Subgroups — #20283689

Order 6784.

380 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811160	C₂	1	1	yes	maximal, characteristic
2	Q₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810702	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810696	C₂	1	2	yes	maximal, complemented
5	#16810688	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810688	C₂	1	2	yes	maximal, complemented
7	#16810683	C₂	1	2	yes	maximal, complemented
8	#16810679	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810580	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719912	C₂²	1	2	yes	—
13	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₅₃	C₂²	1	2	yes	—
16	#13133212	C₄	1	2	yes	—
17	#13133208	C₄	1	1	yes	characteristic
18	#13133206	C₄	1	1	yes	characteristic
19	#13133202	C₄	1	2	yes	—
20	#13133193	C₄	1	1	yes	characteristic
21	#13133192	C₄	1	1	yes	characteristic
22	C₄×D₂₁₂	C₂²	1	2	yes	—
23	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
24	#13133174	C₂²	1	2	yes	—
25	C₄²×D₅₃	C₂²	1	1	yes	characteristic
26	C₄⋊#88322	C₂²	1	1	yes	characteristic
27	C₄⋊#88322	C₂²	1	2	yes	—
28	#13133129	C₂²	1	2	yes	—
29	#13133129	C₂²	1	2	yes	—
30	#13133129	C₂²	1	2	yes	—
31	#13133127	C₂²	1	2	yes	—
32	C₄×#88322	C₂²	1	1	yes	characteristic
33	C₄×#88322	C₂²	1	2	yes	—
34	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
35	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
39	#97×C₅₃	C₂²	1	2	yes	—
40	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
43	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₅₃	C₂³	1	2	yes	—
48	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂³	1	2	yes	—
50	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
53	C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₄²×C₅₃	C₂³	1	2	yes	—
55	#11719921	C₂×C₄	1	2	yes	—
56	#11719921	C₂×C₄	1	2	yes	—
57	#11719921	C₂×C₄	1	2	yes	—
58	#11719921	C₂×C₄	1	2	yes	—
59	#11719921	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
62	#11719919	C₂×C₄	1	2	yes	—
63	#11719919	C₂×C₄	1	2	yes	—
64	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
67	C₂×#88322	C₂³	1	1	yes	characteristic
68	C₂×#88322	C₂³	1	2	yes	—
69	C₂×#88322	C₂³	1	1	yes	characteristic
70	#11719912	—	2	4	no	—
71	C₈×D₅₃	—	2	2	no	—
72	C₈×D₅₃	—	2	2	no	—
73	C₈×D₅₃	—	2	4	no	—
74	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
75	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
86	D₂₁₂	—	2	4	no	—
87	D₂₁₂	—	2	2	no	—
88	C₄×D₅₃	—	2	2	no	—
89	C₄×D₅₃	—	2	2	no	—
90	C₄×D₅₃	—	2	1	no	—
91	C₄×D₅₃	—	2	2	no	—
92	C₄×D₅₃	—	2	1	no	—
93	C₄×D₅₃	—	2	2	no	—
94	C₈×C₅₃	—	2	1	no	—
95	C₈×C₅₃	—	2	2	no	—
96	C₈×C₅₃	#41	1	2	yes	—
97	#88322	—	2	1	no	—
98	#88322	—	2	2	no	—
99	#88322	#41	1	2	yes	—
100	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₅₃	—	2	1	no	—
102	C₂×D₅₃	—	2	1	no	—
103	C₂×D₅₃	—	2	2	no	—
104	C₂×D₅₃	—	2	2	no	—
105	C₂×D₅₃	—	2	2	no	—
106	C₂×D₅₃	#131	1	2	yes	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	2	2	no	—
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	#131	1	2	yes	—
111	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
112	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
113	Dic₅₃	—	2	1	no	—
114	Dic₅₃	—	2	2	no	—
115	Dic₅₃	—	2	1	no	—
116	Dic₅₃	—	2	1	no	—
117	#2966	—	53	1	no	maximal, Sylow
118	C₂×C₅₃	C₄×#41	1	1	yes	characteristic
119	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
120	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
121	D₅₃	—	2	2	no	—
122	D₅₃	—	4	2	no	—
123	#529	—	53	1	no	—
124	#446	D₅₃	1	1	yes	characteristic, complemented
125	C₈×Q₈	—	53	1	no	—
126	#436	—	53	1	no	—
127	#435	—	53	2	no	—
128	C₈×D₄	—	53	1	no	—
129	C₈×D₄	—	53	2	no	—
130	#433	—	53	2	no	—
131	#432	—	53	1	no	—
132	#405	—	53	2	no	—
133	C₂×C₄×C₈	—	53	1	no	—
134	C₅₃	#2966	1	1	yes	characteristic, complemented, Sylow
135	C₂×M₁₆	—	53	2	no	—
136	C₂²×C₈	—	53	1	no	—
137	C₂²×C₈	—	53	1	no	—
138	C₂²×C₈	—	53	2	no	—
139	#126	—	53	2	no	—
140	#125	—	53	1	no	—
141	#124	—	53	1	no	—
142	#123	—	53	2	no	—
143	C₂²⋊Q₈	—	53	1	no	—
144	#121	—	53	1	no	—
145	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
146	C₄×D₄	—	53	2	no	—
147	C₄×D₄	—	53	1	no	—
148	#117	—	53	2	no	—
149	C₂×C₄²	—	53	1	no	—
150	C₄⋊C₈	—	53	1	no	—
151	C₄⋊C₈	—	53	2	no	—
152	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
153	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
154	C₂²⋊C₈	—	53	2	no	—
155	C₂²⋊C₈	—	53	2	no	—
156	C₂²⋊C₈	—	53	2	no	—
157	#97	—	53	2	no	—
158	#97	C₂×D₅₃	1	2	yes	—
159	C₄×C₈	—	53	1	no	—
160	C₄×C₈	—	53	1	no	—
161	C₄×C₈	—	53	1	no	—
162	C₄×C₈	—	53	2	no	—
163	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
164	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
165	C₂×D₄	—	53	2	no	—
166	C₂×D₄	—	53	1	no	—
167	C₂²×C₄	—	53	1	no	—
168	C₂²×C₄	—	53	1	no	—
169	C₂²×C₄	—	53	1	no	—
170	C₂²×C₄	—	53	2	no	—
171	M₁₆	—	106	4	no	—
172	C₂×C₈	—	53	1	no	—
173	C₂×C₈	—	53	1	no	—
174	C₂×C₈	—	53	2	no	—
175	C₂×C₈	—	106	2	no	—
176	C₂×C₈	—	106	4	no	—
177	C₂×C₈	—	106	2	no	—
178	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
179	C₂×C₈	C₂²×D₅₃	1	2	yes	—
180	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
181	C₄⋊C₄	—	53	2	no	—
182	C₄⋊C₄	—	53	2	no	—
183	C₄⋊C₄	—	53	1	no	—
184	C₄⋊C₄	—	53	2	no	—
185	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
186	C₄⋊C₄	C₄×D₅₃	1	2	yes	—
187	C₂²⋊C₄	—	53	2	no	—
188	C₂²⋊C₄	—	53	2	no	—
189	C₂²⋊C₄	—	53	2	no	—
190	C₂²⋊C₄	—	53	2	no	—
191	C₂²⋊C₄	—	53	2	no	—
192	C₄²	—	53	1	no	—
193	C₄²	—	53	1	no	—
194	C₄²	—	53	1	no	—
195	C₄²	C₂²×D₅₃	1	1	yes	characteristic
196	C₄²	C₂²×D₅₃	1	2	yes	—
197	C₂³	—	53	1	no	—
198	C₂³	—	53	2	no	—
199	Q₈	—	2	2	no	—
200	D₄	—	106	4	no	—
201	D₄	—	106	2	no	—
202	C₂×C₄	—	53	1	no	—
203	C₂×C₄	—	53	1	no	—
204	C₂×C₄	—	53	1	no	—
205	C₂×C₄	—	53	2	no	—
206	C₂×C₄	—	106	2	no	—
207	C₂×C₄	—	106	2	no	—
208	C₂×C₄	—	106	1	no	—
209	C₂×C₄	—	106	2	no	—
210	C₂×C₄	—	106	1	no	—
211	C₂×C₄	—	106	2	no	—
212	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
214	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
215	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	2	no	—
219	C₈	—	53	2	no	—
220	C₈	—	106	1	no	—
221	C₈	—	106	2	no	—
222	C₈	#11719949	1	2	yes	—
223	C₂²	—	53	2	no	—
224	C₂²	—	106	1	no	—
225	C₂²	—	106	1	no	—
226	C₂²	—	106	2	no	—
227	C₂²	—	106	2	no	—
228	C₂²	—	106	2	no	—
229	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	106	1	no	—
234	C₄	—	106	2	no	—
235	C₄	—	106	1	no	—
236	C₄	—	106	1	no	—
237	C₄	#13133225	1	2	yes	—
238	C₄	#41×D₅₃	1	1	yes	characteristic
239	C₄	C₂×#11719949	1	1	yes	characteristic
240	C₂	—	106	2	no	—
241	C₂	—	212	2	no	—
242	C₂	C₄×#11719949	1	1	yes	characteristic
243	C₂	#131×D₅₃	1	1	yes	characteristic
244	C₂	C₂×#13133225	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#446	C₂
hypercenter	#446	D₅₃
hypocenter	C₅₃	#2966
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. Q₈×#88322	C₂	semidirect
3. #16810702	C₂	semidirect
4. #16810696	C₂	semidirect
5. #16810688	C₂	semidirect
6. #16810688	C₂	semidirect
7. #16810683	C₂	semidirect
8. #16810679	C₂	semidirect
9. #16810580	C₂	semidirect
10. C₄×C₈×D₅₃	C₂	semidirect
11. C₅₃×#446	C₂	semidirect
124. #446	D₅₃	semidirect
134. C₅₃	#2966	semidirect