Subgroups — C₄×#16810386 #27289273

Order 13568.

1182 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊Dic₅₃	C₂	1	2	yes	maximal
2	C₂×#16810386	C₂	1	4	yes	maximal
3	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₅₃	C₂²	1	4	yes	—
5	C₂²×C₄×Dic₅₃	—	2	2	no	—
6	C₄×C₄⋊Dic₅₃	—	2	4	no	—
7	C₄×C₄⋊Dic₅₃	C₄	1	8	yes	complemented
8	C₂×#13133124	—	2	2	no	—
9	#16810386	C₄	1	16	yes	direct product factor
10	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₃	—	2	4	no	—
13	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	complemented
14	C₂³×Dic₅₃	—	2	4	no	—
15	C₂×C₄⋊Dic₅₃	—	2	4	no	—
16	C₂×C₄⋊Dic₅₃	—	2	8	no	—
17	C₂×C₄⋊Dic₅₃	C₂×C₄	1	16	yes	—
18	C₂×C₄⋊Dic₅₃	C₂×C₄	1	8	yes	—
19	C₂×C₄×Dic₅₃	—	2	8	no	—
20	C₂×C₄×Dic₅₃	—	4	4	no	—
21	C₂×C₄×Dic₅₃	—	4	4	no	—
22	#13133124	—	2	8	no	—
23	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	—	2	4	no	—
27	C₂²×C₈×C₅₃	—	2	4	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₃	D₄	1	4	yes	—
33	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
34	C₄×C₈×C₅₃	—	2	8	no	—
35	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	—	2	4	no	—
37	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	D₈	1	2	yes	—
42	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
43	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	#41	1	4	yes	—
50	C₂²×Dic₅₃	—	2	8	no	—
51	C₂²×Dic₅₃	—	2	16	no	—
52	C₂²×Dic₅₃	—	4	4	no	—
53	C₂²×Dic₅₃	—	4	4	no	—
54	C₂²×Dic₅₃	—	4	8	no	—
55	C₂×C₈×C₅₃	—	2	4	no	—
56	C₂×C₈×C₅₃	—	2	4	no	—
57	C₂×C₈×C₅₃	—	2	16	no	—
58	C₂×C₈×C₅₃	C₄²	1	8	yes	complemented
59	C₄²×C₅₃	C₄²	1	4	yes	—
60	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
61	C₄²×C₅₃	C₄⋊C₄	1	8	yes	—
62	C₄⋊Dic₅₃	—	2	8	no	—
63	C₄⋊Dic₅₃	—	2	16	no	—
64	C₄⋊Dic₅₃	C₄²	1	16	yes	complemented
65	C₄×Dic₅₃	—	4	16	no	—
66	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
67	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂³×C₅₃	C₄×D₄	1	4	yes	—
69	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
70	C₂³×C₅₃	C₂×SD₁₆	1	2	yes	—
71	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
72	C₂³×C₅₃	#135	1	4	yes	—
73	C₂×C₄×C₅₃	—	2	8	no	—
74	C₂×C₄×C₅₃	—	2	4	no	—
75	C₂×C₄×C₅₃	#95	1	4	yes	—
76	C₂×C₄×C₅₃	#95	1	4	yes	—
77	C₂×C₄×C₅₃	#102	1	8	yes	—
78	C₂×C₄×C₅₃	#103	1	8	yes	—
79	C₂×C₄×C₅₃	#106	1	4	yes	—
80	C₂×C₄×C₅₃	#107	1	4	yes	—
81	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
82	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
84	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₅₃	#117	1	4	yes	—
87	C₂×C₄×C₅₃	#117	1	2	yes	—
88	C₂×C₄×C₅₃	C₄×D₄	1	8	yes	—
89	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
90	C₂×Dic₅₃	—	2	16	no	—
91	C₂×Dic₅₃	—	4	8	no	—
92	C₂×Dic₅₃	—	4	8	no	—
93	C₂×Dic₅₃	—	4	16	no	—
94	C₂×Dic₅₃	—	4	16	no	—
95	C₈×C₅₃	—	2	8	no	—
96	C₄×#340	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
98	C₂²×C₅₃	C₄×C₂²⋊C₄	1	4	yes	—
99	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
100	C₂²×C₅₃	C₂×#102	1	2	yes	—
101	C₂²×C₅₃	C₂×#103	1	2	yes	—
102	C₂²×C₅₃	#416	1	4	yes	—
103	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
104	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
105	C₂²×C₅₃	#427	1	2	yes	—
106	C₂²×C₅₃	C₄×D₈	1	4	yes	—
107	C₂²×C₅₃	C₄×SD₁₆	1	8	yes	—
108	C₂²×C₅₃	C₄×Q₁₆	1	4	yes	—
109	C₄×C₅₃	—	2	4	no	—
110	C₄×C₅₃	#340	1	8	yes	complemented
111	C₄×C₅₃	C₄³	1	1	yes	characteristic
112	C₄×C₅₃	#376	1	1	yes	characteristic
113	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
114	C₄×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
115	Dic₅₃	—	4	16	no	—
116	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
117	C₂×C₄×C₄⋊C₄	—	53	2	no	—
118	C₂×#340	—	53	4	no	—
119	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
120	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
121	C₂×C₅₃	#1735	1	1	yes	characteristic
122	C₂×C₅₃	C₄×#102	1	4	yes	—
123	C₂×C₅₃	C₄×#103	1	4	yes	—
124	C₂×C₅₃	C₄×#106	1	2	yes	—
125	C₂×C₅₃	C₄×#107	1	2	yes	—
126	C₂³×C₈	C₂×D₅₃	1	2	yes	—
127	C₂²×C₄⋊C₄	—	53	4	no	—
128	C₂²×C₄²	—	106	2	no	—
129	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
130	C₂×C₄×C₈	—	2	4	no	—
131	C₂×C₄×C₈	Dic₅₃	1	4	yes	complemented
132	C₄×C₄⋊C₄	—	53	8	no	—
133	C₄×C₄⋊C₄	—	106	4	no	—
134	C₂×#95	—	106	2	no	—
135	#340	—	53	16	no	—
136	C₅₃	C₄×#340	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	—	106	4	no	—
138	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
139	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
140	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	—	2	4	no	—
143	C₂²×C₈	C₄×D₅₃	1	8	yes	—
144	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
145	C₂×C₄⋊C₄	—	53	16	no	—
146	C₂×C₄⋊C₄	—	53	8	no	—
147	C₂×C₄⋊C₄	—	106	8	no	—
148	C₂×C₄⋊C₄	—	106	4	no	—
149	C₂×C₄²	—	106	8	no	—
150	C₂×C₄²	—	212	4	no	—
151	C₂×C₄²	—	212	4	no	—
152	C₂×C₄²	Dic₁₀₆	1	2	yes	—
153	C₂×C₄²	C₄×D₅₃	1	4	yes	—
154	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
155	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
156	C₄×C₈	—	2	8	no	—
157	#95	—	106	8	no	—
158	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	106	8	no	—
161	C₂²×C₄	—	106	16	no	—
162	C₂²×C₄	—	212	4	no	—
163	C₂²×C₄	—	212	4	no	—
164	C₂²×C₄	—	212	8	no	—
165	C₂²×C₄	#11719913	1	4	yes	—
166	C₂²×C₄	D₄₂₄	1	2	yes	—
167	C₂²×C₄	Dic₂₁₂	1	2	yes	—
168	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
169	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
170	C₂²×C₄	#11719921	1	4	yes	—
171	C₂²×C₄	#11719921	1	4	yes	—
172	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
174	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
175	C₂²×C₄	#11719945	1	4	yes	—
176	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	16	no	—
181	C₂×C₈	C₄×Dic₅₃	1	8	yes	complemented
182	C₄⋊C₄	—	53	16	no	—
183	C₄⋊C₄	—	106	8	no	—
184	C₄⋊C₄	—	106	16	no	—
185	C₄²	—	212	16	no	—
186	C₄²	C₄×Dic₅₃	1	4	yes	—
187	C₄²	#11719919	1	8	yes	—
188	C₄²	C₂²⋊Dic₅₃	1	4	yes	—
189	C₂³	C₂×#11719921	1	2	yes	—
190	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
191	C₂³	C₄×D₂₁₂	1	4	yes	—
192	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
193	C₂³	C₂×#11719913	1	2	yes	—
194	C₂³	C₂×Dic₂₁₂	1	1	yes	characteristic
195	C₂³	#13133239	1	4	yes	—
196	C₂×C₄	—	2	8	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	106	16	no	—
199	C₂×C₄	—	212	8	no	—
200	C₂×C₄	—	212	8	no	—
201	C₂×C₄	—	212	16	no	—
202	C₂×C₄	—	212	16	no	—
203	C₂×C₄	#13133124	1	4	yes	—
204	C₂×C₄	#13133124	1	4	yes	—
205	C₂×C₄	#13133139	1	8	yes	—
206	C₂×C₄	#13133142	1	8	yes	—
207	C₂×C₄	#13133149	1	4	yes	—
208	C₂×C₄	#13133151	1	4	yes	—
209	C₂×C₄	C₄²×D₅₃	1	4	yes	—
210	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
212	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
213	C₂×C₄	C₂×#11719919	1	2	yes	—
214	C₂×C₄	#13133174	1	4	yes	—
215	C₂×C₄	#13133176	1	2	yes	—
216	C₂×C₄	C₄×C₂²⋊D₅₃	1	8	yes	—
217	C₂×C₄	C₄×Dic₁₀₆	1	4	yes	—
218	C₈	—	2	8	no	—
219	C₂²	C₂×#13133124	1	1	yes	characteristic
220	C₂²	C₄×#11719921	1	4	yes	—
221	C₂²	C₄×C₄⋊Dic₅₃	1	2	yes	—
222	C₂²	C₂×#13133139	1	2	yes	—
223	C₂²	C₂×#13133142	1	2	yes	—
224	C₂²	#16810620	1	4	yes	—
225	C₂²	C₂×#13133149	1	1	yes	characteristic
226	C₂²	C₂×#13133151	1	1	yes	characteristic
227	C₂²	#16810661	1	2	yes	—
228	C₂²	C₄×D₄₂₄	1	4	yes	—
229	C₂²	C₄×#11719913	1	8	yes	—
230	C₂²	C₄×Dic₂₁₂	1	4	yes	—
231	C₄	—	2	4	no	—
232	C₄	—	212	16	no	—
233	C₄	#16810386	1	8	yes	direct product factor
234	C₄	C₄²×Dic₅₃	1	1	yes	characteristic
235	C₄	#16810469	1	1	yes	characteristic
236	C₄	C₄×C₂²⋊Dic₅₃	1	2	yes	—
237	C₄	C₄×#11719919	1	4	yes	—
238	C₂	C₄×#13133124	1	1	yes	characteristic
239	C₂	C₂×#16810386	1	1	yes	characteristic
240	C₂	#20274432	1	1	yes	characteristic
241	C₂	C₄×#13133139	1	4	yes	—
242	C₂	C₄×#13133142	1	4	yes	—
243	C₂	C₄×#13133149	1	2	yes	—
244	C₂	C₄×#13133151	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	D₂₁₂
commutator	C₄×C₅₃	C₄³
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₄×#340
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
7. C₄×C₄⋊Dic₅₃	C₄	semidirect
9. #16810386	C₄	direct (N × H)
13. C₂×C₄×C₈×C₅₃	C₄	semidirect
58. C₂×C₈×C₅₃	C₄²	semidirect
64. C₄⋊Dic₅₃	C₄²	semidirect
110. C₄×C₅₃	#340	semidirect
131. C₂×C₄×C₈	Dic₅₃	semidirect
136. C₅₃	C₄×#340	semidirect
181. C₂×C₈	C₄×Dic₅₃	semidirect
233. C₄	#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
C₂×C₄×C₄⋊Dic₅₃	2
C₂×#16810386	2
C₂²×C₄×C₈×C₅₃	2
C₄×#340	53

Subgroups — C4×#16810386 #27289273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810386 #27289273