Subgroups — C₂²×#16811428 #28297904

Order 13568.

11462 conjugacy classes of subgroups.

The subgroup lattice has 260 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	2	yes	maximal, complemented
2	C₂×#16811428	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133245	C₂	1	6	yes	maximal, complemented
4	C₂²×#13133239	C₂	1	3	yes	maximal, complemented
5	C₂³×D₄₂₄	C₂	1	3	yes	maximal, complemented
6	C₂²×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13133263	—	2	24	no	—
8	C₂×#13133263	C₂²	1	48	yes	complemented
9	C₂²×#11719949	—	2	2	no	—
10	C₂²×#11719945	C₂²	1	6	yes	complemented
11	C₂³×D₂₁₂	C₂²	1	6	yes	complemented
12	C₂²×D₄×D₅₃	—	2	6	no	—
13	#16811428	C₂²	1	256	yes	direct product factor
14	C₂×#13133245	C₂²	1	144	yes	complemented
15	C₂×#13133239	C₂²	1	72	yes	complemented
16	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic, complemented
17	C₂²×#11719913	C₂²	1	6	yes	complemented
18	C₂²×D₄₂₄	C₂²	1	3	yes	complemented
19	C₂²×D₄₂₄	C₂²	1	6	yes	complemented
20	C₂²×D₄₂₄	C₂²	1	72	yes	complemented
21	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#131×C₅₃	C₂²	1	24	yes	complemented
23	C₂²×M₁₆×C₅₃	C₂²	1	3	yes	complemented
24	C₂³×C₈×C₅₃	C₂²	1	3	yes	complemented
25	C₂⁴×D₅₃	—	2	6	no	—
26	#13133263	—	2	192	no	—
27	#13133263	C₂³	1	128	yes	complemented
28	C₂×#11719949	—	2	48	no	—
29	C₂×#11719945	—	2	36	no	—
30	C₂×#11719945	C₂³	1	72	yes	complemented
31	C₂²×Dic₁₀₆	C₂³	1	2	yes	—
32	C₂²×D₂₁₂	—	2	6	no	—
33	C₂²×D₂₁₂	—	2	36	no	—
34	C₂²×D₂₁₂	C₂³	1	6	yes	—
35	C₂²×D₂₁₂	C₂³	1	72	yes	complemented
36	C₂²×C₂²⋊D₅₃	—	2	6	no	—
37	C₂×D₄×D₅₃	—	2	144	no	—
38	C₂²×C₄×D₅₃	—	2	6	no	—
39	#13133245	C₂³	1	384	yes	complemented
40	#13133239	C₂³	1	192	yes	complemented
41	C₂×Dic₂₁₂	C₂³	1	12	yes	complemented
42	C₂×#11719913	C₂³	1	72	yes	complemented
43	C₂×D₄₂₄	C₂³	1	36	yes	complemented
44	C₂×D₄₂₄	C₂³	1	72	yes	complemented
45	C₂×D₄₂₄	C₂³	1	192	yes	complemented
46	C₂×#41×C₅₃	D₄	1	12	yes	—
47	C₂×#41×C₅₃	C₂³	1	12	yes	—
48	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
49	C₂²×D₄×C₅₃	D₄	1	3	yes	—
50	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
51	#131×C₅₃	C₂³	1	64	yes	complemented
52	C₂×M₁₆×C₅₃	C₂³	1	36	yes	complemented
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
55	C₂²×C₈×C₅₃	C₂³	1	36	yes	complemented
56	C₂⁴×C₅₃	C₂×D₄	1	3	yes	—
57	C₂³×D₅₃	—	2	6	no	—
58	C₂³×D₅₃	—	2	72	no	—
59	C₂³×D₅₃	—	8	24	no	—
60	#41×C₅₃	C₂×D₄	1	48	yes	—
61	#41×C₅₃	C₂⁴	1	16	yes	—
62	C₂×Q₈×C₅₃	C₂×D₄	1	12	yes	—
63	C₂×D₄×C₅₃	C₂×D₄	1	36	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	18	yes	—
66	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
68	C₂×C₂²⋊D₅₃	—	2	72	no	—
69	C₂²×Dic₅₃	—	2	2	no	—
70	#11719949	—	2	128	no	—
71	D₄×D₅₃	—	2	384	no	—
72	#11719945	—	2	144	no	—
73	#11719945	C₂⁴	1	96	yes	complemented
74	C₂×D₂₁₂	—	2	72	no	—
75	C₂×D₂₁₂	—	2	18	no	—
76	C₂×D₂₁₂	—	2	144	no	—
77	C₂×D₂₁₂	C₂⁴	1	36	yes	—
78	C₂×D₂₁₂	C₂⁴	1	96	yes	complemented
79	C₂×C₄×D₅₃	—	2	72	no	—
80	C₂×Dic₁₀₆	—	2	6	no	—
81	C₂×Dic₁₀₆	C₂⁴	1	12	yes	—
82	M₁₆×C₅₃	C₂⁴	1	48	yes	complemented
83	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
84	C₂×C₈×C₅₃	C₂⁴	1	18	yes	—
85	C₂×C₈×C₅₃	C₂⁴	1	48	yes	complemented
86	Dic₂₁₂	C₂⁴	1	16	yes	complemented
87	D₄₂₄	C₂⁴	1	48	yes	complemented
88	D₄₂₄	C₂⁴	1	96	yes	complemented
89	#11719913	C₂⁴	1	96	yes	complemented
90	C₂³×C₅₃	—	2	12	no	—
91	C₂³×C₅₃	C₂²×D₄	1	18	yes	—
92	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
93	C₂²×D₅₃	—	2	36	no	—
94	C₂²×D₅₃	—	2	96	no	—
95	C₂²×D₅₃	—	4	24	no	—
96	C₂²×D₅₃	—	8	144	no	—
97	Q₈×C₅₃	C₂²×D₄	1	16	yes	—
98	D₄×C₅₃	C₂²×D₄	1	48	yes	—
99	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
100	C₂×C₄×C₅₃	C₂²×D₄	1	18	yes	—
101	C₂×C₄×C₅₃	C₂²×D₄	1	36	yes	—
102	C₂×C₄×C₅₃	C₂⁵	1	3	yes	—
103	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
104	C₂²⋊D₅₃	—	2	96	no	—
105	C₂×Dic₅₃	—	2	12	no	—
106	D₂₁₂	—	2	36	no	—
107	D₂₁₂	—	2	96	no	—
108	D₂₁₂	C₂⁵	1	24	yes	—
109	C₄×D₅₃	—	2	96	no	—
110	Dic₁₀₆	—	2	12	no	—
111	Dic₁₀₆	C₂⁵	1	8	yes	—
112	C₈×C₅₃	C₂⁵	1	12	yes	—
113	C₈×C₅₃	C₂⁵	1	4	yes	—
114	C₂²×#576	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	—	2	36	no	—
116	C₂²×C₅₃	#576	1	4	yes	complemented
117	C₂²×C₅₃	C₂³×D₄	1	12	yes	—
118	C₂²×C₅₃	C₂³×D₄	1	3	yes	—
119	C₂×D₅₃	—	2	24	no	—
120	C₂×D₅₃	—	4	72	no	—
121	C₂×D₅₃	—	8	96	no	—
122	C₄×C₅₃	C₂³×D₄	1	3	yes	—
123	C₄×C₅₃	C₂³×D₄	1	12	yes	—
124	C₄×C₅₃	C₂⁶	1	1	yes	characteristic
125	Dic₅₃	—	2	8	no	—
126	C₂²×H(5,𝔽₂)	—	53	2	no	—
127	C₂×#576	—	53	48	no	—
128	C₂²×AGL(1,ℤ/8)	—	53	6	no	—
129	C₂²×#135	—	53	3	no	—
130	C₂³×D₈	—	53	3	no	—
131	C₂²×#131	D₅₃	1	1	yes	characteristic, complemented
132	C₂×C₅₃	—	2	12	no	—
133	C₂×C₅₃	C₂×#576	1	6	yes	complemented
134	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
135	D₅₃	—	4	24	no	—
136	C₂×H(5,𝔽₂)	—	53	48	no	—
137	C₂×H(5,𝔽₂)	—	106	24	no	—
138	C₂²×#41	—	53	6	no	—
139	C₂²×#41	—	106	2	no	—
140	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
141	C₂³×D₄	—	53	6	no	—
142	C₂³×D₄	—	106	6	no	—
143	#576	—	53	256	no	—
144	C₂×AGL(1,ℤ/8)	—	53	144	no	—
145	C₂×#135	—	53	72	no	—
146	C₂²×Q₁₆	—	53	1	no	—
147	C₂²×SD₁₆	—	53	6	no	—
148	C₂²×D₈	—	53	3	no	—
149	C₂²×D₈	—	53	6	no	—
150	C₂²×D₈	—	53	72	no	—
151	C₂×#131	C₂×D₅₃	1	24	yes	complemented
152	C₂²×M₁₆	C₂×D₅₃	1	3	yes	complemented
153	C₂³×C₈	C₂×D₅₃	1	3	yes	complemented
154	C₅₃	C₂²×#576	1	1	yes	characteristic, complemented, Sylow
155	C₂⁵	—	106	6	no	—
156	H(5,𝔽₂)	—	53	128	no	—
157	H(5,𝔽₂)	—	106	192	no	—
158	C₂×#41	—	53	72	no	—
159	C₂×#41	—	106	48	no	—
160	C₂×#41	—	106	36	no	—
161	C₂×#41	D₂₁₂	1	12	yes	—
162	C₂×#41	C₂²×D₅₃	1	12	yes	—
163	C₂²×Q₈	—	53	2	no	—
164	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
165	C₂²×D₄	—	53	6	no	—
166	C₂²×D₄	—	53	72	no	—
167	C₂²×D₄	—	106	6	no	—
168	C₂²×D₄	—	106	144	no	—
169	C₂²×D₄	—	106	6	no	—
170	C₂²×D₄	—	106	36	no	—
171	C₂²×D₄	D₂₁₂	1	3	yes	—
172	C₂³×C₄	—	106	6	no	—
173	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
174	AGL(1,ℤ/8)	—	53	384	no	—
175	#135	—	53	192	no	—
176	C₂×Q₁₆	—	53	12	no	—
177	C₂×SD₁₆	—	53	72	no	—
178	C₂×D₈	—	53	36	no	—
179	C₂×D₈	—	53	72	no	—
180	C₂×D₈	—	53	192	no	—
181	#131	C₂²×D₅₃	1	64	yes	complemented
182	C₂×M₁₆	C₂²×D₅₃	1	36	yes	complemented
183	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
185	C₂²×C₈	C₂²×D₅₃	1	36	yes	complemented
186	C₂⁴	—	106	72	no	—
187	C₂⁴	—	106	6	no	—
188	C₂⁴	—	424	24	no	—
189	C₂⁴	C₂×D₂₁₂	1	3	yes	—
190	#41	—	53	96	no	—
191	#41	—	106	128	no	—
192	#41	—	106	144	no	—
193	#41	C₂×D₂₁₂	1	48	yes	—
194	#41	C₂³×D₅₃	1	16	yes	—
195	C₂×Q₈	—	53	12	no	—
196	C₂×Q₈	—	106	6	no	—
197	C₂×Q₈	C₂×D₂₁₂	1	12	yes	—
198	C₂×D₄	—	53	36	no	—
199	C₂×D₄	—	53	96	no	—
200	C₂×D₄	—	106	72	no	—
201	C₂×D₄	—	106	384	no	—
202	C₂×D₄	—	106	72	no	—
203	C₂×D₄	—	106	18	no	—
204	C₂×D₄	—	106	144	no	—
205	C₂×D₄	C₂×D₂₁₂	1	36	yes	—
206	C₂²×C₄	—	106	2	no	—
207	C₂²×C₄	—	106	72	no	—
208	C₂²×C₄	C₂×D₂₁₂	1	18	yes	—
209	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
210	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
211	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
212	Q₁₆	—	53	16	no	—
213	SD₁₆	—	53	96	no	—
214	D₈	—	53	48	no	—
215	D₈	—	53	96	no	—
216	M₁₆	C₂³×D₅₃	1	48	yes	complemented
217	C₂×C₈	C₂³×D₅₃	1	6	yes	—
218	C₂×C₈	C₂³×D₅₃	1	18	yes	—
219	C₂×C₈	C₂³×D₅₃	1	48	yes	complemented
220	C₂³	—	2	12	no	—
221	C₂³	—	106	96	no	—
222	C₂³	—	106	36	no	—
223	C₂³	—	424	144	no	—
224	C₂³	—	212	24	no	—
225	C₂³	C₂²×D₂₁₂	1	18	yes	—
226	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
227	Q₈	—	53	8	no	—
228	Q₈	—	106	12	no	—
229	Q₈	C₂²×D₂₁₂	1	16	yes	—
230	D₄	—	53	24	no	—
231	D₄	—	106	96	no	—
232	D₄	—	106	96	no	—
233	D₄	—	106	36	no	—
234	D₄	C₂²×D₂₁₂	1	48	yes	—
235	C₂×C₄	—	106	12	no	—
236	C₂×C₄	—	106	96	no	—
237	C₂×C₄	C₂²×D₂₁₂	1	18	yes	—
238	C₂×C₄	C₂²×D₂₁₂	1	3	yes	—
239	C₂×C₄	C₂²×D₂₁₂	1	36	yes	—
240	C₂×C₄	C₂⁴×D₅₃	1	3	yes	—
241	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
242	C₈	C₂⁴×D₅₃	1	4	yes	—
243	C₈	C₂⁴×D₅₃	1	12	yes	—
244	C₂²	—	2	36	no	—
245	C₂²	—	106	24	no	—
246	C₂²	—	424	96	no	—
247	C₂²	—	212	72	no	—
248	C₂²	#16811428	1	4	yes	direct product factor
249	C₂²	C₂³×D₂₁₂	1	12	yes	—
250	C₂²	C₂³×D₂₁₂	1	3	yes	—
251	C₄	—	106	8	no	—
252	C₄	C₂³×D₂₁₂	1	12	yes	—
253	C₄	C₂³×D₂₁₂	1	3	yes	—
254	C₄	C₂⁵×D₅₃	1	1	yes	characteristic
255	C₂	—	2	12	no	—
256	C₂	—	212	24	no	—
257	C₂	C₂×#16811428	1	6	yes	direct product factor
258	C₂	C₂⁴×D₂₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₁₂
commutator, cosocle	C₄×C₅₃	C₂⁶
Frattini	C₄	C₂⁵×D₅₃
Fitting	C₂²×#131×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×D₄
hypercenter	C₂²×#131	D₅₃
hypocenter	C₅₃	C₂²×#576
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133263	C₂	semidirect
2. C₂×#16811428	C₂	direct (N × H)
3. C₂²×#13133245	C₂	semidirect
4. C₂²×#13133239	C₂	semidirect
5. C₂³×D₄₂₄	C₂	semidirect
6. C₂²×#131×C₅₃	C₂	semidirect
8. C₂×#13133263	C₂²	semidirect
10. C₂²×#11719945	C₂²	semidirect
11. C₂³×D₂₁₂	C₂²	semidirect
13. #16811428	C₂²	direct (N × H)
14. C₂×#13133245	C₂²	semidirect
15. C₂×#13133239	C₂²	semidirect
16. C₂²×Dic₂₁₂	C₂²	semidirect
17. C₂²×#11719913	C₂²	semidirect
18. C₂²×D₄₂₄	C₂²	semidirect
19. C₂²×D₄₂₄	C₂²	semidirect
20. C₂²×D₄₂₄	C₂²	semidirect
22. C₂×#131×C₅₃	C₂²	semidirect
23. C₂²×M₁₆×C₅₃	C₂²	semidirect
24. C₂³×C₈×C₅₃	C₂²	semidirect
27. #13133263	C₂³	semidirect
30. C₂×#11719945	C₂³	semidirect
35. C₂²×D₂₁₂	C₂³	semidirect
39. #13133245	C₂³	semidirect
40. #13133239	C₂³	semidirect
41. C₂×Dic₂₁₂	C₂³	semidirect
42. C₂×#11719913	C₂³	semidirect
43. C₂×D₄₂₄	C₂³	semidirect
44. C₂×D₄₂₄	C₂³	semidirect
45. C₂×D₄₂₄	C₂³	semidirect
51. #131×C₅₃	C₂³	semidirect
52. C₂×M₁₆×C₅₃	C₂³	semidirect
55. C₂²×C₈×C₅₃	C₂³	semidirect
73. #11719945	C₂⁴	semidirect
78. C₂×D₂₁₂	C₂⁴	semidirect
82. M₁₆×C₅₃	C₂⁴	semidirect
85. C₂×C₈×C₅₃	C₂⁴	semidirect
86. Dic₂₁₂	C₂⁴	semidirect
87. D₄₂₄	C₂⁴	semidirect
88. D₄₂₄	C₂⁴	semidirect
89. #11719913	C₂⁴	semidirect
116. C₂²×C₅₃	#576	semidirect
131. C₂²×#131	D₅₃	semidirect
133. C₂×C₅₃	C₂×#576	semidirect
151. C₂×#131	C₂×D₅₃	semidirect
152. C₂²×M₁₆	C₂×D₅₃	semidirect
153. C₂³×C₈	C₂×D₅₃	semidirect
154. C₅₃	C₂²×#576	semidirect
181. #131	C₂²×D₅₃	semidirect
182. C₂×M₁₆	C₂²×D₅₃	semidirect
185. C₂²×C₈	C₂²×D₅₃	semidirect
216. M₁₆	C₂³×D₅₃	semidirect
219. C₂×C₈	C₂³×D₅₃	semidirect
248. C₂²	#16811428	direct (N × H)
257. C₂	C₂×#16811428	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
C₂×#16811428	2
C₂²×#13133245	2
C₂²×#13133239	2
C₂³×D₄₂₄	2
C₂²×#131×C₅₃	2
C₂²×#576	53

Subgroups — C22×#16811428 #28297904

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811428 #28297904