Subgroups — #20283633

Order 6784.

452 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133239	C₂	1	1	yes	maximal, characteristic
2	#16811389	C₂	1	2	yes	maximal, complemented
3	#16810746	C₂	1	4	yes	maximal, complemented
4	#16810741	C₂	1	4	yes	maximal, complemented
5	C₂×#13133153	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810641	C₂	1	2	yes	maximal, complemented
7	C₅₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719945	C₂²	1	2	yes	—
9	#13133239	C₄	1	2	yes	complemented
10	#13133239	C₄	1	2	yes	complemented
11	#13133239	C₂²	1	2	yes	—
12	#13133239	C₂²	1	2	yes	—
13	C₂×Dic₂₁₂	C₄	1	1	yes	characteristic, complemented
14	C₂×#11719913	C₄	1	2	yes	complemented
15	C₂×D₄₂₄	C₄	1	1	yes	characteristic, complemented
16	#13133225	—	2	4	no	—
17	#13133223	C₂²	1	8	yes	complemented
18	C₂×#11719917	C₂²	1	2	yes	—
19	M₁₆×D₅₃	—	2	4	no	—
20	#13133153	C₂²	1	2	yes	complemented
21	#13133153	C₂²	1	2	yes	complemented
22	#13133145	C₂²	1	8	yes	complemented
23	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	#117×C₅₃	C₂²	1	1	yes	characteristic
26	#97×C₅₃	C₂²	1	2	yes	complemented
27	C₄×C₈×C₅₃	C₂²	1	2	yes	complemented
28	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×C₂²⋊D₅₃	—	2	2	no	—
30	#11719945	—	2	2	no	—
31	#11719945	—	2	2	no	—
32	#11719945	C₂×C₄	1	4	yes	—
33	#11719945	C₂³	1	4	yes	—
34	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₅₃	—	2	2	no	—
36	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
37	M₁₆×C₅₃	D₄	1	2	yes	—
38	M₁₆×C₅₃	D₄	1	2	yes	—
39	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂³	1	2	yes	—
46	C₄⋊C₄×C₅₃	—	2	1	no	—
47	C₂²⋊C₄×C₅₃	—	2	1	no	—
48	C₄²×C₅₃	C₂³	1	2	yes	—
49	#11719917	—	2	2	no	—
50	#11719917	C₂³	1	4	yes	—
51	C₂×#88322	—	2	2	no	—
52	Dic₂₁₂	C₂×C₄	1	2	yes	—
53	Dic₂₁₂	C₂×C₄	1	2	yes	—
54	D₄₂₄	C₂×C₄	1	2	yes	—
55	D₄₂₄	C₂×C₄	1	2	yes	—
56	#11719913	C₂×C₄	1	4	yes	—
57	#11719913	C₂×C₄	1	4	yes	—
58	#11719912	—	2	8	no	—
59	C₈×D₅₃	—	2	8	no	—
60	C₂³×C₅₃	#41	1	1	yes	characteristic
61	C₂²×D₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₂²⋊D₅₃	—	2	4	no	—
70	C₂²⋊D₅₃	—	2	4	no	—
71	C₂×Dic₅₃	—	2	2	no	—
72	D₂₁₂	—	2	2	no	—
73	D₂₁₂	C₂²×C₄	1	4	yes	—
74	C₄×D₅₃	—	2	4	no	—
75	C₄×D₅₃	—	2	4	no	—
76	Dic₁₀₆	—	2	2	no	—
77	Dic₁₀₆	C₂²×C₄	1	4	yes	—
78	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₈×C₅₃	C₂×D₄	1	4	yes	—
83	#88322	—	2	4	no	—
84	C₂²×C₅₃	—	4	1	no	—
85	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
86	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
87	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
88	C₂×D₅₃	—	2	4	no	—
89	C₂×D₅₃	—	8	2	no	—
90	C₄×C₅₃	—	4	2	no	—
91	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
92	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
93	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
94	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
95	Dic₅₃	—	2	4	no	—
96	#2957	—	53	1	no	maximal, Sylow
97	C₂×C₅₃	—	2	1	no	—
98	C₂×C₅₃	—	2	1	no	—
99	C₂×C₅₃	—	2	1	no	—
100	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
101	D₅₃	—	4	4	no	—
102	C₂×#135	—	53	1	no	—
103	#568	—	53	2	no	—
104	#444	—	53	4	no	—
105	#443	—	53	4	no	—
106	C₂×#108	—	53	1	no	—
107	#421	—	53	2	no	—
108	#405	D₅₃	1	1	yes	characteristic, complemented
109	C₅₃	#2957	1	1	yes	characteristic, complemented, Sylow
110	C₂×#41	—	53	2	no	—
111	#135	—	53	2	no	—
112	#135	—	53	2	no	—
113	#135	—	53	2	no	—
114	#135	—	53	2	no	—
115	C₂×Q₁₆	—	53	1	no	—
116	C₂×SD₁₆	—	53	2	no	—
117	C₂×D₈	—	53	1	no	—
118	#131	—	53	8	no	—
119	#131	—	106	4	no	—
120	C₂×M₁₆	—	53	2	no	—
121	C₂×M₁₆	—	106	4	no	—
122	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
123	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
124	#117	C₂×D₅₃	1	1	yes	characteristic
125	#108	—	53	2	no	—
126	#108	—	53	2	no	—
127	#104	—	53	8	no	—
128	#97	C₂×D₅₃	1	2	yes	complemented
129	C₄×C₈	C₂×D₅₃	1	2	yes	complemented
130	#41	—	53	4	no	—
131	#41	—	53	4	no	—
132	#41	—	106	2	no	—
133	#41	—	106	2	no	—
134	C₂×Q₈	—	53	2	no	—
135	C₂×D₄	—	53	2	no	—
136	C₂×D₄	—	106	2	no	—
137	C₂²×C₄	—	106	2	no	—
138	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
139	Q₁₆	—	53	2	no	—
140	Q₁₆	—	53	2	no	—
141	SD₁₆	—	53	4	no	—
142	SD₁₆	—	53	4	no	—
143	D₈	—	53	2	no	—
144	D₈	—	53	2	no	—
145	M₁₆	—	53	4	no	—
146	M₁₆	—	106	2	no	—
147	M₁₆	—	106	8	no	—
148	M₁₆	D₂₁₂	1	2	yes	—
149	M₁₆	D₂₁₂	1	2	yes	—
150	C₂×C₈	—	106	2	no	—
151	C₂×C₈	—	106	8	no	—
152	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
153	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
154	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
155	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
156	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
157	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
158	C₂×C₈	C₂²×D₅₃	1	2	yes	—
159	C₄⋊C₄	—	2	1	no	—
160	C₂²⋊C₄	—	2	1	no	—
161	C₄²	C₂²×D₅₃	1	2	yes	—
162	C₂³	—	106	2	no	—
163	C₂³	#11719945	1	1	yes	characteristic
164	Q₈	—	53	4	no	—
165	Q₈	—	106	2	no	—
166	D₄	—	53	4	no	—
167	D₄	—	106	4	no	—
168	D₄	—	106	4	no	—
169	D₄	—	106	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	106	2	no	—
172	C₂×C₄	—	106	4	no	—
173	C₂×C₄	—	106	4	no	—
174	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
176	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
177	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
178	C₂×C₄	#11719945	1	1	yes	characteristic
179	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
180	C₈	—	106	4	no	—
181	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
182	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
183	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
184	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
185	C₈	C₂×D₂₁₂	1	4	yes	—
186	C₂²	—	4	1	no	—
187	C₂²	—	106	4	no	—
188	C₂²	—	424	2	no	—
189	C₂²	C₄×D₂₁₂	1	1	yes	characteristic
190	C₂²	C₄×D₂₁₂	1	1	yes	characteristic
191	C₂²	C₂×#11719945	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	106	4	no	—
194	C₄	C₄×D₂₁₂	1	1	yes	characteristic
195	C₄	C₄×D₂₁₂	1	1	yes	characteristic
196	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
197	C₄	C₂²×D₂₁₂	1	1	yes	characteristic
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	212	4	no	—
202	C₂	C₂×C₄×D₂₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₂₁₂
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#405	C₂
socle	C₂×C₅₃	C₂×C₄×D₄
hypercenter	#405	D₅₃
hypocenter	C₅₃	#2957
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #16811389	C₂	semidirect
3. #16810746	C₂	semidirect
4. #16810741	C₂	semidirect
5. C₂×#13133153	C₂	semidirect
6. #16810641	C₂	semidirect
7. C₅₃×#405	C₂	semidirect
9. #13133239	C₄	semidirect
10. #13133239	C₄	semidirect
13. C₂×Dic₂₁₂	C₄	semidirect
14. C₂×#11719913	C₄	semidirect
15. C₂×D₄₂₄	C₄	semidirect
17. #13133223	C₂²	semidirect
20. #13133153	C₂²	semidirect
21. #13133153	C₂²	semidirect
22. #13133145	C₂²	semidirect
26. #97×C₅₃	C₂²	semidirect
27. C₄×C₈×C₅₃	C₂²	semidirect
108. #405	D₅₃	semidirect
109. C₅₃	#2957	semidirect
128. #97	C₂×D₅₃	semidirect
129. C₄×C₈	C₂×D₅₃	semidirect