Subgroups — C₅₃×#8655 #27216167

Order 13568.

418 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1834	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1803	C₂	1	2	yes	maximal
4	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1296	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#446	—	2	2	no	—
8	C₅₃×#446	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#103×C₅₃	—	2	2	no	—
13	C₂×#97×C₅₃	—	2	1	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#384	C₂²	1	2	yes	—
16	C₅₃×#340	—	2	1	no	—
17	C₅₃×#340	—	2	1	no	—
18	C₅₃×#340	—	2	2	no	—
19	C₅₃×#335	—	2	2	no	—
20	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	—	2	1	no	—
22	C₂²×C₈×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₄×Q₈×C₅₃	—	2	1	no	—
26	C₄×Q₈×C₅₃	—	2	1	no	—
27	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₄×Q₈×C₅₃	Q₈	1	2	yes	—
29	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	—	2	1	no	—
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	—	2	4	no	—
37	C₄⋊C₈×C₅₃	—	2	1	no	—
38	C₄⋊C₈×C₅₃	—	2	2	no	—
39	C₄⋊C₈×C₅₃	—	2	2	no	—
40	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
41	#103×C₅₃	—	4	4	no	—
42	#97×C₅₃	—	2	4	no	—
43	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₂×Q₈×C₅₃	—	2	1	no	—
46	C₂×Q₈×C₅₃	—	2	2	no	—
47	C₂×Q₈×C₅₃	—	2	2	no	—
48	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	—	4	2	no	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₂×C₈×C₅₃	—	4	2	no	—
61	C₂×C₈×C₅₃	—	4	2	no	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₄⋊C₄×C₅₃	—	2	1	no	—
66	C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₄⋊C₄×C₅₃	—	4	2	no	—
68	C₄⋊C₄×C₅₃	—	4	2	no	—
69	C₄⋊C₄×C₅₃	—	4	2	no	—
70	C₄⋊C₄×C₅₃	—	4	2	no	—
71	C₄⋊C₄×C₅₃	—	8	2	no	—
72	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	—	2	2	no	—
75	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₅₃	#41	1	1	yes	characteristic
79	C₂³×C₅₃	#121	1	1	yes	characteristic
80	Q₈×C₅₃	—	2	4	no	—
81	Q₈×C₅₃	—	4	1	no	—
82	Q₈×C₅₃	—	4	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	4	1	no	—
89	C₂×C₄×C₅₃	—	4	1	no	—
90	C₂×C₄×C₅₃	—	8	2	no	—
91	C₂×C₄×C₅₃	—	8	2	no	—
92	C₂×C₄×C₅₃	—	8	2	no	—
93	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#135	1	2	yes	—
100	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
101	C₂×C₄×C₅₃	#137	1	2	yes	—
102	C₂×C₄×C₅₃	#137	1	2	yes	—
103	C₈×C₅₃	—	2	2	no	—
104	C₈×C₅₃	—	2	2	no	—
105	C₈×C₅₃	—	4	2	no	—
106	C₈×C₅₃	—	8	2	no	—
107	#8655	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₅₃	#380	1	1	yes	characteristic
109	C₂²×C₅₃	#464	1	2	yes	—
110	C₂²×C₅₃	#468	1	2	yes	—
111	C₂²×C₅₃	#472	1	1	yes	characteristic
112	C₂²×C₅₃	#473	1	1	yes	characteristic
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	2	1	no	—
115	C₄×C₅₃	—	4	2	no	—
116	C₄×C₅₃	—	4	2	no	—
117	C₄×C₅₃	—	16	2	no	—
118	C₄×C₅₃	#416	1	1	yes	characteristic
119	C₄×C₅₃	#419	1	1	yes	characteristic
120	C₄×C₅₃	#428	1	2	yes	—
121	C₄×C₅₃	#469	1	1	yes	characteristic
122	C₄×C₅₃	#470	1	1	yes	characteristic
123	C₄×C₅₃	#479	1	2	yes	—
124	C₂×#446	C₂×C₅₃	1	1	yes	characteristic
125	#1834	C₂×C₅₃	1	1	yes	characteristic
126	#1803	C₂×C₅₃	1	2	yes	—
127	C₂×#335	C₂×C₅₃	1	1	yes	characteristic
128	#1296	C₂×C₅₃	1	2	yes	—
129	C₂×C₅₃	#1698	1	2	yes	—
130	C₂×C₅₃	#1700	1	2	yes	—
131	C₂×C₅₃	#1805	1	1	yes	characteristic
132	C₂×C₅₃	#1809	1	1	yes	characteristic
133	C₂×C₅₃	#1816	1	1	yes	characteristic
134	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
135	#446	—	2	2	no	—
136	#446	C₄×C₅₃	1	4	yes	complemented
137	C₂×C₄⋊C₈	—	2	1	no	—
138	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
139	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
140	C₂×#103	—	2	2	no	—
141	C₂×#97	—	2	1	no	—
142	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
143	#384	C₂²×C₅₃	1	2	yes	—
144	#340	—	2	1	no	—
145	#340	—	2	1	no	—
146	#340	—	2	2	no	—
147	#335	—	2	2	no	—
148	C₅₃	#8655	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
153	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	—	2	1	no	—
156	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
157	C₄×Q₈	Q₈×C₅₃	1	2	yes	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
165	C₄⋊C₈	—	2	4	no	—
166	C₄⋊C₈	—	2	1	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
170	#103	—	4	4	no	—
171	#97	—	2	4	no	—
172	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
173	C₄×C₈	D₄×C₅₃	1	2	yes	—
174	C₂×Q₈	—	2	1	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	C₄⋊C₄×C₅₃	1	2	yes	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
183	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
192	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	8	2	no	—
201	C₄⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
205	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
206	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
207	C₄²	#41×C₅₃	1	1	yes	characteristic
208	C₂³	#121×C₅₃	1	1	yes	characteristic
209	Q₈	—	2	4	no	—
210	Q₈	—	4	1	no	—
211	Q₈	—	4	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
224	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
225	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
226	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
227	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
228	C₂×C₄	#135×C₅₃	1	2	yes	—
229	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
230	C₂×C₄	#137×C₅₃	1	2	yes	—
231	C₂×C₄	#137×C₅₃	1	2	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	2	2	no	—
234	C₈	—	4	2	no	—
235	C₈	—	8	2	no	—
236	C₂²	C₅₃×#380	1	1	yes	characteristic
237	C₂²	C₅₃×#464	1	2	yes	—
238	C₂²	C₅₃×#468	1	2	yes	—
239	C₂²	C₅₃×#472	1	1	yes	characteristic
240	C₂²	C₅₃×#473	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	16	2	no	—
246	C₄	C₅₃×#416	1	1	yes	characteristic
247	C₄	C₅₃×#419	1	1	yes	characteristic
248	C₄	C₅₃×#428	1	2	yes	—
249	C₄	C₅₃×#469	1	1	yes	characteristic
250	C₄	C₅₃×#470	1	1	yes	characteristic
251	C₄	C₅₃×#479	1	2	yes	—
252	C₂	C₅₃×#1698	1	2	yes	—
253	C₂	C₅₃×#1700	1	2	yes	—
254	C₂	C₅₃×#1805	1	1	yes	characteristic
255	C₂	C₅₃×#1809	1	1	yes	characteristic
256	C₂	C₅₃×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#121
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅₃×#446	C₄	semidirect
107. #8655	C₅₃	direct (N × H)
136. #446	C₄×C₅₃	semidirect
148. C₅₃	#8655	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×C₅₃×#446	2
C₅₃×#1834	2
C₅₃×#1803	2
C₂×C₅₃×#335	2
C₅₃×#1296	2
#8655	53

Subgroups — C53×#8655 #27216167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8655 #27216167