Subgroups — #20274424

Order 6784.

404 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	C₂²×#11719917	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133176	C₂	1	1	yes	maximal, characteristic
3	#16810382	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	—	2	2	no	—
6	C₂×#11719917	—	2	1	no	—
7	C₂×#11719917	C₄	1	4	yes	complemented
8	C₂×#11719917	C₂²	1	2	yes	—
9	C₂²×#88322	—	2	1	no	—
10	#13133176	—	2	2	no	—
11	#13133176	C₄	1	4	yes	complemented
12	C₂×C₄⋊Dic₅₃	—	2	1	no	—
13	C₂×C₂²⋊Dic₅₃	—	2	1	no	—
14	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
15	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
16	#117×C₅₃	—	2	1	no	—
17	#117×C₅₃	—	2	1	no	—
18	#117×C₅₃	C₄	1	2	yes	complemented
19	#117×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₅₃	C₄	1	2	yes	complemented
21	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
22	C₂×C₄²×C₅₃	—	2	1	no	—
23	C₂⁴×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	—	2	2	no	—
25	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	D₄	1	2	yes	—
31	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₅₃	—	2	2	no	—
36	C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
38	C₂²⋊C₄×C₅₃	—	2	2	no	—
39	C₂²⋊C₄×C₅₃	—	2	2	no	—
40	C₄²×C₅₃	—	2	2	no	—
41	C₄²×C₅₃	—	2	2	no	—
42	C₂²⋊Dic₅₃	—	2	4	no	—
43	C₄⋊Dic₅₃	—	2	4	no	—
44	C₄×Dic₅₃	—	2	2	no	—
45	C₄×Dic₅₃	C₂×C₄	1	4	yes	complemented
46	#11719917	—	2	4	no	—
47	#11719917	—	2	2	no	—
48	#11719917	C₂×C₄	1	4	yes	complemented
49	C₂×#88322	—	2	4	no	—
50	C₂×#88322	—	2	2	no	—
51	C₂³×C₅₃	—	4	2	no	—
52	C₂³×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
56	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	4	no	—
58	C₂×C₄×C₅₃	—	4	4	no	—
59	C₂×C₄×C₅₃	C₄²	1	2	yes	—
60	C₂×C₄×C₅₃	C₄²	1	2	yes	—
61	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
77	C₂×Dic₅₃	—	2	4	no	—
78	C₂×Dic₅₃	—	4	4	no	—
79	#88322	—	2	4	no	—
80	C₂²×C₅₃	—	2	2	no	—
81	C₂²×C₅₃	—	2	2	no	—
82	C₂²×C₅₃	—	2	2	no	—
83	C₂²×C₅₃	—	4	2	no	—
84	C₂²×C₅₃	—	4	2	no	—
85	C₂²×C₅₃	#95	1	2	yes	—
86	C₂²×C₅₃	#95	1	2	yes	—
87	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₅₃	—	4	4	no	—
91	C₄×C₅₃	#95	1	2	yes	—
92	C₄×C₅₃	#95	1	2	yes	—
93	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
94	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	Dic₅₃	—	4	4	no	—
98	#1733	—	53	1	no	maximal, Sylow
99	C₂×C₅₃	—	2	1	no	—
100	C₂×C₅₃	—	2	2	no	—
101	C₂×C₅₃	—	2	2	no	—
102	C₂×C₅₃	—	2	1	no	—
103	C₂×C₅₃	C₂×#95	1	1	yes	characteristic
104	C₂×C₅₃	#421	1	2	yes	—
105	C₂²×M₁₆	—	53	1	no	—
106	C₂×#117	—	53	1	no	—
107	C₂×#117	D₅₃	1	1	yes	characteristic
108	#339	—	53	4	no	—
109	C₅₃	#1733	1	1	yes	characteristic, complemented, Sylow
110	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
111	C₂×M₁₆	—	53	4	no	—
112	C₂×M₁₆	—	53	2	no	—
113	C₂×M₁₆	—	106	2	no	—
114	C₂×M₁₆	—	106	1	no	—
115	C₂²×C₈	—	106	1	no	—
116	#117	—	2	1	no	—
117	#117	—	2	1	no	—
118	#117	—	53	4	no	—
119	#117	—	106	2	no	—
120	#117	Dic₅₃	1	2	yes	complemented
121	#117	C₂×D₅₃	1	2	yes	—
122	C₂×C₄⋊C₄	—	106	1	no	—
123	C₂×C₄⋊C₄	Dic₅₃	1	2	yes	complemented
124	C₂×C₂²⋊C₄	—	2	1	no	—
125	C₂×C₂²⋊C₄	—	106	1	no	—
126	C₂×C₄²	—	2	1	no	—
127	C₂×C₄²	—	53	2	no	—
128	C₂⁴	C₂²⋊D₅₃	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	106	2	no	—
131	C₂²×C₄	Dic₁₀₆	1	2	yes	—
132	C₂²×C₄	C₄×D₅₃	1	2	yes	—
133	C₂²×C₄	C₄×D₅₃	1	2	yes	—
134	C₂²×C₄	D₂₁₂	1	2	yes	—
135	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
137	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
138	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
139	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
140	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
141	M₁₆	—	53	4	no	—
142	M₁₆	—	106	2	no	—
143	M₁₆	—	106	4	no	—
144	C₂×C₈	—	106	2	no	—
145	C₂×C₈	—	106	4	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	106	4	no	—
148	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
149	C₂²⋊C₄	—	2	2	no	—
150	C₂²⋊C₄	—	2	2	no	—
151	C₂²⋊C₄	—	106	4	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	2	no	—
154	C₄²	—	53	4	no	—
155	C₄²	—	106	2	no	—
156	C₂³	—	4	2	no	—
157	C₂³	#11719919	1	2	yes	—
158	C₂³	#11719921	1	2	yes	—
159	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
160	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
161	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	—	106	4	no	—
165	C₂×C₄	—	212	4	no	—
166	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
167	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
168	C₂×C₄	#11719919	1	2	yes	—
169	C₂×C₄	#11719919	1	2	yes	—
170	C₂×C₄	#11719919	1	2	yes	—
171	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
172	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
173	C₂×C₄	#11719921	1	2	yes	—
174	C₂×C₄	#11719921	1	2	yes	—
175	C₂×C₄	#11719921	1	2	yes	—
176	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
177	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
178	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
181	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
182	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
183	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
184	C₈	—	106	4	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	#13133124	1	2	yes	—
191	C₂²	#13133124	1	2	yes	—
192	C₂²	C₂×#11719921	1	1	yes	characteristic
193	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
194	C₂²	C₂×#11719919	1	1	yes	characteristic
195	C₄	—	4	4	no	—
196	C₄	—	212	4	no	—
197	C₄	#13133124	1	2	yes	—
198	C₄	#13133124	1	2	yes	—
199	C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
200	C₄	C₂×#11719921	1	1	yes	characteristic
201	C₄	C₂×#11719919	1	1	yes	characteristic
202	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	2	no	—
205	C₂	—	2	2	no	—
206	C₂	—	2	1	no	—
207	C₂	C₂×#13133124	1	1	yes	characteristic
208	C₂	#16810640	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#117×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₂×#117	D₅₃
hypocenter	C₅₃	#1733
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #16810382	C₂	semidirect
7. C₂×#11719917	C₄	semidirect
11. #13133176	C₄	semidirect
18. #117×C₅₃	C₄	semidirect
20. C₂×C₄⋊C₄×C₅₃	C₄	semidirect
45. C₄×Dic₅₃	C₂×C₄	semidirect
48. #11719917	C₂×C₄	semidirect
109. C₅₃	#1733	semidirect
120. #117	Dic₅₃	semidirect
123. C₂×C₄⋊C₄	Dic₅₃	semidirect