Subgroups — C₂×C₅₃×#3155 #27268153

Order 13568.

3534 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×AGL(1,ℤ/8)×C₅₃	C₂	1	2	yes	maximal, complemented
2	C₂²×#135×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#535	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3155	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₅₃×#496	C₂	1	4	yes	maximal, complemented
7	C₂×C₅₃×#495	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₃×#493	C₂	1	2	yes	maximal, complemented
9	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₅₃	C₂²	1	2	yes	—
11	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
12	C₂×AGL(1,ℤ/8)×C₅₃	—	2	16	no	—
13	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	16	yes	complemented
14	C₂×#135×C₅₃	—	2	8	no	—
15	C₂×#135×C₅₃	C₂²	1	8	yes	complemented
16	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
17	C₂²×SD₁₆×C₅₃	C₂²	1	2	yes	complemented
18	C₂²×SD₁₆×C₅₃	C₂²	1	4	yes	complemented
19	C₂²×D₈×C₅₃	—	2	8	no	—
20	C₂²×D₈×C₅₃	C₂²	1	2	yes	—
21	C₂²×D₈×C₅₃	C₂²	1	2	yes	—
22	C₂²×D₈×C₅₃	C₂²	1	8	yes	complemented
23	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
24	C₂²×D₈×C₅₃	C₂²	1	4	yes	complemented
25	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
26	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#535	—	2	8	no	—
28	C₅₃×#535	C₂²	1	16	yes	complemented
29	C₂×#127×C₅₃	C₂²	1	4	yes	complemented
30	C₂×#124×C₅₃	C₂²	1	4	yes	complemented
31	C₂×#121×C₅₃	—	2	4	no	—
32	C₂×#120×C₅₃	—	2	4	no	—
33	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#496	C₂²	1	32	yes	complemented
35	C₅₃×#495	C₂²	1	16	yes	complemented
36	C₅₃×#493	C₂²	1	16	yes	complemented
37	C₅₃×#405	C₂²	1	8	yes	complemented
38	C₂×#97×C₅₃	C₂²	1	2	yes	complemented
39	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	complemented
40	C₂⁵×C₅₃	—	2	2	no	—
41	C₂×#41×C₅₃	—	2	4	no	—
42	C₂×#41×C₅₃	—	2	8	no	—
43	C₂×#41×C₅₃	—	2	8	no	—
44	C₂×#41×C₅₃	C₂³	1	8	yes	—
45	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
46	C₂²×D₄×C₅₃	—	2	2	no	—
47	C₂²×D₄×C₅₃	—	2	8	no	—
48	C₂²×D₄×C₅₃	—	2	8	no	—
49	C₂²×D₄×C₅₃	—	2	2	no	—
50	C₂²×D₄×C₅₃	—	2	4	no	—
51	C₂²×D₄×C₅₃	—	2	8	no	—
52	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
53	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
54	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
55	C₂³×C₄×C₅₃	—	2	2	no	—
56	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
57	AGL(1,ℤ/8)×C₅₃	—	2	64	no	—
58	#135×C₅₃	—	2	32	no	—
59	C₂×Q₁₆×C₅₃	—	2	4	no	—
60	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	complemented
61	C₂×SD₁₆×C₅₃	—	2	8	no	—
62	C₂×SD₁₆×C₅₃	—	2	16	no	—
63	C₂×SD₁₆×C₅₃	C₂³	1	16	yes	complemented
64	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	complemented
65	C₂×D₈×C₅₃	—	2	8	no	—
66	C₂×D₈×C₅₃	—	2	8	no	—
67	C₂×D₈×C₅₃	—	2	4	no	—
68	C₂×D₈×C₅₃	—	2	16	no	—
69	C₂×D₈×C₅₃	—	2	32	no	—
70	C₂×D₈×C₅₃	C₂³	1	8	yes	—
71	C₂×D₈×C₅₃	C₂³	1	8	yes	—
72	C₂×D₈×C₅₃	C₂³	1	4	yes	complemented
73	C₂×D₈×C₅₃	C₂³	1	16	yes	complemented
74	C₂×M₁₆×C₅₃	D₄	1	8	yes	complemented
75	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
76	C₂²×C₈×C₅₃	D₄	1	8	yes	complemented
77	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
80	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
81	#127×C₅₃	—	2	8	no	—
82	#127×C₅₃	C₂³	1	16	yes	complemented
83	#124×C₅₃	—	2	8	no	—
84	#124×C₅₃	C₂³	1	16	yes	complemented
85	#121×C₅₃	—	2	32	no	—
86	#120×C₅₃	—	2	32	no	—
87	#117×C₅₃	D₄	1	4	yes	—
88	#117×C₅₃	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
90	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
91	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
92	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
93	#97×C₅₃	C₂³	1	8	yes	complemented
94	C₄×C₈×C₅₃	C₂³	1	8	yes	complemented
95	C₂⁴×C₅₃	—	2	4	no	—
96	C₂⁴×C₅₃	—	2	2	no	—
97	C₂⁴×C₅₃	—	2	8	no	—
98	C₂⁴×C₅₃	—	4	8	no	—
99	C₂⁴×C₅₃	—	8	8	no	—
100	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
101	#41×C₅₃	—	2	16	no	—
102	#41×C₅₃	—	2	16	no	—
103	#41×C₅₃	—	4	16	no	—
104	C₂×Q₈×C₅₃	—	2	2	no	—
105	C₂×Q₈×C₅₃	—	2	4	no	—
106	C₂×Q₈×C₅₃	—	2	4	no	—
107	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
108	C₂×D₄×C₅₃	—	2	8	no	—
109	C₂×D₄×C₅₃	—	2	8	no	—
110	C₂×D₄×C₅₃	—	2	16	no	—
111	C₂×D₄×C₅₃	—	2	4	no	—
112	C₂×D₄×C₅₃	—	2	8	no	—
113	C₂×D₄×C₅₃	—	2	4	no	—
114	C₂×D₄×C₅₃	—	2	16	no	—
115	C₂×D₄×C₅₃	—	2	8	no	—
116	C₂×D₄×C₅₃	—	2	4	no	—
117	C₂×D₄×C₅₃	—	2	2	no	—
118	C₂×D₄×C₅₃	—	2	32	no	—
119	C₂×D₄×C₅₃	—	4	8	no	—
120	C₂×D₄×C₅₃	—	4	8	no	—
121	C₂×D₄×C₅₃	—	4	16	no	—
122	C₂×D₄×C₅₃	—	8	16	no	—
123	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
124	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
125	C₂²×C₄×C₅₃	—	2	2	no	—
126	C₂²×C₄×C₅₃	—	2	8	no	—
127	C₂²×C₄×C₅₃	—	4	8	no	—
128	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
129	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
132	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
133	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
134	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
135	Q₁₆×C₅₃	—	2	8	no	—
136	SD₁₆×C₅₃	—	2	16	no	—
137	SD₁₆×C₅₃	—	2	32	no	—
138	D₈×C₅₃	—	2	16	no	—
139	D₈×C₅₃	—	2	16	no	—
140	D₈×C₅₃	—	2	8	no	—
141	D₈×C₅₃	—	2	32	no	—
142	M₁₆×C₅₃	C₂×D₄	1	16	yes	complemented
143	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
144	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
145	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	—
146	C₂×C₈×C₅₃	C₂×D₄	1	16	yes	complemented
147	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
148	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
149	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
150	C₄⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
151	C₂²⋊C₄×C₅₃	—	2	32	no	—
152	C₂²⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
153	C₄²×C₅₃	C₂×D₄	1	4	yes	—
154	C₄²×C₅₃	C₂⁴	1	4	yes	—
155	C₂³×C₅₃	—	2	4	no	—
156	C₂³×C₅₃	—	2	4	no	—
157	C₂³×C₅₃	—	2	8	no	—
158	C₂³×C₅₃	—	4	4	no	—
159	C₂³×C₅₃	—	4	16	no	—
160	C₂³×C₅₃	—	4	8	no	—
161	C₂³×C₅₃	—	4	8	no	—
162	C₂³×C₅₃	—	4	16	no	—
163	C₂³×C₅₃	—	8	16	no	—
164	C₂³×C₅₃	—	8	32	no	—
165	C₂³×C₅₃	#127	1	4	yes	—
166	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
167	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
168	Q₈×C₅₃	—	2	4	no	—
169	Q₈×C₅₃	—	2	4	no	—
170	Q₈×C₅₃	—	4	4	no	—
171	D₄×C₅₃	—	2	8	no	—
172	D₄×C₅₃	—	2	8	no	—
173	D₄×C₅₃	—	2	4	no	—
174	D₄×C₅₃	—	2	4	no	—
175	D₄×C₅₃	—	4	16	no	—
176	D₄×C₅₃	—	4	16	no	—
177	D₄×C₅₃	—	4	8	no	—
178	D₄×C₅₃	—	4	4	no	—
179	D₄×C₅₃	—	8	32	no	—
180	C₂×C₄×C₅₃	—	2	8	no	—
181	C₂×C₄×C₅₃	—	2	4	no	—
182	C₂×C₄×C₅₃	—	4	4	no	—
183	C₂×C₄×C₅₃	—	4	16	no	—
184	C₂×C₄×C₅₃	#127	1	4	yes	—
185	C₂×C₄×C₅₃	#127	1	8	yes	—
186	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
187	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
188	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
190	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
191	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
192	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
193	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
194	C₈×C₅₃	C₂²×D₄	1	4	yes	—
195	C₈×C₅₃	C₂²×D₄	1	4	yes	—
196	C₈×C₅₃	C₂²×D₄	1	8	yes	—
197	C₂×#3155	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂²×C₅₃	—	2	8	no	—
199	C₂²×C₅₃	—	2	4	no	—
200	C₂²×C₅₃	—	4	4	no	—
201	C₂²×C₅₃	—	4	8	no	—
202	C₂²×C₅₃	—	4	8	no	—
203	C₂²×C₅₃	—	4	16	no	—
204	C₂²×C₅₃	—	8	32	no	—
205	C₂²×C₅₃	—	8	8	no	—
206	C₂²×C₅₃	—	8	16	no	—
207	C₂²×C₅₃	C₂×#127	1	4	yes	—
208	C₂²×C₅₃	C₂×#127	1	2	yes	—
209	C₂²×C₅₃	#576	1	4	yes	—
210	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
211	C₄×C₅₃	—	2	8	no	—
212	C₄×C₅₃	—	4	4	no	—
213	C₄×C₅₃	C₂×#127	1	4	yes	—
214	C₄×C₅₃	C₂×#127	1	2	yes	—
215	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
216	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
217	C₂²×AGL(1,ℤ/8)	C₂×C₅₃	1	2	yes	complemented
218	C₂²×#135	C₂×C₅₃	1	1	yes	characteristic, complemented
219	C₂³×D₈	C₂×C₅₃	1	1	yes	characteristic, complemented
220	C₂×#535	C₂×C₅₃	1	2	yes	complemented
221	#3155	C₂×C₅₃	1	16	yes	direct product factor
222	C₂×#496	C₂×C₅₃	1	4	yes	complemented
223	C₂×#495	C₂×C₅₃	1	2	yes	complemented
224	C₂×#493	C₂×C₅₃	1	2	yes	complemented
225	C₂×#405	C₂×C₅₃	1	1	yes	characteristic, complemented
226	C₂×C₅₃	—	2	4	no	—
227	C₂×C₅₃	—	8	4	no	—
228	C₂×C₅₃	—	8	8	no	—
229	C₂×C₅₃	#3155	1	4	yes	direct product factor
230	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
231	C₂×C₅₃	C₂×#576	1	2	yes	—
232	C₂²×#41	C₂²×C₅₃	1	2	yes	—
233	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
234	C₂×AGL(1,ℤ/8)	—	2	16	no	—
235	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	16	yes	complemented
236	C₂×#135	—	2	8	no	—
237	C₂×#135	C₂²×C₅₃	1	8	yes	complemented
238	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic, complemented
239	C₂²×SD₁₆	C₂²×C₅₃	1	4	yes	complemented
240	C₂²×SD₁₆	C₂²×C₅₃	1	2	yes	complemented
241	C₂²×D₈	—	2	8	no	—
242	C₂²×D₈	C₂²×C₅₃	1	2	yes	—
243	C₂²×D₈	C₂²×C₅₃	1	2	yes	—
244	C₂²×D₈	C₂²×C₅₃	1	8	yes	complemented
245	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
246	C₂²×D₈	C₂²×C₅₃	1	4	yes	complemented
247	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
248	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
249	#535	—	2	8	no	—
250	#535	C₂²×C₅₃	1	16	yes	complemented
251	C₂×#127	C₂²×C₅₃	1	4	yes	complemented
252	C₂×#124	C₂²×C₅₃	1	4	yes	complemented
253	C₂×#121	—	2	4	no	—
254	C₂×#120	—	2	4	no	—
255	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
256	#496	C₂²×C₅₃	1	32	yes	complemented
257	#495	C₂²×C₅₃	1	16	yes	complemented
258	#493	C₂²×C₅₃	1	16	yes	complemented
259	#405	C₂²×C₅₃	1	8	yes	complemented
260	C₂×#97	C₂²×C₅₃	1	2	yes	complemented
261	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	complemented
262	C₅₃	C₂×#3155	1	1	yes	characteristic, direct product factor, Sylow
263	C₂⁵	—	2	2	no	—
264	C₂×#41	—	2	4	no	—
265	C₂×#41	—	2	8	no	—
266	C₂×#41	—	2	8	no	—
267	C₂×#41	C₂³×C₅₃	1	8	yes	—
268	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
269	C₂²×D₄	—	2	2	no	—
270	C₂²×D₄	—	2	8	no	—
271	C₂²×D₄	—	2	8	no	—
272	C₂²×D₄	—	2	2	no	—
273	C₂²×D₄	—	2	4	no	—
274	C₂²×D₄	—	2	8	no	—
275	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
276	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
277	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
278	C₂³×C₄	—	2	2	no	—
279	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
280	AGL(1,ℤ/8)	—	2	64	no	—
281	#135	—	2	32	no	—
282	C₂×Q₁₆	—	2	4	no	—
283	C₂×Q₁₆	C₂³×C₅₃	1	4	yes	complemented
284	C₂×SD₁₆	—	2	8	no	—
285	C₂×SD₁₆	—	2	16	no	—
286	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	complemented
287	C₂×SD₁₆	C₂³×C₅₃	1	16	yes	complemented
288	C₂×D₈	—	2	8	no	—
289	C₂×D₈	—	2	8	no	—
290	C₂×D₈	—	2	4	no	—
291	C₂×D₈	—	2	16	no	—
292	C₂×D₈	—	2	32	no	—
293	C₂×D₈	C₂³×C₅₃	1	8	yes	—
294	C₂×D₈	C₂³×C₅₃	1	8	yes	—
295	C₂×D₈	C₂³×C₅₃	1	4	yes	complemented
296	C₂×D₈	C₂³×C₅₃	1	16	yes	complemented
297	C₂×M₁₆	D₄×C₅₃	1	8	yes	complemented
298	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
299	C₂²×C₈	D₄×C₅₃	1	8	yes	complemented
300	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
301	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
302	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
303	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
304	#127	—	2	8	no	—
305	#127	C₂³×C₅₃	1	16	yes	complemented
306	#124	—	2	8	no	—
307	#124	C₂³×C₅₃	1	16	yes	complemented
308	#121	—	2	32	no	—
309	#120	—	2	32	no	—
310	#117	D₄×C₅₃	1	4	yes	—
311	#117	C₂³×C₅₃	1	4	yes	—
312	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
313	C₂×C₂²⋊C₄	—	2	8	no	—
314	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
315	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
316	#97	C₂³×C₅₃	1	8	yes	complemented
317	C₄×C₈	C₂³×C₅₃	1	8	yes	complemented
318	C₂⁴	—	2	4	no	—
319	C₂⁴	—	2	2	no	—
320	C₂⁴	—	2	8	no	—
321	C₂⁴	—	4	8	no	—
322	C₂⁴	—	8	8	no	—
323	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
324	#41	—	2	16	no	—
325	#41	—	2	16	no	—
326	#41	—	4	16	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	2	4	no	—
329	C₂×Q₈	—	2	4	no	—
330	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
331	C₂×D₄	—	2	8	no	—
332	C₂×D₄	—	2	8	no	—
333	C₂×D₄	—	2	16	no	—
334	C₂×D₄	—	2	4	no	—
335	C₂×D₄	—	2	8	no	—
336	C₂×D₄	—	2	4	no	—
337	C₂×D₄	—	2	16	no	—
338	C₂×D₄	—	2	8	no	—
339	C₂×D₄	—	2	4	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	32	no	—
342	C₂×D₄	—	4	8	no	—
343	C₂×D₄	—	4	8	no	—
344	C₂×D₄	—	4	16	no	—
345	C₂×D₄	—	8	16	no	—
346	C₂×D₄	C₂⁴×C₅₃	1	8	yes	—
347	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	8	no	—
350	C₂²×C₄	—	4	8	no	—
351	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
352	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
353	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
354	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
355	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
356	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
357	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
358	Q₁₆	—	2	8	no	—
359	SD₁₆	—	2	16	no	—
360	SD₁₆	—	2	32	no	—
361	D₈	—	2	16	no	—
362	D₈	—	2	16	no	—
363	D₈	—	2	8	no	—
364	D₈	—	2	32	no	—
365	M₁₆	C₂×D₄×C₅₃	1	16	yes	complemented
366	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
367	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
368	C₂×C₈	C₂×D₄×C₅₃	1	8	yes	—
369	C₂×C₈	C₂×D₄×C₅₃	1	16	yes	complemented
370	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
371	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
372	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
373	C₄⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
374	C₂²⋊C₄	—	2	32	no	—
375	C₂²⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
376	C₄²	C₂×D₄×C₅₃	1	4	yes	—
377	C₄²	C₂⁴×C₅₃	1	4	yes	—
378	C₂³	—	2	4	no	—
379	C₂³	—	2	8	no	—
380	C₂³	—	2	4	no	—
381	C₂³	—	4	4	no	—
382	C₂³	—	4	16	no	—
383	C₂³	—	4	8	no	—
384	C₂³	—	4	8	no	—
385	C₂³	—	4	16	no	—
386	C₂³	—	8	16	no	—
387	C₂³	—	8	32	no	—
388	C₂³	#127×C₅₃	1	4	yes	—
389	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
390	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
391	Q₈	—	2	4	no	—
392	Q₈	—	2	4	no	—
393	Q₈	—	4	4	no	—
394	D₄	—	2	8	no	—
395	D₄	—	2	4	no	—
396	D₄	—	2	8	no	—
397	D₄	—	2	4	no	—
398	D₄	—	4	16	no	—
399	D₄	—	4	16	no	—
400	D₄	—	4	8	no	—
401	D₄	—	4	4	no	—
402	D₄	—	8	32	no	—
403	C₂×C₄	—	2	8	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	4	4	no	—
406	C₂×C₄	—	4	16	no	—
407	C₂×C₄	#127×C₅₃	1	4	yes	—
408	C₂×C₄	#127×C₅₃	1	8	yes	—
409	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
410	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
411	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
412	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
413	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
414	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
415	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
416	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
417	C₈	C₂²×D₄×C₅₃	1	4	yes	—
418	C₈	C₂²×D₄×C₅₃	1	4	yes	—
419	C₈	C₂²×D₄×C₅₃	1	8	yes	—
420	C₂²	—	2	4	no	—
421	C₂²	—	2	8	no	—
422	C₂²	—	4	8	no	—
423	C₂²	—	4	16	no	—
424	C₂²	—	4	4	no	—
425	C₂²	—	4	8	no	—
426	C₂²	—	8	32	no	—
427	C₂²	—	8	16	no	—
428	C₂²	—	8	8	no	—
429	C₂²	C₂×#127×C₅₃	1	4	yes	—
430	C₂²	C₂×#127×C₅₃	1	2	yes	—
431	C₂²	C₅₃×#576	1	4	yes	—
432	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
433	C₄	—	2	8	no	—
434	C₄	—	4	4	no	—
435	C₄	C₂×#127×C₅₃	1	4	yes	—
436	C₄	C₂×#127×C₅₃	1	2	yes	—
437	C₄	C₂³×D₄×C₅₃	1	1	yes	characteristic
438	C₄	C₂³×D₄×C₅₃	1	1	yes	characteristic
439	C₂	—	2	4	no	—
440	C₂	—	8	8	no	—
441	C₂	—	8	4	no	—
442	C₂	C₅₃×#3155	1	4	yes	direct product factor
443	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
444	C₂	C₂×C₅₃×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×#117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×AGL(1,ℤ/8)×C₅₃	C₂	semidirect
2. C₂²×#135×C₅₃	C₂	semidirect
3. C₂³×D₈×C₅₃	C₂	semidirect
4. C₂×C₅₃×#535	C₂	semidirect
5. C₅₃×#3155	C₂	direct (N × H)
6. C₂×C₅₃×#496	C₂	semidirect
7. C₂×C₅₃×#495	C₂	semidirect
8. C₂×C₅₃×#493	C₂	semidirect
9. C₂×C₅₃×#405	C₂	semidirect
13. C₂×AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
15. C₂×#135×C₅₃	C₂²	semidirect
16. C₂²×Q₁₆×C₅₃	C₂²	semidirect
17. C₂²×SD₁₆×C₅₃	C₂²	semidirect
18. C₂²×SD₁₆×C₅₃	C₂²	semidirect
22. C₂²×D₈×C₅₃	C₂²	semidirect
23. C₂²×D₈×C₅₃	C₂²	semidirect
24. C₂²×D₈×C₅₃	C₂²	semidirect
28. C₅₃×#535	C₂²	semidirect
29. C₂×#127×C₅₃	C₂²	semidirect
30. C₂×#124×C₅₃	C₂²	semidirect
34. C₅₃×#496	C₂²	semidirect
35. C₅₃×#495	C₂²	semidirect
36. C₅₃×#493	C₂²	semidirect
37. C₅₃×#405	C₂²	semidirect
38. C₂×#97×C₅₃	C₂²	semidirect
39. C₂×C₄×C₈×C₅₃	C₂²	semidirect
60. C₂×Q₁₆×C₅₃	C₂³	semidirect
63. C₂×SD₁₆×C₅₃	C₂³	semidirect
64. C₂×SD₁₆×C₅₃	C₂³	semidirect
72. C₂×D₈×C₅₃	C₂³	semidirect
73. C₂×D₈×C₅₃	C₂³	semidirect
74. C₂×M₁₆×C₅₃	D₄	semidirect
76. C₂²×C₈×C₅₃	D₄	semidirect
82. #127×C₅₃	C₂³	semidirect
84. #124×C₅₃	C₂³	semidirect
93. #97×C₅₃	C₂³	semidirect
94. C₄×C₈×C₅₃	C₂³	semidirect
142. M₁₆×C₅₃	C₂×D₄	semidirect
146. C₂×C₈×C₅₃	C₂×D₄	semidirect
197. C₂×#3155	C₅₃	direct (N × H)
217. C₂²×AGL(1,ℤ/8)	C₂×C₅₃	semidirect
218. C₂²×#135	C₂×C₅₃	semidirect
219. C₂³×D₈	C₂×C₅₃	semidirect
220. C₂×#535	C₂×C₅₃	semidirect
221. #3155	C₂×C₅₃	direct (N × H)
222. C₂×#496	C₂×C₅₃	semidirect
223. C₂×#495	C₂×C₅₃	semidirect
224. C₂×#493	C₂×C₅₃	semidirect
225. C₂×#405	C₂×C₅₃	semidirect
229. C₂×C₅₃	#3155	direct (N × H)
235. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
237. C₂×#135	C₂²×C₅₃	semidirect
238. C₂²×Q₁₆	C₂²×C₅₃	semidirect
239. C₂²×SD₁₆	C₂²×C₅₃	semidirect
240. C₂²×SD₁₆	C₂²×C₅₃	semidirect
244. C₂²×D₈	C₂²×C₅₃	semidirect
245. C₂²×D₈	C₂²×C₅₃	semidirect
246. C₂²×D₈	C₂²×C₅₃	semidirect
250. #535	C₂²×C₅₃	semidirect
251. C₂×#127	C₂²×C₅₃	semidirect
252. C₂×#124	C₂²×C₅₃	semidirect
256. #496	C₂²×C₅₃	semidirect
257. #495	C₂²×C₅₃	semidirect
258. #493	C₂²×C₅₃	semidirect
259. #405	C₂²×C₅₃	semidirect
260. C₂×#97	C₂²×C₅₃	semidirect
261. C₂×C₄×C₈	C₂²×C₅₃	semidirect
262. C₅₃	C₂×#3155	direct (N × H)
283. C₂×Q₁₆	C₂³×C₅₃	semidirect
286. C₂×SD₁₆	C₂³×C₅₃	semidirect
287. C₂×SD₁₆	C₂³×C₅₃	semidirect
295. C₂×D₈	C₂³×C₅₃	semidirect
296. C₂×D₈	C₂³×C₅₃	semidirect
297. C₂×M₁₆	D₄×C₅₃	semidirect
299. C₂²×C₈	D₄×C₅₃	semidirect
305. #127	C₂³×C₅₃	semidirect
307. #124	C₂³×C₅₃	semidirect
316. #97	C₂³×C₅₃	semidirect
317. C₄×C₈	C₂³×C₅₃	semidirect
365. M₁₆	C₂×D₄×C₅₃	semidirect
369. C₂×C₈	C₂×D₄×C₅₃	semidirect
442. C₂	C₅₃×#3155	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×AGL(1,ℤ/8)×C₅₃	2
C₂²×#135×C₅₃	2
C₂³×D₈×C₅₃	2
C₂×C₅₃×#535	2
C₅₃×#3155	2
C₂×C₅₃×#496	2
C₂×C₅₃×#495	2
C₂×C₅₃×#493	2
C₂×C₅₃×#405	2
C₂×#3155	53

Subgroups — C2×C53×#3155 #27268153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3155 #27268153