Subgroups — C₂²×#127×D₅₃ #28290777

Order 13568.

24030 conjugacy classes of subgroups.

The subgroup lattice has 216 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	6	yes	maximal, complemented
2	C₂²×#13133214	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13133213	C₂	1	6	yes	maximal, complemented
4	C₂×#127×D₅₃	C₂	1	48	yes	maximal, direct product factor
5	C₂²×C₄²×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#127×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₅₃	C₂²	1	4	yes	—
8	C₂³×D₂₁₂	C₂²	1	6	yes	complemented
9	C₂³×C₂²⋊D₅₃	C₂²	1	12	yes	complemented
10	C₂²×D₄×D₅₃	—	2	96	no	—
11	C₂²×D₄×D₅₃	C₂²	1	144	yes	complemented
12	C₂³×C₄×D₅₃	C₂²	1	3	yes	—
13	C₂×#13133214	C₂²	1	24	yes	complemented
14	C₂×#13133213	C₂²	1	144	yes	complemented
15	#127×D₅₃	C₂²	1	256	yes	direct product factor
16	C₂²×C₄×Dic₅₃	C₂²	1	3	yes	complemented
17	C₂×C₄²×D₅₃	C₂²	1	24	yes	complemented
18	C₂³×D₄×C₅₃	C₂²	1	6	yes	complemented
19	C₂×#127×C₅₃	C₂²	1	24	yes	complemented
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
21	C₂⁴×D₅₃	—	2	96	no	—
22	C₂⁴×D₅₃	C₂³	1	48	yes	—
23	C₂⁴×D₅₃	C₂³	1	4	yes	—
24	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
25	C₂²×D₂₁₂	—	2	48	no	—
26	C₂²×D₂₁₂	C₂³	1	72	yes	complemented
27	C₂²×C₂²⋊D₅₃	—	2	96	no	—
28	C₂²×C₂²⋊D₅₃	C₂³	1	144	yes	complemented
29	C₂×D₄×D₅₃	—	2	1152	no	—
30	C₂×D₄×D₅₃	C₂³	1	384	yes	complemented
31	C₂³×Dic₅₃	C₂³	1	3	yes	—
32	C₂²×C₄×D₅₃	D₄	1	48	yes	complemented
33	C₂²×C₄×D₅₃	C₂³	1	36	yes	—
34	#13133214	C₂³	1	64	yes	complemented
35	#13133213	C₂³	1	384	yes	complemented
36	C₂×C₄×Dic₅₃	C₂³	1	36	yes	complemented
37	C₄²×D₅₃	C₂³	1	64	yes	complemented
38	C₂⁵×C₅₃	C₂³	1	4	yes	—
39	C₂²×D₄×C₅₃	—	2	48	no	—
40	C₂²×D₄×C₅₃	C₂³	1	72	yes	complemented
41	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
42	#127×C₅₃	C₂³	1	64	yes	complemented
43	C₂×C₄²×C₅₃	C₂³	1	12	yes	complemented
44	C₂⁴×C₅₃	—	2	48	no	—
45	C₂⁴×C₅₃	C₂⁴	1	24	yes	—
46	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
47	C₂³×D₅₃	—	2	48	no	—
48	C₂³×D₅₃	—	2	576	no	—
49	C₂³×D₅₃	—	4	256	no	—
50	C₂³×D₅₃	C₂×D₄	1	24	yes	—
51	C₂³×D₅₃	C₂⁴	1	64	yes	—
52	C₂³×D₅₃	C₂⁴	1	24	yes	—
53	C₂³×D₅₃	C₂⁴	1	6	yes	—
54	C₂×D₄×C₅₃	—	2	288	no	—
55	C₂×D₄×C₅₃	C₂⁴	1	96	yes	complemented
56	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	complemented
57	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
58	C₂×C₂²⋊D₅₃	—	2	576	no	—
59	C₂×C₂²⋊D₅₃	C₂⁴	1	192	yes	complemented
60	C₂²×Dic₅₃	C₂×D₄	1	24	yes	complemented
61	C₂²×Dic₅₃	C₂⁴	1	18	yes	—
62	D₄×D₅₃	—	2	1536	no	—
63	C₂×D₂₁₂	—	2	288	no	—
64	C₂×D₂₁₂	C₂⁴	1	96	yes	complemented
65	C₂×C₄×D₅₃	C₂×D₄	1	288	yes	complemented
66	C₂×C₄×D₅₃	C₂⁴	1	48	yes	—
67	C₄²×C₅₃	C₂⁴	1	16	yes	complemented
68	C₄×Dic₅₃	C₂⁴	1	48	yes	complemented
69	C₂³×C₅₃	—	2	144	no	—
70	C₂³×C₅₃	—	4	64	no	—
71	C₂³×C₅₃	C₂²×D₄	1	12	yes	—
72	C₂³×C₅₃	C₂⁵	1	16	yes	—
73	C₂³×C₅₃	C₂⁵	1	3	yes	—
74	C₂²×D₅₃	—	2	144	no	—
75	C₂²×D₅₃	—	2	384	no	—
76	C₂²×D₅₃	—	4	64	no	—
77	C₂²×D₅₃	—	4	768	no	—
78	C₂²×D₅₃	#127	1	64	yes	direct product factor
79	C₂²×D₅₃	C₂²×D₄	1	72	yes	—
80	C₂²×D₅₃	C₂⁵	1	16	yes	—
81	C₂²×D₅₃	C₂⁵	1	4	yes	—
82	D₄×C₅₃	—	2	192	no	—
83	C₂×C₄×C₅₃	C₂²×D₄	1	72	yes	complemented
84	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
85	C₂²⋊D₅₃	—	2	384	no	—
86	C₂×Dic₅₃	C₂²×D₄	1	72	yes	complemented
87	C₂×Dic₅₃	C₂⁵	1	12	yes	—
88	D₂₁₂	—	2	192	no	—
89	C₄×D₅₃	C₂²×D₄	1	192	yes	complemented
90	C₂³×#127	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	—	2	48	no	—
92	C₂²×C₅₃	—	4	96	no	—
93	C₂²×C₅₃	C₂×#127	1	16	yes	complemented
94	C₂²×C₅₃	C₂³×D₄	1	18	yes	—
95	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
96	C₂×D₅₃	—	2	48	no	—
97	C₂×D₅₃	—	4	96	no	—
98	C₂×D₅₃	—	4	256	no	—
99	C₂×D₅₃	C₂×#127	1	96	yes	direct product factor
100	C₂×D₅₃	C₂³×D₄	1	24	yes	—
101	C₄×C₅₃	C₂³×D₄	1	24	yes	complemented
102	Dic₅₃	C₂³×D₄	1	24	yes	complemented
103	C₂⁴×D₄	—	53	6	no	—
104	C₂²×#127	—	53	1	no	—
105	C₂²×#127	—	53	6	no	—
106	C₂²×#127	—	53	48	no	—
107	C₂²×#127	D₅₃	1	1	yes	characteristic, direct product factor
108	C₂³×C₄²	—	53	1	no	—
109	C₂×C₅₃	—	4	16	no	—
110	C₂×C₅₃	C₂²×#127	1	12	yes	complemented
111	C₂×C₅₃	C₂⁴×D₄	1	3	yes	—
112	D₅₃	—	4	16	no	—
113	D₅₃	C₂²×#127	1	16	yes	direct product factor
114	C₂⁶	—	53	4	no	—
115	C₂³×D₄	—	53	12	no	—
116	C₂³×D₄	—	53	144	no	—
117	C₂³×D₄	—	53	6	no	—
118	C₂³×D₄	—	106	96	no	—
119	C₂³×D₄	C₂×D₅₃	1	6	yes	complemented
120	C₂⁴×C₄	—	53	3	no	—
121	C₂×#127	—	53	144	no	—
122	C₂×#127	—	53	256	no	—
123	C₂×#127	—	53	24	no	—
124	C₂×#127	C₂×D₅₃	1	24	yes	direct product factor
125	C₂²×C₄²	—	53	3	no	—
126	C₂²×C₄²	—	53	24	no	—
127	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
128	C₅₃	C₂³×#127	1	1	yes	characteristic, complemented, Sylow
129	C₂⁵	—	53	1	no	—
130	C₂⁵	—	53	48	no	—
131	C₂⁵	—	53	4	no	—
132	C₂⁵	—	106	96	no	—
133	C₂⁵	C₂²×D₅₃	1	4	yes	—
134	C₂²×D₄	—	2	48	no	—
135	C₂²×D₄	—	53	144	no	—
136	C₂²×D₄	—	53	384	no	—
137	C₂²×D₄	—	53	72	no	—
138	C₂²×D₄	—	106	1152	no	—
139	C₂²×D₄	—	106	96	no	—
140	C₂²×D₄	—	106	48	no	—
141	C₂²×D₄	C₂²×D₅₃	1	72	yes	complemented
142	C₂³×C₄	—	53	3	no	—
143	C₂³×C₄	—	53	48	no	—
144	C₂³×C₄	—	53	36	no	—
145	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
146	#127	—	53	384	no	—
147	#127	—	53	64	no	—
148	#127	C₂²×D₅₃	1	64	yes	direct product factor
149	C₂×C₄²	—	53	36	no	—
150	C₂×C₄²	—	53	64	no	—
151	C₂×C₄²	C₂²×D₅₃	1	12	yes	complemented
152	C₂⁴	—	2	48	no	—
153	C₂⁴	—	53	24	no	—
154	C₂⁴	—	53	6	no	—
155	C₂⁴	—	53	64	no	—
156	C₂⁴	—	53	24	no	—
157	C₂⁴	—	106	576	no	—
158	C₂⁴	—	106	48	no	—
159	C₂⁴	—	212	256	no	—
160	C₂⁴	C₂³×D₅₃	1	24	yes	—
161	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
162	C₂×D₄	—	2	288	no	—
163	C₂×D₄	—	53	192	no	—
164	C₂×D₄	—	53	96	no	—
165	C₂×D₄	—	106	1536	no	—
166	C₂×D₄	—	106	576	no	—
167	C₂×D₄	—	106	288	no	—
168	C₂×D₄	C₂³×D₅₃	1	96	yes	complemented
169	C₂²×C₄	—	53	24	no	—
170	C₂²×C₄	—	53	18	no	—
171	C₂²×C₄	—	53	288	no	—
172	C₂²×C₄	—	53	48	no	—
173	C₂²×C₄	D₄×D₅₃	1	24	yes	complemented
174	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
175	C₄²	—	53	48	no	—
176	C₄²	C₂³×D₅₃	1	16	yes	complemented
177	C₂³	—	2	144	no	—
178	C₂³	—	4	64	no	—
179	C₂³	—	53	64	no	—
180	C₂³	—	53	72	no	—
181	C₂³	—	53	4	no	—
182	C₂³	—	53	16	no	—
183	C₂³	—	106	384	no	—
184	C₂³	—	106	144	no	—
185	C₂³	—	212	768	no	—
186	C₂³	—	212	64	no	—
187	C₂³	C₂×D₄×D₅₃	1	12	yes	—
188	C₂³	C₂⁴×D₅₃	1	16	yes	—
189	C₂³	C₂⁴×D₅₃	1	3	yes	—
190	D₄	—	2	192	no	—
191	D₄	—	106	384	no	—
192	D₄	—	106	192	no	—
193	C₂×C₄	—	53	72	no	—
194	C₂×C₄	—	53	12	no	—
195	C₂×C₄	—	53	192	no	—
196	C₂×C₄	C₂×D₄×D₅₃	1	72	yes	complemented
197	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
198	C₂²	—	2	48	no	—
199	C₂²	—	4	96	no	—
200	C₂²	—	53	24	no	—
201	C₂²	—	53	96	no	—
202	C₂²	—	106	48	no	—
203	C₂²	—	212	256	no	—
204	C₂²	—	212	96	no	—
205	C₂²	#127×D₅₃	1	16	yes	direct product factor
206	C₂²	C₂²×D₄×D₅₃	1	18	yes	—
207	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
208	C₄	—	53	24	no	—
209	C₄	C₂²×D₄×D₅₃	1	24	yes	complemented
210	C₂	—	4	16	no	—
211	C₂	—	53	16	no	—
212	C₂	—	212	16	no	—
213	C₂	C₂×#127×D₅₃	1	12	yes	direct product factor
214	C₂	C₂³×D₄×D₅₃	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	C₂³×#127
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. C₂²×#13133214	C₂	semidirect
3. C₂²×#13133213	C₂	semidirect
4. C₂×#127×D₅₃	C₂	direct (N × H)
5. C₂²×C₄²×D₅₃	C₂	semidirect
6. C₂²×#127×C₅₃	C₂	semidirect
8. C₂³×D₂₁₂	C₂²	semidirect
9. C₂³×C₂²⋊D₅₃	C₂²	semidirect
11. C₂²×D₄×D₅₃	C₂²	semidirect
13. C₂×#13133214	C₂²	semidirect
14. C₂×#13133213	C₂²	semidirect
15. #127×D₅₃	C₂²	direct (N × H)
16. C₂²×C₄×Dic₅₃	C₂²	semidirect
17. C₂×C₄²×D₅₃	C₂²	semidirect
18. C₂³×D₄×C₅₃	C₂²	semidirect
19. C₂×#127×C₅₃	C₂²	semidirect
20. C₂²×C₄²×C₅₃	C₂²	semidirect
26. C₂²×D₂₁₂	C₂³	semidirect
28. C₂²×C₂²⋊D₅₃	C₂³	semidirect
30. C₂×D₄×D₅₃	C₂³	semidirect
32. C₂²×C₄×D₅₃	D₄	semidirect
34. #13133214	C₂³	semidirect
35. #13133213	C₂³	semidirect
36. C₂×C₄×Dic₅₃	C₂³	semidirect
37. C₄²×D₅₃	C₂³	semidirect
40. C₂²×D₄×C₅₃	C₂³	semidirect
42. #127×C₅₃	C₂³	semidirect
43. C₂×C₄²×C₅₃	C₂³	semidirect
55. C₂×D₄×C₅₃	C₂⁴	semidirect
56. C₂²×C₄×C₅₃	C₂×D₄	semidirect
59. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
60. C₂²×Dic₅₃	C₂×D₄	semidirect
64. C₂×D₂₁₂	C₂⁴	semidirect
65. C₂×C₄×D₅₃	C₂×D₄	semidirect
67. C₄²×C₅₃	C₂⁴	semidirect
68. C₄×Dic₅₃	C₂⁴	semidirect
78. C₂²×D₅₃	#127	direct (N × H)
83. C₂×C₄×C₅₃	C₂²×D₄	semidirect
86. C₂×Dic₅₃	C₂²×D₄	semidirect
89. C₄×D₅₃	C₂²×D₄	semidirect
93. C₂²×C₅₃	C₂×#127	semidirect
99. C₂×D₅₃	C₂×#127	direct (N × H)
101. C₄×C₅₃	C₂³×D₄	semidirect
102. Dic₅₃	C₂³×D₄	semidirect
107. C₂²×#127	D₅₃	direct (N × H)
110. C₂×C₅₃	C₂²×#127	semidirect
113. D₅₃	C₂²×#127	direct (N × H)
119. C₂³×D₄	C₂×D₅₃	semidirect
124. C₂×#127	C₂×D₅₃	direct (N × H)
127. C₂²×C₄²	C₂×D₅₃	semidirect
128. C₅₃	C₂³×#127	semidirect
141. C₂²×D₄	C₂²×D₅₃	semidirect
148. #127	C₂²×D₅₃	direct (N × H)
151. C₂×C₄²	C₂²×D₅₃	semidirect
168. C₂×D₄	C₂³×D₅₃	semidirect
173. C₂²×C₄	D₄×D₅₃	semidirect
176. C₄²	C₂³×D₅₃	semidirect
196. C₂×C₄	C₂×D₄×D₅₃	semidirect
205. C₂²	#127×D₅₃	direct (N × H)
209. C₄	C₂²×D₄×D₅₃	semidirect
213. C₂	C₂×#127×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
C₂³×D₄×D₅₃	2
C₂²×#13133214	2
C₂²×#13133213	2
C₂×#127×D₅₃	2
C₂²×C₄²×D₅₃	2
C₂²×#127×C₅₃	2
C₂³×#127	53

Subgroups — C22×#127×D53 #28290777

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#127×D₅₃ #28290777