C₄	—	53	36	no	—
153	C₂³×C₄	—	212	12	no	—
154	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
155	#127	C₂²×D₅₃	1	64	yes	complemented
156	#126	—	53	128	no	—
157	#123	—	53	192	no	—
158	#121	—	53	384	no	—
159	#120	—	53	192	no	—
160	C₂×C₄⋊C₄	—	53	72	no	—
161	C₂×C₂²⋊C₄	—	53	36	no	—
162	C₂×C₂²⋊C₄	—	53	36	no	—
163	C₂×C₂²⋊C₄	—	53	72	no	—
164	C₂×C₄²	C₂²×D₅₃	1	12	yes	complemented
165	C₂⁴	—	2	12	no	—
166	C₂⁴	—	2	24	no	—
167	C₂⁴	—	2	12	no	—
168	C₂⁴	—	53	12	no	—
169	C₂⁴	—	53	16	no	—
170	C₂⁴	—	106	24	no	—
171	C₂⁴	—	212	72	no	—
172	C₂⁴	—	212	64	no	—
173	C₂⁴	C₂³×D₅₃	1	6	yes	—
174	C₂⁴	C₂³×D₅₃	1	18	yes	—
175	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
176	C₂×D₄	—	4	72	no	—
177	C₂×D₄	—	4	72	no	—
178	C₂×D₄	—	53	96	no	—
179	C₂×D₄	—	212	144	no	—
180	C₂×D₄	C₂³×D₅₃	1	48	yes	—
181	C₂×D₄	C₂³×D₅₃	1	48	yes	complemented
182	C₂²×C₄	—	2	12	no	—
183	C₂²×C₄	—	53	36	no	—
184	C₂²×C₄	—	53	48	no	—
185	C₂²×C₄	—	106	24	no	—
186	C₂²×C₄	—	212	72	no	—
187	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
188	C₄⋊C₄	—	53	96	no	—
189	C₂²⋊C₄	—	53	48	no	—
190	C₂²⋊C₄	—	53	48	no	—
191	C₂²⋊C₄	—	53	96	no	—
192	C₄²	C₂³×D₅₃	1	16	yes	complemented
193	C₂³	—	2	36	no	—
194	C₂³	—	2	72	no	—
195	C₂³	—	2	36	no	—
196	C₂³	—	4	48	no	—
197	C₂³	—	4	16	no	—
198	C₂³	—	53	8	no	—
199	C₂³	—	106	72	no	—
200	C₂³	—	212	48	no	—
201	C₂³	—	212	192	no	—
202	C₂³	—	212	32	no	—
203	C₂³	#13133262	1	12	yes	—
204	C₂³	C₂⁴×D₅₃	1	4	yes	—
205	C₂³	C₂⁴×D₅₃	1	12	yes	—
206	C₂³	C₂⁴×D₅₃	1	3	yes	—
207	D₄	—	4	48	no	—
208	D₄	—	4	48	no	—
209	D₄	—	212	96	no	—
210	C₂×C₄	—	2	36	no	—
211	C₂×C₄	—	53	24	no	—
212	C₂×C₄	—	106	72	no	—
213	C₂×C₄	—	212	48	no	—
214	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
215	C₂²	—	2	12	no	—
216	C₂²	—	2	24	no	—
217	C₂²	—	2	12	no	—
218	C₂²	—	4	24	no	—
219	C₂²	—	4	72	no	—
220	C₂²	—	106	24	no	—
221	C₂²	—	212	64	no	—
222	C₂²	—	212	48	no	—
223	C₂²	#16811360	1	16	yes	direct product factor
224	C₂²	C₂×#13133262	1	18	yes	—
225	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
226	C₄	—	2	12	no	—
227	C₄	—	106	24	no	—
228	C₂	—	4	4	no	—
229	C₂	—	4	12	no	—
230	C₂	—	212	8	no	—
231	C₂	C₂×#16811360	1	12	yes	direct product factor
232	C₂	C₂²×#13133262	1	3	yes	—

	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×#560
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. C₂×#16811360	C₂	direct (N × H)
2. C₂²×#13133211	C₂	semidirect
3. C₂²×#13133198	C₂	semidirect
4. C₂²×#13133189	C₂	semidirect
5. C₂²×#13133185	C₂	semidirect
6. C₂²×#127×C₅₃	C₂	semidirect
9. C₂⁴×Dic₅₃	C₂²	semidirect
10. #16811360	C₂²	direct (N × H)
11. C₂×#13133211	C₂²	semidirect
12. C₂×#13133198	C₂²	semidirect
13. C₂×#13133189	C₂²	semidirect
14. C₂×#13133185	C₂²	semidirect
15. C₂²×#11719919	C₂²	semidirect
17. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
20. C₂³×D₄×C₅₃	C₂²	semidirect
21. C₂×#127×C₅₃	C₂²	semidirect
22. C₂²×C₄²×C₅₃	C₂²	semidirect
30. C₂³×Dic₅₃	C₂³	semidirect
31. #13133211	C₂³	semidirect
32. #13133198	C₂³	semidirect
33. #13133189	C₂³	semidirect
34. #13133185	C₂³	semidirect
35. C₂×#11719919	C₂³	semidirect
37. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
44. C₂²×D₄×C₅₃	C₂³	semidirect
46. #127×C₅₃	C₂³	semidirect
47. C₂×C₄²×C₅₃	C₂³	semidirect
62. C₂×D₄×C₅₃	C₂⁴	semidirect
70. C₂²×Dic₅₃	C₂⁴	semidirect
71. C₄²×C₅₃	C₂⁴	semidirect
73. C₂²⋊Dic₅₃	C₂⁴	semidirect
75. #11719919	C₂⁴	semidirect
104. C₂²×C₅₃	#560	semidirect
113. C₂²×#127	D₅₃	semidirect
120. C₂×C₅₃	C₂×#560	semidirect
126. C₂³×D₄	C₂×D₅₃	semidirect
129. C₂×#127	C₂×D₅₃	semidirect
138. C₂²×C₄²	C₂×D₅₃	semidirect
139. C₅₃	C₂²×#560	semidirect
150. C₂²×D₄	C₂²×D₅₃	semidirect
155. #127	C₂²×D₅₃	semidirect
164. C₂×C₄²	C₂²×D₅₃	semidirect
181. C₂×D₄	C₂³×D₅₃	semidirect
192. C₄²	C₂³×D₅₃	semidirect
223. C₂²	#16811360	direct (N × H)
231. C₂	C₂×#16811360	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×#16811360	2
C₂²×#13133211	2
C₂²×#13133198	2
C₂²×#13133189	2
C₂²×#13133185	2
C₂²×#127×C₅₃	2
C₂²×#560	53

Subgroups — C22×#16811360 #28290958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811360 #28290958