Subgroups — #20288592

Order 6784.

1232 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811358	C₂	1	4	yes	maximal, complemented
2	#16811295	C₂	1	4	yes	maximal, complemented
3	#16811264	C₂	1	8	yes	maximal, complemented
4	#16811229	C₂	1	4	yes	maximal, complemented
5	#16811192	C₂	1	2	yes	maximal, complemented
6	#16811174	C₂	1	4	yes	maximal, complemented
7	C₂×#13133214	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811097	C₂	1	2	yes	maximal, complemented
9	C₄×#11719945	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#11719949	C₂²	1	8	yes	complemented
12	C₂×#11719945	C₂²	1	1	yes	characteristic
13	C₂²×D₂₁₂	C₂²	1	2	yes	—
14	C₂²×D₂₁₂	C₂²	1	4	yes	—
15	#13133214	C₂²	1	4	yes	—
16	#13133214	C₂²	1	2	yes	—
17	#13133214	C₂²	1	2	yes	—
18	#13133214	C₂²	1	4	yes	complemented
19	#13133206	C₂²	1	8	yes	complemented
20	#13133205	C₂²	1	4	yes	complemented
21	#13133202	C₂²	1	16	yes	complemented
22	#13133192	C₂²	1	8	yes	complemented
23	#13133192	C₂²	1	16	yes	complemented
24	#13133192	C₂²	1	8	yes	complemented
25	#13133187	C₂²	1	8	yes	—
26	C₄×Dic₁₀₆	C₂²	1	2	yes	complemented
27	C₄×D₂₁₂	C₂²	1	2	yes	complemented
28	C₄×D₂₁₂	C₂²	1	8	yes	complemented
29	#13133179	C₂²	1	8	yes	complemented
30	#13133179	C₂²	1	8	yes	complemented
31	C₄×C₂²⋊D₅₃	C₂²	1	4	yes	—
32	#13133176	C₂²	1	1	yes	characteristic
33	#13133174	C₂²	1	2	yes	complemented
34	C₂²⋊C₄×D₅₃	C₂²	1	8	yes	—
35	C₄²×D₅₃	C₂²	1	2	yes	complemented
36	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
37	#125×C₅₃	C₂²	1	2	yes	complemented
38	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	—
39	C₄×Q₈×C₅₃	C₂²	1	4	yes	complemented
40	#117×C₅₃	C₂²	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
42	C₂³×D₅₃	C₂³	1	4	yes	—
43	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
44	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
47	#11719949	—	4	16	no	—
48	#11719945	—	2	4	no	—
49	C₂×D₂₁₂	—	2	8	no	—
50	C₂×D₂₁₂	—	4	8	no	—
51	C₂×D₂₁₂	C₂³	1	4	yes	—
52	C₂×D₂₁₂	C₂³	1	8	yes	—
53	C₂×D₂₁₂	C₂³	1	4	yes	—
54	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
55	C₂×D₂₁₂	C₂³	1	16	yes	—
56	C₂×D₂₁₂	C₂³	1	8	yes	—
57	C₂×C₄×D₅₃	C₂³	1	2	yes	—
58	C₂×C₄×D₅₃	C₂³	1	4	yes	—
59	C₂×C₄×D₅₃	C₂³	1	16	yes	—
60	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
62	C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
63	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
64	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
65	C₄²×C₅₃	C₂³	1	2	yes	—
66	C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	C₄²×C₅₃	C₂³	1	4	yes	—
69	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
70	#11719921	C₂³	1	16	yes	—
71	#11719921	C₂³	1	4	yes	—
72	#11719921	C₂³	1	16	yes	—
73	C₄⋊Dic₅₃	C₂³	1	2	yes	—
74	#11719919	C₂³	1	4	yes	—
75	C₄×Dic₅₃	C₂³	1	2	yes	—
76	C₄×Dic₅₃	C₂³	1	4	yes	—
77	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
78	C₂²×D₅₃	—	2	8	no	—
79	C₂²×D₅₃	—	4	8	no	—
80	C₂²×D₅₃	C₂⁴	1	2	yes	—
81	C₂²×D₅₃	C₂⁴	1	8	yes	—
82	Q₈×C₅₃	—	2	8	no	—
83	C₂×C₄×C₅₃	—	2	4	no	—
84	C₂×C₄×C₅₃	#41	1	4	yes	complemented
85	C₂×C₄×C₅₃	C₂⁴	1	8	yes	—
86	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
87	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
90	C₂²⋊D₅₃	—	2	4	no	—
91	C₂×Dic₅₃	C₂⁴	1	2	yes	—
92	C₂×Dic₅₃	C₂⁴	1	4	yes	—
93	D₂₁₂	—	2	8	no	—
94	D₂₁₂	—	2	8	no	—
95	D₂₁₂	—	2	2	no	—
96	D₂₁₂	—	2	16	no	—
97	D₂₁₂	—	4	8	no	—
98	D₂₁₂	—	4	16	no	—
99	C₄×D₅₃	—	2	4	no	—
100	C₄×D₅₃	—	2	8	no	—
101	C₄×D₅₃	—	4	16	no	—
102	Dic₁₀₆	—	2	2	no	—
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	C₂×#41	1	2	yes	—
105	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
106	C₂×D₅₃	—	2	2	no	—
107	C₂×D₅₃	—	2	8	no	—
108	C₂×D₅₃	—	2	4	no	—
109	C₂×D₅₃	—	2	16	no	—
110	C₂×D₅₃	—	8	8	no	—
111	C₄×C₅₃	—	2	8	no	—
112	C₄×C₅₃	—	2	2	no	—
113	C₄×C₅₃	C₂×#41	1	2	yes	—
114	C₄×C₅₃	C₂×#41	1	2	yes	—
115	C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
116	Dic₅₃	—	2	2	no	—
117	Dic₅₃	—	2	4	no	—
118	#3508	—	53	1	no	maximal, Sylow
119	C₂×C₅₃	—	2	2	no	—
120	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
121	C₂×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
122	D₅₃	—	4	2	no	—
123	D₅₃	—	4	8	no	—
124	#559	—	53	4	no	—
125	#550	—	53	4	no	—
126	#546	—	53	8	no	—
127	#540	—	53	4	no	—
128	#535	—	53	2	no	—
129	#533	D₅₃	1	1	yes	characteristic, complemented
130	#532	—	53	4	no	—
131	C₂×#127	—	53	1	no	—
132	#518	—	53	2	no	—
133	C₄×#41	—	53	1	no	—
134	C₅₃	#3508	1	1	yes	characteristic, complemented, Sylow
135	C₂×#41	—	53	8	no	—
136	C₂×#41	—	53	1	no	—
137	C₂²×D₄	—	53	2	no	—
138	C₂²×D₄	—	53	4	no	—
139	C₄⋊Q₈	C₂×D₅₃	1	2	yes	complemented
140	#127	—	53	2	no	—
141	#127	—	53	2	no	—
142	#127	—	53	4	no	—
143	#127	—	53	4	no	—
144	#125	C₂×D₅₃	1	2	yes	complemented
145	#124	—	53	8	no	—
146	#124	—	53	4	no	—
147	#123	—	53	16	no	—
148	C₂²⋊Q₈	C₂×D₅₃	1	4	yes	—
149	#121	—	53	8	no	—
150	#121	—	53	8	no	—
151	#121	—	53	16	no	—
152	#120	—	53	8	no	—
153	C₄×Q₈	—	53	2	no	—
154	C₄×Q₈	C₂×D₅₃	1	4	yes	complemented
155	C₄×D₄	—	53	4	no	—
156	C₄×D₄	—	53	8	no	—
157	C₄×D₄	—	53	8	no	—
158	C₄×D₄	—	53	2	no	—
159	C₄×D₄	—	53	8	no	—
160	#117	—	53	1	no	—
161	#117	—	53	2	no	—
162	#117	C₂×D₅₃	1	2	yes	—
163	C₂×C₂²⋊C₄	—	53	8	no	—
164	C₂×C₄²	—	53	2	no	—
165	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
166	C₂⁴	—	53	4	no	—
167	#41	—	106	4	no	—
168	#41	—	212	16	no	—
169	C₂×Q₈	—	53	1	no	—
170	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
171	C₂×D₄	—	53	2	no	—
172	C₂×D₄	—	53	4	no	—
173	C₂×D₄	—	53	4	no	—
174	C₂×D₄	—	53	1	no	—
175	C₂×D₄	—	53	8	no	—
176	C₂×D₄	—	53	16	no	—
177	C₂×D₄	—	53	8	no	—
178	C₂×D₄	—	106	8	no	—
179	C₂×D₄	—	212	8	no	—
180	C₂²×C₄	—	53	2	no	—
181	C₂²×C₄	—	53	4	no	—
182	C₂²×C₄	—	53	16	no	—
183	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
184	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
185	C₄⋊C₄	—	53	2	no	—
186	C₄⋊C₄	—	53	4	no	—
187	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
188	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
189	C₄⋊C₄	C₂²×D₅₃	1	8	yes	—
190	C₂²⋊C₄	—	53	2	no	—
191	C₂²⋊C₄	—	53	16	no	—
192	C₂²⋊C₄	—	53	4	no	—
193	C₂²⋊C₄	—	53	16	no	—
194	C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
195	C₄²	—	53	2	no	—
196	C₄²	—	53	4	no	—
197	C₄²	C₂²×D₅₃	1	1	yes	characteristic
198	C₄²	C₂²×D₅₃	1	1	yes	characteristic
199	C₄²	C₂²×D₅₃	1	2	yes	—
200	C₄²	C₂²×D₅₃	1	4	yes	—
201	C₂³	—	53	8	no	—
202	C₂³	—	53	2	no	—
203	C₂³	—	106	8	no	—
204	C₂³	—	212	8	no	—
205	C₂³	C₂³×D₅₃	1	1	yes	characteristic
206	Q₈	—	2	8	no	—
207	Q₈	—	106	2	no	—
208	D₄	—	106	4	no	—
209	D₄	—	106	8	no	—
210	D₄	—	106	8	no	—
211	D₄	—	106	2	no	—
212	D₄	—	106	16	no	—
213	D₄	—	212	8	no	—
214	D₄	—	212	16	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	53	2	no	—
217	C₂×C₄	—	53	4	no	—
218	C₂×C₄	—	106	4	no	—
219	C₂×C₄	—	106	8	no	—
220	C₂×C₄	—	212	16	no	—
221	C₂×C₄	#11719949	1	4	yes	complemented
222	C₂×C₄	C₂³×D₅₃	1	2	yes	—
223	C₂×C₄	C₂³×D₅₃	1	8	yes	—
224	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
225	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
226	C₂×C₄	C₂³×D₅₃	1	2	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	106	2	no	—
229	C₂²	—	106	8	no	—
230	C₂²	—	106	4	no	—
231	C₂²	—	106	16	no	—
232	C₂²	—	424	8	no	—
233	C₂²	C₂×#11719949	1	2	yes	—
234	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	8	no	—
237	C₄	—	106	2	no	—
238	C₄	—	106	4	no	—
239	C₄	C₂×#11719949	1	2	yes	—
240	C₄	C₂×#11719949	1	2	yes	—
241	C₄	#13133263	1	4	yes	—
242	C₂	—	2	2	no	—
243	C₂	—	212	8	no	—
244	C₂	—	212	2	no	—
245	C₂	C₂²×#11719949	1	1	yes	characteristic
246	C₂	C₂×#13133263	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#533	C₂
hypercenter	#533	D₅₃
hypocenter	C₅₃	#3508
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. #16811358	C₂	semidirect
2. #16811295	C₂	semidirect
3. #16811264	C₂	semidirect
4. #16811229	C₂	semidirect
5. #16811192	C₂	semidirect
6. #16811174	C₂	semidirect
7. C₂×#13133214	C₂	semidirect
8. #16811097	C₂	semidirect
9. C₄×#11719945	C₂	semidirect
10. C₅₃×#533	C₂	semidirect
11. C₂×#11719949	C₂²	semidirect
18. #13133214	C₂²	semidirect
19. #13133206	C₂²	semidirect
20. #13133205	C₂²	semidirect
21. #13133202	C₂²	semidirect
22. #13133192	C₂²	semidirect
23. #13133192	C₂²	semidirect
24. #13133192	C₂²	semidirect
26. C₄×Dic₁₀₆	C₂²	semidirect
27. C₄×D₂₁₂	C₂²	semidirect
28. C₄×D₂₁₂	C₂²	semidirect
29. #13133179	C₂²	semidirect
30. #13133179	C₂²	semidirect
33. #13133174	C₂²	semidirect
35. C₄²×D₅₃	C₂²	semidirect
36. C₄⋊Q₈×C₅₃	C₂²	semidirect
37. #125×C₅₃	C₂²	semidirect
39. C₄×Q₈×C₅₃	C₂²	semidirect
84. C₂×C₄×C₅₃	#41	semidirect
129. #533	D₅₃	semidirect
134. C₅₃	#3508	semidirect
139. C₄⋊Q₈	C₂×D₅₃	semidirect
144. #125	C₂×D₅₃	semidirect
154. C₄×Q₈	C₂×D₅₃	semidirect
221. C₂×C₄	#11719949	semidirect