Subgroups — #20283813

Order 6784.

330 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811177	C₂	1	1	yes	maximal, characteristic
2	#16810751	C₂	1	2	yes	maximal, complemented
3	#16810750	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810747	C₂	1	2	yes	maximal, complemented
7	#16810685	C₂	1	2	yes	maximal
8	#16810683	C₂	1	2	yes	maximal
9	#16810675	C₂	1	1	yes	maximal, characteristic
10	#16810674	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	#13133216	C₄	1	1	yes	characteristic
13	#13133215	C₄	1	1	yes	characteristic
14	#13133207	C₄	1	2	yes	—
15	#13133197	C₄	1	1	yes	characteristic
16	#13133195	C₄	1	2	yes	—
17	#13133193	C₄	1	1	yes	characteristic
18	C₄×Dic₁₀₆	C₂²	1	2	yes	—
19	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
20	#13133174	C₂²	1	2	yes	—
21	C₄²×D₅₃	C₂²	1	1	yes	characteristic
22	C₄⋊#88322	C₂²	1	1	yes	characteristic
23	C₄⋊#88322	C₂²	1	2	yes	—
24	#13133146	C₂²	1	1	yes	characteristic
25	#13133146	C₂²	1	1	yes	characteristic
26	#13133146	C₂²	1	2	yes	—
27	#13133146	C₂²	1	2	yes	—
28	#13133129	C₂²	1	1	yes	characteristic
29	#13133129	C₂²	1	1	yes	characteristic
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	2	yes	—
35	C₈×Dic₅₃	C₂²	1	2	yes	—
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×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
46	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂³	1	2	yes	—
50	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
51	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₄²×C₅₃	C₂³	1	2	yes	—
54	#11719921	C₂×C₄	1	2	yes	—
55	#11719921	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
58	#11719919	C₂×C₄	1	2	yes	—
59	#11719919	C₂×C₄	1	2	yes	—
60	#11719919	C₂×C₄	1	2	yes	—
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	C₂³	1	1	yes	characteristic
66	C₄×Dic₅₃	C₂³	1	2	yes	—
67	C₂×#88322	C₂³	1	2	yes	—
68	C₂×#88322	C₂³	1	1	yes	characteristic
69	C₂×#88322	C₂³	1	1	yes	characteristic
70	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
71	Q₈×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
77	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
78	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
79	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₄×D₅₃	—	2	2	no	—
83	C₄×D₅₃	—	2	1	no	—
84	C₄×D₅₃	—	2	1	no	—
85	C₄×D₅₃	—	2	2	no	—
86	Dic₁₀₆	—	2	2	no	—
87	Dic₁₀₆	—	2	4	no	—
88	C₈×C₅₃	—	2	1	no	—
89	C₈×C₅₃	—	2	2	no	—
90	C₈×C₅₃	—	2	1	no	—
91	#88322	—	2	1	no	—
92	#88322	—	2	2	no	—
93	#88322	—	2	1	no	—
94	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×D₅₃	—	2	1	no	—
96	C₂×D₅₃	—	2	1	no	—
97	C₂×D₅₃	—	2	1	no	—
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	2	no	—
101	C₄×C₅₃	#131	1	2	yes	—
102	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
104	Dic₅₃	—	2	1	no	—
105	Dic₅₃	—	2	2	no	—
106	Dic₅₃	—	2	2	no	—
107	Dic₅₃	—	2	1	no	—
108	Dic₅₃	#131	1	2	yes	—
109	#2986	—	53	1	no	maximal, Sylow
110	C₂×C₅₃	#519	1	1	yes	characteristic
111	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
112	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
113	D₅₃	—	4	1	no	—
114	#533	—	53	1	no	—
115	#446	—	53	1	no	—
116	#446	—	53	2	no	—
117	#446	D₅₃	1	1	yes	characteristic, complemented
118	C₈×Q₈	—	53	1	no	—
119	C₈×Q₈	—	53	2	no	—
120	C₈×Q₈	—	53	1	no	—
121	#433	—	53	2	no	—
122	#433	—	53	2	no	—
123	#431	—	53	1	no	—
124	#431	—	53	1	no	—
125	C₅₃	#2986	1	1	yes	characteristic, complemented, Sylow
126	C₄⋊Q₈	—	53	1	no	—
127	C₄⋊Q₈	—	53	1	no	—
128	#125	—	53	2	no	—
129	C₂²⋊Q₈	—	53	1	no	—
130	C₂²⋊Q₈	—	53	2	no	—
131	C₂²⋊Q₈	—	53	1	no	—
132	C₄×Q₈	—	53	2	no	—
133	C₄×Q₈	—	53	1	no	—
134	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
135	#117	—	53	2	no	—
136	C₂×C₄²	—	53	1	no	—
137	C₄⋊C₈	—	53	1	no	—
138	C₄⋊C₈	—	53	1	no	—
139	C₄⋊C₈	—	53	2	no	—
140	C₄⋊C₈	—	53	2	no	—
141	C₄⋊C₈	—	53	1	no	—
142	C₄⋊C₈	—	53	2	no	—
143	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
144	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
145	C₂²⋊C₈	—	53	2	no	—
146	C₂²⋊C₈	—	53	1	no	—
147	C₂²⋊C₈	—	53	1	no	—
148	#97	—	53	2	no	—
149	#97	C₂×D₅₃	1	2	yes	—
150	C₄×C₈	—	53	2	no	—
151	C₄×C₈	—	53	2	no	—
152	C₄×C₈	—	53	1	no	—
153	C₄×C₈	—	53	2	no	—
154	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
155	C₂×Q₈	—	53	2	no	—
156	C₂×Q₈	—	53	1	no	—
157	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
158	C₂²×C₄	—	53	1	no	—
159	C₂²×C₄	—	53	1	no	—
160	C₂²×C₄	—	53	1	no	—
161	C₂×C₈	—	53	1	no	—
162	C₂×C₈	—	53	2	no	—
163	C₂×C₈	—	53	1	no	—
164	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
165	C₂×C₈	C₂²×D₅₃	1	2	yes	—
166	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
167	C₄⋊C₄	—	53	2	no	—
168	C₄⋊C₄	—	53	2	no	—
169	C₄⋊C₄	—	53	2	no	—
170	C₄⋊C₄	—	53	2	no	—
171	C₄⋊C₄	—	53	2	no	—
172	C₄⋊C₄	—	53	2	no	—
173	C₄⋊C₄	—	53	1	no	—
174	C₄⋊C₄	C₄×D₅₃	1	2	yes	—
175	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
176	C₂²⋊C₄	—	53	2	no	—
177	C₂²⋊C₄	—	53	2	no	—
178	C₄²	—	53	1	no	—
179	C₄²	—	53	1	no	—
180	C₄²	—	53	1	no	—
181	C₄²	—	53	2	no	—
182	C₄²	C₂²×D₅₃	1	1	yes	characteristic
183	C₄²	C₂²×D₅₃	1	2	yes	—
184	C₂³	—	53	1	no	—
185	Q₈	—	2	2	no	—
186	Q₈	—	106	4	no	—
187	Q₈	—	106	2	no	—
188	C₂×C₄	—	53	1	no	—
189	C₂×C₄	—	53	1	no	—
190	C₂×C₄	—	53	1	no	—
191	C₂×C₄	—	53	2	no	—
192	C₂×C₄	—	53	2	no	—
193	C₂×C₄	—	106	2	no	—
194	C₂×C₄	—	106	2	no	—
195	C₂×C₄	—	106	1	no	—
196	C₂×C₄	—	106	1	no	—
197	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
200	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	2	no	—
204	C₈	—	2	1	no	—
205	C₈	—	106	1	no	—
206	C₈	—	106	2	no	—
207	C₈	—	106	1	no	—
208	C₂²	—	106	1	no	—
209	C₂²	—	106	1	no	—
210	C₂²	—	106	1	no	—
211	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	53	2	no	—
216	C₄	—	106	1	no	—
217	C₄	—	106	2	no	—
218	C₄	—	106	1	no	—
219	C₄	—	106	2	no	—
220	C₄	#13133225	1	2	yes	—
221	C₄	#13133264	1	1	yes	characteristic
222	C₄	#13133265	1	1	yes	characteristic
223	C₂	—	212	1	no	—
224	C₂	#16811099	1	1	yes	characteristic
225	C₂	#131×D₅₃	1	1	yes	characteristic
226	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	#2986
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. #16810751	C₂	semidirect
3. #16810750	C₂	semidirect
4. Q₈×#88322	C₂	semidirect
5. C₈×Dic₁₀₆	C₂	semidirect
6. #16810747	C₂	semidirect
11. C₅₃×#446	C₂	semidirect
117. #446	D₅₃	semidirect
125. C₅₃	#2986	semidirect