Subgroups — #20283833

Order 6784.

388 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11719945	C₂	1	1	yes	maximal, characteristic
2	#16810750	C₂	1	2	yes	maximal, complemented
3	#16810700	C₂	1	2	yes	maximal, complemented
4	#16810694	C₂	1	4	yes	maximal, complemented
5	#16810679	C₂	1	2	yes	maximal, complemented
6	#16810676	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810645	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11719945	C₄	1	1	yes	characteristic
10	C₂×#11719912	C₂²	1	4	yes	—
11	C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
12	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
13	C₄×D₂₁₂	C₄	1	1	yes	characteristic
14	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊D₅₃	C₄	1	2	yes	—
16	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
17	#13133176	C₄	1	1	yes	characteristic
18	#13133174	C₄	1	2	yes	—
19	C₄²×D₅₃	C₂²	1	2	yes	—
20	C₄⋊#88322	C₂²	1	1	yes	characteristic
21	C₄⋊#88322	C₂²	1	1	yes	characteristic
22	#13133146	C₂²	1	4	yes	—
23	C₂²⋊#88322	C₂²	1	2	yes	—
24	#13133129	C₂²	1	4	yes	—
25	#13133127	C₂²	1	4	yes	—
26	C₄×#88322	C₂²	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
30	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
31	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
35	#11719945	—	2	4	no	—
36	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₅₃	C₂³	1	2	yes	—
40	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	—	4	2	no	—
42	C₂×C₈×C₅₃	C₂³	1	4	yes	—
43	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
48	#11719921	C₂×C₄	1	2	yes	—
49	#11719921	C₂×C₄	1	2	yes	—
50	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
52	#11719919	C₂×C₄	1	2	yes	—
53	#11719919	C₂×C₄	1	2	yes	—
54	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
55	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
56	C₄×Dic₅₃	C₂³	1	2	yes	—
57	C₂×#88322	C₂³	1	2	yes	—
58	C₂×#88322	C₂³	1	2	yes	—
59	#11719912	—	4	4	no	—
60	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	4	1	no	—
65	C₂×C₄×C₅₃	M₁₆	1	4	yes	complemented
66	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
72	C₂²⋊D₅₃	—	2	4	no	—
73	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
74	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
75	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
76	D₂₁₂	—	2	2	no	—
77	C₄×D₅₃	—	2	4	no	—
78	C₄×D₅₃	—	2	2	no	—
79	C₄×D₅₃	—	2	2	no	—
80	C₄×D₅₃	—	4	2	no	—
81	Dic₁₀₆	—	2	2	no	—
82	C₈×C₅₃	—	2	4	no	—
83	#88322	—	2	2	no	—
84	#88322	—	2	2	no	—
85	C₂²×C₅₃	—	2	1	no	—
86	C₂²×C₅₃	—	2	1	no	—
87	C₂²×C₅₃	C₂×M₁₆	1	2	yes	—
88	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
89	C₂×D₅₃	—	2	2	no	—
90	C₂×D₅₃	—	2	2	no	—
91	C₂×D₅₃	—	2	2	no	—
92	C₄×C₅₃	—	2	1	no	—
93	C₄×C₅₃	—	2	1	no	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
96	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
97	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
98	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
99	Dic₅₃	—	2	2	no	—
100	Dic₅₃	—	2	2	no	—
101	Dic₅₃	—	2	2	no	—
102	#2989	—	53	1	no	maximal, Sylow
103	C₂×C₅₃	—	2	2	no	—
104	C₂×C₅₃	#520	1	1	yes	characteristic
105	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
106	C₂×C₅₃	#568	1	1	yes	characteristic
107	D₅₃	—	4	2	no	—
108	C₄×#41	—	53	1	no	—
109	#446	—	53	2	no	—
110	#436	—	53	2	no	—
111	#435	—	53	4	no	—
112	#432	—	53	2	no	—
113	#431	—	53	1	no	—
114	#423	—	53	2	no	—
115	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic, complemented
116	C₅₃	#2989	1	1	yes	characteristic, complemented, Sylow
117	C₂×#41	—	53	1	no	—
118	C₂×M₁₆	—	53	4	no	—
119	C₂²×C₈	C₂×D₅₃	1	2	yes	—
120	C₄×Q₈	—	53	1	no	—
121	C₄×Q₈	—	53	1	no	—
122	C₄×D₄	—	53	2	no	—
123	C₄×D₄	—	53	2	no	—
124	C₄×D₄	—	53	1	no	—
125	C₄×D₄	—	53	1	no	—
126	#117	—	53	1	no	—
127	#117	—	53	2	no	—
128	C₂×C₄²	—	53	2	no	—
129	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
130	C₄⋊C₈	—	53	4	no	—
131	C₄⋊C₈	—	53	1	no	—
132	C₄⋊C₈	—	53	1	no	—
133	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
134	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
135	C₂²⋊C₈	—	53	2	no	—
136	C₂²⋊C₈	—	53	4	no	—
137	#97	—	53	4	no	—
138	C₄×C₈	—	53	2	no	—
139	#41	—	106	4	no	—
140	C₂×Q₈	—	53	1	no	—
141	C₂×D₄	—	53	2	no	—
142	C₂×D₄	—	53	1	no	—
143	C₂²×C₄	—	53	2	no	—
144	C₂²×C₄	—	53	2	no	—
145	C₂²×C₄	—	53	2	no	—
146	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
147	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
149	M₁₆	—	212	4	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	53	2	no	—
152	C₂×C₈	—	53	2	no	—
153	C₂×C₈	C₂²×D₅₃	1	4	yes	—
154	C₄⋊C₄	—	53	2	no	—
155	C₄⋊C₄	—	53	2	no	—
156	C₄⋊C₄	—	53	1	no	—
157	C₄⋊C₄	—	53	1	no	—
158	C₂²⋊C₄	—	53	2	no	—
159	C₂²⋊C₄	—	53	2	no	—
160	C₂²⋊C₄	—	53	2	no	—
161	C₄²	—	53	2	no	—
162	C₄²	—	53	2	no	—
163	C₄²	—	53	2	no	—
164	C₄²	C₄×D₅₃	1	1	yes	characteristic
165	C₄²	C₄×D₅₃	1	1	yes	characteristic
166	C₄²	C₂²×D₅₃	1	1	yes	characteristic
167	C₄²	C₂²×D₅₃	1	1	yes	characteristic
168	C₂³	—	53	2	no	—
169	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
170	Q₈	—	106	2	no	—
171	D₄	—	106	4	no	—
172	D₄	—	106	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	53	2	no	—
177	C₂×C₄	—	53	2	no	—
178	C₂×C₄	—	53	2	no	—
179	C₂×C₄	—	106	4	no	—
180	C₂×C₄	—	106	2	no	—
181	C₂×C₄	—	106	2	no	—
182	C₂×C₄	—	212	2	no	—
183	C₂×C₄	#11719912	1	4	yes	complemented
184	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
189	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	—	106	2	no	—
192	C₈	—	106	2	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	106	2	no	—
196	C₂²	—	106	2	no	—
197	C₂²	—	106	2	no	—
198	C₂²	C₂×#11719912	1	2	yes	—
199	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	106	2	no	—
204	C₄	—	106	2	no	—
205	C₄	—	106	2	no	—
206	C₄	C₂×#11719912	1	2	yes	—
207	C₄	C₂×#11719912	1	2	yes	—
208	C₄	#13133262	1	1	yes	characteristic
209	C₄	#13133265	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	—	212	2	no	—
212	C₂	#16811104	1	1	yes	characteristic
213	C₂	C₂²×#11719912	1	1	yes	characteristic
214	C₂	#16811389	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#2989
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #16810750	C₂	semidirect
3. #16810700	C₂	semidirect
4. #16810694	C₂	semidirect
5. #16810679	C₂	semidirect
6. #16810676	C₂	semidirect
7. #16810645	C₂	semidirect
8. C₂×C₄⋊C₈×C₅₃	C₂	semidirect
65. C₂×C₄×C₅₃	M₁₆	semidirect
115. C₂×C₄⋊C₈	D₅₃	semidirect
116. C₅₃	#2989	semidirect
183. C₂×C₄	#11719912	semidirect