Subgroups — C₂²×#117×D₅₃ #28290648

Order 13568.

14334 conjugacy classes of subgroups.

The subgroup lattice has 396 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	C₂²×#13133176	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13133175	C₂	1	2	yes	maximal, complemented
4	C₂²×#13133174	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133173	C₂	1	2	yes	maximal, complemented
6	C₂×#117×D₅₃	C₂	1	48	yes	maximal, direct product factor
7	C₂²×C₄⋊C₄×D₅₃	C₂	1	2	yes	maximal, complemented
8	C₂²×C₂²⋊C₄×D₅₃	C₂	1	2	yes	maximal
9	C₂²×C₄²×D₅₃	C₂	1	2	yes	maximal, complemented
10	C₂²×#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
12	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×D₅₃	C₄	1	32	yes	complemented
14	C₂³×C₄×D₅₃	C₂²	1	4	yes	—
15	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
16	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
17	C₂³×C₄×D₅₃	C₂²	1	24	yes	—
18	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
19	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
20	C₂×#13133176	C₂²	1	24	yes	complemented
21	C₂×#13133175	C₂²	1	48	yes	complemented
22	C₂×#13133174	C₂²	1	48	yes	complemented
23	C₂×#13133173	C₂²	1	48	yes	complemented
24	#117×D₅₃	C₂²	1	256	yes	direct product factor
25	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	complemented
26	C₂²×#11719919	C₂²	1	4	yes	complemented
27	C₂×C₄⋊C₄×D₅₃	C₂²	1	48	yes	complemented
28	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	—
29	C₂²×#11719921	C₂²	1	4	yes	—
30	C₂×C₂²⋊C₄×D₅₃	C₂²	1	48	yes	—
31	C₂²×C₄×Dic₅₃	C₂²	1	4	yes	complemented
32	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	complemented
33	C₂×C₄²×D₅₃	C₂²	1	48	yes	complemented
34	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×#117×C₅₃	C₂²	1	24	yes	complemented
36	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
37	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
38	C₂²×C₄²×C₅₃	C₂²	1	2	yes	complemented
39	C₂⁴×D₅₃	—	2	16	no	—
40	C₂⁴×D₅₃	C₂×C₄	1	16	yes	—
41	C₂⁴×D₅₃	C₂³	1	12	yes	—
42	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
43	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
44	C₂³×Dic₅₃	C₂×C₄	1	16	yes	complemented
45	C₂³×Dic₅₃	C₂³	1	4	yes	—
46	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
47	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
48	C₂³×Dic₅₃	C₂³	1	12	yes	—
49	C₂²×C₄×D₅₃	—	2	32	no	—
50	C₂²×C₄×D₅₃	C₂×C₄	1	16	yes	—
51	C₂²×C₄×D₅₃	C₂×C₄	1	16	yes	—
52	C₂²×C₄×D₅₃	C₂×C₄	1	384	yes	complemented
53	C₂²×C₄×D₅₃	C₂×C₄	1	16	yes	complemented
54	C₂²×C₄×D₅₃	C₂×C₄	1	16	yes	complemented
55	C₂²×C₄×D₅₃	C₂³	1	48	yes	—
56	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
57	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
58	C₂²×C₄×D₅₃	C₂³	1	64	yes	—
59	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
60	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
61	#13133176	C₂³	1	64	yes	complemented
62	#13133175	C₂³	1	128	yes	complemented
63	#13133174	C₂³	1	128	yes	complemented
64	#13133173	C₂³	1	128	yes	complemented
65	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	complemented
66	C₂×#11719919	C₂³	1	48	yes	complemented
67	C₄⋊C₄×D₅₃	C₂³	1	128	yes	complemented
68	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	—
69	C₂×#11719921	C₂³	1	48	yes	—
70	C₂²⋊C₄×D₅₃	C₂³	1	128	yes	—
71	C₂×C₄×Dic₅₃	C₂³	1	48	yes	complemented
72	C₂×C₄×Dic₅₃	C₂³	1	24	yes	complemented
73	C₄²×D₅₃	C₂³	1	128	yes	complemented
74	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	complemented
76	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
77	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
80	#117×C₅₃	C₂³	1	64	yes	complemented
81	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	complemented
82	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
83	C₂×C₄²×C₅₃	C₂³	1	24	yes	complemented
84	C₂⁴×C₅₃	—	2	8	no	—
85	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	—
86	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
87	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
88	C₂³×D₅₃	—	2	8	no	—
89	C₂³×D₅₃	—	2	128	no	—
90	C₂³×D₅₃	—	2	96	no	—
91	C₂³×D₅₃	C₂²×C₄	1	8	yes	—
92	C₂³×D₅₃	C₂²×C₄	1	96	yes	—
93	C₂³×D₅₃	C₂²×C₄	1	8	yes	—
94	C₂³×D₅₃	#41	1	16	yes	—
95	C₂³×D₅₃	C₂⁴	1	16	yes	—
96	C₂³×D₅₃	C₂⁴	1	6	yes	—
97	C₂³×D₅₃	C₂⁴	1	6	yes	—
98	C₂²×C₄×C₅₃	—	2	16	no	—
99	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
100	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
101	C₂²×C₄×C₅₃	C₂²×C₄	1	96	yes	complemented
102	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
103	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
104	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
105	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
106	C₂²×Dic₅₃	—	2	16	no	—
107	C₂²×Dic₅₃	C₂²×C₄	1	8	yes	—
108	C₂²×Dic₅₃	C₂²×C₄	1	8	yes	—
109	C₂²×Dic₅₃	C₂²×C₄	1	96	yes	complemented
110	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
111	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
112	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
113	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
114	C₂×C₄×D₅₃	—	2	192	no	—
115	C₂×C₄×D₅₃	C₂²×C₄	1	96	yes	—
116	C₂×C₄×D₅₃	C₂²×C₄	1	96	yes	—
117	C₂×C₄×D₅₃	C₂²×C₄	1	512	yes	complemented
118	C₂×C₄×D₅₃	C₂²×C₄	1	96	yes	complemented
119	C₂×C₄×D₅₃	C₂²×C₄	1	96	yes	complemented
120	C₂×C₄×D₅₃	C₂⁴	1	64	yes	—
121	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
122	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
123	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
124	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
125	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
126	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
127	C₄²×C₅₃	C₂⁴	1	32	yes	complemented
128	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	—
129	#11719921	C₂⁴	1	64	yes	—
130	C₄⋊Dic₅₃	C₂⁴	1	32	yes	complemented
131	#11719919	C₂⁴	1	64	yes	complemented
132	C₄×Dic₅₃	C₂⁴	1	64	yes	complemented
133	C₄×Dic₅₃	C₂⁴	1	32	yes	complemented
134	C₂³×C₅₃	—	2	32	no	—
135	C₂³×C₅₃	—	2	24	no	—
136	C₂³×C₅₃	C₂³×C₄	1	24	yes	—
137	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
138	C₂³×C₅₃	C₂×#41	1	8	yes	—
139	C₂³×C₅₃	C₂⁵	1	4	yes	—
140	C₂³×C₅₃	C₂⁵	1	3	yes	—
141	C₂²×D₅₃	—	2	32	no	—
142	C₂²×D₅₃	—	2	24	no	—
143	C₂²×D₅₃	—	2	384	no	—
144	C₂²×D₅₃	—	2	64	no	—
145	C₂²×D₅₃	#117	1	64	yes	direct product factor
146	C₂²×D₅₃	C₂³×C₄	1	24	yes	—
147	C₂²×D₅₃	C₂³×C₄	1	64	yes	—
148	C₂²×D₅₃	C₂³×C₄	1	24	yes	—
149	C₂²×D₅₃	C₂×#41	1	48	yes	—
150	C₂²×D₅₃	C₂⁵	1	4	yes	—
151	C₂²×D₅₃	C₂⁵	1	4	yes	—
152	C₂×C₄×C₅₃	—	2	48	no	—
153	C₂×C₄×C₅₃	C₂³×C₄	1	24	yes	—
154	C₂×C₄×C₅₃	C₂³×C₄	1	24	yes	—
155	C₂×C₄×C₅₃	C₂³×C₄	1	64	yes	complemented
156	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
157	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
158	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
159	C₂×Dic₅₃	—	2	48	no	—
160	C₂×Dic₅₃	C₂³×C₄	1	24	yes	—
161	C₂×Dic₅₃	C₂³×C₄	1	24	yes	—
162	C₂×Dic₅₃	C₂³×C₄	1	64	yes	complemented
163	C₂×Dic₅₃	C₂⁵	1	16	yes	—
164	C₂×Dic₅₃	C₂⁵	1	4	yes	—
165	C₂×Dic₅₃	C₂⁵	1	4	yes	—
166	C₄×D₅₃	—	2	128	no	—
167	C₄×D₅₃	C₂³×C₄	1	64	yes	—
168	C₄×D₅₃	C₂³×C₄	1	64	yes	—
169	C₄×D₅₃	C₂³×C₄	1	64	yes	complemented
170	C₄×D₅₃	C₂³×C₄	1	64	yes	complemented
171	C₂³×#117	—	53	1	no	maximal, Sylow
172	C₂²×C₅₃	—	2	8	no	—
173	C₂²×C₅₃	—	2	48	no	—
174	C₂²×C₅₃	C₂×#117	1	16	yes	complemented
175	C₂²×C₅₃	C₂⁴×C₄	1	8	yes	—
176	C₂²×C₅₃	C₂⁴×C₄	1	6	yes	—
177	C₂²×C₅₃	C₂²×#41	1	12	yes	—
178	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
179	C₂×D₅₃	—	2	8	no	—
180	C₂×D₅₃	—	2	48	no	—
181	C₂×D₅₃	—	2	128	no	—
182	C₂×D₅₃	C₂×#117	1	96	yes	direct product factor
183	C₂×D₅₃	C₂⁴×C₄	1	8	yes	—
184	C₂×D₅₃	C₂⁴×C₄	1	8	yes	—
185	C₂×D₅₃	C₂²×#41	1	16	yes	—
186	C₄×C₅₃	—	2	16	no	—
187	C₄×C₅₃	C₂⁴×C₄	1	8	yes	—
188	C₄×C₅₃	C₂⁴×C₄	1	8	yes	—
189	Dic₅₃	—	2	16	no	—
190	Dic₅₃	C₂⁴×C₄	1	8	yes	—
191	Dic₅₃	C₂⁴×C₄	1	8	yes	—
192	C₂⁵×C₄	—	53	1	no	—
193	C₂²×#117	—	53	2	no	—
194	C₂²×#117	—	53	1	no	—
195	C₂²×#117	—	53	48	no	—
196	C₂²×#117	—	53	2	no	—
197	C₂²×#117	—	53	2	no	—
198	C₂²×#117	D₅₃	1	1	yes	characteristic, direct product factor
199	C₂³×C₄⋊C₄	—	53	2	no	—
200	C₂³×C₂²⋊C₄	—	53	2	no	—
201	C₂³×C₄²	—	53	2	no	—
202	C₂×C₅₃	—	2	8	no	—
203	C₂×C₅₃	C₂²×#117	1	12	yes	complemented
204	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
205	C₂×C₅₃	C₂³×#41	1	2	yes	—
206	D₅₃	—	2	8	no	—
207	D₅₃	C₂²×#117	1	16	yes	direct product factor
208	C₂⁶	—	53	1	no	—
209	C₂⁴×C₄	—	53	1	no	—
210	C₂⁴×C₄	—	53	32	no	—
211	C₂⁴×C₄	—	53	1	no	—
212	C₂⁴×C₄	—	53	24	no	—
213	C₂⁴×C₄	—	53	1	no	—
214	C₂⁴×C₄	—	53	1	no	—
215	C₂⁴×C₄	—	53	1	no	—
216	C₂⁴×C₄	—	53	4	no	—
217	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
218	C₂×#117	—	53	24	no	—
219	C₂×#117	—	53	256	no	—
220	C₂×#117	—	53	48	no	—
221	C₂×#117	—	53	48	no	—
222	C₂×#117	—	53	48	no	—
223	C₂×#117	C₂×D₅₃	1	24	yes	direct product factor
224	C₂²×C₄⋊C₄	—	53	4	no	—
225	C₂²×C₄⋊C₄	—	53	2	no	—
226	C₂²×C₄⋊C₄	—	53	48	no	—
227	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	complemented
228	C₂²×C₂²⋊C₄	—	53	2	no	—
229	C₂²×C₂²⋊C₄	—	53	4	no	—
230	C₂²×C₂²⋊C₄	—	53	48	no	—
231	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
232	C₂²×C₄²	—	53	4	no	—
233	C₂²×C₄²	—	53	2	no	—
234	C₂²×C₄²	—	53	48	no	—
235	C₂²×C₄²	C₂×D₅₃	1	2	yes	complemented
236	C₅₃	C₂³×#117	1	1	yes	characteristic, complemented, Sylow
237	C₂⁵	—	53	1	no	—
238	C₂⁵	—	53	16	no	—
239	C₂⁵	—	53	12	no	—
240	C₂⁵	—	53	1	no	—
241	C₂⁵	—	106	16	no	—
242	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
243	C₂³×C₄	—	53	16	no	—
244	C₂³×C₄	—	53	1	no	—
245	C₂³×C₄	—	53	12	no	—
246	C₂³×C₄	—	53	1	no	—
247	C₂³×C₄	—	53	16	no	—
248	C₂³×C₄	—	53	16	no	—
249	C₂³×C₄	—	53	16	no	—
250	C₂³×C₄	—	53	16	no	—
251	C₂³×C₄	—	53	384	no	—
252	C₂³×C₄	—	53	12	no	—
253	C₂³×C₄	—	53	12	no	—
254	C₂³×C₄	—	53	12	no	—
255	C₂³×C₄	—	53	64	no	—
256	C₂³×C₄	—	53	12	no	—
257	C₂³×C₄	—	53	4	no	—
258	C₂³×C₄	—	53	48	no	—
259	C₂³×C₄	—	106	32	no	—
260	C₂³×C₄	C₄×D₅₃	1	16	yes	complemented
261	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
262	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
263	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
264	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
265	#117	—	53	64	no	—
266	#117	—	53	128	no	—
267	#117	—	53	128	no	—
268	#117	—	53	128	no	—
269	#117	C₂²×D₅₃	1	64	yes	direct product factor
270	C₂×C₄⋊C₄	—	53	48	no	—
271	C₂×C₄⋊C₄	—	53	24	no	—
272	C₂×C₄⋊C₄	—	53	128	no	—
273	C₂×C₄⋊C₄	C₂²×D₅₃	1	24	yes	complemented
274	C₂×C₂²⋊C₄	—	53	24	no	—
275	C₂×C₂²⋊C₄	—	53	48	no	—
276	C₂×C₂²⋊C₄	—	53	128	no	—
277	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	—
278	C₂×C₄²	—	53	48	no	—
279	C₂×C₄²	—	53	24	no	—
280	C₂×C₄²	—	53	128	no	—
281	C₂×C₄²	C₂²×D₅₃	1	24	yes	complemented
282	C₂⁴	—	2	8	no	—
283	C₂⁴	—	53	8	no	—
284	C₂⁴	—	53	16	no	—
285	C₂⁴	—	53	6	no	—
286	C₂⁴	—	53	96	no	—
287	C₂⁴	—	53	8	no	—
288	C₂⁴	—	53	16	no	—
289	C₂⁴	—	53	6	no	—
290	C₂⁴	—	106	128	no	—
291	C₂⁴	—	106	96	no	—
292	C₂⁴	—	106	8	no	—
293	C₂⁴	C₂×C₄×D₅₃	1	8	yes	—
294	C₂⁴	C₂³×D₅₃	1	6	yes	—
295	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
296	C₂²×C₄	—	2	16	no	—
297	C₂²×C₄	—	53	8	no	—
298	C₂²×C₄	—	53	8	no	—
299	C₂²×C₄	—	53	96	no	—
300	C₂²×C₄	—	53	6	no	—
301	C₂²×C₄	—	53	6	no	—
302	C₂²×C₄	—	53	16	no	—
303	C₂²×C₄	—	53	96	no	—
304	C₂²×C₄	—	53	96	no	—
305	C₂²×C₄	—	53	96	no	—
306	C₂²×C₄	—	53	96	no	—
307	C₂²×C₄	—	53	512	no	—
308	C₂²×C₄	—	53	16	no	—
309	C₂²×C₄	—	53	16	no	—
310	C₂²×C₄	—	53	16	no	—
311	C₂²×C₄	—	53	16	no	—
312	C₂²×C₄	—	53	24	no	—
313	C₂²×C₄	—	53	64	no	—
314	C₂²×C₄	—	106	192	no	—
315	C₂²×C₄	—	106	16	no	—
316	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
317	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
318	C₂²×C₄	C₂×C₄×D₅₃	1	96	yes	complemented
319	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
320	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
321	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
322	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
323	C₄⋊C₄	—	53	64	no	—
324	C₄⋊C₄	—	53	32	no	—
325	C₄⋊C₄	C₂³×D₅₃	1	32	yes	complemented
326	C₂²⋊C₄	—	53	32	no	—
327	C₂²⋊C₄	—	53	64	no	—
328	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	—
329	C₄²	—	53	64	no	—
330	C₄²	—	53	32	no	—
331	C₄²	C₂³×D₅₃	1	32	yes	complemented
332	C₂³	—	2	32	no	—
333	C₂³	—	2	24	no	—
334	C₂³	—	53	64	no	—
335	C₂³	—	53	24	no	—
336	C₂³	—	53	48	no	—
337	C₂³	—	53	4	no	—
338	C₂³	—	53	64	no	—
339	C₂³	—	53	24	no	—
340	C₂³	—	53	4	no	—
341	C₂³	—	106	384	no	—
342	C₂³	—	106	64	no	—
343	C₂³	—	106	32	no	—
344	C₂³	—	106	24	no	—
345	C₂³	C₂²×C₄×D₅₃	1	24	yes	—
346	C₂³	C₂²×C₄×D₅₃	1	4	yes	—
347	C₂³	#41×D₅₃	1	8	yes	—
348	C₂³	C₂⁴×D₅₃	1	4	yes	—
349	C₂³	C₂⁴×D₅₃	1	3	yes	—
350	C₂×C₄	—	2	48	no	—
351	C₂×C₄	—	53	24	no	—
352	C₂×C₄	—	53	24	no	—
353	C₂×C₄	—	53	64	no	—
354	C₂×C₄	—	53	4	no	—
355	C₂×C₄	—	53	4	no	—
356	C₂×C₄	—	53	64	no	—
357	C₂×C₄	—	53	64	no	—
358	C₂×C₄	—	53	64	no	—
359	C₂×C₄	—	53	64	no	—
360	C₂×C₄	—	53	16	no	—
361	C₂×C₄	—	106	128	no	—
362	C₂×C₄	—	106	48	no	—
363	C₂×C₄	C₂²×C₄×D₅₃	1	24	yes	—
364	C₂×C₄	C₂²×C₄×D₅₃	1	24	yes	—
365	C₂×C₄	C₂²×C₄×D₅₃	1	64	yes	complemented
366	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
367	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
368	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
369	C₂²	—	2	48	no	—
370	C₂²	—	2	8	no	—
371	C₂²	—	53	96	no	—
372	C₂²	—	53	8	no	—
373	C₂²	—	53	16	no	—
374	C₂²	—	53	8	no	—
375	C₂²	—	106	128	no	—
376	C₂²	—	106	48	no	—
377	C₂²	—	106	8	no	—
378	C₂²	#117×D₅₃	1	16	yes	direct product factor
379	C₂²	C₂³×C₄×D₅₃	1	8	yes	—
380	C₂²	C₂³×C₄×D₅₃	1	6	yes	—
381	C₂²	C₂×#41×D₅₃	1	12	yes	—
382	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
383	C₄	—	2	16	no	—
384	C₄	—	53	8	no	—
385	C₄	—	53	8	no	—
386	C₄	—	106	16	no	—
387	C₄	C₂³×C₄×D₅₃	1	8	yes	—
388	C₄	C₂³×C₄×D₅₃	1	8	yes	—
389	C₂	—	2	8	no	—
390	C₂	—	53	16	no	—
391	C₂	—	106	8	no	—
392	C₂	C₂×#117×D₅₃	1	12	yes	direct product factor
393	C₂	C₂⁴×C₄×D₅₃	1	1	yes	characteristic
394	C₂	C₂²×#41×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×#117×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁵×D₅₃	C₂²
cosocle	C₂²×C₅₃	C₂⁶
hypercenter	C₂²×#117	D₅₃
hypocenter	C₅₃	C₂³×#117
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
2. C₂²×#13133176	C₂	semidirect
3. C₂²×#13133175	C₂	semidirect
4. C₂²×#13133174	C₂	semidirect
5. C₂²×#13133173	C₂	semidirect
6. C₂×#117×D₅₃	C₂	direct (N × H)
7. C₂²×C₄⋊C₄×D₅₃	C₂	semidirect
9. C₂²×C₄²×D₅₃	C₂	semidirect
10. C₂²×#117×C₅₃	C₂	semidirect
13. C₂³×C₄×D₅₃	C₄	semidirect
20. C₂×#13133176	C₂²	semidirect
21. C₂×#13133175	C₂²	semidirect
22. C₂×#13133174	C₂²	semidirect
23. C₂×#13133173	C₂²	semidirect
24. #117×D₅₃	C₂²	direct (N × H)
25. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
26. C₂²×#11719919	C₂²	semidirect
27. C₂×C₄⋊C₄×D₅₃	C₂²	semidirect
31. C₂²×C₄×Dic₅₃	C₂²	semidirect
32. C₂²×C₄×Dic₅₃	C₂²	semidirect
33. C₂×C₄²×D₅₃	C₂²	semidirect
35. C₂×#117×C₅₃	C₂²	semidirect
36. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
38. C₂²×C₄²×C₅₃	C₂²	semidirect
44. C₂³×Dic₅₃	C₂×C₄	semidirect
52. C₂²×C₄×D₅₃	C₂×C₄	semidirect
53. C₂²×C₄×D₅₃	C₂×C₄	semidirect
54. C₂²×C₄×D₅₃	C₂×C₄	semidirect
61. #13133176	C₂³	semidirect
62. #13133175	C₂³	semidirect
63. #13133174	C₂³	semidirect
64. #13133173	C₂³	semidirect
65. C₂×C₄⋊Dic₅₃	C₂³	semidirect
66. C₂×#11719919	C₂³	semidirect
67. C₄⋊C₄×D₅₃	C₂³	semidirect
71. C₂×C₄×Dic₅₃	C₂³	semidirect
72. C₂×C₄×Dic₅₃	C₂³	semidirect
73. C₄²×D₅₃	C₂³	semidirect
75. C₂³×C₄×C₅₃	C₂×C₄	semidirect
80. #117×C₅₃	C₂³	semidirect
81. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
83. C₂×C₄²×C₅₃	C₂³	semidirect
101. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
109. C₂²×Dic₅₃	C₂²×C₄	semidirect
117. C₂×C₄×D₅₃	C₂²×C₄	semidirect
118. C₂×C₄×D₅₃	C₂²×C₄	semidirect
119. C₂×C₄×D₅₃	C₂²×C₄	semidirect
125. C₄⋊C₄×C₅₃	C₂⁴	semidirect
127. C₄²×C₅₃	C₂⁴	semidirect
130. C₄⋊Dic₅₃	C₂⁴	semidirect
131. #11719919	C₂⁴	semidirect
132. C₄×Dic₅₃	C₂⁴	semidirect
133. C₄×Dic₅₃	C₂⁴	semidirect
145. C₂²×D₅₃	#117	direct (N × H)
155. C₂×C₄×C₅₃	C₂³×C₄	semidirect
162. C₂×Dic₅₃	C₂³×C₄	semidirect
169. C₄×D₅₃	C₂³×C₄	semidirect
170. C₄×D₅₃	C₂³×C₄	semidirect
174. C₂²×C₅₃	C₂×#117	semidirect
182. C₂×D₅₃	C₂×#117	direct (N × H)
198. C₂²×#117	D₅₃	direct (N × H)
203. C₂×C₅₃	C₂²×#117	semidirect
207. D₅₃	C₂²×#117	direct (N × H)
223. C₂×#117	C₂×D₅₃	direct (N × H)
227. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
235. C₂²×C₄²	C₂×D₅₃	semidirect
236. C₅₃	C₂³×#117	semidirect
260. C₂³×C₄	C₄×D₅₃	semidirect
269. #117	C₂²×D₅₃	direct (N × H)
273. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
281. C₂×C₄²	C₂²×D₅₃	semidirect
318. C₂²×C₄	C₂×C₄×D₅₃	semidirect
325. C₄⋊C₄	C₂³×D₅₃	semidirect
331. C₄²	C₂³×D₅₃	semidirect
365. C₂×C₄	C₂²×C₄×D₅₃	semidirect
378. C₂²	#117×D₅₃	direct (N × H)
392. C₂	C₂×#117×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
C₂²×#13133176	2
C₂²×#13133175	2
C₂²×#13133174	2
C₂²×#13133173	2
C₂×#117×D₅₃	2
C₂²×C₄⋊C₄×D₅₃	2
C₂²×C₂²⋊C₄×D₅₃	2
C₂²×C₄²×D₅₃	2
C₂²×#117×C₅₃	2
C₂³×#117	53

Subgroups — C22×#117×D53 #28290648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#117×D₅₃ #28290648