Subgroups — #27402630

Order 13568.

1108 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×#88322	C₂	1	2	yes	maximal
2	#20275434	C₂	1	2	yes	maximal, complemented
3	#20274895	C₂	1	4	yes	maximal, complemented
4	#20274832	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274695	C₂	1	2	yes	maximal, complemented
6	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16810378	C₂	1	2	yes	maximal, complemented
8	C₄²×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×#88322	C₂²	1	4	yes	—
10	D₄×#88322	—	2	8	no	—
11	C₂×C₄⋊#88322	C₂²	1	2	yes	—
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×C₄⋊#88322	C₂²	1	4	yes	—
14	C₂×C₂²⋊#88322	C₂²	1	4	yes	—
15	C₂×C₄×#88322	C₂²	1	2	yes	—
16	#16810378	C₂²	1	1	yes	characteristic
17	#16810378	C₂²	1	2	yes	—
18	#16810378	C₂²	1	1	yes	characteristic
19	#16810378	C₂²	1	2	yes	—
20	#16810378	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
22	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
23	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₄×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
29	C₄×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
30	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
31	C₄³×C₅₃	C₂²	1	1	yes	characteristic
32	C₂²×#88322	—	2	8	no	—
33	C₂²×#88322	—	2	8	no	—
34	C₂²×#88322	—	4	4	no	—
35	C₂²×#88322	C₂³	1	2	yes	—
36	C₂²×#88322	C₂³	1	4	yes	—
37	C₂²×#88322	C₂³	1	2	yes	—
38	C₄⋊#88322	—	2	4	no	—
39	C₄⋊#88322	—	2	4	no	—
40	C₄⋊#88322	—	4	4	no	—
41	C₂²⋊#88322	—	2	8	no	—
42	C₄×#88322	—	2	4	no	—
43	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
45	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
46	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
47	C₄×D₄×C₅₃	—	2	4	no	—
48	C₄×D₄×C₅₃	C₈	1	8	yes	complemented
49	C₄×D₄×C₅₃	D₄	1	4	yes	—
50	C₄×D₄×C₅₃	Q₈	1	4	yes	—
51	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
56	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₅₃	—	4	4	no	—
58	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
61	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	#95×C₅₃	C₂×C₄	1	2	yes	—
69	#95×C₅₃	C₂×C₄	1	2	yes	—
70	#95×C₅₃	C₂×C₄	1	2	yes	—
71	C₂⁴×C₅₃	C₂²×C₄	1	2	yes	—
72	C₂×D₄×C₅₃	C₄⋊C₄	1	4	yes	—
73	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
74	C₂×D₄×C₅₃	M₁₆	1	4	yes	—
75	C₂²×C₄×C₅₃	—	2	4	no	—
76	C₂²×C₄×C₅₃	—	2	4	no	—
77	C₂²×C₄×C₅₃	—	2	4	no	—
78	C₂²×C₄×C₅₃	—	2	4	no	—
79	C₂²×C₄×C₅₃	—	4	2	no	—
80	C₂²×C₄×C₅₃	—	4	2	no	—
81	C₂²×C₄×C₅₃	—	4	2	no	—
82	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	complemented
83	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
93	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
94	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₅₃	—	2	2	no	—
96	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
97	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	complemented
98	C₂²⋊C₄×C₅₃	—	2	4	no	—
99	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
100	C₂²⋊C₄×C₅₃	C₂×C₈	1	8	yes	complemented
101	C₄²×C₅₃	—	2	2	no	—
102	C₄²×C₅₃	—	2	2	no	—
103	C₄²×C₅₃	—	2	2	no	—
104	C₄²×C₅₃	—	2	2	no	—
105	C₄²×C₅₃	—	2	4	no	—
106	C₄²×C₅₃	—	4	2	no	—
107	C₄²×C₅₃	—	4	2	no	—
108	C₄²×C₅₃	—	4	1	no	—
109	C₄²×C₅₃	C₂×C₈	1	4	yes	complemented
110	C₄²×C₅₃	C₂×D₄	1	2	yes	—
111	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
112	C₂×#88322	—	2	2	no	—
113	C₂×#88322	—	2	2	no	—
114	C₂×#88322	—	2	2	no	—
115	C₂×#88322	—	2	4	no	—
116	C₂×#88322	—	2	2	no	—
117	C₂×#88322	—	2	4	no	—
118	C₂×#88322	—	2	2	no	—
119	C₂×#88322	—	2	4	no	—
120	C₂×#88322	—	4	16	no	—
121	C₂×#88322	#41	1	4	yes	—
122	C₂³×C₅₃	—	2	2	no	—
123	C₂³×C₅₃	—	2	2	no	—
124	C₂³×C₅₃	—	2	2	no	—
125	C₂³×C₅₃	—	2	2	no	—
126	C₂³×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
127	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
128	C₂³×C₅₃	C₂×M₁₆	1	4	yes	—
129	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
130	D₄×C₅₃	C₄⋊C₈	1	16	yes	complemented
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	1	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	2	1	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	1	no	—
138	C₂×C₄×C₅₃	—	2	1	no	—
139	C₂×C₄×C₅₃	—	2	1	no	—
140	C₂×C₄×C₅₃	—	2	1	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	8	no	—
144	C₂×C₄×C₅₃	—	2	8	no	—
145	C₂×C₄×C₅₃	—	2	1	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	2	2	no	—
148	C₂×C₄×C₅₃	—	2	1	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	2	1	no	—
151	C₂×C₄×C₅₃	—	4	8	no	—
152	C₂×C₄×C₅₃	—	4	8	no	—
153	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
154	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
155	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
156	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
157	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
158	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
159	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
160	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
161	C₂×C₄×C₅₃	#131	1	2	yes	—
162	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
165	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
166	#88322	—	2	4	no	—
167	#88322	—	4	2	no	—
168	#88322	—	4	4	no	—
169	#22696	—	53	1	no	maximal, Sylow
170	C₂²×C₅₃	—	2	4	no	—
171	C₂²×C₅₃	—	2	4	no	—
172	C₂²×C₅₃	—	2	4	no	—
173	C₂²×C₅₃	—	2	4	no	—
174	C₂²×C₅₃	—	2	4	no	—
175	C₂²×C₅₃	—	2	4	no	—
176	C₂²×C₅₃	C₂×C₄⋊C₈	1	8	yes	—
177	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
178	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
179	C₂²×C₅₃	#518	1	1	yes	characteristic
180	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
181	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
182	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
183	C₂²×C₅₃	#568	1	1	yes	characteristic
184	C₄×C₅₃	—	2	4	no	—
185	C₄×C₅₃	—	2	2	no	—
186	C₄×C₅₃	—	2	2	no	—
187	C₄×C₅₃	—	2	2	no	—
188	C₄×C₅₃	—	2	4	no	—
189	C₄×C₅₃	—	2	2	no	—
190	C₄×C₅₃	—	4	1	no	—
191	C₄×C₅₃	—	4	2	no	—
192	C₄×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
193	C₄×C₅₃	#546	1	2	yes	—
194	C₄×C₅₃	#554	1	2	yes	—
195	C₂×C₈×D₄	—	53	2	no	—
196	C₄²×D₄	D₅₃	1	1	yes	characteristic
197	#1939	—	53	2	no	—
198	#1850	—	53	4	no	—
199	#1831	—	53	1	no	—
200	#1797	—	53	2	no	—
201	C₄×C₄⋊C₈	—	53	1	no	—
202	C₂×#336	—	53	2	no	—
203	C₂×C₅₃	—	2	8	no	—
204	C₂×C₅₃	#2349	1	1	yes	characteristic
205	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
206	C₂×C₅₃	#2906	1	1	yes	characteristic
207	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
208	C₂×C₅₃	#2970	1	1	yes	characteristic
209	C₂×C₅₃	#2972	1	1	yes	characteristic
210	C₂×C₅₃	#2974	1	1	yes	characteristic
211	C₂³×C₈	—	53	4	no	—
212	C₂×C₄×D₄	Dic₅₃	1	1	yes	characteristic
213	C₂×C₄×D₄	Dic₅₃	1	1	yes	characteristic
214	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
215	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
216	C₈×D₄	—	106	8	no	—
217	C₂×C₄⋊C₈	—	53	2	no	—
218	C₂×C₄⋊C₈	—	53	2	no	—
219	C₂×C₄⋊C₈	—	53	4	no	—
220	C₂×C₂²⋊C₈	—	53	4	no	—
221	C₂×C₄×C₈	—	53	2	no	—
222	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
223	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
224	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
225	C₄×C₂²⋊C₄	Dic₅₃	1	2	yes	—
226	C₄×C₂²⋊C₄	Dic₅₃	1	2	yes	—
227	C₄×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
228	C₄³	C₂×D₅₃	1	1	yes	characteristic
229	#336	—	53	1	no	—
230	#336	—	53	2	no	—
231	#336	—	53	2	no	—
232	#336	—	53	4	no	—
233	#336	—	53	1	no	—
234	C₅₃	#22696	1	1	yes	characteristic, complemented, Sylow
235	C₂²×D₄	C₂×Dic₅₃	1	1	yes	characteristic
236	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
237	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
238	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
239	C₂²×C₈	—	53	2	no	—
240	C₂²×C₈	—	53	4	no	—
241	C₂²×C₈	—	53	2	no	—
242	C₂²×C₈	—	106	8	no	—
243	C₂²×C₈	—	106	8	no	—
244	C₂²×C₈	—	212	4	no	—
245	C₄×D₄	—	2	4	no	—
246	C₄×D₄	#88322	1	8	yes	complemented
247	C₄×D₄	Dic₁₀₆	1	4	yes	—
248	C₄×D₄	D₂₁₂	1	4	yes	—
249	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
250	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
251	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
252	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
253	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
254	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
255	C₂×C₄²	—	4	4	no	—
256	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
257	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
258	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
259	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
260	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
261	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
262	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
263	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
264	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
265	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
266	C₄⋊C₈	—	106	4	no	—
267	C₄⋊C₈	—	106	4	no	—
268	C₄⋊C₈	—	212	4	no	—
269	C₂²⋊C₈	—	106	8	no	—
270	C₄×C₈	—	106	4	no	—
271	#95	C₂×Dic₅₃	1	2	yes	—
272	#95	C₂×Dic₅₃	1	2	yes	—
273	#95	C₂×Dic₅₃	1	2	yes	—
274	C₂⁴	C₂²×Dic₅₃	1	2	yes	—
275	C₂×D₄	C₂×#88322	1	4	yes	—
276	C₂×D₄	#11719917	1	4	yes	—
277	C₂×D₄	C₄⋊Dic₅₃	1	4	yes	—
278	C₂²×C₄	—	2	4	no	—
279	C₂²×C₄	—	2	4	no	—
280	C₂²×C₄	—	2	4	no	—
281	C₂²×C₄	—	2	4	no	—
282	C₂²×C₄	—	4	2	no	—
283	C₂²×C₄	—	4	2	no	—
284	C₂²×C₄	—	4	2	no	—
285	C₂²×C₄	C₂×#88322	1	8	yes	complemented
286	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
287	C₂²×C₄	C₂×D₂₁₂	1	4	yes	—
288	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
289	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
290	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
291	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
292	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
293	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
294	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
295	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
296	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
297	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
298	C₂×C₈	—	53	4	no	—
299	C₂×C₈	—	106	2	no	—
300	C₂×C₈	—	106	2	no	—
301	C₂×C₈	—	106	2	no	—
302	C₂×C₈	—	106	4	no	—
303	C₂×C₈	—	106	2	no	—
304	C₂×C₈	—	106	4	no	—
305	C₂×C₈	—	106	2	no	—
306	C₂×C₈	—	106	4	no	—
307	C₂×C₈	—	212	16	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	C₂×#88322	1	4	yes	complemented
310	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
311	C₂²⋊C₄	—	2	4	no	—
312	C₂²⋊C₄	C₂×#88322	1	8	yes	complemented
313	C₂²⋊C₄	C₄⋊Dic₅₃	1	8	yes	—
314	C₄²	—	2	2	no	—
315	C₄²	—	2	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	2	4	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	4	2	no	—
320	C₄²	—	4	2	no	—
321	C₄²	—	4	1	no	—
322	C₄²	C₂×#88322	1	4	yes	complemented
323	C₄²	C₂×Dic₁₀₆	1	2	yes	—
324	C₄²	C₂×D₂₁₂	1	2	yes	—
325	C₂³	—	2	2	no	—
326	C₂³	—	2	2	no	—
327	C₂³	—	2	2	no	—
328	C₂³	—	2	2	no	—
329	C₂³	C₂×C₄⋊Dic₅₃	1	4	yes	—
330	C₂³	C₂²×#88322	1	4	yes	—
331	C₂³	C₂×#11719917	1	4	yes	—
332	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
333	D₄	C₄⋊#88322	1	16	yes	complemented
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	8	no	—
347	C₂×C₄	—	2	8	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	4	8	no	—
355	C₂×C₄	—	4	8	no	—
356	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
357	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
358	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
359	C₂×C₄	C₂²×#88322	1	2	yes	—
360	C₂×C₄	C₂²×#88322	1	2	yes	—
361	C₂×C₄	C₂²×#88322	1	4	yes	—
362	C₂×C₄	C₂²×#88322	1	2	yes	—
363	C₂×C₄	C₂×#11719917	1	2	yes	—
364	C₂×C₄	#13133224	1	2	yes	—
365	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
366	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
367	C₂×C₄	#41×D₅₃	1	2	yes	—
368	C₂×C₄	#13133262	1	2	yes	—
369	C₈	—	106	4	no	—
370	C₈	—	212	2	no	—
371	C₈	—	212	4	no	—
372	C₂²	—	2	4	no	—
373	C₂²	—	2	4	no	—
374	C₂²	—	2	4	no	—
375	C₂²	—	2	4	no	—
376	C₂²	—	2	4	no	—
377	C₂²	—	2	4	no	—
378	C₂²	C₂×C₄⋊#88322	1	8	yes	—
379	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
380	C₂²	#41×Dic₅₃	1	1	yes	characteristic
381	C₂²	#16811098	1	1	yes	characteristic
382	C₂²	C₂³×#88322	1	1	yes	characteristic
383	C₂²	C₂²×#11719917	1	1	yes	characteristic
384	C₂²	C₂×#13133224	1	1	yes	characteristic
385	C₂²	#16811390	1	1	yes	characteristic
386	C₄	—	2	2	no	—
387	C₄	—	2	2	no	—
388	C₄	—	2	4	no	—
389	C₄	—	2	2	no	—
390	C₄	—	2	4	no	—
391	C₄	—	2	2	no	—
392	C₄	—	4	1	no	—
393	C₄	—	4	2	no	—
394	C₄	C₂×C₄⋊#88322	1	4	yes	—
395	C₄	#16811268	1	2	yes	—
396	C₄	#16811332	1	2	yes	—
397	C₂	—	2	8	no	—
398	C₂	#20277625	1	1	yes	characteristic
399	C₂	C₂²×C₄⋊#88322	1	1	yes	characteristic
400	C₂	#20283283	1	1	yes	characteristic
401	C₂	#41×#88322	1	1	yes	characteristic
402	C₂	#20283729	1	1	yes	characteristic
403	C₂	#20283739	1	1	yes	characteristic
404	C₂	#20283750	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Fitting	C₄²×D₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂²×D₄×C₅₃	C₂×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄²×D₄	D₅₃
hypocenter	C₅₃	#22696
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄

Complements

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

N	H ≅ G/N	type
2. #20275434	C₂	semidirect
3. #20274895	C₂	semidirect
4. #20274832	C₂	semidirect
5. #20274695	C₂	semidirect
6. C₄×C₄⋊#88322	C₂	semidirect
7. C₂×#16810378	C₂	semidirect
48. C₄×D₄×C₅₃	C₈	semidirect
82. C₂²×C₄×C₅₃	C₂×C₈	semidirect
97. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
100. C₂²⋊C₄×C₅₃	C₂×C₈	semidirect
109. C₄²×C₅₃	C₂×C₈	semidirect
130. D₄×C₅₃	C₄⋊C₈	semidirect
234. C₅₃	#22696	semidirect
246. C₄×D₄	#88322	semidirect
285. C₂²×C₄	C₂×#88322	semidirect
309. C₄⋊C₄	C₂×#88322	semidirect
312. C₂²⋊C₄	C₂×#88322	semidirect
322. C₄²	C₂×#88322	semidirect
333. D₄	C₄⋊#88322	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×#88322	2
#20275434	2
#20274895	2
#20274832	2
#20274695	2
C₄×C₄⋊#88322	2
C₂×#16810378	2
C₄²×D₄×C₅₃	2
#22696	53