Subgroups — #20283781

Order 6784.

410 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811173	C₂	1	1	yes	maximal, characteristic
2	#16810702	C₂	1	2	yes	maximal, complemented
3	#16810700	C₂	1	2	yes	maximal, complemented
4	C₈×D₂₁₂	C₂	1	2	yes	maximal, complemented
5	#16810688	C₂	1	2	yes	maximal, complemented
6	#16810676	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810588	C₂	1	4	yes	maximal, complemented
8	C₅₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11719945	C₂²	1	2	yes	—
10	C₂×#11719912	C₂²	1	4	yes	—
11	C₂×C₈×D₅₃	C₂²	1	4	yes	—
12	#13133217	C₄	1	1	yes	characteristic
13	#13133214	C₄	1	1	yes	characteristic
14	#13133205	C₄	1	2	yes	—
15	#13133191	C₄	1	2	yes	—
16	#13133191	C₄	1	2	yes	—
17	C₄×D₂₁₂	C₂²	1	2	yes	—
18	C₄×D₂₁₂	C₂²	1	2	yes	—
19	C₄⋊#88322	C₂²	1	1	yes	characteristic
20	C₄⋊#88322	C₂²	1	1	yes	characteristic
21	C₂²⋊#88322	C₂²	1	2	yes	—
22	#13133129	C₂²	1	4	yes	—
23	#13133129	C₂²	1	4	yes	—
24	C₄×#88322	C₂²	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
29	C₄×C₈×C₅₃	C₂²	1	2	yes	—
30	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×C₂²⋊D₅₃	C₂×C₄	1	4	yes	—
34	#11719945	—	4	2	no	—
35	#11719945	—	4	2	no	—
36	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
37	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
38	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₅₃	C₂³	1	4	yes	—
40	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
41	C₂×C₈×C₅₃	C₂³	1	2	yes	—
42	C₂×C₈×C₅₃	C₂³	1	2	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	#11719921	C₂×C₄	1	4	yes	—
48	#11719921	C₂×C₄	1	4	yes	—
49	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
50	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
51	C₂×#88322	C₂³	1	2	yes	—
52	C₂×#88322	C₂³	1	2	yes	—
53	#11719912	—	4	4	no	—
54	C₈×D₅₃	—	4	4	no	—
55	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×D₅₃	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
67	C₂²⋊D₅₃	—	4	4	no	—
68	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
69	D₂₁₂	—	2	4	no	—
70	D₂₁₂	—	2	4	no	—
71	D₂₁₂	—	4	2	no	—
72	C₄×D₅₃	—	2	4	no	—
73	C₄×D₅₃	—	2	4	no	—
74	Dic₁₀₆	—	4	2	no	—
75	C₈×C₅₃	—	2	2	no	—
76	C₈×C₅₃	—	2	2	no	—
77	#88322	—	2	2	no	—
78	#88322	—	2	2	no	—
79	C₂²×C₅₃	—	2	1	no	—
80	C₂²×C₅₃	—	2	1	no	—
81	C₂²×C₅₃	—	2	1	no	—
82	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
83	C₂×D₅₃	—	2	4	no	—
84	C₂×D₅₃	—	2	4	no	—
85	C₂×D₅₃	—	2	4	no	—
86	C₄×C₅₃	—	2	1	no	—
87	C₄×C₅₃	—	2	1	no	—
88	C₄×C₅₃	—	2	1	no	—
89	C₄×C₅₃	#131	1	2	yes	—
90	C₄×C₅₃	#131	1	2	yes	—
91	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
92	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
93	Dic₅₃	—	2	4	no	—
94	#2979	—	53	1	no	maximal, Sylow
95	C₂×C₅₃	—	4	1	no	—
96	C₂×C₅₃	#518	1	1	yes	characteristic
97	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
98	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
99	D₅₃	—	4	4	no	—
100	#532	—	53	1	no	—
101	#436	—	53	2	no	—
102	#436	—	53	2	no	—
103	C₈×D₄	—	53	2	no	—
104	C₈×D₄	—	53	2	no	—
105	#431	—	53	1	no	—
106	#431	D₅₃	1	1	yes	characteristic, complemented
107	#408	—	53	4	no	—
108	C₅₃	#2979	1	1	yes	characteristic, complemented, Sylow
109	C₂×#41	—	53	2	no	—
110	C₂×M₁₆	—	53	4	no	—
111	C₂²×C₈	—	53	4	no	—
112	C₄⋊Q₈	—	53	1	no	—
113	#127	—	53	1	no	—
114	#124	—	53	2	no	—
115	#121	—	53	2	no	—
116	#121	—	53	2	no	—
117	C₄×D₄	—	53	2	no	—
118	C₄×D₄	—	53	2	no	—
119	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
120	C₄⋊C₈	—	53	1	no	—
121	C₄⋊C₈	—	53	1	no	—
122	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
123	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
124	C₂²⋊C₈	—	53	2	no	—
125	C₂²⋊C₈	—	53	4	no	—
126	C₂²⋊C₈	—	53	4	no	—
127	C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
128	C₄×C₈	—	53	2	no	—
129	C₄×C₈	C₂×D₅₃	1	2	yes	—
130	#41	—	212	2	no	—
131	#41	—	212	2	no	—
132	C₂×Q₈	—	53	2	no	—
133	C₂×D₄	—	53	4	no	—
134	C₂×D₄	—	53	2	no	—
135	C₂×D₄	—	53	2	no	—
136	C₂×D₄	—	53	2	no	—
137	C₂²×C₄	—	53	4	no	—
138	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
139	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
140	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
141	M₁₆	—	212	4	no	—
142	C₂×C₈	—	53	2	no	—
143	C₂×C₈	—	53	2	no	—
144	C₂×C₈	—	212	4	no	—
145	C₂×C₈	C₂²×D₅₃	1	2	yes	—
146	C₂×C₈	C₂²×D₅₃	1	2	yes	—
147	C₄⋊C₄	—	53	2	no	—
148	C₄⋊C₄	—	53	2	no	—
149	C₂²⋊C₄	—	53	4	no	—
150	C₂²⋊C₄	—	53	4	no	—
151	C₄²	C₄×D₅₃	1	1	yes	characteristic
152	C₄²	C₄×D₅₃	1	1	yes	characteristic
153	C₄²	C₂²×D₅₃	1	1	yes	characteristic
154	C₄²	C₂²×D₅₃	1	1	yes	characteristic
155	C₂³	—	53	4	no	—
156	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
157	Q₈	—	212	2	no	—
158	D₄	—	106	4	no	—
159	D₄	—	106	4	no	—
160	D₄	—	212	2	no	—
161	D₄	—	212	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	53	4	no	—
167	C₂×C₄	—	106	4	no	—
168	C₂×C₄	—	106	4	no	—
169	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
174	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	2	2	no	—
177	C₈	—	106	2	no	—
178	C₈	—	106	2	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	106	4	no	—
183	C₂²	—	106	4	no	—
184	C₂²	—	106	4	no	—
185	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	106	4	no	—
190	C₄	#13133223	1	2	yes	—
191	C₄	#13133225	1	2	yes	—
192	C₄	#13133263	1	1	yes	characteristic
193	C₄	#13133263	1	1	yes	characteristic
194	C₂	—	4	1	no	—
195	C₂	—	212	4	no	—
196	C₂	#16811097	1	1	yes	characteristic
197	C₂	C₂×#13133223	1	1	yes	characteristic
198	C₂	C₂×#13133225	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#431	C₂
hypercenter	#431	D₅₃
hypocenter	C₅₃	#2979
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #16810702	C₂	semidirect
3. #16810700	C₂	semidirect
4. C₈×D₂₁₂	C₂	semidirect
5. #16810688	C₂	semidirect
6. #16810676	C₂	semidirect
7. #16810588	C₂	semidirect
8. C₅₃×#431	C₂	semidirect
106. #431	D₅₃	semidirect
108. C₅₃	#2979	semidirect