Subgroups — C₂²×#16810386 #27381488

Order 13568.

3006 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊Dic₅₃	C₂	1	2	yes	maximal
2	C₂×#16810386	C₂	1	12	yes	maximal, direct product factor
3	C₂⁴×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂⁴×Dic₅₃	—	2	2	no	—
5	C₂²×C₄⋊Dic₅₃	—	2	6	no	—
6	C₂²×C₄⋊Dic₅₃	—	2	8	no	—
7	C₂²×C₄⋊Dic₅₃	C₄	1	16	yes	complemented
8	C₂²×C₄⋊Dic₅₃	C₂²	1	12	yes	—
9	#16810386	C₂²	1	16	yes	direct product factor
10	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₅₃	—	2	8	no	—
12	C₂³×C₈×C₅₃	C₄	1	8	yes	complemented
13	C₂³×C₈×C₅₃	C₂²	1	6	yes	—
14	C₂³×Dic₅₃	—	2	16	no	—
15	C₂³×Dic₅₃	—	2	12	no	—
16	C₂³×Dic₅₃	—	4	8	no	—
17	C₂³×Dic₅₃	—	4	8	no	—
18	C₂×C₄⋊Dic₅₃	—	2	48	no	—
19	C₂×C₄⋊Dic₅₃	—	2	24	no	—
20	C₂×C₄⋊Dic₅₃	—	2	12	no	—
21	C₂×C₄⋊Dic₅₃	C₂×C₄	1	48	yes	complemented
22	C₂×C₄⋊Dic₅₃	C₂³	1	8	yes	—
23	C₂⁵×C₅₃	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
26	C₂³×C₄×C₅₃	D₄	1	3	yes	—
27	C₂³×C₄×C₅₃	D₄	1	8	yes	—
28	C₂³×C₄×C₅₃	Q₈	1	4	yes	—
29	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
30	C₂²×C₈×C₅₃	—	2	32	no	—
31	C₂²×C₈×C₅₃	—	2	24	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	24	yes	complemented
33	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
34	C₂⁴×C₅₃	C₂²⋊C₄	1	8	yes	—
35	C₂⁴×C₅₃	C₄⋊C₄	1	4	yes	—
36	C₂⁴×C₅₃	D₈	1	4	yes	—
37	C₂⁴×C₅₃	SD₁₆	1	8	yes	—
38	C₂⁴×C₅₃	Q₁₆	1	4	yes	—
39	C₂⁴×C₅₃	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₅₃	C₄²	1	16	yes	—
41	C₂²×C₄×C₅₃	C₂²⋊C₄	1	24	yes	—
42	C₂²×C₄×C₅₃	C₂²⋊C₄	1	16	yes	—
43	C₂²×C₄×C₅₃	C₄⋊C₄	1	12	yes	—
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	32	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
49	C₂²×C₄×C₅₃	C₂×Q₈	1	6	yes	—
50	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	2	48	no	—
52	C₂²×Dic₅₃	—	2	8	no	—
53	C₂²×Dic₅₃	—	4	64	no	—
54	C₂²×Dic₅₃	—	4	24	no	—
55	C₂²×Dic₅₃	—	4	24	no	—
56	C₂×C₈×C₅₃	—	2	48	no	—
57	C₂×C₈×C₅₃	—	2	8	no	—
58	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	complemented
59	C₄⋊Dic₅₃	—	2	48	no	—
60	C₄⋊Dic₅₃	—	2	8	no	—
61	C₄⋊Dic₅₃	C₂²×C₄	1	16	yes	complemented
62	C₂³×C₅₃	#95	1	16	yes	—
63	C₂³×C₅₃	#102	1	32	yes	—
64	C₂³×C₅₃	#103	1	32	yes	—
65	C₂³×C₅₃	#106	1	16	yes	—
66	C₂³×C₅₃	#107	1	16	yes	—
67	C₂³×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
68	C₂³×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
69	C₂³×C₅₃	C₂×D₈	1	6	yes	—
70	C₂³×C₅₃	C₂×SD₁₆	1	12	yes	—
71	C₂³×C₅₃	C₂×Q₁₆	1	6	yes	—
72	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#95	1	48	yes	—
74	C₂×C₄×C₅₃	C₂×C₄²	1	12	yes	—
75	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
76	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	24	yes	—
78	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
79	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
80	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
82	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
83	C₂×Dic₅₃	—	2	16	no	—
84	C₂×Dic₅₃	—	4	8	no	—
85	C₂×Dic₅₃	—	4	8	no	—
86	C₂×Dic₅₃	—	4	96	no	—
87	C₈×C₅₃	—	2	8	no	—
88	C₂²×#340	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	#340	1	64	yes	complemented
90	C₂²×C₅₃	C₂×#95	1	12	yes	—
91	C₂²×C₅₃	C₂×#102	1	24	yes	—
92	C₂²×C₅₃	C₂×#103	1	24	yes	—
93	C₂²×C₅₃	C₂×#106	1	12	yes	—
94	C₂²×C₅₃	C₂×#107	1	12	yes	—
95	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	2	yes	—
96	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
98	C₂²×C₅₃	C₂²×SD₁₆	1	2	yes	—
99	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
100	C₄×C₅₃	C₂×#95	1	12	yes	—
101	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
102	C₄×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
103	C₄×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
104	Dic₅₃	—	4	16	no	—
105	C₂⁴×C₈	D₅₃	1	1	yes	characteristic
106	C₂³×C₄⋊C₄	—	53	2	no	—
107	C₂×#340	—	53	12	no	—
108	C₂×C₅₃	C₂×#340	1	24	yes	complemented
109	C₂×C₅₃	C₂²×#95	1	1	yes	characteristic
110	C₂×C₅₃	C₂²×#102	1	2	yes	—
111	C₂×C₅₃	C₂²×#103	1	2	yes	—
112	C₂×C₅₃	C₂²×#106	1	1	yes	characteristic
113	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
114	C₂⁴×C₄	—	106	2	no	—
115	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
116	C₂³×C₈	—	2	8	no	—
117	C₂³×C₈	Dic₅₃	1	8	yes	complemented
118	C₂³×C₈	C₂×D₅₃	1	6	yes	—
119	C₂²×C₄⋊C₄	—	53	16	no	—
120	C₂²×C₄⋊C₄	—	53	12	no	—
121	C₂²×C₄⋊C₄	—	106	6	no	—
122	C₂²×C₄⋊C₄	—	106	8	no	—
123	#340	—	53	16	no	—
124	C₅₃	C₂²×#340	1	1	yes	characteristic, complemented, Sylow
125	C₂⁵	D₂₁₂	1	1	yes	characteristic
126	C₂³×C₄	—	106	12	no	—
127	C₂³×C₄	—	106	16	no	—
128	C₂³×C₄	—	212	8	no	—
129	C₂³×C₄	—	212	8	no	—
130	C₂³×C₄	Dic₁₀₆	1	4	yes	—
131	C₂³×C₄	C₄×D₅₃	1	8	yes	—
132	C₂³×C₄	D₂₁₂	1	3	yes	—
133	C₂³×C₄	C₂×Dic₅₃	1	4	yes	—
134	C₂³×C₄	C₂²⋊D₅₃	1	8	yes	—
135	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
136	C₂²×C₈	—	2	24	no	—
137	C₂²×C₈	—	2	32	no	—
138	C₂²×C₈	C₂×Dic₅₃	1	24	yes	complemented
139	C₂²×C₈	C₂²×D₅₃	1	4	yes	—
140	C₂×C₄⋊C₄	—	53	48	no	—
141	C₂×C₄⋊C₄	—	53	8	no	—
142	C₂×C₄⋊C₄	—	106	12	no	—
143	C₂×C₄⋊C₄	—	106	24	no	—
144	C₂×C₄⋊C₄	—	106	48	no	—
145	C₂⁴	#11719913	1	8	yes	—
146	C₂⁴	D₄₂₄	1	4	yes	—
147	C₂⁴	Dic₂₁₂	1	4	yes	—
148	C₂⁴	C₄⋊Dic₅₃	1	4	yes	—
149	C₂⁴	#11719921	1	8	yes	—
150	C₂⁴	C₂×D₂₁₂	1	3	yes	—
151	C₂²×C₄	—	106	48	no	—
152	C₂²×C₄	—	106	8	no	—
153	C₂²×C₄	—	212	64	no	—
154	C₂²×C₄	—	212	24	no	—
155	C₂²×C₄	—	212	24	no	—
156	C₂²×C₄	C₄×Dic₅₃	1	16	yes	—
157	C₂²×C₄	#11719919	1	32	yes	—
158	C₂²×C₄	C₄⋊Dic₅₃	1	12	yes	—
159	C₂²×C₄	#11719921	1	24	yes	—
160	C₂²×C₄	C₂²⋊Dic₅₃	1	16	yes	—
161	C₂²×C₄	C₂×Dic₁₀₆	1	6	yes	—
162	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
163	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
164	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
165	C₂²×C₄	C₂×C₂²⋊D₅₃	1	12	yes	—
166	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
167	C₂×C₈	—	2	48	no	—
168	C₂×C₈	—	2	8	no	—
169	C₂×C₈	C₂²×Dic₅₃	1	8	yes	complemented
170	C₄⋊C₄	—	53	16	no	—
171	C₄⋊C₄	—	106	48	no	—
172	C₄⋊C₄	—	106	8	no	—
173	C₂³	#13133124	1	16	yes	—
174	C₂³	#13133139	1	32	yes	—
175	C₂³	#13133142	1	32	yes	—
176	C₂³	#13133149	1	16	yes	—
177	C₂³	#13133151	1	16	yes	—
178	C₂³	C₂×#11719921	1	12	yes	—
179	C₂³	C₂×C₄⋊Dic₅₃	1	6	yes	—
180	C₂³	C₂×D₄₂₄	1	6	yes	—
181	C₂³	C₂×#11719913	1	12	yes	—
182	C₂³	C₂×Dic₂₁₂	1	6	yes	—
183	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
184	C₂×C₄	—	106	16	no	—
185	C₂×C₄	—	212	96	no	—
186	C₂×C₄	—	212	8	no	—
187	C₂×C₄	—	212	8	no	—
188	C₂×C₄	#13133124	1	48	yes	—
189	C₂×C₄	C₂×C₄×Dic₅₃	1	12	yes	—
190	C₂×C₄	C₂×#11719921	1	12	yes	—
191	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	12	yes	—
192	C₂×C₄	C₂×#11719919	1	24	yes	—
193	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
194	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
195	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
196	C₂×C₄	C₂²×C₂²⋊D₅₃	1	2	yes	—
197	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
198	C₈	—	2	8	no	—
199	C₂²	#16810386	1	64	yes	direct product factor
200	C₂²	C₂×#13133124	1	12	yes	—
201	C₂²	C₂×#13133139	1	24	yes	—
202	C₂²	C₂×#13133142	1	24	yes	—
203	C₂²	C₂×#13133149	1	12	yes	—
204	C₂²	C₂×#13133151	1	12	yes	—
205	C₂²	C₂²×#11719921	1	2	yes	—
206	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
207	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
208	C₂²	C₂²×#11719913	1	2	yes	—
209	C₂²	C₂²×Dic₂₁₂	1	1	yes	characteristic
210	C₄	—	212	16	no	—
211	C₄	C₂×#13133124	1	12	yes	—
212	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
213	C₄	C₂²×C₂²⋊Dic₅₃	1	1	yes	characteristic
214	C₄	C₂²×#11719919	1	2	yes	—
215	C₂	C₂×#16810386	1	24	yes	direct product factor
216	C₂	C₂²×#13133124	1	1	yes	characteristic
217	C₂	C₂²×#13133139	1	2	yes	—
218	C₂	C₂²×#13133142	1	2	yes	—
219	C₂	C₂²×#13133149	1	1	yes	characteristic
220	C₂	C₂²×#13133151	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	D₂₁₂
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂⁴×C₈×C₅₃	C₂
socle, prime radical	C₂⁵×C₅₃	D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂⁴×C₈	D₅₃
hypocenter	C₅₃	C₂²×#340
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16810386	C₂	direct (N × H)
7. C₂²×C₄⋊Dic₅₃	C₄	semidirect
9. #16810386	C₂²	direct (N × H)
12. C₂³×C₈×C₅₃	C₄	semidirect
21. C₂×C₄⋊Dic₅₃	C₂×C₄	semidirect
32. C₂²×C₈×C₅₃	C₂×C₄	semidirect
58. C₂×C₈×C₅₃	C₂²×C₄	semidirect
61. C₄⋊Dic₅₃	C₂²×C₄	semidirect
89. C₂²×C₅₃	#340	semidirect
108. C₂×C₅₃	C₂×#340	semidirect
117. C₂³×C₈	Dic₅₃	semidirect
124. C₅₃	C₂²×#340	semidirect
138. C₂²×C₈	C₂×Dic₅₃	semidirect
169. C₂×C₈	C₂²×Dic₅₃	semidirect
199. C₂²	#16810386	direct (N × H)
215. C₂	C₂×#16810386	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×C₄⋊Dic₅₃	2
C₂×#16810386	2
C₂⁴×C₈×C₅₃	2
C₂²×#340	53

Subgroups — C22×#16810386 #27381488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810386 #27381488