Subgroups — #20283663

Order 6784.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 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	#16810750	C₂	1	2	yes	maximal, complemented
5	Q₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810648	C₂	1	2	yes	maximal
8	C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
9	#16810580	C₂	1	2	yes	maximal
10	C₄×#11719912	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719912	C₂²	1	1	yes	characteristic
13	C₂×#11719912	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₅₃	C₂²	1	2	yes	—
15	#13133216	C₄	1	1	yes	characteristic
16	#13133215	C₄	1	1	yes	characteristic
17	#13133207	C₄	1	2	yes	—
18	#13133197	C₄	1	1	yes	characteristic
19	#13133195	C₄	1	2	yes	—
20	#13133193	C₄	1	1	yes	characteristic
21	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
22	C₄×Dic₁₀₆	C₂²	1	2	yes	—
23	#13133174	C₂²	1	2	yes	—
24	C₄²×D₅₃	C₂²	1	1	yes	characteristic
25	C₄⋊#88322	C₂²	1	1	yes	characteristic
26	C₄⋊#88322	C₂²	1	2	yes	—
27	#13133146	C₂²	1	2	yes	—
28	#13133146	C₂²	1	2	yes	—
29	#13133146	C₂²	1	2	yes	—
30	#13133127	C₂²	1	1	yes	characteristic
31	#13133127	C₂²	1	1	yes	characteristic
32	#13133127	C₂²	1	2	yes	—
33	C₄×#88322	C₂²	1	1	yes	characteristic
34	C₄×#88322	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	1	yes	characteristic
46	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
47	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
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	2	yes	—
66	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
67	C₂×#88322	C₂³	1	2	yes	—
68	C₂×#88322	C₂³	1	1	yes	characteristic
69	C₂×#88322	C₂³	1	1	yes	characteristic
70	#11719912	—	2	2	no	—
71	#11719912	Q₈	1	4	yes	complemented
72	C₈×D₅₃	—	2	4	no	—
73	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
74	Q₈×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
82	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
83	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄×D₅₃	—	2	2	no	—
86	C₄×D₅₃	—	4	1	no	—
87	C₄×D₅₃	C₂×Q₈	1	2	yes	—
88	C₄×D₅₃	#41	1	2	yes	—
89	Dic₁₀₆	—	2	2	no	—
90	Dic₁₀₆	—	4	2	no	—
91	C₈×C₅₃	—	2	2	no	—
92	C₈×C₅₃	—	2	1	no	—
93	C₈×C₅₃	C₂×Q₈	1	2	yes	—
94	#88322	—	2	2	no	—
95	#88322	—	2	1	no	—
96	#88322	C₂×Q₈	1	2	yes	—
97	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₅₃	—	2	1	no	—
99	C₂×D₅₃	—	2	1	no	—
100	C₂×D₅₃	C₄×Q₈	1	2	yes	—
101	C₄×C₅₃	—	2	2	no	—
102	C₄×C₅₃	—	2	2	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	#131	1	2	yes	—
105	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
106	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
107	Dic₅₃	—	2	2	no	—
108	Dic₅₃	—	2	1	no	—
109	Dic₅₃	—	2	1	no	—
110	Dic₅₃	—	2	2	no	—
111	Dic₅₃	C₄×Q₈	1	2	yes	—
112	#2962	—	53	1	no	maximal, Sylow
113	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
114	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
115	C₂×C₅₃	#568	1	1	yes	characteristic
116	D₅₃	—	2	2	no	—
117	#533	—	53	1	no	—
118	#446	—	53	1	no	—
119	#446	—	53	2	no	—
120	#446	—	53	2	no	—
121	#446	D₅₃	1	1	yes	characteristic, complemented
122	C₈×Q₈	—	53	1	no	—
123	C₈×Q₈	—	53	1	no	—
124	#424	—	53	2	no	—
125	C₂×C₄⋊C₈	—	53	1	no	—
126	#405	—	53	2	no	—
127	C₄×M₁₆	—	53	1	no	—
128	C₅₃	#2962	1	1	yes	characteristic, complemented, Sylow
129	C₂×M₁₆	—	53	1	no	—
130	C₂×M₁₆	—	53	1	no	—
131	C₂²×C₈	—	53	2	no	—
132	C₄⋊Q₈	—	53	1	no	—
133	C₄⋊Q₈	—	53	1	no	—
134	#125	—	53	2	no	—
135	C₂²⋊Q₈	—	53	1	no	—
136	C₂²⋊Q₈	—	53	2	no	—
137	C₂²⋊Q₈	—	53	1	no	—
138	C₄×Q₈	—	53	1	no	—
139	C₄×Q₈	—	53	2	no	—
140	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
141	#117	—	53	2	no	—
142	C₂×C₄²	—	53	1	no	—
143	C₄⋊C₈	—	53	2	no	—
144	C₄⋊C₈	—	53	2	no	—
145	C₄⋊C₈	—	53	2	no	—
146	C₄⋊C₈	—	53	1	no	—
147	C₄⋊C₈	—	53	2	no	—
148	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
149	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
150	#97	—	53	1	no	—
151	#97	—	53	1	no	—
152	#97	—	53	2	no	—
153	#97	C₂×D₅₃	1	2	yes	—
154	C₄×C₈	—	53	2	no	—
155	C₄×C₈	—	53	1	no	—
156	C₄×C₈	—	53	2	no	—
157	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
158	C₂×Q₈	—	53	1	no	—
159	C₂×Q₈	—	53	2	no	—
160	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
161	C₂²×C₄	—	53	1	no	—
162	C₂²×C₄	—	53	1	no	—
163	C₂²×C₄	—	53	1	no	—
164	M₁₆	—	53	4	no	—
165	M₁₆	—	106	2	no	—
166	C₂×C₈	—	53	1	no	—
167	C₂×C₈	—	53	2	no	—
168	C₂×C₈	—	53	1	no	—
169	C₂×C₈	—	106	4	no	—
170	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₅₃	1	2	yes	—
172	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
173	C₄⋊C₄	—	53	2	no	—
174	C₄⋊C₄	—	53	2	no	—
175	C₄⋊C₄	—	53	2	no	—
176	C₄⋊C₄	—	53	2	no	—
177	C₄⋊C₄	—	53	2	no	—
178	C₄⋊C₄	—	53	2	no	—
179	C₄⋊C₄	—	53	1	no	—
180	C₄⋊C₄	C₄×D₅₃	1	2	yes	—
181	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
182	C₂²⋊C₄	—	53	2	no	—
183	C₂²⋊C₄	—	53	2	no	—
184	C₄²	—	53	1	no	—
185	C₄²	—	53	1	no	—
186	C₄²	—	53	2	no	—
187	C₄²	—	53	1	no	—
188	C₄²	C₂²×D₅₃	1	1	yes	characteristic
189	C₄²	C₂²×D₅₃	1	2	yes	—
190	C₂³	—	53	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	106	2	no	—
193	Q₈	—	212	2	no	—
194	C₂×C₄	—	53	1	no	—
195	C₂×C₄	—	53	2	no	—
196	C₂×C₄	—	53	2	no	—
197	C₂×C₄	—	53	1	no	—
198	C₂×C₄	—	53	1	no	—
199	C₂×C₄	—	53	2	no	—
200	C₂×C₄	—	53	2	no	—
201	C₂×C₄	—	106	2	no	—
202	C₂×C₄	—	212	1	no	—
203	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	1	no	—
210	C₈	—	53	2	no	—
211	C₈	—	106	2	no	—
212	C₈	—	106	1	no	—
213	C₈	Q₈×D₅₃	1	2	yes	—
214	C₂²	—	53	2	no	—
215	C₂²	—	106	1	no	—
216	C₂²	—	106	1	no	—
217	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	53	2	no	—
222	C₄	—	106	1	no	—
223	C₄	—	106	2	no	—
224	C₄	—	106	1	no	—
225	C₄	—	106	2	no	—
226	C₄	#13133225	1	2	yes	—
227	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
228	C₄	#41×D₅₃	1	1	yes	characteristic
229	C₂	—	106	2	no	—
230	C₂	C₄×Q₈×D₅₃	1	1	yes	characteristic
231	C₂	C₂×#13133225	1	1	yes	characteristic
232	C₂	#16811388	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	#2962
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. #16810750	C₂	semidirect
5. Q₈×#88322	C₂	semidirect
6. C₈×Dic₁₀₆	C₂	semidirect
11. C₅₃×#446	C₂	semidirect
71. #11719912	Q₈	semidirect
121. #446	D₅₃	semidirect
128. C₅₃	#2962	semidirect