Subgroups — C₂×C₅₃×#3152 #27268149

Order 13568.

3262 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#137×C₅₃	C₂	1	4	yes	maximal, complemented
2	C₂×C₅₃×#532	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3152	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₅₃×#497	C₂	1	4	yes	maximal, complemented
6	C₂×C₅₃×#494	C₂	1	2	yes	maximal, complemented
7	C₂×C₅₃×#492	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₅₃	C₂²	1	2	yes	—
10	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
11	C₂×#137×C₅₃	—	2	32	no	—
12	C₂×#137×C₅₃	C₂²	1	32	yes	complemented
13	C₂²×Q₁₆×C₅₃	C₂²	1	8	yes	—
14	C₂²×SD₁₆×C₅₃	C₂²	1	8	yes	—
15	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#532	—	2	4	no	—
17	C₅₃×#532	C₂²	1	8	yes	—
18	C₂×C₄⋊Q₈×C₅₃	—	2	2	no	—
19	C₂×C₄⋊Q₈×C₅₃	—	2	4	no	—
20	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄⋊Q₈×C₅₃	C₂²	1	8	yes	complemented
24	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#124×C₅₃	C₂²	1	2	yes	—
26	C₂×#121×C₅₃	—	2	2	no	—
27	C₂×C₄×D₄×C₅₃	—	2	2	no	—
28	C₂²×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#497	C₂²	1	32	yes	complemented
31	C₅₃×#494	C₂²	1	16	yes	complemented
32	C₅₃×#492	C₂²	1	16	yes	complemented
33	C₄×M₁₆×C₅₃	C₂²	1	8	yes	complemented
34	C₂×#97×C₅₃	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
36	C₂×#41×C₅₃	—	2	4	no	—
37	C₂×#41×C₅₃	—	2	16	no	—
38	C₂×#41×C₅₃	C₂³	1	8	yes	—
39	C₂²×Q₈×C₅₃	—	2	4	no	—
40	C₂²×Q₈×C₅₃	—	2	2	no	—
41	C₂²×Q₈×C₅₃	—	2	16	no	—
42	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
43	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
44	C₂²×Q₈×C₅₃	C₂³	1	8	yes	—
45	C₂²×D₄×C₅₃	—	2	2	no	—
46	C₂²×D₄×C₅₃	—	2	2	no	—
47	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
48	C₂³×C₄×C₅₃	—	2	2	no	—
49	C₂³×C₄×C₅₃	—	2	2	no	—
50	C₂³×C₄×C₅₃	D₄	1	2	yes	—
51	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
52	#137×C₅₃	—	2	128	no	—
53	C₂×Q₁₆×C₅₃	—	2	32	no	—
54	C₂×Q₁₆×C₅₃	C₂³	1	32	yes	—
55	C₂×SD₁₆×C₅₃	—	2	32	no	—
56	C₂×SD₁₆×C₅₃	C₂³	1	32	yes	—
57	C₂×M₁₆×C₅₃	D₄	1	16	yes	complemented
58	C₂×M₁₆×C₅₃	C₂³	1	8	yes	—
59	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
60	C₄⋊Q₈×C₅₃	—	2	4	no	—
61	C₄⋊Q₈×C₅₃	—	2	4	no	—
62	C₄⋊Q₈×C₅₃	—	2	16	no	—
63	C₄⋊Q₈×C₅₃	—	2	2	no	—
64	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	—
65	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
66	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	—
67	#127×C₅₃	—	2	2	no	—
68	#127×C₅₃	C₂³	1	4	yes	—
69	#124×C₅₃	—	2	4	no	—
70	#124×C₅₃	C₂³	1	8	yes	—
71	#121×C₅₃	—	2	16	no	—
72	C₄×D₄×C₅₃	—	2	16	no	—
73	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
74	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
75	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
76	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
78	C₂×C₄²×C₅₃	D₄	1	2	yes	—
79	C₂×C₄²×C₅₃	D₄	1	4	yes	—
80	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
83	#97×C₅₃	C₂³	1	8	yes	—
84	C₄×C₈×C₅₃	C₂³	1	8	yes	—
85	C₂⁴×C₅₃	—	2	2	no	—
86	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
87	#41×C₅₃	—	2	32	no	—
88	#41×C₅₃	—	4	16	no	—
89	C₂×Q₈×C₅₃	—	2	2	no	—
90	C₂×Q₈×C₅₃	—	2	8	no	—
91	C₂×Q₈×C₅₃	—	2	4	no	—
92	C₂×Q₈×C₅₃	—	2	16	no	—
93	C₂×Q₈×C₅₃	—	2	8	no	—
94	C₂×Q₈×C₅₃	—	2	32	no	—
95	C₂×Q₈×C₅₃	—	4	8	no	—
96	C₂×Q₈×C₅₃	—	4	16	no	—
97	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
98	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
99	C₂×D₄×C₅₃	—	2	8	no	—
100	C₂×D₄×C₅₃	—	2	8	no	—
101	C₂×D₄×C₅₃	—	2	8	no	—
102	C₂×D₄×C₅₃	—	2	2	no	—
103	C₂×D₄×C₅₃	—	4	8	no	—
104	C₂×D₄×C₅₃	—	4	8	no	—
105	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
106	C₂²×C₄×C₅₃	—	2	8	no	—
107	C₂²×C₄×C₅₃	—	2	8	no	—
108	C₂²×C₄×C₅₃	—	2	2	no	—
109	C₂²×C₄×C₅₃	—	2	4	no	—
110	C₂²×C₄×C₅₃	—	2	8	no	—
111	C₂²×C₄×C₅₃	—	4	8	no	—
112	C₂²×C₄×C₅₃	—	4	8	no	—
113	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
114	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
118	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
120	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
121	Q₁₆×C₅₃	—	2	64	no	—
122	SD₁₆×C₅₃	—	2	64	no	—
123	M₁₆×C₅₃	C₂×D₄	1	32	yes	complemented
124	C₂×C₈×C₅₃	C₂×D₄	1	16	yes	—
125	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
126	C₄⋊C₄×C₅₃	—	2	16	no	—
127	C₄⋊C₄×C₅₃	—	2	16	no	—
128	C₄⋊C₄×C₅₃	—	2	8	no	—
129	C₂²⋊C₄×C₅₃	—	2	16	no	—
130	C₄²×C₅₃	C₂×D₄	1	8	yes	—
131	C₄²×C₅₃	C₂×D₄	1	2	yes	—
132	C₄²×C₅₃	C₂×D₄	1	2	yes	—
133	C₄²×C₅₃	C₂⁴	1	2	yes	—
134	C₄²×C₅₃	C₂⁴	1	2	yes	—
135	C₂³×C₅₃	—	2	4	no	—
136	C₂³×C₅₃	—	2	4	no	—
137	C₂³×C₅₃	—	4	4	no	—
138	C₂³×C₅₃	—	4	8	no	—
139	C₂³×C₅₃	#127	1	4	yes	—
140	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
141	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
142	Q₈×C₅₃	—	2	16	no	—
143	Q₈×C₅₃	—	2	8	no	—
144	Q₈×C₅₃	—	4	16	no	—
145	Q₈×C₅₃	—	4	8	no	—
146	Q₈×C₅₃	—	4	4	no	—
147	D₄×C₅₃	—	2	8	no	—
148	D₄×C₅₃	—	4	16	no	—
149	D₄×C₅₃	—	4	4	no	—
150	D₄×C₅₃	—	4	16	no	—
151	C₂×C₄×C₅₃	—	2	4	no	—
152	C₂×C₄×C₅₃	—	2	16	no	—
153	C₂×C₄×C₅₃	—	2	4	no	—
154	C₂×C₄×C₅₃	—	2	8	no	—
155	C₂×C₄×C₅₃	—	4	4	no	—
156	C₂×C₄×C₅₃	—	4	16	no	—
157	C₂×C₄×C₅₃	—	4	8	no	—
158	C₂×C₄×C₅₃	—	4	16	no	—
159	C₂×C₄×C₅₃	#127	1	4	yes	—
160	C₂×C₄×C₅₃	#127	1	8	yes	—
161	C₂×C₄×C₅₃	#137	1	8	yes	complemented
162	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
163	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
164	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
166	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
167	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
168	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
169	C₈×C₅₃	C₂²×D₄	1	16	yes	—
170	C₂×#3152	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂²×C₅₃	—	2	8	no	—
172	C₂²×C₅₃	—	2	4	no	—
173	C₂²×C₅₃	—	4	4	no	—
174	C₂²×C₅₃	—	4	8	no	—
175	C₂²×C₅₃	—	8	8	no	—
176	C₂²×C₅₃	C₂×#127	1	4	yes	—
177	C₂²×C₅₃	C₂×#127	1	2	yes	—
178	C₂²×C₅₃	C₂×#137	1	4	yes	—
179	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
180	C₄×C₅₃	—	2	4	no	—
181	C₄×C₅₃	—	4	4	no	—
182	C₄×C₅₃	—	4	8	no	—
183	C₄×C₅₃	C₂×#127	1	4	yes	—
184	C₄×C₅₃	C₂×#127	1	2	yes	—
185	C₄×C₅₃	C₂×#137	1	8	yes	complemented
186	C₄×C₅₃	C₂³×D₄	1	2	yes	—
187	C₂²×#137	C₂×C₅₃	1	4	yes	complemented
188	C₂×#532	C₂×C₅₃	1	1	yes	characteristic
189	C₂²×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
190	#3152	C₂×C₅₃	1	16	yes	direct product factor
191	C₂×#497	C₂×C₅₃	1	4	yes	complemented
192	C₂×#494	C₂×C₅₃	1	2	yes	complemented
193	C₂×#492	C₂×C₅₃	1	2	yes	complemented
194	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
195	C₂×C₅₃	—	2	4	no	—
196	C₂×C₅₃	—	8	4	no	—
197	C₂×C₅₃	#3152	1	4	yes	direct product factor
198	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
199	C₂×C₅₃	C₂²×#137	1	2	yes	—
200	C₂²×#41	C₂²×C₅₃	1	2	yes	—
201	C₂³×Q₈	C₂²×C₅₃	1	2	yes	—
202	C₂×#137	—	2	32	no	—
203	C₂×#137	C₂²×C₅₃	1	32	yes	complemented
204	C₂²×Q₁₆	C₂²×C₅₃	1	8	yes	—
205	C₂²×SD₁₆	C₂²×C₅₃	1	8	yes	—
206	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
207	#532	—	2	4	no	—
208	#532	C₂²×C₅₃	1	8	yes	—
209	C₂×C₄⋊Q₈	—	2	2	no	—
210	C₂×C₄⋊Q₈	—	2	4	no	—
211	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
212	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
213	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
214	C₂×C₄⋊Q₈	C₂²×C₅₃	1	8	yes	complemented
215	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
216	C₂×#124	C₂²×C₅₃	1	2	yes	—
217	C₂×#121	—	2	2	no	—
218	C₂×C₄×D₄	—	2	2	no	—
219	C₂²×C₄⋊C₄	—	2	2	no	—
220	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
221	#497	C₂²×C₅₃	1	32	yes	complemented
222	#494	C₂²×C₅₃	1	16	yes	complemented
223	#492	C₂²×C₅₃	1	16	yes	complemented
224	C₄×M₁₆	C₂²×C₅₃	1	8	yes	complemented
225	C₂×#97	C₂²×C₅₃	1	2	yes	—
226	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
227	C₅₃	C₂×#3152	1	1	yes	characteristic, direct product factor, Sylow
228	C₂×#41	—	2	4	no	—
229	C₂×#41	—	2	16	no	—
230	C₂×#41	C₂³×C₅₃	1	8	yes	—
231	C₂²×Q₈	—	2	2	no	—
232	C₂²×Q₈	—	2	4	no	—
233	C₂²×Q₈	—	2	16	no	—
234	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
235	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
236	C₂²×Q₈	C₂³×C₅₃	1	8	yes	—
237	C₂²×D₄	—	2	2	no	—
238	C₂²×D₄	—	2	2	no	—
239	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
240	C₂³×C₄	—	2	2	no	—
241	C₂³×C₄	—	2	2	no	—
242	C₂³×C₄	D₄×C₅₃	1	2	yes	—
243	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
244	#137	—	2	128	no	—
245	C₂×Q₁₆	—	2	32	no	—
246	C₂×Q₁₆	C₂³×C₅₃	1	32	yes	—
247	C₂×SD₁₆	—	2	32	no	—
248	C₂×SD₁₆	C₂³×C₅₃	1	32	yes	—
249	C₂×M₁₆	D₄×C₅₃	1	16	yes	complemented
250	C₂×M₁₆	C₂³×C₅₃	1	8	yes	—
251	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
252	C₄⋊Q₈	—	2	4	no	—
253	C₄⋊Q₈	—	2	4	no	—
254	C₄⋊Q₈	—	2	16	no	—
255	C₄⋊Q₈	—	2	2	no	—
256	C₄⋊Q₈	C₂³×C₅₃	1	8	yes	—
257	C₄⋊Q₈	C₂³×C₅₃	1	8	yes	—
258	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
259	#127	—	2	2	no	—
260	#127	C₂³×C₅₃	1	4	yes	—
261	#124	—	2	4	no	—
262	#124	C₂³×C₅₃	1	8	yes	—
263	#121	—	2	16	no	—
264	C₄×D₄	—	2	16	no	—
265	C₂×C₄⋊C₄	—	2	4	no	—
266	C₂×C₄⋊C₄	—	2	4	no	—
267	C₂×C₄⋊C₄	—	2	16	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₂²⋊C₄	—	2	4	no	—
270	C₂×C₄²	D₄×C₅₃	1	2	yes	—
271	C₂×C₄²	D₄×C₅₃	1	4	yes	—
272	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
273	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
274	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
275	#97	C₂³×C₅₃	1	8	yes	—
276	C₄×C₈	C₂³×C₅₃	1	8	yes	—
277	C₂⁴	—	2	2	no	—
278	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
279	#41	—	2	32	no	—
280	#41	—	4	16	no	—
281	C₂×Q₈	—	2	2	no	—
282	C₂×Q₈	—	2	8	no	—
283	C₂×Q₈	—	2	4	no	—
284	C₂×Q₈	—	2	16	no	—
285	C₂×Q₈	—	2	8	no	—
286	C₂×Q₈	—	2	32	no	—
287	C₂×Q₈	—	4	8	no	—
288	C₂×Q₈	—	4	16	no	—
289	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
290	C₂×Q₈	C₂⁴×C₅₃	1	8	yes	—
291	C₂×D₄	—	2	8	no	—
292	C₂×D₄	—	2	8	no	—
293	C₂×D₄	—	2	8	no	—
294	C₂×D₄	—	2	2	no	—
295	C₂×D₄	—	4	8	no	—
296	C₂×D₄	—	4	8	no	—
297	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
298	C₂²×C₄	—	2	8	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	8	no	—
301	C₂²×C₄	—	2	4	no	—
302	C₂²×C₄	—	2	8	no	—
303	C₂²×C₄	—	4	8	no	—
304	C₂²×C₄	—	4	8	no	—
305	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
306	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
307	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
309	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
310	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
311	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
312	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
313	Q₁₆	—	2	64	no	—
314	SD₁₆	—	2	64	no	—
315	M₁₆	C₂×D₄×C₅₃	1	32	yes	complemented
316	C₂×C₈	C₂×D₄×C₅₃	1	16	yes	—
317	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
318	C₄⋊C₄	—	2	16	no	—
319	C₄⋊C₄	—	2	16	no	—
320	C₄⋊C₄	—	2	8	no	—
321	C₂²⋊C₄	—	2	16	no	—
322	C₄²	C₂×D₄×C₅₃	1	8	yes	—
323	C₄²	C₂×D₄×C₅₃	1	2	yes	—
324	C₄²	C₂×D₄×C₅₃	1	2	yes	—
325	C₄²	C₂⁴×C₅₃	1	2	yes	—
326	C₄²	C₂⁴×C₅₃	1	2	yes	—
327	C₂³	—	2	4	no	—
328	C₂³	—	2	4	no	—
329	C₂³	—	4	4	no	—
330	C₂³	—	4	8	no	—
331	C₂³	#127×C₅₃	1	4	yes	—
332	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
333	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
334	Q₈	—	2	16	no	—
335	Q₈	—	2	8	no	—
336	Q₈	—	4	16	no	—
337	Q₈	—	4	8	no	—
338	Q₈	—	4	4	no	—
339	D₄	—	2	8	no	—
340	D₄	—	4	16	no	—
341	D₄	—	4	4	no	—
342	D₄	—	4	16	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	16	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	2	8	no	—
347	C₂×C₄	—	4	4	no	—
348	C₂×C₄	—	4	16	no	—
349	C₂×C₄	—	4	8	no	—
350	C₂×C₄	—	4	16	no	—
351	C₂×C₄	#127×C₅₃	1	4	yes	—
352	C₂×C₄	#127×C₅₃	1	8	yes	—
353	C₂×C₄	#137×C₅₃	1	8	yes	complemented
354	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
355	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
356	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
357	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
358	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
359	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
360	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
361	C₈	C₂²×D₄×C₅₃	1	16	yes	—
362	C₂²	—	2	4	no	—
363	C₂²	—	2	8	no	—
364	C₂²	—	4	8	no	—
365	C₂²	—	4	4	no	—
366	C₂²	—	8	8	no	—
367	C₂²	C₂×#127×C₅₃	1	4	yes	—
368	C₂²	C₂×#127×C₅₃	1	2	yes	—
369	C₂²	C₂×#137×C₅₃	1	4	yes	—
370	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
371	C₄	—	2	4	no	—
372	C₄	—	4	4	no	—
373	C₄	—	4	8	no	—
374	C₄	C₂×#127×C₅₃	1	4	yes	—
375	C₄	C₂×#127×C₅₃	1	2	yes	—
376	C₄	C₂×#137×C₅₃	1	8	yes	complemented
377	C₄	C₂³×D₄×C₅₃	1	2	yes	—
378	C₂	—	2	4	no	—
379	C₂	—	8	4	no	—
380	C₂	C₅₃×#3152	1	4	yes	direct product factor
381	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
382	C₂	C₂²×#137×C₅₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₂×C₅₃×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#137×C₅₃	C₂	semidirect
3. C₂²×C₄⋊Q₈×C₅₃	C₂	semidirect
4. C₅₃×#3152	C₂	direct (N × H)
5. C₂×C₅₃×#497	C₂	semidirect
6. C₂×C₅₃×#494	C₂	semidirect
7. C₂×C₅₃×#492	C₂	semidirect
8. C₂×C₄×M₁₆×C₅₃	C₂	semidirect
12. C₂×#137×C₅₃	C₂²	semidirect
23. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
30. C₅₃×#497	C₂²	semidirect
31. C₅₃×#494	C₂²	semidirect
32. C₅₃×#492	C₂²	semidirect
33. C₄×M₁₆×C₅₃	C₂²	semidirect
57. C₂×M₁₆×C₅₃	D₄	semidirect
123. M₁₆×C₅₃	C₂×D₄	semidirect
161. C₂×C₄×C₅₃	#137	semidirect
170. C₂×#3152	C₅₃	direct (N × H)
185. C₄×C₅₃	C₂×#137	semidirect
187. C₂²×#137	C₂×C₅₃	semidirect
189. C₂²×C₄⋊Q₈	C₂×C₅₃	semidirect
190. #3152	C₂×C₅₃	direct (N × H)
191. C₂×#497	C₂×C₅₃	semidirect
192. C₂×#494	C₂×C₅₃	semidirect
193. C₂×#492	C₂×C₅₃	semidirect
194. C₂×C₄×M₁₆	C₂×C₅₃	semidirect
197. C₂×C₅₃	#3152	direct (N × H)
203. C₂×#137	C₂²×C₅₃	semidirect
214. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
221. #497	C₂²×C₅₃	semidirect
222. #494	C₂²×C₅₃	semidirect
223. #492	C₂²×C₅₃	semidirect
224. C₄×M₁₆	C₂²×C₅₃	semidirect
227. C₅₃	C₂×#3152	direct (N × H)
249. C₂×M₁₆	D₄×C₅₃	semidirect
315. M₁₆	C₂×D₄×C₅₃	semidirect
353. C₂×C₄	#137×C₅₃	semidirect
376. C₄	C₂×#137×C₅₃	semidirect
380. C₂	C₅₃×#3152	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×#137×C₅₃	2
C₂×C₅₃×#532	2
C₂²×C₄⋊Q₈×C₅₃	2
C₅₃×#3152	2
C₂×C₅₃×#497	2
C₂×C₅₃×#494	2
C₂×C₅₃×#492	2
C₂×C₄×M₁₆×C₅₃	2
C₂×#3152	53

Subgroups — C2×C53×#3152 #27268149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3152 #27268149