Subgroups — C₂²×#16811204 #28290856

Order 13568.

12270 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719945	C₂	1	2	yes	maximal, complemented
2	C₂×#16811204	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133197	C₂	1	4	yes	maximal, complemented
4	C₂²×#13133192	C₂	1	4	yes	maximal, complemented
5	C₂²×#13133191	C₂	1	4	yes	maximal, complemented
6	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11719945	—	2	32	no	—
8	C₂²×#11719945	C₂²	1	48	yes	complemented
9	C₂³×Dic₁₀₆	C₂²	1	2	yes	—
10	C₂³×D₂₁₂	C₂²	1	2	yes	—
11	C₂³×D₂₁₂	C₂²	1	4	yes	—
12	C₂³×C₂²⋊D₅₃	C₂²	1	4	yes	—
13	C₂³×C₄×D₅₃	C₂²	1	4	yes	—
14	#16811204	C₂²	1	256	yes	direct product factor
15	C₂×#13133197	C₂²	1	96	yes	complemented
16	C₂×#13133192	C₂²	1	96	yes	complemented
17	C₂×#13133191	C₂²	1	96	yes	complemented
18	C₂²×C₄⋊Dic₅₃	C₂²	1	4	yes	—
19	C₂²×#11719921	C₂²	1	8	yes	—
20	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
21	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
23	C₂²×C₄⋊C₄×C₅₃	C₂²	1	24	yes	complemented
24	C₂⁴×D₅₃	C₂³	1	4	yes	—
25	C₂×#11719945	—	2	384	no	—
26	C₂×#11719945	C₂³	1	128	yes	complemented
27	C₂²×Dic₁₀₆	—	2	16	no	—
28	C₂²×Dic₁₀₆	C₂³	1	24	yes	—
29	C₂²×D₂₁₂	—	2	16	no	—
30	C₂²×D₂₁₂	—	4	16	no	—
31	C₂²×D₂₁₂	C₂³	1	24	yes	—
32	C₂²×D₂₁₂	C₂³	1	48	yes	—
33	C₂²×C₂²⋊D₅₃	—	2	32	no	—
34	C₂²×C₂²⋊D₅₃	C₂³	1	48	yes	—
35	C₂³×Dic₅₃	C₂³	1	4	yes	—
36	C₂²×C₄×D₅₃	—	2	32	no	—
37	C₂²×C₄×D₅₃	C₂³	1	48	yes	—
38	#13133197	C₂³	1	256	yes	complemented
39	#13133192	C₂³	1	256	yes	complemented
40	#13133191	C₂³	1	256	yes	complemented
41	C₂×C₄⋊Dic₅₃	C₂³	1	48	yes	—
42	C₂×#11719921	C₂³	1	96	yes	—
43	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	—	4	8	no	—
45	C₂³×C₄×C₅₃	D₄	1	16	yes	complemented
46	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
47	C₂³×C₄×C₅₃	C₂³	1	24	yes	—
48	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
49	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
50	C₂×C₄⋊C₄×C₅₃	C₂³	1	48	yes	—
51	C₂×C₄⋊C₄×C₅₃	C₂³	1	64	yes	complemented
52	C₂⁴×C₅₃	—	2	4	no	—
53	C₂⁴×C₅₃	—	2	4	no	—
54	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
55	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
56	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
57	C₂³×D₅₃	—	2	16	no	—
58	C₂³×D₅₃	—	2	16	no	—
59	C₂³×D₅₃	—	2	16	no	—
60	C₂³×D₅₃	C₂⁴	1	24	yes	—
61	C₂²×C₄×C₅₃	—	2	16	no	—
62	C₂²×C₄×C₅₃	—	4	48	no	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	16	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	96	yes	complemented
65	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
66	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	—
67	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
68	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
69	C₂×C₂²⋊D₅₃	—	2	192	no	—
70	C₂×C₂²⋊D₅₃	C₂⁴	1	64	yes	—
71	C₂²×Dic₅₃	—	2	16	no	—
72	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
73	#11719945	—	2	512	no	—
74	C₂×D₂₁₂	—	2	96	no	—
75	C₂×D₂₁₂	—	4	96	no	—
76	C₂×D₂₁₂	C₂⁴	1	32	yes	—
77	C₂×D₂₁₂	C₂⁴	1	64	yes	—
78	C₂×C₄×D₅₃	—	2	192	no	—
79	C₂×C₄×D₅₃	C₂⁴	1	64	yes	—
80	C₂×Dic₁₀₆	—	2	96	no	—
81	C₂×Dic₁₀₆	C₂⁴	1	32	yes	—
82	C₄⋊C₄×C₅₃	C₂⁴	1	64	yes	—
83	#11719921	C₂⁴	1	128	yes	—
84	C₄⋊Dic₅₃	C₂⁴	1	64	yes	—
85	C₂³×C₅₃	—	2	32	no	—
86	C₂³×C₅₃	—	2	12	no	—
87	C₂³×C₅₃	—	2	12	no	—
88	C₂³×C₅₃	C₂²×D₄	1	24	yes	—
89	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
90	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
91	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
92	C₂³×C₅₃	C₂⁵	1	4	yes	—
93	C₂³×C₅₃	C₂⁵	1	3	yes	—
94	C₂²×D₅₃	—	2	48	no	—
95	C₂²×D₅₃	—	2	48	no	—
96	C₂²×D₅₃	—	2	48	no	—
97	C₂²×D₅₃	—	4	64	no	—
98	C₂²×D₅₃	C₂⁵	1	16	yes	—
99	C₂×C₄×C₅₃	—	2	48	no	—
100	C₂×C₄×C₅₃	—	4	32	no	—
101	C₂×C₄×C₅₃	C₂²×D₄	1	48	yes	—
102	C₂×C₄×C₅₃	C₂²×D₄	1	64	yes	complemented
103	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
104	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
105	C₂²⋊D₅₃	—	2	128	no	—
106	C₂×Dic₅₃	—	2	48	no	—
107	C₂×Dic₅₃	C₂⁵	1	16	yes	—
108	D₂₁₂	—	2	64	no	—
109	D₂₁₂	—	4	64	no	—
110	C₄×D₅₃	—	2	128	no	—
111	Dic₁₀₆	—	2	64	no	—
112	C₂²×#537	—	53	1	no	maximal, Sylow
113	C₂²×C₅₃	—	2	4	no	—
114	C₂²×C₅₃	—	2	4	no	—
115	C₂²×C₅₃	—	2	48	no	—
116	C₂²×C₅₃	#537	1	16	yes	complemented
117	C₂²×C₅₃	C₂³×D₄	1	8	yes	—
118	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
119	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
120	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	6	yes	—
121	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
122	C₂×D₅₃	—	2	16	no	—
123	C₂×D₅₃	—	2	16	no	—
124	C₂×D₅₃	—	2	16	no	—
125	C₂×D₅₃	—	4	96	no	—
126	C₄×C₅₃	—	2	16	no	—
127	C₄×C₅₃	C₂³×D₄	1	16	yes	—
128	Dic₅₃	—	2	16	no	—
129	C₂³×#41	—	53	2	no	—
130	C₂×#537	—	53	48	no	—
131	C₂²×C₂²⋊Q₈	—	53	4	no	—
132	C₂²×#121	—	53	4	no	—
133	C₂²×#121	—	53	4	no	—
134	C₂³×C₄⋊C₄	D₅₃	1	1	yes	characteristic, complemented
135	C₂×C₅₃	—	2	8	no	—
136	C₂×C₅₃	C₂×#537	1	12	yes	complemented
137	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
138	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₅₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
140	D₅₃	—	4	16	no	—
141	C₂²×#41	—	53	48	no	—
142	C₂²×#41	—	106	32	no	—
143	C₂³×Q₈	—	53	2	no	—
144	C₂³×D₄	—	53	4	no	—
145	C₂³×D₄	—	53	2	no	—
146	C₂³×D₄	—	53	4	no	—
147	C₂⁴×C₄	—	53	4	no	—
148	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
149	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
150	#537	—	53	256	no	—
151	C₂×C₂²⋊Q₈	—	53	96	no	—
152	C₂×#121	—	53	96	no	—
153	C₂×#121	—	53	96	no	—
154	C₂²×C₄⋊C₄	—	53	4	no	—
155	C₂²×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
156	C₂²×C₄⋊C₄	C₂×D₅₃	1	24	yes	complemented
157	C₂²×C₂²⋊C₄	—	53	8	no	—
158	C₅₃	C₂²×#537	1	1	yes	characteristic, complemented, Sylow
159	C₂⁵	—	53	4	no	—
160	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
161	C₂×#41	—	53	128	no	—
162	C₂×#41	—	106	384	no	—
163	C₂²×Q₈	—	53	24	no	—
164	C₂²×Q₈	—	106	16	no	—
165	C₂²×D₄	—	53	48	no	—
166	C₂²×D₄	—	53	24	no	—
167	C₂²×D₄	—	53	48	no	—
168	C₂²×D₄	—	106	32	no	—
169	C₂²×D₄	—	106	16	no	—
170	C₂²×D₄	—	212	16	no	—
171	C₂³×C₄	—	4	8	no	—
172	C₂³×C₄	—	53	4	no	—
173	C₂³×C₄	—	53	48	no	—
174	C₂³×C₄	—	106	32	no	—
175	C₂³×C₄	D₂₁₂	1	16	yes	complemented
176	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
177	C₂³×C₄	C₂²×D₅₃	1	24	yes	—
178	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
179	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
180	C₂²⋊Q₈	—	53	256	no	—
181	#121	—	53	256	no	—
182	#121	—	53	256	no	—
183	C₂×C₄⋊C₄	—	53	48	no	—
184	C₂×C₄⋊C₄	C₂²×D₅₃	1	48	yes	—
185	C₂×C₄⋊C₄	C₂²×D₅₃	1	64	yes	complemented
186	C₂×C₂²⋊C₄	—	53	96	no	—
187	C₂⁴	—	2	4	no	—
188	C₂⁴	—	2	4	no	—
189	C₂⁴	—	53	24	no	—
190	C₂⁴	—	106	16	no	—
191	C₂⁴	—	106	16	no	—
192	C₂⁴	—	106	16	no	—
193	C₂⁴	C₂×D₂₁₂	1	8	yes	—
194	C₂⁴	C₂³×D₅₃	1	6	yes	—
195	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
196	#41	—	106	512	no	—
197	C₂×Q₈	—	53	32	no	—
198	C₂×Q₈	—	106	96	no	—
199	C₂×D₄	—	53	64	no	—
200	C₂×D₄	—	53	32	no	—
201	C₂×D₄	—	53	64	no	—
202	C₂×D₄	—	106	192	no	—
203	C₂×D₄	—	106	96	no	—
204	C₂×D₄	—	212	96	no	—
205	C₂²×C₄	—	2	16	no	—
206	C₂²×C₄	—	4	48	no	—
207	C₂²×C₄	—	53	24	no	—
208	C₂²×C₄	—	53	64	no	—
209	C₂²×C₄	—	106	16	no	—
210	C₂²×C₄	—	106	192	no	—
211	C₂²×C₄	C₂×D₂₁₂	1	16	yes	—
212	C₂²×C₄	C₂×D₂₁₂	1	96	yes	complemented
213	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
214	C₂²×C₄	C₂³×D₅₃	1	32	yes	—
215	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
216	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
217	C₄⋊C₄	—	53	64	no	—
218	C₄⋊C₄	C₂³×D₅₃	1	64	yes	—
219	C₂²⋊C₄	—	53	128	no	—
220	C₂³	—	2	12	no	—
221	C₂³	—	2	12	no	—
222	C₂³	—	2	32	no	—
223	C₂³	—	53	16	no	—
224	C₂³	—	106	48	no	—
225	C₂³	—	106	48	no	—
226	C₂³	—	106	48	no	—
227	C₂³	—	212	64	no	—
228	C₂³	C₂²×D₂₁₂	1	24	yes	—
229	C₂³	C₂²×D₂₁₂	1	4	yes	—
230	C₂³	#13133262	1	4	yes	—
231	C₂³	#13133265	1	4	yes	—
232	C₂³	C₂⁴×D₅₃	1	4	yes	—
233	C₂³	C₂⁴×D₅₃	1	3	yes	—
234	Q₈	—	106	64	no	—
235	D₄	—	106	128	no	—
236	D₄	—	106	64	no	—
237	D₄	—	212	64	no	—
238	C₂×C₄	—	2	48	no	—
239	C₂×C₄	—	4	32	no	—
240	C₂×C₄	—	53	16	no	—
241	C₂×C₄	—	106	48	no	—
242	C₂×C₄	—	106	128	no	—
243	C₂×C₄	C₂²×D₂₁₂	1	48	yes	—
244	C₂×C₄	C₂²×D₂₁₂	1	64	yes	complemented
245	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
246	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
247	C₂²	—	2	48	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	106	16	no	—
251	C₂²	—	106	16	no	—
252	C₂²	—	106	16	no	—
253	C₂²	—	212	96	no	—
254	C₂²	#16811204	1	16	yes	direct product factor
255	C₂²	C₂³×D₂₁₂	1	8	yes	—
256	C₂²	C₂³×D₂₁₂	1	6	yes	—
257	C₂²	C₂×#13133262	1	6	yes	—
258	C₂²	C₂×#13133265	1	6	yes	—
259	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
260	C₄	—	2	16	no	—
261	C₄	—	106	16	no	—
262	C₄	C₂³×D₂₁₂	1	16	yes	—
263	C₂	—	2	8	no	—
264	C₂	—	212	16	no	—
265	C₂	C₂×#16811204	1	12	yes	direct product factor
266	C₂	C₂⁴×D₂₁₂	1	1	yes	characteristic
267	C₂	C₂²×#13133262	1	1	yes	characteristic
268	C₂	C₂²×#13133265	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×C₄⋊C₄×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂³×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂²×#537
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#11719945	C₂	semidirect
2. C₂×#16811204	C₂	direct (N × H)
3. C₂²×#13133197	C₂	semidirect
4. C₂²×#13133192	C₂	semidirect
5. C₂²×#13133191	C₂	semidirect
6. C₂³×C₄⋊C₄×C₅₃	C₂	semidirect
8. C₂²×#11719945	C₂²	semidirect
14. #16811204	C₂²	direct (N × H)
15. C₂×#13133197	C₂²	semidirect
16. C₂×#13133192	C₂²	semidirect
17. C₂×#13133191	C₂²	semidirect
23. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
26. C₂×#11719945	C₂³	semidirect
38. #13133197	C₂³	semidirect
39. #13133192	C₂³	semidirect
40. #13133191	C₂³	semidirect
45. C₂³×C₄×C₅₃	D₄	semidirect
51. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
64. C₂²×C₄×C₅₃	C₂×D₄	semidirect
102. C₂×C₄×C₅₃	C₂²×D₄	semidirect
116. C₂²×C₅₃	#537	semidirect
134. C₂³×C₄⋊C₄	D₅₃	semidirect
136. C₂×C₅₃	C₂×#537	semidirect
156. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
158. C₅₃	C₂²×#537	semidirect
175. C₂³×C₄	D₂₁₂	semidirect
185. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
212. C₂²×C₄	C₂×D₂₁₂	semidirect
244. C₂×C₄	C₂²×D₂₁₂	semidirect
254. C₂²	#16811204	direct (N × H)
265. C₂	C₂×#16811204	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×#11719945	2
C₂×#16811204	2
C₂²×#13133197	2
C₂²×#13133192	2
C₂²×#13133191	2
C₂³×C₄⋊C₄×C₅₃	2
C₂²×#537	53

Subgroups — C22×#16811204 #28290856

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811204 #28290856