Subgroups — C₂²×#124×D₅₃ #28290752

Order 13568.

14334 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₄×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13133206	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#13133205	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13133204	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13133203	C₂	1	4	yes	maximal, complemented
7	C₂×#124×D₅₃	C₂	1	48	yes	maximal, direct product factor
8	C₂²×C₂²⋊C₄×D₅₃	C₂	1	4	yes	maximal, complemented
9	C₂²×C₄²×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#124×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵×D₅₃	C₂²	1	2	yes	—
12	C₂³×Dic₁₀₆	C₂²	1	2	yes	complemented
13	C₂³×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
14	C₂²×Q₈×D₅₃	—	2	16	no	—
15	C₂²×Q₈×D₅₃	C₂²	1	24	yes	complemented
16	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
17	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
18	C₂²×D₄×D₅₃	—	2	16	no	—
19	C₂²×D₄×D₅₃	C₂²	1	24	yes	—
20	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
21	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
22	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
23	C₂×#13133206	C₂²	1	24	yes	complemented
24	C₂×#13133205	C₂²	1	24	yes	complemented
25	C₂×#13133204	C₂²	1	24	yes	complemented
26	C₂×#13133203	C₂²	1	96	yes	complemented
27	#124×D₅₃	C₂²	1	256	yes	direct product factor
28	C₂²×C₂²⋊Dic₅₃	C₂²	1	4	yes	complemented
29	C₂²×#11719921	C₂²	1	4	yes	complemented
30	C₂²×#11719921	C₂²	1	4	yes	complemented
31	C₂×C₂²⋊C₄×D₅₃	C₂²	1	96	yes	complemented
32	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	complemented
33	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₄²×D₅₃	C₂²	1	24	yes	complemented
35	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
36	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×#124×C₅₃	C₂²	1	24	yes	complemented
38	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
39	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
40	C₂⁴×D₅₃	—	2	16	no	—
41	C₂⁴×D₅₃	—	2	32	no	—
42	C₂⁴×D₅₃	C₂³	1	24	yes	—
43	C₂⁴×D₅₃	C₂³	1	2	yes	—
44	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₀₆	—	2	8	no	—
46	C₂²×Dic₁₀₆	—	2	16	no	—
47	C₂²×Dic₁₀₆	C₂³	1	24	yes	complemented
48	C₂²×Dic₁₀₆	C₂³	1	12	yes	complemented
49	C₂×Q₈×D₅₃	—	2	192	no	—
50	C₂×Q₈×D₅₃	C₂³	1	64	yes	complemented
51	C₂²×D₂₁₂	—	2	8	no	—
52	C₂²×D₂₁₂	C₂³	1	12	yes	—
53	C₂²×C₂²⋊D₅₃	—	2	16	no	—
54	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
55	C₂×D₄×D₅₃	—	2	192	no	—
56	C₂×D₄×D₅₃	C₂³	1	64	yes	—
57	C₂³×Dic₅₃	C₂³	1	2	yes	—
58	C₂³×Dic₅₃	C₂³	1	2	yes	—
59	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×D₅₃	—	2	16	no	—
61	C₂²×C₄×D₅₃	—	2	16	no	—
62	C₂²×C₄×D₅₃	D₄	1	16	yes	—
63	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
64	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
65	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
66	#13133206	C₂³	1	64	yes	complemented
67	#13133205	C₂³	1	64	yes	complemented
68	#13133204	C₂³	1	64	yes	complemented
69	#13133203	C₂³	1	256	yes	complemented
70	C₂×C₂²⋊Dic₅₃	C₂³	1	48	yes	complemented
71	C₂×#11719921	C₂³	1	48	yes	complemented
72	C₂×#11719921	C₂³	1	48	yes	complemented
73	C₂²⋊C₄×D₅₃	C₂³	1	256	yes	complemented
74	C₂×C₄×Dic₅₃	C₂³	1	24	yes	complemented
75	C₂×C₄×Dic₅₃	C₂³	1	12	yes	complemented
76	C₄²×D₅₃	C₂³	1	64	yes	complemented
77	C₂⁵×C₅₃	C₂³	1	2	yes	—
78	C₂²×Q₈×C₅₃	—	2	8	no	—
79	C₂²×Q₈×C₅₃	C₂³	1	12	yes	complemented
80	C₂²×D₄×C₅₃	—	2	8	no	—
81	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
82	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
83	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
84	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
85	#124×C₅₃	C₂³	1	64	yes	complemented
86	C₂×C₂²⋊C₄×C₅₃	C₂³	1	48	yes	complemented
87	C₂×C₄²×C₅₃	C₂³	1	12	yes	complemented
88	C₂⁴×C₅₃	—	2	8	no	—
89	C₂⁴×C₅₃	—	2	16	no	—
90	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
91	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
92	C₂³×D₅₃	—	2	8	no	—
93	C₂³×D₅₃	—	2	16	no	—
94	C₂³×D₅₃	—	2	96	no	—
95	C₂³×D₅₃	—	2	192	no	—
96	C₂³×D₅₃	—	4	128	no	—
97	C₂³×D₅₃	C₂×D₄	1	8	yes	—
98	C₂³×D₅₃	#41	1	16	yes	—
99	C₂³×D₅₃	C₂⁴	1	32	yes	—
100	C₂³×D₅₃	C₂⁴	1	12	yes	—
101	C₂³×D₅₃	C₂⁴	1	6	yes	—
102	C₂×Q₈×C₅₃	—	2	48	no	—
103	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	complemented
104	C₂×D₄×C₅₃	—	2	48	no	—
105	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
106	C₂²×C₄×C₅₃	—	2	8	no	—
107	C₂²×C₄×C₅₃	—	2	8	no	—
108	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
109	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
110	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
111	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
112	C₂×C₂²⋊D₅₃	—	2	96	no	—
113	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
114	C₂²×Dic₅₃	—	2	8	no	—
115	C₂²×Dic₅₃	—	2	8	no	—
116	C₂²×Dic₅₃	C₂×D₄	1	8	yes	—
117	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
118	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
119	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
120	Q₈×D₅₃	—	2	256	no	—
121	D₄×D₅₃	—	2	256	no	—
122	C₂×D₂₁₂	—	2	48	no	—
123	C₂×D₂₁₂	C₂⁴	1	16	yes	—
124	C₂×C₄×D₅₃	—	2	96	no	—
125	C₂×C₄×D₅₃	—	2	96	no	—
126	C₂×C₄×D₅₃	C₂×D₄	1	96	yes	—
127	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
128	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
129	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
130	C₂×Dic₁₀₆	—	2	48	no	—
131	C₂×Dic₁₀₆	—	2	96	no	—
132	C₂×Dic₁₀₆	C₂⁴	1	32	yes	complemented
133	C₂×Dic₁₀₆	C₂⁴	1	16	yes	complemented
134	C₂²⋊C₄×C₅₃	C₂⁴	1	64	yes	complemented
135	C₄²×C₅₃	C₂⁴	1	16	yes	complemented
136	C₂²⋊Dic₅₃	C₂⁴	1	64	yes	complemented
137	#11719921	C₂⁴	1	64	yes	complemented
138	#11719921	C₂⁴	1	64	yes	complemented
139	C₄×Dic₅₃	C₂⁴	1	32	yes	complemented
140	C₄×Dic₅₃	C₂⁴	1	16	yes	complemented
141	C₂³×C₅₃	—	2	24	no	—
142	C₂³×C₅₃	—	2	48	no	—
143	C₂³×C₅₃	—	4	32	no	—
144	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
145	C₂³×C₅₃	C₂×#41	1	8	yes	—
146	C₂³×C₅₃	C₂⁵	1	8	yes	—
147	C₂³×C₅₃	C₂⁵	1	3	yes	—
148	C₂²×D₅₃	—	2	24	no	—
149	C₂²×D₅₃	—	2	48	no	—
150	C₂²×D₅₃	—	2	64	no	—
151	C₂²×D₅₃	—	2	128	no	—
152	C₂²×D₅₃	—	4	32	no	—
153	C₂²×D₅₃	—	4	384	no	—
154	C₂²×D₅₃	#124	1	64	yes	direct product factor
155	C₂²×D₅₃	C₂²×D₄	1	24	yes	—
156	C₂²×D₅₃	C₂×#41	1	48	yes	—
157	C₂²×D₅₃	C₂⁵	1	8	yes	—
158	C₂²×D₅₃	C₂⁵	1	4	yes	—
159	Q₈×C₅₃	—	2	32	no	—
160	D₄×C₅₃	—	2	32	no	—
161	C₂×C₄×C₅₃	—	2	24	no	—
162	C₂×C₄×C₅₃	—	2	24	no	—
163	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
164	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
165	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
166	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
167	C₂²⋊D₅₃	—	2	64	no	—
168	C₂×Dic₅₃	—	2	24	no	—
169	C₂×Dic₅₃	—	2	24	no	—
170	C₂×Dic₅₃	C₂²×D₄	1	24	yes	—
171	C₂×Dic₅₃	C₂⁵	1	8	yes	—
172	C₂×Dic₅₃	C₂⁵	1	8	yes	—
173	C₂×Dic₅₃	C₂⁵	1	4	yes	—
174	D₂₁₂	—	2	32	no	—
175	C₄×D₅₃	—	2	64	no	—
176	C₄×D₅₃	—	2	64	no	—
177	C₄×D₅₃	C₂²×D₄	1	64	yes	—
178	Dic₁₀₆	—	2	32	no	—
179	Dic₁₀₆	—	2	64	no	—
180	C₂³×#124	—	53	1	no	maximal, Sylow
181	C₂²×C₅₃	—	2	8	no	—
182	C₂²×C₅₃	—	2	16	no	—
183	C₂²×C₅₃	—	4	48	no	—
184	C₂²×C₅₃	C₂×#124	1	16	yes	complemented
185	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
186	C₂²×C₅₃	C₂²×#41	1	12	yes	—
187	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
188	C₂×D₅₃	—	2	8	no	—
189	C₂×D₅₃	—	2	16	no	—
190	C₂×D₅₃	—	4	48	no	—
191	C₂×D₅₃	—	4	128	no	—
192	C₂×D₅₃	C₂×#124	1	96	yes	direct product factor
193	C₂×D₅₃	C₂³×D₄	1	8	yes	—
194	C₂×D₅₃	C₂²×#41	1	16	yes	—
195	C₄×C₅₃	—	2	8	no	—
196	C₄×C₅₃	—	2	8	no	—
197	C₄×C₅₃	C₂³×D₄	1	8	yes	—
198	Dic₅₃	—	2	8	no	—
199	Dic₅₃	—	2	8	no	—
200	Dic₅₃	C₂³×D₄	1	8	yes	—
201	C₂⁴×Q₈	—	53	1	no	—
202	C₂⁴×D₄	—	53	1	no	—
203	C₂²×#124	—	53	1	no	—
204	C₂²×#124	—	53	1	no	—
205	C₂²×#124	—	53	1	no	—
206	C₂²×#124	—	53	48	no	—
207	C₂²×#124	—	53	4	no	—
208	C₂²×#124	D₅₃	1	1	yes	characteristic, direct product factor
209	C₂³×C₂²⋊C₄	—	53	4	no	—
210	C₂³×C₄²	—	53	1	no	—
211	C₂×C₅₃	—	4	8	no	—
212	C₂×C₅₃	C₂²×#124	1	12	yes	complemented
213	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
214	C₂×C₅₃	C₂³×#41	1	2	yes	—
215	D₅₃	—	4	8	no	—
216	D₅₃	C₂²×#124	1	16	yes	direct product factor
217	C₂⁶	—	53	2	no	—
218	C₂³×Q₈	—	53	24	no	—
219	C₂³×Q₈	—	53	2	no	—
220	C₂³×Q₈	—	53	1	no	—
221	C₂³×Q₈	—	106	16	no	—
222	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
223	C₂³×D₄	—	53	2	no	—
224	C₂³×D₄	—	53	24	no	—
225	C₂³×D₄	—	53	1	no	—
226	C₂³×D₄	—	106	16	no	—
227	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
228	C₂⁴×C₄	—	53	1	no	—
229	C₂⁴×C₄	—	53	2	no	—
230	C₂⁴×C₄	—	53	2	no	—
231	C₂×#124	—	53	24	no	—
232	C₂×#124	—	53	24	no	—
233	C₂×#124	—	53	256	no	—
234	C₂×#124	—	53	96	no	—
235	C₂×#124	—	53	24	no	—
236	C₂×#124	C₂×D₅₃	1	24	yes	direct product factor
237	C₂²×C₂²⋊C₄	—	53	4	no	—
238	C₂²×C₂²⋊C₄	—	53	4	no	—
239	C₂²×C₂²⋊C₄	—	53	4	no	—
240	C₂²×C₂²⋊C₄	—	53	96	no	—
241	C₂²×C₂²⋊C₄	C₂×D₅₃	1	4	yes	complemented
242	C₂²×C₄²	—	53	2	no	—
243	C₂²×C₄²	—	53	1	no	—
244	C₂²×C₄²	—	53	24	no	—
245	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
246	C₅₃	C₂³×#124	1	1	yes	characteristic, complemented, Sylow
247	C₂⁵	—	53	1	no	—
248	C₂⁵	—	53	24	no	—
249	C₂⁵	—	53	2	no	—
250	C₂⁵	—	106	16	no	—
251	C₂⁵	—	106	32	no	—
252	C₂⁵	C₂²×D₅₃	1	2	yes	—
253	C₂²×Q₈	—	2	8	no	—
254	C₂²×Q₈	—	53	64	no	—
255	C₂²×Q₈	—	53	24	no	—
256	C₂²×Q₈	—	53	12	no	—
257	C₂²×Q₈	—	106	16	no	—
258	C₂²×Q₈	—	106	192	no	—
259	C₂²×Q₈	—	106	8	no	—
260	C₂²×Q₈	C₂²×D₅₃	1	12	yes	complemented
261	C₂²×D₄	—	2	8	no	—
262	C₂²×D₄	—	53	24	no	—
263	C₂²×D₄	—	53	64	no	—
264	C₂²×D₄	—	53	12	no	—
265	C₂²×D₄	—	106	192	no	—
266	C₂²×D₄	—	106	16	no	—
267	C₂²×D₄	—	106	8	no	—
268	C₂²×D₄	C₂²×D₅₃	1	12	yes	—
269	C₂³×C₄	—	53	1	no	—
270	C₂³×C₄	—	53	16	no	—
271	C₂³×C₄	—	53	12	no	—
272	C₂³×C₄	—	53	2	no	—
273	C₂³×C₄	—	53	2	no	—
274	C₂³×C₄	—	53	24	no	—
275	C₂³×C₄	—	53	24	no	—
276	C₂³×C₄	—	106	16	no	—
277	C₂³×C₄	—	106	16	no	—
278	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
279	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
280	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
281	#124	—	53	64	no	—
282	#124	—	53	64	no	—
283	#124	—	53	256	no	—
284	#124	—	53	64	no	—
285	#124	C₂²×D₅₃	1	64	yes	direct product factor
286	C₂×C₂²⋊C₄	—	53	48	no	—
287	C₂×C₂²⋊C₄	—	53	48	no	—
288	C₂×C₂²⋊C₄	—	53	48	no	—
289	C₂×C₂²⋊C₄	—	53	256	no	—
290	C₂×C₂²⋊C₄	C₂²×D₅₃	1	48	yes	complemented
291	C₂×C₄²	—	53	24	no	—
292	C₂×C₄²	—	53	12	no	—
293	C₂×C₄²	—	53	64	no	—
294	C₂×C₄²	C₂²×D₅₃	1	12	yes	complemented
295	C₂⁴	—	2	8	no	—
296	C₂⁴	—	2	16	no	—
297	C₂⁴	—	53	8	no	—
298	C₂⁴	—	53	16	no	—
299	C₂⁴	—	53	6	no	—
300	C₂⁴	—	53	32	no	—
301	C₂⁴	—	53	12	no	—
302	C₂⁴	—	106	96	no	—
303	C₂⁴	—	106	192	no	—
304	C₂⁴	—	106	8	no	—
305	C₂⁴	—	106	16	no	—
306	C₂⁴	—	212	128	no	—
307	C₂⁴	C₂³×D₅₃	1	12	yes	—
308	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
309	C₂×Q₈	—	2	48	no	—
310	C₂×Q₈	—	53	32	no	—
311	C₂×Q₈	—	53	16	no	—
312	C₂×Q₈	—	106	96	no	—
313	C₂×Q₈	—	106	256	no	—
314	C₂×Q₈	—	106	48	no	—
315	C₂×Q₈	C₂³×D₅₃	1	16	yes	complemented
316	C₂×D₄	—	2	48	no	—
317	C₂×D₄	—	53	32	no	—
318	C₂×D₄	—	53	16	no	—
319	C₂×D₄	—	106	256	no	—
320	C₂×D₄	—	106	96	no	—
321	C₂×D₄	—	106	48	no	—
322	C₂×D₄	C₂³×D₅₃	1	16	yes	—
323	C₂²×C₄	—	2	8	no	—
324	C₂²×C₄	—	2	8	no	—
325	C₂²×C₄	—	53	8	no	—
326	C₂²×C₄	—	53	6	no	—
327	C₂²×C₄	—	53	96	no	—
328	C₂²×C₄	—	53	16	no	—
329	C₂²×C₄	—	53	12	no	—
330	C₂²×C₄	—	53	12	no	—
331	C₂²×C₄	—	53	32	no	—
332	C₂²×C₄	—	53	32	no	—
333	C₂²×C₄	—	106	96	no	—
334	C₂²×C₄	—	106	96	no	—
335	C₂²×C₄	—	106	8	no	—
336	C₂²×C₄	—	106	8	no	—
337	C₂²×C₄	D₄×D₅₃	1	8	yes	—
338	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
339	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
340	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
341	C₂²⋊C₄	—	53	64	no	—
342	C₂²⋊C₄	—	53	64	no	—
343	C₂²⋊C₄	—	53	64	no	—
344	C₂²⋊C₄	C₂³×D₅₃	1	64	yes	complemented
345	C₄²	—	53	32	no	—
346	C₄²	—	53	16	no	—
347	C₄²	C₂³×D₅₃	1	16	yes	complemented
348	C₂³	—	2	24	no	—
349	C₂³	—	2	48	no	—
350	C₂³	—	4	32	no	—
351	C₂³	—	53	64	no	—
352	C₂³	—	53	24	no	—
353	C₂³	—	53	48	no	—
354	C₂³	—	53	4	no	—
355	C₂³	—	53	8	no	—
356	C₂³	—	106	64	no	—
357	C₂³	—	106	128	no	—
358	C₂³	—	106	24	no	—
359	C₂³	—	106	48	no	—
360	C₂³	—	212	384	no	—
361	C₂³	—	212	32	no	—
362	C₂³	C₂×D₄×D₅₃	1	4	yes	—
363	C₂³	#41×D₅₃	1	8	yes	—
364	C₂³	C₂⁴×D₅₃	1	8	yes	—
365	C₂³	C₂⁴×D₅₃	1	3	yes	—
366	Q₈	—	2	32	no	—
367	Q₈	—	106	64	no	—
368	Q₈	—	106	32	no	—
369	D₄	—	2	32	no	—
370	D₄	—	106	64	no	—
371	D₄	—	106	32	no	—
372	C₂×C₄	—	2	24	no	—
373	C₂×C₄	—	2	24	no	—
374	C₂×C₄	—	53	24	no	—
375	C₂×C₄	—	53	4	no	—
376	C₂×C₄	—	53	64	no	—
377	C₂×C₄	—	53	8	no	—
378	C₂×C₄	—	53	8	no	—
379	C₂×C₄	—	106	64	no	—
380	C₂×C₄	—	106	64	no	—
381	C₂×C₄	—	106	24	no	—
382	C₂×C₄	—	106	24	no	—
383	C₂×C₄	C₂×D₄×D₅₃	1	24	yes	—
384	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
385	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
386	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
387	C₂²	—	2	8	no	—
388	C₂²	—	2	16	no	—
389	C₂²	—	4	48	no	—
390	C₂²	—	53	96	no	—
391	C₂²	—	53	8	no	—
392	C₂²	—	53	16	no	—
393	C₂²	—	106	8	no	—
394	C₂²	—	106	16	no	—
395	C₂²	—	212	128	no	—
396	C₂²	—	212	48	no	—
397	C₂²	#124×D₅₃	1	16	yes	direct product factor
398	C₂²	C₂²×D₄×D₅₃	1	6	yes	—
399	C₂²	C₂×#41×D₅₃	1	12	yes	—
400	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
401	C₄	—	2	8	no	—
402	C₄	—	2	8	no	—
403	C₄	—	53	8	no	—
404	C₄	—	106	8	no	—
405	C₄	—	106	8	no	—
406	C₄	C₂²×D₄×D₅₃	1	8	yes	—
407	C₂	—	4	8	no	—
408	C₂	—	53	16	no	—
409	C₂	—	212	8	no	—
410	C₂	C₂×#124×D₅₃	1	12	yes	direct product factor
411	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
412	C₂	C₂²×#41×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×#124×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂³×D₄×D₅₃	C₂
hypercenter	C₂²×#124	D₅₃
hypocenter	C₅₃	C₂³×#124
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#124

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₈×D₅₃	C₂	semidirect
3. C₂²×#13133206	C₂	semidirect
4. C₂²×#13133205	C₂	semidirect
5. C₂²×#13133204	C₂	semidirect
6. C₂²×#13133203	C₂	semidirect
7. C₂×#124×D₅₃	C₂	direct (N × H)
8. C₂²×C₂²⋊C₄×D₅₃	C₂	semidirect
9. C₂²×C₄²×D₅₃	C₂	semidirect
10. C₂²×#124×C₅₃	C₂	semidirect
12. C₂³×Dic₁₀₆	C₂²	semidirect
13. C₂³×Dic₁₀₆	C₂²	semidirect
15. C₂²×Q₈×D₅₃	C₂²	semidirect
23. C₂×#13133206	C₂²	semidirect
24. C₂×#13133205	C₂²	semidirect
25. C₂×#13133204	C₂²	semidirect
26. C₂×#13133203	C₂²	semidirect
27. #124×D₅₃	C₂²	direct (N × H)
28. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
29. C₂²×#11719921	C₂²	semidirect
30. C₂²×#11719921	C₂²	semidirect
31. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
32. C₂²×C₄×Dic₅₃	C₂²	semidirect
33. C₂²×C₄×Dic₅₃	C₂²	semidirect
34. C₂×C₄²×D₅₃	C₂²	semidirect
35. C₂³×Q₈×C₅₃	C₂²	semidirect
37. C₂×#124×C₅₃	C₂²	semidirect
38. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
39. C₂²×C₄²×C₅₃	C₂²	semidirect
47. C₂²×Dic₁₀₆	C₂³	semidirect
48. C₂²×Dic₁₀₆	C₂³	semidirect
50. C₂×Q₈×D₅₃	C₂³	semidirect
66. #13133206	C₂³	semidirect
67. #13133205	C₂³	semidirect
68. #13133204	C₂³	semidirect
69. #13133203	C₂³	semidirect
70. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
71. C₂×#11719921	C₂³	semidirect
72. C₂×#11719921	C₂³	semidirect
73. C₂²⋊C₄×D₅₃	C₂³	semidirect
74. C₂×C₄×Dic₅₃	C₂³	semidirect
75. C₂×C₄×Dic₅₃	C₂³	semidirect
76. C₄²×D₅₃	C₂³	semidirect
79. C₂²×Q₈×C₅₃	C₂³	semidirect
85. #124×C₅₃	C₂³	semidirect
86. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
87. C₂×C₄²×C₅₃	C₂³	semidirect
103. C₂×Q₈×C₅₃	C₂⁴	semidirect
132. C₂×Dic₁₀₆	C₂⁴	semidirect
133. C₂×Dic₁₀₆	C₂⁴	semidirect
134. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
135. C₄²×C₅₃	C₂⁴	semidirect
136. C₂²⋊Dic₅₃	C₂⁴	semidirect
137. #11719921	C₂⁴	semidirect
138. #11719921	C₂⁴	semidirect
139. C₄×Dic₅₃	C₂⁴	semidirect
140. C₄×Dic₅₃	C₂⁴	semidirect
154. C₂²×D₅₃	#124	direct (N × H)
184. C₂²×C₅₃	C₂×#124	semidirect
192. C₂×D₅₃	C₂×#124	direct (N × H)
208. C₂²×#124	D₅₃	direct (N × H)
212. C₂×C₅₃	C₂²×#124	semidirect
216. D₅₃	C₂²×#124	direct (N × H)
222. C₂³×Q₈	C₂×D₅₃	semidirect
236. C₂×#124	C₂×D₅₃	direct (N × H)
241. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
245. C₂²×C₄²	C₂×D₅₃	semidirect
246. C₅₃	C₂³×#124	semidirect
260. C₂²×Q₈	C₂²×D₅₃	semidirect
285. #124	C₂²×D₅₃	direct (N × H)
290. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
294. C₂×C₄²	C₂²×D₅₃	semidirect
315. C₂×Q₈	C₂³×D₅₃	semidirect
344. C₂²⋊C₄	C₂³×D₅₃	semidirect
347. C₄²	C₂³×D₅₃	semidirect
397. C₂²	#124×D₅₃	direct (N × H)
410. C₂	C₂×#124×D₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₈×D₅₃	2
C₂³×D₄×D₅₃	2
C₂²×#13133206	2
C₂²×#13133205	2
C₂²×#13133204	2
C₂²×#13133203	2
C₂×#124×D₅₃	2
C₂²×C₂²⋊C₄×D₅₃	2
C₂²×C₄²×D₅₃	2
C₂²×#124×C₅₃	2
C₂³×#124	53

Subgroups — C22×#124×D53 #28290752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#124×D₅₃ #28290752