Subgroups — C₂×C₅₃×#3529 #27270203

Order 13568.

7418 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3529	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₅₃×#563	C₂	1	3	yes	maximal, complemented
3	C₂×C₅₃×#562	C₂	1	6	yes	maximal, complemented
4	C₂×C₅₃×H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×#560	C₂	1	6	yes	maximal, complemented
6	C₂×C₅₃×#532	C₂	1	3	yes	maximal, complemented
7	C₂×C₅₃×#529	C₂	1	6	yes	maximal, complemented
8	C₂×C₅₃×#525	C₂	1	6	yes	maximal, complemented
9	C₂²×#41×C₅₃	C₂²	1	3	yes	—
10	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic, complemented
11	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	complemented
12	C₅₃×#563	C₂²	1	48	yes	complemented
13	C₅₃×#562	C₂²	1	96	yes	complemented
14	C₅₃×H(3,𝔽₄)	C₂²	1	16	yes	complemented
15	C₅₃×#560	C₂²	1	96	yes	complemented
16	C₅₃×#532	C₂²	1	48	yes	complemented
17	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	complemented
18	C₂×#127×C₅₃	C₂²	1	3	yes	—
19	C₅₃×#529	C₂²	1	96	yes	complemented
20	C₂×#126×C₅₃	C₂²	1	12	yes	complemented
21	C₂×#125×C₅₃	C₂²	1	6	yes	complemented
22	C₂×#124×C₅₃	C₂²	1	6	yes	—
23	C₂×#124×C₅₃	C₂²	1	3	yes	—
24	C₂×#124×C₅₃	C₂²	1	3	yes	complemented
25	C₅₃×#525	C₂²	1	96	yes	complemented
26	C₂×#123×C₅₃	C₂²	1	6	yes	—
27	C₂×#123×C₅₃	C₂²	1	12	yes	complemented
28	C₂×#123×C₅₃	C₂²	1	6	yes	complemented
29	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	6	yes	complemented
30	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	12	yes	complemented
31	C₂×#121×C₅₃	C₂²	1	6	yes	—
32	C₂×#121×C₅₃	C₂²	1	12	yes	—
33	C₂×#121×C₅₃	C₂²	1	12	yes	complemented
34	C₂×#120×C₅₃	C₂²	1	6	yes	—
35	C₂×#120×C₅₃	C₂²	1	6	yes	complemented
36	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
37	C₂×C₄×D₄×C₅₃	C₂²	1	6	yes	—
38	C₂×C₄×D₄×C₅₃	C₂²	1	12	yes	complemented
39	C₂×#117×C₅₃	C₂²	1	6	yes	complemented
40	C₂²×C₄²×C₅₃	C₂²	1	3	yes	complemented
41	C₂⁵×C₅₃	C₂³	1	2	yes	—
42	C₂×#41×C₅₃	—	4	12	no	—
43	C₂×#41×C₅₃	—	4	4	no	—
44	C₂×#41×C₅₃	C₂³	1	24	yes	—
45	C₂×#41×C₅₃	C₂³	1	8	yes	complemented
46	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
47	C₂²×Q₈×C₅₃	C₂³	1	3	yes	—
48	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₅₃	C₂³	1	6	yes	—
50	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
51	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
52	C₂²×D₄×C₅₃	C₂³	1	6	yes	—
53	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
54	C₂³×C₄×C₅₃	—	4	8	no	—
55	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
56	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
57	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
58	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
59	C₂³×C₄×C₅₃	C₂³	1	16	yes	complemented
60	C₄⋊Q₈×C₅₃	C₂³	1	24	yes	complemented
61	#127×C₅₃	C₂³	1	24	yes	—
62	#126×C₅₃	C₂³	1	96	yes	complemented
63	#125×C₅₃	C₂³	1	48	yes	complemented
64	#124×C₅₃	C₂³	1	48	yes	—
65	#124×C₅₃	C₂³	1	24	yes	—
66	#124×C₅₃	C₂³	1	24	yes	complemented
67	#123×C₅₃	C₂³	1	48	yes	—
68	#123×C₅₃	C₂³	1	48	yes	complemented
69	#123×C₅₃	C₂³	1	96	yes	complemented
70	C₂²⋊Q₈×C₅₃	C₂³	1	96	yes	complemented
71	C₂²⋊Q₈×C₅₃	C₂³	1	48	yes	complemented
72	#121×C₅₃	C₂³	1	96	yes	—
73	#121×C₅₃	C₂³	1	48	yes	—
74	#121×C₅₃	C₂³	1	96	yes	complemented
75	#120×C₅₃	C₂³	1	48	yes	—
76	#120×C₅₃	C₂³	1	48	yes	complemented
77	C₄×Q₈×C₅₃	C₂³	1	16	yes	complemented
78	C₄×D₄×C₅₃	C₂³	1	48	yes	—
79	C₄×D₄×C₅₃	C₂³	1	96	yes	complemented
80	#117×C₅₃	C₂³	1	48	yes	complemented
81	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
82	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
83	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
84	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
85	C₂×C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
86	C₂×C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
87	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	—
88	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	—
89	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	—
90	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
91	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
92	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
93	C₂×C₄²×C₅₃	C₂³	1	24	yes	complemented
94	C₂⁴×C₅₃	—	4	4	no	—
95	C₂⁴×C₅₃	—	4	8	no	—
96	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
97	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
98	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
99	#41×C₅₃	—	4	48	no	—
100	#41×C₅₃	—	4	16	no	—
101	C₂×Q₈×C₅₃	—	4	4	no	—
102	C₂×Q₈×C₅₃	—	4	6	no	—
103	C₂×Q₈×C₅₃	—	4	2	no	—
104	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
105	C₂×Q₈×C₅₃	C₂⁴	1	12	yes	—
106	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
107	C₂×D₄×C₅₃	—	4	24	no	—
108	C₂×D₄×C₅₃	—	4	12	no	—
109	C₂×D₄×C₅₃	—	4	12	no	—
110	C₂×D₄×C₅₃	—	4	6	no	—
111	C₂×D₄×C₅₃	—	4	6	no	—
112	C₂×D₄×C₅₃	C₂⁴	1	24	yes	—
113	C₂×D₄×C₅₃	C₂⁴	1	48	yes	—
114	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
115	C₂×D₄×C₅₃	C₂⁴	1	24	yes	—
116	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
117	C₂²×C₄×C₅₃	—	2	24	no	—
118	C₂²×C₄×C₅₃	—	2	12	no	—
119	C₂²×C₄×C₅₃	—	4	32	no	—
120	C₂²×C₄×C₅₃	—	4	12	no	—
121	C₂²×C₄×C₅₃	—	4	12	no	—
122	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
123	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
124	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
125	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
126	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
127	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
128	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
129	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
130	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
131	C₄⋊C₄×C₅₃	C₂⁴	1	48	yes	—
132	C₄⋊C₄×C₅₃	C₂⁴	1	24	yes	—
133	C₄⋊C₄×C₅₃	C₂⁴	1	48	yes	—
134	C₄⋊C₄×C₅₃	C₂⁴	1	24	yes	—
135	C₂²⋊C₄×C₅₃	C₂⁴	1	24	yes	—
136	C₂²⋊C₄×C₅₃	C₂⁴	1	24	yes	—
137	C₂²⋊C₄×C₅₃	C₂⁴	1	48	yes	—
138	C₂²⋊C₄×C₅₃	C₂⁴	1	48	yes	—
139	C₂²⋊C₄×C₅₃	C₂⁴	1	48	yes	—
140	C₄²×C₅₃	C₂⁴	1	24	yes	—
141	C₄²×C₅₃	C₂⁴	1	12	yes	—
142	C₄²×C₅₃	C₂⁴	1	12	yes	—
143	C₂³×C₅₃	—	2	12	no	—
144	C₂³×C₅₃	—	2	6	no	—
145	C₂³×C₅₃	—	2	12	no	—
146	C₂³×C₅₃	—	2	24	no	—
147	C₂³×C₅₃	—	4	8	no	—
148	C₂³×C₅₃	—	4	16	no	—
149	C₂³×C₅₃	—	4	32	no	—
150	C₂³×C₅₃	C₂⁵	1	6	yes	—
151	C₂³×C₅₃	C₂⁵	1	12	yes	—
152	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
153	Q₈×C₅₃	—	4	8	no	—
154	Q₈×C₅₃	—	4	12	no	—
155	Q₈×C₅₃	—	4	4	no	—
156	D₄×C₅₃	—	4	48	no	—
157	D₄×C₅₃	—	4	24	no	—
158	D₄×C₅₃	—	4	24	no	—
159	D₄×C₅₃	—	4	12	no	—
160	D₄×C₅₃	—	4	12	no	—
161	C₂×C₄×C₅₃	—	2	12	no	—
162	C₂×C₄×C₅₃	—	2	6	no	—
163	C₂×C₄×C₅₃	—	2	12	no	—
164	C₂×C₄×C₅₃	—	2	12	no	—
165	C₂×C₄×C₅₃	—	2	48	no	—
166	C₂×C₄×C₅₃	—	2	24	no	—
167	C₂×C₄×C₅₃	—	4	24	no	—
168	C₂×C₄×C₅₃	—	4	24	no	—
169	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
170	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
171	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
172	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
173	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
174	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
175	C₂×#3529	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂²×C₅₃	—	2	12	no	—
177	C₂²×C₅₃	—	2	6	no	—
178	C₂²×C₅₃	—	2	12	no	—
179	C₂²×C₅₃	—	2	24	no	—
180	C₂²×C₅₃	—	4	32	no	—
181	C₂²×C₅₃	—	4	12	no	—
182	C₂²×C₅₃	—	4	24	no	—
183	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
184	C₂²×C₅₃	#585	1	4	yes	—
185	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
186	C₄×C₅₃	—	2	12	no	—
187	C₄×C₅₃	—	2	6	no	—
188	C₄×C₅₃	—	2	12	no	—
189	C₄×C₅₃	—	2	12	no	—
190	C₄×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
191	#3529	C₂×C₅₃	1	32	yes	direct product factor
192	C₂×#563	C₂×C₅₃	1	3	yes	complemented
193	C₂×#562	C₂×C₅₃	1	6	yes	complemented
194	C₂×H(3,𝔽₄)	C₂×C₅₃	1	1	yes	characteristic, complemented
195	C₂×#560	C₂×C₅₃	1	6	yes	complemented
196	C₂×#532	C₂×C₅₃	1	3	yes	complemented
197	C₂×#529	C₂×C₅₃	1	6	yes	complemented
198	C₂×#525	C₂×C₅₃	1	6	yes	complemented
199	C₂×C₅₃	—	4	6	no	—
200	C₂×C₅₃	—	4	12	no	—
201	C₂×C₅₃	#3529	1	4	yes	direct product factor
202	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
203	C₂×C₅₃	C₂×#585	1	2	yes	—
204	C₂²×#41	C₂²×C₅₃	1	3	yes	—
205	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic, complemented
206	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	complemented
207	#563	C₂²×C₅₃	1	48	yes	complemented
208	#562	C₂²×C₅₃	1	96	yes	complemented
209	H(3,𝔽₄)	C₂²×C₅₃	1	16	yes	complemented
210	#560	C₂²×C₅₃	1	96	yes	complemented
211	#532	C₂²×C₅₃	1	48	yes	complemented
212	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	complemented
213	C₂×#127	C₂²×C₅₃	1	3	yes	—
214	#529	C₂²×C₅₃	1	96	yes	complemented
215	C₂×#126	C₂²×C₅₃	1	12	yes	complemented
216	C₂×#125	C₂²×C₅₃	1	6	yes	complemented
217	C₂×#124	C₂²×C₅₃	1	6	yes	—
218	C₂×#124	C₂²×C₅₃	1	3	yes	—
219	C₂×#124	C₂²×C₅₃	1	3	yes	complemented
220	#525	C₂²×C₅₃	1	96	yes	complemented
221	C₂×#123	C₂²×C₅₃	1	6	yes	—
222	C₂×#123	C₂²×C₅₃	1	6	yes	complemented
223	C₂×#123	C₂²×C₅₃	1	12	yes	complemented
224	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	12	yes	complemented
225	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	6	yes	complemented
226	C₂×#121	C₂²×C₅₃	1	12	yes	—
227	C₂×#121	C₂²×C₅₃	1	6	yes	—
228	C₂×#121	C₂²×C₅₃	1	12	yes	complemented
229	C₂×#120	C₂²×C₅₃	1	6	yes	—
230	C₂×#120	C₂²×C₅₃	1	6	yes	complemented
231	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
232	C₂×C₄×D₄	C₂²×C₅₃	1	6	yes	—
233	C₂×C₄×D₄	C₂²×C₅₃	1	12	yes	complemented
234	C₂×#117	C₂²×C₅₃	1	6	yes	complemented
235	C₂²×C₄²	C₂²×C₅₃	1	3	yes	complemented
236	C₅₃	C₂×#3529	1	1	yes	characteristic, direct product factor, Sylow
237	C₂⁵	C₂³×C₅₃	1	2	yes	—
238	C₂×#41	—	4	12	no	—
239	C₂×#41	—	4	4	no	—
240	C₂×#41	C₂³×C₅₃	1	24	yes	—
241	C₂×#41	C₂³×C₅₃	1	8	yes	complemented
242	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
243	C₂²×Q₈	C₂³×C₅₃	1	3	yes	—
244	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
245	C₂²×D₄	C₂³×C₅₃	1	6	yes	—
246	C₂²×D₄	C₂³×C₅₃	1	12	yes	—
247	C₂²×D₄	C₂³×C₅₃	1	3	yes	—
248	C₂²×D₄	C₂³×C₅₃	1	6	yes	—
249	C₂²×D₄	C₂³×C₅₃	1	3	yes	—
250	C₂³×C₄	—	4	8	no	—
251	C₂³×C₄	C₂³×C₅₃	1	6	yes	—
252	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
253	C₂³×C₄	C₂³×C₅₃	1	6	yes	—
254	C₂³×C₄	C₂³×C₅₃	1	6	yes	—
255	C₂³×C₄	C₂³×C₅₃	1	16	yes	complemented
256	C₄⋊Q₈	C₂³×C₅₃	1	24	yes	complemented
257	#127	C₂³×C₅₃	1	24	yes	—
258	#126	C₂³×C₅₃	1	96	yes	complemented
259	#125	C₂³×C₅₃	1	48	yes	complemented
260	#124	C₂³×C₅₃	1	48	yes	—
261	#124	C₂³×C₅₃	1	24	yes	—
262	#124	C₂³×C₅₃	1	24	yes	complemented
263	#123	C₂³×C₅₃	1	48	yes	—
264	#123	C₂³×C₅₃	1	48	yes	complemented
265	#123	C₂³×C₅₃	1	96	yes	complemented
266	C₂²⋊Q₈	C₂³×C₅₃	1	96	yes	complemented
267	C₂²⋊Q₈	C₂³×C₅₃	1	48	yes	complemented
268	#121	C₂³×C₅₃	1	96	yes	—
269	#121	C₂³×C₅₃	1	48	yes	—
270	#121	C₂³×C₅₃	1	96	yes	complemented
271	#120	C₂³×C₅₃	1	48	yes	—
272	#120	C₂³×C₅₃	1	48	yes	complemented
273	C₄×Q₈	C₂³×C₅₃	1	16	yes	complemented
274	C₄×D₄	C₂³×C₅₃	1	48	yes	—
275	C₄×D₄	C₂³×C₅₃	1	96	yes	complemented
276	#117	C₂³×C₅₃	1	48	yes	complemented
277	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
278	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
279	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
280	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
281	C₂×C₂²⋊C₄	C₂³×C₅₃	1	6	yes	—
282	C₂×C₂²⋊C₄	C₂³×C₅₃	1	6	yes	—
283	C₂×C₂²⋊C₄	C₂³×C₅₃	1	12	yes	—
284	C₂×C₂²⋊C₄	C₂³×C₅₃	1	12	yes	—
285	C₂×C₂²⋊C₄	C₂³×C₅₃	1	12	yes	—
286	C₂×C₄²	C₂³×C₅₃	1	6	yes	—
287	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
288	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
289	C₂×C₄²	C₂³×C₅₃	1	24	yes	complemented
290	C₂⁴	—	4	4	no	—
291	C₂⁴	—	4	8	no	—
292	C₂⁴	C₂⁴×C₅₃	1	6	yes	—
293	C₂⁴	C₂⁴×C₅₃	1	3	yes	—
294	C₂⁴	C₂⁴×C₅₃	1	8	yes	—
295	#41	—	4	48	no	—
296	#41	—	4	16	no	—
297	C₂×Q₈	—	4	4	no	—
298	C₂×Q₈	—	4	6	no	—
299	C₂×Q₈	—	4	2	no	—
300	C₂×Q₈	C₂⁴×C₅₃	1	8	yes	—
301	C₂×Q₈	C₂⁴×C₅₃	1	12	yes	—
302	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
303	C₂×D₄	—	4	24	no	—
304	C₂×D₄	—	4	12	no	—
305	C₂×D₄	—	4	12	no	—
306	C₂×D₄	—	4	6	no	—
307	C₂×D₄	—	4	6	no	—
308	C₂×D₄	C₂⁴×C₅₃	1	24	yes	—
309	C₂×D₄	C₂⁴×C₅₃	1	48	yes	—
310	C₂×D₄	C₂⁴×C₅₃	1	12	yes	—
311	C₂×D₄	C₂⁴×C₅₃	1	24	yes	—
312	C₂×D₄	C₂⁴×C₅₃	1	12	yes	—
313	C₂²×C₄	—	2	24	no	—
314	C₂²×C₄	—	2	12	no	—
315	C₂²×C₄	—	4	32	no	—
316	C₂²×C₄	—	4	12	no	—
317	C₂²×C₄	—	4	12	no	—
318	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
319	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
320	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
321	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
322	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
323	C₂²×C₄	C₂⁴×C₅₃	1	24	yes	—
324	C₂²×C₄	C₂⁴×C₅₃	1	24	yes	—
325	C₂²×C₄	C₂⁴×C₅₃	1	24	yes	—
326	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
327	C₄⋊C₄	C₂⁴×C₅₃	1	48	yes	—
328	C₄⋊C₄	C₂⁴×C₅₃	1	24	yes	—
329	C₄⋊C₄	C₂⁴×C₅₃	1	48	yes	—
330	C₄⋊C₄	C₂⁴×C₅₃	1	24	yes	—
331	C₂²⋊C₄	C₂⁴×C₅₃	1	24	yes	—
332	C₂²⋊C₄	C₂⁴×C₅₃	1	24	yes	—
333	C₂²⋊C₄	C₂⁴×C₅₃	1	48	yes	—
334	C₂²⋊C₄	C₂⁴×C₅₃	1	48	yes	—
335	C₂²⋊C₄	C₂⁴×C₅₃	1	48	yes	—
336	C₄²	C₂⁴×C₅₃	1	24	yes	—
337	C₄²	C₂⁴×C₅₃	1	12	yes	—
338	C₄²	C₂⁴×C₅₃	1	12	yes	—
339	C₂³	—	2	12	no	—
340	C₂³	—	2	6	no	—
341	C₂³	—	2	12	no	—
342	C₂³	—	2	24	no	—
343	C₂³	—	4	8	no	—
344	C₂³	—	4	16	no	—
345	C₂³	—	4	32	no	—
346	C₂³	C₂⁵×C₅₃	1	12	yes	—
347	C₂³	C₂⁵×C₅₃	1	6	yes	—
348	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
349	Q₈	—	4	8	no	—
350	Q₈	—	4	12	no	—
351	Q₈	—	4	4	no	—
352	D₄	—	4	48	no	—
353	D₄	—	4	24	no	—
354	D₄	—	4	24	no	—
355	D₄	—	4	12	no	—
356	D₄	—	4	12	no	—
357	C₂×C₄	—	2	12	no	—
358	C₂×C₄	—	2	6	no	—
359	C₂×C₄	—	2	12	no	—
360	C₂×C₄	—	2	12	no	—
361	C₂×C₄	—	2	48	no	—
362	C₂×C₄	—	2	24	no	—
363	C₂×C₄	—	4	24	no	—
364	C₂×C₄	—	4	24	no	—
365	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
366	C₂×C₄	C₂⁵×C₅₃	1	12	yes	—
367	C₂×C₄	C₂⁵×C₅₃	1	12	yes	—
368	C₂×C₄	C₂⁵×C₅₃	1	6	yes	—
369	C₂×C₄	C₂⁵×C₅₃	1	12	yes	—
370	C₂×C₄	C₂⁵×C₅₃	1	2	yes	—
371	C₂²	—	2	12	no	—
372	C₂²	—	2	6	no	—
373	C₂²	—	2	12	no	—
374	C₂²	—	2	24	no	—
375	C₂²	—	4	32	no	—
376	C₂²	—	4	24	no	—
377	C₂²	—	4	12	no	—
378	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
379	C₂²	C₅₃×#585	1	4	yes	—
380	C₂²	C₂⁶×C₅₃	1	1	yes	characteristic
381	C₄	—	2	12	no	—
382	C₄	—	2	6	no	—
383	C₄	—	2	12	no	—
384	C₄	—	2	12	no	—
385	C₄	C₂×H(5,𝔽₂)×C₅₃	1	4	yes	—
386	C₂	—	4	12	no	—
387	C₂	—	4	6	no	—
388	C₂	C₅₃×#3529	1	4	yes	direct product factor
389	C₂	C₂²×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
390	C₂	C₂×C₅₃×#585	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₅₃
socle	C₂³×C₅₃	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3529	C₂	direct (N × H)
2. C₂×C₅₃×#563	C₂	semidirect
3. C₂×C₅₃×#562	C₂	semidirect
4. C₂×C₅₃×H(3,𝔽₄)	C₂	semidirect
5. C₂×C₅₃×#560	C₂	semidirect
6. C₂×C₅₃×#532	C₂	semidirect
7. C₂×C₅₃×#529	C₂	semidirect
8. C₂×C₅₃×#525	C₂	semidirect
10. C₂²×#41×C₅₃	C₂²	semidirect
11. C₂⁴×C₄×C₅₃	C₂²	semidirect
12. C₅₃×#563	C₂²	semidirect
13. C₅₃×#562	C₂²	semidirect
14. C₅₃×H(3,𝔽₄)	C₂²	semidirect
15. C₅₃×#560	C₂²	semidirect
16. C₅₃×#532	C₂²	semidirect
17. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
19. C₅₃×#529	C₂²	semidirect
20. C₂×#126×C₅₃	C₂²	semidirect
21. C₂×#125×C₅₃	C₂²	semidirect
24. C₂×#124×C₅₃	C₂²	semidirect
25. C₅₃×#525	C₂²	semidirect
27. C₂×#123×C₅₃	C₂²	semidirect
28. C₂×#123×C₅₃	C₂²	semidirect
29. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
30. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
33. C₂×#121×C₅₃	C₂²	semidirect
35. C₂×#120×C₅₃	C₂²	semidirect
36. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
38. C₂×C₄×D₄×C₅₃	C₂²	semidirect
39. C₂×#117×C₅₃	C₂²	semidirect
40. C₂²×C₄²×C₅₃	C₂²	semidirect
45. C₂×#41×C₅₃	C₂³	semidirect
59. C₂³×C₄×C₅₃	C₂³	semidirect
60. C₄⋊Q₈×C₅₃	C₂³	semidirect
62. #126×C₅₃	C₂³	semidirect
63. #125×C₅₃	C₂³	semidirect
66. #124×C₅₃	C₂³	semidirect
68. #123×C₅₃	C₂³	semidirect
69. #123×C₅₃	C₂³	semidirect
70. C₂²⋊Q₈×C₅₃	C₂³	semidirect
71. C₂²⋊Q₈×C₅₃	C₂³	semidirect
74. #121×C₅₃	C₂³	semidirect
76. #120×C₅₃	C₂³	semidirect
77. C₄×Q₈×C₅₃	C₂³	semidirect
79. C₄×D₄×C₅₃	C₂³	semidirect
80. #117×C₅₃	C₂³	semidirect
93. C₂×C₄²×C₅₃	C₂³	semidirect
175. C₂×#3529	C₅₃	direct (N × H)
191. #3529	C₂×C₅₃	direct (N × H)
192. C₂×#563	C₂×C₅₃	semidirect
193. C₂×#562	C₂×C₅₃	semidirect
194. C₂×H(3,𝔽₄)	C₂×C₅₃	semidirect
195. C₂×#560	C₂×C₅₃	semidirect
196. C₂×#532	C₂×C₅₃	semidirect
197. C₂×#529	C₂×C₅₃	semidirect
198. C₂×#525	C₂×C₅₃	semidirect
201. C₂×C₅₃	#3529	direct (N × H)
205. C₂²×#41	C₂²×C₅₃	semidirect
206. C₂⁴×C₄	C₂²×C₅₃	semidirect
207. #563	C₂²×C₅₃	semidirect
208. #562	C₂²×C₅₃	semidirect
209. H(3,𝔽₄)	C₂²×C₅₃	semidirect
210. #560	C₂²×C₅₃	semidirect
211. #532	C₂²×C₅₃	semidirect
212. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
214. #529	C₂²×C₅₃	semidirect
215. C₂×#126	C₂²×C₅₃	semidirect
216. C₂×#125	C₂²×C₅₃	semidirect
219. C₂×#124	C₂²×C₅₃	semidirect
220. #525	C₂²×C₅₃	semidirect
222. C₂×#123	C₂²×C₅₃	semidirect
223. C₂×#123	C₂²×C₅₃	semidirect
224. C₂×C₂²⋊Q₈	C₂²×C₅₃	semidirect
225. C₂×C₂²⋊Q₈	C₂²×C₅₃	semidirect
228. C₂×#121	C₂²×C₅₃	semidirect
230. C₂×#120	C₂²×C₅₃	semidirect
231. C₂×C₄×Q₈	C₂²×C₅₃	semidirect
233. C₂×C₄×D₄	C₂²×C₅₃	semidirect
234. C₂×#117	C₂²×C₅₃	semidirect
235. C₂²×C₄²	C₂²×C₅₃	semidirect
236. C₅₃	C₂×#3529	direct (N × H)
241. C₂×#41	C₂³×C₅₃	semidirect
255. C₂³×C₄	C₂³×C₅₃	semidirect
256. C₄⋊Q₈	C₂³×C₅₃	semidirect
258. #126	C₂³×C₅₃	semidirect
259. #125	C₂³×C₅₃	semidirect
262. #124	C₂³×C₅₃	semidirect
264. #123	C₂³×C₅₃	semidirect
265. #123	C₂³×C₅₃	semidirect
266. C₂²⋊Q₈	C₂³×C₅₃	semidirect
267. C₂²⋊Q₈	C₂³×C₅₃	semidirect
270. #121	C₂³×C₅₃	semidirect
272. #120	C₂³×C₅₃	semidirect
273. C₄×Q₈	C₂³×C₅₃	semidirect
275. C₄×D₄	C₂³×C₅₃	semidirect
276. #117	C₂³×C₅₃	semidirect
289. C₂×C₄²	C₂³×C₅₃	semidirect
388. C₂	C₅₃×#3529	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3529	2
C₂×C₅₃×#563	2
C₂×C₅₃×#562	2
C₂×C₅₃×H(3,𝔽₄)	2
C₂×C₅₃×#560	2
C₂×C₅₃×#532	2
C₂×C₅₃×#529	2
C₂×C₅₃×#525	2
C₂×#3529	53

Subgroups — C2×C53×#3529 #27270203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3529 #27270203