Subgroups — #28328571

Order 13568.

536 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291132	C₂	1	2	yes	maximal, complemented
2	#20291098	C₂	1	2	yes	maximal, complemented
3	#20291097	C₂	1	2	yes	maximal, complemented
4	#20291093	C₂	1	2	yes	maximal, complemented
5	#20291079	C₂	1	1	yes	maximal, characteristic, complemented
6	#20291076	C₂	1	1	yes	maximal, characteristic, complemented
7	#20291075	C₂	1	1	yes	maximal, characteristic, complemented
8	#20291048	C₂	1	1	yes	maximal, characteristic, complemented
9	D₅₃⋊(C₄⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊#11719910	C₂	1	1	yes	maximal, characteristic, complemented
11	#20283671	C₂	1	1	yes	maximal, characteristic
12	C₂×#13133292	C₂²	1	2	yes	—
13	C₂×D₅₃⋊C₁₆	C₂²	1	2	yes	—
14	C₂²×#11719910	C₂²	1	2	yes	—
15	#16811544	C₂²	1	2	yes	—
16	#16811544	C₂²	1	2	yes	—
17	C₄⋊#11719910	C₂²	1	1	yes	characteristic
18	C₄⋊#11719910	C₂²	1	1	yes	characteristic
19	C₄⋊#11719910	C₂²	1	1	yes	characteristic
20	C₄⋊#11719910	C₂²	1	1	yes	characteristic
21	C₄⋊#11719910	C₂²	1	2	yes	—
22	#16811518	C₂²	1	2	yes	—
23	#16811518	C₂²	1	2	yes	—
24	C₂²⋊#11719910	C₂²	1	2	yes	—
25	#16811512	C₂²	1	2	yes	—
26	#16811511	C₂²	1	2	yes	—
27	#16811509	C₂²	1	2	yes	—
28	C₄×#11719945	C₄	1	1	yes	characteristic
29	#16810747	C₄	1	1	yes	characteristic
30	#16810747	C₂²	1	1	yes	characteristic
31	#16810688	C₄	1	1	yes	characteristic
32	#16810688	C₂²	1	1	yes	characteristic
33	#16810686	C₄	1	1	yes	characteristic
34	#16810686	C₄	1	1	yes	characteristic
35	#16810686	C₂²	1	1	yes	characteristic
36	#16810686	C₂²	1	1	yes	characteristic
37	#16810674	C₄	1	1	yes	characteristic
38	#16810674	C₄	1	1	yes	characteristic
39	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
40	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
41	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
42	C₅₃×#431	C₄	1	1	yes	characteristic
43	#13133292	—	4	2	no	—
44	D₅₃⋊C₁₆	—	4	2	no	—
45	C₂×#11719910	—	4	2	no	—
46	C₂×#11719910	C₂³	1	2	yes	—
47	C₂×#11719910	C₂³	1	2	yes	—
48	C₂×#11719910	C₂³	1	2	yes	—
49	C₂×#11719910	C₂³	1	2	yes	—
50	C₂×#11719945	C₈	1	1	yes	characteristic
51	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
52	C₂²×#88322	C₂³	1	1	yes	characteristic
53	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
57	C₄×Dic₁₀₆	C₈	1	1	yes	characteristic
58	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
59	C₄×D₂₁₂	C₈	1	1	yes	characteristic
60	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
61	C₄×C₂²⋊D₅₃	C₈	1	1	yes	characteristic
62	C₄×C₂²⋊D₅₃	C₈	1	1	yes	characteristic
63	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
65	#13133176	C₈	1	1	yes	characteristic
66	#13133174	C₈	1	1	yes	characteristic
67	#13133174	C₈	1	1	yes	characteristic
68	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
69	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
70	#13133146	C₂×C₄	1	1	yes	characteristic
71	#13133146	C₂×C₄	1	1	yes	characteristic
72	#13133146	C₂×C₄	1	1	yes	characteristic
73	#13133146	C₂×C₄	1	1	yes	characteristic
74	#13133129	C₂×C₄	1	1	yes	characteristic
75	#13133129	C₂×C₄	1	1	yes	characteristic
76	#13133129	C₂×C₄	1	1	yes	characteristic
77	#13133129	C₂×C₄	1	1	yes	characteristic
78	C₄×#88322	C₂×C₄	1	1	yes	characteristic
79	C₄×#88322	C₂×C₄	1	1	yes	characteristic
80	C₄×#88322	C₂³	1	1	yes	characteristic
81	C₄×#88322	C₂³	1	1	yes	characteristic
82	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
85	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
97	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
98	#11719945	—	2	4	no	—
99	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
100	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
101	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
102	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
103	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
104	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
112	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
113	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂²⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
116	C₂²⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
117	#11719921	C₂×C₈	1	1	yes	characteristic
118	#11719921	C₂×C₈	1	1	yes	characteristic
119	#11719921	C₂×C₈	1	1	yes	characteristic
120	#11719921	C₂×C₈	1	1	yes	characteristic
121	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
122	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
123	#11719919	C₂×C₈	1	1	yes	characteristic
124	#11719919	C₂×C₈	1	1	yes	characteristic
125	#11719919	C₂×C₈	1	1	yes	characteristic
126	#11719919	C₂×C₈	1	1	yes	characteristic
127	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
128	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
129	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
130	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
131	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
133	C₂×#88322	—	2	1	no	—
134	C₂×#88322	—	2	1	no	—
135	C₂×#88322	—	4	1	no	—
136	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
137	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
138	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
139	C₂×#88322	C₂⁴	1	1	yes	characteristic
140	C₈×D₅₃	—	2	1	no	—
141	C₈×D₅₃	—	2	1	no	—
142	C₈×D₅₃	—	2	1	no	—
143	C₈×D₅₃	—	2	1	no	—
144	C₈×D₅₃	—	4	1	no	—
145	C₈×D₅₃	—	4	1	no	—
146	#11719910	—	2	2	no	—
147	#11719910	—	2	2	no	—
148	#11719910	—	2	2	no	—
149	#11719910	—	2	2	no	—
150	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
151	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
152	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
153	C₂×C₄×C₅₃	—	2	2	no	—
154	C₂×C₄×C₅₃	—	2	1	no	—
155	C₂×C₄×C₅₃	—	2	1	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
163	C₂²⋊D₅₃	—	2	2	no	—
164	C₂²⋊D₅₃	M₃₂	1	4	yes	complemented
165	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
166	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
167	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
168	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
169	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
170	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
171	D₂₁₂	—	2	2	no	—
172	C₄×D₅₃	—	2	2	no	—
173	C₄×D₅₃	—	2	1	no	—
174	C₄×D₅₃	—	2	1	no	—
175	C₄×D₅₃	—	2	1	no	—
176	C₄×D₅₃	—	2	1	no	—
177	C₄×D₅₃	—	2	2	no	—
178	C₄×D₅₃	—	2	2	no	—
179	C₄×D₅₃	—	2	2	no	—
180	Dic₁₀₆	—	2	2	no	—
181	C₈×C₅₃	—	2	1	no	—
182	C₈×C₅₃	—	2	1	no	—
183	C₈×C₅₃	—	2	1	no	—
184	C₈×C₅₃	—	2	1	no	—
185	#88322	—	2	1	no	—
186	#88322	—	2	1	no	—
187	#88322	—	2	1	no	—
188	#88322	H(5,𝔽₂)	1	1	yes	characteristic
189	#88322	2₋¹⁺⁴	1	1	yes	characteristic
190	#33504	—	53	1	no	maximal, Sylow
191	C₂²×C₅₃	—	2	1	no	—
192	C₂²×C₅₃	—	2	1	no	—
193	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
194	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
195	C₂×D₅₃	—	2	1	no	—
196	C₂×D₅₃	—	2	1	no	—
197	C₂×D₅₃	—	2	1	no	—
198	C₂×D₅₃	—	2	1	no	—
199	C₂×D₅₃	—	2	1	no	—
200	C₂×D₅₃	C₂×M₃₂	1	2	yes	—
201	C₄×C₅₃	—	2	1	no	—
202	C₄×C₅₃	—	2	1	no	—
203	C₄×C₅₃	—	2	1	no	—
204	C₄×C₅₃	—	2	1	no	—
205	C₄×C₅₃	#504	1	2	yes	—
206	C₄×C₅₃	#520	1	1	yes	characteristic
207	C₄×C₅₃	#568	1	1	yes	characteristic
208	Dic₅₃	—	2	1	no	—
209	Dic₅₃	—	2	1	no	—
210	Dic₅₃	—	2	1	no	—
211	Dic₅₃	—	2	1	no	—
212	Dic₅₃	—	2	1	no	—
213	Dic₅₃	C₂×M₃₂	1	2	yes	—
214	C₈×#41	—	53	1	no	—
215	#2183	—	53	2	no	—
216	#2168	—	53	2	no	—
217	#2168	—	53	2	no	—
218	D₄×C₁₆	—	53	2	no	—
219	#2163	—	53	1	no	—
220	#2163	—	53	1	no	—
221	#2162	—	53	1	no	—
222	#2150	—	53	1	no	—
223	C₂×C₄⋊C₁₆	—	53	1	no	—
224	C₂×C₄⋊C₁₆	—	53	1	no	—
225	C₂×C₅₃	—	2	2	no	—
226	C₂×C₅₃	#2988	1	1	yes	characteristic
227	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
228	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
229	D₅₃	—	2	2	no	—
230	D₅₃	—	4	1	no	—
231	C₄×#41	—	53	1	no	—
232	C₂×M₃₂	—	53	2	no	—
233	C₂²×C₁₆	—	53	2	no	—
234	C₂²×C₁₆	—	53	2	no	—
235	C₈×Q₈	—	53	1	no	—
236	C₈×Q₈	—	53	1	no	—
237	C₈×D₄	—	53	1	no	—
238	C₈×D₄	—	53	1	no	—
239	C₈×D₄	—	53	1	no	—
240	C₈×D₄	—	53	1	no	—
241	C₈×D₄	—	53	1	no	—
242	C₈×D₄	—	53	1	no	—
243	#431	—	53	1	no	—
244	#431	—	53	1	no	—
245	#431	#6157	1	1	yes	characteristic
246	C₂×C₄×C₈	—	53	1	no	—
247	C₂×C₄×C₈	—	53	1	no	—
248	C₂×C₄×C₈	—	53	1	no	—
249	C₄⋊C₁₆	—	53	1	no	—
250	C₄⋊C₁₆	—	53	1	no	—
251	C₄⋊C₁₆	—	53	1	no	—
252	C₄⋊C₁₆	—	53	2	no	—
253	C₄⋊C₁₆	—	53	1	no	—
254	C₄⋊C₁₆	—	53	2	no	—
255	C₄⋊C₁₆	—	53	2	no	—
256	C₂²⋊C₁₆	—	53	2	no	—
257	C₂²⋊C₁₆	—	53	2	no	—
258	C₂²⋊C₁₆	—	53	2	no	—
259	#346	—	53	2	no	—
260	#346	—	53	2	no	—
261	C₄×C₁₆	—	53	2	no	—
262	C₅₃	#33504	1	1	yes	characteristic, complemented, Sylow
263	C₂×#41	—	53	1	no	—
264	C₂²×C₈	—	53	1	no	—
265	C₂²×C₈	—	53	1	no	—
266	C₂²×C₈	—	53	1	no	—
267	C₂²×C₈	—	53	1	no	—
268	C₂²×C₈	—	53	1	no	—
269	C₂²×C₈	—	53	1	no	—
270	C₄×Q₈	—	53	1	no	—
271	C₄×Q₈	—	53	1	no	—
272	C₄×D₄	—	53	1	no	—
273	C₄×D₄	—	53	1	no	—
274	C₄×D₄	—	53	1	no	—
275	C₄×D₄	—	53	1	no	—
276	C₄×D₄	—	53	1	no	—
277	C₄×D₄	—	53	1	no	—
278	#117	—	53	1	no	—
279	#117	—	53	1	no	—
280	#117	—	53	1	no	—
281	C₂×C₄²	—	53	1	no	—
282	C₂×C₄²	—	53	1	no	—
283	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
284	M₃₂	—	212	2	no	—
285	C₂×C₁₆	—	53	2	no	—
286	C₂×C₁₆	—	53	2	no	—
287	C₂×C₁₆	—	53	2	no	—
288	C₂×C₁₆	—	53	2	no	—
289	C₂×C₁₆	—	212	2	no	—
290	C₂×C₁₆	—	212	2	no	—
291	C₄⋊C₈	—	53	1	no	—
292	C₄⋊C₈	—	53	1	no	—
293	C₄⋊C₈	—	53	1	no	—
294	C₄⋊C₈	—	53	1	no	—
295	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
296	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
297	C₂²⋊C₈	—	53	1	no	—
298	C₂²⋊C₈	—	53	1	no	—
299	C₂²⋊C₈	—	53	1	no	—
300	C₂²⋊C₈	—	53	1	no	—
301	C₂²⋊C₈	C₂×#6157	1	1	yes	characteristic
302	C₂²⋊C₈	C₂×#6157	1	1	yes	characteristic
303	C₄×C₈	—	53	1	no	—
304	C₄×C₈	—	53	1	no	—
305	C₄×C₈	—	53	1	no	—
306	C₄×C₈	—	53	1	no	—
307	C₄×C₈	—	53	1	no	—
308	C₄×C₈	—	53	1	no	—
309	C₄×C₈	—	53	1	no	—
310	C₄×C₈	—	53	1	no	—
311	C₄×C₈	C₂×#6157	1	1	yes	characteristic
312	C₄×C₈	C₂×#6157	1	1	yes	characteristic
313	#41	—	106	4	no	—
314	C₂×Q₈	—	53	1	no	—
315	C₂×D₄	—	53	1	no	—
316	C₂×D₄	—	53	1	no	—
317	C₂×D₄	—	53	1	no	—
318	C₂²×C₄	—	53	1	no	—
319	C₂²×C₄	—	53	1	no	—
320	C₂²×C₄	—	53	1	no	—
321	C₂²×C₄	—	53	1	no	—
322	C₂²×C₄	—	53	1	no	—
323	C₂²×C₄	—	53	1	no	—
324	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
325	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
326	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
327	C₂×C₈	—	53	1	no	—
328	C₂×C₈	—	53	1	no	—
329	C₂×C₈	—	53	1	no	—
330	C₂×C₈	—	53	1	no	—
331	C₂×C₈	—	106	1	no	—
332	C₂×C₈	—	106	1	no	—
333	C₂×C₈	—	106	1	no	—
334	C₂×C₈	—	106	1	no	—
335	C₂×C₈	—	106	1	no	—
336	C₂×C₈	—	106	1	no	—
337	C₂×C₈	—	212	1	no	—
338	C₂×C₈	—	212	1	no	—
339	C₂×C₈	—	212	1	no	—
340	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
341	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
342	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
343	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
344	C₄⋊C₄	—	53	1	no	—
345	C₄⋊C₄	—	53	1	no	—
346	C₄⋊C₄	—	53	1	no	—
347	C₄⋊C₄	—	53	1	no	—
348	C₄⋊C₄	—	53	1	no	—
349	C₄⋊C₄	—	53	1	no	—
350	C₂²⋊C₄	—	53	1	no	—
351	C₂²⋊C₄	—	53	1	no	—
352	C₂²⋊C₄	—	53	1	no	—
353	C₂²⋊C₄	—	53	1	no	—
354	C₂²⋊C₄	—	53	1	no	—
355	C₂²⋊C₄	—	53	1	no	—
356	C₄²	—	53	1	no	—
357	C₄²	—	53	1	no	—
358	C₄²	—	53	1	no	—
359	C₄²	—	53	1	no	—
360	C₄²	—	53	1	no	—
361	C₄²	—	53	1	no	—
362	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
363	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
364	C₄²	C₂²×#6157	1	1	yes	characteristic
365	C₄²	C₂²×#6157	1	1	yes	characteristic
366	C₁₆	—	106	2	no	—
367	C₁₆	—	106	2	no	—
368	C₁₆	—	106	2	no	—
369	C₁₆	—	106	2	no	—
370	C₂³	—	53	1	no	—
371	C₂³	—	53	1	no	—
372	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
373	Q₈	—	106	2	no	—
374	D₄	—	53	4	no	—
375	D₄	—	106	2	no	—
376	D₄	—	106	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	53	1	no	—
382	C₂×C₄	—	53	1	no	—
383	C₂×C₄	—	53	1	no	—
384	C₂×C₄	—	53	1	no	—
385	C₂×C₄	—	53	1	no	—
386	C₂×C₄	—	53	1	no	—
387	C₂×C₄	—	106	2	no	—
388	C₂×C₄	—	106	2	no	—
389	C₂×C₄	—	106	2	no	—
390	C₂×C₄	—	106	2	no	—
391	C₂×C₄	—	106	1	no	—
392	C₂×C₄	—	106	1	no	—
393	C₂×C₄	—	106	1	no	—
394	C₂×C₄	—	106	1	no	—
395	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
396	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
397	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
398	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
399	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
400	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
401	C₈	—	2	1	no	—
402	C₈	—	2	1	no	—
403	C₈	—	2	1	no	—
404	C₈	—	2	1	no	—
405	C₈	—	53	1	no	—
406	C₈	—	53	1	no	—
407	C₈	—	106	1	no	—
408	C₈	—	106	1	no	—
409	C₈	—	106	1	no	—
410	C₂²	—	2	1	no	—
411	C₂²	—	2	1	no	—
412	C₂²	—	53	2	no	—
413	C₂²	—	106	1	no	—
414	C₂²	—	106	1	no	—
415	C₂²	—	106	1	no	—
416	C₂²	—	106	1	no	—
417	C₂²	—	106	1	no	—
418	C₂²	D₅₃⋊M₃₂	1	2	yes	—
419	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
420	C₄	—	2	1	no	—
421	C₄	—	2	1	no	—
422	C₄	—	2	1	no	—
423	C₄	—	2	1	no	—
424	C₄	—	53	2	no	—
425	C₄	—	106	1	no	—
426	C₄	—	106	1	no	—
427	C₄	—	106	1	no	—
428	C₄	—	106	1	no	—
429	C₄	—	106	1	no	—
430	C₄	#16811682	1	2	yes	—
431	C₄	#16811692	1	1	yes	characteristic
432	C₄	#16811702	1	1	yes	characteristic
433	C₂	—	2	2	no	—
434	C₂	—	106	2	no	—
435	C₂	—	212	1	no	—
436	C₂	#20291632	1	1	yes	characteristic
437	C₂	C₂×D₅₃⋊M₃₂	1	1	yes	characteristic
438	C₂	C₂×#16811682	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#431	C₄
prime radical	C₂×#11719945	C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	#431	#6157
hypocenter	C₅₃	#33504
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
1. #20291132	C₂	semidirect
2. #20291098	C₂	semidirect
3. #20291097	C₂	semidirect
4. #20291093	C₂	semidirect
5. #20291079	C₂	semidirect
6. #20291076	C₂	semidirect
7. #20291075	C₂	semidirect
8. #20291048	C₂	semidirect
9. D₅₃⋊(C₄⋊C₁₆)	C₂	semidirect
10. C₂×C₄⋊#11719910	C₂	semidirect
164. C₂²⋊D₅₃	M₃₂	semidirect
262. C₅₃	#33504	semidirect

Maximal subgroups

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

subgroup	index
#20291132	2
#20291098	2
#20291097	2
#20291093	2
#20291079	2
#20291076	2
#20291075	2
#20291048	2
D₅₃⋊(C₄⋊C₁₆)	2
C₂×C₄⋊#11719910	2
#20283671	2
#33504	53