Subgroups — #20283802

Order 6784.

364 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#16810753	C₂	1	2	yes	maximal, complemented
3	#16810752	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₁₀₆	C₂	1	2	yes	maximal, complemented
5	#16810680	C₂	1	1	yes	maximal, characteristic
6	#16810677	C₂	1	2	yes	maximal
7	#16810673	C₂	1	2	yes	maximal
8	C₅₃×#431	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₀₆	C₄	1	1	yes	characteristic
10	C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
11	C₄×Dic₁₀₆	C₄	1	2	yes	—
12	C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
13	C₄×Dic₁₀₆	C₂²	1	2	yes	—
14	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
15	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
17	C₂×#11719919	C₄	1	2	yes	—
18	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
19	C₄⋊#88322	C₂²	1	2	yes	—
20	#13133146	C₂²	1	4	yes	—
21	#13133146	C₂²	1	4	yes	—
22	C₂²⋊#88322	C₂²	1	2	yes	—
23	#13133128	C₂²	1	2	yes	—
24	#13133127	C₂²	1	4	yes	—
25	C₈×Dic₅₃	C₂²	1	4	yes	—
26	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
28	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²⋊C₈×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	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
35	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
36	C₂×Dic₁₀₆	—	2	4	no	—
37	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
38	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
39	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂³	1	2	yes	—
41	C₂×C₈×C₅₃	C₂³	1	2	yes	—
42	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
47	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
49	#11719919	C₂×C₄	1	2	yes	—
50	#11719919	C₂×C₄	1	2	yes	—
51	#11719919	C₂×C₄	1	2	yes	—
52	#11719919	C₂×C₄	1	2	yes	—
53	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
54	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
55	C₄×Dic₅₃	C₂³	1	2	yes	—
56	C₄×Dic₅₃	C₂³	1	2	yes	—
57	C₂×#88322	C₂³	1	4	yes	—
58	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₅₃	—	2	4	no	—
69	C₂×Dic₅₃	—	2	4	no	—
70	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
71	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
72	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
73	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
74	Dic₁₀₆	—	2	2	no	—
75	Dic₁₀₆	—	2	4	no	—
76	Dic₁₀₆	M₁₆	1	4	yes	—
77	C₈×C₅₃	—	2	2	no	—
78	C₈×C₅₃	—	2	2	no	—
79	#88322	—	2	4	no	—
80	C₂²×C₅₃	—	2	1	no	—
81	C₂²×C₅₃	—	2	1	no	—
82	C₂²×C₅₃	#131	1	2	yes	—
83	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
84	C₄×C₅₃	—	2	1	no	—
85	C₄×C₅₃	—	2	1	no	—
86	C₄×C₅₃	—	2	1	no	—
87	C₄×C₅₃	—	2	1	no	—
88	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
89	C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
90	Dic₅₃	—	2	2	no	—
91	Dic₅₃	—	2	2	no	—
92	Dic₅₃	—	2	2	no	—
93	Dic₅₃	C₂×M₁₆	1	4	yes	—
94	#2984	—	53	1	no	maximal, Sylow
95	C₂×C₅₃	—	2	2	no	—
96	C₂×C₅₃	#519	1	1	yes	characteristic
97	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
98	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
99	C₂×C₄×Q₈	—	53	1	no	—
100	#446	—	53	4	no	—
101	#446	—	53	2	no	—
102	C₈×Q₈	—	53	2	no	—
103	#432	—	53	1	no	—
104	#432	—	53	2	no	—
105	#431	—	53	2	no	—
106	#431	D₅₃	1	1	yes	characteristic
107	C₅₃	#2984	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₈	—	53	1	no	—
109	C₄×Q₈	—	53	1	no	—
110	C₄×Q₈	—	53	2	no	—
111	C₄×Q₈	—	53	1	no	—
112	C₄×Q₈	—	53	1	no	—
113	C₄×Q₈	—	53	2	no	—
114	C₄×Q₈	—	53	1	no	—
115	C₂×C₄⋊C₄	—	53	2	no	—
116	C₂×C₄⋊C₄	—	53	1	no	—
117	C₂×C₄²	—	53	2	no	—
118	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
119	C₄⋊C₈	—	53	4	no	—
120	C₄⋊C₈	—	53	4	no	—
121	C₄⋊C₈	—	53	2	no	—
122	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
123	C₂²⋊C₈	—	53	2	no	—
124	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
125	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
126	#97	—	53	4	no	—
127	#97	—	53	2	no	—
128	C₄×C₈	—	53	4	no	—
129	C₄×C₈	C₂×D₅₃	1	2	yes	—
130	C₂×Q₈	—	53	1	no	—
131	C₂×Q₈	—	53	2	no	—
132	C₂×Q₈	—	53	1	no	—
133	C₂×Q₈	—	106	4	no	—
134	C₂²×C₄	—	53	2	no	—
135	C₂²×C₄	—	53	2	no	—
136	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
137	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
138	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
139	C₂×C₈	—	53	4	no	—
140	C₂×C₈	C₂²×D₅₃	1	2	yes	—
141	C₂×C₈	C₂²×D₅₃	1	2	yes	—
142	C₄⋊C₄	—	53	2	no	—
143	C₄⋊C₄	—	53	2	no	—
144	C₄⋊C₄	—	53	2	no	—
145	C₄⋊C₄	—	53	2	no	—
146	C₄⋊C₄	—	53	2	no	—
147	C₄⋊C₄	—	53	1	no	—
148	C₄⋊C₄	—	53	1	no	—
149	C₄²	—	53	2	no	—
150	C₄²	—	53	2	no	—
151	C₄²	—	53	2	no	—
152	C₄²	—	53	2	no	—
153	C₄²	C₄×D₅₃	1	1	yes	characteristic
154	C₄²	C₄×D₅₃	1	1	yes	characteristic
155	C₄²	C₂²×D₅₃	1	1	yes	characteristic
156	C₄²	C₂²×D₅₃	1	1	yes	characteristic
157	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
158	Q₈	—	53	4	no	—
159	Q₈	—	106	4	no	—
160	Q₈	—	106	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	53	2	no	—
165	C₂×C₄	—	53	2	no	—
166	C₂×C₄	—	53	2	no	—
167	C₂×C₄	—	53	2	no	—
168	C₂×C₄	—	106	4	no	—
169	C₂×C₄	—	106	4	no	—
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	C₂×C₄×D₅₃	1	1	yes	characteristic
175	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	2	2	no	—
178	C₈	—	106	4	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	#13133223	1	2	yes	—
182	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	53	4	no	—
188	C₄	—	106	2	no	—
189	C₄	—	106	2	no	—
190	C₄	—	106	2	no	—
191	C₄	M₁₆×D₅₃	1	2	yes	—
192	C₄	#13133264	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	#16811100	1	1	yes	characteristic
195	C₂	C₂×M₁₆×D₅₃	1	1	yes	characteristic
196	C₂	C₂×#13133223	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	#2984
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. #16810753	C₂	semidirect
3. #16810752	C₂	semidirect
4. C₈×Dic₁₀₆	C₂	semidirect
107. C₅₃	#2984	semidirect