Subgroups — #20283856

Order 6784.

356 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811157	C₂	1	1	yes	maximal, characteristic
2	#16810751	C₂	1	2	yes	maximal, complemented
3	#16810696	C₂	1	2	yes	maximal, complemented
4	#16810691	C₂	1	2	yes	maximal, complemented
5	#16810686	C₂	1	2	yes	maximal, complemented
6	#16810683	C₂	1	2	yes	maximal, complemented
7	#16810648	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11719912	C₂²	1	2	yes	—
11	C₂²×#88322	C₂²	1	1	yes	characteristic
12	C₂²×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₅₃	C₂²	1	2	yes	—
14	#13133211	C₄	1	2	yes	—
15	#13133209	C₄	1	1	yes	characteristic
16	#13133205	C₄	1	1	yes	characteristic
17	#13133200	C₄	1	2	yes	—
18	#13133196	C₄	1	1	yes	characteristic
19	#13133191	C₄	1	1	yes	characteristic
20	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
21	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
23	#13133174	C₂²	1	2	yes	—
24	C₄⋊#88322	C₂²	1	2	yes	—
25	C₄⋊#88322	C₂²	1	2	yes	—
26	#13133146	C₂²	1	2	yes	—
27	#13133146	C₂²	1	2	yes	—
28	#13133129	C₂²	1	2	yes	—
29	#13133129	C₂²	1	2	yes	—
30	#13133127	C₂²	1	2	yes	—
31	C₈×Dic₅₃	C₂²	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
36	#97×C₅₃	C₂²	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
41	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₅₃	C₂³	1	2	yes	—
43	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂³	1	2	yes	—
45	C₂×C₈×C₅₃	C₂³	1	2	yes	—
46	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
51	#11719921	C₂×C₄	1	2	yes	—
52	#11719921	C₂×C₄	1	2	yes	—
53	#11719921	C₂×C₄	1	2	yes	—
54	#11719921	C₂×C₄	1	2	yes	—
55	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
57	#11719919	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	C₄×Dic₅₃	C₂³	1	2	yes	—
62	C₂×#88322	—	4	1	no	—
63	C₂×#88322	—	4	1	no	—
64	C₂×#88322	C₂³	1	2	yes	—
65	C₂×#88322	C₂³	1	2	yes	—
66	#11719912	—	4	2	no	—
67	C₈×D₅₃	—	4	2	no	—
68	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	4	1	no	—
74	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₅₃	—	2	4	no	—
81	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
82	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
83	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
84	D₂₁₂	—	4	1	no	—
85	C₄×D₅₃	—	2	2	no	—
86	C₄×D₅₃	—	2	2	no	—
87	Dic₁₀₆	—	4	1	no	—
88	C₈×C₅₃	—	2	2	no	—
89	C₈×C₅₃	—	2	2	no	—
90	#88322	—	2	2	no	—
91	#88322	—	2	2	no	—
92	C₂²×C₅₃	—	2	1	no	—
93	C₂²×C₅₃	—	2	1	no	—
94	C₂²×C₅₃	#131	1	2	yes	—
95	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₅₃	—	2	2	no	—
97	C₂×D₅₃	—	2	2	no	—
98	C₂×D₅₃	—	2	2	no	—
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
105	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
106	Dic₅₃	—	2	2	no	—
107	Dic₅₃	—	2	2	no	—
108	Dic₅₃	—	2	2	no	—
109	#2992	—	53	1	no	maximal, Sylow
110	C₂×C₅₃	—	2	2	no	—
111	C₂×C₅₃	#520	1	1	yes	characteristic
112	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
113	C₂×C₅₃	#568	1	1	yes	characteristic
114	D₅₃	—	4	2	no	—
115	#529	—	53	1	no	—
116	#446	—	53	2	no	—
117	#435	—	53	2	no	—
118	#435	—	53	2	no	—
119	C₈×D₄	—	53	2	no	—
120	#433	—	53	2	no	—
121	#432	D₅₃	1	1	yes	characteristic, complemented
122	#424	—	53	2	no	—
123	C₂×C₄⋊C₈	—	53	1	no	—
124	C₅₃	#2992	1	1	yes	characteristic, complemented, Sylow
125	C₂×M₁₆	—	53	2	no	—
126	C₂²×C₈	—	53	1	no	—
127	C₂²×C₈	—	53	1	no	—
128	C₂²×C₈	—	53	2	no	—
129	#126	—	53	2	no	—
130	#125	—	53	1	no	—
131	#124	—	53	1	no	—
132	#123	—	53	2	no	—
133	C₂²⋊Q₈	—	53	1	no	—
134	#121	—	53	1	no	—
135	C₄×Q₈	—	53	1	no	—
136	C₄×D₄	—	53	2	no	—
137	C₄×D₄	—	53	1	no	—
138	#117	—	53	2	no	—
139	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
140	C₄⋊C₈	—	53	2	no	—
141	C₄⋊C₈	—	53	2	no	—
142	C₄⋊C₈	—	53	2	no	—
143	C₄⋊C₈	—	53	2	no	—
144	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
145	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
146	C₂²⋊C₈	—	53	2	no	—
147	C₂²⋊C₈	—	53	2	no	—
148	C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
149	#97	—	53	2	no	—
150	#97	C₂×D₅₃	1	2	yes	—
151	C₄×C₈	—	53	2	no	—
152	C₂×Q₈	—	53	1	no	—
153	C₂×D₄	—	53	2	no	—
154	C₂×D₄	—	53	1	no	—
155	C₂²×C₄	—	53	2	no	—
156	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
157	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
159	M₁₆	—	212	2	no	—
160	C₂×C₈	—	53	2	no	—
161	C₂×C₈	—	53	2	no	—
162	C₂×C₈	—	212	2	no	—
163	C₂×C₈	—	212	1	no	—
164	C₂×C₈	—	212	1	no	—
165	C₂×C₈	C₂²×D₅₃	1	2	yes	—
166	C₂×C₈	C₂²×D₅₃	1	2	yes	—
167	C₄⋊C₄	—	53	2	no	—
168	C₄⋊C₄	—	53	2	no	—
169	C₄⋊C₄	—	53	1	no	—
170	C₄⋊C₄	—	53	2	no	—
171	C₄⋊C₄	—	53	1	no	—
172	C₄⋊C₄	—	53	2	no	—
173	C₂²⋊C₄	—	53	2	no	—
174	C₂²⋊C₄	—	53	2	no	—
175	C₂²⋊C₄	—	53	2	no	—
176	C₂²⋊C₄	—	53	2	no	—
177	C₂²⋊C₄	—	53	2	no	—
178	C₄²	—	53	2	no	—
179	C₄²	C₄×D₅₃	1	1	yes	characteristic
180	C₄²	C₄×D₅₃	1	1	yes	characteristic
181	C₄²	C₂²×D₅₃	1	1	yes	characteristic
182	C₄²	C₂²×D₅₃	1	1	yes	characteristic
183	C₂³	—	53	2	no	—
184	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
185	Q₈	—	212	1	no	—
186	D₄	—	106	4	no	—
187	D₄	—	212	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	53	2	no	—
193	C₂×C₄	—	53	2	no	—
194	C₂×C₄	—	53	2	no	—
195	C₂×C₄	—	106	2	no	—
196	C₂×C₄	—	106	2	no	—
197	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	106	2	no	—
206	C₈	—	106	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	106	2	no	—
210	C₂²	—	106	2	no	—
211	C₂²	—	106	2	no	—
212	C₂²	#13133225	1	2	yes	—
213	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	106	2	no	—
220	C₄	—	106	2	no	—
221	C₄	—	106	2	no	—
222	C₄	#13133263	1	1	yes	characteristic
223	C₄	#13133264	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	212	2	no	—
226	C₂	#16811103	1	1	yes	characteristic
227	C₂	C₂×#13133225	1	1	yes	characteristic
228	C₂	#16811389	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#432	C₂
hypercenter	#432	D₅₃
hypocenter	C₅₃	#2992
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #16810751	C₂	semidirect
3. #16810696	C₂	semidirect
4. #16810691	C₂	semidirect
5. #16810686	C₂	semidirect
6. #16810683	C₂	semidirect
7. #16810648	C₂	semidirect
8. C₂×C₄⋊#88322	C₂	semidirect
9. C₅₃×#432	C₂	semidirect
121. #432	D₅₃	semidirect
124. C₅₃	#2992	semidirect