Subgroups — #20283723

Order 6784.

406 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11719947	C₂	1	1	yes	maximal, characteristic
2	#16810753	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810750	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810694	C₂	1	2	yes	maximal, complemented
5	#16810691	C₂	1	2	yes	maximal, complemented
6	D₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810679	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810677	C₂	1	2	yes	maximal, complemented
9	C₄×#11719912	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133127	C₂	1	2	yes	maximal, complemented
11	C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719947	C₄	1	1	yes	characteristic
13	C₂×#11719912	C₂²	1	1	yes	characteristic
14	C₂×#11719912	C₂²	1	1	yes	characteristic
15	C₂²×#88322	C₂²	1	2	yes	—
16	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
17	#13133182	C₄	1	1	yes	characteristic
18	D₄×Dic₅₃	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
20	#13133177	C₄	1	2	yes	—
21	#13133175	C₄	1	1	yes	characteristic
22	#13133173	C₄	1	2	yes	—
23	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
24	C₄²×D₅₃	C₂²	1	1	yes	characteristic
25	C₄⋊#88322	C₂²	1	1	yes	characteristic
26	#13133146	C₂²	1	2	yes	—
27	#13133146	C₂²	1	2	yes	—
28	C₂²⋊#88322	C₂²	1	2	yes	—
29	#13133129	C₂²	1	2	yes	—
30	#13133127	C₂²	1	2	yes	—
31	#13133127	C₂²	1	1	yes	characteristic
32	#13133127	C₂²	1	2	yes	—
33	#13133127	C₂²	1	2	yes	—
34	#13133127	C₂²	1	1	yes	characteristic
35	C₄×#88322	C₂²	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
37	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
40	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
44	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
45	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
46	#11719947	—	2	4	no	—
47	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
50	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	—	2	4	no	—
52	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
57	C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
59	#11719921	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
61	#11719919	C₂×C₄	1	2	yes	—
62	#11719919	C₂×C₄	1	2	yes	—
63	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
66	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
67	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
68	C₄×Dic₅₃	C₂³	1	2	yes	—
69	C₂×#88322	—	2	4	no	—
70	C₂×#88322	C₂³	1	1	yes	characteristic
71	C₂×#88322	C₂³	1	2	yes	—
72	C₂×#88322	C₂³	1	1	yes	characteristic
73	#11719912	—	2	2	no	—
74	#11719912	—	2	2	no	—
75	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
77	D₄×C₅₃	M₁₆	1	4	yes	complemented
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
84	C₂²⋊D₅₃	—	2	4	no	—
85	C₂×Dic₅₃	—	2	4	no	—
86	C₂×Dic₅₃	—	4	2	no	—
87	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
88	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
92	C₄×D₅₃	—	2	2	no	—
93	C₄×D₅₃	—	2	1	no	—
94	C₄×D₅₃	—	2	1	no	—
95	C₄×D₅₃	—	4	1	no	—
96	Dic₁₀₆	—	2	2	no	—
97	C₈×C₅₃	—	2	1	no	—
98	C₈×C₅₃	—	2	2	no	—
99	C₈×C₅₃	#41	1	2	yes	complemented
100	#88322	—	2	1	no	—
101	#88322	—	2	2	no	—
102	#88322	#41	1	2	yes	complemented
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	C₂×M₁₆	1	4	yes	—
106	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×D₅₃	—	2	1	no	—
108	C₂×D₅₃	—	2	1	no	—
109	C₂×D₅₃	—	2	1	no	—
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
113	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
114	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
115	Dic₅₃	—	2	1	no	—
116	Dic₅₃	—	2	2	no	—
117	Dic₅₃	—	2	1	no	—
118	Dic₅₃	—	2	1	no	—
119	Dic₅₃	—	2	2	no	—
120	#2970	—	53	1	no	maximal, Sylow
121	C₂×C₅₃	—	2	4	no	—
122	C₂×C₅₃	C₄×#41	1	1	yes	characteristic
123	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
124	C₂×C₅₃	#568	1	1	yes	characteristic
125	D₅₃	—	4	1	no	—
126	C₄×#41	—	53	1	no	—
127	#446	—	53	1	no	—
128	#446	—	53	1	no	—
129	#435	—	53	2	no	—
130	#435	—	53	2	no	—
131	C₈×D₄	—	53	1	no	—
132	C₈×D₄	D₅₃	1	1	yes	characteristic, complemented
133	#432	—	53	1	no	—
134	#432	—	53	2	no	—
135	C₄×M₁₆	—	53	1	no	—
136	C₂×#97	—	53	2	no	—
137	C₅₃	#2970	1	1	yes	characteristic, complemented, Sylow
138	C₂×#41	—	53	1	no	—
139	C₂×M₁₆	—	53	1	no	—
140	C₂×M₁₆	—	53	1	no	—
141	C₂²×C₈	—	53	2	no	—
142	C₂²×C₈	C₂×D₅₃	1	2	yes	—
143	C₄×Q₈	—	53	1	no	—
144	C₄×Q₈	—	53	1	no	—
145	C₄×D₄	—	53	1	no	—
146	C₄×D₄	—	53	2	no	—
147	C₄×D₄	—	53	2	no	—
148	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
149	#117	—	53	1	no	—
150	#117	—	53	2	no	—
151	C₂×C₄²	—	53	2	no	—
152	C₂×C₄²	—	53	1	no	—
153	C₄⋊C₈	—	53	2	no	—
154	C₄⋊C₈	—	53	2	no	—
155	C₄⋊C₈	—	53	1	no	—
156	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
157	C₂²⋊C₈	—	53	2	no	—
158	C₂²⋊C₈	—	53	2	no	—
159	C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
160	#97	—	53	1	no	—
161	#97	—	53	2	no	—
162	#97	—	53	2	no	—
163	#97	—	53	2	no	—
164	#97	—	53	1	no	—
165	C₄×C₈	—	53	1	no	—
166	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
167	#41	—	106	4	no	—
168	C₂×Q₈	—	53	1	no	—
169	C₂×D₄	—	53	2	no	—
170	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
171	C₂²×C₄	—	53	2	no	—
172	C₂²×C₄	—	53	2	no	—
173	C₂²×C₄	—	53	1	no	—
174	C₂²×C₄	—	53	1	no	—
175	C₂²×C₄	—	53	1	no	—
176	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
177	M₁₆	—	106	2	no	—
178	M₁₆	—	106	2	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	53	1	no	—
181	C₂×C₈	—	53	1	no	—
182	C₂×C₈	—	53	2	no	—
183	C₂×C₈	—	106	4	no	—
184	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
185	C₂×C₈	C₂²×D₅₃	1	2	yes	—
186	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
187	C₄⋊C₄	—	53	2	no	—
188	C₄⋊C₄	—	53	2	no	—
189	C₄⋊C₄	—	53	1	no	—
190	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
191	C₂²⋊C₄	—	53	2	no	—
192	C₂²⋊C₄	—	53	2	no	—
193	C₂²⋊C₄	C₄×D₅₃	1	2	yes	—
194	C₄²	—	53	2	no	—
195	C₄²	—	53	1	no	—
196	C₄²	—	53	1	no	—
197	C₄²	—	53	2	no	—
198	C₄²	—	53	1	no	—
199	C₄²	—	53	2	no	—
200	C₄²	C₂²×D₅₃	1	1	yes	characteristic
201	C₂³	—	53	1	no	—
202	C₂³	C₂×C₄×D₅₃	1	2	yes	—
203	Q₈	—	106	2	no	—
204	D₄	—	106	4	no	—
205	D₄	#11719912	1	4	yes	complemented
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	53	1	no	—
209	C₂×C₄	—	53	1	no	—
210	C₂×C₄	—	53	1	no	—
211	C₂×C₄	—	53	2	no	—
212	C₂×C₄	—	53	2	no	—
213	C₂×C₄	—	106	4	no	—
214	C₂×C₄	—	106	2	no	—
215	C₂×C₄	—	106	1	no	—
216	C₂×C₄	—	106	1	no	—
217	C₂×C₄	—	212	1	no	—
218	C₂×C₄	—	212	2	no	—
219	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
220	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
222	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	2	no	—
225	C₈	—	53	2	no	—
226	C₈	—	106	2	no	—
227	C₈	—	106	1	no	—
228	C₈	#11719947	1	2	yes	complemented
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	106	1	no	—
232	C₂²	—	106	1	no	—
233	C₂²	—	106	1	no	—
234	C₂²	C₂×#11719912	1	4	yes	—
235	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	106	1	no	—
239	C₄	—	106	1	no	—
240	C₄	—	106	2	no	—
241	C₄	—	106	1	no	—
242	C₄	—	106	2	no	—
243	C₄	C₂×#11719912	1	2	yes	—
244	C₄	#41×D₅₃	1	1	yes	characteristic
245	C₄	C₂×#11719947	1	1	yes	characteristic
246	C₂	—	2	4	no	—
247	C₂	—	212	1	no	—
248	C₂	C₄×#11719947	1	1	yes	characteristic
249	C₂	C₂²×#11719912	1	1	yes	characteristic
250	C₂	#16811388	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₈×D₄×C₅₃	C₂
hypercenter	C₈×D₄	D₅₃
hypocenter	C₅₃	#2970
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #16810753	C₂	semidirect
3. #16810750	C₂	semidirect
4. #16810694	C₂	semidirect
5. #16810691	C₂	semidirect
6. D₄×#88322	C₂	semidirect
7. #16810679	C₂	semidirect
8. #16810677	C₂	semidirect
9. C₄×#11719912	C₂	semidirect
10. C₂×#13133127	C₂	semidirect
11. C₈×D₄×C₅₃	C₂	semidirect
77. D₄×C₅₃	M₁₆	semidirect
99. C₈×C₅₃	#41	semidirect
102. #88322	#41	semidirect
132. C₈×D₄	D₅₃	semidirect
137. C₅₃	#2970	semidirect
205. D₄	#11719912	semidirect
228. C₈	#11719947	semidirect