Subgroups — C₂×#20272616 #27274379

Order 13568.

810 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272616	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
6	C₂²×C₄×Dic₅₃	—	2	1	no	—
7	#16810676	—	2	1	no	—
8	#16810676	—	2	1	no	—
9	#16810676	C₄	1	2	yes	complemented
10	#16810676	C₂²	1	2	yes	—
11	C₂×C₄⋊#88322	C₄	1	2	yes	complemented
12	C₂×C₂²⋊#88322	—	2	1	no	—
13	C₂×C₄×#88322	—	2	1	no	—
14	C₄×C₄⋊Dic₅₃	—	2	1	no	—
15	C₄×C₄⋊Dic₅₃	—	2	1	no	—
16	C₄×C₄⋊Dic₅₃	C₄	1	2	yes	—
17	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
18	C₂×#13133124	C₄	1	2	yes	—
19	C₂²×M₁₆×C₅₃	—	2	1	no	—
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#423	—	2	1	no	—
22	C₅₃×#423	—	2	1	no	—
23	C₅₃×#423	C₄	1	2	yes	complemented
24	C₅₃×#423	C₂²	1	2	yes	—
25	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
26	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	complemented
27	C₂³×Dic₅₃	—	2	2	no	—
28	C₂²×#88322	—	2	2	no	—
29	C₂×C₄⋊Dic₅₃	—	2	8	no	—
30	C₂×C₄⋊Dic₅₃	—	2	2	no	—
31	C₂×C₄⋊Dic₅₃	—	2	2	no	—
32	C₂×C₄×Dic₅₃	—	2	2	no	—
33	C₂×C₄×Dic₅₃	—	2	2	no	—
34	C₂×C₄×Dic₅₃	—	4	2	no	—
35	C₂×C₄×Dic₅₃	—	4	2	no	—
36	C₄⋊#88322	—	2	2	no	—
37	C₄⋊#88322	C₂×C₄	1	4	yes	complemented
38	C₂²⋊#88322	—	2	2	no	—
39	C₂²⋊#88322	—	2	2	no	—
40	C₄×#88322	—	2	2	no	—
41	C₄×#88322	—	2	2	no	—
42	#13133124	—	2	2	no	—
43	#13133124	C₂×C₄	1	4	yes	—
44	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
47	C₂×M₁₆×C₅₃	—	2	2	no	—
48	C₂×M₁₆×C₅₃	—	2	2	no	—
49	C₂×M₁₆×C₅₃	—	4	2	no	—
50	C₂×M₁₆×C₅₃	—	4	2	no	—
51	C₂²×C₈×C₅₃	—	2	2	no	—
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	C₄⋊C₈×C₅₃	—	2	2	no	—
65	C₄⋊C₈×C₅₃	—	2	2	no	—
66	C₄⋊C₈×C₅₃	—	2	2	no	—
67	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
68	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	—	2	4	no	—
70	C₂²×C₄×C₅₃	—	4	2	no	—
71	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₄²	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	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	D₈	1	2	yes	—
87	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
88	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
89	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
90	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×Dic₅₃	—	2	8	no	—
94	C₂²×Dic₅₃	—	2	4	no	—
95	C₂²×Dic₅₃	—	4	4	no	—
96	C₂²×Dic₅₃	—	4	4	no	—
97	M₁₆×C₅₃	—	4	4	no	—
98	M₁₆×C₅₃	—	4	2	no	—
99	M₁₆×C₅₃	—	4	2	no	—
100	C₂×C₈×C₅₃	—	2	4	no	—
101	C₂×C₈×C₅₃	—	4	4	no	—
102	C₄²×C₅₃	C₄²	1	1	yes	characteristic
103	C₄²×C₅₃	C₄²	1	1	yes	characteristic
104	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
118	C₄⋊Dic₅₃	—	2	8	no	—
119	C₄⋊Dic₅₃	—	2	8	no	—
120	C₄×Dic₅₃	—	4	4	no	—
121	C₄×Dic₅₃	—	4	2	no	—
122	C₄×Dic₅₃	—	4	2	no	—
123	C₂×#88322	—	2	4	no	—
124	C₂×#88322	—	4	4	no	—
125	C₂³×C₅₃	—	2	2	no	—
126	C₂³×C₅₃	—	2	2	no	—
127	C₂³×C₅₃	#95	1	1	yes	characteristic
128	C₂³×C₅₃	#95	1	1	yes	characteristic
129	C₂³×C₅₃	#102	1	2	yes	—
130	C₂³×C₅₃	#103	1	2	yes	—
131	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	4	1	no	—
140	C₂×C₄×C₅₃	—	4	1	no	—
141	C₂×C₄×C₅₃	—	4	1	no	—
142	C₂×C₄×C₅₃	—	4	1	no	—
143	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
144	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
146	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
147	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
148	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
149	C₂×C₄×C₅₃	#102	1	2	yes	—
150	C₂×C₄×C₅₃	#102	1	2	yes	—
151	C₂×C₄×C₅₃	#102	1	2	yes	—
152	C₂×C₄×C₅₃	#103	1	2	yes	—
153	C₂×C₄×C₅₃	#103	1	2	yes	—
154	C₂×C₄×C₅₃	#103	1	2	yes	—
155	C₂×C₄×C₅₃	#104	1	4	yes	—
156	C₂×C₄×C₅₃	#106	1	2	yes	—
157	C₂×C₄×C₅₃	#106	1	2	yes	—
158	C₂×C₄×C₅₃	#107	1	2	yes	—
159	C₂×C₄×C₅₃	#107	1	2	yes	—
160	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
170	C₂×Dic₅₃	—	2	8	no	—
171	C₂×Dic₅₃	—	4	16	no	—
172	C₂×Dic₅₃	—	4	4	no	—
173	C₂×Dic₅₃	—	4	4	no	—
174	C₈×C₅₃	—	4	4	no	—
175	#88322	—	4	4	no	—
176	C₂×#1303	—	53	1	no	maximal, Sylow
177	C₂²×C₅₃	—	2	4	no	—
178	C₂²×C₅₃	—	2	4	no	—
179	C₂²×C₅₃	—	2	1	no	—
180	C₂²×C₅₃	—	2	1	no	—
181	C₂²×C₅₃	—	2	1	no	—
182	C₂²×C₅₃	—	2	1	no	—
183	C₂²×C₅₃	#339	1	2	yes	—
184	C₂²×C₅₃	#340	1	2	yes	—
185	C₂²×C₅₃	#340	1	2	yes	—
186	C₂²×C₅₃	#344	1	2	yes	—
187	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
188	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
189	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
190	C₄×C₅₃	—	2	1	no	—
191	C₄×C₅₃	—	2	1	no	—
192	C₄×C₅₃	—	2	1	no	—
193	C₄×C₅₃	—	2	1	no	—
194	C₄×C₅₃	—	2	1	no	—
195	C₄×C₅₃	—	2	1	no	—
196	C₄×C₅₃	#339	1	2	yes	—
197	C₄×C₅₃	#340	1	2	yes	—
198	C₄×C₅₃	#340	1	2	yes	—
199	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
200	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
201	C₄×C₅₃	C₂×#104	1	2	yes	—
202	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
203	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
204	Dic₅₃	—	4	8	no	—
205	C₂×#431	—	53	1	no	—
206	C₂×#423	D₅₃	1	1	yes	characteristic
207	C₂×C₄×C₄⋊C₄	—	53	1	no	—
208	#1303	—	53	4	no	—
209	C₂×C₅₃	—	2	1	no	—
210	C₂×C₅₃	—	2	2	no	—
211	C₂×C₅₃	—	2	1	no	—
212	C₂×C₅₃	#1303	1	4	yes	complemented
213	C₂×C₅₃	C₂×#339	1	1	yes	characteristic
214	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
215	C₂×C₅₃	C₂×#344	1	1	yes	characteristic
216	C₂²×M₁₆	—	2	1	no	—
217	C₂²×C₄⋊C₄	—	106	1	no	—
218	C₂²×C₄²	—	106	1	no	—
219	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
220	#431	—	53	2	no	—
221	#431	—	53	2	no	—
222	#431	—	106	1	no	—
223	#431	—	106	1	no	—
224	#423	—	2	1	no	—
225	#423	—	2	1	no	—
226	#423	Dic₅₃	1	2	yes	complemented
227	#423	C₂×D₅₃	1	2	yes	—
228	C₂×C₄⋊C₈	—	2	1	no	—
229	C₂×C₄⋊C₈	—	53	2	no	—
230	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	complemented
231	C₂×C₂²⋊C₈	—	106	1	no	—
232	C₂×C₄×C₈	—	106	1	no	—
233	C₄×C₄⋊C₄	—	53	2	no	—
234	C₄×C₄⋊C₄	—	53	2	no	—
235	C₄×C₄⋊C₄	—	106	1	no	—
236	C₄×C₄⋊C₄	—	106	1	no	—
237	C₂×#95	—	53	2	no	—
238	C₅₃	C₂×#1303	1	1	yes	characteristic, complemented, Sylow
239	C₂³×C₄	—	106	2	no	—
240	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
241	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
242	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
243	C₂×M₁₆	—	2	2	no	—
244	C₂×M₁₆	—	2	2	no	—
245	C₂×M₁₆	—	4	2	no	—
246	C₂×M₁₆	—	4	2	no	—
247	C₂²×C₈	—	2	2	no	—
248	C₂²×C₈	—	106	2	no	—
249	C₂×C₄⋊C₄	—	106	8	no	—
250	C₂×C₄⋊C₄	—	106	2	no	—
251	C₂×C₄⋊C₄	—	106	2	no	—
252	C₂×C₄²	—	106	2	no	—
253	C₂×C₄²	—	106	2	no	—
254	C₂×C₄²	—	212	2	no	—
255	C₂×C₄²	—	212	2	no	—
256	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
257	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
258	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
259	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
260	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
261	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
262	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
263	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
264	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
265	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
266	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
267	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
268	C₄⋊C₈	—	2	2	no	—
269	C₄⋊C₈	—	2	2	no	—
270	C₄⋊C₈	—	2	2	no	—
271	C₄⋊C₈	—	53	4	no	—
272	C₄⋊C₈	—	106	2	no	—
273	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	complemented
274	C₂²⋊C₈	—	106	2	no	—
275	C₂²⋊C₈	—	106	2	no	—
276	C₄×C₈	—	106	2	no	—
277	C₄×C₈	—	106	2	no	—
278	#95	—	53	4	no	—
279	#95	—	106	2	no	—
280	C₂⁴	#11719921	1	1	yes	characteristic
281	C₂²×C₄	—	2	4	no	—
282	C₂²×C₄	—	4	2	no	—
283	C₂²×C₄	—	106	8	no	—
284	C₂²×C₄	—	106	4	no	—
285	C₂²×C₄	—	212	4	no	—
286	C₂²×C₄	—	212	4	no	—
287	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
288	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
289	C₂²×C₄	#11719919	1	1	yes	characteristic
290	C₂²×C₄	#11719919	1	1	yes	characteristic
291	C₂²×C₄	#11719919	1	1	yes	characteristic
292	C₂²×C₄	#11719919	1	1	yes	characteristic
293	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
294	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
295	C₂²×C₄	#11719921	1	1	yes	characteristic
296	C₂²×C₄	#11719921	1	1	yes	characteristic
297	C₂²×C₄	#11719921	1	1	yes	characteristic
298	C₂²×C₄	#11719921	1	1	yes	characteristic
299	C₂²×C₄	#11719921	1	1	yes	characteristic
300	C₂²×C₄	#11719922	1	2	yes	—
301	C₂²×C₄	#11719923	1	2	yes	—
302	C₂²×C₄	#11719924	1	2	yes	—
303	C₂²×C₄	#11719925	1	2	yes	—
304	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
305	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
306	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
307	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
308	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
309	M₁₆	—	4	4	no	—
310	M₁₆	—	4	2	no	—
311	M₁₆	—	4	2	no	—
312	C₂×C₈	—	2	4	no	—
313	C₂×C₈	—	4	4	no	—
314	C₂×C₈	—	106	4	no	—
315	C₂×C₈	—	212	4	no	—
316	C₄⋊C₄	—	106	8	no	—
317	C₄⋊C₄	—	106	8	no	—
318	C₄²	—	212	4	no	—
319	C₄²	—	212	2	no	—
320	C₄²	—	212	2	no	—
321	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
322	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
323	C₄²	#11719919	1	1	yes	characteristic
324	C₄²	#11719919	1	1	yes	characteristic
325	C₄²	#11719919	1	1	yes	characteristic
326	C₄²	#11719919	1	1	yes	characteristic
327	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
328	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
329	C₄²	#11719921	1	1	yes	characteristic
330	C₄²	#11719921	1	1	yes	characteristic
331	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
332	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
333	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
334	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
335	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
336	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
337	C₂³	—	2	2	no	—
338	C₂³	—	2	2	no	—
339	C₂³	#13133124	1	1	yes	characteristic
340	C₂³	#13133124	1	1	yes	characteristic
341	C₂³	#13133137	1	2	yes	—
342	C₂³	#13133140	1	2	yes	—
343	C₂³	C₂×#11719921	1	1	yes	characteristic
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	4	1	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	4	1	no	—
355	C₂×C₄	—	106	8	no	—
356	C₂×C₄	—	212	4	no	—
357	C₂×C₄	—	212	16	no	—
358	C₂×C₄	—	212	4	no	—
359	C₂×C₄	#13133124	1	1	yes	characteristic
360	C₂×C₄	#13133124	1	1	yes	characteristic
361	C₂×C₄	#13133124	1	1	yes	characteristic
362	C₂×C₄	#13133124	1	1	yes	characteristic
363	C₂×C₄	#13133124	1	1	yes	characteristic
364	C₂×C₄	#13133124	1	1	yes	characteristic
365	C₂×C₄	#13133137	1	2	yes	—
366	C₂×C₄	#13133138	1	2	yes	—
367	C₂×C₄	#13133138	1	2	yes	—
368	C₂×C₄	#13133140	1	2	yes	—
369	C₂×C₄	#13133141	1	2	yes	—
370	C₂×C₄	#13133141	1	2	yes	—
371	C₂×C₄	#13133143	1	4	yes	—
372	C₂×C₄	#13133148	1	2	yes	—
373	C₂×C₄	#13133148	1	2	yes	—
374	C₂×C₄	#13133150	1	2	yes	—
375	C₂×C₄	#13133150	1	2	yes	—
376	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
377	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
378	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
379	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
380	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
381	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
382	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
383	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
384	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
385	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
386	C₈	—	4	4	no	—
387	C₈	—	212	4	no	—
388	C₂²	—	2	1	no	—
389	C₂²	—	2	1	no	—
390	C₂²	—	2	4	no	—
391	C₂²	—	2	4	no	—
392	C₂²	—	2	1	no	—
393	C₂²	—	2	1	no	—
394	C₂²	#16810384	1	2	yes	—
395	C₂²	#16810385	1	2	yes	—
396	C₂²	#16810385	1	2	yes	—
397	C₂²	#16810394	1	2	yes	—
398	C₂²	C₂×#13133124	1	1	yes	characteristic
399	C₂²	C₂×#13133137	1	1	yes	characteristic
400	C₂²	C₂×#13133140	1	1	yes	characteristic
401	C₄	—	2	1	no	—
402	C₄	—	2	1	no	—
403	C₄	—	2	1	no	—
404	C₄	—	2	1	no	—
405	C₄	—	2	1	no	—
406	C₄	—	2	1	no	—
407	C₄	—	212	8	no	—
408	C₄	#16810384	1	2	yes	—
409	C₄	#16810385	1	2	yes	—
410	C₄	#16810385	1	2	yes	—
411	C₄	C₂×#13133138	1	1	yes	characteristic
412	C₄	C₂×#13133141	1	1	yes	characteristic
413	C₄	C₂×#13133143	1	2	yes	—
414	C₄	C₂×#13133148	1	1	yes	characteristic
415	C₄	C₂×#13133150	1	1	yes	characteristic
416	C₂	—	2	1	no	—
417	C₂	—	2	1	no	—
418	C₂	—	2	2	no	—
419	C₂	#20272616	1	4	yes	direct product factor
420	C₂	C₂×#16810384	1	1	yes	characteristic
421	C₂	C₂×#16810385	1	1	yes	characteristic
422	C₂	C₂×#16810394	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#423	C₂
socle	C₂³×C₅₃	#95
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#423	D₅₃
hypocenter	C₅₃	C₂×#1303
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #20272616	C₂	direct (N × H)
9. #16810676	C₄	semidirect
11. C₂×C₄⋊#88322	C₄	semidirect
23. C₅₃×#423	C₄	semidirect
26. C₂×C₄⋊C₈×C₅₃	C₄	semidirect
37. C₄⋊#88322	C₂×C₄	semidirect
67. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
212. C₂×C₅₃	#1303	semidirect
226. #423	Dic₅₃	semidirect
230. C₂×C₄⋊C₈	Dic₅₃	semidirect
238. C₅₃	C₂×#1303	semidirect
273. C₄⋊C₈	C₂×Dic₅₃	semidirect
419. C₂	#20272616	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
C₂×C₄×C₄⋊Dic₅₃	2
#20272616	2
C₂×C₅₃×#423	2
C₂×#1303	53

Subgroups — C2×#20272616 #27274379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272616 #27274379