Subgroups — C₂×C₅₃×#3334 #27268737

Order 13568.

2374 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#547	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3334	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#480	C₂	1	4	yes	maximal, complemented
5	C₂×C₅₃×#474	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#466	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#107×C₅₃	C₂	1	2	yes	maximal, complemented
10	C₂²×#41×C₅₃	—	2	2	no	—
11	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#547	—	2	8	no	—
14	C₅₃×#547	C₂²	1	16	yes	complemented
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
16	C₂×#123×C₅₃	—	2	4	no	—
17	C₂×C₂²⋊Q₈×C₅₃	—	2	2	no	—
18	C₂×#121×C₅₃	C₂²	1	4	yes	—
19	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
22	C₅₃×#500	C₂²	1	8	yes	complemented
23	C₅₃×#480	C₂²	1	32	yes	complemented
24	C₅₃×#474	C₂²	1	16	yes	complemented
25	C₅₃×#466	C₂²	1	16	yes	complemented
26	C₄×D₈×C₅₃	C₂²	1	8	yes	complemented
27	C₈×D₄×C₅₃	C₂²	1	8	yes	complemented
28	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×#107×C₅₃	C₂²	1	2	yes	—
30	C₂×#107×C₅₃	C₂²	1	4	yes	—
31	C₂×#107×C₅₃	C₂²	1	2	yes	—
32	C₂×#107×C₅₃	C₂²	1	16	yes	complemented
33	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#102×C₅₃	C₂²	1	2	yes	—
35	C₂×#102×C₅₃	C₂²	1	4	yes	—
36	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×#41×C₅₃	—	2	16	no	—
39	C₂×#41×C₅₃	—	4	16	no	—
40	C₂²×Q₈×C₅₃	—	2	2	no	—
41	C₂²×D₄×C₅₃	—	2	4	no	—
42	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
43	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
44	C₂³×C₄×C₅₃	—	2	4	no	—
45	C₂³×C₄×C₅₃	—	2	2	no	—
46	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
47	C₂×D₈×C₅₃	—	2	4	no	—
48	C₂×D₈×C₅₃	C₂³	1	4	yes	—
49	C₂²×C₈×C₅₃	—	2	8	no	—
50	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂³	1	8	yes	—
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₄⋊Q₈×C₅₃	—	2	4	no	—
55	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	—
56	#123×C₅₃	—	2	32	no	—
57	C₂²⋊Q₈×C₅₃	—	2	16	no	—
58	#121×C₅₃	—	2	8	no	—
59	#121×C₅₃	C₂³	1	16	yes	—
60	C₄×D₄×C₅₃	—	2	4	no	—
61	C₄×D₄×C₅₃	D₄	1	4	yes	—
62	C₄×D₄×C₅₃	C₂³	1	8	yes	—
63	C₄×D₄×C₅₃	C₂³	1	4	yes	—
64	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
65	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
66	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
70	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
71	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
73	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
74	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
75	#107×C₅₃	C₂³	1	8	yes	—
76	#107×C₅₃	C₂³	1	8	yes	—
77	#107×C₅₃	C₂³	1	4	yes	—
78	#107×C₅₃	C₂³	1	16	yes	—
79	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
80	#102×C₅₃	C₂³	1	8	yes	—
81	#102×C₅₃	C₂³	1	16	yes	—
82	C₂²⋊C₈×C₅₃	C₂³	1	8	yes	—
83	C₄×C₈×C₅₃	C₂³	1	4	yes	—
84	C₂⁴×C₅₃	—	2	2	no	—
85	C₂⁴×C₅₃	C₂×D₄	1	2	yes	—
86	#41×C₅₃	—	4	64	no	—
87	C₂×Q₈×C₅₃	—	2	8	no	—
88	C₂×Q₈×C₅₃	—	4	8	no	—
89	C₂×D₄×C₅₃	—	2	16	no	—
90	C₂×D₄×C₅₃	—	2	2	no	—
91	C₂×D₄×C₅₃	—	2	4	no	—
92	C₂×D₄×C₅₃	—	2	4	no	—
93	C₂×D₄×C₅₃	—	4	16	no	—
94	C₂×D₄×C₅₃	D₈	1	8	yes	complemented
95	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
96	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
97	C₂²×C₄×C₅₃	—	2	4	no	—
98	C₂²×C₄×C₅₃	—	2	4	no	—
99	C₂²×C₄×C₅₃	—	2	16	no	—
100	C₂²×C₄×C₅₃	—	2	2	no	—
101	C₂²×C₄×C₅₃	—	2	4	no	—
102	C₂²×C₄×C₅₃	—	2	8	no	—
103	C₂²×C₄×C₅₃	—	4	16	no	—
104	C₂²×C₄×C₅₃	—	4	8	no	—
105	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
106	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
110	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
111	D₈×C₅₃	—	2	8	no	—
112	C₂×C₈×C₅₃	—	2	4	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	2	16	no	—
115	C₂×C₈×C₅₃	#41	1	4	yes	complemented
116	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
117	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
118	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
119	C₄⋊C₄×C₅₃	—	2	2	no	—
120	C₄⋊C₄×C₅₃	—	2	4	no	—
121	C₄⋊C₄×C₅₃	—	2	16	no	—
122	C₄⋊C₄×C₅₃	—	2	8	no	—
123	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
124	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
125	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
126	C₂²⋊C₄×C₅₃	—	2	8	no	—
127	C₂²⋊C₄×C₅₃	—	2	16	no	—
128	C₂²⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
129	C₄²×C₅₃	C₂×D₄	1	2	yes	—
130	C₄²×C₅₃	C₂⁴	1	2	yes	—
131	C₂³×C₅₃	—	2	4	no	—
132	C₂³×C₅₃	—	2	4	no	—
133	C₂³×C₅₃	—	2	4	no	—
134	C₂³×C₅₃	—	4	4	no	—
135	C₂³×C₅₃	—	4	4	no	—
136	C₂³×C₅₃	—	4	4	no	—
137	C₂³×C₅₃	C₂×D₈	1	8	yes	—
138	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
139	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
140	Q₈×C₅₃	—	4	16	no	—
141	D₄×C₅₃	—	2	4	no	—
142	D₄×C₅₃	—	2	4	no	—
143	D₄×C₅₃	—	4	32	no	—
144	D₄×C₅₃	—	4	4	no	—
145	D₄×C₅₃	C₂×D₈	1	16	yes	complemented
146	C₂×C₄×C₅₃	—	2	4	no	—
147	C₂×C₄×C₅₃	—	2	2	no	—
148	C₂×C₄×C₅₃	—	2	8	no	—
149	C₂×C₄×C₅₃	—	2	4	no	—
150	C₂×C₄×C₅₃	—	2	4	no	—
151	C₂×C₄×C₅₃	—	2	4	no	—
152	C₂×C₄×C₅₃	—	2	8	no	—
153	C₂×C₄×C₅₃	—	4	4	no	—
154	C₂×C₄×C₅₃	—	4	8	no	—
155	C₂×C₄×C₅₃	—	4	32	no	—
156	C₂×C₄×C₅₃	—	4	16	no	—
157	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
158	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
159	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
160	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
162	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
163	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
164	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
165	C₈×C₅₃	—	2	4	no	—
166	C₈×C₅₃	—	2	2	no	—
167	C₈×C₅₃	C₂×#41	1	4	yes	complemented
168	C₂×#3334	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₂²×C₅₃	—	2	16	no	—
170	C₂²×C₅₃	—	2	4	no	—
171	C₂²×C₅₃	—	2	4	no	—
172	C₂²×C₅₃	—	4	4	no	—
173	C₂²×C₅₃	—	4	4	no	—
174	C₂²×C₅₃	—	4	4	no	—
175	C₂²×C₅₃	—	8	8	no	—
176	C₂²×C₅₃	#547	1	2	yes	—
177	C₂²×C₅₃	C₂²×D₈	1	8	yes	—
178	C₂²×C₅₃	C₂²×D₈	1	2	yes	—
179	C₂²×C₅₃	#578	1	2	yes	—
180	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
181	C₄×C₅₃	—	2	4	no	—
182	C₄×C₅₃	—	2	2	no	—
183	C₄×C₅₃	—	4	4	no	—
184	C₄×C₅₃	—	4	8	no	—
185	C₄×C₅₃	#547	1	2	yes	—
186	C₄×C₅₃	C₂²×D₈	1	4	yes	—
187	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
188	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
189	C₂×#547	C₂×C₅₃	1	2	yes	complemented
190	#3334	C₂×C₅₃	1	16	yes	direct product factor
191	C₂×#500	C₂×C₅₃	1	1	yes	characteristic, complemented
192	C₂×#480	C₂×C₅₃	1	4	yes	complemented
193	C₂×#474	C₂×C₅₃	1	2	yes	complemented
194	C₂×#466	C₂×C₅₃	1	2	yes	complemented
195	C₂×C₄×D₈	C₂×C₅₃	1	1	yes	characteristic, complemented
196	C₂×C₈×D₄	C₂×C₅₃	1	1	yes	characteristic, complemented
197	C₂²×#107	C₂×C₅₃	1	2	yes	complemented
198	C₂×C₅₃	—	2	8	no	—
199	C₂×C₅₃	—	8	4	no	—
200	C₂×C₅₃	#3334	1	4	yes	direct product factor
201	C₂×C₅₃	C₂×#547	1	1	yes	characteristic
202	C₂×C₅₃	C₂³×D₈	1	1	yes	characteristic
203	C₂×C₅₃	C₂×#578	1	1	yes	characteristic
204	C₂²×#41	—	2	2	no	—
205	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
206	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
207	#547	—	2	8	no	—
208	#547	C₂²×C₅₃	1	16	yes	complemented
209	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
210	C₂×#123	—	2	4	no	—
211	C₂×C₂²⋊Q₈	—	2	2	no	—
212	C₂×#121	C₂²×C₅₃	1	4	yes	—
213	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
214	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
215	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
216	#500	C₂²×C₅₃	1	8	yes	complemented
217	#480	C₂²×C₅₃	1	32	yes	complemented
218	#474	C₂²×C₅₃	1	16	yes	complemented
219	#466	C₂²×C₅₃	1	16	yes	complemented
220	C₄×D₈	C₂²×C₅₃	1	8	yes	complemented
221	C₈×D₄	C₂²×C₅₃	1	8	yes	complemented
222	C₂×#107	C₂²×C₅₃	1	2	yes	—
223	C₂×#107	C₂²×C₅₃	1	4	yes	—
224	C₂×#107	C₂²×C₅₃	1	2	yes	—
225	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
226	C₂×#107	C₂²×C₅₃	1	16	yes	complemented
227	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
228	C₂×#102	C₂²×C₅₃	1	2	yes	—
229	C₂×#102	C₂²×C₅₃	1	4	yes	—
230	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
231	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
232	C₅₃	C₂×#3334	1	1	yes	characteristic, direct product factor, Sylow
233	C₂×#41	—	2	16	no	—
234	C₂×#41	—	4	16	no	—
235	C₂²×Q₈	—	2	2	no	—
236	C₂²×D₄	—	2	4	no	—
237	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
238	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
239	C₂³×C₄	—	2	4	no	—
240	C₂³×C₄	—	2	2	no	—
241	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
242	C₂×D₈	—	2	4	no	—
243	C₂×D₈	C₂³×C₅₃	1	4	yes	—
244	C₂²×C₈	—	2	8	no	—
245	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
246	C₂²×C₈	C₂³×C₅₃	1	8	yes	—
247	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
248	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
249	C₄⋊Q₈	—	2	4	no	—
250	C₄⋊Q₈	C₂³×C₅₃	1	8	yes	—
251	#123	—	2	32	no	—
252	C₂²⋊Q₈	—	2	16	no	—
253	#121	—	2	8	no	—
254	#121	C₂³×C₅₃	1	16	yes	—
255	C₄×D₄	—	2	4	no	—
256	C₄×D₄	D₄×C₅₃	1	4	yes	—
257	C₄×D₄	C₂³×C₅₃	1	8	yes	—
258	C₄×D₄	C₂³×C₅₃	1	4	yes	—
259	C₂×C₄⋊C₄	—	2	4	no	—
260	C₂×C₄⋊C₄	—	2	8	no	—
261	C₂×C₄⋊C₄	—	2	2	no	—
262	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
263	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
264	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
265	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
266	C₂×C₂²⋊C₄	—	2	2	no	—
267	C₂×C₂²⋊C₄	—	2	4	no	—
268	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
269	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
270	#107	C₂³×C₅₃	1	4	yes	—
271	#107	C₂³×C₅₃	1	8	yes	—
272	#107	C₂³×C₅₃	1	16	yes	—
273	#107	C₂³×C₅₃	1	8	yes	—
274	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
275	#102	C₂³×C₅₃	1	8	yes	—
276	#102	C₂³×C₅₃	1	16	yes	—
277	C₂²⋊C₈	C₂³×C₅₃	1	8	yes	—
278	C₄×C₈	C₂³×C₅₃	1	4	yes	—
279	C₂⁴	—	2	2	no	—
280	C₂⁴	C₂×D₄×C₅₃	1	2	yes	—
281	#41	—	4	64	no	—
282	C₂×Q₈	—	2	8	no	—
283	C₂×Q₈	—	4	8	no	—
284	C₂×D₄	—	2	16	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	2	4	no	—
287	C₂×D₄	—	2	4	no	—
288	C₂×D₄	—	4	16	no	—
289	C₂×D₄	D₈×C₅₃	1	8	yes	complemented
290	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
291	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
292	C₂²×C₄	—	2	4	no	—
293	C₂²×C₄	—	2	4	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	16	no	—
297	C₂²×C₄	—	2	8	no	—
298	C₂²×C₄	—	4	16	no	—
299	C₂²×C₄	—	4	8	no	—
300	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
301	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
302	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
303	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
304	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
305	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
306	D₈	—	2	8	no	—
307	C₂×C₈	—	2	4	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	16	no	—
310	C₂×C₈	#41×C₅₃	1	4	yes	complemented
311	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
312	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
313	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	4	no	—
316	C₄⋊C₄	—	2	16	no	—
317	C₄⋊C₄	—	2	8	no	—
318	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
319	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
320	C₄⋊C₄	C₂⁴×C₅₃	1	8	yes	—
321	C₂²⋊C₄	—	2	8	no	—
322	C₂²⋊C₄	—	2	16	no	—
323	C₂²⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
324	C₄²	C₂×D₄×C₅₃	1	2	yes	—
325	C₄²	C₂⁴×C₅₃	1	2	yes	—
326	C₂³	—	2	4	no	—
327	C₂³	—	2	4	no	—
328	C₂³	—	2	4	no	—
329	C₂³	—	4	4	no	—
330	C₂³	—	4	4	no	—
331	C₂³	—	4	4	no	—
332	C₂³	C₂×D₈×C₅₃	1	8	yes	—
333	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
334	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
335	Q₈	—	4	16	no	—
336	D₄	—	2	4	no	—
337	D₄	—	2	4	no	—
338	D₄	—	4	32	no	—
339	D₄	—	4	4	no	—
340	D₄	C₂×D₈×C₅₃	1	16	yes	complemented
341	C₂×C₄	—	2	4	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	8	no	—
344	C₂×C₄	—	2	4	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	8	no	—
348	C₂×C₄	—	4	4	no	—
349	C₂×C₄	—	4	8	no	—
350	C₂×C₄	—	4	32	no	—
351	C₂×C₄	—	4	16	no	—
352	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
353	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
354	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
355	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
357	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
358	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
359	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
360	C₈	—	2	4	no	—
361	C₈	—	2	2	no	—
362	C₈	C₂×#41×C₅₃	1	4	yes	complemented
363	C₂²	—	2	4	no	—
364	C₂²	—	2	4	no	—
365	C₂²	—	2	16	no	—
366	C₂²	—	4	4	no	—
367	C₂²	—	4	4	no	—
368	C₂²	—	4	4	no	—
369	C₂²	—	8	8	no	—
370	C₂²	C₅₃×#547	1	2	yes	—
371	C₂²	C₂²×D₈×C₅₃	1	8	yes	—
372	C₂²	C₂²×D₈×C₅₃	1	2	yes	—
373	C₂²	C₅₃×#578	1	2	yes	—
374	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
375	C₄	—	2	4	no	—
376	C₄	—	2	2	no	—
377	C₄	—	4	4	no	—
378	C₄	—	4	8	no	—
379	C₄	C₅₃×#547	1	2	yes	—
380	C₄	C₂²×D₈×C₅₃	1	4	yes	—
381	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
382	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
383	C₂	—	2	8	no	—
384	C₂	—	8	4	no	—
385	C₂	C₅₃×#3334	1	4	yes	direct product factor
386	C₂	C₂×C₅₃×#547	1	1	yes	characteristic
387	C₂	C₂³×D₈×C₅₃	1	1	yes	characteristic
388	C₂	C₂×C₅₃×#578	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#547	C₂	semidirect
2. C₅₃×#3334	C₂	direct (N × H)
3. C₂×C₅₃×#500	C₂	semidirect
4. C₂×C₅₃×#480	C₂	semidirect
5. C₂×C₅₃×#474	C₂	semidirect
6. C₂×C₅₃×#466	C₂	semidirect
7. C₂×C₄×D₈×C₅₃	C₂	semidirect
8. C₂×C₈×D₄×C₅₃	C₂	semidirect
9. C₂²×#107×C₅₃	C₂	semidirect
14. C₅₃×#547	C₂²	semidirect
22. C₅₃×#500	C₂²	semidirect
23. C₅₃×#480	C₂²	semidirect
24. C₅₃×#474	C₂²	semidirect
25. C₅₃×#466	C₂²	semidirect
26. C₄×D₈×C₅₃	C₂²	semidirect
27. C₈×D₄×C₅₃	C₂²	semidirect
32. C₂×#107×C₅₃	C₂²	semidirect
94. C₂×D₄×C₅₃	D₈	semidirect
115. C₂×C₈×C₅₃	#41	semidirect
145. D₄×C₅₃	C₂×D₈	semidirect
167. C₈×C₅₃	C₂×#41	semidirect
168. C₂×#3334	C₅₃	direct (N × H)
189. C₂×#547	C₂×C₅₃	semidirect
190. #3334	C₂×C₅₃	direct (N × H)
191. C₂×#500	C₂×C₅₃	semidirect
192. C₂×#480	C₂×C₅₃	semidirect
193. C₂×#474	C₂×C₅₃	semidirect
194. C₂×#466	C₂×C₅₃	semidirect
195. C₂×C₄×D₈	C₂×C₅₃	semidirect
196. C₂×C₈×D₄	C₂×C₅₃	semidirect
197. C₂²×#107	C₂×C₅₃	semidirect
200. C₂×C₅₃	#3334	direct (N × H)
208. #547	C₂²×C₅₃	semidirect
216. #500	C₂²×C₅₃	semidirect
217. #480	C₂²×C₅₃	semidirect
218. #474	C₂²×C₅₃	semidirect
219. #466	C₂²×C₅₃	semidirect
220. C₄×D₈	C₂²×C₅₃	semidirect
221. C₈×D₄	C₂²×C₅₃	semidirect
226. C₂×#107	C₂²×C₅₃	semidirect
232. C₅₃	C₂×#3334	direct (N × H)
289. C₂×D₄	D₈×C₅₃	semidirect
310. C₂×C₈	#41×C₅₃	semidirect
340. D₄	C₂×D₈×C₅₃	semidirect
362. C₈	C₂×#41×C₅₃	semidirect
385. C₂	C₅₃×#3334	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#547	2
C₅₃×#3334	2
C₂×C₅₃×#500	2
C₂×C₅₃×#480	2
C₂×C₅₃×#474	2
C₂×C₅₃×#466	2
C₂×C₄×D₈×C₅₃	2
C₂×C₈×D₄×C₅₃	2
C₂²×#107×C₅₃	2
C₂×#3334	53

Subgroups — C2×C53×#3334 #27268737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3334 #27268737