Subgroups — #20283379

Order 6784.

392 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811091	C₂	1	1	yes	maximal, characteristic
2	#16810685	C₂	1	2	yes	maximal, complemented
3	#16810682	C₂	1	2	yes	maximal, complemented
4	#16810678	C₂	1	2	yes	maximal, complemented
5	#16810674	C₂	1	2	yes	maximal, complemented
6	#16810646	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13133146	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810591	C₂	1	1	yes	maximal, characteristic
9	#16810589	C₂	1	1	yes	maximal, characteristic
10	#16810579	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	1	yes	characteristic
13	C₂×#11719917	C₂²	1	1	yes	characteristic
14	C₂²×#88322	C₂²	1	1	yes	characteristic
15	#13133182	C₄	1	1	yes	characteristic
16	#13133182	C₄	1	1	yes	characteristic
17	#13133179	C₄	1	1	yes	characteristic
18	#13133179	C₄	1	1	yes	characteristic
19	#13133177	C₄	1	2	yes	—
20	#13133177	C₄	1	2	yes	—
21	#13133174	C₂²	1	2	yes	—
22	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
23	C₄²×D₅₃	C₂²	1	2	yes	—
24	#13133146	C₂²	1	1	yes	characteristic
25	#13133146	C₂²	1	1	yes	characteristic
26	#13133146	C₂²	1	1	yes	characteristic
27	#13133146	C₂²	1	1	yes	characteristic
28	#13133146	C₂²	1	2	yes	—
29	#13133146	C₂²	1	2	yes	—
30	#13133129	C₂²	1	2	yes	—
31	#13133129	C₂²	1	2	yes	—
32	#13133129	C₂²	1	2	yes	—
33	#13133129	C₂²	1	2	yes	—
34	#13133128	C₂²	1	2	yes	—
35	C₄×#88322	C₂²	1	2	yes	—
36	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
38	#117×C₅₃	C₂²	1	1	yes	characteristic
39	#97×C₅₃	C₂²	1	2	yes	—
40	C₄×C₈×C₅₃	C₂²	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
43	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
45	#11719945	—	2	2	no	—
46	#11719945	—	2	2	no	—
47	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₅₃	C₂³	1	2	yes	—
51	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
52	M₁₆×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂³	1	2	yes	—
56	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
60	C₄²×C₅₃	C₂³	1	2	yes	—
61	#11719921	C₂×C₄	1	2	yes	—
62	#11719921	C₂×C₄	1	2	yes	—
63	#11719919	C₂×C₄	1	2	yes	—
64	#11719919	C₂×C₄	1	2	yes	—
65	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
66	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
67	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
68	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
70	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
71	#11719917	—	2	2	no	—
72	C₂×#88322	—	2	2	no	—
73	C₂×#88322	C₂³	1	1	yes	characteristic
74	C₂×#88322	C₂³	1	2	yes	—
75	C₂×#88322	C₂³	1	1	yes	characteristic
76	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#41	1	2	yes	—
82	C₂×C₄×C₅₃	#41	1	2	yes	—
83	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
84	C₂²⋊D₅₃	—	2	4	no	—
85	C₂×Dic₅₃	—	2	2	no	—
86	C₂×Dic₅₃	—	4	1	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	1	yes	characteristic
92	D₂₁₂	—	2	2	no	—
93	C₄×D₅₃	—	2	2	no	—
94	C₄×D₅₃	—	2	2	no	—
95	C₄×D₅₃	—	2	4	no	—
96	C₄×D₅₃	—	4	2	no	—
97	Dic₁₀₆	—	2	2	no	—
98	C₈×C₅₃	—	2	1	no	—
99	C₈×C₅₃	—	2	1	no	—
100	C₈×C₅₃	—	2	2	no	—
101	#88322	—	2	1	no	—
102	#88322	—	2	1	no	—
103	#88322	—	2	2	no	—
104	C₂²×C₅₃	—	2	1	no	—
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	#117	1	2	yes	—
107	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×D₅₃	—	2	2	no	—
109	C₂×D₅₃	—	2	2	no	—
110	C₂×D₅₃	—	2	2	no	—
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	—	2	2	no	—
113	C₄×C₅₃	#117	1	2	yes	—
114	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
115	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
116	Dic₅₃	—	2	2	no	—
117	Dic₅₃	—	2	1	no	—
118	Dic₅₃	—	2	1	no	—
119	Dic₅₃	—	2	1	no	—
120	Dic₅₃	#131	1	2	yes	—
121	#2925	—	53	1	no	maximal, Sylow
122	C₂×C₅₃	—	2	2	no	—
123	C₂×C₅₃	C₂×#117	1	1	yes	characteristic
124	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
125	C₂×C₅₃	#568	1	1	yes	characteristic
126	D₅₃	—	4	2	no	—
127	C₄×#41	—	53	1	no	—
128	#433	—	53	2	no	—
129	#433	—	53	2	no	—
130	#432	—	53	2	no	—
131	#431	—	53	2	no	—
132	#423	—	53	1	no	—
133	C₂×C₄⋊C₈	—	53	1	no	—
134	#408	—	53	1	no	—
135	#408	—	53	1	no	—
136	#405	—	53	1	no	—
137	#405	D₅₃	1	1	yes	characteristic, complemented
138	C₅₃	#2925	1	1	yes	characteristic, complemented, Sylow
139	C₂×#41	—	53	1	no	—
140	C₂×M₁₆	—	53	1	no	—
141	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
142	C₂²×C₈	—	53	1	no	—
143	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
144	C₄×Q₈	—	53	1	no	—
145	C₄×Q₈	—	53	1	no	—
146	C₄×D₄	—	53	1	no	—
147	C₄×D₄	—	53	1	no	—
148	C₄×D₄	—	53	2	no	—
149	C₄×D₄	—	53	2	no	—
150	#117	—	53	2	no	—
151	#117	C₂×D₅₃	1	1	yes	characteristic
152	C₂×C₄²	—	53	1	no	—
153	C₂×C₄²	—	53	2	no	—
154	C₄⋊C₈	—	53	1	no	—
155	C₄⋊C₈	—	53	1	no	—
156	C₄⋊C₈	—	53	1	no	—
157	C₄⋊C₈	—	53	1	no	—
158	C₄⋊C₈	—	53	2	no	—
159	C₄⋊C₈	—	53	2	no	—
160	C₂²⋊C₈	—	53	2	no	—
161	C₂²⋊C₈	—	53	2	no	—
162	C₂²⋊C₈	—	53	2	no	—
163	C₂²⋊C₈	—	53	2	no	—
164	#97	—	53	2	no	—
165	#97	C₂×D₅₃	1	2	yes	—
166	C₄×C₈	—	53	2	no	—
167	C₄×C₈	C₂×D₅₃	1	2	yes	—
168	#41	—	106	2	no	—
169	#41	—	106	2	no	—
170	C₂×Q₈	—	53	1	no	—
171	C₂×D₄	—	53	2	no	—
172	C₂×D₄	—	53	1	no	—
173	C₂²×C₄	—	53	1	no	—
174	C₂²×C₄	—	53	1	no	—
175	C₂²×C₄	—	53	2	no	—
176	C₂²×C₄	—	53	2	no	—
177	C₂²×C₄	—	53	2	no	—
178	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
179	M₁₆	—	2	2	no	—
180	M₁₆	—	106	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	53	1	no	—
183	C₂×C₈	—	53	2	no	—
184	C₂×C₈	—	53	1	no	—
185	C₂×C₈	—	106	2	no	—
186	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
187	C₂×C₈	C₂²×D₅₃	1	2	yes	—
188	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
189	C₄⋊C₄	—	53	2	no	—
190	C₄⋊C₄	—	53	2	no	—
191	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
192	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
193	C₂²⋊C₄	—	53	2	no	—
194	C₂²⋊C₄	—	53	2	no	—
195	C₂²⋊C₄	C₄×D₅₃	1	2	yes	—
196	C₄²	—	53	1	no	—
197	C₄²	—	53	2	no	—
198	C₄²	—	53	1	no	—
199	C₄²	—	53	2	no	—
200	C₄²	—	53	1	no	—
201	C₄²	—	53	1	no	—
202	C₄²	C₂²×D₅₃	1	2	yes	—
203	C₂³	—	53	2	no	—
204	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
205	Q₈	—	106	2	no	—
206	D₄	—	106	4	no	—
207	D₄	—	106	2	no	—
208	C₂×C₄	—	53	1	no	—
209	C₂×C₄	—	53	1	no	—
210	C₂×C₄	—	53	1	no	—
211	C₂×C₄	—	53	1	no	—
212	C₂×C₄	—	53	2	no	—
213	C₂×C₄	—	106	4	no	—
214	C₂×C₄	—	106	2	no	—
215	C₂×C₄	—	106	2	no	—
216	C₂×C₄	—	106	2	no	—
217	C₂×C₄	—	212	2	no	—
218	C₂×C₄	—	212	1	no	—
219	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
220	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
221	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
222	C₂×C₄	#11719945	1	2	yes	—
223	C₂×C₄	#11719945	1	2	yes	—
224	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	2	2	no	—
228	C₈	—	106	1	no	—
229	C₈	—	106	1	no	—
230	C₈	—	106	2	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	106	2	no	—
234	C₂²	—	106	2	no	—
235	C₂²	—	106	2	no	—
236	C₂²	#13133174	1	2	yes	—
237	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	53	2	no	—
241	C₄	—	106	2	no	—
242	C₄	—	106	1	no	—
243	C₄	—	106	1	no	—
244	C₄	—	106	1	no	—
245	C₄	#13133174	1	2	yes	—
246	C₄	C₂×#11719945	1	1	yes	characteristic
247	C₄	C₂×#11719945	1	1	yes	characteristic
248	C₂	—	2	2	no	—
249	C₂	—	212	2	no	—
250	C₂	C₂×#13133174	1	1	yes	characteristic
251	C₂	#131×D₅₃	1	1	yes	characteristic
252	C₂	#16811388	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#405	C₂
hypercenter	#405	D₅₃
hypocenter	C₅₃	#2925
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #16810685	C₂	semidirect
3. #16810682	C₂	semidirect
4. #16810678	C₂	semidirect
5. #16810674	C₂	semidirect
6. #16810646	C₂	semidirect
7. C₂×#13133146	C₂	semidirect
10. #16810579	C₂	semidirect
11. C₅₃×#405	C₂	semidirect
137. #405	D₅₃	semidirect
138. C₅₃	#2925	semidirect