Subgroups — C₂³×#13133263 #28297979

Order 13568.

65726 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133263	C₂	1	112	yes	maximal, direct product factor
2	C₂³×#11719949	C₂	1	2	yes	maximal, complemented
3	C₂³×#11719945	C₂	1	3	yes	maximal, complemented
4	C₂⁴×D₂₁₂	C₂	1	3	yes	maximal, complemented
5	C₂³×D₄×D₅₃	C₂	1	6	yes	maximal, complemented
6	C₂³×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₅₃	C₂²	1	6	yes	complemented
8	C₂×#13133263	C₂²	1	1792	yes	direct product factor
9	C₂²×#11719949	C₂²	1	112	yes	complemented
10	C₂²×#11719945	C₂²	1	168	yes	complemented
11	C₂³×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
12	C₂³×D₂₁₂	C₂²	1	6	yes	complemented
13	C₂³×D₂₁₂	C₂²	1	3	yes	complemented
14	C₂³×D₂₁₂	C₂²	1	168	yes	complemented
15	C₂³×C₂²⋊D₅₃	C₂²	1	6	yes	complemented
16	C₂²×D₄×D₅₃	C₂²	1	336	yes	complemented
17	C₂³×C₄×D₅₃	C₂²	1	6	yes	complemented
18	C₂²×#41×C₅₃	C₂²	1	56	yes	complemented
19	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
20	C₂³×D₄×C₅₃	C₂²	1	3	yes	complemented
21	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	complemented
22	C₂⁴×D₅₃	—	4	48	no	—
23	C₂⁴×D₅₃	C₂³	1	6	yes	—
24	C₂⁴×D₅₃	C₂³	1	168	yes	complemented
25	#13133263	C₂³	1	4096	yes	direct product factor
26	C₂×#11719949	C₂³	1	896	yes	complemented
27	C₂×#11719945	C₂³	1	1344	yes	complemented
28	C₂²×Dic₁₀₆	C₂³	1	28	yes	complemented
29	C₂²×D₂₁₂	C₂³	1	168	yes	complemented
30	C₂²×D₂₁₂	C₂³	1	84	yes	complemented
31	C₂²×D₂₁₂	C₂³	1	1344	yes	complemented
32	C₂²×C₂²⋊D₅₃	C₂³	1	168	yes	complemented
33	C₂×D₄×D₅₃	C₂³	1	2688	yes	complemented
34	C₂³×Dic₅₃	C₂³	1	2	yes	—
35	C₂²×C₄×D₅₃	C₂³	1	168	yes	complemented
36	C₂⁵×C₅₃	C₂³	1	3	yes	—
37	C₂×#41×C₅₃	C₂³	1	448	yes	complemented
38	C₂²×Q₈×C₅₃	C₂³	1	28	yes	complemented
39	C₂²×D₄×C₅₃	C₂³	1	84	yes	complemented
40	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
42	C₂³×C₄×C₅₃	C₂³	1	84	yes	complemented
43	C₂⁴×C₅₃	—	2	24	no	—
44	C₂⁴×C₅₃	C₂⁴	1	42	yes	—
45	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₂³×D₅₃	—	2	48	no	—
47	C₂³×D₅₃	—	4	672	no	—
48	C₂³×D₅₃	C₂⁴	1	84	yes	—
49	C₂³×D₅₃	C₂⁴	1	672	yes	complemented
50	#41×C₅₃	C₂⁴	1	512	yes	complemented
51	C₂×Q₈×C₅₃	C₂⁴	1	112	yes	complemented
52	C₂×D₄×C₅₃	C₂⁴	1	336	yes	complemented
53	C₂²×C₄×C₅₃	C₂⁴	1	14	yes	—
54	C₂²×C₄×C₅₃	C₂⁴	1	42	yes	—
55	C₂²×C₄×C₅₃	C₂⁴	1	336	yes	complemented
56	C₂×C₂²⋊D₅₃	C₂⁴	1	672	yes	complemented
57	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
58	#11719949	C₂⁴	1	1024	yes	complemented
59	D₄×D₅₃	C₂⁴	1	3072	yes	complemented
60	#11719945	C₂⁴	1	1536	yes	complemented
61	C₂×D₂₁₂	C₂⁴	1	672	yes	complemented
62	C₂×D₂₁₂	C₂⁴	1	336	yes	complemented
63	C₂×D₂₁₂	C₂⁴	1	1536	yes	complemented
64	C₂×C₄×D₅₃	C₂⁴	1	672	yes	complemented
65	C₂×Dic₁₀₆	C₂⁴	1	112	yes	complemented
66	C₂³×C₅₃	—	2	168	no	—
67	C₂³×C₅₃	H(5,𝔽₂)	1	8	yes	complemented
68	C₂³×C₅₃	C₂⁵	1	84	yes	—
69	C₂³×C₅₃	C₂⁵	1	7	yes	—
70	C₂²×D₅₃	—	2	336	no	—
71	C₂²×D₅₃	—	4	1344	no	—
72	C₂²×D₅₃	C₂⁵	1	168	yes	—
73	C₂²×D₅₃	C₂⁵	1	384	yes	complemented
74	Q₈×C₅₃	C₂⁵	1	64	yes	complemented
75	D₄×C₅₃	C₂⁵	1	192	yes	complemented
76	C₂×C₄×C₅₃	C₂⁵	1	28	yes	—
77	C₂×C₄×C₅₃	C₂⁵	1	84	yes	—
78	C₂×C₄×C₅₃	C₂⁵	1	192	yes	complemented
79	C₂²⋊D₅₃	C₂⁵	1	384	yes	complemented
80	C₂×Dic₅₃	C₂⁵	1	56	yes	—
81	D₂₁₂	C₂⁵	1	192	yes	complemented
82	D₂₁₂	C₂⁵	1	384	yes	complemented
83	C₄×D₅₃	C₂⁵	1	384	yes	complemented
84	Dic₁₀₆	C₂⁵	1	64	yes	complemented
85	C₂³×H(5,𝔽₂)	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	—	2	168	no	—
87	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	28	yes	complemented
88	C₂²×C₅₃	C₂⁶	1	24	yes	—
89	C₂²×C₅₃	C₂⁶	1	7	yes	—
90	C₂×D₅₃	—	2	336	no	—
91	C₂×D₅₃	—	4	384	no	—
92	C₂×D₅₃	C₂⁶	1	48	yes	—
93	C₄×C₅₃	C₂⁶	1	8	yes	—
94	C₄×C₅₃	C₂⁶	1	24	yes	—
95	Dic₅₃	C₂⁶	1	16	yes	—
96	C₂²×H(5,𝔽₂)	—	53	112	no	—
97	C₂³×#41	—	53	2	no	—
98	C₂³×#41	—	53	3	no	—
99	C₂³×#41	D₅₃	1	1	yes	characteristic, complemented
100	C₂⁴×D₄	—	53	6	no	—
101	C₂⁴×D₄	—	53	3	no	—
102	C₂×C₅₃	—	2	24	no	—
103	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	14	yes	complemented
104	C₂×C₅₃	C₂⁷	1	1	yes	characteristic
105	D₅₃	—	2	48	no	—
106	C₂⁶	—	53	6	no	—
107	C₂×H(5,𝔽₂)	—	53	1792	no	—
108	C₂²×#41	—	53	112	no	—
109	C₂²×#41	—	53	168	no	—
110	C₂²×#41	C₂×D₅₃	1	56	yes	complemented
111	C₂³×Q₈	—	53	1	no	—
112	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
113	C₂³×D₄	—	53	6	no	—
114	C₂³×D₄	—	53	336	no	—
115	C₂³×D₄	—	53	6	no	—
116	C₂³×D₄	—	53	3	no	—
117	C₂³×D₄	—	53	168	no	—
118	C₂³×D₄	C₂×D₅₃	1	3	yes	complemented
119	C₂⁴×C₄	—	53	6	no	—
120	C₂⁴×C₄	C₂×D₅₃	1	3	yes	complemented
121	C₅₃	C₂³×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
122	C₂⁵	—	53	6	no	—
123	C₂⁵	—	53	168	no	—
124	C₂⁵	—	212	48	no	—
125	C₂⁵	C₂²×D₅₃	1	3	yes	—
126	H(5,𝔽₂)	—	53	4096	no	—
127	C₂×#41	—	53	896	no	—
128	C₂×#41	—	53	1344	no	—
129	C₂×#41	C₂²×D₅₃	1	448	yes	complemented
130	C₂²×Q₈	—	53	28	no	—
131	C₂²×Q₈	C₂²×D₅₃	1	28	yes	complemented
132	C₂²×D₄	—	53	168	no	—
133	C₂²×D₄	—	53	2688	no	—
134	C₂²×D₄	—	53	168	no	—
135	C₂²×D₄	—	53	84	no	—
136	C₂²×D₄	—	53	1344	no	—
137	C₂²×D₄	C₂²×D₅₃	1	84	yes	complemented
138	C₂³×C₄	—	53	2	no	—
139	C₂³×C₄	—	53	168	no	—
140	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
141	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
142	C₂³×C₄	C₂²×D₅₃	1	84	yes	complemented
143	C₂⁴	—	2	24	no	—
144	C₂⁴	—	53	84	no	—
145	C₂⁴	—	53	672	no	—
146	C₂⁴	—	106	48	no	—
147	C₂⁴	—	212	672	no	—
148	C₂⁴	C₂³×D₅₃	1	42	yes	—
149	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
150	#41	—	53	1024	no	—
151	#41	—	53	1536	no	—
152	#41	C₂³×D₅₃	1	512	yes	complemented
153	C₂×Q₈	—	53	112	no	—
154	C₂×Q₈	C₂³×D₅₃	1	112	yes	complemented
155	C₂×D₄	—	53	672	no	—
156	C₂×D₄	—	53	3072	no	—
157	C₂×D₄	—	53	672	no	—
158	C₂×D₄	—	53	336	no	—
159	C₂×D₄	—	53	1536	no	—
160	C₂×D₄	C₂³×D₅₃	1	336	yes	complemented
161	C₂²×C₄	—	53	28	no	—
162	C₂²×C₄	—	53	672	no	—
163	C₂²×C₄	C₂³×D₅₃	1	42	yes	—
164	C₂²×C₄	C₂³×D₅₃	1	14	yes	—
165	C₂²×C₄	C₂³×D₅₃	1	336	yes	complemented
166	C₂³	—	2	168	no	—
167	C₂³	—	53	168	no	—
168	C₂³	—	53	384	no	—
169	C₂³	—	106	336	no	—
170	C₂³	—	212	1344	no	—
171	C₂³	#13133263	1	8	yes	direct product factor
172	C₂³	C₂⁴×D₅₃	1	84	yes	—
173	C₂³	C₂⁴×D₅₃	1	7	yes	—
174	Q₈	—	53	64	no	—
175	Q₈	C₂⁴×D₅₃	1	64	yes	complemented
176	D₄	—	53	384	no	—
177	D₄	—	53	384	no	—
178	D₄	—	53	192	no	—
179	D₄	C₂⁴×D₅₃	1	192	yes	complemented
180	C₂×C₄	—	53	56	no	—
181	C₂×C₄	—	53	384	no	—
182	C₂×C₄	C₂⁴×D₅₃	1	84	yes	—
183	C₂×C₄	C₂⁴×D₅₃	1	28	yes	—
184	C₂×C₄	C₂⁴×D₅₃	1	192	yes	complemented
185	C₂²	—	2	168	no	—
186	C₂²	—	53	48	no	—
187	C₂²	—	106	336	no	—
188	C₂²	—	212	384	no	—
189	C₂²	C₂×#13133263	1	28	yes	direct product factor
190	C₂²	C₂⁵×D₅₃	1	24	yes	—
191	C₂²	C₂⁵×D₅₃	1	7	yes	—
192	C₄	—	53	16	no	—
193	C₄	C₂⁵×D₅₃	1	24	yes	—
194	C₄	C₂⁵×D₅₃	1	8	yes	—
195	C₂	—	2	24	no	—
196	C₂	—	106	48	no	—
197	C₂	C₂²×#13133263	1	14	yes	direct product factor
198	C₂	C₂⁶×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂×C₅₃	C₂⁷
Frattini	C₂	C₂⁶×D₅₃
Fitting	C₂³×#41×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂³×#41	D₅₃
hypocenter	C₅₃	C₂³×H(5,𝔽₂)
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133263	C₂	direct (N × H)
2. C₂³×#11719949	C₂	semidirect
3. C₂³×#11719945	C₂	semidirect
4. C₂⁴×D₂₁₂	C₂	semidirect
5. C₂³×D₄×D₅₃	C₂	semidirect
6. C₂³×#41×C₅₃	C₂	semidirect
7. C₂⁵×D₅₃	C₂²	semidirect
8. C₂×#13133263	C₂²	direct (N × H)
9. C₂²×#11719949	C₂²	semidirect
10. C₂²×#11719945	C₂²	semidirect
11. C₂³×Dic₁₀₆	C₂²	semidirect
12. C₂³×D₂₁₂	C₂²	semidirect
13. C₂³×D₂₁₂	C₂²	semidirect
14. C₂³×D₂₁₂	C₂²	semidirect
15. C₂³×C₂²⋊D₅₃	C₂²	semidirect
16. C₂²×D₄×D₅₃	C₂²	semidirect
17. C₂³×C₄×D₅₃	C₂²	semidirect
18. C₂²×#41×C₅₃	C₂²	semidirect
19. C₂³×Q₈×C₅₃	C₂²	semidirect
20. C₂³×D₄×C₅₃	C₂²	semidirect
21. C₂⁴×C₄×C₅₃	C₂²	semidirect
24. C₂⁴×D₅₃	C₂³	semidirect
25. #13133263	C₂³	direct (N × H)
26. C₂×#11719949	C₂³	semidirect
27. C₂×#11719945	C₂³	semidirect
28. C₂²×Dic₁₀₆	C₂³	semidirect
29. C₂²×D₂₁₂	C₂³	semidirect
30. C₂²×D₂₁₂	C₂³	semidirect
31. C₂²×D₂₁₂	C₂³	semidirect
32. C₂²×C₂²⋊D₅₃	C₂³	semidirect
33. C₂×D₄×D₅₃	C₂³	semidirect
35. C₂²×C₄×D₅₃	C₂³	semidirect
37. C₂×#41×C₅₃	C₂³	semidirect
38. C₂²×Q₈×C₅₃	C₂³	semidirect
39. C₂²×D₄×C₅₃	C₂³	semidirect
42. C₂³×C₄×C₅₃	C₂³	semidirect
49. C₂³×D₅₃	C₂⁴	semidirect
50. #41×C₅₃	C₂⁴	semidirect
51. C₂×Q₈×C₅₃	C₂⁴	semidirect
52. C₂×D₄×C₅₃	C₂⁴	semidirect
55. C₂²×C₄×C₅₃	C₂⁴	semidirect
56. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
58. #11719949	C₂⁴	semidirect
59. D₄×D₅₃	C₂⁴	semidirect
60. #11719945	C₂⁴	semidirect
61. C₂×D₂₁₂	C₂⁴	semidirect
62. C₂×D₂₁₂	C₂⁴	semidirect
63. C₂×D₂₁₂	C₂⁴	semidirect
64. C₂×C₄×D₅₃	C₂⁴	semidirect
65. C₂×Dic₁₀₆	C₂⁴	semidirect
67. C₂³×C₅₃	H(5,𝔽₂)	semidirect
73. C₂²×D₅₃	C₂⁵	semidirect
74. Q₈×C₅₃	C₂⁵	semidirect
75. D₄×C₅₃	C₂⁵	semidirect
78. C₂×C₄×C₅₃	C₂⁵	semidirect
79. C₂²⋊D₅₃	C₂⁵	semidirect
81. D₂₁₂	C₂⁵	semidirect
82. D₂₁₂	C₂⁵	semidirect
83. C₄×D₅₃	C₂⁵	semidirect
84. Dic₁₀₆	C₂⁵	semidirect
87. C₂²×C₅₃	C₂×H(5,𝔽₂)	semidirect
99. C₂³×#41	D₅₃	semidirect
103. C₂×C₅₃	C₂²×H(5,𝔽₂)	semidirect
110. C₂²×#41	C₂×D₅₃	semidirect
112. C₂³×Q₈	C₂×D₅₃	semidirect
118. C₂³×D₄	C₂×D₅₃	semidirect
120. C₂⁴×C₄	C₂×D₅₃	semidirect
121. C₅₃	C₂³×H(5,𝔽₂)	semidirect
129. C₂×#41	C₂²×D₅₃	semidirect
131. C₂²×Q₈	C₂²×D₅₃	semidirect
137. C₂²×D₄	C₂²×D₅₃	semidirect
142. C₂³×C₄	C₂²×D₅₃	semidirect
152. #41	C₂³×D₅₃	semidirect
154. C₂×Q₈	C₂³×D₅₃	semidirect
160. C₂×D₄	C₂³×D₅₃	semidirect
165. C₂²×C₄	C₂³×D₅₃	semidirect
171. C₂³	#13133263	direct (N × H)
175. Q₈	C₂⁴×D₅₃	semidirect
179. D₄	C₂⁴×D₅₃	semidirect
184. C₂×C₄	C₂⁴×D₅₃	semidirect
189. C₂²	C₂×#13133263	direct (N × H)
197. C₂	C₂²×#13133263	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133263	2
C₂³×#11719949	2
C₂³×#11719945	2
C₂⁴×D₂₁₂	2
C₂³×D₄×D₅₃	2
C₂³×#41×C₅₃	2
C₂³×H(5,𝔽₂)	53

Subgroups — C23×#13133263 #28297979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133263 #28297979