Subgroups — C₅₃×#24297 #27231809

Order 13568.

1334 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#576	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3067	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3058	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3019	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2955	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#2048	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2008	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1978	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1976	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1814	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1795	C₂	1	2	yes	maximal, complemented
12	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#576	—	2	2	no	—
14	C₅₃×#576	—	2	2	no	—
15	C₅₃×#576	—	2	2	no	—
16	C₅₃×#576	—	2	2	no	—
17	C₅₃×#575	C₂²	1	1	yes	characteristic
18	C₅₃×#575	C₂²	1	2	yes	complemented
19	C₂×AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
20	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	2	yes	—
21	C₂×#135×C₅₃	—	2	1	no	—
22	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×D₈×C₅₃	—	2	1	no	—
24	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic, complemented
27	C₅₃×#535	C₂²	1	2	yes	complemented
28	C₅₃×#473	—	2	1	no	—
29	C₅₃×#472	—	2	1	no	—
30	C₅₃×#471	—	2	2	no	—
31	C₅₃×#469	—	2	2	no	—
32	C₅₃×#466	—	2	2	no	—
33	C₅₃×#455	—	2	4	no	—
34	C₅₃×#453	—	2	4	no	—
35	C₅₃×#444	—	2	2	no	—
36	C₅₃×#443	—	2	2	no	—
37	C₅₃×#430	C₂²	1	1	yes	characteristic, complemented
38	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic, complemented
39	C₅₃×#427	C₂²	1	1	yes	characteristic, complemented
40	C₅₃×#421	C₂²	1	1	yes	characteristic
41	C₅₃×#421	C₂²	1	2	yes	—
42	C₅₃×#421	C₂²	1	1	yes	characteristic, complemented
43	C₅₃×#421	C₂²	1	2	yes	complemented
44	C₅₃×#418	C₂²	1	2	yes	complemented
45	C₅₃×#418	C₂²	1	2	yes	complemented
46	C₅₃×#413	C₂²	1	2	yes	complemented
47	C₂×#100×C₅₃	C₂²	1	2	yes	complemented
48	C₅₃×#410	C₂²	1	2	yes	complemented
49	C₅₃×#405	C₂²	1	1	yes	characteristic, complemented
50	C₅₃×#344	C₂²	1	2	yes	complemented
51	H(5,𝔽₂)×C₅₃	—	2	4	no	—
52	H(5,𝔽₂)×C₅₃	—	2	4	no	—
53	H(5,𝔽₂)×C₅₃	—	2	4	no	—
54	H(5,𝔽₂)×C₅₃	—	4	2	no	—
55	C₂×#41×C₅₃	—	2	2	no	—
56	C₂×#41×C₅₃	—	2	2	no	—
57	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×#41×C₅₃	C₂³	1	2	yes	—
59	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₅₃	—	2	2	no	—
61	C₂²×D₄×C₅₃	—	2	2	no	—
62	C₂²×D₄×C₅₃	—	4	2	no	—
63	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
64	#137×C₅₃	—	2	2	no	—
65	#137×C₅₃	—	2	2	no	—
66	#137×C₅₃	—	2	1	no	—
67	#137×C₅₃	D₄	1	2	yes	—
68	AGL(1,ℤ/8)×C₅₃	—	2	8	no	—
69	AGL(1,ℤ/8)×C₅₃	—	2	8	no	—
70	AGL(1,ℤ/8)×C₅₃	—	2	4	no	—
71	AGL(1,ℤ/8)×C₅₃	—	2	4	no	—
72	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
73	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
74	AGL(1,ℤ/8)×C₅₃	—	2	1	no	—
75	AGL(1,ℤ/8)×C₅₃	D₄	1	2	yes	—
76	#135×C₅₃	—	2	4	no	—
77	#135×C₅₃	—	2	4	no	—
78	#135×C₅₃	—	2	1	no	—
79	#135×C₅₃	—	2	1	no	—
80	#135×C₅₃	—	2	1	no	—
81	#135×C₅₃	—	2	1	no	—
82	#135×C₅₃	—	2	2	no	—
83	#135×C₅₃	—	4	2	no	—
84	C₂×Q₁₆×C₅₃	D₄	1	1	yes	characteristic, complemented
85	C₂×SD₁₆×C₅₃	—	2	2	no	—
86	C₂×SD₁₆×C₅₃	D₄	1	2	yes	complemented
87	C₂×D₈×C₅₃	—	2	4	no	—
88	C₂×D₈×C₅₃	—	2	4	no	—
89	C₂×D₈×C₅₃	—	2	1	no	—
90	C₂×D₈×C₅₃	—	2	1	no	—
91	C₂×D₈×C₅₃	—	2	1	no	—
92	C₂×D₈×C₅₃	—	2	1	no	—
93	C₂×D₈×C₅₃	—	2	2	no	—
94	C₂×D₈×C₅₃	D₄	1	2	yes	—
95	C₂×D₈×C₅₃	D₄	1	2	yes	—
96	C₂×D₈×C₅₃	D₄	1	1	yes	characteristic, complemented
97	#131×C₅₃	—	2	1	no	—
98	#131×C₅₃	—	2	1	no	—
99	#131×C₅₃	—	2	1	no	—
100	#131×C₅₃	—	2	1	no	—
101	#131×C₅₃	—	2	1	no	—
102	#131×C₅₃	—	2	1	no	—
103	#131×C₅₃	D₄	1	2	yes	complemented
104	#131×C₅₃	D₄	1	2	yes	complemented
105	C₂×M₁₆×C₅₃	—	2	1	no	—
106	C₂×M₁₆×C₅₃	—	2	1	no	—
107	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
108	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
109	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
110	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
111	C₂²×C₈×C₅₃	—	2	1	no	—
112	C₂²×C₈×C₅₃	—	2	1	no	—
113	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
114	#127×C₅₃	—	2	2	no	—
115	#124×C₅₃	—	2	2	no	—
116	#121×C₅₃	—	2	4	no	—
117	#120×C₅₃	—	4	2	no	—
118	#117×C₅₃	C₂³	1	2	yes	—
119	#117×C₅₃	C₂³	1	1	yes	characteristic
120	#108×C₅₃	—	2	2	no	—
121	#108×C₅₃	—	2	1	no	—
122	#108×C₅₃	—	2	1	no	—
123	#107×C₅₃	—	2	1	no	—
124	#106×C₅₃	—	2	1	no	—
125	#104×C₅₃	—	2	2	no	—
126	#104×C₅₃	—	2	4	no	—
127	#104×C₅₃	—	2	2	no	—
128	#104×C₅₃	—	2	4	no	—
129	#102×C₅₃	—	2	4	no	—
130	#102×C₅₃	—	4	2	no	—
131	#101×C₅₃	—	2	2	no	—
132	#100×C₅₃	—	2	4	no	—
133	#100×C₅₃	—	2	2	no	—
134	#99×C₅₃	—	4	2	no	—
135	#97×C₅₃	—	2	1	no	—
136	C₄×C₈×C₅₃	—	2	1	no	—
137	C₂⁴×C₅₃	—	2	2	no	—
138	C₂⁴×C₅₃	—	4	2	no	—
139	#41×C₅₃	—	2	1	no	—
140	#41×C₅₃	—	2	1	no	—
141	#41×C₅₃	—	2	8	no	—
142	#41×C₅₃	—	2	1	no	—
143	#41×C₅₃	—	2	2	no	—
144	#41×C₅₃	—	2	1	no	—
145	#41×C₅₃	—	2	2	no	—
146	#41×C₅₃	—	2	2	no	—
147	#41×C₅₃	—	2	1	no	—
148	#41×C₅₃	—	2	1	no	—
149	#41×C₅₃	—	2	2	no	—
150	#41×C₅₃	—	4	4	no	—
151	#41×C₅₃	—	4	4	no	—
152	#41×C₅₃	—	4	4	no	—
153	#41×C₅₃	C₂×D₄	1	2	yes	—
154	#41×C₅₃	C₂×D₄	1	2	yes	—
155	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
156	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
157	C₂×Q₈×C₅₃	#41	1	1	yes	characteristic
158	C₂×D₄×C₅₃	—	2	1	no	—
159	C₂×D₄×C₅₃	—	2	8	no	—
160	C₂×D₄×C₅₃	—	2	2	no	—
161	C₂×D₄×C₅₃	—	2	1	no	—
162	C₂×D₄×C₅₃	—	2	2	no	—
163	C₂×D₄×C₅₃	—	2	2	no	—
164	C₂×D₄×C₅₃	—	2	2	no	—
165	C₂×D₄×C₅₃	—	2	2	no	—
166	C₂×D₄×C₅₃	—	2	2	no	—
167	C₂×D₄×C₅₃	—	4	4	no	—
168	C₂×D₄×C₅₃	—	4	2	no	—
169	C₂×D₄×C₅₃	—	4	2	no	—
170	C₂×D₄×C₅₃	—	4	4	no	—
171	C₂×D₄×C₅₃	—	4	8	no	—
172	C₂×D₄×C₅₃	—	4	8	no	—
173	C₂×D₄×C₅₃	—	4	4	no	—
174	C₂×D₄×C₅₃	—	4	4	no	—
175	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
176	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
177	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
178	C₂×D₄×C₅₃	#41	1	1	yes	characteristic
179	C₂²×C₄×C₅₃	—	2	1	no	—
180	C₂²×C₄×C₅₃	—	2	1	no	—
181	C₂²×C₄×C₅₃	—	2	2	no	—
182	C₂²×C₄×C₅₃	—	4	2	no	—
183	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
184	Q₁₆×C₅₃	—	2	2	no	—
185	Q₁₆×C₅₃	—	2	1	no	—
186	Q₁₆×C₅₃	—	2	1	no	—
187	Q₁₆×C₅₃	—	4	2	no	—
188	SD₁₆×C₅₃	—	2	8	no	—
189	SD₁₆×C₅₃	—	2	2	no	—
190	SD₁₆×C₅₃	—	2	2	no	—
191	SD₁₆×C₅₃	—	2	2	no	—
192	SD₁₆×C₅₃	—	2	2	no	—
193	SD₁₆×C₅₃	—	4	2	no	—
194	SD₁₆×C₅₃	—	4	2	no	—
195	D₈×C₅₃	—	2	8	no	—
196	D₈×C₅₃	—	2	2	no	—
197	D₈×C₅₃	—	2	2	no	—
198	D₈×C₅₃	—	2	2	no	—
199	D₈×C₅₃	—	2	1	no	—
200	D₈×C₅₃	—	2	1	no	—
201	D₈×C₅₃	—	2	2	no	—
202	D₈×C₅₃	—	2	2	no	—
203	D₈×C₅₃	—	4	2	no	—
204	M₁₆×C₅₃	—	2	1	no	—
205	M₁₆×C₅₃	—	2	1	no	—
206	M₁₆×C₅₃	—	2	2	no	—
207	M₁₆×C₅₃	—	2	2	no	—
208	M₁₆×C₅₃	—	2	1	no	—
209	M₁₆×C₅₃	—	2	2	no	—
210	M₁₆×C₅₃	—	2	2	no	—
211	M₁₆×C₅₃	—	2	1	no	—
212	M₁₆×C₅₃	—	4	1	no	—
213	M₁₆×C₅₃	—	4	1	no	—
214	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
215	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
216	C₂×C₈×C₅₃	—	2	1	no	—
217	C₂×C₈×C₅₃	—	2	1	no	—
218	C₂×C₈×C₅₃	—	2	2	no	—
219	C₂×C₈×C₅₃	—	2	2	no	—
220	C₂×C₈×C₅₃	—	2	1	no	—
221	C₂×C₈×C₅₃	—	2	2	no	—
222	C₂×C₈×C₅₃	—	4	1	no	—
223	C₂×C₈×C₅₃	—	4	1	no	—
224	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
225	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
226	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
227	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
228	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
229	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
230	C₄⋊C₄×C₅₃	—	2	2	no	—
231	C₄⋊C₄×C₅₃	—	2	1	no	—
232	C₂²⋊C₄×C₅₃	—	2	1	no	—
233	C₂²⋊C₄×C₅₃	—	2	2	no	—
234	C₂²⋊C₄×C₅₃	—	4	4	no	—
235	C₄²×C₅₃	—	2	1	no	—
236	C₄²×C₅₃	—	2	2	no	—
237	C₂³×C₅₃	—	2	1	no	—
238	C₂³×C₅₃	—	2	1	no	—
239	C₂³×C₅₃	—	2	4	no	—
240	C₂³×C₅₃	—	2	2	no	—
241	C₂³×C₅₃	—	4	8	no	—
242	C₂³×C₅₃	—	4	2	no	—
243	C₂³×C₅₃	—	4	2	no	—
244	C₂³×C₅₃	—	16	2	no	—
245	C₂³×C₅₃	—	16	4	no	—
246	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
247	Q₈×C₅₃	—	2	1	no	—
248	Q₈×C₅₃	—	2	2	no	—
249	Q₈×C₅₃	—	2	1	no	—
250	Q₈×C₅₃	—	2	2	no	—
251	Q₈×C₅₃	—	2	1	no	—
252	Q₈×C₅₃	#121	1	2	yes	—
253	D₄×C₅₃	—	2	2	no	—
254	D₄×C₅₃	—	2	2	no	—
255	D₄×C₅₃	—	2	2	no	—
256	D₄×C₅₃	—	2	4	no	—
257	D₄×C₅₃	—	2	1	no	—
258	D₄×C₅₃	—	2	2	no	—
259	D₄×C₅₃	—	2	1	no	—
260	D₄×C₅₃	—	2	1	no	—
261	D₄×C₅₃	—	2	4	no	—
262	D₄×C₅₃	—	4	8	no	—
263	D₄×C₅₃	—	4	8	no	—
264	D₄×C₅₃	—	4	2	no	—
265	D₄×C₅₃	—	4	1	no	—
266	D₄×C₅₃	—	4	2	no	—
267	D₄×C₅₃	—	4	1	no	—
268	D₄×C₅₃	—	4	2	no	—
269	D₄×C₅₃	—	4	2	no	—
270	D₄×C₅₃	—	16	4	no	—
271	D₄×C₅₃	#121	1	2	yes	—
272	C₂×C₄×C₅₃	—	2	1	no	—
273	C₂×C₄×C₅₃	—	2	2	no	—
274	C₂×C₄×C₅₃	—	2	2	no	—
275	C₂×C₄×C₅₃	—	2	1	no	—
276	C₂×C₄×C₅₃	—	2	2	no	—
277	C₂×C₄×C₅₃	—	4	8	no	—
278	C₂×C₄×C₅₃	—	4	1	no	—
279	C₂×C₄×C₅₃	—	4	2	no	—
280	C₂×C₄×C₅₃	—	4	2	no	—
281	C₂×C₄×C₅₃	—	4	1	no	—
282	C₂×C₄×C₅₃	—	4	1	no	—
283	C₂×C₄×C₅₃	—	4	2	no	—
284	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
285	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
286	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
288	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
289	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
290	C₈×C₅₃	—	2	2	no	—
291	C₈×C₅₃	—	2	2	no	—
292	C₈×C₅₃	—	4	1	no	—
293	C₈×C₅₃	—	4	2	no	—
294	C₈×C₅₃	#120	1	1	yes	characteristic
295	C₈×C₅₃	#120	1	1	yes	characteristic
296	C₈×C₅₃	#120	1	1	yes	characteristic
297	C₈×C₅₃	#120	1	1	yes	characteristic
298	#24297	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
299	C₂²×C₅₃	—	2	2	no	—
300	C₂²×C₅₃	—	4	1	no	—
301	C₂²×C₅₃	—	4	2	no	—
302	C₂²×C₅₃	—	4	2	no	—
303	C₂²×C₅₃	—	4	4	no	—
304	C₂²×C₅₃	—	4	2	no	—
305	C₂²×C₅₃	—	4	1	no	—
306	C₂²×C₅₃	—	8	4	no	—
307	C₂²×C₅₃	—	8	2	no	—
308	C₂²×C₅₃	—	8	4	no	—
309	C₂²×C₅₃	—	16	2	no	—
310	C₂²×C₅₃	—	16	8	no	—
311	C₂²×C₅₃	#535	1	1	yes	characteristic
312	C₂²×C₅₃	D₄²	1	1	yes	characteristic
313	C₂²×C₅₃	#546	1	1	yes	characteristic
314	C₄×C₅₃	—	2	2	no	—
315	C₄×C₅₃	—	4	2	no	—
316	C₄×C₅₃	—	4	1	no	—
317	C₄×C₅₃	—	8	1	no	—
318	C₄×C₅₃	—	8	2	no	—
319	C₄×C₅₃	C₂×#120	1	1	yes	characteristic
320	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
321	C₄×C₅₃	D₄²	1	1	yes	characteristic
322	C₄×C₅₃	#546	1	1	yes	characteristic
323	C₂×#576	C₂×C₅₃	1	1	yes	characteristic, complemented
324	#3067	C₂×C₅₃	1	1	yes	characteristic, complemented
325	#3058	C₂×C₅₃	1	1	yes	characteristic, complemented
326	#3019	C₂×C₅₃	1	2	yes	complemented
327	#2955	C₂×C₅₃	1	1	yes	characteristic, complemented
328	#2048	C₂×C₅₃	1	2	yes	complemented
329	#2008	C₂×C₅₃	1	2	yes	complemented
330	#1978	C₂×C₅₃	1	1	yes	characteristic, complemented
331	#1976	C₂×C₅₃	1	1	yes	characteristic, complemented
332	#1814	C₂×C₅₃	1	1	yes	characteristic, complemented
333	#1795	C₂×C₅₃	1	2	yes	complemented
334	C₂×C₅₃	—	2	1	no	—
335	C₂×C₅₃	—	2	1	no	—
336	C₂×C₅₃	—	2	1	no	—
337	C₂×C₅₃	—	4	2	no	—
338	C₂×C₅₃	—	8	1	no	—
339	C₂×C₅₃	—	8	4	no	—
340	C₂×C₅₃	—	8	2	no	—
341	C₂×C₅₃	#2408	1	1	yes	characteristic
342	C₂×H(5,𝔽₂)	C₂²×C₅₃	1	2	yes	—
343	#576	—	2	2	no	—
344	#576	—	2	2	no	—
345	#576	—	2	2	no	—
346	#576	—	2	2	no	—
347	#575	C₂²×C₅₃	1	1	yes	characteristic
348	#575	C₂²×C₅₃	1	2	yes	complemented
349	C₂×AGL(1,ℤ/8)	—	2	2	no	—
350	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	2	yes	—
351	C₂×#135	—	2	1	no	—
352	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
353	C₂²×D₈	—	2	1	no	—
354	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
355	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
356	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic, complemented
357	#535	C₂²×C₅₃	1	2	yes	complemented
358	#473	—	2	1	no	—
359	#472	—	2	1	no	—
360	#471	—	2	2	no	—
361	#469	—	2	2	no	—
362	#466	—	2	2	no	—
363	#455	—	2	4	no	—
364	#453	—	2	4	no	—
365	#444	—	2	2	no	—
366	#443	—	2	2	no	—
367	#430	C₂²×C₅₃	1	1	yes	characteristic, complemented
368	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic, complemented
369	#427	C₂²×C₅₃	1	1	yes	characteristic, complemented
370	#421	C₂²×C₅₃	1	1	yes	characteristic
371	#421	C₂²×C₅₃	1	2	yes	—
372	#421	C₂²×C₅₃	1	2	yes	complemented
373	#421	C₂²×C₅₃	1	1	yes	characteristic, complemented
374	#418	C₂²×C₅₃	1	2	yes	complemented
375	#418	C₂²×C₅₃	1	2	yes	complemented
376	#413	C₂²×C₅₃	1	2	yes	complemented
377	C₂×#100	C₂²×C₅₃	1	2	yes	complemented
378	#410	C₂²×C₅₃	1	2	yes	complemented
379	#405	C₂²×C₅₃	1	1	yes	characteristic, complemented
380	#344	C₂²×C₅₃	1	2	yes	complemented
381	C₅₃	#24297	1	1	yes	characteristic, direct product factor, Sylow
382	H(5,𝔽₂)	—	2	4	no	—
383	H(5,𝔽₂)	—	2	4	no	—
384	H(5,𝔽₂)	—	2	4	no	—
385	H(5,𝔽₂)	—	4	2	no	—
386	C₂×#41	—	2	2	no	—
387	C₂×#41	—	2	2	no	—
388	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
389	C₂×#41	C₂³×C₅₃	1	2	yes	—
390	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
391	C₂²×D₄	—	2	2	no	—
392	C₂²×D₄	—	2	2	no	—
393	C₂²×D₄	—	4	2	no	—
394	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
395	#137	—	2	1	no	—
396	#137	—	2	2	no	—
397	#137	—	2	2	no	—
398	#137	D₄×C₅₃	1	2	yes	—
399	AGL(1,ℤ/8)	—	2	8	no	—
400	AGL(1,ℤ/8)	—	2	8	no	—
401	AGL(1,ℤ/8)	—	2	4	no	—
402	AGL(1,ℤ/8)	—	2	1	no	—
403	AGL(1,ℤ/8)	—	2	2	no	—
404	AGL(1,ℤ/8)	—	2	2	no	—
405	AGL(1,ℤ/8)	—	2	4	no	—
406	AGL(1,ℤ/8)	D₄×C₅₃	1	2	yes	—
407	#135	—	2	4	no	—
408	#135	—	2	4	no	—
409	#135	—	2	2	no	—
410	#135	—	2	1	no	—
411	#135	—	2	1	no	—
412	#135	—	2	1	no	—
413	#135	—	2	1	no	—
414	#135	—	4	2	no	—
415	C₂×Q₁₆	D₄×C₅₃	1	1	yes	characteristic, complemented
416	C₂×SD₁₆	—	2	2	no	—
417	C₂×SD₁₆	D₄×C₅₃	1	2	yes	complemented
418	C₂×D₈	—	2	4	no	—
419	C₂×D₈	—	2	4	no	—
420	C₂×D₈	—	2	2	no	—
421	C₂×D₈	—	2	1	no	—
422	C₂×D₈	—	2	1	no	—
423	C₂×D₈	—	2	1	no	—
424	C₂×D₈	—	2	1	no	—
425	C₂×D₈	D₄×C₅₃	1	2	yes	—
426	C₂×D₈	D₄×C₅₃	1	2	yes	—
427	C₂×D₈	D₄×C₅₃	1	1	yes	characteristic, complemented
428	#131	—	2	1	no	—
429	#131	—	2	1	no	—
430	#131	—	2	1	no	—
431	#131	—	2	1	no	—
432	#131	—	2	1	no	—
433	#131	—	2	1	no	—
434	#131	D₄×C₅₃	1	2	yes	complemented
435	#131	D₄×C₅₃	1	2	yes	complemented
436	C₂×M₁₆	—	2	1	no	—
437	C₂×M₁₆	—	2	1	no	—
438	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
439	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
440	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
441	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
442	C₂²×C₈	—	2	1	no	—
443	C₂²×C₈	—	2	1	no	—
444	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
445	#127	—	2	2	no	—
446	#124	—	2	2	no	—
447	#121	—	2	4	no	—
448	#120	—	4	2	no	—
449	#117	C₂³×C₅₃	1	2	yes	—
450	#117	C₂³×C₅₃	1	1	yes	characteristic
451	#108	—	2	2	no	—
452	#108	—	2	1	no	—
453	#108	—	2	1	no	—
454	#107	—	2	1	no	—
455	#106	—	2	1	no	—
456	#104	—	2	2	no	—
457	#104	—	2	4	no	—
458	#104	—	2	2	no	—
459	#104	—	2	4	no	—
460	#102	—	2	4	no	—
461	#102	—	4	2	no	—
462	#101	—	2	2	no	—
463	#100	—	2	2	no	—
464	#100	—	2	4	no	—
465	#99	—	4	2	no	—
466	#97	—	2	1	no	—
467	C₄×C₈	—	2	1	no	—
468	C₂⁴	—	2	2	no	—
469	C₂⁴	—	4	2	no	—
470	#41	—	2	1	no	—
471	#41	—	2	1	no	—
472	#41	—	2	8	no	—
473	#41	—	2	1	no	—
474	#41	—	2	2	no	—
475	#41	—	2	1	no	—
476	#41	—	2	2	no	—
477	#41	—	2	2	no	—
478	#41	—	2	1	no	—
479	#41	—	2	1	no	—
480	#41	—	2	2	no	—
481	#41	—	4	4	no	—
482	#41	—	4	4	no	—
483	#41	—	4	4	no	—
484	#41	C₂×D₄×C₅₃	1	2	yes	—
485	#41	C₂×D₄×C₅₃	1	2	yes	—
486	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
487	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
488	C₂×Q₈	#41×C₅₃	1	1	yes	characteristic
489	C₂×D₄	—	2	1	no	—
490	C₂×D₄	—	2	8	no	—
491	C₂×D₄	—	2	2	no	—
492	C₂×D₄	—	2	1	no	—
493	C₂×D₄	—	2	2	no	—
494	C₂×D₄	—	2	2	no	—
495	C₂×D₄	—	2	2	no	—
496	C₂×D₄	—	2	2	no	—
497	C₂×D₄	—	2	2	no	—
498	C₂×D₄	—	4	4	no	—
499	C₂×D₄	—	4	2	no	—
500	C₂×D₄	—	4	2	no	—
501	C₂×D₄	—	4	4	no	—
502	C₂×D₄	—	4	8	no	—
503	C₂×D₄	—	4	8	no	—
504	C₂×D₄	—	4	4	no	—
505	C₂×D₄	—	4	4	no	—
506	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
507	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
508	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
509	C₂×D₄	#41×C₅₃	1	1	yes	characteristic
510	C₂²×C₄	—	2	1	no	—
511	C₂²×C₄	—	2	1	no	—
512	C₂²×C₄	—	2	2	no	—
513	C₂²×C₄	—	4	2	no	—
514	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
515	Q₁₆	—	2	2	no	—
516	Q₁₆	—	2	1	no	—
517	Q₁₆	—	2	1	no	—
518	Q₁₆	—	4	2	no	—
519	SD₁₆	—	2	8	no	—
520	SD₁₆	—	2	2	no	—
521	SD₁₆	—	2	2	no	—
522	SD₁₆	—	2	2	no	—
523	SD₁₆	—	2	2	no	—
524	SD₁₆	—	4	2	no	—
525	SD₁₆	—	4	2	no	—
526	D₈	—	2	8	no	—
527	D₈	—	2	2	no	—
528	D₈	—	2	2	no	—
529	D₈	—	2	2	no	—
530	D₈	—	2	1	no	—
531	D₈	—	2	1	no	—
532	D₈	—	2	2	no	—
533	D₈	—	2	2	no	—
534	D₈	—	4	2	no	—
535	M₁₆	—	2	1	no	—
536	M₁₆	—	2	1	no	—
537	M₁₆	—	2	2	no	—
538	M₁₆	—	2	2	no	—
539	M₁₆	—	2	1	no	—
540	M₁₆	—	2	2	no	—
541	M₁₆	—	2	2	no	—
542	M₁₆	—	2	1	no	—
543	M₁₆	—	4	1	no	—
544	M₁₆	—	4	1	no	—
545	M₁₆	C₂×D₄×C₅₃	1	2	yes	—
546	M₁₆	C₂×D₄×C₅₃	1	2	yes	—
547	C₂×C₈	—	2	1	no	—
548	C₂×C₈	—	2	1	no	—
549	C₂×C₈	—	2	2	no	—
550	C₂×C₈	—	2	2	no	—
551	C₂×C₈	—	2	1	no	—
552	C₂×C₈	—	2	2	no	—
553	C₂×C₈	—	4	1	no	—
554	C₂×C₈	—	4	1	no	—
555	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
556	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
557	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
558	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
559	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
560	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
561	C₄⋊C₄	—	2	2	no	—
562	C₄⋊C₄	—	2	1	no	—
563	C₂²⋊C₄	—	2	1	no	—
564	C₂²⋊C₄	—	2	2	no	—
565	C₂²⋊C₄	—	4	4	no	—
566	C₄²	—	2	1	no	—
567	C₄²	—	2	2	no	—
568	C₂³	—	2	1	no	—
569	C₂³	—	2	1	no	—
570	C₂³	—	2	4	no	—
571	C₂³	—	2	2	no	—
572	C₂³	—	4	8	no	—
573	C₂³	—	4	2	no	—
574	C₂³	—	4	2	no	—
575	C₂³	—	16	2	no	—
576	C₂³	—	16	4	no	—
577	C₂³	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
578	Q₈	—	2	1	no	—
579	Q₈	—	2	2	no	—
580	Q₈	—	2	1	no	—
581	Q₈	—	2	1	no	—
582	Q₈	—	2	2	no	—
583	Q₈	#121×C₅₃	1	2	yes	—
584	D₄	—	2	2	no	—
585	D₄	—	2	2	no	—
586	D₄	—	2	2	no	—
587	D₄	—	2	4	no	—
588	D₄	—	2	1	no	—
589	D₄	—	2	1	no	—
590	D₄	—	2	2	no	—
591	D₄	—	2	1	no	—
592	D₄	—	2	4	no	—
593	D₄	—	4	8	no	—
594	D₄	—	4	8	no	—
595	D₄	—	4	2	no	—
596	D₄	—	4	1	no	—
597	D₄	—	4	2	no	—
598	D₄	—	4	1	no	—
599	D₄	—	4	2	no	—
600	D₄	—	4	2	no	—
601	D₄	—	16	4	no	—
602	D₄	#121×C₅₃	1	2	yes	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	4	8	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	2	no	—
611	C₂×C₄	—	4	2	no	—
612	C₂×C₄	—	4	1	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	4	2	no	—
615	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
616	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
617	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
618	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
619	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
620	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
621	C₈	—	2	2	no	—
622	C₈	—	2	2	no	—
623	C₈	—	4	1	no	—
624	C₈	—	4	2	no	—
625	C₈	#120×C₅₃	1	1	yes	characteristic
626	C₈	#120×C₅₃	1	1	yes	characteristic
627	C₈	#120×C₅₃	1	1	yes	characteristic
628	C₈	#120×C₅₃	1	1	yes	characteristic
629	C₂²	—	2	2	no	—
630	C₂²	—	4	1	no	—
631	C₂²	—	4	2	no	—
632	C₂²	—	4	2	no	—
633	C₂²	—	4	4	no	—
634	C₂²	—	4	2	no	—
635	C₂²	—	4	1	no	—
636	C₂²	—	8	4	no	—
637	C₂²	—	8	2	no	—
638	C₂²	—	8	4	no	—
639	C₂²	—	16	2	no	—
640	C₂²	—	16	8	no	—
641	C₂²	C₅₃×#535	1	1	yes	characteristic
642	C₂²	D₄²×C₅₃	1	1	yes	characteristic
643	C₂²	C₅₃×#546	1	1	yes	characteristic
644	C₄	—	2	2	no	—
645	C₄	—	4	2	no	—
646	C₄	—	4	1	no	—
647	C₄	—	8	2	no	—
648	C₄	—	8	1	no	—
649	C₄	C₂×#120×C₅₃	1	1	yes	characteristic
650	C₄	C₂×#121×C₅₃	1	1	yes	characteristic
651	C₄	D₄²×C₅₃	1	1	yes	characteristic
652	C₄	C₅₃×#546	1	1	yes	characteristic
653	C₂	—	2	1	no	—
654	C₂	—	2	1	no	—
655	C₂	—	2	1	no	—
656	C₂	—	4	2	no	—
657	C₂	—	8	1	no	—
658	C₂	—	8	4	no	—
659	C₂	—	8	2	no	—
660	C₂	C₅₃×#2408	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#2408
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#576	C₂	semidirect
2. C₅₃×#3067	C₂	semidirect
3. C₅₃×#3058	C₂	semidirect
4. C₅₃×#3019	C₂	semidirect
5. C₅₃×#2955	C₂	semidirect
6. C₅₃×#2048	C₂	semidirect
7. C₅₃×#2008	C₂	semidirect
8. C₅₃×#1978	C₂	semidirect
9. C₅₃×#1976	C₂	semidirect
10. C₅₃×#1814	C₂	semidirect
11. C₅₃×#1795	C₂	semidirect
18. C₅₃×#575	C₂²	semidirect
26. C₂×#131×C₅₃	C₂²	semidirect
27. C₅₃×#535	C₂²	semidirect
37. C₅₃×#430	C₂²	semidirect
38. C₂×#108×C₅₃	C₂²	semidirect
39. C₅₃×#427	C₂²	semidirect
42. C₅₃×#421	C₂²	semidirect
43. C₅₃×#421	C₂²	semidirect
44. C₅₃×#418	C₂²	semidirect
45. C₅₃×#418	C₂²	semidirect
46. C₅₃×#413	C₂²	semidirect
47. C₂×#100×C₅₃	C₂²	semidirect
48. C₅₃×#410	C₂²	semidirect
49. C₅₃×#405	C₂²	semidirect
50. C₅₃×#344	C₂²	semidirect
84. C₂×Q₁₆×C₅₃	D₄	semidirect
86. C₂×SD₁₆×C₅₃	D₄	semidirect
96. C₂×D₈×C₅₃	D₄	semidirect
103. #131×C₅₃	D₄	semidirect
104. #131×C₅₃	D₄	semidirect
298. #24297	C₅₃	direct (N × H)
323. C₂×#576	C₂×C₅₃	semidirect
324. #3067	C₂×C₅₃	semidirect
325. #3058	C₂×C₅₃	semidirect
326. #3019	C₂×C₅₃	semidirect
327. #2955	C₂×C₅₃	semidirect
328. #2048	C₂×C₅₃	semidirect
329. #2008	C₂×C₅₃	semidirect
330. #1978	C₂×C₅₃	semidirect
331. #1976	C₂×C₅₃	semidirect
332. #1814	C₂×C₅₃	semidirect
333. #1795	C₂×C₅₃	semidirect
348. #575	C₂²×C₅₃	semidirect
356. C₂×#131	C₂²×C₅₃	semidirect
357. #535	C₂²×C₅₃	semidirect
367. #430	C₂²×C₅₃	semidirect
368. C₂×#108	C₂²×C₅₃	semidirect
369. #427	C₂²×C₅₃	semidirect
372. #421	C₂²×C₅₃	semidirect
373. #421	C₂²×C₅₃	semidirect
374. #418	C₂²×C₅₃	semidirect
375. #418	C₂²×C₅₃	semidirect
376. #413	C₂²×C₅₃	semidirect
377. C₂×#100	C₂²×C₅₃	semidirect
378. #410	C₂²×C₅₃	semidirect
379. #405	C₂²×C₅₃	semidirect
380. #344	C₂²×C₅₃	semidirect
381. C₅₃	#24297	direct (N × H)
415. C₂×Q₁₆	D₄×C₅₃	semidirect
417. C₂×SD₁₆	D₄×C₅₃	semidirect
427. C₂×D₈	D₄×C₅₃	semidirect
434. #131	D₄×C₅₃	semidirect
435. #131	D₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#576	2
C₅₃×#3067	2
C₅₃×#3058	2
C₅₃×#3019	2
C₅₃×#2955	2
C₅₃×#2048	2
C₅₃×#2008	2
C₅₃×#1978	2
C₅₃×#1976	2
C₅₃×#1814	2
C₅₃×#1795	2
#24297	53

Subgroups — C53×#24297 #27231809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#24297 #27231809