Subgroups — #20274441

Order 6784.

372 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11719917	C₂	1	1	yes	maximal, characteristic
2	C₂²×#11719917	C₂	1	1	yes	maximal, characteristic
3	#16810387	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	—	2	2	no	—
6	C₂×#11719917	—	2	2	no	—
7	C₂×#11719917	—	2	1	no	—
8	C₂×#11719917	—	2	2	no	—
9	C₂×#11719917	C₄	1	4	yes	—
10	C₂×#11719917	C₄	1	2	yes	—
11	C₂×#11719917	C₄	1	2	yes	—
12	C₂×#11719917	C₂²	1	2	yes	—
13	C₂²×#88322	—	2	1	no	—
14	C₂²×#88322	C₂²	1	2	yes	—
15	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₅₃	—	2	1	no	—
17	C₂×M₁₆×C₅₃	—	2	2	no	—
18	C₂×M₁₆×C₅₃	C₄	1	4	yes	—
19	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
20	C₂²×C₈×C₅₃	—	2	1	no	—
21	C₂⁴×C₅₃	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₅₃	D₄	1	2	yes	—
27	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	D₄	1	2	yes	—
30	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	M₁₆×C₅₃	—	2	4	no	—
33	M₁₆×C₅₃	—	2	2	no	—
34	M₁₆×C₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₈×C₅₃	—	2	4	no	—
36	C₂×C₈×C₅₃	—	2	2	no	—
37	#11719917	—	2	4	no	—
38	#11719917	—	2	4	no	—
39	#11719917	—	2	4	no	—
40	#11719917	—	2	2	no	—
41	#11719917	C₂×C₄	1	4	yes	—
42	C₂×#88322	—	2	4	no	—
43	C₂×#88322	—	2	2	no	—
44	C₂×#88322	—	2	2	no	—
45	C₂×#88322	—	2	2	no	—
46	C₂×#88322	C₂×C₄	1	4	yes	—
47	C₂³×C₅₃	—	4	2	no	—
48	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
51	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₄²	1	2	yes	—
54	C₂×C₄×C₅₃	C₄²	1	2	yes	—
55	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₈×C₅₃	—	2	4	no	—
72	#88322	—	2	4	no	—
73	#88322	—	2	4	no	—
74	C₂²×C₅₃	—	2	2	no	—
75	C₂²×C₅₃	—	2	2	no	—
76	C₂²×C₅₃	—	2	2	no	—
77	C₂²×C₅₃	—	4	2	no	—
78	C₂²×C₅₃	—	4	2	no	—
79	C₂²×C₅₃	#95	1	2	yes	—
80	C₂²×C₅₃	#95	1	2	yes	—
81	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₅₃	#95	1	2	yes	—
85	C₄×C₅₃	#95	1	2	yes	—
86	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
87	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	#1738	—	53	1	no	maximal, Sylow
91	C₂×C₅₃	—	2	2	no	—
92	C₂×C₅₃	—	2	2	no	—
93	C₂×C₅₃	—	2	1	no	—
94	C₂×C₅₃	—	2	1	no	—
95	C₂×C₅₃	C₂×#95	1	1	yes	characteristic
96	C₂×C₅₃	#430	1	2	yes	—
97	C₂²×M₁₆	—	53	1	no	—
98	C₂²×M₁₆	—	53	1	no	—
99	C₂²×M₁₆	D₅₃	1	1	yes	characteristic
100	#341	—	53	4	no	—
101	C₅₃	#1738	1	1	yes	characteristic, complemented, Sylow
102	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
103	C₂×M₁₆	—	2	1	no	—
104	C₂×M₁₆	—	2	2	no	—
105	C₂×M₁₆	—	53	2	no	—
106	C₂×M₁₆	—	53	2	no	—
107	C₂×M₁₆	—	53	4	no	—
108	C₂×M₁₆	—	53	2	no	—
109	C₂×M₁₆	—	106	2	no	—
110	C₂×M₁₆	—	106	2	no	—
111	C₂×M₁₆	—	106	2	no	—
112	C₂×M₁₆	—	106	1	no	—
113	C₂×M₁₆	Dic₅₃	1	4	yes	—
114	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	—	53	2	no	—
117	C₂²×C₈	—	106	1	no	—
118	C₂⁴	Dic₁₀₆	1	1	yes	characteristic
119	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
120	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
121	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
122	C₂²×C₄	C₄×D₅₃	1	2	yes	—
123	C₂²×C₄	D₂₁₂	1	2	yes	—
124	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
125	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
126	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
129	M₁₆	—	2	2	no	—
130	M₁₆	—	2	4	no	—
131	M₁₆	—	53	4	no	—
132	M₁₆	—	106	2	no	—
133	M₁₆	—	106	4	no	—
134	M₁₆	—	106	4	no	—
135	M₁₆	—	106	4	no	—
136	M₁₆	C₂×Dic₅₃	1	4	yes	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	53	4	no	—
140	C₂×C₈	—	106	2	no	—
141	C₂×C₈	—	106	2	no	—
142	C₂×C₈	—	106	2	no	—
143	C₂×C₈	—	106	4	no	—
144	C₂³	—	4	2	no	—
145	C₂³	#11719919	1	1	yes	characteristic
146	C₂³	#11719919	1	1	yes	characteristic
147	C₂³	#11719919	1	2	yes	—
148	C₂³	C₄⋊Dic₅₃	1	2	yes	—
149	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
150	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
151	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
152	C₂×C₄	#11719919	1	2	yes	—
153	C₂×C₄	#11719919	1	1	yes	characteristic
154	C₂×C₄	#11719919	1	1	yes	characteristic
155	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
156	C₂×C₄	#11719921	1	2	yes	—
157	C₂×C₄	#11719921	1	2	yes	—
158	C₂×C₄	#11719921	1	2	yes	—
159	C₂×C₄	#11719921	1	2	yes	—
160	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
161	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
162	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
163	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
164	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
165	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
166	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
167	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
168	C₈	—	2	4	no	—
169	C₈	—	106	4	no	—
170	C₈	—	106	4	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	4	2	no	—
175	C₂²	—	4	2	no	—
176	C₂²	#13133124	1	2	yes	—
177	C₂²	#13133124	1	2	yes	—
178	C₂²	C₂×#11719919	1	1	yes	characteristic
179	C₂²	C₂×#11719919	1	1	yes	characteristic
180	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
181	C₄	#13133124	1	2	yes	—
182	C₄	#13133124	1	2	yes	—
183	C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
184	C₄	C₂×#11719921	1	1	yes	characteristic
185	C₄	C₂×#11719921	1	1	yes	characteristic
186	C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
187	C₂	—	2	2	no	—
188	C₂	—	2	2	no	—
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	C₂×#13133124	1	1	yes	characteristic
192	C₂	#16810671	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂²×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₂²×M₁₆	D₅₃
hypocenter	C₅₃	#1738
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #16810387	C₂	semidirect
101. C₅₃	#1738	semidirect