Subgroups — C₄×#95×D₅₃ #27305159

Order 13568.

4462 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×D₅₃	C₂	1	3	yes	maximal
2	C₂×#95×D₅₃	C₂	1	4	yes	maximal
3	C₄×#13133124	C₂	1	3	yes	maximal, complemented
4	#95×Dic₅₃	C₂	1	4	yes	maximal, complemented
5	C₄×#95×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₄×D₅₃	C₂²	1	6	yes	—
7	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₅₃	C₂²	1	6	yes	—
9	C₂²×C₄×Dic₅₃	C₂²	1	3	yes	—
10	C₂×C₄²×D₅₃	—	2	24	no	—
11	C₂×C₄²×D₅₃	C₄	1	24	yes	complemented
12	C₂×#13133124	C₂²	1	12	yes	—
13	#95×D₅₃	C₄	1	32	yes	direct product factor
14	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
15	C₂×#95×C₅₃	C₂²	1	4	yes	—
16	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
17	C₂³×Dic₅₃	C₂³	1	6	yes	—
18	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
19	C₂²×C₄×D₅₃	—	2	24	no	—
20	C₂²×C₄×D₅₃	—	2	24	no	—
21	C₂²×C₄×D₅₃	C₂×C₄	1	24	yes	—
22	C₂²×C₄×D₅₃	C₂×C₄	1	48	yes	—
23	C₂²×C₄×D₅₃	C₂×C₄	1	12	yes	—
24	C₂²×C₄×D₅₃	D₄	1	12	yes	—
25	C₂²×C₄×D₅₃	Q₈	1	4	yes	—
26	C₂×C₄×Dic₅₃	—	2	12	no	—
27	C₂×C₄×Dic₅₃	—	2	24	no	—
28	C₂×C₄×Dic₅₃	C₂×C₄	1	12	yes	complemented
29	C₂×C₄×Dic₅₃	C₂×C₄	1	24	yes	complemented
30	C₄²×D₅₃	—	2	96	no	—
31	#13133124	C₂×C₄	1	48	yes	complemented
32	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
33	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	—	2	12	no	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	complemented
36	#95×C₅₃	C₂×C₄	1	16	yes	complemented
37	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
38	C₂³×D₅₃	C₂²×C₄	1	6	yes	—
39	C₂³×D₅₃	C₂²×C₄	1	8	yes	—
40	C₂³×D₅₃	C₂×D₄	1	6	yes	—
41	C₂³×D₅₃	C₂×Q₈	1	2	yes	—
42	C₂³×D₅₃	#41	1	8	yes	—
43	C₂²×C₄×C₅₃	—	2	12	no	—
44	C₂²×C₄×C₅₃	—	2	12	no	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
49	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
50	C₂²×Dic₅₃	—	2	12	no	—
51	C₂²×Dic₅₃	—	2	12	no	—
52	C₂²×Dic₅₃	C₂²×C₄	1	24	yes	—
53	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
54	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
55	C₂²×Dic₅₃	C₂×D₄	1	6	yes	—
56	C₂²×Dic₅₃	C₂×Q₈	1	2	yes	—
57	C₂×C₄×D₅₃	—	2	48	no	—
58	C₂×C₄×D₅₃	—	2	48	no	—
59	C₂×C₄×D₅₃	—	2	192	no	—
60	C₂×C₄×D₅₃	C₄²	1	16	yes	—
61	C₂×C₄×D₅₃	C₄²	1	96	yes	complemented
62	C₂×C₄×D₅₃	C₂²⋊C₄	1	48	yes	—
63	C₂×C₄×D₅₃	C₄⋊C₄	1	48	yes	—
64	C₄²×C₅₃	—	2	24	no	—
65	C₄×Dic₅₃	—	2	24	no	—
66	C₄×Dic₅₃	—	2	48	no	—
67	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
68	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
69	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
70	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
71	C₂³×C₅₃	C₂×#41	1	4	yes	—
72	C₂²×D₅₃	C₂×C₄²	1	4	yes	—
73	C₂²×D₅₃	C₂×C₄²	1	24	yes	—
74	C₂²×D₅₃	C₂×C₂²⋊C₄	1	12	yes	—
75	C₂²×D₅₃	C₂×C₄⋊C₄	1	12	yes	—
76	C₂²×D₅₃	#117	1	24	yes	—
77	C₂²×D₅₃	C₄×D₄	1	48	yes	—
78	C₂²×D₅₃	C₄×Q₈	1	16	yes	—
79	C₂×C₄×C₅₃	—	2	12	no	—
80	C₂×C₄×C₅₃	—	2	12	no	—
81	C₂×C₄×C₅₃	—	2	48	no	—
82	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
83	C₂×C₄×C₅₃	C₂×C₄²	1	24	yes	complemented
84	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
85	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
86	C₂×Dic₅₃	—	2	12	no	—
87	C₂×Dic₅₃	—	2	12	no	—
88	C₂×Dic₅₃	—	2	48	no	—
89	C₂×Dic₅₃	C₂×C₄²	1	4	yes	—
90	C₂×Dic₅₃	C₂×C₄²	1	24	yes	complemented
91	C₂×Dic₅₃	C₂×C₂²⋊C₄	1	12	yes	—
92	C₂×Dic₅₃	C₂×C₄⋊C₄	1	12	yes	—
93	C₄×D₅₃	—	2	192	no	—
94	C₄×D₅₃	#95	1	64	yes	direct product factor
95	C₂×C₄×#95	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
97	C₂²×C₅₃	C₂²×C₄²	1	6	yes	—
98	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	3	yes	—
99	C₂²×C₅₃	C₂²×C₄⋊C₄	1	3	yes	—
100	C₂²×C₅₃	C₂×#117	1	6	yes	—
101	C₂²×C₅₃	C₂×C₄×D₄	1	12	yes	—
102	C₂²×C₅₃	C₂×C₄×Q₈	1	4	yes	—
103	C₂×D₅₃	C₄³	1	8	yes	—
104	C₂×D₅₃	C₂×#95	1	8	yes	—
105	C₂×D₅₃	#376	1	8	yes	—
106	C₂×D₅₃	C₄×C₂²⋊C₄	1	48	yes	—
107	C₂×D₅₃	C₄×C₄⋊C₄	1	48	yes	—
108	C₄×C₅₃	—	2	24	no	—
109	C₄×C₅₃	C₂×#95	1	8	yes	complemented
110	Dic₅₃	—	2	24	no	—
111	Dic₅₃	C₂×#95	1	8	yes	complemented
112	C₂³×C₄²	—	53	3	no	—
113	C₂²×#95	—	53	4	no	—
114	C₄×#95	—	53	3	no	—
115	C₄×#95	—	53	4	no	—
116	C₄×#95	D₅₃	1	1	yes	characteristic, direct product factor
117	C₂×C₅₃	C₂×C₄³	1	1	yes	characteristic
118	C₂×C₅₃	C₂²×#95	1	1	yes	characteristic
119	C₂×C₅₃	C₂×#376	1	1	yes	characteristic
120	C₂×C₅₃	C₂×C₄×C₂²⋊C₄	1	6	yes	—
121	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	6	yes	—
122	D₅₃	C₄×#95	1	16	yes	direct product factor
123	C₂⁴×C₄	—	53	1	no	—
124	C₂⁴×C₄	—	53	6	no	—
125	C₂²×C₄²	—	53	6	no	—
126	C₂²×C₄²	—	53	3	no	—
127	C₂²×C₄²	—	53	24	no	—
128	C₂²×C₄²	—	106	24	no	—
129	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
130	C₂×#95	—	53	12	no	—
131	C₂×#95	—	53	32	no	—
132	C₂×#95	C₂×D₅₃	1	4	yes	—
133	C₅₃	C₂×C₄×#95	1	1	yes	characteristic, complemented, Sylow
134	C₂⁵	—	53	1	no	—
135	C₂³×C₄	—	53	1	no	—
136	C₂³×C₄	—	53	12	no	—
137	C₂³×C₄	—	53	12	no	—
138	C₂³×C₄	—	53	4	no	—
139	C₂³×C₄	—	53	6	no	—
140	C₂³×C₄	—	53	48	no	—
141	C₂³×C₄	—	53	24	no	—
142	C₂³×C₄	—	106	24	no	—
143	C₂³×C₄	—	106	24	no	—
144	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
145	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
146	C₂×C₄²	—	2	12	no	—
147	C₂×C₄²	—	53	24	no	—
148	C₂×C₄²	—	53	12	no	—
149	C₂×C₄²	—	106	96	no	—
150	C₂×C₄²	—	106	24	no	—
151	C₂×C₄²	—	106	12	no	—
152	C₂×C₄²	C₄×D₅₃	1	12	yes	complemented
153	#95	—	53	48	no	—
154	#95	C₄×D₅₃	1	16	yes	direct product factor
155	C₂⁴	—	53	6	no	—
156	C₂⁴	—	53	8	no	—
157	C₂⁴	—	53	6	no	—
158	C₂⁴	—	53	2	no	—
159	C₂⁴	—	53	8	no	—
160	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
161	C₂²×C₄	—	2	12	no	—
162	C₂²×C₄	—	2	12	no	—
163	C₂²×C₄	—	53	6	no	—
164	C₂²×C₄	—	53	6	no	—
165	C₂²×C₄	—	53	2	no	—
166	C₂²×C₄	—	53	16	no	—
167	C₂²×C₄	—	53	48	no	—
168	C₂²×C₄	—	53	48	no	—
169	C₂²×C₄	—	53	24	no	—
170	C₂²×C₄	—	53	12	no	—
171	C₂²×C₄	—	53	96	no	—
172	C₂²×C₄	—	106	48	no	—
173	C₂²×C₄	—	106	48	no	—
174	C₂²×C₄	—	106	192	no	—
175	C₂²×C₄	—	106	12	no	—
176	C₂²×C₄	—	106	12	no	—
177	C₂²×C₄	C₂×C₄×D₅₃	1	24	yes	—
178	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
179	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
180	C₂²×C₄	D₄×D₅₃	1	6	yes	—
181	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
182	C₄²	—	2	24	no	—
183	C₄²	—	106	48	no	—
184	C₄²	—	106	24	no	—
185	C₂³	—	53	4	no	—
186	C₂³	—	53	24	no	—
187	C₂³	—	53	12	no	—
188	C₂³	—	53	12	no	—
189	C₂³	—	53	24	no	—
190	C₂³	—	53	48	no	—
191	C₂³	—	53	16	no	—
192	C₂³	C₂²×C₄×D₅₃	1	3	yes	—
193	C₂³	C₂²×C₄×D₅₃	1	4	yes	—
194	C₂³	C₂×D₄×D₅₃	1	3	yes	—
195	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
196	C₂³	#41×D₅₃	1	4	yes	—
197	C₂×C₄	—	2	12	no	—
198	C₂×C₄	—	2	12	no	—
199	C₂×C₄	—	2	48	no	—
200	C₂×C₄	—	53	4	no	—
201	C₂×C₄	—	53	12	no	—
202	C₂×C₄	—	53	12	no	—
203	C₂×C₄	—	53	64	no	—
204	C₂×C₄	—	53	24	no	—
205	C₂×C₄	—	106	192	no	—
206	C₂×C₄	—	106	12	no	—
207	C₂×C₄	—	106	12	no	—
208	C₂×C₄	—	106	48	no	—
209	C₂×C₄	C₄²×D₅₃	1	4	yes	—
210	C₂×C₄	C₄²×D₅₃	1	24	yes	complemented
211	C₂×C₄	C₂²⋊C₄×D₅₃	1	12	yes	—
212	C₂×C₄	C₄⋊C₄×D₅₃	1	12	yes	—
213	C₂²	—	53	8	no	—
214	C₂²	—	53	8	no	—
215	C₂²	—	53	8	no	—
216	C₂²	—	53	48	no	—
217	C₂²	—	53	48	no	—
218	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
219	C₂²	C₂×C₄²×D₅₃	1	6	yes	—
220	C₂²	C₂×C₂²⋊C₄×D₅₃	1	3	yes	—
221	C₂²	C₂×C₄⋊C₄×D₅₃	1	3	yes	—
222	C₂²	#117×D₅₃	1	6	yes	—
223	C₂²	C₄×D₄×D₅₃	1	12	yes	—
224	C₂²	C₄×Q₈×D₅₃	1	4	yes	—
225	C₄	—	2	24	no	—
226	C₄	—	53	8	no	—
227	C₄	—	106	24	no	—
228	C₄	#95×D₅₃	1	8	yes	direct product factor
229	C₂	—	53	16	no	—
230	C₂	C₄³×D₅₃	1	1	yes	characteristic
231	C₂	C₂×#95×D₅₃	1	1	yes	characteristic
232	C₂	#376×D₅₃	1	1	yes	characteristic
233	C₂	C₄×C₂²⋊C₄×D₅₃	1	6	yes	—
234	C₂	C₄×C₄⋊C₄×D₅₃	1	6	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂×C₄³
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₄×#95×C₅₃	C₂
socle, cosocle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁴×D₅₃	C₂³
hypercenter	C₄×#95	D₅₃
hypocenter	C₅₃	C₂×C₄×#95
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₄×#95

Complements

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

N	H ≅ G/N	type
3. C₄×#13133124	C₂	semidirect
4. #95×Dic₅₃	C₂	semidirect
5. C₄×#95×C₅₃	C₂	semidirect
11. C₂×C₄²×D₅₃	C₄	semidirect
13. #95×D₅₃	C₄	direct (N × H)
28. C₂×C₄×Dic₅₃	C₂×C₄	semidirect
29. C₂×C₄×Dic₅₃	C₂×C₄	semidirect
31. #13133124	C₂×C₄	semidirect
35. C₂×C₄²×C₅₃	C₂×C₄	semidirect
36. #95×C₅₃	C₂×C₄	semidirect
61. C₂×C₄×D₅₃	C₄²	semidirect
83. C₂×C₄×C₅₃	C₂×C₄²	semidirect
90. C₂×Dic₅₃	C₂×C₄²	semidirect
94. C₄×D₅₃	#95	direct (N × H)
109. C₄×C₅₃	C₂×#95	semidirect
111. Dic₅₃	C₂×#95	semidirect
116. C₄×#95	D₅₃	direct (N × H)
122. D₅₃	C₄×#95	direct (N × H)
133. C₅₃	C₂×C₄×#95	semidirect
152. C₂×C₄²	C₄×D₅₃	semidirect
154. #95	C₄×D₅₃	direct (N × H)
210. C₂×C₄	C₄²×D₅₃	semidirect
228. C₄	#95×D₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄²×D₅₃	2
C₂×#95×D₅₃	2
C₄×#13133124	2
#95×Dic₅₃	2
C₄×#95×C₅₃	2
C₂×C₄×#95	53

Subgroups — C4×#95×D53 #27305159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#95×D₅₃ #27305159