Subgroups — C₂²×C₅₃×#548 #27270210

Order 13568.

12142 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#548	C₂	1	48	yes	maximal, direct product factor
4	C₂²×#124×C₅₃	C₂	1	3	yes	maximal, complemented
5	C₂²×C₂²⋊Q₈×C₅₃	C₂	1	3	yes	maximal, complemented
6	C₂²×#121×C₅₃	C₂	1	3	yes	maximal, complemented
7	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₄×C₅₃	C₂	1	3	yes	maximal, complemented
9	C₂²×#41×C₅₃	—	2	16	no	—
10	C₂²×#41×C₅₃	C₂²	1	24	yes	complemented
11	C₂³×Q₈×C₅₃	—	2	16	no	—
12	C₂³×Q₈×C₅₃	C₂²	1	3	yes	—
13	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂³×Q₈×C₅₃	C₂²	1	24	yes	complemented
15	C₂³×D₄×C₅₃	C₂²	1	3	yes	—
16	C₂³×D₄×C₅₃	C₂²	1	3	yes	—
17	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
18	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
19	C₅₃×#548	C₂²	1	256	yes	direct product factor
20	C₂×#124×C₅₃	C₂²	1	72	yes	complemented
21	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	72	yes	complemented
22	C₂×#121×C₅₃	C₂²	1	72	yes	complemented
23	C₂×C₄×Q₈×C₅₃	C₂²	1	24	yes	complemented
24	C₂×C₄×D₄×C₅₃	C₂²	1	72	yes	complemented
25	C₂²×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
26	C₂²×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
27	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
29	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	6	yes	—
30	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
31	C₂⁵×C₅₃	C₂³	1	3	yes	—
32	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×#41×C₅₃	—	2	192	no	—
34	C₂×#41×C₅₃	C₂³	1	64	yes	complemented
35	C₂²×Q₈×C₅₃	—	2	192	no	—
36	C₂²×Q₈×C₅₃	—	2	24	no	—
37	C₂²×Q₈×C₅₃	D₄	1	16	yes	complemented
38	C₂²×Q₈×C₅₃	C₂³	1	36	yes	—
39	C₂²×Q₈×C₅₃	C₂³	1	12	yes	—
40	C₂²×Q₈×C₅₃	C₂³	1	64	yes	complemented
41	C₂²×D₄×C₅₃	—	2	24	no	—
42	C₂²×D₄×C₅₃	—	2	24	no	—
43	C₂²×D₄×C₅₃	C₂³	1	36	yes	—
44	C₂²×D₄×C₅₃	C₂³	1	36	yes	—
45	C₂³×C₄×C₅₃	—	2	24	no	—
46	C₂³×C₄×C₅₃	—	2	24	no	—
47	C₂³×C₄×C₅₃	C₂³	1	36	yes	—
48	C₂³×C₄×C₅₃	C₂³	1	36	yes	—
49	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
51	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
52	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
54	#124×C₅₃	C₂³	1	192	yes	complemented
55	C₂²⋊Q₈×C₅₃	C₂³	1	192	yes	complemented
56	#121×C₅₃	C₂³	1	192	yes	complemented
57	C₄×Q₈×C₅₃	C₂³	1	64	yes	complemented
58	C₄×D₄×C₅₃	C₂³	1	192	yes	complemented
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	36	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	36	yes	—
61	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	—
62	C₂×C₂²⋊C₄×C₅₃	C₂³	1	36	yes	—
63	C₂×C₂²⋊C₄×C₅₃	C₂³	1	72	yes	—
64	C₂×C₄²×C₅₃	C₂³	1	36	yes	—
65	C₂⁴×C₅₃	—	2	4	no	—
66	C₂⁴×C₅₃	—	2	4	no	—
67	C₂⁴×C₅₃	—	2	12	no	—
68	C₂⁴×C₅₃	—	2	12	no	—
69	C₂⁴×C₅₃	—	2	12	no	—
70	C₂⁴×C₅₃	#41	1	8	yes	complemented
71	C₂⁴×C₅₃	C₂⁴	1	18	yes	—
72	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
73	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
74	#41×C₅₃	—	2	256	no	—
75	C₂×Q₈×C₅₃	—	2	256	no	—
76	C₂×Q₈×C₅₃	—	2	144	no	—
77	C₂×Q₈×C₅₃	C₂×D₄	1	96	yes	complemented
78	C₂×Q₈×C₅₃	C₂⁴	1	48	yes	—
79	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	—
80	C₂×D₄×C₅₃	—	2	144	no	—
81	C₂×D₄×C₅₃	—	2	144	no	—
82	C₂×D₄×C₅₃	C₂⁴	1	48	yes	—
83	C₂×D₄×C₅₃	C₂⁴	1	48	yes	—
84	C₂²×C₄×C₅₃	—	2	4	no	—
85	C₂²×C₄×C₅₃	—	2	12	no	—
86	C₂²×C₄×C₅₃	—	2	12	no	—
87	C₂²×C₄×C₅₃	—	2	4	no	—
88	C₂²×C₄×C₅₃	—	2	144	no	—
89	C₂²×C₄×C₅₃	—	2	144	no	—
90	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
91	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
92	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
93	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
94	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
95	C₂²×C₄×C₅₃	C₂⁴	1	48	yes	—
96	C₂²×C₄×C₅₃	C₂⁴	1	48	yes	—
97	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
98	C₄⋊C₄×C₅₃	C₂⁴	1	48	yes	—
99	C₄⋊C₄×C₅₃	C₂⁴	1	48	yes	—
100	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
101	C₂²⋊C₄×C₅₃	C₂⁴	1	48	yes	—
102	C₂²⋊C₄×C₅₃	C₂⁴	1	96	yes	—
103	C₄²×C₅₃	C₂⁴	1	48	yes	—
104	C₂³×C₅₃	—	2	32	no	—
105	C₂³×C₅₃	—	2	12	no	—
106	C₂³×C₅₃	—	2	12	no	—
107	C₂³×C₅₃	—	2	36	no	—
108	C₂³×C₅₃	—	2	36	no	—
109	C₂³×C₅₃	—	2	36	no	—
110	C₂³×C₅₃	—	4	48	no	—
111	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
112	C₂³×C₅₃	C₂×#41	1	4	yes	—
113	C₂³×C₅₃	C₂×#41	1	24	yes	complemented
114	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
115	C₂³×C₅₃	C₂⁵	1	12	yes	—
116	C₂³×C₅₃	C₂⁵	1	4	yes	—
117	C₂³×C₅₃	C₂⁵	1	3	yes	—
118	Q₈×C₅₃	—	2	96	no	—
119	Q₈×C₅₃	C₂²×D₄	1	64	yes	complemented
120	D₄×C₅₃	—	2	96	no	—
121	D₄×C₅₃	—	2	96	no	—
122	C₂×C₄×C₅₃	—	2	12	no	—
123	C₂×C₄×C₅₃	—	2	36	no	—
124	C₂×C₄×C₅₃	—	2	36	no	—
125	C₂×C₄×C₅₃	—	2	12	no	—
126	C₂×C₄×C₅₃	—	2	96	no	—
127	C₂×C₄×C₅₃	—	2	96	no	—
128	C₂×C₄×C₅₃	C₂²×D₄	1	72	yes	—
129	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
130	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
131	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
132	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
133	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
134	C₂²×#548	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂²×C₅₃	—	2	4	no	—
136	C₂²×C₅₃	—	2	4	no	—
137	C₂²×C₅₃	—	2	48	no	—
138	C₂²×C₅₃	—	2	12	no	—
139	C₂²×C₅₃	—	2	12	no	—
140	C₂²×C₅₃	—	2	12	no	—
141	C₂²×C₅₃	—	4	72	no	—
142	C₂²×C₅₃	#548	1	16	yes	direct product factor
143	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
144	C₂²×C₅₃	C₂²×#41	1	6	yes	—
145	C₂²×C₅₃	C₂²×#41	1	8	yes	complemented
146	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	6	yes	—
147	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
148	C₄×C₅₃	—	2	4	no	—
149	C₄×C₅₃	—	2	12	no	—
150	C₄×C₅₃	—	2	12	no	—
151	C₄×C₅₃	—	2	4	no	—
152	C₄×C₅₃	C₂³×D₄	1	24	yes	—
153	C₂³×#41	C₂×C₅₃	1	1	yes	characteristic, complemented
154	C₂⁴×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
155	C₂×#548	C₂×C₅₃	1	48	yes	direct product factor
156	C₂²×#124	C₂×C₅₃	1	3	yes	complemented
157	C₂²×C₂²⋊Q₈	C₂×C₅₃	1	3	yes	complemented
158	C₂²×#121	C₂×C₅₃	1	3	yes	complemented
159	C₂²×C₄×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
160	C₂²×C₄×D₄	C₂×C₅₃	1	3	yes	complemented
161	C₂×C₅₃	—	2	8	no	—
162	C₂×C₅₃	—	4	12	no	—
163	C₂×C₅₃	C₂×#548	1	12	yes	direct product factor
164	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
165	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
166	C₂×C₅₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
167	C₂²×#41	—	2	16	no	—
168	C₂²×#41	C₂²×C₅₃	1	24	yes	complemented
169	C₂³×Q₈	—	2	16	no	—
170	C₂³×Q₈	C₂²×C₅₃	1	3	yes	—
171	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
172	C₂³×Q₈	C₂²×C₅₃	1	24	yes	complemented
173	C₂³×D₄	C₂²×C₅₃	1	3	yes	—
174	C₂³×D₄	C₂²×C₅₃	1	3	yes	—
175	C₂⁴×C₄	C₂²×C₅₃	1	3	yes	—
176	C₂⁴×C₄	C₂²×C₅₃	1	3	yes	—
177	#548	C₂²×C₅₃	1	256	yes	direct product factor
178	C₂×#124	C₂²×C₅₃	1	72	yes	complemented
179	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	72	yes	complemented
180	C₂×#121	C₂²×C₅₃	1	72	yes	complemented
181	C₂×C₄×Q₈	C₂²×C₅₃	1	24	yes	complemented
182	C₂×C₄×D₄	C₂²×C₅₃	1	72	yes	complemented
183	C₂²×C₄⋊C₄	C₂²×C₅₃	1	3	yes	—
184	C₂²×C₄⋊C₄	C₂²×C₅₃	1	3	yes	—
185	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
186	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	3	yes	—
187	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	6	yes	—
188	C₂²×C₄²	C₂²×C₅₃	1	3	yes	—
189	C₅₃	C₂²×#548	1	1	yes	characteristic, direct product factor, Sylow
190	C₂⁵	C₂³×C₅₃	1	3	yes	—
191	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
192	C₂×#41	—	2	192	no	—
193	C₂×#41	C₂³×C₅₃	1	64	yes	complemented
194	C₂²×Q₈	—	2	192	no	—
195	C₂²×Q₈	—	2	24	no	—
196	C₂²×Q₈	D₄×C₅₃	1	16	yes	complemented
197	C₂²×Q₈	C₂³×C₅₃	1	36	yes	—
198	C₂²×Q₈	C₂³×C₅₃	1	12	yes	—
199	C₂²×Q₈	C₂³×C₅₃	1	64	yes	complemented
200	C₂²×D₄	—	2	24	no	—
201	C₂²×D₄	—	2	24	no	—
202	C₂²×D₄	C₂³×C₅₃	1	36	yes	—
203	C₂²×D₄	C₂³×C₅₃	1	36	yes	—
204	C₂³×C₄	—	2	24	no	—
205	C₂³×C₄	—	2	24	no	—
206	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
207	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
208	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
209	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
210	C₂³×C₄	C₂³×C₅₃	1	36	yes	—
211	C₂³×C₄	C₂³×C₅₃	1	36	yes	—
212	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
213	#124	C₂³×C₅₃	1	192	yes	complemented
214	C₂²⋊Q₈	C₂³×C₅₃	1	192	yes	complemented
215	#121	C₂³×C₅₃	1	192	yes	complemented
216	C₄×Q₈	C₂³×C₅₃	1	64	yes	complemented
217	C₄×D₄	C₂³×C₅₃	1	192	yes	complemented
218	C₂×C₄⋊C₄	C₂³×C₅₃	1	36	yes	—
219	C₂×C₄⋊C₄	C₂³×C₅₃	1	36	yes	—
220	C₂×C₂²⋊C₄	C₂³×C₅₃	1	12	yes	—
221	C₂×C₂²⋊C₄	C₂³×C₅₃	1	36	yes	—
222	C₂×C₂²⋊C₄	C₂³×C₅₃	1	72	yes	—
223	C₂×C₄²	C₂³×C₅₃	1	36	yes	—
224	C₂⁴	—	2	4	no	—
225	C₂⁴	—	2	4	no	—
226	C₂⁴	—	2	12	no	—
227	C₂⁴	—	2	12	no	—
228	C₂⁴	—	2	12	no	—
229	C₂⁴	#41×C₅₃	1	8	yes	complemented
230	C₂⁴	C₂⁴×C₅₃	1	18	yes	—
231	C₂⁴	C₂⁴×C₅₃	1	6	yes	—
232	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
233	#41	—	2	256	no	—
234	C₂×Q₈	—	2	256	no	—
235	C₂×Q₈	—	2	144	no	—
236	C₂×Q₈	C₂×D₄×C₅₃	1	96	yes	complemented
237	C₂×Q₈	C₂⁴×C₅₃	1	48	yes	—
238	C₂×Q₈	C₂⁴×C₅₃	1	16	yes	—
239	C₂×D₄	—	2	144	no	—
240	C₂×D₄	—	2	144	no	—
241	C₂×D₄	C₂⁴×C₅₃	1	48	yes	—
242	C₂×D₄	C₂⁴×C₅₃	1	48	yes	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	12	no	—
245	C₂²×C₄	—	2	12	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	144	no	—
248	C₂²×C₄	—	2	144	no	—
249	C₂²×C₄	C₂×D₄×C₅₃	1	24	yes	—
250	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
251	C₂²×C₄	C₂⁴×C₅₃	1	18	yes	—
252	C₂²×C₄	C₂⁴×C₅₃	1	18	yes	—
253	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
254	C₂²×C₄	C₂⁴×C₅₃	1	48	yes	—
255	C₂²×C₄	C₂⁴×C₅₃	1	48	yes	—
256	C₂²×C₄	C₂⁴×C₅₃	1	18	yes	—
257	C₄⋊C₄	C₂⁴×C₅₃	1	48	yes	—
258	C₄⋊C₄	C₂⁴×C₅₃	1	48	yes	—
259	C₂²⋊C₄	C₂⁴×C₅₃	1	16	yes	—
260	C₂²⋊C₄	C₂⁴×C₅₃	1	48	yes	—
261	C₂²⋊C₄	C₂⁴×C₅₃	1	96	yes	—
262	C₄²	C₂⁴×C₅₃	1	48	yes	—
263	C₂³	—	2	32	no	—
264	C₂³	—	2	12	no	—
265	C₂³	—	2	12	no	—
266	C₂³	—	2	36	no	—
267	C₂³	—	2	36	no	—
268	C₂³	—	2	36	no	—
269	C₂³	—	4	48	no	—
270	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
271	C₂³	C₂×#41×C₅₃	1	4	yes	—
272	C₂³	C₂×#41×C₅₃	1	24	yes	complemented
273	C₂³	2₋¹⁺⁴×C₅₃	1	4	yes	—
274	C₂³	C₂⁵×C₅₃	1	12	yes	—
275	C₂³	C₂⁵×C₅₃	1	4	yes	—
276	C₂³	C₂⁵×C₅₃	1	3	yes	—
277	Q₈	—	2	96	no	—
278	Q₈	C₂²×D₄×C₅₃	1	64	yes	complemented
279	D₄	—	2	96	no	—
280	D₄	—	2	96	no	—
281	C₂×C₄	—	2	12	no	—
282	C₂×C₄	—	2	36	no	—
283	C₂×C₄	—	2	36	no	—
284	C₂×C₄	—	2	12	no	—
285	C₂×C₄	—	2	96	no	—
286	C₂×C₄	—	2	96	no	—
287	C₂×C₄	C₂²×D₄×C₅₃	1	72	yes	—
288	C₂×C₄	C₂⁵×C₅₃	1	4	yes	—
289	C₂×C₄	C₂⁵×C₅₃	1	12	yes	—
290	C₂×C₄	C₂⁵×C₅₃	1	12	yes	—
291	C₂×C₄	C₂⁵×C₅₃	1	4	yes	—
292	C₂×C₄	C₂⁵×C₅₃	1	12	yes	—
293	C₂²	—	2	48	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	2	12	no	—
297	C₂²	—	2	12	no	—
298	C₂²	—	2	12	no	—
299	C₂²	—	4	72	no	—
300	C₂²	C₅₃×#548	1	16	yes	direct product factor
301	C₂²	C₂³×D₄×C₅₃	1	6	yes	—
302	C₂²	C₂²×#41×C₅₃	1	6	yes	—
303	C₂²	C₂²×#41×C₅₃	1	8	yes	complemented
304	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	6	yes	—
305	C₂²	C₂⁶×C₅₃	1	1	yes	characteristic
306	C₄	—	2	4	no	—
307	C₄	—	2	12	no	—
308	C₄	—	2	12	no	—
309	C₄	—	2	4	no	—
310	C₄	C₂³×D₄×C₅₃	1	24	yes	—
311	C₂	—	2	8	no	—
312	C₂	—	4	12	no	—
313	C₂	C₂×C₅₃×#548	1	12	yes	direct product factor
314	C₂	C₂⁴×D₄×C₅₃	1	1	yes	characteristic
315	C₂	C₂³×#41×C₅₃	1	1	yes	characteristic
316	C₂	C₂²×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂³×#41×C₅₃	C₂	semidirect
2. C₂⁴×Q₈×C₅₃	C₂	semidirect
3. C₂×C₅₃×#548	C₂	direct (N × H)
4. C₂²×#124×C₅₃	C₂	semidirect
5. C₂²×C₂²⋊Q₈×C₅₃	C₂	semidirect
6. C₂²×#121×C₅₃	C₂	semidirect
7. C₂²×C₄×Q₈×C₅₃	C₂	semidirect
8. C₂²×C₄×D₄×C₅₃	C₂	semidirect
10. C₂²×#41×C₅₃	C₂²	semidirect
14. C₂³×Q₈×C₅₃	C₂²	semidirect
19. C₅₃×#548	C₂²	direct (N × H)
20. C₂×#124×C₅₃	C₂²	semidirect
21. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
22. C₂×#121×C₅₃	C₂²	semidirect
23. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
24. C₂×C₄×D₄×C₅₃	C₂²	semidirect
34. C₂×#41×C₅₃	C₂³	semidirect
37. C₂²×Q₈×C₅₃	D₄	semidirect
40. C₂²×Q₈×C₅₃	C₂³	semidirect
54. #124×C₅₃	C₂³	semidirect
55. C₂²⋊Q₈×C₅₃	C₂³	semidirect
56. #121×C₅₃	C₂³	semidirect
57. C₄×Q₈×C₅₃	C₂³	semidirect
58. C₄×D₄×C₅₃	C₂³	semidirect
70. C₂⁴×C₅₃	#41	semidirect
77. C₂×Q₈×C₅₃	C₂×D₄	semidirect
113. C₂³×C₅₃	C₂×#41	semidirect
119. Q₈×C₅₃	C₂²×D₄	semidirect
134. C₂²×#548	C₅₃	direct (N × H)
142. C₂²×C₅₃	#548	direct (N × H)
145. C₂²×C₅₃	C₂²×#41	semidirect
153. C₂³×#41	C₂×C₅₃	semidirect
154. C₂⁴×Q₈	C₂×C₅₃	semidirect
155. C₂×#548	C₂×C₅₃	direct (N × H)
156. C₂²×#124	C₂×C₅₃	semidirect
157. C₂²×C₂²⋊Q₈	C₂×C₅₃	semidirect
158. C₂²×#121	C₂×C₅₃	semidirect
159. C₂²×C₄×Q₈	C₂×C₅₃	semidirect
160. C₂²×C₄×D₄	C₂×C₅₃	semidirect
163. C₂×C₅₃	C₂×#548	direct (N × H)
168. C₂²×#41	C₂²×C₅₃	semidirect
172. C₂³×Q₈	C₂²×C₅₃	semidirect
177. #548	C₂²×C₅₃	direct (N × H)
178. C₂×#124	C₂²×C₅₃	semidirect
179. C₂×C₂²⋊Q₈	C₂²×C₅₃	semidirect
180. C₂×#121	C₂²×C₅₃	semidirect
181. C₂×C₄×Q₈	C₂²×C₅₃	semidirect
182. C₂×C₄×D₄	C₂²×C₅₃	semidirect
189. C₅₃	C₂²×#548	direct (N × H)
193. C₂×#41	C₂³×C₅₃	semidirect
196. C₂²×Q₈	D₄×C₅₃	semidirect
199. C₂²×Q₈	C₂³×C₅₃	semidirect
213. #124	C₂³×C₅₃	semidirect
214. C₂²⋊Q₈	C₂³×C₅₃	semidirect
215. #121	C₂³×C₅₃	semidirect
216. C₄×Q₈	C₂³×C₅₃	semidirect
217. C₄×D₄	C₂³×C₅₃	semidirect
229. C₂⁴	#41×C₅₃	semidirect
236. C₂×Q₈	C₂×D₄×C₅₃	semidirect
272. C₂³	C₂×#41×C₅₃	semidirect
278. Q₈	C₂²×D₄×C₅₃	semidirect
300. C₂²	C₅₃×#548	direct (N × H)
303. C₂²	C₂²×#41×C₅₃	semidirect
313. C₂	C₂×C₅₃×#548	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×#41×C₅₃	2
C₂⁴×Q₈×C₅₃	2
C₂×C₅₃×#548	2
C₂²×#124×C₅₃	2
C₂²×C₂²⋊Q₈×C₅₃	2
C₂²×#121×C₅₃	2
C₂²×C₄×Q₈×C₅₃	2
C₂²×C₄×D₄×C₅₃	2
C₂²×#548	53

Subgroups — C22×C53×#548 #27270210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#548 #27270210