Subgroups — C₂²×#16810736 #28248138

Order 13568.

3654 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₂₁₂	C₂	1	2	yes	maximal, complemented
3	C₂×#16810736	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13133149	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13133139	C₂	1	2	yes	maximal, complemented
6	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₂₁₂	C₂²	1	2	yes	—
8	C₂³×C₄×D₅₃	—	2	2	no	—
9	C₂²×D₄₂₄	C₄	1	16	yes	complemented
10	C₂²×D₄₂₄	C₂²	1	12	yes	—
11	C₂×C₄×D₂₁₂	—	2	12	no	—
12	C₂×C₄×D₂₁₂	C₂²	1	24	yes	complemented
13	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
14	C₂²×#11719921	—	2	2	no	—
15	#16810736	C₂²	1	64	yes	direct product factor
16	C₂×#13133149	C₂²	1	12	yes	complemented
17	C₂×#13133139	C₂²	1	24	yes	complemented
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#97×C₅₃	C₂²	1	12	yes	complemented
22	C₂⁴×D₅₃	—	2	2	no	—
23	C₂²×D₂₁₂	—	2	6	no	—
24	C₂²×D₂₁₂	—	2	8	no	—
25	C₂²×D₂₁₂	C₂×C₄	1	16	yes	complemented
26	C₂²×D₂₁₂	C₂³	1	12	yes	—
27	C₂³×Dic₅₃	—	2	2	no	—
28	C₂²×C₄×D₅₃	—	2	24	no	—
29	C₂²×C₄×D₅₃	—	4	16	no	—
30	C₂×D₄₂₄	C₂×C₄	1	96	yes	complemented
31	C₂×D₄₂₄	C₂³	1	16	yes	—
32	C₄×D₂₁₂	—	2	48	no	—
33	C₄×D₂₁₂	C₂³	1	32	yes	complemented
34	C₂×C₄⋊Dic₅₃	—	2	6	no	—
35	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
36	C₂×#11719921	—	2	24	no	—
37	#13133149	C₂³	1	16	yes	complemented
38	#13133139	C₂³	1	32	yes	complemented
39	C₂³×C₄×C₅₃	D₄	1	2	yes	—
40	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	—	2	4	no	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	complemented
43	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
45	C₂×C₄²×C₅₃	D₄	1	6	yes	—
46	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
47	#97×C₅₃	C₂³	1	16	yes	complemented
48	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂³×D₅₃	—	2	16	no	—
50	C₂³×D₅₃	—	2	12	no	—
51	C₂³×D₅₃	—	4	8	no	—
52	C₂³×D₅₃	—	4	8	no	—
53	C₂²×C₄×C₅₃	—	2	8	no	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₅₃	#41	1	4	yes	—
58	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
59	C₂²×Dic₅₃	—	2	12	no	—
60	C₂²×Dic₅₃	—	4	8	no	—
61	C₂×D₂₁₂	—	2	48	no	—
62	C₂×D₂₁₂	—	2	12	no	—
63	C₂×D₂₁₂	—	2	24	no	—
64	C₂×D₂₁₂	C₂²×C₄	1	48	yes	complemented
65	C₂×D₂₁₂	C₂⁴	1	8	yes	—
66	C₂×C₄×D₅₃	—	2	32	no	—
67	C₂×C₄×D₅₃	—	4	96	no	—
68	C₂×C₈×C₅₃	—	2	12	no	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	24	yes	complemented
70	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
71	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
72	C₄²×C₅₃	C₂×D₄	1	12	yes	—
73	C₄²×C₅₃	C₂⁴	1	4	yes	—
74	#11719921	—	2	32	no	—
75	C₄⋊Dic₅₃	—	2	12	no	—
76	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
77	D₄₂₄	C₂²×C₄	1	64	yes	complemented
78	C₂³×C₅₃	C₄×D₄	1	4	yes	—
79	C₂³×C₅₃	AGL(1,ℤ/8)	1	8	yes	—
80	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
81	C₂²×D₅₃	—	2	48	no	—
82	C₂²×D₅₃	—	2	8	no	—
83	C₂²×D₅₃	—	4	24	no	—
84	C₂²×D₅₃	—	4	24	no	—
85	C₂²×D₅₃	—	4	64	no	—
86	C₂×C₄×C₅₃	—	2	24	no	—
87	C₂×C₄×C₅₃	C₄×D₄	1	12	yes	—
88	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
89	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
90	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
91	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
92	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
93	C₂×Dic₅₃	—	2	8	no	—
94	C₂×Dic₅₃	—	4	24	no	—
95	D₂₁₂	—	2	48	no	—
96	D₂₁₂	—	2	8	no	—
97	D₂₁₂	C₂³×C₄	1	16	yes	complemented
98	C₄×D₅₃	—	4	64	no	—
99	C₈×C₅₃	—	2	4	no	—
100	C₈×C₅₃	C₂³×C₄	1	8	yes	complemented
101	C₂²×#442	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	#442	1	16	yes	complemented
103	C₂²×C₅₃	C₂×C₄×D₄	1	6	yes	—
104	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	12	yes	—
105	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
106	C₂×D₅₃	—	2	16	no	—
107	C₂×D₅₃	—	4	96	no	—
108	C₂×D₅₃	—	4	8	no	—
109	C₂×D₅₃	—	4	8	no	—
110	C₄×C₅₃	—	2	8	no	—
111	C₄×C₅₃	C₂×C₄×D₄	1	6	yes	—
112	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
113	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
114	Dic₅₃	—	4	8	no	—
115	C₂³×D₈	—	53	1	no	—
116	C₂²×C₄×D₄	—	53	2	no	—
117	C₂×#442	—	53	24	no	—
118	C₂²×#106	—	53	1	no	—
119	C₂²×#102	—	53	2	no	—
120	C₂²×#97	D₅₃	1	1	yes	characteristic, complemented
121	C₂×C₅₃	C₂×#442	1	12	yes	complemented
122	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
123	C₂×C₅₃	C₂²×AGL(1,ℤ/8)	1	2	yes	—
124	D₅₃	—	4	16	no	—
125	C₂³×D₄	—	53	2	no	—
126	C₂⁴×C₄	—	106	2	no	—
127	C₂²×D₈	—	53	16	no	—
128	C₂²×D₈	—	53	12	no	—
129	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
130	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
131	C₂×C₄×D₄	—	53	24	no	—
132	C₂×C₄×D₄	—	106	12	no	—
133	C₂²×C₄⋊C₄	—	53	2	no	—
134	C₂²×C₂²⋊C₄	—	106	2	no	—
135	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
136	#442	—	53	64	no	—
137	C₂×#106	—	53	12	no	—
138	C₂×#102	—	53	24	no	—
139	C₂×#97	C₂×D₅₃	1	12	yes	complemented
140	C₅₃	C₂²×#442	1	1	yes	characteristic, complemented, Sylow
141	C₂⁵	—	106	2	no	—
142	C₂²×D₄	—	53	16	no	—
143	C₂²×D₄	—	53	12	no	—
144	C₂²×D₄	—	106	6	no	—
145	C₂²×D₄	—	106	8	no	—
146	C₂³×C₄	—	106	2	no	—
147	C₂³×C₄	—	106	24	no	—
148	C₂³×C₄	—	212	16	no	—
149	C₂³×C₄	D₂₁₂	1	2	yes	—
150	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
151	C₂×D₈	—	53	96	no	—
152	C₂×D₈	—	53	16	no	—
153	C₂²×C₈	—	2	4	no	—
154	C₂²×C₈	C₄×D₅₃	1	8	yes	complemented
155	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
156	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
157	C₄×D₄	—	53	32	no	—
158	C₄×D₄	—	106	48	no	—
159	C₂×C₄⋊C₄	—	53	12	no	—
160	C₂×C₄⋊C₄	—	106	6	no	—
161	C₂×C₂²⋊C₄	—	106	24	no	—
162	C₂×C₄²	D₂₁₂	1	6	yes	—
163	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
164	#106	—	53	16	no	—
165	#102	—	53	32	no	—
166	#97	C₂²×D₅₃	1	16	yes	complemented
167	C₂⁴	—	106	16	no	—
168	C₂⁴	—	106	12	no	—
169	C₂⁴	—	212	8	no	—
170	C₂⁴	—	212	8	no	—
171	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
172	C₂×D₄	—	53	48	no	—
173	C₂×D₄	—	53	8	no	—
174	C₂×D₄	—	106	48	no	—
175	C₂×D₄	—	106	12	no	—
176	C₂×D₄	—	106	24	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	106	12	no	—
179	C₂²×C₄	—	106	32	no	—
180	C₂²×C₄	—	212	96	no	—
181	C₂²×C₄	—	212	8	no	—
182	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
183	C₂²×C₄	C₂×D₂₁₂	1	12	yes	—
184	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
185	C₂²×C₄	#11719945	1	4	yes	—
186	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
187	D₈	—	53	64	no	—
188	C₂×C₈	—	2	12	no	—
189	C₂×C₈	C₂×C₄×D₅₃	1	24	yes	complemented
190	C₂×C₈	C₂³×D₅₃	1	4	yes	—
191	C₂×C₈	C₂³×D₅₃	1	4	yes	—
192	C₄⋊C₄	—	53	8	no	—
193	C₄⋊C₄	—	106	12	no	—
194	C₂²⋊C₄	—	106	32	no	—
195	C₄²	C₂×D₂₁₂	1	12	yes	—
196	C₄²	C₂³×D₅₃	1	4	yes	—
197	C₂³	—	106	48	no	—
198	C₂³	—	106	8	no	—
199	C₂³	—	212	24	no	—
200	C₂³	—	212	24	no	—
201	C₂³	—	212	64	no	—
202	C₂³	C₄×D₂₁₂	1	4	yes	—
203	C₂³	#13133245	1	8	yes	—
204	C₂³	C₂²×D₂₁₂	1	3	yes	—
205	D₄	—	53	16	no	—
206	D₄	—	106	48	no	—
207	D₄	—	106	8	no	—
208	C₂×C₄	—	2	24	no	—
209	C₂×C₄	—	106	8	no	—
210	C₂×C₄	—	212	64	no	—
211	C₂×C₄	—	212	24	no	—
212	C₂×C₄	C₄×D₂₁₂	1	12	yes	—
213	C₂×C₄	C₂²×C₄×D₅₃	1	6	yes	—
214	C₂×C₄	C₂²×D₂₁₂	1	8	yes	—
215	C₂×C₄	C₂²×D₂₁₂	1	3	yes	—
216	C₂×C₄	C₂×#11719945	1	6	yes	—
217	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
218	C₈	—	2	4	no	—
219	C₈	C₂²×C₄×D₅₃	1	8	yes	complemented
220	C₂²	—	106	16	no	—
221	C₂²	—	212	96	no	—
222	C₂²	—	212	8	no	—
223	C₂²	—	212	8	no	—
224	C₂²	#16810736	1	16	yes	direct product factor
225	C₂²	C₂×C₄×D₂₁₂	1	6	yes	—
226	C₂²	C₂×#13133245	1	12	yes	—
227	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
228	C₄	—	2	8	no	—
229	C₄	—	212	8	no	—
230	C₄	C₂×C₄×D₂₁₂	1	6	yes	—
231	C₄	C₂³×C₄×D₅₃	1	1	yes	characteristic
232	C₄	C₂²×#11719945	1	1	yes	characteristic
233	C₂	—	212	16	no	—
234	C₂	C₂×#16810736	1	12	yes	direct product factor
235	C₂	C₂²×C₄×D₂₁₂	1	1	yes	characteristic
236	C₂	C₂²×#13133245	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₁₂
commutator	C₄×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×#97×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂×D₄
prime radical	C₂³×D₄₂₄	C₂
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂²×#97	D₅₃
hypocenter	C₅₃	C₂²×#442
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₂₁₂	C₂	semidirect
3. C₂×#16810736	C₂	direct (N × H)
4. C₂²×#13133149	C₂	semidirect
5. C₂²×#13133139	C₂	semidirect
6. C₂²×#97×C₅₃	C₂	semidirect
9. C₂²×D₄₂₄	C₄	semidirect
12. C₂×C₄×D₂₁₂	C₂²	semidirect
15. #16810736	C₂²	direct (N × H)
16. C₂×#13133149	C₂²	semidirect
17. C₂×#13133139	C₂²	semidirect
21. C₂×#97×C₅₃	C₂²	semidirect
25. C₂²×D₂₁₂	C₂×C₄	semidirect
30. C₂×D₄₂₄	C₂×C₄	semidirect
33. C₄×D₂₁₂	C₂³	semidirect
37. #13133149	C₂³	semidirect
38. #13133139	C₂³	semidirect
42. C₂²×C₈×C₅₃	C₂×C₄	semidirect
47. #97×C₅₃	C₂³	semidirect
64. C₂×D₂₁₂	C₂²×C₄	semidirect
69. C₂×C₈×C₅₃	C₂²×C₄	semidirect
77. D₄₂₄	C₂²×C₄	semidirect
97. D₂₁₂	C₂³×C₄	semidirect
100. C₈×C₅₃	C₂³×C₄	semidirect
102. C₂²×C₅₃	#442	semidirect
120. C₂²×#97	D₅₃	semidirect
121. C₂×C₅₃	C₂×#442	semidirect
139. C₂×#97	C₂×D₅₃	semidirect
140. C₅₃	C₂²×#442	semidirect
154. C₂²×C₈	C₄×D₅₃	semidirect
166. #97	C₂²×D₅₃	semidirect
189. C₂×C₈	C₂×C₄×D₅₃	semidirect
219. C₈	C₂²×C₄×D₅₃	semidirect
224. C₂²	#16810736	direct (N × H)
234. C₂	C₂×#16810736	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×D₄₂₄	2
C₂²×C₄×D₂₁₂	2
C₂×#16810736	2
C₂²×#13133149	2
C₂²×#13133139	2
C₂²×#97×C₅₃	2
C₂²×#442	53

Subgroups — C22×#16810736 #28248138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810736 #28248138