Subgroups — C₂²×#16811097 #28290672

Order 13568.

14590 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811097	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₄×D₂₁₂	C₂	1	4	yes	maximal, complemented
4	C₂²×#13133179	C₂	1	4	yes	maximal, complemented
5	C₂²×#13133176	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₂²⋊C₄×D₅₃	C₂	1	4	yes	maximal, complemented
7	C₂²×#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁵×D₅₃	C₂²	1	2	yes	—
9	C₂³×D₂₁₂	C₄	1	32	yes	complemented
10	C₂³×D₂₁₂	C₂²	1	2	yes	—
11	C₂³×D₂₁₂	C₂²	1	2	yes	—
12	C₂³×D₂₁₂	C₂²	1	24	yes	—
13	C₂³×C₄×D₅₃	C₂²	1	8	yes	complemented
14	#16811097	C₂²	1	256	yes	direct product factor
15	C₂×C₄×D₂₁₂	C₂²	1	96	yes	complemented
16	C₂×#13133179	C₂²	1	96	yes	complemented
17	C₂×#13133176	C₂²	1	24	yes	complemented
18	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	complemented
19	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	—
20	C₂²×#11719921	C₂²	1	8	yes	complemented
21	C₂×C₂²⋊C₄×D₅₃	C₂²	1	96	yes	complemented
22	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	complemented
23	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#117×C₅₃	C₂²	1	24	yes	complemented
25	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
26	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
27	C₂²×C₄²×C₅₃	C₂²	1	2	yes	complemented
28	C₂⁴×D₅₃	—	4	16	no	—
29	C₂⁴×D₅₃	C₂×C₄	1	32	yes	complemented
30	C₂⁴×D₅₃	C₂³	1	4	yes	—
31	C₂⁴×D₅₃	C₂³	1	24	yes	—
32	C₂²×D₂₁₂	C₂×C₄	1	32	yes	complemented
33	C₂²×D₂₁₂	C₂×C₄	1	32	yes	complemented
34	C₂²×D₂₁₂	C₂×C₄	1	384	yes	complemented
35	C₂²×D₂₁₂	C₂³	1	24	yes	—
36	C₂²×D₂₁₂	C₂³	1	24	yes	—
37	C₂²×D₂₁₂	C₂³	1	64	yes	—
38	C₂³×Dic₅₃	C₂³	1	4	yes	—
39	C₂²×C₄×D₅₃	—	4	32	no	—
40	C₂²×C₄×D₅₃	C₂³	1	96	yes	complemented
41	C₄×D₂₁₂	C₂³	1	256	yes	complemented
42	#13133179	C₂³	1	256	yes	complemented
43	#13133176	C₂³	1	64	yes	complemented
44	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	complemented
45	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	—
46	C₂×#11719921	C₂³	1	96	yes	complemented
47	C₂²⋊C₄×D₅₃	C₂³	1	256	yes	complemented
48	C₂×C₄×Dic₅₃	C₂³	1	24	yes	complemented
49	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	complemented
51	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
52	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
55	#117×C₅₃	C₂³	1	64	yes	complemented
56	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	complemented
57	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
58	C₂×C₄²×C₅₃	C₂³	1	24	yes	complemented
59	C₂⁴×C₅₃	—	2	8	no	—
60	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	—
61	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
62	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
63	C₂³×D₅₃	—	2	32	no	—
64	C₂³×D₅₃	—	4	128	no	—
65	C₂³×D₅₃	—	4	96	no	—
66	C₂³×D₅₃	C₂²×C₄	1	32	yes	—
67	C₂³×D₅₃	C₂²×C₄	1	192	yes	complemented
68	C₂³×D₅₃	C₂⁴	1	24	yes	—
69	C₂³×D₅₃	C₂⁴	1	32	yes	—
70	C₂²×C₄×C₅₃	—	2	16	no	—
71	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
73	C₂²×C₄×C₅₃	C₂²×C₄	1	96	yes	complemented
74	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
75	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
76	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
77	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
78	C₂²×Dic₅₃	—	2	16	no	—
79	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
80	C₂×D₂₁₂	C₂²×C₄	1	192	yes	complemented
81	C₂×D₂₁₂	C₂²×C₄	1	192	yes	complemented
82	C₂×D₂₁₂	C₂²×C₄	1	512	yes	complemented
83	C₂×D₂₁₂	C₂⁴	1	32	yes	—
84	C₂×D₂₁₂	C₂⁴	1	32	yes	—
85	C₂×C₄×D₅₃	—	4	192	no	—
86	C₂×C₄×D₅₃	C₂⁴	1	128	yes	complemented
87	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
88	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
89	C₄²×C₅₃	C₂⁴	1	32	yes	complemented
90	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	—
91	#11719921	C₂⁴	1	128	yes	complemented
92	C₄⋊Dic₅₃	C₂⁴	1	32	yes	complemented
93	C₄×Dic₅₃	C₂⁴	1	32	yes	complemented
94	C₂³×C₅₃	—	2	32	no	—
95	C₂³×C₅₃	—	2	24	no	—
96	C₂³×C₅₃	C₂³×C₄	1	24	yes	—
97	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
98	C₂³×C₅₃	H(5,𝔽₂)	1	8	yes	—
99	C₂³×C₅₃	C₂⁵	1	4	yes	—
100	C₂³×C₅₃	C₂⁵	1	3	yes	—
101	C₂²×D₅₃	—	2	128	no	—
102	C₂²×D₅₃	—	2	96	no	—
103	C₂²×D₅₃	—	4	384	no	—
104	C₂²×D₅₃	—	4	64	no	—
105	C₂²×D₅₃	C₂³×C₄	1	96	yes	—
106	C₂²×D₅₃	C₂³×C₄	1	128	yes	complemented
107	C₂²×D₅₃	C₂⁵	1	16	yes	—
108	C₂×C₄×C₅₃	—	2	48	no	—
109	C₂×C₄×C₅₃	C₂³×C₄	1	24	yes	—
110	C₂×C₄×C₅₃	C₂³×C₄	1	24	yes	—
111	C₂×C₄×C₅₃	C₂³×C₄	1	64	yes	complemented
112	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
113	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
114	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
115	C₂×Dic₅₃	—	2	48	no	—
116	C₂×Dic₅₃	C₂⁵	1	16	yes	—
117	D₂₁₂	C₂³×C₄	1	128	yes	complemented
118	D₂₁₂	C₂³×C₄	1	128	yes	complemented
119	C₄×D₅₃	—	4	128	no	—
120	C₂²×#518	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	—	2	8	no	—
122	C₂²×C₅₃	—	2	48	no	—
123	C₂²×C₅₃	#518	1	16	yes	complemented
124	C₂²×C₅₃	C₂⁴×C₄	1	8	yes	—
125	C₂²×C₅₃	C₂⁴×C₄	1	6	yes	—
126	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	12	yes	—
127	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
128	C₂×D₅₃	—	2	32	no	—
129	C₂×D₅₃	—	2	192	no	—
130	C₂×D₅₃	—	4	128	no	—
131	C₂×D₅₃	C₂⁴×C₄	1	32	yes	—
132	C₄×C₅₃	—	2	16	no	—
133	C₄×C₅₃	C₂⁴×C₄	1	8	yes	—
134	C₄×C₅₃	C₂⁴×C₄	1	8	yes	—
135	Dic₅₃	—	2	16	no	—
136	C₂⁴×D₄	—	53	1	no	—
137	C₂×#518	—	53	48	no	—
138	C₂²×C₄×D₄	—	53	4	no	—
139	C₂²×C₄×D₄	—	53	4	no	—
140	C₂²×#117	—	53	1	no	—
141	C₂²×#117	D₅₃	1	1	yes	characteristic, complemented
142	C₂³×C₂²⋊C₄	—	53	4	no	—
143	C₂×C₅₃	—	2	8	no	—
144	C₂×C₅₃	C₂×#518	1	12	yes	complemented
145	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
146	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	2	yes	—
147	D₅₃	—	2	32	no	—
148	C₂⁶	—	53	2	no	—
149	C₂³×D₄	—	53	32	no	—
150	C₂³×D₄	—	53	2	no	—
151	C₂³×D₄	—	53	2	no	—
152	C₂³×D₄	—	53	24	no	—
153	C₂⁴×C₄	—	53	8	no	—
154	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
155	#518	—	53	256	no	—
156	C₂×C₄×D₄	—	53	96	no	—
157	C₂×C₄×D₄	—	53	96	no	—
158	C₂×#117	—	53	24	no	—
159	C₂×#117	C₂×D₅₃	1	24	yes	complemented
160	C₂²×C₄⋊C₄	—	53	2	no	—
161	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	complemented
162	C₂²×C₂²⋊C₄	—	53	2	no	—
163	C₂²×C₂²⋊C₄	—	53	8	no	—
164	C₂²×C₂²⋊C₄	—	53	96	no	—
165	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
166	C₂²×C₄²	—	53	2	no	—
167	C₂²×C₄²	C₂×D₅₃	1	2	yes	complemented
168	C₅₃	C₂²×#518	1	1	yes	characteristic, complemented, Sylow
169	C₂⁵	—	53	4	no	—
170	C₂⁵	—	53	32	no	—
171	C₂⁵	—	53	24	no	—
172	C₂⁵	—	212	16	no	—
173	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
174	C₂²×D₄	—	53	32	no	—
175	C₂²×D₄	—	53	32	no	—
176	C₂²×D₄	—	53	384	no	—
177	C₂²×D₄	—	53	24	no	—
178	C₂²×D₄	—	53	24	no	—
179	C₂²×D₄	—	53	64	no	—
180	C₂³×C₄	—	53	4	no	—
181	C₂³×C₄	—	53	96	no	—
182	C₂³×C₄	—	212	32	no	—
183	C₂³×C₄	C₄×D₅₃	1	16	yes	complemented
184	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
185	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
186	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
187	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
188	C₄×D₄	—	53	256	no	—
189	C₄×D₄	—	53	256	no	—
190	#117	—	53	64	no	—
191	#117	C₂²×D₅₃	1	64	yes	complemented
192	C₂×C₄⋊C₄	—	53	24	no	—
193	C₂×C₄⋊C₄	C₂²×D₅₃	1	24	yes	complemented
194	C₂×C₂²⋊C₄	—	53	24	no	—
195	C₂×C₂²⋊C₄	—	53	96	no	—
196	C₂×C₂²⋊C₄	—	53	256	no	—
197	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	—
198	C₂×C₄²	—	53	24	no	—
199	C₂×C₄²	C₂²×D₅₃	1	24	yes	complemented
200	C₂⁴	—	2	8	no	—
201	C₂⁴	—	53	32	no	—
202	C₂⁴	—	53	24	no	—
203	C₂⁴	—	53	192	no	—
204	C₂⁴	—	53	32	no	—
205	C₂⁴	—	106	32	no	—
206	C₂⁴	—	212	128	no	—
207	C₂⁴	—	212	96	no	—
208	C₂⁴	C₂×C₄×D₅₃	1	8	yes	—
209	C₂⁴	C₂³×D₅₃	1	6	yes	—
210	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
211	C₂×D₄	—	53	192	no	—
212	C₂×D₄	—	53	192	no	—
213	C₂×D₄	—	53	512	no	—
214	C₂×D₄	—	53	32	no	—
215	C₂×D₄	—	53	32	no	—
216	C₂²×C₄	—	2	16	no	—
217	C₂²×C₄	—	53	24	no	—
218	C₂²×C₄	—	53	128	no	—
219	C₂²×C₄	—	106	16	no	—
220	C₂²×C₄	—	212	192	no	—
221	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
222	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
223	C₂²×C₄	C₂×C₄×D₅₃	1	96	yes	complemented
224	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
225	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
226	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
227	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
228	C₄⋊C₄	—	53	32	no	—
229	C₄⋊C₄	C₂³×D₅₃	1	32	yes	complemented
230	C₂²⋊C₄	—	53	32	no	—
231	C₂²⋊C₄	—	53	128	no	—
232	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	—
233	C₄²	—	53	32	no	—
234	C₄²	C₂³×D₅₃	1	32	yes	complemented
235	C₂³	—	2	32	no	—
236	C₂³	—	2	24	no	—
237	C₂³	—	53	96	no	—
238	C₂³	—	53	16	no	—
239	C₂³	—	53	128	no	—
240	C₂³	—	106	128	no	—
241	C₂³	—	106	96	no	—
242	C₂³	—	212	384	no	—
243	C₂³	—	212	64	no	—
244	C₂³	C₂²×C₄×D₅₃	1	24	yes	—
245	C₂³	C₂²×C₄×D₅₃	1	4	yes	—
246	C₂³	#13133263	1	8	yes	—
247	C₂³	C₂⁴×D₅₃	1	4	yes	—
248	C₂³	C₂⁴×D₅₃	1	3	yes	—
249	D₄	—	53	128	no	—
250	D₄	—	53	128	no	—
251	C₂×C₄	—	2	48	no	—
252	C₂×C₄	—	53	16	no	—
253	C₂×C₄	—	106	48	no	—
254	C₂×C₄	—	212	128	no	—
255	C₂×C₄	C₂²×C₄×D₅₃	1	24	yes	—
256	C₂×C₄	C₂²×C₄×D₅₃	1	24	yes	—
257	C₂×C₄	C₂²×C₄×D₅₃	1	64	yes	complemented
258	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
259	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
260	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
261	C₂²	—	2	48	no	—
262	C₂²	—	2	8	no	—
263	C₂²	—	53	32	no	—
264	C₂²	—	106	192	no	—
265	C₂²	—	106	32	no	—
266	C₂²	—	212	128	no	—
267	C₂²	#16811097	1	16	yes	direct product factor
268	C₂²	C₂³×C₄×D₅₃	1	8	yes	—
269	C₂²	C₂³×C₄×D₅₃	1	6	yes	—
270	C₂²	C₂×#13133263	1	12	yes	—
271	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
272	C₄	—	2	16	no	—
273	C₄	—	106	16	no	—
274	C₄	C₂³×C₄×D₅₃	1	8	yes	—
275	C₄	C₂³×C₄×D₅₃	1	8	yes	—
276	C₂	—	2	8	no	—
277	C₂	—	106	32	no	—
278	C₂	C₂×#16811097	1	12	yes	direct product factor
279	C₂	C₂⁴×C₄×D₅₃	1	1	yes	characteristic
280	C₂	C₂²×#13133263	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×#117×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁴×D₂₁₂	C₂
cosocle	C₂²×C₅₃	C₂⁶
hypercenter	C₂²×#117	D₅₃
hypocenter	C₅₃	C₂²×#518
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
2. C₂×#16811097	C₂	direct (N × H)
3. C₂²×C₄×D₂₁₂	C₂	semidirect
4. C₂²×#13133179	C₂	semidirect
5. C₂²×#13133176	C₂	semidirect
6. C₂²×C₂²⋊C₄×D₅₃	C₂	semidirect
7. C₂²×#117×C₅₃	C₂	semidirect
9. C₂³×D₂₁₂	C₄	semidirect
13. C₂³×C₄×D₅₃	C₂²	semidirect
14. #16811097	C₂²	direct (N × H)
15. C₂×C₄×D₂₁₂	C₂²	semidirect
16. C₂×#13133179	C₂²	semidirect
17. C₂×#13133176	C₂²	semidirect
18. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
20. C₂²×#11719921	C₂²	semidirect
21. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
22. C₂²×C₄×Dic₅₃	C₂²	semidirect
24. C₂×#117×C₅₃	C₂²	semidirect
25. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
27. C₂²×C₄²×C₅₃	C₂²	semidirect
29. C₂⁴×D₅₃	C₂×C₄	semidirect
32. C₂²×D₂₁₂	C₂×C₄	semidirect
33. C₂²×D₂₁₂	C₂×C₄	semidirect
34. C₂²×D₂₁₂	C₂×C₄	semidirect
40. C₂²×C₄×D₅₃	C₂³	semidirect
41. C₄×D₂₁₂	C₂³	semidirect
42. #13133179	C₂³	semidirect
43. #13133176	C₂³	semidirect
44. C₂×C₄⋊Dic₅₃	C₂³	semidirect
46. C₂×#11719921	C₂³	semidirect
47. C₂²⋊C₄×D₅₃	C₂³	semidirect
48. C₂×C₄×Dic₅₃	C₂³	semidirect
50. C₂³×C₄×C₅₃	C₂×C₄	semidirect
55. #117×C₅₃	C₂³	semidirect
56. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
58. C₂×C₄²×C₅₃	C₂³	semidirect
67. C₂³×D₅₃	C₂²×C₄	semidirect
73. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
80. C₂×D₂₁₂	C₂²×C₄	semidirect
81. C₂×D₂₁₂	C₂²×C₄	semidirect
82. C₂×D₂₁₂	C₂²×C₄	semidirect
86. C₂×C₄×D₅₃	C₂⁴	semidirect
87. C₄⋊C₄×C₅₃	C₂⁴	semidirect
89. C₄²×C₅₃	C₂⁴	semidirect
91. #11719921	C₂⁴	semidirect
92. C₄⋊Dic₅₃	C₂⁴	semidirect
93. C₄×Dic₅₃	C₂⁴	semidirect
106. C₂²×D₅₃	C₂³×C₄	semidirect
111. C₂×C₄×C₅₃	C₂³×C₄	semidirect
117. D₂₁₂	C₂³×C₄	semidirect
118. D₂₁₂	C₂³×C₄	semidirect
123. C₂²×C₅₃	#518	semidirect
141. C₂²×#117	D₅₃	semidirect
144. C₂×C₅₃	C₂×#518	semidirect
159. C₂×#117	C₂×D₅₃	semidirect
161. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
167. C₂²×C₄²	C₂×D₅₃	semidirect
168. C₅₃	C₂²×#518	semidirect
183. C₂³×C₄	C₄×D₅₃	semidirect
191. #117	C₂²×D₅₃	semidirect
193. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
199. C₂×C₄²	C₂²×D₅₃	semidirect
223. C₂²×C₄	C₂×C₄×D₅₃	semidirect
229. C₄⋊C₄	C₂³×D₅₃	semidirect
234. C₄²	C₂³×D₅₃	semidirect
257. C₂×C₄	C₂²×C₄×D₅₃	semidirect
267. C₂²	#16811097	direct (N × H)
278. C₂	C₂×#16811097	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₂₁₂	2
C₂×#16811097	2
C₂²×C₄×D₂₁₂	2
C₂²×#13133179	2
C₂²×#13133176	2
C₂²×C₂²⋊C₄×D₅₃	2
C₂²×#117×C₅₃	2
C₂²×#518	53

Subgroups — C22×#16811097 #28290672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811097 #28290672