Subgroups — C₅₃×#33607 #27241119

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3058	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2218	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2215	C₂	1	4	yes	maximal, complemented
6	C₅₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2140	C₂	1	2	yes	maximal, complemented
8	C₂²×D₈×C₅₃	C₂²	1	2	yes	—
9	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#535	—	2	2	no	—
11	C₂×D₁₆×C₅₃	—	2	4	no	—
12	C₂×D₁₆×C₅₃	C₂²	1	2	yes	—
13	C₂×D₁₆×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#504	—	2	4	no	—
15	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₅₃	C₂²	1	2	yes	complemented
17	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	complemented
19	C₅₃×#469	—	2	4	no	—
20	C₅₃×#466	C₂²	1	8	yes	complemented
21	C₅₃×#427	C₂²	1	2	yes	—
22	C₅₃×#418	—	2	2	no	—
23	C₅₃×#367	C₂²	1	2	yes	complemented
24	C₅₃×#366	C₂²	1	2	yes	complemented
25	C₅₃×#357	C₂²	1	8	yes	complemented
26	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₅₃	—	2	2	no	—
28	C₂×D₈×C₅₃	—	2	2	no	—
29	C₂×D₈×C₅₃	—	2	4	no	—
30	C₂×D₈×C₅₃	—	2	4	no	—
31	C₂×D₈×C₅₃	C₂³	1	4	yes	—
32	#131×C₅₃	—	2	2	no	—
33	#131×C₅₃	—	2	2	no	—
34	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
35	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	#127×C₅₃	—	4	2	no	—
39	#124×C₅₃	—	4	2	no	—
40	#121×C₅₃	—	2	8	no	—
41	#120×C₅₃	—	4	4	no	—
42	#117×C₅₃	—	2	2	no	—
43	D₁₆×C₅₃	—	2	4	no	—
44	D₁₆×C₅₃	—	2	4	no	—
45	M₃₂×C₅₃	—	2	2	no	—
46	M₃₂×C₅₃	—	2	4	no	—
47	C₂×C₁₆×C₅₃	—	2	4	no	—
48	C₂×C₁₆×C₅₃	D₄	1	4	yes	complemented
49	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
50	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
52	#107×C₅₃	C₂³	1	4	yes	—
53	#106×C₅₃	—	2	2	no	—
54	#102×C₅₃	—	2	8	no	—
55	C₂⁴×C₅₃	—	4	2	no	—
56	#41×C₅₃	—	2	2	no	—
57	#41×C₅₃	—	2	2	no	—
58	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
59	C₂×D₄×C₅₃	—	2	4	no	—
60	C₂×D₄×C₅₃	—	4	8	no	—
61	C₂×D₄×C₅₃	—	4	4	no	—
62	C₂×D₄×C₅₃	—	4	2	no	—
63	C₂×D₄×C₅₃	—	4	4	no	—
64	C₂×D₄×C₅₃	D₈	1	2	yes	—
65	C₂×D₄×C₅₃	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	D₈×C₅₃	—	2	4	no	—
69	D₈×C₅₃	—	2	4	no	—
70	D₈×C₅₃	—	4	4	no	—
71	M₁₆×C₅₃	—	2	2	no	—
72	M₁₆×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₅₃	#41	1	2	yes	—
79	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₄×C₅₃	—	2	4	no	—
81	C₂²⋊C₄×C₅₃	—	4	2	no	—
82	C₂²⋊C₄×C₅₃	—	4	8	no	—
83	C₄²×C₅₃	—	4	2	no	—
84	C₁₆×C₅₃	—	2	2	no	—
85	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
86	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
87	C₂³×C₅₃	—	4	4	no	—
88	C₂³×C₅₃	—	8	4	no	—
89	C₂³×C₅₃	—	16	4	no	—
90	C₂³×C₅₃	C₂×D₈	1	2	yes	—
91	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
92	Q₈×C₅₃	—	2	2	no	—
93	D₄×C₅₃	—	2	4	no	—
94	D₄×C₅₃	—	2	2	no	—
95	D₄×C₅₃	—	4	4	no	—
96	D₄×C₅₃	—	4	4	no	—
97	D₄×C₅₃	—	8	4	no	—
98	D₄×C₅₃	—	16	8	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	4	4	no	—
102	C₂×C₄×C₅₃	#121	1	2	yes	—
103	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#135	1	2	yes	—
106	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₅₃	—	2	2	no	—
108	C₈×C₅₃	#121	1	2	yes	—
109	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
110	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
111	#33607	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	2	4	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	—	8	8	no	—
116	C₂²×C₅₃	—	8	4	no	—
117	C₂²×C₅₃	—	16	8	no	—
118	C₂²×C₅₃	#466	1	2	yes	—
119	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	8	4	no	—
122	C₄×C₅₃	#466	1	2	yes	—
123	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
124	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
125	C₂²×D₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
126	C₂×#504	C₂×C₅₃	1	1	yes	characteristic, complemented
127	#3058	C₂×C₅₃	1	2	yes	complemented
128	#2218	C₂×C₅₃	1	4	yes	complemented
129	#2215	C₂×C₅₃	1	4	yes	complemented
130	#2158	C₂×C₅₃	1	1	yes	characteristic, complemented
131	#2140	C₂×C₅₃	1	2	yes	complemented
132	C₂×C₅₃	—	2	2	no	—
133	C₂×C₅₃	—	4	2	no	—
134	C₂×C₅₃	—	16	4	no	—
135	C₂×C₅₃	C₂×#466	1	1	yes	characteristic
136	C₂×C₅₃	#3415	1	2	yes	—
137	C₂²×D₈	C₂²×C₅₃	1	2	yes	—
138	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
139	#535	—	2	2	no	—
140	C₂×D₁₆	—	2	4	no	—
141	C₂×D₁₆	C₂²×C₅₃	1	2	yes	—
142	C₂×D₁₆	C₂²×C₅₃	1	2	yes	—
143	#504	—	2	4	no	—
144	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
145	C₂×M₃₂	C₂²×C₅₃	1	2	yes	complemented
146	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
147	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	complemented
148	#469	—	2	4	no	—
149	#466	C₂²×C₅₃	1	8	yes	complemented
150	#427	C₂²×C₅₃	1	2	yes	—
151	#418	—	2	2	no	—
152	#367	C₂²×C₅₃	1	2	yes	complemented
153	#366	C₂²×C₅₃	1	2	yes	complemented
154	#357	C₂²×C₅₃	1	8	yes	complemented
155	C₅₃	#33607	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
157	C₂²×D₄	—	2	2	no	—
158	C₂×D₈	—	2	4	no	—
159	C₂×D₈	—	2	2	no	—
160	C₂×D₈	—	2	4	no	—
161	C₂×D₈	C₂³×C₅₃	1	4	yes	—
162	#131	—	2	2	no	—
163	#131	—	2	2	no	—
164	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
165	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
166	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
167	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
168	#127	—	4	2	no	—
169	#124	—	4	2	no	—
170	#121	—	2	8	no	—
171	#120	—	4	4	no	—
172	#117	—	2	2	no	—
173	D₁₆	—	2	4	no	—
174	D₁₆	—	2	4	no	—
175	M₃₂	—	2	2	no	—
176	M₃₂	—	2	4	no	—
177	C₂×C₁₆	—	2	4	no	—
178	C₂×C₁₆	D₄×C₅₃	1	4	yes	complemented
179	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
180	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
181	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
182	#107	C₂³×C₅₃	1	4	yes	—
183	#106	—	2	2	no	—
184	#102	—	2	8	no	—
185	C₂⁴	—	4	2	no	—
186	#41	—	2	2	no	—
187	#41	—	2	2	no	—
188	C₂×Q₈	D₈×C₅₃	1	1	yes	characteristic
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	4	8	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂×D₄	—	4	4	no	—
194	C₂×D₄	D₈×C₅₃	1	2	yes	—
195	C₂×D₄	D₈×C₅₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
198	D₈	—	2	4	no	—
199	D₈	—	2	4	no	—
200	D₈	—	4	4	no	—
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	4	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
206	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
208	C₂×C₈	#41×C₅₃	1	2	yes	—
209	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
210	C₄⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₂²⋊C₄	—	4	8	no	—
213	C₄²	—	4	2	no	—
214	C₁₆	—	2	2	no	—
215	C₁₆	C₂×D₄×C₅₃	1	2	yes	—
216	C₁₆	C₂×D₄×C₅₃	1	2	yes	—
217	C₂³	—	4	4	no	—
218	C₂³	—	8	4	no	—
219	C₂³	—	16	4	no	—
220	C₂³	C₂×D₈×C₅₃	1	2	yes	—
221	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	D₄	—	2	4	no	—
224	D₄	—	2	2	no	—
225	D₄	—	4	4	no	—
226	D₄	—	4	4	no	—
227	D₄	—	8	4	no	—
228	D₄	—	16	8	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	#121×C₅₃	1	2	yes	—
233	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
234	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
235	C₂×C₄	#135×C₅₃	1	2	yes	—
236	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	#121×C₅₃	1	2	yes	—
239	C₈	C₂²×D₄×C₅₃	1	1	yes	characteristic
240	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	8	4	no	—
245	C₂²	—	8	8	no	—
246	C₂²	—	16	8	no	—
247	C₂²	C₅₃×#466	1	2	yes	—
248	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	8	4	no	—
251	C₄	C₅₃×#466	1	2	yes	—
252	C₄	C₂×#121×C₅₃	1	1	yes	characteristic
253	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	4	2	no	—
256	C₂	—	16	4	no	—
257	C₂	C₂×C₅₃×#466	1	1	yes	characteristic
258	C₂	C₅₃×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×#41×C₅₃ ◁ C₂×#131×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₆×C₅₃	C₂	semidirect
2. C₂×C₅₃×#504	C₂	semidirect
3. C₅₃×#3058	C₂	semidirect
4. C₅₃×#2218	C₂	semidirect
5. C₅₃×#2215	C₂	semidirect
6. C₅₃×#2158	C₂	semidirect
7. C₅₃×#2140	C₂	semidirect
16. C₂×M₃₂×C₅₃	C₂²	semidirect
18. C₂²×C₁₆×C₅₃	C₂²	semidirect
20. C₅₃×#466	C₂²	semidirect
23. C₅₃×#367	C₂²	semidirect
24. C₅₃×#366	C₂²	semidirect
25. C₅₃×#357	C₂²	semidirect
48. C₂×C₁₆×C₅₃	D₄	semidirect
111. #33607	C₅₃	direct (N × H)
125. C₂²×D₁₆	C₂×C₅₃	semidirect
126. C₂×#504	C₂×C₅₃	semidirect
127. #3058	C₂×C₅₃	semidirect
128. #2218	C₂×C₅₃	semidirect
129. #2215	C₂×C₅₃	semidirect
130. #2158	C₂×C₅₃	semidirect
131. #2140	C₂×C₅₃	semidirect
145. C₂×M₃₂	C₂²×C₅₃	semidirect
147. C₂²×C₁₆	C₂²×C₅₃	semidirect
149. #466	C₂²×C₅₃	semidirect
152. #367	C₂²×C₅₃	semidirect
153. #366	C₂²×C₅₃	semidirect
154. #357	C₂²×C₅₃	semidirect
155. C₅₃	#33607	direct (N × H)
178. C₂×C₁₆	D₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₆×C₅₃	2
C₂×C₅₃×#504	2
C₅₃×#3058	2
C₅₃×#2218	2
C₅₃×#2215	2
C₅₃×#2158	2
C₅₃×#2140	2
#33607	53

Subgroups — C53×#33607 #27241119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33607 #27241119