Subgroups — #27402138

Order 13568.

1116 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810695	C₂	1	2	yes	maximal
2	C₂×#16810647	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275439	C₂	1	2	yes	maximal, complemented
5	#20274900	C₂	1	2	yes	maximal, complemented
6	#20274895	C₂	1	2	yes	maximal, complemented
7	#20274867	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274695	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274695	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274568	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄²×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×#11719917	C₂²	1	2	yes	—
13	C₂³×#88322	C₂²	1	2	yes	—
14	#16810695	—	2	8	no	—
15	#16810647	—	2	4	no	—
16	C₂×C₄⋊#88322	—	2	4	no	—
17	C₂×C₄⋊#88322	C₂²	1	2	yes	—
18	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#88322	C₂²	1	2	yes	—
22	C₂×C₄⋊#88322	C₂²	1	2	yes	—
23	C₂×C₄⋊#88322	C₂²	1	2	yes	—
24	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
26	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
27	C₂×#13133128	C₂²	1	2	yes	—
28	#16810378	C₂²	1	2	yes	—
29	#16810378	C₂²	1	2	yes	—
30	#16810378	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
32	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
33	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
34	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
35	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
40	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
41	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
42	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
43	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
44	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
45	C₄³×C₅₃	C₂²	1	1	yes	characteristic
46	C₂×#11719917	—	2	4	no	—
47	C₂×#11719917	—	2	4	no	—
48	C₂×#11719917	—	4	2	no	—
49	C₂²×#88322	—	2	4	no	—
50	C₂²×#88322	—	2	4	no	—
51	C₂²×#88322	—	4	2	no	—
52	C₂²×#88322	C₂³	1	2	yes	—
53	C₂²×#88322	C₂³	1	2	yes	—
54	C₂²×#88322	C₂³	1	2	yes	—
55	C₂²×#88322	C₂³	1	2	yes	—
56	C₄⋊#88322	—	2	2	no	—
57	C₄⋊#88322	—	2	2	no	—
58	C₄⋊#88322	—	2	4	no	—
59	C₄⋊#88322	—	2	2	no	—
60	C₄⋊#88322	—	2	4	no	—
61	C₄⋊#88322	—	2	4	no	—
62	C₄⋊#88322	—	4	2	no	—
63	C₄⋊#88322	D₄	1	4	yes	complemented
64	C₂²⋊#88322	—	2	4	no	—
65	C₂²⋊#88322	—	2	4	no	—
66	#13133128	—	2	4	no	—
67	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×D₄×C₅₃	—	2	4	no	—
75	C₄×D₄×C₅₃	—	2	4	no	—
76	C₄×D₄×C₅₃	D₄	1	4	yes	—
77	C₄×D₄×C₅₃	Q₈	1	4	yes	—
78	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄²×C₅₃	—	2	2	no	—
88	C₂×C₄²×C₅₃	—	2	2	no	—
89	C₂×C₄²×C₅₃	—	2	2	no	—
90	C₂×C₄²×C₅₃	—	2	2	no	—
91	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
98	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
99	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
104	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
105	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
106	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
107	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
108	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
109	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
110	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×D₄×C₅₃	—	2	2	no	—
113	C₂×D₄×C₅₃	—	2	2	no	—
114	C₂×D₄×C₅₃	C₄⋊C₄	1	4	yes	—
115	C₂²×C₄×C₅₃	—	2	2	no	—
116	C₂²×C₄×C₅₃	—	2	2	no	—
117	C₂²×C₄×C₅₃	—	2	1	no	—
118	C₂²×C₄×C₅₃	—	2	1	no	—
119	C₂²×C₄×C₅₃	—	2	1	no	—
120	C₂²×C₄×C₅₃	—	2	1	no	—
121	C₂²×C₄×C₅₃	—	2	1	no	—
122	C₂²×C₄×C₅₃	—	2	1	no	—
123	C₂²×C₄×C₅₃	—	2	1	no	—
124	C₂²×C₄×C₅₃	—	2	1	no	—
125	C₂²×C₄×C₅₃	—	2	2	no	—
126	C₂²×C₄×C₅₃	—	2	2	no	—
127	C₂²×C₄×C₅₃	—	2	2	no	—
128	C₂²×C₄×C₅₃	—	2	2	no	—
129	C₂²×C₄×C₅₃	—	2	2	no	—
130	C₂²×C₄×C₅₃	—	2	2	no	—
131	C₂²×C₄×C₅₃	—	2	2	no	—
132	C₂²×C₄×C₅₃	—	4	1	no	—
133	C₂²×C₄×C₅₃	—	4	1	no	—
134	C₂²×C₄×C₅₃	M₁₆	1	4	yes	complemented
135	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
148	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
149	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
150	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
151	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
152	C₄⋊C₄×C₅₃	—	2	2	no	—
153	C₄⋊C₄×C₅₃	—	2	2	no	—
154	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
155	C₂²⋊C₄×C₅₃	—	2	2	no	—
156	C₂²⋊C₄×C₅₃	—	2	2	no	—
157	C₂²⋊C₄×C₅₃	—	2	2	no	—
158	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
159	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
160	C₂²⋊C₄×C₅₃	M₁₆	1	4	yes	complemented
161	C₄²×C₅₃	—	2	2	no	—
162	C₄²×C₅₃	—	2	2	no	—
163	C₄²×C₅₃	—	2	2	no	—
164	C₄²×C₅₃	—	2	2	no	—
165	C₄²×C₅₃	—	2	2	no	—
166	C₄²×C₅₃	—	2	2	no	—
167	C₄²×C₅₃	—	2	2	no	—
168	C₄²×C₅₃	—	2	1	no	—
169	C₄²×C₅₃	—	2	1	no	—
170	C₄²×C₅₃	—	2	1	no	—
171	C₄²×C₅₃	—	2	1	no	—
172	C₄²×C₅₃	—	2	1	no	—
173	C₄²×C₅₃	—	2	1	no	—
174	C₄²×C₅₃	—	4	1	no	—
175	C₄²×C₅₃	C₂×D₄	1	2	yes	—
176	C₄²×C₅₃	C₂×D₄	1	2	yes	—
177	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
178	C₄²×C₅₃	#41	1	2	yes	—
179	#11719917	—	4	8	no	—
180	C₂×#88322	—	2	2	no	—
181	C₂×#88322	—	2	2	no	—
182	C₂×#88322	—	2	2	no	—
183	C₂×#88322	—	2	2	no	—
184	C₂×#88322	—	2	2	no	—
185	C₂×#88322	—	2	2	no	—
186	C₂×#88322	—	2	2	no	—
187	C₂×#88322	—	2	2	no	—
188	C₂×#88322	—	2	2	no	—
189	C₂×#88322	—	2	2	no	—
190	C₂×#88322	—	2	2	no	—
191	C₂×#88322	—	4	8	no	—
192	C₂×#88322	C₂×D₄	1	4	yes	—
193	C₂³×C₅₃	—	2	1	no	—
194	C₂³×C₅₃	—	2	1	no	—
195	C₂³×C₅₃	—	2	1	no	—
196	C₂³×C₅₃	—	2	1	no	—
197	C₂³×C₅₃	—	2	1	no	—
198	C₂³×C₅₃	—	2	1	no	—
199	C₂³×C₅₃	—	2	1	no	—
200	C₂³×C₅₃	—	2	1	no	—
201	C₂³×C₅₃	—	2	1	no	—
202	C₂³×C₅₃	—	2	1	no	—
203	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
204	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
205	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
206	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
207	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
208	D₄×C₅₃	—	2	8	no	—
209	C₂×C₄×C₅₃	—	2	1	no	—
210	C₂×C₄×C₅₃	—	2	1	no	—
211	C₂×C₄×C₅₃	—	2	1	no	—
212	C₂×C₄×C₅₃	—	2	1	no	—
213	C₂×C₄×C₅₃	—	2	1	no	—
214	C₂×C₄×C₅₃	—	2	1	no	—
215	C₂×C₄×C₅₃	—	2	1	no	—
216	C₂×C₄×C₅₃	—	2	1	no	—
217	C₂×C₄×C₅₃	—	2	1	no	—
218	C₂×C₄×C₅₃	—	2	1	no	—
219	C₂×C₄×C₅₃	—	2	1	no	—
220	C₂×C₄×C₅₃	—	2	1	no	—
221	C₂×C₄×C₅₃	—	2	1	no	—
222	C₂×C₄×C₅₃	—	2	1	no	—
223	C₂×C₄×C₅₃	—	2	1	no	—
224	C₂×C₄×C₅₃	—	2	1	no	—
225	C₂×C₄×C₅₃	—	2	2	no	—
226	C₂×C₄×C₅₃	—	2	2	no	—
227	C₂×C₄×C₅₃	—	2	2	no	—
228	C₂×C₄×C₅₃	—	2	2	no	—
229	C₂×C₄×C₅₃	—	2	2	no	—
230	C₂×C₄×C₅₃	—	2	2	no	—
231	C₂×C₄×C₅₃	—	2	2	no	—
232	C₂×C₄×C₅₃	—	2	2	no	—
233	C₂×C₄×C₅₃	—	2	8	no	—
234	C₂×C₄×C₅₃	—	2	1	no	—
235	C₂×C₄×C₅₃	—	2	1	no	—
236	C₂×C₄×C₅₃	—	2	1	no	—
237	C₂×C₄×C₅₃	—	2	1	no	—
238	C₂×C₄×C₅₃	—	2	1	no	—
239	C₂×C₄×C₅₃	—	2	1	no	—
240	C₂×C₄×C₅₃	—	2	1	no	—
241	C₂×C₄×C₅₃	—	2	1	no	—
242	C₂×C₄×C₅₃	—	2	1	no	—
243	C₂×C₄×C₅₃	—	4	4	no	—
244	C₂×C₄×C₅₃	—	4	4	no	—
245	C₂×C₄×C₅₃	—	4	4	no	—
246	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
247	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
248	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
249	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
250	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
251	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
252	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
253	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
254	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
255	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
256	C₂×C₄×C₅₃	#131	1	2	yes	—
257	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
260	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
261	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
262	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
263	#88322	—	2	4	no	—
264	#88322	—	4	2	no	—
265	#88322	—	4	2	no	—
266	#88322	—	4	2	no	—
267	#22648	—	53	1	no	maximal, Sylow
268	C₂²×C₅₃	—	2	2	no	—
269	C₂²×C₅₃	—	2	2	no	—
270	C₂²×C₅₃	—	2	2	no	—
271	C₂²×C₅₃	—	2	2	no	—
272	C₂²×C₅₃	—	2	2	no	—
273	C₂²×C₅₃	—	2	2	no	—
274	C₂²×C₅₃	—	2	2	no	—
275	C₂²×C₅₃	—	2	2	no	—
276	C₂²×C₅₃	—	2	2	no	—
277	C₂²×C₅₃	—	4	1	no	—
278	C₂²×C₅₃	—	4	1	no	—
279	C₂²×C₅₃	—	4	1	no	—
280	C₂²×C₅₃	—	4	1	no	—
281	C₂²×C₅₃	#423	1	4	yes	complemented
282	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
283	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
284	C₂²×C₅₃	#520	1	1	yes	characteristic
285	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
286	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
287	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
288	C₂²×C₅₃	#568	1	1	yes	characteristic
289	C₄×C₅₃	—	2	2	no	—
290	C₄×C₅₃	—	2	2	no	—
291	C₄×C₅₃	—	2	2	no	—
292	C₄×C₅₃	—	2	2	no	—
293	C₄×C₅₃	—	2	2	no	—
294	C₄×C₅₃	—	2	2	no	—
295	C₄×C₅₃	—	2	2	no	—
296	C₄×C₅₃	—	2	2	no	—
297	C₄×C₅₃	—	4	1	no	—
298	C₄×C₅₃	—	4	1	no	—
299	C₄×C₅₃	—	4	1	no	—
300	C₄×C₅₃	#435	1	4	yes	complemented
301	C₄×C₅₃	D₄²	1	1	yes	characteristic
302	C₄×C₅₃	#547	1	1	yes	characteristic
303	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
304	C₄×C₅₃	#554	1	1	yes	characteristic
305	C₂×#435	—	53	2	no	—
306	C₂×#423	—	53	1	no	—
307	C₂²×C₄⋊C₈	—	53	1	no	—
308	C₄²×D₄	D₅₃	1	1	yes	characteristic
309	#1940	—	53	2	no	—
310	#1851	—	53	2	no	—
311	#1850	—	53	2	no	—
312	#1842	—	53	1	no	—
313	#1797	—	53	1	no	—
314	#1797	—	53	1	no	—
315	#1767	—	53	1	no	—
316	C₂×C₅₃	—	2	4	no	—
317	C₂×C₅₃	—	4	2	no	—
318	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
319	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
320	C₂×C₅₃	#2906	1	1	yes	characteristic
321	C₂×C₅₃	C₂×#435	1	1	yes	characteristic
322	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
323	C₂×C₅₃	#2989	1	1	yes	characteristic
324	C₂×C₅₃	#2990	1	1	yes	characteristic
325	C₂²×M₁₆	—	53	2	no	—
326	C₂³×C₈	—	53	2	no	—
327	C₂×C₄×D₄	Dic₅₃	1	1	yes	characteristic
328	C₂×C₄×D₄	Dic₅₃	1	1	yes	characteristic
329	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
330	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
331	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
332	#435	—	106	8	no	—
333	#423	—	106	4	no	—
334	C₂×C₄⋊C₈	—	53	1	no	—
335	C₂×C₄⋊C₈	—	53	1	no	—
336	C₂×C₄⋊C₈	—	53	1	no	—
337	C₂×C₄⋊C₈	—	53	1	no	—
338	C₂×C₄⋊C₈	—	53	2	no	—
339	C₂×C₄⋊C₈	—	53	2	no	—
340	C₂×C₄⋊C₈	—	53	2	no	—
341	C₂×C₄⋊C₈	—	53	2	no	—
342	C₂×C₄⋊C₈	—	106	4	no	—
343	C₂×C₂²⋊C₈	—	53	2	no	—
344	C₂×C₂²⋊C₈	—	53	2	no	—
345	C₂×#97	—	53	2	no	—
346	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
347	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
348	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
349	C₄×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
350	C₄×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
351	C₄×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
352	C₄×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
353	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
354	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
355	C₄³	C₂×D₅₃	1	1	yes	characteristic
356	#336	—	53	2	no	—
357	#336	—	53	2	no	—
358	#336	—	53	2	no	—
359	C₅₃	#22648	1	1	yes	characteristic, complemented, Sylow
360	C₂²×D₄	C₂×Dic₅₃	1	1	yes	characteristic
361	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
362	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
363	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
364	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
365	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
366	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
367	C₂×M₁₆	—	106	4	no	—
368	C₂×M₁₆	—	106	4	no	—
369	C₂×M₁₆	—	212	2	no	—
370	C₂²×C₈	—	53	2	no	—
371	C₂²×C₈	—	53	2	no	—
372	C₂²×C₈	—	53	2	no	—
373	C₂²×C₈	—	53	2	no	—
374	C₂²×C₈	—	106	4	no	—
375	C₂²×C₈	—	106	4	no	—
376	C₂²×C₈	—	212	2	no	—
377	C₄×D₄	—	2	4	no	—
378	C₄×D₄	—	2	4	no	—
379	C₄×D₄	Dic₁₀₆	1	4	yes	—
380	C₄×D₄	D₂₁₂	1	4	yes	—
381	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
383	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
384	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
385	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
386	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
387	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
388	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
389	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
390	C₂×C₄²	—	2	2	no	—
391	C₂×C₄²	—	2	2	no	—
392	C₂×C₄²	—	2	2	no	—
393	C₂×C₄²	—	2	2	no	—
394	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
395	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
396	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
397	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
398	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
399	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
400	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
401	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
402	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
403	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
404	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
405	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
406	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
407	C₄⋊C₈	—	53	4	no	—
408	C₄⋊C₈	—	106	4	no	—
409	C₄⋊C₈	—	106	4	no	—
410	C₄⋊C₈	—	106	4	no	—
411	C₄⋊C₈	—	106	2	no	—
412	C₄⋊C₈	—	106	2	no	—
413	C₄⋊C₈	—	106	2	no	—
414	C₄⋊C₈	—	212	2	no	—
415	C₂²⋊C₈	—	106	4	no	—
416	C₂²⋊C₈	—	106	4	no	—
417	#97	—	106	4	no	—
418	#95	C₂×Dic₅₃	1	1	yes	characteristic
419	#95	C₂×Dic₅₃	1	1	yes	characteristic
420	#95	C₂×Dic₅₃	1	1	yes	characteristic
421	#95	C₂×Dic₅₃	1	1	yes	characteristic
422	#95	C₂×Dic₅₃	1	1	yes	characteristic
423	#95	C₂×Dic₅₃	1	1	yes	characteristic
424	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
425	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
426	C₂×D₄	—	2	2	no	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	C₄⋊Dic₅₃	1	4	yes	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	2	2	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	4	1	no	—
447	C₂²×C₄	—	4	1	no	—
448	C₂²×C₄	#11719917	1	4	yes	complemented
449	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
450	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
451	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
452	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
453	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
454	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
455	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
456	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
457	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
458	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
459	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
460	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
461	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
462	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
463	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
464	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
465	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
466	M₁₆	—	212	8	no	—
467	C₂×C₈	—	53	4	no	—
468	C₂×C₈	—	106	2	no	—
469	C₂×C₈	—	106	2	no	—
470	C₂×C₈	—	106	2	no	—
471	C₂×C₈	—	106	2	no	—
472	C₂×C₈	—	106	2	no	—
473	C₂×C₈	—	106	2	no	—
474	C₂×C₈	—	106	2	no	—
475	C₂×C₈	—	106	2	no	—
476	C₂×C₈	—	106	2	no	—
477	C₂×C₈	—	106	2	no	—
478	C₂×C₈	—	106	2	no	—
479	C₂×C₈	—	212	8	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
483	C₂²⋊C₄	—	2	2	no	—
484	C₂²⋊C₄	—	2	2	no	—
485	C₂²⋊C₄	—	2	2	no	—
486	C₂²⋊C₄	#11719917	1	4	yes	complemented
487	C₂²⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
488	C₂²⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
489	C₄²	—	2	2	no	—
490	C₄²	—	2	2	no	—
491	C₄²	—	2	2	no	—
492	C₄²	—	2	2	no	—
493	C₄²	—	2	2	no	—
494	C₄²	—	2	2	no	—
495	C₄²	—	2	2	no	—
496	C₄²	—	2	1	no	—
497	C₄²	—	2	1	no	—
498	C₄²	—	2	1	no	—
499	C₄²	—	2	1	no	—
500	C₄²	—	2	1	no	—
501	C₄²	—	2	1	no	—
502	C₄²	—	4	1	no	—
503	C₄²	C₂×Dic₁₀₆	1	2	yes	—
504	C₄²	C₂×D₂₁₂	1	2	yes	—
505	C₄²	D₄×D₅₃	1	2	yes	—
506	C₄²	#11719947	1	2	yes	—
507	C₂³	—	2	1	no	—
508	C₂³	—	2	1	no	—
509	C₂³	—	2	1	no	—
510	C₂³	—	2	1	no	—
511	C₂³	—	2	1	no	—
512	C₂³	—	2	1	no	—
513	C₂³	—	2	1	no	—
514	C₂³	—	2	1	no	—
515	C₂³	—	2	1	no	—
516	C₂³	—	2	1	no	—
517	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
518	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
519	C₂³	C₂×#11719917	1	2	yes	—
520	C₂³	C₂×#11719917	1	2	yes	—
521	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
522	D₄	—	2	8	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	8	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	4	4	no	—
558	C₂×C₄	—	4	4	no	—
559	C₂×C₄	—	4	4	no	—
560	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
561	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
562	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
563	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
564	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
565	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
566	C₂×C₄	C₂×#11719917	1	2	yes	—
567	C₂×C₄	C₂×#11719917	1	2	yes	—
568	C₂×C₄	C₂×#11719917	1	2	yes	—
569	C₂×C₄	C₂×#11719917	1	2	yes	—
570	C₂×C₄	#13133224	1	2	yes	—
571	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
572	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
573	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
574	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
575	C₂×C₄	#13133263	1	1	yes	characteristic
576	C₂×C₄	#13133264	1	1	yes	characteristic
577	C₈	—	106	4	no	—
578	C₈	—	212	2	no	—
579	C₈	—	212	2	no	—
580	C₈	—	212	2	no	—
581	C₂²	—	2	2	no	—
582	C₂²	—	2	2	no	—
583	C₂²	—	2	2	no	—
584	C₂²	—	2	2	no	—
585	C₂²	—	2	2	no	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	2	no	—
588	C₂²	—	2	2	no	—
589	C₂²	—	2	2	no	—
590	C₂²	—	4	1	no	—
591	C₂²	—	4	1	no	—
592	C₂²	—	4	1	no	—
593	C₂²	—	4	1	no	—
594	C₂²	#16810647	1	4	yes	complemented
595	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
596	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
597	C₂²	#16811106	1	1	yes	characteristic
598	C₂²	C₂²×#11719917	1	1	yes	characteristic
599	C₂²	C₂²×#11719917	1	1	yes	characteristic
600	C₂²	C₂×#13133224	1	1	yes	characteristic
601	C₂²	#16811390	1	1	yes	characteristic
602	C₄	—	2	2	no	—
603	C₄	—	2	2	no	—
604	C₄	—	2	2	no	—
605	C₄	—	2	2	no	—
606	C₄	—	2	2	no	—
607	C₄	—	2	2	no	—
608	C₄	—	2	2	no	—
609	C₄	—	2	2	no	—
610	C₄	—	4	1	no	—
611	C₄	—	4	1	no	—
612	C₄	—	4	1	no	—
613	C₄	#16810695	1	4	yes	complemented
614	C₄	D₄×D₂₁₂	1	1	yes	characteristic
615	C₄	#16811277	1	1	yes	characteristic
616	C₄	D₄×Dic₁₀₆	1	1	yes	characteristic
617	C₄	#16811330	1	1	yes	characteristic
618	C₂	—	2	4	no	—
619	C₂	—	4	2	no	—
620	C₂	D₄×C₄⋊Dic₅₃	1	1	yes	characteristic
621	C₂	C₂×#16810647	1	1	yes	characteristic
622	C₂	#20283283	1	1	yes	characteristic
623	C₂	C₂×#16810695	1	1	yes	characteristic
624	C₂	D₄×#11719917	1	1	yes	characteristic
625	C₂	#20283832	1	1	yes	characteristic
626	C₂	#20283840	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₄²×D₄×C₅₃	C₂
prime radical	C₂²×D₄×C₅₃	C₂×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄²×D₄	D₅₃
hypocenter	C₅₃	#22648
LCS	G ▷ C₂³×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. C₂×#16810647	C₂	semidirect
3. C₂²×C₄⋊#88322	C₂	semidirect
4. #20275439	C₂	semidirect
5. #20274900	C₂	semidirect
6. #20274895	C₂	semidirect
7. #20274867	C₂	semidirect
8. #20274695	C₂	semidirect
9. #20274695	C₂	semidirect
10. #20274568	C₂	semidirect
63. C₄⋊#88322	D₄	semidirect
134. C₂²×C₄×C₅₃	M₁₆	semidirect
160. C₂²⋊C₄×C₅₃	M₁₆	semidirect
281. C₂²×C₅₃	#423	semidirect
300. C₄×C₅₃	#435	semidirect
359. C₅₃	#22648	semidirect
448. C₂²×C₄	#11719917	semidirect
486. C₂²⋊C₄	#11719917	semidirect
594. C₂²	#16810647	semidirect
613. C₄	#16810695	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810695	2
C₂×#16810647	2
C₂²×C₄⋊#88322	2
#20275439	2
#20274900	2
#20274895	2
#20274867	2
#20274695	2
#20274568	2
C₄²×D₄×C₅₃	2
#22648	53