Subgroups — C₂×D₄×C₄⋊C₄×C₅₃ #27241622

Order 13568.

4718 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₅₃	C₂	1	2	yes	maximal
2	C₂²×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄⋊C₄×C₅₃	C₂	1	2	yes	maximal, complemented
4	D₄×C₄⋊C₄×C₅₃	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₅₃×#389	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#385	C₂	1	4	yes	maximal, complemented
7	C₂×C₅₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#380	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₅₃	C₂²	1	4	yes	—
12	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₅₃	—	2	16	no	—
14	C₂×C₄×D₄×C₅₃	C₄	1	16	yes	complemented
15	C₂×C₄×D₄×C₅₃	C₂²	1	16	yes	—
16	C₂×C₄×D₄×C₅₃	C₂²	1	8	yes	—
17	C₂²×C₄⋊C₄×C₅₃	—	2	16	no	—
18	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
22	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₅₃	C₂²	1	16	yes	complemented
26	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
27	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
29	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#389	C₂²	1	16	yes	complemented
31	C₅₃×#385	C₂²	1	32	yes	complemented
32	C₅₃×#384	C₂²	1	8	yes	complemented
33	C₅₃×#380	C₂²	1	16	yes	complemented
34	C₄×C₄⋊C₄×C₅₃	C₂²	1	8	yes	complemented
35	C₂×#95×C₅₃	C₂²	1	2	yes	—
36	C₂×#95×C₅₃	C₂²	1	4	yes	—
37	C₂⁵×C₅₃	C₂³	1	2	yes	—
38	C₂²×D₄×C₅₃	C₂×C₄	1	8	yes	—
39	C₂²×D₄×C₅₃	D₄	1	8	yes	—
40	C₂²×D₄×C₅₃	Q₈	1	8	yes	—
41	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
42	C₂³×C₄×C₅₃	—	2	8	no	—
43	C₂³×C₄×C₅₃	—	2	8	no	—
44	C₂³×C₄×C₅₃	—	2	16	no	—
45	C₂³×C₄×C₅₃	—	2	16	no	—
46	C₂³×C₄×C₅₃	—	4	8	no	—
47	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	complemented
48	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
49	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
50	C₂³×C₄×C₅₃	C₂³	1	16	yes	—
51	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
53	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
54	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
55	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
57	C₄×D₄×C₅₃	—	2	64	no	—
58	C₄×D₄×C₅₃	C₂×C₄	1	64	yes	complemented
59	C₂×C₄⋊C₄×C₅₃	—	2	64	no	—
60	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
61	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
62	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
63	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
64	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
65	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	complemented
66	C₂×C₄⋊C₄×C₅₃	D₄	1	8	yes	direct product factor
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
69	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
71	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
72	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
73	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
74	C₂×C₂²⋊C₄×C₅₃	—	2	16	no	—
75	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	16	yes	complemented
76	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
77	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
78	C₂×C₄²×C₅₃	—	2	8	no	—
79	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	complemented
80	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
81	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
82	#95×C₅₃	C₂³	1	8	yes	—
83	#95×C₅₃	C₂³	1	16	yes	—
84	C₂⁴×C₅₃	—	2	4	no	—
85	C₂⁴×C₅₃	—	2	4	no	—
86	C₂⁴×C₅₃	—	2	4	no	—
87	C₂⁴×C₅₃	—	2	4	no	—
88	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	—
89	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
90	C₂⁴×C₅₃	C₂×Q₈	1	8	yes	—
91	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
92	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₂×D₄×C₅₃	C₄⋊C₄	1	64	yes	direct product factor
94	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	—
95	C₂×D₄×C₅₃	C₂×D₄	1	16	yes	—
96	C₂×D₄×C₅₃	C₂×Q₈	1	16	yes	—
97	C₂²×C₄×C₅₃	—	2	4	no	—
98	C₂²×C₄×C₅₃	—	2	4	no	—
99	C₂²×C₄×C₅₃	—	2	4	no	—
100	C₂²×C₄×C₅₃	—	2	2	no	—
101	C₂²×C₄×C₅₃	—	2	4	no	—
102	C₂²×C₄×C₅₃	—	2	2	no	—
103	C₂²×C₄×C₅₃	—	2	64	no	—
104	C₂²×C₄×C₅₃	—	2	16	no	—
105	C₂²×C₄×C₅₃	—	2	16	no	—
106	C₂²×C₄×C₅₃	—	2	4	no	—
107	C₂²×C₄×C₅₃	—	2	8	no	—
108	C₂²×C₄×C₅₃	—	2	8	no	—
109	C₂²×C₄×C₅₃	—	2	32	no	—
110	C₂²×C₄×C₅₃	—	2	4	no	—
111	C₂²×C₄×C₅₃	—	2	8	no	—
112	C₂²×C₄×C₅₃	—	2	4	no	—
113	C₂²×C₄×C₅₃	—	2	32	no	—
114	C₂²×C₄×C₅₃	—	4	16	no	—
115	C₂²×C₄×C₅₃	—	4	64	no	—
116	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
117	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
118	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
119	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
120	C₂²×C₄×C₅₃	C₂²×C₄	1	32	yes	complemented
121	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
122	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
123	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
124	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
125	C₂²×C₄×C₅₃	#41	1	4	yes	—
126	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
127	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
128	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
129	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
131	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
132	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
133	C₄⋊C₄×C₅₃	—	2	32	no	—
134	C₄⋊C₄×C₅₃	—	2	16	no	—
135	C₄⋊C₄×C₅₃	—	2	16	no	—
136	C₄⋊C₄×C₅₃	—	2	8	no	—
137	C₄⋊C₄×C₅₃	—	4	8	no	—
138	C₄⋊C₄×C₅₃	C₂²×C₄	1	16	yes	complemented
139	C₄⋊C₄×C₅₃	C₂×D₄	1	16	yes	direct product factor
140	C₂²⋊C₄×C₅₃	—	2	32	no	—
141	C₂²⋊C₄×C₅₃	C₂²×C₄	1	32	yes	complemented
142	C₄²×C₅₃	—	2	16	no	—
143	C₄²×C₅₃	C₂²×C₄	1	16	yes	complemented
144	C₂³×C₅₃	—	2	16	no	—
145	C₂³×C₅₃	—	2	16	no	—
146	C₂³×C₅₃	—	2	16	no	—
147	C₂³×C₅₃	—	2	16	no	—
148	C₂³×C₅₃	—	2	16	no	—
149	C₂³×C₅₃	—	2	16	no	—
150	C₂³×C₅₃	—	2	4	no	—
151	C₂³×C₅₃	—	2	4	no	—
152	C₂³×C₅₃	—	2	4	no	—
153	C₂³×C₅₃	—	2	4	no	—
154	C₂³×C₅₃	C₂×C₄⋊C₄	1	32	yes	complemented
155	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
156	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
157	C₂³×C₅₃	C₂²×D₄	1	8	yes	—
158	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
159	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
160	C₂³×C₅₃	C₂²×Q₈	1	8	yes	—
161	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
162	C₂³×C₅₃	C₂×#41	1	2	yes	—
163	C₂³×C₅₃	H(5,𝔽₂)	1	2	yes	—
164	C₂³×C₅₃	2₋¹⁺⁴	1	2	yes	—
165	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
166	D₄×C₅₃	C₂×C₄⋊C₄	1	64	yes	direct product factor
167	C₂×C₄×C₅₃	—	2	16	no	—
168	C₂×C₄×C₅₃	—	2	8	no	—
169	C₂×C₄×C₅₃	—	2	16	no	—
170	C₂×C₄×C₅₃	—	2	4	no	—
171	C₂×C₄×C₅₃	—	2	4	no	—
172	C₂×C₄×C₅₃	—	2	4	no	—
173	C₂×C₄×C₅₃	—	2	4	no	—
174	C₂×C₄×C₅₃	—	2	2	no	—
175	C₂×C₄×C₅₃	—	2	2	no	—
176	C₂×C₄×C₅₃	—	2	64	no	—
177	C₂×C₄×C₅₃	—	2	8	no	—
178	C₂×C₄×C₅₃	—	2	8	no	—
179	C₂×C₄×C₅₃	—	2	4	no	—
180	C₂×C₄×C₅₃	—	2	4	no	—
181	C₂×C₄×C₅₃	—	2	8	no	—
182	C₂×C₄×C₅₃	—	2	4	no	—
183	C₂×C₄×C₅₃	—	4	64	no	—
184	C₂×C₄×C₅₃	—	4	16	no	—
185	C₂×C₄×C₅₃	—	4	8	no	—
186	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	16	yes	complemented
187	C₂×C₄×C₅₃	C₄×D₄	1	16	yes	complemented
188	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
189	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
190	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
191	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
192	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
193	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
194	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
195	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
196	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
197	C₂×D₄×C₄⋊C₄	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂²×C₅₃	—	2	64	no	—
199	C₂²×C₅₃	—	2	8	no	—
200	C₂²×C₅₃	—	2	8	no	—
201	C₂²×C₅₃	—	2	8	no	—
202	C₂²×C₅₃	—	2	8	no	—
203	C₂²×C₅₃	—	2	8	no	—
204	C₂²×C₅₃	—	2	8	no	—
205	C₂²×C₅₃	C₂²×C₄⋊C₄	1	4	yes	—
206	C₂²×C₅₃	C₂²×C₄⋊C₄	1	16	yes	complemented
207	C₂²×C₅₃	C₂×C₄×D₄	1	4	yes	—
208	C₂²×C₅₃	#520	1	4	yes	—
209	C₂²×C₅₃	D₄²	1	4	yes	—
210	C₂²×C₅₃	#547	1	4	yes	—
211	C₂²×C₅₃	D₄×Q₈	1	4	yes	—
212	C₂²×C₅₃	#554	1	4	yes	—
213	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
214	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
215	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
216	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
217	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
218	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
219	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
220	C₄×C₅₃	—	2	8	no	—
221	C₄×C₅₃	—	2	4	no	—
222	C₄×C₅₃	—	2	8	no	—
223	C₄×C₅₃	—	4	8	no	—
224	C₄×C₅₃	—	4	4	no	—
225	C₄×C₅₃	C₂²×C₄⋊C₄	1	8	yes	complemented
226	C₄×C₅₃	C₂×C₄×D₄	1	8	yes	complemented
227	C₂²×C₄×D₄	C₂×C₅₃	1	2	yes	—
228	C₂²×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic
229	C₂³×C₄⋊C₄	C₂×C₅₃	1	2	yes	complemented
230	D₄×C₄⋊C₄	C₂×C₅₃	1	16	yes	direct product factor
231	C₂×#389	C₂×C₅₃	1	2	yes	complemented
232	C₂×#385	C₂×C₅₃	1	4	yes	complemented
233	C₂×#384	C₂×C₅₃	1	1	yes	characteristic, complemented
234	C₂×#380	C₂×C₅₃	1	2	yes	complemented
235	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
236	C₂×C₅₃	—	2	16	no	—
237	C₂×C₅₃	D₄×C₄⋊C₄	1	8	yes	direct product factor
238	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
239	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
240	C₂×C₅₃	C₂×#520	1	1	yes	characteristic
241	C₂×C₅₃	C₂×D₄²	1	1	yes	characteristic
242	C₂×C₅₃	C₂×#547	1	1	yes	characteristic
243	C₂×C₅₃	C₂×D₄×Q₈	1	1	yes	characteristic
244	C₂×C₅₃	C₂×#554	1	1	yes	characteristic
245	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
246	C₂⁴×C₄	C₂²×C₅₃	1	4	yes	—
247	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
248	C₂×C₄×D₄	—	2	16	no	—
249	C₂×C₄×D₄	C₄×C₅₃	1	16	yes	complemented
250	C₂×C₄×D₄	C₂²×C₅₃	1	16	yes	—
251	C₂×C₄×D₄	C₂²×C₅₃	1	8	yes	—
252	C₂²×C₄⋊C₄	—	2	16	no	—
253	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
254	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
255	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
256	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
257	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
258	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
259	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
260	C₂²×C₄⋊C₄	C₂²×C₅₃	1	16	yes	complemented
261	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
262	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
263	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
264	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
265	#389	C₂²×C₅₃	1	16	yes	complemented
266	#385	C₂²×C₅₃	1	32	yes	complemented
267	#384	C₂²×C₅₃	1	8	yes	complemented
268	#380	C₂²×C₅₃	1	16	yes	complemented
269	C₄×C₄⋊C₄	C₂²×C₅₃	1	8	yes	complemented
270	C₂×#95	C₂²×C₅₃	1	2	yes	—
271	C₂×#95	C₂²×C₅₃	1	4	yes	—
272	C₅₃	C₂×D₄×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
273	C₂⁵	C₂³×C₅₃	1	2	yes	—
274	C₂²×D₄	C₂×C₄×C₅₃	1	8	yes	—
275	C₂²×D₄	D₄×C₅₃	1	8	yes	—
276	C₂²×D₄	Q₈×C₅₃	1	8	yes	—
277	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
278	C₂³×C₄	—	2	8	no	—
279	C₂³×C₄	—	2	8	no	—
280	C₂³×C₄	—	2	16	no	—
281	C₂³×C₄	—	2	16	no	—
282	C₂³×C₄	—	4	8	no	—
283	C₂³×C₄	C₂×C₄×C₅₃	1	16	yes	complemented
284	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
285	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
286	C₂³×C₄	C₂³×C₅₃	1	16	yes	—
287	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
288	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
289	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
290	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
291	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
292	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
293	C₄×D₄	—	2	64	no	—
294	C₄×D₄	C₂×C₄×C₅₃	1	64	yes	complemented
295	C₂×C₄⋊C₄	—	2	16	no	—
296	C₂×C₄⋊C₄	—	2	8	no	—
297	C₂×C₄⋊C₄	—	2	8	no	—
298	C₂×C₄⋊C₄	—	2	4	no	—
299	C₂×C₄⋊C₄	—	2	64	no	—
300	C₂×C₄⋊C₄	—	4	4	no	—
301	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	complemented
302	C₂×C₄⋊C₄	D₄×C₅₃	1	8	yes	direct product factor
303	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
304	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
305	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
306	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
307	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
308	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
309	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
310	C₂×C₂²⋊C₄	—	2	16	no	—
311	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	16	yes	complemented
312	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
313	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
314	C₂×C₄²	—	2	8	no	—
315	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	complemented
316	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
317	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
318	#95	C₂³×C₅₃	1	8	yes	—
319	#95	C₂³×C₅₃	1	16	yes	—
320	C₂⁴	—	2	4	no	—
321	C₂⁴	—	2	4	no	—
322	C₂⁴	—	2	4	no	—
323	C₂⁴	—	2	4	no	—
324	C₂⁴	C₂²×C₄×C₅₃	1	8	yes	—
325	C₂⁴	C₂×D₄×C₅₃	1	8	yes	—
326	C₂⁴	C₂×Q₈×C₅₃	1	8	yes	—
327	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
328	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
329	C₂×D₄	C₄⋊C₄×C₅₃	1	64	yes	direct product factor
330	C₂×D₄	C₂²×C₄×C₅₃	1	16	yes	—
331	C₂×D₄	C₂×D₄×C₅₃	1	16	yes	—
332	C₂×D₄	C₂×Q₈×C₅₃	1	16	yes	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	4	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	64	no	—
340	C₂²×C₄	—	2	16	no	—
341	C₂²×C₄	—	2	16	no	—
342	C₂²×C₄	—	2	4	no	—
343	C₂²×C₄	—	2	8	no	—
344	C₂²×C₄	—	2	8	no	—
345	C₂²×C₄	—	2	32	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	2	8	no	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	2	32	no	—
350	C₂²×C₄	—	4	16	no	—
351	C₂²×C₄	—	4	64	no	—
352	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
353	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
354	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
355	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
356	C₂²×C₄	C₂²×C₄×C₅₃	1	32	yes	complemented
357	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
358	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
359	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
360	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
361	C₂²×C₄	#41×C₅₃	1	4	yes	—
362	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
363	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
364	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
365	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
366	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
367	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
368	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
369	C₄⋊C₄	—	2	32	no	—
370	C₄⋊C₄	—	2	16	no	—
371	C₄⋊C₄	—	2	16	no	—
372	C₄⋊C₄	—	2	8	no	—
373	C₄⋊C₄	—	4	8	no	—
374	C₄⋊C₄	C₂²×C₄×C₅₃	1	16	yes	complemented
375	C₄⋊C₄	C₂×D₄×C₅₃	1	16	yes	direct product factor
376	C₂²⋊C₄	—	2	32	no	—
377	C₂²⋊C₄	C₂²×C₄×C₅₃	1	32	yes	complemented
378	C₄²	—	2	16	no	—
379	C₄²	C₂²×C₄×C₅₃	1	16	yes	complemented
380	C₂³	—	2	16	no	—
381	C₂³	—	2	16	no	—
382	C₂³	—	2	16	no	—
383	C₂³	—	2	16	no	—
384	C₂³	—	2	16	no	—
385	C₂³	—	2	16	no	—
386	C₂³	—	2	4	no	—
387	C₂³	—	2	4	no	—
388	C₂³	—	2	4	no	—
389	C₂³	—	2	4	no	—
390	C₂³	C₂×C₄⋊C₄×C₅₃	1	32	yes	complemented
391	C₂³	C₂³×C₄×C₅₃	1	8	yes	—
392	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
393	C₂³	C₂²×D₄×C₅₃	1	8	yes	—
394	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
395	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
396	C₂³	C₂²×Q₈×C₅₃	1	8	yes	—
397	C₂³	C₂²×Q₈×C₅₃	1	2	yes	—
398	C₂³	C₂×#41×C₅₃	1	2	yes	—
399	C₂³	H(5,𝔽₂)×C₅₃	1	2	yes	—
400	C₂³	2₋¹⁺⁴×C₅₃	1	2	yes	—
401	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
402	D₄	C₂×C₄⋊C₄×C₅₃	1	64	yes	direct product factor
403	C₂×C₄	—	2	16	no	—
404	C₂×C₄	—	2	8	no	—
405	C₂×C₄	—	2	16	no	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	64	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	8	no	—
415	C₂×C₄	—	2	8	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	8	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	4	64	no	—
420	C₂×C₄	—	4	16	no	—
421	C₂×C₄	—	4	8	no	—
422	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	16	yes	complemented
423	C₂×C₄	C₄×D₄×C₅₃	1	16	yes	complemented
424	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
425	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
426	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
427	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
428	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
429	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
430	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
431	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
432	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
433	C₂²	—	2	8	no	—
434	C₂²	—	2	8	no	—
435	C₂²	—	2	8	no	—
436	C₂²	—	2	8	no	—
437	C₂²	—	2	64	no	—
438	C₂²	—	2	8	no	—
439	C₂²	—	2	8	no	—
440	C₂²	C₂²×C₄⋊C₄×C₅₃	1	4	yes	—
441	C₂²	C₂²×C₄⋊C₄×C₅₃	1	16	yes	complemented
442	C₂²	C₂×C₄×D₄×C₅₃	1	4	yes	—
443	C₂²	C₅₃×#520	1	4	yes	—
444	C₂²	D₄²×C₅₃	1	4	yes	—
445	C₂²	C₅₃×#547	1	4	yes	—
446	C₂²	D₄×Q₈×C₅₃	1	4	yes	—
447	C₂²	C₅₃×#554	1	4	yes	—
448	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
449	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
450	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
451	C₂²	C₂³×Q₈×C₅₃	1	1	yes	characteristic
452	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
453	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
454	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
455	C₄	—	2	8	no	—
456	C₄	—	2	8	no	—
457	C₄	—	2	4	no	—
458	C₄	—	4	8	no	—
459	C₄	—	4	4	no	—
460	C₄	C₂²×C₄⋊C₄×C₅₃	1	8	yes	complemented
461	C₄	C₂×C₄×D₄×C₅₃	1	8	yes	complemented
462	C₂	—	2	16	no	—
463	C₂	D₄×C₄⋊C₄×C₅₃	1	8	yes	direct product factor
464	C₂	C₂³×C₄⋊C₄×C₅₃	1	1	yes	characteristic
465	C₂	C₂²×C₄×D₄×C₅₃	1	1	yes	characteristic
466	C₂	C₂×C₅₃×#520	1	1	yes	characteristic
467	C₂	C₂×D₄²×C₅₃	1	1	yes	characteristic
468	C₂	C₂×C₅₃×#547	1	1	yes	characteristic
469	C₂	C₂×D₄×Q₈×C₅₃	1	1	yes	characteristic
470	C₂	C₂×C₅₃×#554	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂³×D₄×C₅₃	C₂²
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
3. C₂³×C₄⋊C₄×C₅₃	C₂	semidirect
4. D₄×C₄⋊C₄×C₅₃	C₂	direct (N × H)
5. C₂×C₅₃×#389	C₂	semidirect
6. C₂×C₅₃×#385	C₂	semidirect
7. C₂×C₅₃×#384	C₂	semidirect
8. C₂×C₅₃×#380	C₂	semidirect
9. C₂×C₄×C₄⋊C₄×C₅₃	C₂	semidirect
14. C₂×C₄×D₄×C₅₃	C₄	semidirect
25. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
30. C₅₃×#389	C₂²	semidirect
31. C₅₃×#385	C₂²	semidirect
32. C₅₃×#384	C₂²	semidirect
33. C₅₃×#380	C₂²	semidirect
34. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
47. C₂³×C₄×C₅₃	C₂×C₄	semidirect
58. C₄×D₄×C₅₃	C₂×C₄	semidirect
65. C₂×C₄⋊C₄×C₅₃	C₂×C₄	semidirect
66. C₂×C₄⋊C₄×C₅₃	D₄	direct (N × H)
75. C₂×C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
79. C₂×C₄²×C₅₃	C₂×C₄	semidirect
93. C₂×D₄×C₅₃	C₄⋊C₄	direct (N × H)
120. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
138. C₄⋊C₄×C₅₃	C₂²×C₄	semidirect
139. C₄⋊C₄×C₅₃	C₂×D₄	direct (N × H)
141. C₂²⋊C₄×C₅₃	C₂²×C₄	semidirect
143. C₄²×C₅₃	C₂²×C₄	semidirect
154. C₂³×C₅₃	C₂×C₄⋊C₄	semidirect
166. D₄×C₅₃	C₂×C₄⋊C₄	direct (N × H)
186. C₂×C₄×C₅₃	C₂×C₄⋊C₄	semidirect
187. C₂×C₄×C₅₃	C₄×D₄	semidirect
197. C₂×D₄×C₄⋊C₄	C₅₃	direct (N × H)
206. C₂²×C₅₃	C₂²×C₄⋊C₄	semidirect
225. C₄×C₅₃	C₂²×C₄⋊C₄	semidirect
226. C₄×C₅₃	C₂×C₄×D₄	semidirect
229. C₂³×C₄⋊C₄	C₂×C₅₃	semidirect
230. D₄×C₄⋊C₄	C₂×C₅₃	direct (N × H)
231. C₂×#389	C₂×C₅₃	semidirect
232. C₂×#385	C₂×C₅₃	semidirect
233. C₂×#384	C₂×C₅₃	semidirect
234. C₂×#380	C₂×C₅₃	semidirect
235. C₂×C₄×C₄⋊C₄	C₂×C₅₃	semidirect
237. C₂×C₅₃	D₄×C₄⋊C₄	direct (N × H)
249. C₂×C₄×D₄	C₄×C₅₃	semidirect
260. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
265. #389	C₂²×C₅₃	semidirect
266. #385	C₂²×C₅₃	semidirect
267. #384	C₂²×C₅₃	semidirect
268. #380	C₂²×C₅₃	semidirect
269. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
272. C₅₃	C₂×D₄×C₄⋊C₄	direct (N × H)
283. C₂³×C₄	C₂×C₄×C₅₃	semidirect
294. C₄×D₄	C₂×C₄×C₅₃	semidirect
301. C₂×C₄⋊C₄	C₂×C₄×C₅₃	semidirect
302. C₂×C₄⋊C₄	D₄×C₅₃	direct (N × H)
311. C₂×C₂²⋊C₄	C₂×C₄×C₅₃	semidirect
315. C₂×C₄²	C₂×C₄×C₅₃	semidirect
329. C₂×D₄	C₄⋊C₄×C₅₃	direct (N × H)
356. C₂²×C₄	C₂²×C₄×C₅₃	semidirect
374. C₄⋊C₄	C₂²×C₄×C₅₃	semidirect
375. C₄⋊C₄	C₂×D₄×C₅₃	direct (N × H)
377. C₂²⋊C₄	C₂²×C₄×C₅₃	semidirect
379. C₄²	C₂²×C₄×C₅₃	semidirect
390. C₂³	C₂×C₄⋊C₄×C₅₃	semidirect
402. D₄	C₂×C₄⋊C₄×C₅₃	direct (N × H)
422. C₂×C₄	C₂×C₄⋊C₄×C₅₃	semidirect
423. C₂×C₄	C₄×D₄×C₅₃	semidirect
441. C₂²	C₂²×C₄⋊C₄×C₅₃	semidirect
460. C₄	C₂²×C₄⋊C₄×C₅₃	semidirect
461. C₄	C₂×C₄×D₄×C₅₃	semidirect
463. C₂	D₄×C₄⋊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
C₂²×C₄×D₄×C₅₃	2
C₂³×C₄⋊C₄×C₅₃	2
D₄×C₄⋊C₄×C₅₃	2
C₂×C₅₃×#389	2
C₂×C₅₃×#385	2
C₂×C₅₃×#384	2
C₂×C₅₃×#380	2
C₂×C₄×C₄⋊C₄×C₅₃	2
C₂×D₄×C₄⋊C₄	53

Subgroups — C2×D4×C4⋊C4×C53 #27241622

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×C₄⋊C₄×C₅₃ #27241622