Subgroups — C₂²×C₅₃×#532 #27270178

Order 13568.

12878 conjugacy classes of subgroups.

The subgroup lattice has 300 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	2	yes	maximal, complemented
2	C₂×C₅₃×#532	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#127×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#124×C₅₃	C₂	1	2	yes	maximal, complemented
6	C₂²×#121×C₅₃	C₂	1	4	yes	maximal, complemented
7	C₂²×C₄×D₄×C₅₃	C₂	1	4	yes	maximal, complemented
8	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₅₃	—	2	32	no	—
10	C₂²×#41×C₅₃	C₂²	1	48	yes	complemented
11	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
12	C₂³×D₄×C₅₃	C₂²	1	4	yes	—
13	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
14	C₂³×D₄×C₅₃	C₂²	1	4	yes	—
15	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₅₃	C₂²	1	4	yes	—
17	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#532	C₂²	1	256	yes	direct product factor
19	C₂×C₄⋊Q₈×C₅₃	C₂²	1	24	yes	complemented
20	C₂×#127×C₅₃	C₂²	1	24	yes	complemented
21	C₂×#124×C₅₃	C₂²	1	48	yes	complemented
22	C₂×#121×C₅₃	C₂²	1	96	yes	complemented
23	C₂×C₄×D₄×C₅₃	C₂²	1	96	yes	complemented
24	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
25	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
26	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
29	C₂²×C₄²×C₅₃	C₂²	1	24	yes	complemented
30	C₂⁵×C₅₃	C₂³	1	4	yes	—
31	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×#41×C₅₃	—	2	384	no	—
33	C₂×#41×C₅₃	C₂³	1	128	yes	complemented
34	C₂²×Q₈×C₅₃	—	2	16	no	—
35	C₂²×Q₈×C₅₃	C₂³	1	24	yes	—
36	C₂²×D₄×C₅₃	—	2	32	no	—
37	C₂²×D₄×C₅₃	—	2	16	no	—
38	C₂²×D₄×C₅₃	—	2	32	no	—
39	C₂²×D₄×C₅₃	C₂³	1	48	yes	—
40	C₂²×D₄×C₅₃	C₂³	1	48	yes	—
41	C₂²×D₄×C₅₃	C₂³	1	24	yes	—
42	C₂³×C₄×C₅₃	—	2	32	no	—
43	C₂³×C₄×C₅₃	—	2	16	no	—
44	C₂³×C₄×C₅₃	D₄	1	16	yes	complemented
45	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
46	C₂³×C₄×C₅₃	C₂³	1	24	yes	—
47	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
48	C₂³×C₄×C₅₃	C₂³	1	48	yes	—
49	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
51	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
52	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₅₃	C₂³	1	64	yes	complemented
54	#127×C₅₃	C₂³	1	64	yes	complemented
55	#124×C₅₃	C₂³	1	128	yes	complemented
56	#121×C₅₃	C₂³	1	256	yes	complemented
57	C₄×D₄×C₅₃	C₂³	1	256	yes	complemented
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	48	yes	—
59	C₂×C₂²⋊C₄×C₅₃	C₂³	1	96	yes	—
60	C₂×C₄²×C₅₃	C₂³	1	24	yes	—
61	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
62	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
63	C₂×C₄²×C₅₃	C₂³	1	64	yes	complemented
64	C₂⁴×C₅₃	—	2	8	no	—
65	C₂⁴×C₅₃	—	2	16	no	—
66	C₂⁴×C₅₃	—	2	32	no	—
67	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
68	C₂⁴×C₅₃	C₂⁴	1	24	yes	—
69	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
70	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
71	#41×C₅₃	—	2	512	no	—
72	C₂×Q₈×C₅₃	—	2	96	no	—
73	C₂×Q₈×C₅₃	C₂⁴	1	32	yes	—
74	C₂×D₄×C₅₃	—	2	192	no	—
75	C₂×D₄×C₅₃	—	2	96	no	—
76	C₂×D₄×C₅₃	—	2	192	no	—
77	C₂×D₄×C₅₃	C₂⁴	1	64	yes	—
78	C₂×D₄×C₅₃	C₂⁴	1	64	yes	—
79	C₂×D₄×C₅₃	C₂⁴	1	32	yes	—
80	C₂²×C₄×C₅₃	—	2	16	no	—
81	C₂²×C₄×C₅₃	—	2	8	no	—
82	C₂²×C₄×C₅₃	—	2	192	no	—
83	C₂²×C₄×C₅₃	—	2	96	no	—
84	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
85	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
86	C₂²×C₄×C₅₃	C₂×D₄	1	96	yes	complemented
87	C₂²×C₄×C₅₃	#41	1	16	yes	complemented
88	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
89	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	—
90	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
91	C₂²×C₄×C₅₃	C₂⁴	1	64	yes	—
92	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
93	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
94	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
95	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
96	C₄⋊C₄×C₅₃	C₂⁴	1	64	yes	—
97	C₂²⋊C₄×C₅₃	C₂⁴	1	128	yes	—
98	C₄²×C₅₃	C₂⁴	1	32	yes	—
99	C₄²×C₅₃	C₂⁴	1	16	yes	—
100	C₄²×C₅₃	C₂⁴	1	16	yes	—
101	C₂³×C₅₃	—	2	32	no	—
102	C₂³×C₅₃	—	2	24	no	—
103	C₂³×C₅₃	—	2	48	no	—
104	C₂³×C₅₃	—	2	96	no	—
105	C₂³×C₅₃	—	4	64	no	—
106	C₂³×C₅₃	C₂²×D₄	1	24	yes	—
107	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
108	C₂³×C₅₃	C₂×#41	1	8	yes	—
109	C₂³×C₅₃	C₂⁵	1	16	yes	—
110	C₂³×C₅₃	C₂⁵	1	4	yes	—
111	C₂³×C₅₃	C₂⁵	1	3	yes	—
112	Q₈×C₅₃	—	2	64	no	—
113	D₄×C₅₃	—	2	128	no	—
114	D₄×C₅₃	—	2	64	no	—
115	D₄×C₅₃	—	2	128	no	—
116	C₂×C₄×C₅₃	—	2	48	no	—
117	C₂×C₄×C₅₃	—	2	24	no	—
118	C₂×C₄×C₅₃	—	2	128	no	—
119	C₂×C₄×C₅₃	—	2	64	no	—
120	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
121	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
122	C₂×C₄×C₅₃	C₂²×D₄	1	64	yes	complemented
123	C₂×C₄×C₅₃	C₂×#41	1	48	yes	complemented
124	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
125	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
126	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
127	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
128	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
129	C₂²×#532	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₅₃	—	2	8	no	—
131	C₂²×C₅₃	—	2	48	no	—
132	C₂²×C₅₃	—	2	16	no	—
133	C₂²×C₅₃	—	2	32	no	—
134	C₂²×C₅₃	—	4	96	no	—
135	C₂²×C₅₃	#532	1	16	yes	direct product factor
136	C₂²×C₅₃	C₂³×D₄	1	8	yes	—
137	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
138	C₂²×C₅₃	C₂²×#41	1	12	yes	—
139	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
140	C₄×C₅₃	—	2	16	no	—
141	C₄×C₅₃	—	2	8	no	—
142	C₄×C₅₃	C₂³×D₄	1	8	yes	—
143	C₄×C₅₃	C₂³×D₄	1	8	yes	—
144	C₄×C₅₃	C₂²×#41	1	16	yes	complemented
145	C₂³×#41	C₂×C₅₃	1	2	yes	complemented
146	C₂×#532	C₂×C₅₃	1	48	yes	direct product factor
147	C₂²×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
148	C₂²×#127	C₂×C₅₃	1	1	yes	characteristic, complemented
149	C₂²×#124	C₂×C₅₃	1	2	yes	complemented
150	C₂²×#121	C₂×C₅₃	1	4	yes	complemented
151	C₂²×C₄×D₄	C₂×C₅₃	1	4	yes	complemented
152	C₂³×C₄²	C₂×C₅₃	1	1	yes	characteristic, complemented
153	C₂×C₅₃	—	2	8	no	—
154	C₂×C₅₃	—	4	16	no	—
155	C₂×C₅₃	C₂×#532	1	12	yes	direct product factor
156	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
157	C₂×C₅₃	C₂³×#41	1	2	yes	—
158	C₂²×#41	—	2	32	no	—
159	C₂²×#41	C₂²×C₅₃	1	48	yes	complemented
160	C₂³×Q₈	C₂²×C₅₃	1	2	yes	—
161	C₂³×D₄	C₂²×C₅₃	1	4	yes	—
162	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
163	C₂³×D₄	C₂²×C₅₃	1	4	yes	—
164	C₂⁴×C₄	C₂²×C₅₃	1	4	yes	—
165	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
166	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
167	#532	C₂²×C₅₃	1	256	yes	direct product factor
168	C₂×C₄⋊Q₈	C₂²×C₅₃	1	24	yes	complemented
169	C₂×#127	C₂²×C₅₃	1	24	yes	complemented
170	C₂×#124	C₂²×C₅₃	1	48	yes	complemented
171	C₂×#121	C₂²×C₅₃	1	96	yes	complemented
172	C₂×C₄×D₄	C₂²×C₅₃	1	96	yes	complemented
173	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
174	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
175	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
176	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
177	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
178	C₂²×C₄²	C₂²×C₅₃	1	24	yes	complemented
179	C₅₃	C₂²×#532	1	1	yes	characteristic, direct product factor, Sylow
180	C₂⁵	C₂³×C₅₃	1	4	yes	—
181	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
182	C₂×#41	—	2	384	no	—
183	C₂×#41	C₂³×C₅₃	1	128	yes	complemented
184	C₂²×Q₈	—	2	16	no	—
185	C₂²×Q₈	C₂³×C₅₃	1	24	yes	—
186	C₂²×D₄	—	2	32	no	—
187	C₂²×D₄	—	2	16	no	—
188	C₂²×D₄	—	2	32	no	—
189	C₂²×D₄	C₂³×C₅₃	1	48	yes	—
190	C₂²×D₄	C₂³×C₅₃	1	48	yes	—
191	C₂²×D₄	C₂³×C₅₃	1	24	yes	—
192	C₂³×C₄	—	2	32	no	—
193	C₂³×C₄	—	2	16	no	—
194	C₂³×C₄	D₄×C₅₃	1	16	yes	complemented
195	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
196	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
197	C₂³×C₄	C₂³×C₅₃	1	48	yes	—
198	C₂³×C₄	C₂³×C₅₃	1	24	yes	—
199	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
200	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
201	C₂³×C₄	C₂³×C₅₃	1	12	yes	—
202	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
203	C₄⋊Q₈	C₂³×C₅₃	1	64	yes	complemented
204	#127	C₂³×C₅₃	1	64	yes	complemented
205	#124	C₂³×C₅₃	1	128	yes	complemented
206	#121	C₂³×C₅₃	1	256	yes	complemented
207	C₄×D₄	C₂³×C₅₃	1	256	yes	complemented
208	C₂×C₄⋊C₄	C₂³×C₅₃	1	48	yes	—
209	C₂×C₂²⋊C₄	C₂³×C₅₃	1	96	yes	—
210	C₂×C₄²	C₂³×C₅₃	1	24	yes	—
211	C₂×C₄²	C₂³×C₅₃	1	12	yes	—
212	C₂×C₄²	C₂³×C₅₃	1	12	yes	—
213	C₂×C₄²	C₂³×C₅₃	1	64	yes	complemented
214	C₂⁴	—	2	8	no	—
215	C₂⁴	—	2	16	no	—
216	C₂⁴	—	2	32	no	—
217	C₂⁴	C₂×D₄×C₅₃	1	8	yes	—
218	C₂⁴	C₂⁴×C₅₃	1	24	yes	—
219	C₂⁴	C₂⁴×C₅₃	1	6	yes	—
220	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
221	#41	—	2	512	no	—
222	C₂×Q₈	—	2	96	no	—
223	C₂×Q₈	C₂⁴×C₅₃	1	32	yes	—
224	C₂×D₄	—	2	192	no	—
225	C₂×D₄	—	2	96	no	—
226	C₂×D₄	—	2	192	no	—
227	C₂×D₄	C₂⁴×C₅₃	1	64	yes	—
228	C₂×D₄	C₂⁴×C₅₃	1	64	yes	—
229	C₂×D₄	C₂⁴×C₅₃	1	32	yes	—
230	C₂²×C₄	—	2	16	no	—
231	C₂²×C₄	—	2	8	no	—
232	C₂²×C₄	—	2	192	no	—
233	C₂²×C₄	—	2	96	no	—
234	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
235	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
236	C₂²×C₄	C₂×D₄×C₅₃	1	96	yes	complemented
237	C₂²×C₄	#41×C₅₃	1	16	yes	complemented
238	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
239	C₂²×C₄	C₂⁴×C₅₃	1	32	yes	—
240	C₂²×C₄	C₂⁴×C₅₃	1	24	yes	—
241	C₂²×C₄	C₂⁴×C₅₃	1	64	yes	—
242	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
243	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
244	C₂²×C₄	C₂⁴×C₅₃	1	16	yes	—
245	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
246	C₄⋊C₄	C₂⁴×C₅₃	1	64	yes	—
247	C₂²⋊C₄	C₂⁴×C₅₃	1	128	yes	—
248	C₄²	C₂⁴×C₅₃	1	32	yes	—
249	C₄²	C₂⁴×C₅₃	1	16	yes	—
250	C₄²	C₂⁴×C₅₃	1	16	yes	—
251	C₂³	—	2	32	no	—
252	C₂³	—	2	24	no	—
253	C₂³	—	2	48	no	—
254	C₂³	—	2	96	no	—
255	C₂³	—	4	64	no	—
256	C₂³	C₂²×D₄×C₅₃	1	24	yes	—
257	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
258	C₂³	C₂×#41×C₅₃	1	8	yes	—
259	C₂³	C₂⁵×C₅₃	1	16	yes	—
260	C₂³	C₂⁵×C₅₃	1	4	yes	—
261	C₂³	C₂⁵×C₅₃	1	3	yes	—
262	Q₈	—	2	64	no	—
263	D₄	—	2	128	no	—
264	D₄	—	2	64	no	—
265	D₄	—	2	128	no	—
266	C₂×C₄	—	2	48	no	—
267	C₂×C₄	—	2	24	no	—
268	C₂×C₄	—	2	128	no	—
269	C₂×C₄	—	2	64	no	—
270	C₂×C₄	C₂²×D₄×C₅₃	1	24	yes	—
271	C₂×C₄	C₂²×D₄×C₅₃	1	24	yes	—
272	C₂×C₄	C₂²×D₄×C₅₃	1	64	yes	complemented
273	C₂×C₄	C₂×#41×C₅₃	1	48	yes	complemented
274	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
275	C₂×C₄	C₂⁵×C₅₃	1	16	yes	—
276	C₂×C₄	C₂⁵×C₅₃	1	4	yes	—
277	C₂×C₄	C₂⁵×C₅₃	1	4	yes	—
278	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
279	C₂²	—	2	8	no	—
280	C₂²	—	2	48	no	—
281	C₂²	—	2	16	no	—
282	C₂²	—	2	32	no	—
283	C₂²	—	4	96	no	—
284	C₂²	C₅₃×#532	1	16	yes	direct product factor
285	C₂²	C₂³×D₄×C₅₃	1	8	yes	—
286	C₂²	C₂³×D₄×C₅₃	1	6	yes	—
287	C₂²	C₂²×#41×C₅₃	1	12	yes	—
288	C₂²	C₂⁶×C₅₃	1	1	yes	characteristic
289	C₄	—	2	16	no	—
290	C₄	—	2	8	no	—
291	C₄	C₂³×D₄×C₅₃	1	8	yes	—
292	C₄	C₂³×D₄×C₅₃	1	8	yes	—
293	C₄	C₂²×#41×C₅₃	1	16	yes	complemented
294	C₂	—	2	8	no	—
295	C₂	—	4	16	no	—
296	C₂	C₂×C₅₃×#532	1	12	yes	direct product factor
297	C₂	C₂⁴×D₄×C₅₃	1	1	yes	characteristic
298	C₂	C₂³×#41×C₅₃	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₅₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₅₃
socle	C₂⁴×C₅₃	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×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. C₂×C₅₃×#532	C₂	direct (N × H)
3. C₂²×C₄⋊Q₈×C₅₃	C₂	semidirect
4. C₂²×#127×C₅₃	C₂	semidirect
5. C₂²×#124×C₅₃	C₂	semidirect
6. C₂²×#121×C₅₃	C₂	semidirect
7. C₂²×C₄×D₄×C₅₃	C₂	semidirect
8. C₂³×C₄²×C₅₃	C₂	semidirect
10. C₂²×#41×C₅₃	C₂²	semidirect
18. C₅₃×#532	C₂²	direct (N × H)
19. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
20. C₂×#127×C₅₃	C₂²	semidirect
21. C₂×#124×C₅₃	C₂²	semidirect
22. C₂×#121×C₅₃	C₂²	semidirect
23. C₂×C₄×D₄×C₅₃	C₂²	semidirect
29. C₂²×C₄²×C₅₃	C₂²	semidirect
33. C₂×#41×C₅₃	C₂³	semidirect
44. C₂³×C₄×C₅₃	D₄	semidirect
53. C₄⋊Q₈×C₅₃	C₂³	semidirect
54. #127×C₅₃	C₂³	semidirect
55. #124×C₅₃	C₂³	semidirect
56. #121×C₅₃	C₂³	semidirect
57. C₄×D₄×C₅₃	C₂³	semidirect
63. C₂×C₄²×C₅₃	C₂³	semidirect
86. C₂²×C₄×C₅₃	C₂×D₄	semidirect
87. C₂²×C₄×C₅₃	#41	semidirect
122. C₂×C₄×C₅₃	C₂²×D₄	semidirect
123. C₂×C₄×C₅₃	C₂×#41	semidirect
129. C₂²×#532	C₅₃	direct (N × H)
135. C₂²×C₅₃	#532	direct (N × H)
144. C₄×C₅₃	C₂²×#41	semidirect
145. C₂³×#41	C₂×C₅₃	semidirect
146. C₂×#532	C₂×C₅₃	direct (N × H)
147. C₂²×C₄⋊Q₈	C₂×C₅₃	semidirect
148. C₂²×#127	C₂×C₅₃	semidirect
149. C₂²×#124	C₂×C₅₃	semidirect
150. C₂²×#121	C₂×C₅₃	semidirect
151. C₂²×C₄×D₄	C₂×C₅₃	semidirect
152. C₂³×C₄²	C₂×C₅₃	semidirect
155. C₂×C₅₃	C₂×#532	direct (N × H)
159. C₂²×#41	C₂²×C₅₃	semidirect
167. #532	C₂²×C₅₃	direct (N × H)
168. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
169. C₂×#127	C₂²×C₅₃	semidirect
170. C₂×#124	C₂²×C₅₃	semidirect
171. C₂×#121	C₂²×C₅₃	semidirect
172. C₂×C₄×D₄	C₂²×C₅₃	semidirect
178. C₂²×C₄²	C₂²×C₅₃	semidirect
179. C₅₃	C₂²×#532	direct (N × H)
183. C₂×#41	C₂³×C₅₃	semidirect
194. C₂³×C₄	D₄×C₅₃	semidirect
203. C₄⋊Q₈	C₂³×C₅₃	semidirect
204. #127	C₂³×C₅₃	semidirect
205. #124	C₂³×C₅₃	semidirect
206. #121	C₂³×C₅₃	semidirect
207. C₄×D₄	C₂³×C₅₃	semidirect
213. C₂×C₄²	C₂³×C₅₃	semidirect
236. C₂²×C₄	C₂×D₄×C₅₃	semidirect
237. C₂²×C₄	#41×C₅₃	semidirect
272. C₂×C₄	C₂²×D₄×C₅₃	semidirect
273. C₂×C₄	C₂×#41×C₅₃	semidirect
284. C₂²	C₅₃×#532	direct (N × H)
293. C₄	C₂²×#41×C₅₃	semidirect
296. C₂	C₂×C₅₃×#532	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
C₂×C₅₃×#532	2
C₂²×C₄⋊Q₈×C₅₃	2
C₂²×#127×C₅₃	2
C₂²×#124×C₅₃	2
C₂²×#121×C₅₃	2
C₂²×C₄×D₄×C₅₃	2
C₂³×C₄²×C₅₃	2
C₂²×#532	53

Subgroups — C22×C53×#532 #27270178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#532 #27270178