Subgroups — C₂²×#16810402 #27536756

Order 13568.

1470 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810402	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₄×D₅₃	C₄	1	1	yes	characteristic
6	C₂³×#88322	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅₃	—	2	4	no	—
8	C₂²×C₈×D₅₃	—	2	4	no	—
9	C₂²×C₈×D₅₃	C₄	1	6	yes	—
10	C₂²×C₈×D₅₃	C₂²	1	6	yes	—
11	C₂²×#11719908	—	2	4	no	—
12	C₂²×#11719908	C₂²	1	6	yes	complemented
13	#16810402	C₂²	1	16	yes	direct product factor
14	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₅₃	—	2	4	no	—
16	C₂²×C₁₆×C₅₃	C₂²	1	6	yes	complemented
17	C₂⁴×D₅₃	C₈	1	1	yes	characteristic
18	C₂³×Dic₅₃	C₈	1	1	yes	characteristic
19	C₂²×C₄×D₅₃	—	2	4	no	—
20	C₂²×C₄×D₅₃	—	2	4	no	—
21	C₂²×C₄×D₅₃	C₈	1	6	yes	—
22	C₂²×C₄×D₅₃	C₂×C₄	1	6	yes	—
23	C₂²×#88322	—	2	4	no	—
24	C₂²×#88322	C₂×C₄	1	6	yes	—
25	C₂×C₈×D₅₃	—	2	24	no	—
26	C₂×C₈×D₅₃	—	2	12	no	—
27	C₂×C₈×D₅₃	—	2	12	no	—
28	C₂×C₈×D₅₃	C₂×C₄	1	12	yes	—
29	C₂×C₈×D₅₃	C₂³	1	4	yes	—
30	C₂×#11719908	—	2	12	no	—
31	C₂×#11719908	C₂³	1	4	yes	complemented
32	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
34	C₂²×C₈×C₅₃	D₄	1	4	yes	—
35	C₂²×C₈×C₅₃	D₄	1	4	yes	—
36	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
37	C₂×C₁₆×C₅₃	—	2	12	no	—
38	C₂×C₁₆×C₅₃	C₂³	1	4	yes	complemented
39	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂³×D₅₃	—	2	4	no	—
41	C₂³×D₅₃	—	2	4	no	—
42	C₂³×D₅₃	C₁₆	1	8	yes	complemented
43	C₂³×D₅₃	C₂×C₈	1	6	yes	—
44	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
46	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
48	C₂²×Dic₅₃	—	2	4	no	—
49	C₂²×Dic₅₃	C₂×C₈	1	6	yes	—
50	C₂×C₄×D₅₃	—	2	24	no	—
51	C₂×C₄×D₅₃	—	2	12	no	—
52	C₂×C₄×D₅₃	—	2	12	no	—
53	C₂×C₄×D₅₃	C₂×C₈	1	12	yes	—
54	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	12	yes	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
57	C₂×C₈×C₅₃	C₂×D₄	1	6	yes	—
58	C₂×C₈×C₅₃	C₂×D₄	1	6	yes	—
59	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
60	C₂×#88322	—	2	12	no	—
61	C₂×#88322	C₂²×C₄	1	4	yes	—
62	C₈×D₅₃	—	2	24	no	—
63	C₈×D₅₃	—	2	4	no	—
64	C₈×D₅₃	—	2	4	no	—
65	C₁₆×C₅₃	—	2	4	no	—
66	#11719908	—	2	4	no	—
67	C₂³×C₅₃	C₂²⋊C₈	1	4	yes	—
68	C₂³×C₅₃	C₂×C₁₆	1	4	yes	—
69	C₂³×C₅₃	M₃₂	1	4	yes	—
70	C₂³×C₅₃	C₂²×C₈	1	3	yes	—
71	C₂²×D₅₃	—	2	32	no	—
72	C₂²×D₅₃	—	2	12	no	—
73	C₂²×D₅₃	—	2	12	no	—
74	C₂²×D₅₃	C₂×C₁₆	1	24	yes	complemented
75	C₂²×D₅₃	C₂²×C₈	1	4	yes	—
76	C₂×C₄×C₅₃	C₂²⋊C₈	1	12	yes	—
77	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
78	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
79	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
80	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
81	C₂×Dic₅₃	—	2	12	no	—
82	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
83	C₄×D₅₃	—	2	4	no	—
84	C₄×D₅₃	—	2	4	no	—
85	C₄×D₅₃	—	2	24	no	—
86	C₈×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
87	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
88	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
89	#88322	—	2	4	no	—
90	C₂²×C₂²⋊C₁₆	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	C₂²⋊C₁₆	1	16	yes	complemented
92	C₂²×C₅₃	C₂×C₂²⋊C₈	1	6	yes	—
93	C₂²×C₅₃	C₂²×C₁₆	1	6	yes	—
94	C₂²×C₅₃	C₂×M₃₂	1	6	yes	—
95	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
96	C₂×D₅₃	—	2	4	no	—
97	C₂×D₅₃	—	2	48	no	—
98	C₂×D₅₃	—	2	4	no	—
99	C₂×D₅₃	C₂²×C₁₆	1	8	yes	complemented
100	C₄×C₅₃	C₂×C₂²⋊C₈	1	6	yes	—
101	C₄×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
102	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
103	Dic₅₃	—	2	4	no	—
104	C₂⁴×C₈	—	53	1	no	—
105	C₂³×C₁₆	—	53	1	no	—
106	C₂³×C₁₆	D₅₃	1	1	yes	characteristic, complemented
107	C₂×C₂²⋊C₁₆	—	53	12	no	—
108	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	12	yes	complemented
109	C₂×C₅₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
110	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
111	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
112	D₅₃	—	2	8	no	—
113	C₂⁴×C₄	—	53	1	no	—
114	C₂³×C₈	—	53	1	no	—
115	C₂³×C₈	—	53	6	no	—
116	C₂³×C₈	—	53	6	no	—
117	C₂³×C₈	—	106	4	no	—
118	C₂³×C₈	—	106	4	no	—
119	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
120	C₂²×C₁₆	—	2	4	no	—
121	C₂²×C₁₆	—	53	6	no	—
122	C₂²×C₁₆	—	106	4	no	—
123	C₂²×C₁₆	C₂×D₅₃	1	6	yes	complemented
124	C₂²⋊C₁₆	—	53	16	no	—
125	C₅₃	C₂²×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
126	C₂⁵	—	53	1	no	—
127	C₂³×C₄	—	53	1	no	—
128	C₂³×C₄	—	53	6	no	—
129	C₂³×C₄	—	53	6	no	—
130	C₂³×C₄	—	106	4	no	—
131	C₂³×C₄	—	106	4	no	—
132	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
133	C₂²×C₈	—	53	6	no	—
134	C₂²×C₈	—	53	12	no	—
135	C₂²×C₈	—	53	4	no	—
136	C₂²×C₈	—	106	4	no	—
137	C₂²×C₈	—	106	12	no	—
138	C₂²×C₈	—	106	12	no	—
139	C₂²×C₈	—	106	24	no	—
140	C₂²×C₈	C₄×D₅₃	1	3	yes	—
141	C₂²×C₈	D₂₁₂	1	4	yes	—
142	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
143	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
144	C₂×C₁₆	—	2	12	no	—
145	C₂×C₁₆	—	53	4	no	—
146	C₂×C₁₆	—	106	12	no	—
147	C₂×C₁₆	C₂²×D₅₃	1	4	yes	complemented
148	C₂⁴	—	53	8	no	—
149	C₂⁴	—	53	6	no	—
150	C₂⁴	—	106	4	no	—
151	C₂⁴	—	106	4	no	—
152	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
153	C₂²×C₄	—	53	6	no	—
154	C₂²×C₄	—	53	12	no	—
155	C₂²×C₄	—	53	4	no	—
156	C₂²×C₄	—	106	4	no	—
157	C₂²×C₄	—	106	24	no	—
158	C₂²×C₄	—	106	12	no	—
159	C₂²×C₄	—	106	12	no	—
160	C₂²×C₄	C₈×D₅₃	1	3	yes	—
161	C₂²×C₄	#11719912	1	4	yes	—
162	C₂²×C₄	#11719921	1	4	yes	—
163	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
164	C₂×C₈	—	53	4	no	—
165	C₂×C₈	—	106	12	no	—
166	C₂×C₈	—	106	24	no	—
167	C₂×C₈	—	106	4	no	—
168	C₂×C₈	—	106	4	no	—
169	C₂×C₈	#11719921	1	12	yes	—
170	C₂×C₈	C₂×C₄×D₅₃	1	3	yes	—
171	C₂×C₈	C₂×D₂₁₂	1	6	yes	—
172	C₂×C₈	C₂×C₂²⋊D₅₃	1	6	yes	—
173	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
174	C₁₆	—	2	4	no	—
175	C₁₆	—	106	4	no	—
176	C₂³	—	53	24	no	—
177	C₂³	—	53	4	no	—
178	C₂³	—	106	12	no	—
179	C₂³	—	106	12	no	—
180	C₂³	—	106	32	no	—
181	C₂³	#13133129	1	4	yes	—
182	C₂³	C₁₆×D₅₃	1	4	yes	—
183	C₂³	#13133156	1	4	yes	—
184	C₂³	C₂×C₈×D₅₃	1	3	yes	—
185	C₂×C₄	—	53	4	no	—
186	C₂×C₄	—	106	12	no	—
187	C₂×C₄	—	106	4	no	—
188	C₂×C₄	—	106	24	no	—
189	C₂×C₄	—	106	4	no	—
190	C₂×C₄	#13133129	1	12	yes	—
191	C₂×C₄	C₂×#11719921	1	6	yes	—
192	C₂×C₄	C₂×C₈×D₅₃	1	3	yes	—
193	C₂×C₄	C₂×#11719912	1	6	yes	—
194	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
195	C₈	—	106	4	no	—
196	C₈	C₂×#11719921	1	6	yes	—
197	C₈	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
198	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
199	C₂²	—	53	8	no	—
200	C₂²	—	106	4	no	—
201	C₂²	—	106	4	no	—
202	C₂²	—	106	48	no	—
203	C₂²	#16810402	1	16	yes	direct product factor
204	C₂²	C₂×#13133129	1	6	yes	—
205	C₂²	C₂×C₁₆×D₅₃	1	6	yes	—
206	C₂²	C₂×#13133156	1	6	yes	—
207	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
208	C₄	—	106	4	no	—
209	C₄	C₂×#13133129	1	6	yes	—
210	C₄	C₂²×#11719921	1	1	yes	characteristic
211	C₄	C₂²×#11719912	1	1	yes	characteristic
212	C₂	—	106	8	no	—
213	C₂	C₂×#16810402	1	12	yes	direct product factor
214	C₂	C₂²×#13133129	1	1	yes	characteristic
215	C₂	C₂²×C₁₆×D₅₃	1	1	yes	characteristic
216	C₂	C₂²×#13133156	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂³×C₁₆×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂×C₈
prime radical	C₂⁴×D₅₃	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂³×C₁₆	D₅₃
hypocenter	C₅₃	C₂²×C₂²⋊C₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂³×#11719908	C₂	semidirect
3. C₂×#16810402	C₂	direct (N × H)
4. C₂³×C₁₆×C₅₃	C₂	semidirect
12. C₂²×#11719908	C₂²	semidirect
13. #16810402	C₂²	direct (N × H)
16. C₂²×C₁₆×C₅₃	C₂²	semidirect
31. C₂×#11719908	C₂³	semidirect
38. C₂×C₁₆×C₅₃	C₂³	semidirect
42. C₂³×D₅₃	C₁₆	semidirect
74. C₂²×D₅₃	C₂×C₁₆	semidirect
91. C₂²×C₅₃	C₂²⋊C₁₆	semidirect
99. C₂×D₅₃	C₂²×C₁₆	semidirect
106. C₂³×C₁₆	D₅₃	semidirect
108. C₂×C₅₃	C₂×C₂²⋊C₁₆	semidirect
123. C₂²×C₁₆	C₂×D₅₃	semidirect
125. C₅₃	C₂²×C₂²⋊C₁₆	semidirect
147. C₂×C₁₆	C₂²×D₅₃	semidirect
203. C₂²	#16810402	direct (N × H)
213. C₂	C₂×#16810402	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₈×D₅₃	2
C₂³×#11719908	2
C₂×#16810402	2
C₂³×C₁₆×C₅₃	2
C₂²×C₂²⋊C₁₆	53

Subgroups — C22×#16810402 #27536756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810402 #27536756