Subgroups — C₂×D₄×C₂²⋊C₄×C₅₃ #27241610

Order 13568.

8774 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₄×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₂³×C₂²⋊C₄×C₅₃	C₂	1	2	yes	maximal, complemented
4	D₄×C₂²⋊C₄×C₅₃	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₅₃×#390	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#386	C₂	1	4	yes	maximal, complemented
7	C₂×C₅₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#379	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁶×C₅₃	C₂²	1	2	yes	—
11	C₂³×D₄×C₅₃	—	2	16	no	—
12	C₂³×D₄×C₅₃	C₄	1	16	yes	complemented
13	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
14	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂³×D₄×C₅₃	C₂²	1	8	yes	—
16	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₅₃	C₂²	1	4	yes	complemented
19	C₂×C₄×D₄×C₅₃	—	2	16	no	—
20	C₂×C₄×D₄×C₅₃	C₂²	1	16	yes	complemented
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
22	C₂²×C₂²⋊C₄×C₅₃	—	2	16	no	—
23	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	16	yes	complemented
27	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
28	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
29	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
30	C₂²×C₄²×C₅₃	C₂²	1	2	yes	complemented
31	C₅₃×#390	C₂²	1	16	yes	complemented
32	C₅₃×#386	C₂²	1	32	yes	complemented
33	C₅₃×#380	C₂²	1	8	yes	complemented
34	C₅₃×#379	C₂²	1	16	yes	complemented
35	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	complemented
36	C₂×#95×C₅₃	C₂²	1	2	yes	complemented
37	C₂×#95×C₅₃	C₂²	1	2	yes	complemented
38	C₂⁵×C₅₃	—	2	8	no	—
39	C₂⁵×C₅₃	—	2	16	no	—
40	C₂⁵×C₅₃	—	2	8	no	—
41	C₂⁵×C₅₃	—	4	8	no	—
42	C₂⁵×C₅₃	C₂×C₄	1	16	yes	complemented
43	C₂⁵×C₅₃	C₂³	1	2	yes	—
44	C₂⁵×C₅₃	C₂³	1	8	yes	—
45	C₂⁵×C₅₃	C₂³	1	2	yes	—
46	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₅₃	—	2	16	no	—
48	C₂²×D₄×C₅₃	—	2	128	no	—
49	C₂²×D₄×C₅₃	—	2	64	no	—
50	C₂²×D₄×C₅₃	—	2	8	no	—
51	C₂²×D₄×C₅₃	C₂×C₄	1	8	yes	—
52	C₂²×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
53	C₂²×D₄×C₅₃	C₂×C₄	1	8	yes	complemented
54	C₂²×D₄×C₅₃	C₂×C₄	1	64	yes	complemented
55	C₂²×D₄×C₅₃	D₄	1	16	yes	—
56	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
57	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
58	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
59	C₂³×C₄×C₅₃	—	2	8	no	—
60	C₂³×C₄×C₅₃	—	2	16	no	—
61	C₂³×C₄×C₅₃	—	2	16	no	—
62	C₂³×C₄×C₅₃	—	4	8	no	—
63	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	complemented
64	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
65	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
66	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
68	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
69	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₅₃	C₂³	1	16	yes	complemented
71	C₄×D₄×C₅₃	—	2	64	no	—
72	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
73	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	complemented
74	C₂×C₂²⋊C₄×C₅₃	—	2	64	no	—
75	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
76	C₂×C₂²⋊C₄×C₅₃	—	2	16	no	—
77	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
78	C₂×C₂²⋊C₄×C₅₃	—	2	16	no	—
79	C₂×C₂²⋊C₄×C₅₃	—	4	8	no	—
80	C₂×C₂²⋊C₄×C₅₃	D₄	1	8	yes	direct product factor
81	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
82	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
83	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
84	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	complemented
85	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	complemented
86	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	complemented
87	C₂×C₄²×C₅₃	—	2	8	no	—
88	C₂×C₄²×C₅₃	C₂³	1	8	yes	complemented
89	#95×C₅₃	C₂³	1	8	yes	complemented
90	#95×C₅₃	C₂³	1	8	yes	complemented
91	C₂⁴×C₅₃	—	2	4	no	—
92	C₂⁴×C₅₃	—	2	4	no	—
93	C₂⁴×C₅₃	—	2	4	no	—
94	C₂⁴×C₅₃	—	2	8	no	—
95	C₂⁴×C₅₃	—	2	4	no	—
96	C₂⁴×C₅₃	—	2	64	no	—
97	C₂⁴×C₅₃	—	2	64	no	—
98	C₂⁴×C₅₃	—	2	16	no	—
99	C₂⁴×C₅₃	—	2	4	no	—
100	C₂⁴×C₅₃	—	2	32	no	—
101	C₂⁴×C₅₃	—	2	8	no	—
102	C₂⁴×C₅₃	—	2	16	no	—
103	C₂⁴×C₅₃	—	2	4	no	—
104	C₂⁴×C₅₃	—	2	8	no	—
105	C₂⁴×C₅₃	—	4	32	no	—
106	C₂⁴×C₅₃	—	4	16	no	—
107	C₂⁴×C₅₃	—	4	64	no	—
108	C₂⁴×C₅₃	—	4	64	no	—
109	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	—
110	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	—
111	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
112	C₂⁴×C₅₃	C₂²×C₄	1	32	yes	complemented
113	C₂⁴×C₅₃	C₂×D₄	1	16	yes	—
114	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
115	C₂⁴×C₅₃	#41	1	4	yes	—
116	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
117	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
118	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
119	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
120	C₂×D₄×C₅₃	—	2	64	no	—
121	C₂×D₄×C₅₃	—	2	32	no	—
122	C₂×D₄×C₅₃	—	2	256	no	—
123	C₂×D₄×C₅₃	—	2	32	no	—
124	C₂×D₄×C₅₃	—	2	16	no	—
125	C₂×D₄×C₅₃	C₂²⋊C₄	1	64	yes	direct product factor
126	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	—
127	C₂×D₄×C₅₃	C₂²×C₄	1	32	yes	complemented
128	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	complemented
129	C₂×D₄×C₅₃	C₂×D₄	1	32	yes	—
130	C₂²×C₄×C₅₃	—	2	4	no	—
131	C₂²×C₄×C₅₃	—	2	4	no	—
132	C₂²×C₄×C₅₃	—	2	4	no	—
133	C₂²×C₄×C₅₃	—	2	32	no	—
134	C₂²×C₄×C₅₃	—	2	16	no	—
135	C₂²×C₄×C₅₃	—	2	4	no	—
136	C₂²×C₄×C₅₃	—	2	8	no	—
137	C₂²×C₄×C₅₃	—	2	8	no	—
138	C₂²×C₄×C₅₃	—	2	32	no	—
139	C₂²×C₄×C₅₃	—	2	4	no	—
140	C₂²×C₄×C₅₃	—	2	4	no	—
141	C₂²×C₄×C₅₃	—	2	8	no	—
142	C₂²×C₄×C₅₃	—	2	32	no	—
143	C₂²×C₄×C₅₃	—	4	16	no	—
144	C₂²×C₄×C₅₃	—	4	64	no	—
145	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
146	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
147	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	complemented
148	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
149	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	complemented
150	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
151	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
152	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
155	C₄⋊C₄×C₅₃	—	2	16	no	—
156	C₂²⋊C₄×C₅₃	—	2	16	no	—
157	C₂²⋊C₄×C₅₃	—	2	32	no	—
158	C₂²⋊C₄×C₅₃	—	2	8	no	—
159	C₂²⋊C₄×C₅₃	—	2	32	no	—
160	C₂²⋊C₄×C₅₃	—	4	16	no	—
161	C₂²⋊C₄×C₅₃	C₂×D₄	1	16	yes	direct product factor
162	C₄²×C₅₃	—	2	16	no	—
163	C₂³×C₅₃	—	2	8	no	—
164	C₂³×C₅₃	—	2	16	no	—
165	C₂³×C₅₃	—	2	8	no	—
166	C₂³×C₅₃	—	2	16	no	—
167	C₂³×C₅₃	—	2	32	no	—
168	C₂³×C₅₃	—	2	16	no	—
169	C₂³×C₅₃	—	2	16	no	—
170	C₂³×C₅₃	—	2	32	no	—
171	C₂³×C₅₃	—	2	4	no	—
172	C₂³×C₅₃	—	2	4	no	—
173	C₂³×C₅₃	—	2	4	no	—
174	C₂³×C₅₃	—	2	8	no	—
175	C₂³×C₅₃	—	2	4	no	—
176	C₂³×C₅₃	—	2	16	no	—
177	C₂³×C₅₃	—	2	64	no	—
178	C₂³×C₅₃	—	2	16	no	—
179	C₂³×C₅₃	—	2	64	no	—
180	C₂³×C₅₃	—	2	16	no	—
181	C₂³×C₅₃	—	2	8	no	—
182	C₂³×C₅₃	—	2	4	no	—
183	C₂³×C₅₃	—	2	4	no	—
184	C₂³×C₅₃	—	2	8	no	—
185	C₂³×C₅₃	—	4	32	no	—
186	C₂³×C₅₃	—	4	256	no	—
187	C₂³×C₅₃	—	4	64	no	—
188	C₂³×C₅₃	—	4	64	no	—
189	C₂³×C₅₃	—	4	16	no	—
190	C₂³×C₅₃	—	4	16	no	—
191	C₂³×C₅₃	C₂×C₂²⋊C₄	1	32	yes	complemented
192	C₂³×C₅₃	C₄×D₄	1	16	yes	complemented
193	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
194	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
195	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
196	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
197	C₂³×C₅₃	C₂²×D₄	1	16	yes	—
198	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
199	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
200	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
201	C₂³×C₅₃	C₂×#41	1	4	yes	—
202	C₂³×C₅₃	C₂×#41	1	2	yes	—
203	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
204	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
205	D₄×C₅₃	—	2	64	no	—
206	D₄×C₅₃	—	2	32	no	—
207	D₄×C₅₃	C₂×C₂²⋊C₄	1	64	yes	direct product factor
208	C₂×C₄×C₅₃	—	2	8	no	—
209	C₂×C₄×C₅₃	—	2	16	no	—
210	C₂×C₄×C₅₃	—	2	4	no	—
211	C₂×C₄×C₅₃	—	2	4	no	—
212	C₂×C₄×C₅₃	—	2	4	no	—
213	C₂×C₄×C₅₃	—	2	32	no	—
214	C₂×C₄×C₅₃	—	2	8	no	—
215	C₂×C₄×C₅₃	—	2	8	no	—
216	C₂×C₄×C₅₃	—	2	4	no	—
217	C₂×C₄×C₅₃	—	2	4	no	—
218	C₂×C₄×C₅₃	—	2	8	no	—
219	C₂×C₄×C₅₃	—	2	4	no	—
220	C₂×C₄×C₅₃	—	4	64	no	—
221	C₂×C₄×C₅₃	—	4	16	no	—
222	C₂×C₄×C₅₃	—	4	8	no	—
223	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	16	yes	complemented
224	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
225	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
226	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
227	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	complemented
228	C₂×D₄×C₂²⋊C₄	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
229	C₂²×C₅₃	—	2	32	no	—
230	C₂²×C₅₃	—	2	64	no	—
231	C₂²×C₅₃	—	2	4	no	—
232	C₂²×C₅₃	—	2	8	no	—
233	C₂²×C₅₃	—	2	4	no	—
234	C₂²×C₅₃	—	2	8	no	—
235	C₂²×C₅₃	—	2	16	no	—
236	C₂²×C₅₃	—	2	8	no	—
237	C₂²×C₅₃	—	2	8	no	—
238	C₂²×C₅₃	—	2	16	no	—
239	C₂²×C₅₃	—	2	8	no	—
240	C₂²×C₅₃	—	2	8	no	—
241	C₂²×C₅₃	—	2	8	no	—
242	C₂²×C₅₃	—	4	128	no	—
243	C₂²×C₅₃	—	4	32	no	—
244	C₂²×C₅₃	—	4	8	no	—
245	C₂²×C₅₃	—	4	8	no	—
246	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	4	yes	—
247	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	16	yes	complemented
248	C₂²×C₅₃	C₂×C₄×D₄	1	4	yes	—
249	C₂²×C₅₃	C₂×C₄×D₄	1	8	yes	complemented
250	C₂²×C₅₃	#518	1	4	yes	—
251	C₂²×C₅₃	D₄²	1	8	yes	—
252	C₂²×C₅₃	#546	1	8	yes	—
253	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
254	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
255	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
256	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
257	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
258	C₄×C₅₃	—	2	4	no	—
259	C₄×C₅₃	—	2	8	no	—
260	C₄×C₅₃	—	4	8	no	—
261	C₄×C₅₃	—	4	4	no	—
262	C₄×C₅₃	C₂²×C₂²⋊C₄	1	8	yes	complemented
263	C₂⁴×D₄	C₂×C₅₃	1	1	yes	characteristic
264	C₂²×C₄×D₄	C₂×C₅₃	1	2	yes	complemented
265	C₂³×C₂²⋊C₄	C₂×C₅₃	1	2	yes	complemented
266	D₄×C₂²⋊C₄	C₂×C₅₃	1	16	yes	direct product factor
267	C₂×#390	C₂×C₅₃	1	2	yes	complemented
268	C₂×#386	C₂×C₅₃	1	4	yes	complemented
269	C₂×#380	C₂×C₅₃	1	1	yes	characteristic, complemented
270	C₂×#379	C₂×C₅₃	1	2	yes	complemented
271	C₂×C₄×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
272	C₂×C₅₃	—	2	8	no	—
273	C₂×C₅₃	—	2	16	no	—
274	C₂×C₅₃	—	4	8	no	—
275	C₂×C₅₃	D₄×C₂²⋊C₄	1	8	yes	direct product factor
276	C₂×C₅₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
277	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
278	C₂×C₅₃	C₂×#518	1	1	yes	characteristic
279	C₂×C₅₃	C₂×D₄²	1	2	yes	—
280	C₂×C₅₃	C₂×#546	1	2	yes	—
281	C₂⁶	C₂²×C₅₃	1	2	yes	—
282	C₂³×D₄	—	2	16	no	—
283	C₂³×D₄	C₄×C₅₃	1	16	yes	complemented
284	C₂³×D₄	C₂²×C₅₃	1	8	yes	—
285	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
286	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
287	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
288	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
289	C₂⁴×C₄	C₂²×C₅₃	1	4	yes	complemented
290	C₂×C₄×D₄	—	2	16	no	—
291	C₂×C₄×D₄	C₂²×C₅₃	1	16	yes	complemented
292	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
293	C₂²×C₂²⋊C₄	—	2	16	no	—
294	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
295	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
296	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
297	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	16	yes	complemented
298	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
299	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
300	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
301	C₂²×C₄²	C₂²×C₅₃	1	2	yes	complemented
302	#390	C₂²×C₅₃	1	16	yes	complemented
303	#386	C₂²×C₅₃	1	32	yes	complemented
304	#380	C₂²×C₅₃	1	8	yes	complemented
305	#379	C₂²×C₅₃	1	16	yes	complemented
306	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	complemented
307	C₂×#95	C₂²×C₅₃	1	2	yes	complemented
308	C₂×#95	C₂²×C₅₃	1	2	yes	complemented
309	C₅₃	C₂×D₄×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
310	C₂⁵	—	2	8	no	—
311	C₂⁵	—	2	16	no	—
312	C₂⁵	—	2	8	no	—
313	C₂⁵	—	4	8	no	—
314	C₂⁵	C₂×C₄×C₅₃	1	16	yes	complemented
315	C₂⁵	C₂³×C₅₃	1	2	yes	—
316	C₂⁵	C₂³×C₅₃	1	8	yes	—
317	C₂⁵	C₂³×C₅₃	1	2	yes	—
318	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
319	C₂²×D₄	—	2	16	no	—
320	C₂²×D₄	—	2	64	no	—
321	C₂²×D₄	—	2	128	no	—
322	C₂²×D₄	—	2	8	no	—
323	C₂²×D₄	C₂×C₄×C₅₃	1	8	yes	—
324	C₂²×D₄	C₂×C₄×C₅₃	1	16	yes	complemented
325	C₂²×D₄	C₂×C₄×C₅₃	1	8	yes	complemented
326	C₂²×D₄	C₂×C₄×C₅₃	1	64	yes	complemented
327	C₂²×D₄	D₄×C₅₃	1	16	yes	—
328	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
329	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
330	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
331	C₂³×C₄	—	2	8	no	—
332	C₂³×C₄	—	2	16	no	—
333	C₂³×C₄	—	2	16	no	—
334	C₂³×C₄	—	4	8	no	—
335	C₂³×C₄	C₂×C₄×C₅₃	1	8	yes	complemented
336	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
337	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
338	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
339	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
340	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
341	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
342	C₂³×C₄	C₂³×C₅₃	1	16	yes	complemented
343	C₄×D₄	—	2	64	no	—
344	C₂×C₄⋊C₄	—	2	8	no	—
345	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	complemented
346	C₂×C₂²⋊C₄	—	2	8	no	—
347	C₂×C₂²⋊C₄	—	2	16	no	—
348	C₂×C₂²⋊C₄	—	2	4	no	—
349	C₂×C₂²⋊C₄	—	2	16	no	—
350	C₂×C₂²⋊C₄	—	2	64	no	—
351	C₂×C₂²⋊C₄	—	4	8	no	—
352	C₂×C₂²⋊C₄	D₄×C₅₃	1	8	yes	direct product factor
353	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	complemented
357	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	complemented
358	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	complemented
359	C₂×C₄²	—	2	8	no	—
360	C₂×C₄²	C₂³×C₅₃	1	8	yes	complemented
361	#95	C₂³×C₅₃	1	8	yes	complemented
362	#95	C₂³×C₅₃	1	8	yes	complemented
363	C₂⁴	—	2	4	no	—
364	C₂⁴	—	2	4	no	—
365	C₂⁴	—	2	4	no	—
366	C₂⁴	—	2	8	no	—
367	C₂⁴	—	2	4	no	—
368	C₂⁴	—	2	64	no	—
369	C₂⁴	—	2	64	no	—
370	C₂⁴	—	2	16	no	—
371	C₂⁴	—	2	4	no	—
372	C₂⁴	—	2	32	no	—
373	C₂⁴	—	2	8	no	—
374	C₂⁴	—	2	16	no	—
375	C₂⁴	—	2	4	no	—
376	C₂⁴	—	2	8	no	—
377	C₂⁴	—	4	32	no	—
378	C₂⁴	—	4	16	no	—
379	C₂⁴	—	4	64	no	—
380	C₂⁴	—	4	64	no	—
381	C₂⁴	C₂²×C₄×C₅₃	1	8	yes	—
382	C₂⁴	C₂²×C₄×C₅₃	1	8	yes	—
383	C₂⁴	C₂²×C₄×C₅₃	1	4	yes	—
384	C₂⁴	C₂²×C₄×C₅₃	1	32	yes	complemented
385	C₂⁴	C₂×D₄×C₅₃	1	16	yes	—
386	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
387	C₂⁴	#41×C₅₃	1	4	yes	—
388	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
389	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
390	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
391	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
392	C₂×D₄	—	2	64	no	—
393	C₂×D₄	—	2	32	no	—
394	C₂×D₄	—	2	256	no	—
395	C₂×D₄	—	2	32	no	—
396	C₂×D₄	—	2	16	no	—
397	C₂×D₄	C₂²⋊C₄×C₅₃	1	64	yes	direct product factor
398	C₂×D₄	C₂²×C₄×C₅₃	1	16	yes	—
399	C₂×D₄	C₂²×C₄×C₅₃	1	32	yes	complemented
400	C₂×D₄	C₂²×C₄×C₅₃	1	16	yes	complemented
401	C₂×D₄	C₂×D₄×C₅₃	1	32	yes	—
402	C₂²×C₄	—	2	4	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	32	no	—
406	C₂²×C₄	—	2	16	no	—
407	C₂²×C₄	—	2	4	no	—
408	C₂²×C₄	—	2	8	no	—
409	C₂²×C₄	—	2	8	no	—
410	C₂²×C₄	—	2	32	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	8	no	—
414	C₂²×C₄	—	2	32	no	—
415	C₂²×C₄	—	4	16	no	—
416	C₂²×C₄	—	4	64	no	—
417	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
418	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
419	C₂²×C₄	C₂²×C₄×C₅₃	1	16	yes	complemented
420	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
421	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	complemented
422	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
423	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
424	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
425	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
426	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
427	C₄⋊C₄	—	2	16	no	—
428	C₂²⋊C₄	—	2	16	no	—
429	C₂²⋊C₄	—	2	32	no	—
430	C₂²⋊C₄	—	2	8	no	—
431	C₂²⋊C₄	—	2	32	no	—
432	C₂²⋊C₄	—	4	16	no	—
433	C₂²⋊C₄	C₂×D₄×C₅₃	1	16	yes	direct product factor
434	C₄²	—	2	16	no	—
435	C₂³	—	2	8	no	—
436	C₂³	—	2	16	no	—
437	C₂³	—	2	8	no	—
438	C₂³	—	2	16	no	—
439	C₂³	—	2	32	no	—
440	C₂³	—	2	16	no	—
441	C₂³	—	2	16	no	—
442	C₂³	—	2	32	no	—
443	C₂³	—	2	4	no	—
444	C₂³	—	2	4	no	—
445	C₂³	—	2	4	no	—
446	C₂³	—	2	8	no	—
447	C₂³	—	2	4	no	—
448	C₂³	—	2	16	no	—
449	C₂³	—	2	64	no	—
450	C₂³	—	2	16	no	—
451	C₂³	—	2	64	no	—
452	C₂³	—	2	16	no	—
453	C₂³	—	2	4	no	—
454	C₂³	—	2	8	no	—
455	C₂³	—	2	4	no	—
456	C₂³	—	2	8	no	—
457	C₂³	—	4	32	no	—
458	C₂³	—	4	64	no	—
459	C₂³	—	4	64	no	—
460	C₂³	—	4	256	no	—
461	C₂³	—	4	16	no	—
462	C₂³	—	4	16	no	—
463	C₂³	C₂×C₂²⋊C₄×C₅₃	1	32	yes	complemented
464	C₂³	C₄×D₄×C₅₃	1	16	yes	complemented
465	C₂³	C₂³×C₄×C₅₃	1	8	yes	—
466	C₂³	C₂³×C₄×C₅₃	1	8	yes	—
467	C₂³	C₂³×C₄×C₅₃	1	4	yes	—
468	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
469	C₂³	C₂²×D₄×C₅₃	1	16	yes	—
470	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
471	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
472	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
473	C₂³	C₂×#41×C₅₃	1	4	yes	—
474	C₂³	C₂×#41×C₅₃	1	2	yes	—
475	C₂³	H(5,𝔽₂)×C₅₃	1	4	yes	—
476	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
477	D₄	—	2	64	no	—
478	D₄	—	2	32	no	—
479	D₄	C₂×C₂²⋊C₄×C₅₃	1	64	yes	direct product factor
480	C₂×C₄	—	2	8	no	—
481	C₂×C₄	—	2	16	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	32	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	8	no	—
488	C₂×C₄	—	2	8	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	2	8	no	—
492	C₂×C₄	—	4	64	no	—
493	C₂×C₄	—	4	16	no	—
494	C₂×C₄	—	4	8	no	—
495	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	16	yes	complemented
496	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
497	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
498	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
499	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	complemented
500	C₂²	—	2	16	no	—
501	C₂²	—	2	8	no	—
502	C₂²	—	2	8	no	—
503	C₂²	—	2	16	no	—
504	C₂²	—	2	32	no	—
505	C₂²	—	2	64	no	—
506	C₂²	—	2	4	no	—
507	C₂²	—	2	8	no	—
508	C₂²	—	2	4	no	—
509	C₂²	—	2	8	no	—
510	C₂²	—	2	8	no	—
511	C₂²	—	2	8	no	—
512	C₂²	—	2	8	no	—
513	C₂²	—	4	128	no	—
514	C₂²	—	4	8	no	—
515	C₂²	—	4	8	no	—
516	C₂²	—	4	32	no	—
517	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	4	yes	—
518	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	16	yes	complemented
519	C₂²	C₂×C₄×D₄×C₅₃	1	4	yes	—
520	C₂²	C₂×C₄×D₄×C₅₃	1	8	yes	complemented
521	C₂²	C₅₃×#518	1	4	yes	—
522	C₂²	D₄²×C₅₃	1	8	yes	—
523	C₂²	C₅₃×#546	1	8	yes	—
524	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
525	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
526	C₂²	C₂³×D₄×C₅₃	1	2	yes	—
527	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
528	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
529	C₄	—	2	8	no	—
530	C₄	—	2	4	no	—
531	C₄	—	4	8	no	—
532	C₄	—	4	4	no	—
533	C₄	C₂²×C₂²⋊C₄×C₅₃	1	8	yes	complemented
534	C₂	—	2	8	no	—
535	C₂	—	2	16	no	—
536	C₂	—	4	8	no	—
537	C₂	D₄×C₂²⋊C₄×C₅₃	1	8	yes	direct product factor
538	C₂	C₂³×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
539	C₂	C₂²×C₄×D₄×C₅₃	1	1	yes	characteristic
540	C₂	C₂×C₅₃×#518	1	1	yes	characteristic
541	C₂	C₂×D₄²×C₅₃	1	2	yes	—
542	C₂	C₂×C₅₃×#546	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂⁴×D₄×C₅₃	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₄×C₅₃	C₂	semidirect
3. C₂³×C₂²⋊C₄×C₅₃	C₂	semidirect
4. D₄×C₂²⋊C₄×C₅₃	C₂	direct (N × H)
5. C₂×C₅₃×#390	C₂	semidirect
6. C₂×C₅₃×#386	C₂	semidirect
7. C₂×C₅₃×#380	C₂	semidirect
8. C₂×C₅₃×#379	C₂	semidirect
9. C₂×C₄×C₂²⋊C₄×C₅₃	C₂	semidirect
12. C₂³×D₄×C₅₃	C₄	semidirect
18. C₂⁴×C₄×C₅₃	C₂²	semidirect
20. C₂×C₄×D₄×C₅₃	C₂²	semidirect
21. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
26. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
27. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
28. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
29. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
30. C₂²×C₄²×C₅₃	C₂²	semidirect
31. C₅₃×#390	C₂²	semidirect
32. C₅₃×#386	C₂²	semidirect
33. C₅₃×#380	C₂²	semidirect
34. C₅₃×#379	C₂²	semidirect
35. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
36. C₂×#95×C₅₃	C₂²	semidirect
37. C₂×#95×C₅₃	C₂²	semidirect
42. C₂⁵×C₅₃	C₂×C₄	semidirect
52. C₂²×D₄×C₅₃	C₂×C₄	semidirect
53. C₂²×D₄×C₅₃	C₂×C₄	semidirect
54. C₂²×D₄×C₅₃	C₂×C₄	semidirect
63. C₂³×C₄×C₅₃	C₂×C₄	semidirect
70. C₂³×C₄×C₅₃	C₂³	semidirect
73. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
80. C₂×C₂²⋊C₄×C₅₃	D₄	direct (N × H)
84. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
85. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
86. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
88. C₂×C₄²×C₅₃	C₂³	semidirect
89. #95×C₅₃	C₂³	semidirect
90. #95×C₅₃	C₂³	semidirect
112. C₂⁴×C₅₃	C₂²×C₄	semidirect
125. C₂×D₄×C₅₃	C₂²⋊C₄	direct (N × H)
127. C₂×D₄×C₅₃	C₂²×C₄	semidirect
128. C₂×D₄×C₅₃	C₂²×C₄	semidirect
147. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
149. C₂²×C₄×C₅₃	C₂×D₄	semidirect
161. C₂²⋊C₄×C₅₃	C₂×D₄	direct (N × H)
191. C₂³×C₅₃	C₂×C₂²⋊C₄	semidirect
192. C₂³×C₅₃	C₄×D₄	semidirect
207. D₄×C₅₃	C₂×C₂²⋊C₄	direct (N × H)
223. C₂×C₄×C₅₃	C₂×C₂²⋊C₄	semidirect
227. C₂×C₄×C₅₃	C₂²×D₄	semidirect
228. C₂×D₄×C₂²⋊C₄	C₅₃	direct (N × H)
247. C₂²×C₅₃	C₂²×C₂²⋊C₄	semidirect
249. C₂²×C₅₃	C₂×C₄×D₄	semidirect
262. C₄×C₅₃	C₂²×C₂²⋊C₄	semidirect
264. C₂²×C₄×D₄	C₂×C₅₃	semidirect
265. C₂³×C₂²⋊C₄	C₂×C₅₃	semidirect
266. D₄×C₂²⋊C₄	C₂×C₅₃	direct (N × H)
267. C₂×#390	C₂×C₅₃	semidirect
268. C₂×#386	C₂×C₅₃	semidirect
269. C₂×#380	C₂×C₅₃	semidirect
270. C₂×#379	C₂×C₅₃	semidirect
271. C₂×C₄×C₂²⋊C₄	C₂×C₅₃	semidirect
275. C₂×C₅₃	D₄×C₂²⋊C₄	direct (N × H)
283. C₂³×D₄	C₄×C₅₃	semidirect
289. C₂⁴×C₄	C₂²×C₅₃	semidirect
291. C₂×C₄×D₄	C₂²×C₅₃	semidirect
292. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
297. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
298. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
299. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
300. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
301. C₂²×C₄²	C₂²×C₅₃	semidirect
302. #390	C₂²×C₅₃	semidirect
303. #386	C₂²×C₅₃	semidirect
304. #380	C₂²×C₅₃	semidirect
305. #379	C₂²×C₅₃	semidirect
306. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
307. C₂×#95	C₂²×C₅₃	semidirect
308. C₂×#95	C₂²×C₅₃	semidirect
309. C₅₃	C₂×D₄×C₂²⋊C₄	direct (N × H)
314. C₂⁵	C₂×C₄×C₅₃	semidirect
324. C₂²×D₄	C₂×C₄×C₅₃	semidirect
325. C₂²×D₄	C₂×C₄×C₅₃	semidirect
326. C₂²×D₄	C₂×C₄×C₅₃	semidirect
335. C₂³×C₄	C₂×C₄×C₅₃	semidirect
342. C₂³×C₄	C₂³×C₅₃	semidirect
345. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
352. C₂×C₂²⋊C₄	D₄×C₅₃	direct (N × H)
356. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
357. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
358. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
360. C₂×C₄²	C₂³×C₅₃	semidirect
361. #95	C₂³×C₅₃	semidirect
362. #95	C₂³×C₅₃	semidirect
384. C₂⁴	C₂²×C₄×C₅₃	semidirect
397. C₂×D₄	C₂²⋊C₄×C₅₃	direct (N × H)
399. C₂×D₄	C₂²×C₄×C₅₃	semidirect
400. C₂×D₄	C₂²×C₄×C₅₃	semidirect
419. C₂²×C₄	C₂²×C₄×C₅₃	semidirect
421. C₂²×C₄	C₂×D₄×C₅₃	semidirect
433. C₂²⋊C₄	C₂×D₄×C₅₃	direct (N × H)
463. C₂³	C₂×C₂²⋊C₄×C₅₃	semidirect
464. C₂³	C₄×D₄×C₅₃	semidirect
479. D₄	C₂×C₂²⋊C₄×C₅₃	direct (N × H)
495. C₂×C₄	C₂×C₂²⋊C₄×C₅₃	semidirect
499. C₂×C₄	C₂²×D₄×C₅₃	semidirect
518. C₂²	C₂²×C₂²⋊C₄×C₅₃	semidirect
520. C₂²	C₂×C₄×D₄×C₅₃	semidirect
533. C₄	C₂²×C₂²⋊C₄×C₅₃	semidirect
537. C₂	D₄×C₂²⋊C₄×C₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₅₃	2
C₂²×C₄×D₄×C₅₃	2
C₂³×C₂²⋊C₄×C₅₃	2
D₄×C₂²⋊C₄×C₅₃	2
C₂×C₅₃×#390	2
C₂×C₅₃×#386	2
C₂×C₅₃×#380	2
C₂×C₅₃×#379	2
C₂×C₄×C₂²⋊C₄×C₅₃	2
C₂×D₄×C₂²⋊C₄	53

Subgroups — C2×D4×C22⋊C4×C53 #27241610

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×C₂²⋊C₄×C₅₃ #27241610