Subgroups — C₂²×C₂²⋊C₈×D₅₃ #28247517

Order 13568.

7646 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₈×D₅₃	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#13133129	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₈×D₅₃	C₂	1	24	yes	maximal, direct product factor
6	C₂²×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₅₃	C₄	1	1	yes	characteristic
8	C₂⁴×Dic₅₃	C₄	1	1	yes	characteristic
9	C₂³×C₄×D₅₃	—	2	16	no	—
10	C₂³×C₄×D₅₃	C₄	1	1	yes	characteristic
11	C₂³×C₄×D₅₃	C₄	1	1	yes	characteristic
12	C₂³×C₄×D₅₃	C₄	1	12	yes	—
13	C₂³×C₄×D₅₃	C₂²	1	12	yes	—
14	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
16	C₂³×#88322	C₂²	1	2	yes	complemented
17	C₂²×C₈×D₅₃	—	2	16	no	—
18	C₂²×C₈×D₅₃	C₂²	1	24	yes	complemented
19	C₂×C₂²⋊#88322	C₂²	1	12	yes	complemented
20	C₂×#13133129	C₂²	1	24	yes	complemented
21	C₂²⋊C₈×D₅₃	C₂²	1	64	yes	direct product factor
22	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂³×C₈×C₅₃	C₂²	1	2	yes	complemented
24	C₂×C₂²⋊C₈×C₅₃	C₂²	1	12	yes	complemented
25	C₂⁴×D₅₃	—	2	8	no	—
26	C₂⁴×D₅₃	—	2	8	no	—
27	C₂⁴×D₅₃	C₈	1	16	yes	complemented
28	C₂⁴×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂⁴×D₅₃	C₂×C₄	1	12	yes	—
30	C₂⁴×D₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂³×Dic₅₃	—	2	8	no	—
32	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×Dic₅₃	C₂×C₄	1	12	yes	—
34	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×D₅₃	—	2	8	no	—
36	C₂²×C₄×D₅₃	—	2	8	no	—
37	C₂²×C₄×D₅₃	—	2	8	no	—
38	C₂²×C₄×D₅₃	—	2	96	no	—
39	C₂²×C₄×D₅₃	—	2	96	no	—
40	C₂²×C₄×D₅₃	C₂×C₄	1	12	yes	—
41	C₂²×C₄×D₅₃	C₂×C₄	1	6	yes	—
42	C₂²×C₄×D₅₃	C₂×C₄	1	12	yes	—
43	C₂²×C₄×D₅₃	C₂×C₄	1	48	yes	—
44	C₂²×C₄×D₅₃	C₂×C₄	1	6	yes	—
45	C₂²×C₄×D₅₃	D₄	1	16	yes	—
46	C₂²×C₄×D₅₃	C₂³	1	16	yes	—
47	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
48	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
49	C₂²×#88322	—	2	8	no	—
50	C₂²×#88322	C₂³	1	12	yes	complemented
51	C₂×C₈×D₅₃	—	2	96	no	—
52	C₂×C₈×D₅₃	C₂³	1	32	yes	complemented
53	C₂²⋊#88322	C₂³	1	16	yes	complemented
54	#13133129	C₂³	1	32	yes	complemented
55	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂³×C₄×C₅₃	—	2	8	no	—
57	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂³×C₄×C₅₃	C₂×C₄	1	6	yes	—
59	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
60	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	—	2	8	no	—
62	C₂²×C₈×C₅₃	C₂³	1	12	yes	complemented
63	C₂²⋊C₈×C₅₃	C₂³	1	16	yes	complemented
64	C₂⁴×C₅₃	—	2	4	no	—
65	C₂⁴×C₅₃	—	2	4	no	—
66	C₂⁴×C₅₃	C₂×C₈	1	8	yes	complemented
67	C₂⁴×C₅₃	C₂²×C₄	1	6	yes	—
68	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂³×D₅₃	—	2	4	no	—
70	C₂³×D₅₃	—	2	4	no	—
71	C₂³×D₅₃	—	2	128	no	—
72	C₂³×D₅₃	—	2	48	no	—
73	C₂³×D₅₃	—	2	48	no	—
74	C₂³×D₅₃	C₂²⋊C₄	1	8	yes	—
75	C₂³×D₅₃	C₂×C₈	1	8	yes	—
76	C₂³×D₅₃	C₂×C₈	1	96	yes	complemented
77	C₂³×D₅₃	C₂×C₈	1	8	yes	complemented
78	C₂³×D₅₃	M₁₆	1	8	yes	—
79	C₂³×D₅₃	C₂²×C₄	1	6	yes	—
80	C₂³×D₅₃	C₂²×C₄	1	16	yes	—
81	C₂³×D₅₃	C₂²×C₄	1	6	yes	—
82	C₂²×C₄×C₅₃	—	2	4	no	—
83	C₂²×C₄×C₅₃	—	2	24	no	—
84	C₂²×C₄×C₅₃	—	2	24	no	—
85	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
86	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
87	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
88	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
89	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
90	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
91	C₂²×Dic₅₃	—	2	4	no	—
92	C₂²×Dic₅₃	—	2	48	no	—
93	C₂²×Dic₅₃	C₂²⋊C₄	1	8	yes	—
94	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
95	C₂²×Dic₅₃	C₂²×C₄	1	16	yes	—
96	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
97	C₂×C₄×D₅₃	—	2	48	no	—
98	C₂×C₄×D₅₃	—	2	24	no	—
99	C₂×C₄×D₅₃	—	2	24	no	—
100	C₂×C₄×D₅₃	—	2	48	no	—
101	C₂×C₄×D₅₃	—	2	192	no	—
102	C₂×C₄×D₅₃	—	2	64	no	—
103	C₂×C₄×D₅₃	C₂²⋊C₄	1	48	yes	—
104	C₂×C₄×D₅₃	C₂²×C₄	1	16	yes	—
105	C₂×C₄×D₅₃	C₂²×C₄	1	12	yes	—
106	C₂×C₄×D₅₃	C₂²×C₄	1	16	yes	—
107	C₂×C₄×D₅₃	C₂²×C₄	1	12	yes	—
108	C₂×C₄×D₅₃	C₂×D₄	1	48	yes	—
109	C₂×C₄×D₅₃	C₂⁴	1	4	yes	—
110	C₂×C₄×D₅₃	C₂⁴	1	4	yes	—
111	C₂×C₈×C₅₃	—	2	24	no	—
112	C₂×C₈×C₅₃	C₂⁴	1	8	yes	complemented
113	C₂×#88322	—	2	24	no	—
114	C₂×#88322	C₂⁴	1	8	yes	complemented
115	C₈×D₅₃	—	2	64	no	—
116	C₂³×C₅₃	—	2	12	no	—
117	C₂³×C₅₃	—	2	12	no	—
118	C₂³×C₅₃	—	2	32	no	—
119	C₂³×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
120	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
121	C₂³×C₅₃	C₂²×C₈	1	24	yes	complemented
122	C₂³×C₅₃	C₂×M₁₆	1	4	yes	—
123	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
124	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
125	C₂²×D₅₃	—	2	12	no	—
126	C₂²×D₅₃	—	2	12	no	—
127	C₂²×D₅₃	—	2	32	no	—
128	C₂²×D₅₃	—	2	32	no	—
129	C₂²×D₅₃	—	2	32	no	—
130	C₂²×D₅₃	—	2	384	no	—
131	C₂²×D₅₃	C₂²⋊C₈	1	64	yes	direct product factor
132	C₂²×D₅₃	C₂×C₂²⋊C₄	1	24	yes	—
133	C₂²×D₅₃	C₂²×C₈	1	24	yes	—
134	C₂²×D₅₃	C₂²×C₈	1	64	yes	complemented
135	C₂²×D₅₃	C₂²×C₈	1	24	yes	complemented
136	C₂²×D₅₃	C₂×M₁₆	1	24	yes	—
137	C₂²×D₅₃	C₂³×C₄	1	4	yes	—
138	C₂²×D₅₃	C₂³×C₄	1	4	yes	—
139	C₂×C₄×C₅₃	—	2	12	no	—
140	C₂×C₄×C₅₃	—	2	24	no	—
141	C₂×C₄×C₅₃	—	2	8	no	—
142	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
143	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
144	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
145	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
146	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
147	C₂×Dic₅₃	—	2	12	no	—
148	C₂×Dic₅₃	—	2	32	no	—
149	C₂×Dic₅₃	C₂×C₂²⋊C₄	1	24	yes	—
150	C₂×Dic₅₃	C₂³×C₄	1	4	yes	—
151	C₂×Dic₅₃	C₂³×C₄	1	4	yes	—
152	C₄×D₅₃	—	2	32	no	—
153	C₄×D₅₃	—	2	24	no	—
154	C₄×D₅₃	—	2	8	no	—
155	C₄×D₅₃	—	2	32	no	—
156	C₄×D₅₃	C₂×C₂²⋊C₄	1	48	yes	—
157	C₄×D₅₃	C₂²×D₄	1	16	yes	—
158	C₈×C₅₃	—	2	8	no	—
159	#88322	—	2	8	no	—
160	C₂³×C₂²⋊C₈	—	53	1	no	maximal, Sylow
161	C₂²×C₅₃	—	2	48	no	—
162	C₂²×C₅₃	—	2	4	no	—
163	C₂²×C₅₃	—	2	4	no	—
164	C₂²×C₅₃	C₂×C₂²⋊C₈	1	16	yes	complemented
165	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	6	yes	—
166	C₂²×C₅₃	C₂³×C₈	1	6	yes	—
167	C₂²×C₅₃	C₂³×C₈	1	8	yes	complemented
168	C₂²×C₅₃	C₂²×M₁₆	1	6	yes	—
169	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
170	C₂×D₅₃	—	2	48	no	—
171	C₂×D₅₃	—	2	4	no	—
172	C₂×D₅₃	—	2	4	no	—
173	C₂×D₅₃	—	2	128	no	—
174	C₂×D₅₃	C₂×C₂²⋊C₈	1	96	yes	direct product factor
175	C₂×D₅₃	C₂²×C₂²⋊C₄	1	8	yes	—
176	C₂×D₅₃	C₂³×C₈	1	8	yes	—
177	C₂×D₅₃	C₂³×C₈	1	8	yes	complemented
178	C₂×D₅₃	C₂²×M₁₆	1	8	yes	—
179	C₄×C₅₃	—	2	4	no	—
180	C₄×C₅₃	C₂²×C₂²⋊C₄	1	6	yes	—
181	C₄×C₅₃	C₂³×D₄	1	2	yes	—
182	Dic₅₃	—	2	4	no	—
183	Dic₅₃	C₂²×C₂²⋊C₄	1	8	yes	—
184	C₂⁵×C₄	—	53	1	no	—
185	C₂⁴×C₈	—	53	2	no	—
186	C₂²×C₂²⋊C₈	—	53	24	no	—
187	C₂²×C₂²⋊C₈	—	53	1	no	—
188	C₂²×C₂²⋊C₈	—	53	2	no	—
189	C₂²×C₂²⋊C₈	D₅₃	1	1	yes	characteristic, direct product factor
190	C₂×C₅₃	—	2	8	no	—
191	C₂×C₅₃	C₂²×C₂²⋊C₈	1	12	yes	complemented
192	C₂×C₅₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
193	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
194	C₂×C₅₃	C₂³×M₁₆	1	1	yes	characteristic
195	D₅₃	—	2	8	no	—
196	D₅₃	C₂²×C₂²⋊C₈	1	16	yes	direct product factor
197	C₂⁶	—	53	1	no	—
198	C₂⁴×C₄	—	53	1	no	—
199	C₂⁴×C₄	—	53	1	no	—
200	C₂⁴×C₄	—	53	1	no	—
201	C₂⁴×C₄	—	53	12	no	—
202	C₂⁴×C₄	—	53	1	no	—
203	C₂⁴×C₄	—	53	12	no	—
204	C₂⁴×C₄	—	53	1	no	—
205	C₂⁴×C₄	—	106	16	no	—
206	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
207	C₂³×C₈	—	53	2	no	—
208	C₂³×C₈	—	53	24	no	—
209	C₂³×C₈	—	106	16	no	—
210	C₂³×C₈	C₂×D₅₃	1	2	yes	complemented
211	C₂×C₂²⋊C₈	—	53	12	no	—
212	C₂×C₂²⋊C₈	—	53	24	no	—
213	C₂×C₂²⋊C₈	—	53	64	no	—
214	C₂×C₂²⋊C₈	C₂×D₅₃	1	12	yes	direct product factor
215	C₅₃	C₂³×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
216	C₂⁵	—	53	1	no	—
217	C₂⁵	—	53	12	no	—
218	C₂⁵	—	53	1	no	—
219	C₂⁵	—	53	16	no	—
220	C₂⁵	—	106	8	no	—
221	C₂⁵	—	106	8	no	—
222	C₂⁵	C₄×D₅₃	1	1	yes	characteristic
223	C₂³×C₄	—	2	8	no	—
224	C₂³×C₄	—	53	1	no	—
225	C₂³×C₄	—	53	6	no	—
226	C₂³×C₄	—	53	16	no	—
227	C₂³×C₄	—	53	6	no	—
228	C₂³×C₄	—	53	12	no	—
229	C₂³×C₄	—	53	1	no	—
230	C₂³×C₄	—	53	12	no	—
231	C₂³×C₄	—	53	48	no	—
232	C₂³×C₄	—	53	12	no	—
233	C₂³×C₄	—	53	6	no	—
234	C₂³×C₄	—	53	16	no	—
235	C₂³×C₄	—	53	6	no	—
236	C₂³×C₄	—	106	8	no	—
237	C₂³×C₄	—	106	96	no	—
238	C₂³×C₄	—	106	96	no	—
239	C₂³×C₄	—	106	8	no	—
240	C₂³×C₄	—	106	8	no	—
241	C₂³×C₄	—	106	8	no	—
242	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
243	C₂³×C₄	C₄×D₅₃	1	6	yes	—
244	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
245	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
246	C₂²×C₈	—	2	8	no	—
247	C₂²×C₈	—	53	12	no	—
248	C₂²×C₈	—	53	32	no	—
249	C₂²×C₈	—	106	96	no	—
250	C₂²×C₈	—	106	8	no	—
251	C₂²×C₈	C₂²×D₅₃	1	12	yes	complemented
252	C₂²⋊C₈	—	53	16	no	—
253	C₂²⋊C₈	—	53	32	no	—
254	C₂²⋊C₈	C₂²×D₅₃	1	16	yes	direct product factor
255	C₂⁴	—	2	4	no	—
256	C₂⁴	—	2	4	no	—
257	C₂⁴	—	53	8	no	—
258	C₂⁴	—	53	8	no	—
259	C₂⁴	—	53	8	no	—
260	C₂⁴	—	53	6	no	—
261	C₂⁴	—	53	8	no	—
262	C₂⁴	—	53	96	no	—
263	C₂⁴	—	53	16	no	—
264	C₂⁴	—	53	6	no	—
265	C₂⁴	—	106	48	no	—
266	C₂⁴	—	106	128	no	—
267	C₂⁴	—	106	48	no	—
268	C₂⁴	—	106	4	no	—
269	C₂⁴	—	106	4	no	—
270	C₂⁴	C₈×D₅₃	1	8	yes	complemented
271	C₂⁴	C₂×C₄×D₅₃	1	6	yes	—
272	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
273	C₂²×C₄	—	2	4	no	—
274	C₂²×C₄	—	2	24	no	—
275	C₂²×C₄	—	2	24	no	—
276	C₂²×C₄	—	53	8	no	—
277	C₂²×C₄	—	53	6	no	—
278	C₂²×C₄	—	53	48	no	—
279	C₂²×C₄	—	53	12	no	—
280	C₂²×C₄	—	53	48	no	—
281	C₂²×C₄	—	53	4	no	—
282	C₂²×C₄	—	53	16	no	—
283	C₂²×C₄	—	53	6	no	—
284	C₂²×C₄	—	53	16	no	—
285	C₂²×C₄	—	53	16	no	—
286	C₂²×C₄	—	53	12	no	—
287	C₂²×C₄	—	53	4	no	—
288	C₂²×C₄	—	106	48	no	—
289	C₂²×C₄	—	106	192	no	—
290	C₂²×C₄	—	106	64	no	—
291	C₂²×C₄	—	106	48	no	—
292	C₂²×C₄	—	106	48	no	—
293	C₂²×C₄	—	106	4	no	—
294	C₂²×C₄	—	106	24	no	—
295	C₂²×C₄	—	106	24	no	—
296	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
297	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
298	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
299	C₂²×C₄	D₄×D₅₃	1	8	yes	—
300	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
301	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
302	C₂×C₈	—	2	24	no	—
303	C₂×C₈	—	53	8	no	—
304	C₂×C₈	—	106	64	no	—
305	C₂×C₈	—	106	24	no	—
306	C₂×C₈	C₂³×D₅₃	1	8	yes	complemented
307	C₂³	—	2	12	no	—
308	C₂³	—	2	32	no	—
309	C₂³	—	2	12	no	—
310	C₂³	—	53	24	no	—
311	C₂³	—	53	64	no	—
312	C₂³	—	53	24	no	—
313	C₂³	—	53	24	no	—
314	C₂³	—	53	4	no	—
315	C₂³	—	53	24	no	—
316	C₂³	—	53	64	no	—
317	C₂³	—	53	4	no	—
318	C₂³	—	106	32	no	—
319	C₂³	—	106	384	no	—
320	C₂³	—	106	32	no	—
321	C₂³	—	106	12	no	—
322	C₂³	—	106	32	no	—
323	C₂³	—	106	12	no	—
324	C₂³	C₂²⋊C₄×D₅₃	1	4	yes	—
325	C₂³	C₂×C₈×D₅₃	1	4	yes	—
326	C₂³	C₂×C₈×D₅₃	1	24	yes	complemented
327	C₂³	M₁₆×D₅₃	1	4	yes	—
328	C₂³	C₂²×C₄×D₅₃	1	4	yes	—
329	C₂³	C₂²×C₄×D₅₃	1	3	yes	—
330	C₂×C₄	—	2	12	no	—
331	C₂×C₄	—	2	24	no	—
332	C₂×C₄	—	2	8	no	—
333	C₂×C₄	—	53	24	no	—
334	C₂×C₄	—	53	4	no	—
335	C₂×C₄	—	53	48	no	—
336	C₂×C₄	—	53	16	no	—
337	C₂×C₄	—	53	4	no	—
338	C₂×C₄	—	106	32	no	—
339	C₂×C₄	—	106	32	no	—
340	C₂×C₄	—	106	32	no	—
341	C₂×C₄	—	106	12	no	—
342	C₂×C₄	—	106	24	no	—
343	C₂×C₄	—	106	8	no	—
344	C₂×C₄	C₂²⋊C₄×D₅₃	1	12	yes	—
345	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
346	C₂×C₄	C₂²×C₄×D₅₃	1	3	yes	—
347	C₂×C₄	C₂×D₄×D₅₃	1	12	yes	—
348	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
349	C₈	—	2	8	no	—
350	C₈	—	106	8	no	—
351	C₂²	—	2	48	no	—
352	C₂²	—	2	4	no	—
353	C₂²	—	2	4	no	—
354	C₂²	—	53	96	no	—
355	C₂²	—	53	8	no	—
356	C₂²	—	53	8	no	—
357	C₂²	—	53	8	no	—
358	C₂²	—	53	8	no	—
359	C₂²	—	106	128	no	—
360	C₂²	—	106	48	no	—
361	C₂²	—	106	4	no	—
362	C₂²	—	106	4	no	—
363	C₂²	C₂²⋊C₈×D₅₃	1	16	yes	direct product factor
364	C₂²	C₂×C₂²⋊C₄×D₅₃	1	6	yes	—
365	C₂²	C₂²×C₈×D₅₃	1	6	yes	—
366	C₂²	C₂²×C₈×D₅₃	1	8	yes	complemented
367	C₂²	C₂×M₁₆×D₅₃	1	6	yes	—
368	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
369	C₄	—	2	4	no	—
370	C₄	—	53	8	no	—
371	C₄	—	106	4	no	—
372	C₄	C₂×C₂²⋊C₄×D₅₃	1	6	yes	—
373	C₄	C₂²×D₄×D₅₃	1	2	yes	—
374	C₂	—	2	8	no	—
375	C₂	—	53	16	no	—
376	C₂	—	106	8	no	—
377	C₂	C₂×C₂²⋊C₈×D₅₃	1	12	yes	direct product factor
378	C₂	C₂²×C₂²⋊C₄×D₅₃	1	1	yes	characteristic
379	C₂	C₂³×C₈×D₅₃	1	1	yes	characteristic
380	C₂	C₂²×M₁₆×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×C₂²⋊C₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂²×C₄
prime radical	C₂⁵×D₅₃	C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂²×C₂²⋊C₈	D₅₃
hypocenter	C₅₃	C₂³×C₂²⋊C₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂³×C₈×D₅₃	C₂	semidirect
3. C₂²×C₂²⋊#88322	C₂	semidirect
4. C₂²×#13133129	C₂	semidirect
5. C₂×C₂²⋊C₈×D₅₃	C₂	direct (N × H)
6. C₂²×C₂²⋊C₈×C₅₃	C₂	semidirect
16. C₂³×#88322	C₂²	semidirect
18. C₂²×C₈×D₅₃	C₂²	semidirect
19. C₂×C₂²⋊#88322	C₂²	semidirect
20. C₂×#13133129	C₂²	semidirect
21. C₂²⋊C₈×D₅₃	C₂²	direct (N × H)
23. C₂³×C₈×C₅₃	C₂²	semidirect
24. C₂×C₂²⋊C₈×C₅₃	C₂²	semidirect
27. C₂⁴×D₅₃	C₈	semidirect
50. C₂²×#88322	C₂³	semidirect
52. C₂×C₈×D₅₃	C₂³	semidirect
53. C₂²⋊#88322	C₂³	semidirect
54. #13133129	C₂³	semidirect
62. C₂²×C₈×C₅₃	C₂³	semidirect
63. C₂²⋊C₈×C₅₃	C₂³	semidirect
66. C₂⁴×C₅₃	C₂×C₈	semidirect
76. C₂³×D₅₃	C₂×C₈	semidirect
77. C₂³×D₅₃	C₂×C₈	semidirect
112. C₂×C₈×C₅₃	C₂⁴	semidirect
114. C₂×#88322	C₂⁴	semidirect
121. C₂³×C₅₃	C₂²×C₈	semidirect
131. C₂²×D₅₃	C₂²⋊C₈	direct (N × H)
134. C₂²×D₅₃	C₂²×C₈	semidirect
135. C₂²×D₅₃	C₂²×C₈	semidirect
164. C₂²×C₅₃	C₂×C₂²⋊C₈	semidirect
167. C₂²×C₅₃	C₂³×C₈	semidirect
174. C₂×D₅₃	C₂×C₂²⋊C₈	direct (N × H)
177. C₂×D₅₃	C₂³×C₈	semidirect
189. C₂²×C₂²⋊C₈	D₅₃	direct (N × H)
191. C₂×C₅₃	C₂²×C₂²⋊C₈	semidirect
196. D₅₃	C₂²×C₂²⋊C₈	direct (N × H)
210. C₂³×C₈	C₂×D₅₃	semidirect
214. C₂×C₂²⋊C₈	C₂×D₅₃	direct (N × H)
215. C₅₃	C₂³×C₂²⋊C₈	semidirect
251. C₂²×C₈	C₂²×D₅₃	semidirect
254. C₂²⋊C₈	C₂²×D₅₃	direct (N × H)
270. C₂⁴	C₈×D₅₃	semidirect
306. C₂×C₈	C₂³×D₅₃	semidirect
326. C₂³	C₂×C₈×D₅₃	semidirect
363. C₂²	C₂²⋊C₈×D₅₃	direct (N × H)
366. C₂²	C₂²×C₈×D₅₃	semidirect
377. C₂	C₂×C₂²⋊C₈×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
C₂⁴×C₄×D₅₃	2
C₂³×C₈×D₅₃	2
C₂²×C₂²⋊#88322	2
C₂²×#13133129	2
C₂×C₂²⋊C₈×D₅₃	2
C₂²×C₂²⋊C₈×C₅₃	2
C₂³×C₂²⋊C₈	53

Subgroups — C22×C22⋊C8×D53 #28247517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊C₈×D₅₃ #28247517