Subgroups — #28293110

Order 13568.

7130 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20288518	C₂	1	16	yes	maximal, complemented
2	#20288513	C₂	1	32	yes	maximal, complemented
3	#20288120	C₂	1	4	yes	maximal, complemented
4	C₂×#16811249	C₂	1	8	yes	maximal, complemented
5	C₂×#16811098	C₂	1	2	yes	maximal
6	C₂×C₅₃×#518	C₂	1	1	yes	maximal, characteristic
7	#16811332	C₂²	1	128	yes	complemented
8	#16811330	C₂²	1	64	yes	complemented
9	D₄×Dic₁₀₆	C₂²	1	64	yes	complemented
10	#16811254	C₂²	1	16	yes	complemented
11	#16811251	C₂²	1	32	yes	complemented
12	#16811249	C₂²	1	16	yes	—
13	#16811249	C₂²	1	64	yes	complemented
14	#16811249	C₂²	1	32	yes	complemented
15	#16811180	C₂²	1	16	yes	complemented
16	#16811176	C₂²	1	32	yes	complemented
17	C₂×#13133196	C₂²	1	32	yes	complemented
18	C₂×#13133194	C₂²	1	64	yes	complemented
19	#16811098	C₂²	1	16	yes	—
20	#16811098	C₂²	1	16	yes	—
21	C₂×D₄×Dic₅₃	C₂²	1	16	yes	—
22	C₂×#13133176	C₂²	1	4	yes	—
23	C₂²×C₂²⋊Dic₅₃	C₂²	1	8	yes	—
24	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#518	C₂²	1	16	yes	—
26	C₂×C₄×D₄×C₅₃	C₂²	1	8	yes	—
27	C₂×#117×C₅₃	C₂²	1	2	yes	—
28	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
29	C₂²×Dic₁₀₆	C₂³	1	32	yes	complemented
30	C₂³×Dic₅₃	C₂³	1	16	yes	—
31	#13133217	C₂³	1	32	yes	complemented
32	#13133216	C₂³	1	64	yes	complemented
33	#13133209	C₂³	1	32	yes	complemented
34	#13133208	C₂³	1	64	yes	complemented
35	#13133196	C₂³	1	64	yes	—
36	#13133196	C₂³	1	128	yes	complemented
37	#13133194	C₂³	1	128	yes	—
38	#13133194	C₂³	1	256	yes	complemented
39	C₄×Dic₁₀₆	C₂³	1	64	yes	complemented
40	D₄×Dic₅₃	C₂³	1	128	yes	—
41	#13133176	C₂³	1	16	yes	—
42	#13133176	C₂³	1	16	yes	—
43	C₂×C₄⋊Dic₅₃	C₂³	1	8	yes	—
44	C₂×C₄⋊Dic₅₃	C₂³	1	32	yes	complemented
45	C₂×#11719919	C₂³	1	64	yes	complemented
46	C₂×C₂²⋊Dic₅₃	C₂³	1	8	yes	—
47	C₂×C₂²⋊Dic₅₃	C₂³	1	16	yes	—
48	C₂×C₂²⋊Dic₅₃	C₂³	1	32	yes	—
49	C₂×C₂²⋊Dic₅₃	C₂³	1	32	yes	—
50	C₂×C₄×Dic₅₃	C₂³	1	8	yes	—
51	C₂⁵×C₅₃	C₂³	1	2	yes	—
52	C₂²×D₄×C₅₃	Q₈	1	16	yes	complemented
53	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
54	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
55	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
56	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
57	C₄×D₄×C₅₃	C₂³	1	64	yes	—
58	#117×C₅₃	C₂³	1	16	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
60	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
61	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
62	C₂×C₂²⋊C₄×C₅₃	C₂³	1	32	yes	—
63	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
64	C₂⁴×C₅₃	—	8	4	no	—
65	C₂⁴×C₅₃	C₂×Q₈	1	16	yes	complemented
66	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
67	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
68	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
69	C₂×D₄×C₅₃	C₂×Q₈	1	32	yes	complemented
70	C₂×D₄×C₅₃	C₂×Q₈	1	64	yes	complemented
71	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
72	C₂²×C₄×C₅₃	—	8	8	no	—
73	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	complemented
74	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
75	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	—
76	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
77	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
78	C₂²×Dic₅₃	—	8	16	no	—
79	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
80	C₂²×Dic₅₃	C₂⁴	1	64	yes	—
81	C₂×Dic₁₀₆	—	8	32	no	—
82	C₂×Dic₁₀₆	C₂⁴	1	64	yes	—
83	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
84	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
85	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
86	C₄²×C₅₃	C₂⁴	1	16	yes	—
87	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
88	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
89	C₂²⋊Dic₅₃	C₂⁴	1	64	yes	—
90	C₄⋊Dic₅₃	C₂⁴	1	32	yes	—
91	C₄⋊Dic₅₃	C₂⁴	1	64	yes	—
92	#11719919	C₂⁴	1	128	yes	—
93	C₄×Dic₅₃	C₂⁴	1	32	yes	—
94	C₂³×C₅₃	—	4	16	no	—
95	C₂³×C₅₃	—	4	8	no	—
96	C₂³×C₅₃	—	4	2	no	—
97	C₂³×C₅₃	—	8	16	no	—
98	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
99	C₂³×C₅₃	C₂²×Q₈	1	32	yes	—
100	C₂³×C₅₃	C₂²×Q₈	1	16	yes	—
101	C₂³×C₅₃	C₂⁵	1	8	yes	—
102	C₂³×C₅₃	C₂⁵	1	2	yes	—
103	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
104	D₄×C₅₃	C₂²×Q₈	1	64	yes	—
105	C₂×C₄×C₅₃	—	4	32	no	—
106	C₂×C₄×C₅₃	—	4	8	no	—
107	C₂×C₄×C₅₃	C₂²×Q₈	1	8	yes	—
108	C₂×C₄×C₅₃	C₂²×Q₈	1	16	yes	—
109	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
110	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
111	C₂×Dic₅₃	—	4	64	no	—
112	C₂×Dic₅₃	—	4	16	no	—
113	C₂×Dic₅₃	C₂⁵	1	32	yes	—
114	Dic₁₀₆	—	4	64	no	—
115	#62843	—	53	1	no	maximal, Sylow
116	C₂²×C₅₃	—	2	16	no	—
117	C₂²×C₅₃	—	2	4	no	—
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	—	4	4	no	—
120	C₂²×C₅₃	—	4	32	no	—
121	C₂²×C₅₃	—	4	16	no	—
122	C₂²×C₅₃	C₂³×Q₈	1	4	yes	—
123	C₂²×C₅₃	C₂³×Q₈	1	2	yes	—
124	C₂²×C₅₃	C₂³×Q₈	1	16	yes	—
125	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
126	C₄×C₅₃	—	2	16	no	—
127	C₄×C₅₃	C₂³×Q₈	1	8	yes	—
128	Dic₅₃	—	2	32	no	—
129	#3501	—	53	32	no	—
130	#3501	—	53	16	no	—
131	#3458	—	53	4	no	—
132	C₂×#543	—	53	8	no	—
133	C₂×#518	—	53	2	no	—
134	C₂×#518	D₅₃	1	1	yes	characteristic
135	C₂×C₅₃	—	2	4	no	—
136	C₂×C₅₃	—	2	2	no	—
137	C₂×C₅₃	—	2	16	no	—
138	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
139	C₂×C₅₃	H(7,𝔽₂)	1	2	yes	—
140	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
141	#554	—	53	64	no	—
142	#554	—	53	128	no	—
143	D₄×Q₈	—	53	64	no	—
144	#544	—	53	32	no	—
145	#544	—	53	16	no	—
146	#543	—	53	16	no	—
147	#543	—	53	64	no	—
148	#543	—	53	32	no	—
149	#533	—	53	32	no	—
150	#533	—	53	16	no	—
151	C₂×C₂²⋊Q₈	—	53	32	no	—
152	C₂×C₂²⋊Q₈	—	53	64	no	—
153	#518	—	53	16	no	—
154	#518	—	53	16	no	—
155	#518	C₂×D₅₃	1	16	yes	—
156	C₂×C₄×D₄	—	53	16	no	—
157	C₂×C₄×D₄	C₂×D₅₃	1	8	yes	—
158	C₂×#117	—	53	4	no	—
159	C₂×#117	C₂×D₅₃	1	2	yes	—
160	C₂²×C₂²⋊C₄	—	53	8	no	—
161	C₂²×C₂²⋊C₄	C₂×D₅₃	1	4	yes	—
162	C₅₃	#62843	1	1	yes	characteristic, complemented, Sylow
163	C₂⁵	C₂²×D₅₃	1	2	yes	—
164	C₂²×Q₈	—	53	32	no	—
165	C₂²×D₄	Dic₁₀₆	1	16	yes	complemented
166	C₂²×D₄	C₂²×D₅₃	1	8	yes	—
167	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
168	C₂³×C₄	—	53	16	no	—
169	C₂³×C₄	C₂²×D₅₃	1	8	yes	—
170	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
171	C₄⋊Q₈	—	53	64	no	—
172	C₄⋊Q₈	—	53	32	no	—
173	#125	—	53	64	no	—
174	#125	—	53	32	no	—
175	C₂²⋊Q₈	—	53	64	no	—
176	C₂²⋊Q₈	—	53	128	no	—
177	C₂²⋊Q₈	—	53	128	no	—
178	C₂²⋊Q₈	—	53	256	no	—
179	C₄×Q₈	—	53	64	no	—
180	C₄×D₄	—	53	128	no	—
181	C₄×D₄	C₂²×D₅₃	1	64	yes	—
182	#117	—	53	16	no	—
183	#117	—	53	16	no	—
184	#117	C₂²×D₅₃	1	16	yes	—
185	C₂×C₄⋊C₄	—	53	8	no	—
186	C₂×C₄⋊C₄	—	53	64	no	—
187	C₂×C₄⋊C₄	—	53	32	no	—
188	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
189	C₂×C₂²⋊C₄	—	53	8	no	—
190	C₂×C₂²⋊C₄	—	53	16	no	—
191	C₂×C₂²⋊C₄	—	53	32	no	—
192	C₂×C₂²⋊C₄	—	53	32	no	—
193	C₂×C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
194	C₂×C₂²⋊C₄	C₂²×D₅₃	1	8	yes	—
195	C₂×C₂²⋊C₄	C₂²×D₅₃	1	32	yes	—
196	C₂×C₄²	—	53	8	no	—
197	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
198	C₂⁴	—	8	4	no	—
199	C₂⁴	C₂×Dic₁₀₆	1	16	yes	complemented
200	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
201	C₂⁴	C₂³×D₅₃	1	4	yes	—
202	C₂⁴	C₂³×D₅₃	1	8	yes	—
203	C₂×Q₈	—	53	64	no	—
204	C₂×Q₈	—	424	32	no	—
205	C₂×D₄	C₂×Dic₁₀₆	1	64	yes	complemented
206	C₂×D₄	C₂×Dic₁₀₆	1	32	yes	complemented
207	C₂×D₄	C₂³×D₅₃	1	16	yes	—
208	C₂²×C₄	—	8	8	no	—
209	C₂²×C₄	—	53	16	no	—
210	C₂²×C₄	—	53	64	no	—
211	C₂²×C₄	—	424	16	no	—
212	C₂²×C₄	C₂×Dic₁₀₆	1	8	yes	complemented
213	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
214	C₂²×C₄	C₂³×D₅₃	1	32	yes	—
215	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
216	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
217	C₄⋊C₄	—	53	128	no	—
218	C₄⋊C₄	—	53	64	no	—
219	C₄⋊C₄	—	53	32	no	—
220	C₄⋊C₄	C₂³×D₅₃	1	16	yes	—
221	C₂²⋊C₄	—	53	16	no	—
222	C₂²⋊C₄	—	53	16	no	—
223	C₂²⋊C₄	—	53	64	no	—
224	C₂²⋊C₄	C₂³×D₅₃	1	16	yes	—
225	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	—
226	C₄²	—	53	32	no	—
227	C₄²	C₂³×D₅₃	1	16	yes	—
228	C₂³	—	4	8	no	—
229	C₂³	—	4	16	no	—
230	C₂³	—	4	2	no	—
231	C₂³	—	8	16	no	—
232	C₂³	C₂²×Dic₁₀₆	1	4	yes	—
233	C₂³	C₂²×Dic₁₀₆	1	16	yes	—
234	C₂³	C₂²×Dic₁₀₆	1	32	yes	—
235	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
236	C₂³	C₂⁴×D₅₃	1	2	yes	—
237	C₂³	C₂⁴×D₅₃	1	8	yes	—
238	Q₈	—	212	64	no	—
239	D₄	C₂²×Dic₁₀₆	1	64	yes	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	4	32	no	—
242	C₂×C₄	—	53	32	no	—
243	C₂×C₄	—	212	16	no	—
244	C₂×C₄	—	212	64	no	—
245	C₂×C₄	C₂²×Dic₁₀₆	1	16	yes	—
246	C₂×C₄	C₂²×Dic₁₀₆	1	8	yes	—
247	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
248	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	16	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	16	no	—
254	C₂²	—	4	32	no	—
255	C₂²	C₂³×Dic₁₀₆	1	2	yes	—
256	C₂²	C₂³×Dic₁₀₆	1	4	yes	—
257	C₂²	C₂³×Dic₁₀₆	1	16	yes	—
258	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
259	C₄	—	2	16	no	—
260	C₄	—	106	32	no	—
261	C₄	C₂³×Dic₁₀₆	1	8	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	2	4	no	—
264	C₂	—	2	16	no	—
265	C₂	C₂⁴×Dic₁₀₆	1	1	yes	characteristic
266	C₂	#20289525	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁶
Fitting	C₂×C₅₃×#518	C₂
prime radical	C₂³×D₄×C₅₃	C₂²
hypercenter	C₂×#518	D₅₃
hypocenter	C₅₃	#62843
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×#518

Complements

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

N	H ≅ G/N	type
1. #20288518	C₂	semidirect
2. #20288513	C₂	semidirect
3. #20288120	C₂	semidirect
4. C₂×#16811249	C₂	semidirect
7. #16811332	C₂²	semidirect
8. #16811330	C₂²	semidirect
9. D₄×Dic₁₀₆	C₂²	semidirect
10. #16811254	C₂²	semidirect
11. #16811251	C₂²	semidirect
13. #16811249	C₂²	semidirect
14. #16811249	C₂²	semidirect
15. #16811180	C₂²	semidirect
16. #16811176	C₂²	semidirect
17. C₂×#13133196	C₂²	semidirect
18. C₂×#13133194	C₂²	semidirect
29. C₂²×Dic₁₀₆	C₂³	semidirect
31. #13133217	C₂³	semidirect
32. #13133216	C₂³	semidirect
33. #13133209	C₂³	semidirect
34. #13133208	C₂³	semidirect
36. #13133196	C₂³	semidirect
38. #13133194	C₂³	semidirect
39. C₄×Dic₁₀₆	C₂³	semidirect
44. C₂×C₄⋊Dic₅₃	C₂³	semidirect
45. C₂×#11719919	C₂³	semidirect
52. C₂²×D₄×C₅₃	Q₈	semidirect
65. C₂⁴×C₅₃	C₂×Q₈	semidirect
69. C₂×D₄×C₅₃	C₂×Q₈	semidirect
70. C₂×D₄×C₅₃	C₂×Q₈	semidirect
73. C₂²×C₄×C₅₃	C₂×Q₈	semidirect
162. C₅₃	#62843	semidirect
165. C₂²×D₄	Dic₁₀₆	semidirect
199. C₂⁴	C₂×Dic₁₀₆	semidirect
205. C₂×D₄	C₂×Dic₁₀₆	semidirect
206. C₂×D₄	C₂×Dic₁₀₆	semidirect
212. C₂²×C₄	C₂×Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20288518	2
#20288513	2
#20288120	2
C₂×#16811249	2
C₂×#16811098	2
C₂×C₅₃×#518	2
#62843	53