Subgroups — C₂×C₅₃×#3147 #27268144

Order 13568.

3366 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₅₃	C₂	1	4	yes	maximal, complemented
2	C₂×C₅₃×#532	C₂	1	2	yes	maximal, complemented
3	C₅₃×#3147	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#495	C₂	1	4	yes	maximal, complemented
5	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#492	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₅₃	C₂²	1	4	yes	—
10	C₂×#135×C₅₃	—	2	32	no	—
11	C₂×#135×C₅₃	C₂²	1	32	yes	complemented
12	C₂²×Q₁₆×C₅₃	C₂²	1	4	yes	—
13	C₂²×SD₁₆×C₅₃	C₂²	1	8	yes	—
14	C₂²×D₈×C₅₃	C₂²	1	4	yes	—
15	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#532	—	2	8	no	—
17	C₅₃×#532	C₂²	1	16	yes	complemented
18	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
19	C₂×#127×C₅₃	C₂²	1	2	yes	—
20	C₂×#124×C₅₃	C₂²	1	4	yes	—
21	C₂×#121×C₅₃	—	2	4	no	—
22	C₂×C₄×D₄×C₅₃	—	2	4	no	—
23	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#495	C₂²	1	32	yes	complemented
25	C₅₃×#494	C₂²	1	8	yes	complemented
26	C₅₃×#493	C₂²	1	8	yes	complemented
27	C₅₃×#492	C₂²	1	16	yes	complemented
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	8	yes	complemented
32	C₂×#41×C₅₃	—	2	8	no	—
33	C₂×#41×C₅₃	—	2	32	no	—
34	C₂×#41×C₅₃	C₂³	1	16	yes	—
35	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
36	C₂²×D₄×C₅₃	—	2	4	no	—
37	C₂²×D₄×C₅₃	—	2	4	no	—
38	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
39	C₂³×C₄×C₅₃	—	2	4	no	—
40	C₂³×C₄×C₅₃	D₄	1	2	yes	—
41	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	#135×C₅₃	—	2	128	no	—
43	C₂×Q₁₆×C₅₃	—	2	16	no	—
44	C₂×Q₁₆×C₅₃	C₂³	1	16	yes	—
45	C₂×SD₁₆×C₅₃	—	2	32	no	—
46	C₂×SD₁₆×C₅₃	C₂³	1	32	yes	—
47	C₂×D₈×C₅₃	—	2	16	no	—
48	C₂×D₈×C₅₃	C₂³	1	16	yes	—
49	C₂²×C₈×C₅₃	D₄	1	16	yes	complemented
50	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂³	1	8	yes	—
53	C₄⋊Q₈×C₅₃	—	2	4	no	—
54	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	—
55	#127×C₅₃	—	2	4	no	—
56	#127×C₅₃	C₂³	1	8	yes	—
57	#124×C₅₃	—	2	8	no	—
58	#124×C₅₃	C₂³	1	16	yes	—
59	#121×C₅₃	—	2	32	no	—
60	C₄×D₄×C₅₃	—	2	32	no	—
61	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
62	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
63	C₂×C₄²×C₅₃	D₄	1	4	yes	—
64	C₂×C₄²×C₅₃	D₄	1	2	yes	—
65	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
68	C₄×C₈×C₅₃	C₂³	1	4	yes	—
69	C₄×C₈×C₅₃	C₂³	1	4	yes	—
70	C₄×C₈×C₅₃	C₂³	1	8	yes	—
71	C₂⁴×C₅₃	—	2	4	no	—
72	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
73	#41×C₅₃	—	2	64	no	—
74	#41×C₅₃	—	4	32	no	—
75	C₂×Q₈×C₅₃	—	2	4	no	—
76	C₂×Q₈×C₅₃	—	2	16	no	—
77	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
78	C₂×D₄×C₅₃	—	2	16	no	—
79	C₂×D₄×C₅₃	—	2	16	no	—
80	C₂×D₄×C₅₃	—	2	16	no	—
81	C₂×D₄×C₅₃	—	2	4	no	—
82	C₂×D₄×C₅₃	—	4	16	no	—
83	C₂×D₄×C₅₃	—	4	16	no	—
84	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
85	C₂²×C₄×C₅₃	—	2	8	no	—
86	C₂²×C₄×C₅₃	—	2	4	no	—
87	C₂²×C₄×C₅₃	—	2	16	no	—
88	C₂²×C₄×C₅₃	—	4	16	no	—
89	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
91	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
93	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
97	Q₁₆×C₅₃	—	2	32	no	—
98	SD₁₆×C₅₃	—	2	64	no	—
99	D₈×C₅₃	—	2	32	no	—
100	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	—
101	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	—
102	C₂×C₈×C₅₃	C₂×D₄	1	32	yes	complemented
103	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
104	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
105	C₄⋊C₄×C₅₃	—	2	16	no	—
106	C₂²⋊C₄×C₅₃	—	2	32	no	—
107	C₄²×C₅₃	C₂×D₄	1	8	yes	—
108	C₄²×C₅₃	C₂×D₄	1	2	yes	—
109	C₄²×C₅₃	C₂×D₄	1	2	yes	—
110	C₄²×C₅₃	C₂⁴	1	2	yes	—
111	C₄²×C₅₃	C₂⁴	1	2	yes	—
112	C₂³×C₅₃	—	2	4	no	—
113	C₂³×C₅₃	—	2	8	no	—
114	C₂³×C₅₃	—	4	8	no	—
115	C₂³×C₅₃	—	4	16	no	—
116	C₂³×C₅₃	#127	1	4	yes	—
117	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
118	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
119	Q₈×C₅₃	—	2	16	no	—
120	Q₈×C₅₃	—	4	8	no	—
121	D₄×C₅₃	—	2	16	no	—
122	D₄×C₅₃	—	4	32	no	—
123	D₄×C₅₃	—	4	8	no	—
124	D₄×C₅₃	—	4	32	no	—
125	C₂×C₄×C₅₃	—	2	4	no	—
126	C₂×C₄×C₅₃	—	2	16	no	—
127	C₂×C₄×C₅₃	—	2	8	no	—
128	C₂×C₄×C₅₃	—	4	8	no	—
129	C₂×C₄×C₅₃	—	4	32	no	—
130	C₂×C₄×C₅₃	#127	1	4	yes	—
131	C₂×C₄×C₅₃	#127	1	8	yes	—
132	C₂×C₄×C₅₃	#135	1	8	yes	complemented
133	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
134	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
135	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
137	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
138	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
139	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
140	C₈×C₅₃	C₂²×D₄	1	8	yes	—
141	C₈×C₅₃	C₂²×D₄	1	8	yes	—
142	C₂×#3147	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂²×C₅₃	—	2	8	no	—
144	C₂²×C₅₃	—	2	4	no	—
145	C₂²×C₅₃	—	4	8	no	—
146	C₂²×C₅₃	—	4	16	no	—
147	C₂²×C₅₃	—	8	16	no	—
148	C₂²×C₅₃	C₂×#127	1	4	yes	—
149	C₂²×C₅₃	C₂×#127	1	2	yes	—
150	C₂²×C₅₃	C₂×#135	1	4	yes	—
151	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
152	C₄×C₅₃	—	2	4	no	—
153	C₄×C₅₃	—	4	8	no	—
154	C₄×C₅₃	C₂×#127	1	4	yes	—
155	C₄×C₅₃	C₂×#127	1	2	yes	—
156	C₄×C₅₃	C₂×#135	1	8	yes	complemented
157	C₄×C₅₃	C₂³×D₄	1	2	yes	—
158	C₂²×#135	C₂×C₅₃	1	4	yes	complemented
159	C₂×#532	C₂×C₅₃	1	2	yes	complemented
160	#3147	C₂×C₅₃	1	16	yes	direct product factor
161	C₂×#495	C₂×C₅₃	1	4	yes	complemented
162	C₂×#494	C₂×C₅₃	1	1	yes	characteristic, complemented
163	C₂×#493	C₂×C₅₃	1	1	yes	characteristic, complemented
164	C₂×#492	C₂×C₅₃	1	2	yes	complemented
165	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
166	C₂×C₅₃	—	2	4	no	—
167	C₂×C₅₃	—	8	8	no	—
168	C₂×C₅₃	#3147	1	4	yes	direct product factor
169	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
170	C₂×C₅₃	C₂²×#135	1	2	yes	—
171	C₂²×#41	C₂²×C₅₃	1	4	yes	—
172	C₂×#135	—	2	32	no	—
173	C₂×#135	C₂²×C₅₃	1	32	yes	complemented
174	C₂²×Q₁₆	C₂²×C₅₃	1	4	yes	—
175	C₂²×SD₁₆	C₂²×C₅₃	1	8	yes	—
176	C₂²×D₈	C₂²×C₅₃	1	4	yes	—
177	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
178	#532	—	2	8	no	—
179	#532	C₂²×C₅₃	1	16	yes	complemented
180	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
181	C₂×#127	C₂²×C₅₃	1	2	yes	—
182	C₂×#124	C₂²×C₅₃	1	4	yes	—
183	C₂×#121	—	2	4	no	—
184	C₂×C₄×D₄	—	2	4	no	—
185	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
186	#495	C₂²×C₅₃	1	32	yes	complemented
187	#494	C₂²×C₅₃	1	8	yes	complemented
188	#493	C₂²×C₅₃	1	8	yes	complemented
189	#492	C₂²×C₅₃	1	16	yes	complemented
190	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
191	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
192	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
193	C₂×C₄×C₈	C₂²×C₅₃	1	8	yes	complemented
194	C₅₃	C₂×#3147	1	1	yes	characteristic, direct product factor, Sylow
195	C₂×#41	—	2	8	no	—
196	C₂×#41	—	2	32	no	—
197	C₂×#41	C₂³×C₅₃	1	16	yes	—
198	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
199	C₂²×D₄	—	2	4	no	—
200	C₂²×D₄	—	2	4	no	—
201	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
202	C₂³×C₄	—	2	4	no	—
203	C₂³×C₄	D₄×C₅₃	1	2	yes	—
204	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
205	#135	—	2	128	no	—
206	C₂×Q₁₆	—	2	16	no	—
207	C₂×Q₁₆	C₂³×C₅₃	1	16	yes	—
208	C₂×SD₁₆	—	2	32	no	—
209	C₂×SD₁₆	C₂³×C₅₃	1	32	yes	—
210	C₂×D₈	—	2	16	no	—
211	C₂×D₈	C₂³×C₅₃	1	16	yes	—
212	C₂²×C₈	D₄×C₅₃	1	16	yes	complemented
213	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
214	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
215	C₂²×C₈	C₂³×C₅₃	1	8	yes	—
216	C₄⋊Q₈	—	2	4	no	—
217	C₄⋊Q₈	C₂³×C₅₃	1	8	yes	—
218	#127	—	2	4	no	—
219	#127	C₂³×C₅₃	1	8	yes	—
220	#124	—	2	8	no	—
221	#124	C₂³×C₅₃	1	16	yes	—
222	#121	—	2	32	no	—
223	C₄×D₄	—	2	32	no	—
224	C₂×C₄⋊C₄	—	2	4	no	—
225	C₂×C₂²⋊C₄	—	2	8	no	—
226	C₂×C₄²	D₄×C₅₃	1	2	yes	—
227	C₂×C₄²	D₄×C₅₃	1	4	yes	—
228	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
229	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
230	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
231	C₄×C₈	C₂³×C₅₃	1	4	yes	—
232	C₄×C₈	C₂³×C₅₃	1	4	yes	—
233	C₄×C₈	C₂³×C₅₃	1	8	yes	—
234	C₂⁴	—	2	4	no	—
235	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
236	#41	—	2	64	no	—
237	#41	—	4	32	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	2	16	no	—
240	C₂×Q₈	C₂⁴×C₅₃	1	8	yes	—
241	C₂×D₄	—	2	16	no	—
242	C₂×D₄	—	2	16	no	—
243	C₂×D₄	—	2	16	no	—
244	C₂×D₄	—	2	4	no	—
245	C₂×D₄	—	4	16	no	—
246	C₂×D₄	—	4	16	no	—
247	C₂×D₄	C₂⁴×C₅₃	1	8	yes	—
248	C₂²×C₄	—	2	8	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	16	no	—
251	C₂²×C₄	—	4	16	no	—
252	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
254	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
255	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
257	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
258	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
259	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
260	Q₁₆	—	2	32	no	—
261	SD₁₆	—	2	64	no	—
262	D₈	—	2	32	no	—
263	C₂×C₈	C₂×D₄×C₅₃	1	8	yes	—
264	C₂×C₈	C₂×D₄×C₅₃	1	8	yes	—
265	C₂×C₈	C₂×D₄×C₅₃	1	32	yes	complemented
266	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
267	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
268	C₄⋊C₄	—	2	16	no	—
269	C₂²⋊C₄	—	2	32	no	—
270	C₄²	C₂×D₄×C₅₃	1	8	yes	—
271	C₄²	C₂×D₄×C₅₃	1	2	yes	—
272	C₄²	C₂×D₄×C₅₃	1	2	yes	—
273	C₄²	C₂⁴×C₅₃	1	2	yes	—
274	C₄²	C₂⁴×C₅₃	1	2	yes	—
275	C₂³	—	2	4	no	—
276	C₂³	—	2	8	no	—
277	C₂³	—	4	8	no	—
278	C₂³	—	4	16	no	—
279	C₂³	#127×C₅₃	1	4	yes	—
280	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
281	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
282	Q₈	—	2	16	no	—
283	Q₈	—	4	8	no	—
284	D₄	—	2	16	no	—
285	D₄	—	4	32	no	—
286	D₄	—	4	8	no	—
287	D₄	—	4	32	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	16	no	—
290	C₂×C₄	—	2	8	no	—
291	C₂×C₄	—	4	8	no	—
292	C₂×C₄	—	4	32	no	—
293	C₂×C₄	#127×C₅₃	1	4	yes	—
294	C₂×C₄	#127×C₅₃	1	8	yes	—
295	C₂×C₄	#135×C₅₃	1	8	yes	complemented
296	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
297	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
298	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
300	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
301	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
302	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
303	C₈	C₂²×D₄×C₅₃	1	8	yes	—
304	C₈	C₂²×D₄×C₅₃	1	8	yes	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	8	no	—
307	C₂²	—	4	16	no	—
308	C₂²	—	4	8	no	—
309	C₂²	—	8	16	no	—
310	C₂²	C₂×#127×C₅₃	1	4	yes	—
311	C₂²	C₂×#127×C₅₃	1	2	yes	—
312	C₂²	C₂×#135×C₅₃	1	4	yes	—
313	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
314	C₄	—	2	4	no	—
315	C₄	—	4	8	no	—
316	C₄	C₂×#127×C₅₃	1	4	yes	—
317	C₄	C₂×#127×C₅₃	1	2	yes	—
318	C₄	C₂×#135×C₅₃	1	8	yes	complemented
319	C₄	C₂³×D₄×C₅₃	1	2	yes	—
320	C₂	—	2	4	no	—
321	C₂	—	8	8	no	—
322	C₂	C₅₃×#3147	1	4	yes	direct product factor
323	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
324	C₂	C₂²×#135×C₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
socle	C₂³×C₅₃	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#135×C₅₃	C₂	semidirect
2. C₂×C₅₃×#532	C₂	semidirect
3. C₅₃×#3147	C₂	direct (N × H)
4. C₂×C₅₃×#495	C₂	semidirect
5. C₂×C₅₃×#494	C₂	semidirect
6. C₂×C₅₃×#493	C₂	semidirect
7. C₂×C₅₃×#492	C₂	semidirect
8. C₂²×C₄×C₈×C₅₃	C₂	semidirect
11. C₂×#135×C₅₃	C₂²	semidirect
17. C₅₃×#532	C₂²	semidirect
24. C₅₃×#495	C₂²	semidirect
25. C₅₃×#494	C₂²	semidirect
26. C₅₃×#493	C₂²	semidirect
27. C₅₃×#492	C₂²	semidirect
31. C₂×C₄×C₈×C₅₃	C₂²	semidirect
49. C₂²×C₈×C₅₃	D₄	semidirect
102. C₂×C₈×C₅₃	C₂×D₄	semidirect
132. C₂×C₄×C₅₃	#135	semidirect
142. C₂×#3147	C₅₃	direct (N × H)
156. C₄×C₅₃	C₂×#135	semidirect
158. C₂²×#135	C₂×C₅₃	semidirect
159. C₂×#532	C₂×C₅₃	semidirect
160. #3147	C₂×C₅₃	direct (N × H)
161. C₂×#495	C₂×C₅₃	semidirect
162. C₂×#494	C₂×C₅₃	semidirect
163. C₂×#493	C₂×C₅₃	semidirect
164. C₂×#492	C₂×C₅₃	semidirect
165. C₂²×C₄×C₈	C₂×C₅₃	semidirect
168. C₂×C₅₃	#3147	direct (N × H)
173. C₂×#135	C₂²×C₅₃	semidirect
179. #532	C₂²×C₅₃	semidirect
186. #495	C₂²×C₅₃	semidirect
187. #494	C₂²×C₅₃	semidirect
188. #493	C₂²×C₅₃	semidirect
189. #492	C₂²×C₅₃	semidirect
193. C₂×C₄×C₈	C₂²×C₅₃	semidirect
194. C₅₃	C₂×#3147	direct (N × H)
212. C₂²×C₈	D₄×C₅₃	semidirect
265. C₂×C₈	C₂×D₄×C₅₃	semidirect
295. C₂×C₄	#135×C₅₃	semidirect
318. C₄	C₂×#135×C₅₃	semidirect
322. C₂	C₅₃×#3147	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#135×C₅₃	2
C₂×C₅₃×#532	2
C₅₃×#3147	2
C₂×C₅₃×#495	2
C₂×C₅₃×#494	2
C₂×C₅₃×#493	2
C₂×C₅₃×#492	2
C₂²×C₄×C₈×C₅₃	2
C₂×#3147	53

Subgroups — C2×C53×#3147 #27268144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3147 #27268144