Subgroups — #20283733

Order 6784.

402 conjugacy classes of subgroups.

The subgroup lattice has 210 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	C₈×Dic₁₀₆	C₂	1	2	yes	maximal, complemented
3	#16810702	C₂	1	2	yes	maximal, complemented
4	#16810699	C₂	1	4	yes	maximal, complemented
5	#16810680	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810678	C₂	1	2	yes	maximal, complemented
7	#16810575	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×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	2	yes	—
21	#13133146	C₂²	1	4	yes	—
22	C₂²⋊#88322	C₂²	1	2	yes	—
23	#13133129	C₂²	1	4	yes	—
24	#13133128	C₂²	1	2	yes	—
25	C₈×Dic₅₃	C₂²	1	4	yes	—
26	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	#97×C₅₃	C₂²	1	1	yes	characteristic
29	#97×C₅₃	C₂²	1	1	yes	characteristic
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	M₁₆×C₅₃	—	2	4	no	—
42	C₂×C₈×C₅₃	C₂³	1	2	yes	—
43	C₂×C₈×C₅₃	C₂³	1	2	yes	—
44	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
49	#11719921	C₂×C₄	1	2	yes	—
50	#11719921	C₂×C₄	1	2	yes	—
51	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
53	#11719919	C₂×C₄	1	2	yes	—
54	#11719919	C₂×C₄	1	2	yes	—
55	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
56	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
57	C₄×Dic₅₃	C₂³	1	2	yes	—
58	C₂×#88322	C₂³	1	4	yes	—
59	#11719912	—	2	8	no	—
60	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	4	1	no	—
65	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
71	C₂²⋊D₅₃	—	2	4	no	—
72	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
73	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
74	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
75	D₂₁₂	—	2	2	no	—
76	C₄×D₅₃	—	2	4	no	—
77	C₄×D₅₃	—	2	4	no	—
78	C₄×D₅₃	—	4	2	no	—
79	Dic₁₀₆	M₁₆	1	4	yes	complemented
80	C₈×C₅₃	—	2	2	no	—
81	C₈×C₅₃	#41	1	4	yes	complemented
82	#88322	—	2	4	no	—
83	C₂²×C₅₃	—	2	1	no	—
84	C₂²×C₅₃	—	2	1	no	—
85	C₂²×C₅₃	—	2	1	no	—
86	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
87	C₂×D₅₃	—	2	2	no	—
88	C₂×D₅₃	—	2	2	no	—
89	C₂×D₅₃	—	2	2	no	—
90	C₄×C₅₃	—	2	1	no	—
91	C₄×C₅₃	—	2	1	no	—
92	C₄×C₅₃	—	2	1	no	—
93	C₄×C₅₃	—	2	1	no	—
94	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
95	C₄×C₅₃	C₂×#41	1	2	yes	—
96	Dic₅₃	—	2	2	no	—
97	Dic₅₃	—	2	2	no	—
98	Dic₅₃	C₂×M₁₆	1	4	yes	—
99	#2971	—	53	1	no	maximal, Sylow
100	C₂×C₅₃	—	4	1	no	—
101	C₂×C₅₃	C₄×#41	1	1	yes	characteristic
102	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
103	C₂×C₅₃	#568	1	1	yes	characteristic
104	D₅₃	—	4	2	no	—
105	C₄×#41	—	53	1	no	—
106	C₈×Q₈	—	53	2	no	—
107	#436	—	53	4	no	—
108	#436	—	53	2	no	—
109	#432	—	53	1	no	—
110	#432	—	53	2	no	—
111	C₄×M₁₆	—	53	2	no	—
112	C₄×M₁₆	D₅₃	1	1	yes	characteristic, complemented
113	C₅₃	#2971	1	1	yes	characteristic, complemented, Sylow
114	C₂×#41	—	53	1	no	—
115	C₂×M₁₆	—	53	4	no	—
116	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
117	C₄×Q₈	—	53	1	no	—
118	C₄×Q₈	—	53	1	no	—
119	C₄×D₄	—	53	2	no	—
120	C₄×D₄	—	53	2	no	—
121	C₄×D₄	—	53	1	no	—
122	C₄×D₄	—	53	1	no	—
123	#117	—	53	1	no	—
124	#117	—	53	2	no	—
125	C₂×C₄²	—	53	2	no	—
126	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
127	C₄⋊C₈	—	53	4	no	—
128	C₄⋊C₈	—	53	2	no	—
129	C₂²⋊C₈	—	53	2	no	—
130	C₂²⋊C₈	—	53	4	no	—
131	#97	—	53	2	no	—
132	#97	C₂×D₅₃	1	1	yes	characteristic
133	#97	C₂×D₅₃	1	1	yes	characteristic
134	C₄×C₈	—	53	4	no	—
135	C₄×C₈	C₂×D₅₃	1	2	yes	—
136	#41	—	106	4	no	—
137	C₂×Q₈	—	53	1	no	—
138	C₂×D₄	—	53	2	no	—
139	C₂×D₄	—	53	1	no	—
140	C₂²×C₄	—	53	2	no	—
141	C₂²×C₄	—	53	2	no	—
142	C₂²×C₄	—	53	2	no	—
143	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
146	M₁₆	—	2	4	no	—
147	M₁₆	—	106	8	no	—
148	C₂×C₈	—	53	4	no	—
149	C₂×C₈	C₂²×D₅₃	1	2	yes	—
150	C₂×C₈	C₂²×D₅₃	1	2	yes	—
151	C₄⋊C₄	—	53	2	no	—
152	C₄⋊C₄	—	53	2	no	—
153	C₄⋊C₄	—	53	1	no	—
154	C₄⋊C₄	—	53	1	no	—
155	C₂²⋊C₄	—	53	2	no	—
156	C₂²⋊C₄	—	53	2	no	—
157	C₂²⋊C₄	—	53	2	no	—
158	C₄²	—	53	2	no	—
159	C₄²	—	53	2	no	—
160	C₄²	—	53	2	no	—
161	C₄²	C₄×D₅₃	1	1	yes	characteristic
162	C₄²	C₄×D₅₃	1	1	yes	characteristic
163	C₄²	C₂²×D₅₃	1	1	yes	characteristic
164	C₄²	C₂²×D₅₃	1	1	yes	characteristic
165	C₂³	—	53	2	no	—
166	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
167	Q₈	—	53	4	no	—
168	D₄	—	106	4	no	—
169	D₄	—	106	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	53	2	no	—
174	C₂×C₄	—	53	2	no	—
175	C₂×C₄	—	53	2	no	—
176	C₂×C₄	—	106	4	no	—
177	C₂×C₄	—	106	4	no	—
178	C₂×C₄	—	212	2	no	—
179	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	106	4	no	—
187	C₈	#11719945	1	4	yes	complemented
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	106	2	no	—
192	C₂²	—	106	2	no	—
193	C₂²	—	106	2	no	—
194	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	53	4	no	—
200	C₄	—	106	2	no	—
201	C₄	—	106	2	no	—
202	C₄	M₁₆×D₅₃	1	2	yes	—
203	C₄	C₂×#11719945	1	2	yes	—
204	C₂	—	4	1	no	—
205	C₂	—	212	2	no	—
206	C₂	C₄×#11719945	1	1	yes	characteristic
207	C₂	C₂×M₁₆×D₅₃	1	1	yes	characteristic
208	C₂	#16811389	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₄×M₁₆×C₅₃	C₂
hypercenter	C₄×M₁₆	D₅₃
hypocenter	C₅₃	#2971
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₁₀₆	C₂	semidirect
3. #16810702	C₂	semidirect
4. #16810699	C₂	semidirect
5. #16810680	C₂	semidirect
6. #16810678	C₂	semidirect
7. #16810575	C₂	semidirect
8. C₄×M₁₆×C₅₃	C₂	semidirect
79. Dic₁₀₆	M₁₆	semidirect
81. C₈×C₅₃	#41	semidirect
112. C₄×M₁₆	D₅₃	semidirect
113. C₅₃	#2971	semidirect
187. C₈	#11719945	semidirect