Subgroups — C₂²×C₅₃×#495 #27268143

Order 13568.

3518 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×SD₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₂³×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#124×C₅₃	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#495	C₂	1	24	yes	maximal, direct product factor
6	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
8	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
9	C₂²×Q₁₆×C₅₃	—	2	8	no	—
10	C₂²×Q₁₆×C₅₃	C₂²	1	12	yes	complemented
11	C₂²×SD₁₆×C₅₃	—	2	16	no	—
12	C₂²×SD₁₆×C₅₃	C₂²	1	24	yes	complemented
13	C₂²×D₈×C₅₃	—	2	8	no	—
14	C₂²×D₈×C₅₃	C₂²	1	12	yes	—
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#124×C₅₃	—	2	12	no	—
18	C₂×#124×C₅₃	C₂²	1	24	yes	complemented
19	C₂²×C₂²⋊C₄×C₅₃	—	2	4	no	—
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#495	C₂²	1	64	yes	direct product factor
22	C₂×C₄×C₈×C₅₃	C₂²	1	12	yes	complemented
23	C₂⁵×C₅₃	—	2	2	no	—
24	C₂²×Q₈×C₅₃	—	2	6	no	—
25	C₂²×Q₈×C₅₃	—	2	8	no	—
26	C₂²×Q₈×C₅₃	—	2	8	no	—
27	C₂²×Q₈×C₅₃	C₂³	1	12	yes	—
28	C₂²×D₄×C₅₃	—	2	8	no	—
29	C₂²×D₄×C₅₃	—	2	8	no	—
30	C₂²×D₄×C₅₃	—	2	6	no	—
31	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
32	C₂³×C₄×C₅₃	—	2	2	no	—
33	C₂³×C₄×C₅₃	D₄	1	2	yes	—
34	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×Q₁₆×C₅₃	—	2	48	no	—
36	C₂×Q₁₆×C₅₃	C₂³	1	16	yes	complemented
37	C₂×SD₁₆×C₅₃	—	2	96	no	—
38	C₂×SD₁₆×C₅₃	C₂³	1	32	yes	complemented
39	C₂×D₈×C₅₃	—	2	48	no	—
40	C₂×D₈×C₅₃	C₂³	1	16	yes	—
41	C₂²×C₈×C₅₃	D₄	1	8	yes	—
42	C₂²×C₈×C₅₃	D₄	1	8	yes	—
43	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
45	#124×C₅₃	—	2	48	no	—
46	#124×C₅₃	C₂³	1	32	yes	complemented
47	C₂×C₂²⋊C₄×C₅₃	—	2	48	no	—
48	C₂×C₄²×C₅₃	D₄	1	6	yes	—
49	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
50	C₄×C₈×C₅₃	C₂³	1	16	yes	complemented
51	C₂⁴×C₅₃	—	2	12	no	—
52	C₂⁴×C₅₃	—	4	8	no	—
53	C₂⁴×C₅₃	—	4	16	no	—
54	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×Q₈×C₅₃	—	2	12	no	—
56	C₂×Q₈×C₅₃	—	2	24	no	—
57	C₂×Q₈×C₅₃	—	2	24	no	—
58	C₂×Q₈×C₅₃	—	4	24	no	—
59	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
60	C₂×D₄×C₅₃	—	2	24	no	—
61	C₂×D₄×C₅₃	—	2	24	no	—
62	C₂×D₄×C₅₃	—	2	12	no	—
63	C₂×D₄×C₅₃	—	4	24	no	—
64	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
65	C₂²×C₄×C₅₃	—	2	8	no	—
66	C₂²×C₄×C₅₃	—	2	12	no	—
67	C₂²×C₄×C₅₃	—	4	8	no	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
69	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
73	Q₁₆×C₅₃	—	2	32	no	—
74	SD₁₆×C₅₃	—	2	64	no	—
75	D₈×C₅₃	—	2	32	no	—
76	C₂×C₈×C₅₃	C₂×D₄	1	24	yes	—
77	C₂×C₈×C₅₃	C₂×D₄	1	24	yes	—
78	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
79	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
80	C₂²⋊C₄×C₅₃	—	2	64	no	—
81	C₄²×C₅₃	C₂×D₄	1	12	yes	—
82	C₄²×C₅₃	C₂⁴	1	4	yes	—
83	C₂³×C₅₃	—	2	8	no	—
84	C₂³×C₅₃	—	4	24	no	—
85	C₂³×C₅₃	—	4	48	no	—
86	C₂³×C₅₃	—	8	32	no	—
87	C₂³×C₅₃	#127	1	4	yes	—
88	C₂³×C₅₃	#135	1	8	yes	—
89	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
90	Q₈×C₅₃	—	2	8	no	—
91	Q₈×C₅₃	—	2	8	no	—
92	Q₈×C₅₃	—	4	24	no	—
93	D₄×C₅₃	—	2	8	no	—
94	D₄×C₅₃	—	2	8	no	—
95	D₄×C₅₃	—	4	24	no	—
96	C₂×C₄×C₅₃	—	2	24	no	—
97	C₂×C₄×C₅₃	—	2	8	no	—
98	C₂×C₄×C₅₃	—	4	24	no	—
99	C₂×C₄×C₅₃	#127	1	12	yes	—
100	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
101	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
102	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
103	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
104	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
105	C₈×C₅₃	C₂²×D₄	1	8	yes	—
106	C₈×C₅₃	C₂²×D₄	1	8	yes	—
107	C₂²×#495	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₅₃	—	4	8	no	—
109	C₂²×C₅₃	—	4	16	no	—
110	C₂²×C₅₃	—	8	48	no	—
111	C₂²×C₅₃	#495	1	16	yes	direct product factor
112	C₂²×C₅₃	C₂×#127	1	6	yes	—
113	C₂²×C₅₃	C₂×#135	1	12	yes	—
114	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
115	C₄×C₅₃	—	2	8	no	—
116	C₄×C₅₃	—	4	8	no	—
117	C₄×C₅₃	C₂×#127	1	6	yes	—
118	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
119	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
120	C₂³×Q₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
121	C₂³×SD₁₆	C₂×C₅₃	1	2	yes	complemented
122	C₂³×D₈	C₂×C₅₃	1	1	yes	characteristic
123	C₂²×#124	C₂×C₅₃	1	2	yes	complemented
124	C₂×#495	C₂×C₅₃	1	24	yes	direct product factor
125	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
126	C₂×C₅₃	—	8	8	no	—
127	C₂×C₅₃	C₂×#495	1	12	yes	direct product factor
128	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
129	C₂×C₅₃	C₂²×#135	1	2	yes	—
130	C₂³×Q₈	C₂²×C₅₃	1	2	yes	—
131	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
132	C₂²×Q₁₆	—	2	8	no	—
133	C₂²×Q₁₆	C₂²×C₅₃	1	12	yes	complemented
134	C₂²×SD₁₆	—	2	16	no	—
135	C₂²×SD₁₆	C₂²×C₅₃	1	24	yes	complemented
136	C₂²×D₈	—	2	8	no	—
137	C₂²×D₈	C₂²×C₅₃	1	12	yes	—
138	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
139	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
140	C₂×#124	—	2	12	no	—
141	C₂×#124	C₂²×C₅₃	1	24	yes	complemented
142	C₂²×C₂²⋊C₄	—	2	4	no	—
143	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
144	#495	C₂²×C₅₃	1	64	yes	direct product factor
145	C₂×C₄×C₈	C₂²×C₅₃	1	12	yes	complemented
146	C₅₃	C₂²×#495	1	1	yes	characteristic, direct product factor, Sylow
147	C₂⁵	—	2	2	no	—
148	C₂²×Q₈	—	2	6	no	—
149	C₂²×Q₈	—	2	8	no	—
150	C₂²×Q₈	—	2	8	no	—
151	C₂²×Q₈	C₂³×C₅₃	1	12	yes	—
152	C₂²×D₄	—	2	8	no	—
153	C₂²×D₄	—	2	8	no	—
154	C₂²×D₄	—	2	6	no	—
155	C₂²×D₄	C₂³×C₅₃	1	12	yes	—
156	C₂³×C₄	—	2	2	no	—
157	C₂³×C₄	D₄×C₅₃	1	2	yes	—
158	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
159	C₂×Q₁₆	—	2	48	no	—
160	C₂×Q₁₆	C₂³×C₅₃	1	16	yes	complemented
161	C₂×SD₁₆	—	2	96	no	—
162	C₂×SD₁₆	C₂³×C₅₃	1	32	yes	complemented
163	C₂×D₈	—	2	48	no	—
164	C₂×D₈	C₂³×C₅₃	1	16	yes	—
165	C₂²×C₈	D₄×C₅₃	1	8	yes	—
166	C₂²×C₈	D₄×C₅₃	1	8	yes	—
167	C₂²×C₈	C₂³×C₅₃	1	6	yes	—
168	C₂²×C₈	C₂³×C₅₃	1	6	yes	—
169	#124	—	2	48	no	—
170	#124	C₂³×C₅₃	1	32	yes	complemented
171	C₂×C₂²⋊C₄	—	2	48	no	—
172	C₂×C₄²	D₄×C₅₃	1	6	yes	—
173	C₂×C₄²	C₂³×C₅₃	1	6	yes	—
174	C₄×C₈	C₂³×C₅₃	1	16	yes	complemented
175	C₂⁴	—	2	12	no	—
176	C₂⁴	—	4	8	no	—
177	C₂⁴	—	4	16	no	—
178	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
179	C₂×Q₈	—	2	12	no	—
180	C₂×Q₈	—	2	24	no	—
181	C₂×Q₈	—	2	24	no	—
182	C₂×Q₈	—	4	24	no	—
183	C₂×Q₈	C₂⁴×C₅₃	1	8	yes	—
184	C₂×D₄	—	2	24	no	—
185	C₂×D₄	—	2	24	no	—
186	C₂×D₄	—	2	12	no	—
187	C₂×D₄	—	4	24	no	—
188	C₂×D₄	C₂⁴×C₅₃	1	8	yes	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	—	2	12	no	—
191	C₂²×C₄	—	4	8	no	—
192	C₂²×C₄	C₂×D₄×C₅₃	1	12	yes	—
193	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
194	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
195	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
196	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
197	Q₁₆	—	2	32	no	—
198	SD₁₆	—	2	64	no	—
199	D₈	—	2	32	no	—
200	C₂×C₈	C₂×D₄×C₅₃	1	24	yes	—
201	C₂×C₈	C₂×D₄×C₅₃	1	24	yes	—
202	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
203	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
204	C₂²⋊C₄	—	2	64	no	—
205	C₄²	C₂×D₄×C₅₃	1	12	yes	—
206	C₄²	C₂⁴×C₅₃	1	4	yes	—
207	C₂³	—	2	8	no	—
208	C₂³	—	4	24	no	—
209	C₂³	—	4	48	no	—
210	C₂³	—	8	32	no	—
211	C₂³	#127×C₅₃	1	4	yes	—
212	C₂³	#135×C₅₃	1	8	yes	—
213	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
214	Q₈	—	2	8	no	—
215	Q₈	—	2	8	no	—
216	Q₈	—	4	24	no	—
217	D₄	—	2	8	no	—
218	D₄	—	2	8	no	—
219	D₄	—	4	24	no	—
220	C₂×C₄	—	2	24	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	4	24	no	—
223	C₂×C₄	#127×C₅₃	1	12	yes	—
224	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
225	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
226	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
227	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
228	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
229	C₈	C₂²×D₄×C₅₃	1	8	yes	—
230	C₈	C₂²×D₄×C₅₃	1	8	yes	—
231	C₂²	—	4	16	no	—
232	C₂²	—	4	8	no	—
233	C₂²	—	8	48	no	—
234	C₂²	C₅₃×#495	1	16	yes	direct product factor
235	C₂²	C₂×#127×C₅₃	1	6	yes	—
236	C₂²	C₂×#135×C₅₃	1	12	yes	—
237	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
238	C₄	—	2	8	no	—
239	C₄	—	4	8	no	—
240	C₄	C₂×#127×C₅₃	1	6	yes	—
241	C₄	C₂³×D₄×C₅₃	1	1	yes	characteristic
242	C₄	C₂³×D₄×C₅₃	1	1	yes	characteristic
243	C₂	—	8	8	no	—
244	C₂	C₂×C₅₃×#495	1	12	yes	direct product factor
245	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
246	C₂	C₂²×#135×C₅₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₂³×D₈×C₅₃	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₁₆×C₅₃	C₂	semidirect
2. C₂³×SD₁₆×C₅₃	C₂	semidirect
4. C₂²×#124×C₅₃	C₂	semidirect
5. C₂×C₅₃×#495	C₂	direct (N × H)
6. C₂²×C₄×C₈×C₅₃	C₂	semidirect
10. C₂²×Q₁₆×C₅₃	C₂²	semidirect
12. C₂²×SD₁₆×C₅₃	C₂²	semidirect
18. C₂×#124×C₅₃	C₂²	semidirect
21. C₅₃×#495	C₂²	direct (N × H)
22. C₂×C₄×C₈×C₅₃	C₂²	semidirect
36. C₂×Q₁₆×C₅₃	C₂³	semidirect
38. C₂×SD₁₆×C₅₃	C₂³	semidirect
46. #124×C₅₃	C₂³	semidirect
50. C₄×C₈×C₅₃	C₂³	semidirect
107. C₂²×#495	C₅₃	direct (N × H)
111. C₂²×C₅₃	#495	direct (N × H)
120. C₂³×Q₁₆	C₂×C₅₃	semidirect
121. C₂³×SD₁₆	C₂×C₅₃	semidirect
123. C₂²×#124	C₂×C₅₃	semidirect
124. C₂×#495	C₂×C₅₃	direct (N × H)
125. C₂²×C₄×C₈	C₂×C₅₃	semidirect
127. C₂×C₅₃	C₂×#495	direct (N × H)
133. C₂²×Q₁₆	C₂²×C₅₃	semidirect
135. C₂²×SD₁₆	C₂²×C₅₃	semidirect
141. C₂×#124	C₂²×C₅₃	semidirect
144. #495	C₂²×C₅₃	direct (N × H)
145. C₂×C₄×C₈	C₂²×C₅₃	semidirect
146. C₅₃	C₂²×#495	direct (N × H)
160. C₂×Q₁₆	C₂³×C₅₃	semidirect
162. C₂×SD₁₆	C₂³×C₅₃	semidirect
170. #124	C₂³×C₅₃	semidirect
174. C₄×C₈	C₂³×C₅₃	semidirect
234. C₂²	C₅₃×#495	direct (N × H)
244. C₂	C₂×C₅₃×#495	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×Q₁₆×C₅₃	2
C₂³×SD₁₆×C₅₃	2
C₂³×D₈×C₅₃	2
C₂²×#124×C₅₃	2
C₂×C₅₃×#495	2
C₂²×C₄×C₈×C₅₃	2
C₂²×#495	53

Subgroups — C22×C53×#495 #27268143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#495 #27268143