Subgroups — #27402882

Order 13568.

896 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810747	C₂	1	2	yes	maximal
2	C₄²×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
3	#20275431	C₂	1	2	yes	maximal
4	#20275430	C₂	1	4	yes	maximal
5	#20274866	C₂	1	2	yes	maximal
6	C₄⋊C₈×Dic₅₃	C₂	1	2	yes	maximal
7	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
12	#16810747	—	2	8	no	—
13	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
15	C₂×#13133146	C₂²	1	4	yes	—
16	C₂×#13133146	C₂²	1	4	yes	—
17	C₂×C₄×#88322	C₂²	1	2	yes	—
18	C₂×C₈×Dic₅₃	C₂²	1	4	yes	—
19	C₄×#11719919	C₄	1	2	yes	—
20	C₄×#11719919	C₄	1	2	yes	—
21	C₄×#11719919	C₂²	1	2	yes	—
22	C₄×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
23	C₄×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
24	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
25	C₄²×Dic₅₃	C₂²	1	2	yes	—
26	#16810378	C₂²	1	2	yes	—
27	#16810377	C₂²	1	4	yes	—
28	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
31	C₄³×C₅₃	C₂²	1	1	yes	characteristic
32	C₂²×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×#88322	C₂³	1	2	yes	—
34	C₂²×#88322	C₂³	1	2	yes	—
35	C₄×Dic₁₀₆	—	2	4	no	—
36	C₄×Dic₁₀₆	C₈	1	8	yes	complemented
37	C₄×Dic₁₀₆	D₄	1	4	yes	—
38	C₄×Dic₁₀₆	Q₈	1	4	yes	—
39	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×#11719919	C₂×C₄	1	2	yes	—
43	C₂×#11719919	C₂×C₄	1	2	yes	—
44	C₂×#11719919	C₂×C₄	1	2	yes	—
45	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
50	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
51	C₄⋊#88322	—	2	2	no	—
52	C₄⋊#88322	Q₈	1	4	yes	complemented
53	#13133146	—	2	8	no	—
54	#13133146	—	4	4	no	—
55	C₄×#88322	—	2	4	no	—
56	C₈×Dic₅₃	—	2	8	no	—
57	#13133124	C₂×C₄	1	2	yes	—
58	#13133124	C₂×C₄	1	2	yes	—
59	#13133124	C₂×C₄	1	2	yes	—
60	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
61	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₅₃	—	2	4	no	—
69	C₄⋊C₈×C₅₃	—	2	4	no	—
70	C₄⋊C₈×C₅₃	—	4	2	no	—
71	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
79	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
80	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
81	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
82	C₂×Dic₁₀₆	C₄⋊C₄	1	4	yes	—
83	C₂×Dic₁₀₆	C₂×C₈	1	4	yes	—
84	C₂×Dic₁₀₆	M₁₆	1	4	yes	—
85	C₂×C₈×C₅₃	—	2	4	no	—
86	C₂×C₈×C₅₃	—	2	4	no	—
87	C₂×C₈×C₅₃	—	2	4	no	—
88	C₄²×C₅₃	—	2	2	no	—
89	C₄²×C₅₃	—	2	2	no	—
90	C₄²×C₅₃	—	2	1	no	—
91	C₄²×C₅₃	—	2	1	no	—
92	C₄²×C₅₃	—	4	1	no	—
93	C₄²×C₅₃	C₂×C₈	1	4	yes	—
94	C₄²×C₅₃	C₂×D₄	1	2	yes	—
95	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
96	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
97	C₄²×C₅₃	#41	1	2	yes	—
98	C₄⋊Dic₅₃	—	2	2	no	—
99	C₄⋊Dic₅₃	C₄⋊C₄	1	4	yes	—
100	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
101	#11719919	—	2	4	no	—
102	#11719919	C₄⋊C₄	1	8	yes	—
103	#11719919	C₂×C₈	1	8	yes	—
104	C₄×Dic₅₃	—	2	4	no	—
105	C₄×Dic₅₃	—	2	2	no	—
106	C₄×Dic₅₃	—	2	2	no	—
107	C₄×Dic₅₃	—	2	4	no	—
108	C₄×Dic₅₃	—	4	2	no	—
109	C₄×Dic₅₃	C₂×C₈	1	8	yes	—
110	C₄×Dic₅₃	C₂×D₄	1	4	yes	—
111	C₄×Dic₅₃	C₂×Q₈	1	4	yes	—
112	C₂×#88322	—	2	2	no	—
113	C₂×#88322	—	2	2	no	—
114	C₂×#88322	—	2	2	no	—
115	C₂×#88322	—	2	2	no	—
116	C₂×#88322	—	2	2	no	—
117	C₂×#88322	C₂×Q₈	1	4	yes	—
118	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	1	no	—
123	C₂×C₄×C₅₃	—	2	1	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	2	1	no	—
127	C₂×C₄×C₅₃	—	2	1	no	—
128	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
132	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
133	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
136	C₂×C₄×C₅₃	#131	1	2	yes	—
137	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂×Dic₅₃	—	2	2	no	—
144	C₂×Dic₅₃	—	2	2	no	—
145	C₂×Dic₅₃	—	2	2	no	—
146	C₂×Dic₅₃	—	2	2	no	—
147	C₂×Dic₅₃	—	2	2	no	—
148	C₂×Dic₅₃	—	2	2	no	—
149	C₂×Dic₅₃	—	2	2	no	—
150	C₂×Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
151	C₂×Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
152	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
153	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
154	C₂×Dic₅₃	C₂×M₁₆	1	4	yes	—
155	Dic₁₀₆	C₄⋊C₈	1	16	yes	complemented
156	C₈×C₅₃	—	4	4	no	—
157	#88322	—	2	4	no	—
158	#88322	—	4	2	no	—
159	Q₈×C₄⋊C₈	—	53	1	no	maximal, Sylow
160	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
161	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
162	C₂²×C₅₃	#520	1	1	yes	characteristic
163	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
164	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
165	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
166	C₂²×C₅₃	#568	1	1	yes	characteristic
167	C₄×C₅₃	—	2	2	no	—
168	C₄×C₅₃	—	2	2	no	—
169	C₄×C₅₃	—	2	2	no	—
170	C₄×C₅₃	—	4	1	no	—
171	C₄×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
172	C₄×C₅₃	C₈×Q₈	1	4	yes	complemented
173	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
174	C₄×C₅₃	#550	1	1	yes	characteristic
175	C₄×C₅₃	#557	1	1	yes	characteristic
176	C₄×C₅₃	Q₈²	1	1	yes	characteristic
177	Dic₅₃	—	2	4	no	—
178	Dic₅₃	—	2	4	no	—
179	Dic₅₃	—	4	2	no	—
180	Dic₅₃	C₂×C₄⋊C₈	1	8	yes	—
181	C₂×C₈×Q₈	—	53	2	no	—
182	C₄²×Q₈	—	53	1	no	—
183	#1939	—	53	4	no	—
184	#1939	—	53	2	no	—
185	#1842	—	53	2	no	—
186	#1842	D₅₃	1	1	yes	characteristic
187	C₄×C₄⋊C₈	—	53	2	no	—
188	C₄×C₄⋊C₈	—	53	1	no	—
189	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
191	C₂×C₅₃	#2906	1	1	yes	characteristic
192	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
193	C₂×C₅₃	Q₈×M₁₆	1	1	yes	characteristic
194	C₂×C₅₃	#2988	1	1	yes	characteristic
195	C₂×C₅₃	#2991	1	1	yes	characteristic
196	C₂×C₄×Q₈	—	53	1	no	—
197	C₂×C₄×Q₈	—	53	1	no	—
198	C₂×C₄×Q₈	—	53	1	no	—
199	C₈×Q₈	—	106	8	no	—
200	C₂×C₄⋊C₈	—	53	1	no	—
201	C₂×C₄⋊C₈	—	53	4	no	—
202	C₂×C₄⋊C₈	—	53	1	no	—
203	C₂×C₄⋊C₈	—	53	4	no	—
204	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
205	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
206	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
207	C₂×C₄×C₈	—	53	4	no	—
208	C₂×C₄×C₈	—	53	2	no	—
209	C₄×C₄⋊C₄	—	53	1	no	—
210	C₄×C₄⋊C₄	—	53	1	no	—
211	C₄×C₄⋊C₄	—	53	2	no	—
212	C₄×C₄⋊C₄	—	53	2	no	—
213	C₄×C₄⋊C₄	—	53	1	no	—
214	C₄×C₄⋊C₄	—	53	2	no	—
215	C₄³	—	53	2	no	—
216	C₄³	C₂×D₅₃	1	1	yes	characteristic
217	#336	—	53	4	no	—
218	#336	—	53	2	no	—
219	C₅₃	Q₈×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
220	C₂²×Q₈	—	53	1	no	—
221	C₂²×C₈	—	53	2	no	—
222	C₂²×C₈	—	53	2	no	—
223	C₂²×C₈	C₂²×D₅₃	1	4	yes	—
224	C₄×Q₈	—	53	4	no	—
225	C₄×Q₈	—	53	4	no	—
226	C₄×Q₈	—	53	8	no	—
227	C₄×Q₈	—	106	4	no	—
228	C₂×C₄⋊C₄	—	53	2	no	—
229	C₂×C₄⋊C₄	—	53	2	no	—
230	C₂×C₄⋊C₄	—	53	1	no	—
231	C₂×C₄⋊C₄	—	53	1	no	—
232	C₂×C₄⋊C₄	—	53	2	no	—
233	C₂×C₄⋊C₄	—	53	1	no	—
234	C₂×C₄²	—	53	2	no	—
235	C₂×C₄²	—	53	2	no	—
236	C₂×C₄²	—	53	2	no	—
237	C₂×C₄²	—	53	2	no	—
238	C₂×C₄²	—	53	2	no	—
239	C₂×C₄²	—	53	2	no	—
240	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
241	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
242	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
243	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
244	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
245	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
246	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
247	C₄⋊C₈	—	2	4	no	—
248	C₄⋊C₈	—	2	4	no	—
249	C₄⋊C₈	—	4	2	no	—
250	C₄⋊C₈	—	53	4	no	—
251	C₄⋊C₈	—	106	8	no	—
252	C₄⋊C₈	—	106	2	no	—
253	C₄⋊C₈	—	212	4	no	—
254	C₄×C₈	—	106	8	no	—
255	C₄×C₈	—	106	4	no	—
256	#95	—	53	2	no	—
257	#95	—	53	2	no	—
258	#95	—	53	2	no	—
259	C₂×Q₈	—	53	4	no	—
260	C₂×Q₈	—	53	4	no	—
261	C₂×Q₈	—	53	4	no	—
262	C₂²×C₄	—	53	2	no	—
263	C₂²×C₄	—	53	2	no	—
264	C₂²×C₄	—	53	2	no	—
265	C₂²×C₄	—	53	2	no	—
266	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
271	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
272	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
273	C₂×C₈	—	2	4	no	—
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	—	53	4	no	—
277	C₂×C₈	—	106	2	no	—
278	C₂×C₈	—	106	2	no	—
279	C₂×C₈	—	106	2	no	—
280	C₂×C₈	—	106	2	no	—
281	C₂×C₈	—	106	2	no	—
282	C₄⋊C₄	—	53	8	no	—
283	C₄⋊C₄	—	53	4	no	—
284	C₄⋊C₄	—	53	8	no	—
285	C₄⋊C₄	—	53	4	no	—
286	C₄⋊C₄	—	106	4	no	—
287	C₄⋊C₄	—	106	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	1	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	4	1	no	—
293	C₄²	—	53	8	no	—
294	C₄²	—	53	4	no	—
295	C₄²	—	53	4	no	—
296	C₄²	—	106	4	no	—
297	C₄²	—	106	4	no	—
298	C₄²	—	106	2	no	—
299	C₄²	—	106	2	no	—
300	C₄²	—	212	2	no	—
301	C₄²	C₈×D₅₃	1	4	yes	—
302	C₄²	D₄×D₅₃	1	2	yes	—
303	C₄²	#11719947	1	2	yes	—
304	C₄²	Q₈×D₅₃	1	2	yes	—
305	C₄²	Q₈×D₅₃	1	2	yes	—
306	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
307	Q₈	—	53	16	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	53	4	no	—
318	C₂×C₄	—	53	4	no	—
319	C₂×C₄	—	53	4	no	—
320	C₂×C₄	—	53	4	no	—
321	C₂×C₄	—	53	4	no	—
322	C₂×C₄	—	106	2	no	—
323	C₂×C₄	—	106	2	no	—
324	C₂×C₄	—	106	2	no	—
325	C₂×C₄	—	106	2	no	—
326	C₂×C₄	—	106	2	no	—
327	C₂×C₄	—	106	2	no	—
328	C₂×C₄	—	106	2	no	—
329	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
330	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
331	C₂×C₄	#13133182	1	2	yes	—
332	C₂×C₄	#13133182	1	2	yes	—
333	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
334	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
335	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
336	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
337	C₂×C₄	#13133225	1	2	yes	—
338	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
339	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
340	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
341	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
342	C₂×C₄	#13133262	1	1	yes	characteristic
343	C₂×C₄	#13133265	1	1	yes	characteristic
344	C₈	—	4	4	no	—
345	C₈	—	106	4	no	—
346	C₈	—	212	2	no	—
347	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
348	C₂²	C₂×#13133182	1	1	yes	characteristic
349	C₂²	#16811104	1	1	yes	characteristic
350	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
351	C₂²	C₂×M₁₆×D₅₃	1	1	yes	characteristic
352	C₂²	C₂×#13133225	1	1	yes	characteristic
353	C₂²	#16811389	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	2	no	—
357	C₄	—	4	1	no	—
358	C₄	—	53	8	no	—
359	C₄	—	106	4	no	—
360	C₄	—	106	4	no	—
361	C₄	—	212	2	no	—
362	C₄	C₄⋊C₈×D₅₃	1	4	yes	—
363	C₄	#16810747	1	4	yes	complemented
364	C₄	#16811288	1	1	yes	characteristic
365	C₄	#16811293	1	1	yes	characteristic
366	C₄	#16811348	1	1	yes	characteristic
367	C₄	#16811353	1	1	yes	characteristic
368	C₂	#20277627	1	1	yes	characteristic
369	C₂	C₂×C₄⋊C₈×D₅₃	1	1	yes	characteristic
370	C₂	#20283284	1	1	yes	characteristic
371	C₂	C₂×#16810747	1	1	yes	characteristic
372	C₂	#20283666	1	1	yes	characteristic
373	C₂	#20283824	1	1	yes	characteristic
374	C₂	#20283845	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#1842	C₂
socle, prime radical	C₂³×C₅₃	C₂³×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1842	D₅₃
hypocenter	C₅₃	Q₈×C₄⋊C₈
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1842

Complements

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

N	H ≅ G/N	type
36. C₄×Dic₁₀₆	C₈	semidirect
52. C₄⋊#88322	Q₈	semidirect
155. Dic₁₀₆	C₄⋊C₈	semidirect
172. C₄×C₅₃	C₈×Q₈	semidirect
219. C₅₃	Q₈×C₄⋊C₈	semidirect
363. C₄	#16810747	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810747	2
C₄²×Dic₁₀₆	2
#20275431	2
#20275430	2
#20274866	2
C₄⋊C₈×Dic₅₃	2
C₄×C₄⋊#88322	2
C₅₃×#1842	2
Q₈×C₄⋊C₈	53