Subgroups — C₂×C₅₃×#3362 #27268811

Order 13568.

2174 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#550	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3362	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#486	C₂	1	3	yes	maximal, complemented
5	C₂×C₅₃×#476	C₂	1	3	yes	maximal, complemented
6	C₂×C₅₃×#460	C₂	1	3	yes	maximal, complemented
7	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×SD₁₆×C₅₃	C₂	1	3	yes	maximal, complemented
9	C₂²×#41×C₅₃	—	2	1	no	—
10	C₂²×SD₁₆×C₅₃	C₂²	1	3	yes	—
11	Q₈²×C₅₃	—	2	4	no	—
12	Q₈²×C₅₃	C₂²	1	8	yes	complemented
13	C₅₃×#550	—	2	4	no	—
14	C₅₃×#550	C₂²	1	8	yes	—
15	C₂×C₄⋊Q₈×C₅₃	—	2	3	no	—
16	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
17	C₂×#127×C₅₃	C₂²	1	3	yes	—
18	C₂×#121×C₅₃	—	2	3	no	—
19	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
20	C₂×C₄×Q₈×C₅₃	C₂²	1	3	yes	—
21	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₅₃	C₂²	1	3	yes	—
23	C₅₃×#486	C₂²	1	24	yes	complemented
24	C₅₃×#476	C₂²	1	24	yes	complemented
25	C₅₃×#460	C₂²	1	24	yes	complemented
26	C₈×Q₈×C₅₃	C₂²	1	8	yes	complemented
27	C₄×SD₁₆×C₅₃	C₂²	1	24	yes	complemented
28	C₂×#106×C₅₃	C₂²	1	3	yes	—
29	C₂×C₄⋊C₈×C₅₃	C₂²	1	3	yes	—
30	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×#102×C₅₃	C₂²	1	3	yes	—
32	C₂×#102×C₅₃	C₂²	1	6	yes	—
33	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
34	C₂×#41×C₅₃	—	2	8	no	—
35	C₂×#41×C₅₃	—	4	8	no	—
36	C₂²×Q₈×C₅₃	—	2	3	no	—
37	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²×D₄×C₅₃	—	2	3	no	—
40	C₂²×D₄×C₅₃	—	2	3	no	—
41	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
42	C₂³×C₄×C₅₃	—	2	3	no	—
43	C₂×SD₁₆×C₅₃	—	2	12	no	—
44	C₂×SD₁₆×C₅₃	C₂³	1	12	yes	—
45	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
46	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₅₃	—	2	24	no	—
48	C₄⋊Q₈×C₅₃	—	2	6	no	—
49	C₄⋊Q₈×C₅₃	C₂³	1	12	yes	—
50	#127×C₅₃	—	2	6	no	—
51	#127×C₅₃	C₂³	1	12	yes	—
52	#121×C₅₃	—	2	24	no	—
53	C₄×Q₈×C₅₃	—	2	6	no	—
54	C₄×Q₈×C₅₃	—	2	8	no	—
55	C₄×Q₈×C₅₃	D₄	1	4	yes	—
56	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
57	C₄×Q₈×C₅₃	C₂³	1	12	yes	—
58	C₄×D₄×C₅₃	—	2	6	no	—
59	C₄×D₄×C₅₃	C₂³	1	12	yes	—
60	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
61	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
62	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
63	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
65	C₂×C₂²⋊C₄×C₅₃	—	2	3	no	—
66	C₂×C₄²×C₅₃	—	2	3	no	—
67	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
68	#106×C₅₃	C₂³	1	12	yes	—
69	C₄⋊C₈×C₅₃	C₂³	1	12	yes	—
70	#103×C₅₃	C₂³	1	4	yes	—
71	#102×C₅₃	C₂³	1	12	yes	—
72	#102×C₅₃	C₂³	1	24	yes	—
73	C₄×C₈×C₅₃	C₂³	1	12	yes	—
74	C₂⁴×C₅₃	—	2	3	no	—
75	#41×C₅₃	—	4	32	no	—
76	C₂×Q₈×C₅₃	—	2	1	no	—
77	C₂×Q₈×C₅₃	—	2	12	no	—
78	C₂×Q₈×C₅₃	—	2	2	no	—
79	C₂×Q₈×C₅₃	—	8	6	no	—
80	C₂×Q₈×C₅₃	SD₁₆	1	8	yes	complemented
81	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
82	C₂×Q₈×C₅₃	#41	1	4	yes	complemented
83	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
84	C₂×D₄×C₅₃	—	2	3	no	—
85	C₂×D₄×C₅₃	—	2	12	no	—
86	C₂×D₄×C₅₃	—	2	12	no	—
87	C₂×D₄×C₅₃	—	2	6	no	—
88	C₂×D₄×C₅₃	—	2	6	no	—
89	C₂×D₄×C₅₃	—	4	12	no	—
90	C₂×D₄×C₅₃	—	8	6	no	—
91	C₂×D₄×C₅₃	C₂⁴	1	6	yes	—
92	C₂²×C₄×C₅₃	—	2	1	no	—
93	C₂²×C₄×C₅₃	—	2	3	no	—
94	C₂²×C₄×C₅₃	—	2	1	no	—
95	C₂²×C₄×C₅₃	—	2	12	no	—
96	C₂²×C₄×C₅₃	—	4	12	no	—
97	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
98	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
99	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
100	SD₁₆×C₅₃	—	2	24	no	—
101	C₂×C₈×C₅₃	—	2	6	no	—
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
104	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
105	C₄⋊C₄×C₅₃	—	2	12	no	—
106	C₄⋊C₄×C₅₃	—	2	12	no	—
107	C₄⋊C₄×C₅₃	—	2	1	no	—
108	C₄⋊C₄×C₅₃	—	2	3	no	—
109	C₄⋊C₄×C₅₃	C₂×D₄	1	12	yes	—
110	C₄⋊C₄×C₅₃	C₂⁴	1	6	yes	—
111	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
112	C₂²⋊C₄×C₅₃	—	2	12	no	—
113	C₄²×C₅₃	—	2	12	no	—
114	C₄²×C₅₃	C₂×D₄	1	6	yes	—
115	C₄²×C₅₃	C₂⁴	1	6	yes	—
116	C₂³×C₅₃	—	2	6	no	—
117	C₂³×C₅₃	—	4	6	no	—
118	C₂³×C₅₃	—	4	6	no	—
119	C₂³×C₅₃	—	4	6	no	—
120	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
121	Q₈×C₅₃	—	2	4	no	—
122	Q₈×C₅₃	—	2	2	no	—
123	Q₈×C₅₃	—	8	12	no	—
124	Q₈×C₅₃	C₂×SD₁₆	1	16	yes	complemented
125	Q₈×C₅₃	C₂×#41	1	4	yes	complemented
126	D₄×C₅₃	—	2	6	no	—
127	D₄×C₅₃	—	2	6	no	—
128	D₄×C₅₃	—	4	6	no	—
129	D₄×C₅₃	—	4	24	no	—
130	D₄×C₅₃	—	8	12	no	—
131	C₂×C₄×C₅₃	—	2	4	no	—
132	C₂×C₄×C₅₃	—	2	6	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	6	no	—
136	C₂×C₄×C₅₃	—	4	2	no	—
137	C₂×C₄×C₅₃	—	4	6	no	—
138	C₂×C₄×C₅₃	—	4	24	no	—
139	C₂×C₄×C₅₃	C₂×SD₁₆	1	12	yes	—
140	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
141	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
143	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
144	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
145	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
146	C₈×C₅₃	—	2	6	no	—
147	C₈×C₅₃	—	2	2	no	—
148	C₂×#3362	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂²×C₅₃	—	4	6	no	—
150	C₂²×C₅₃	—	4	6	no	—
151	C₂²×C₅₃	—	4	6	no	—
152	C₂²×C₅₃	—	8	12	no	—
153	C₂²×C₅₃	#548	1	2	yes	—
154	C₂²×C₅₃	C₂²×SD₁₆	1	2	yes	—
155	C₂²×C₅₃	#576	1	2	yes	—
156	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
157	C₄×C₅₃	—	2	4	no	—
158	C₄×C₅₃	—	2	6	no	—
159	C₄×C₅₃	—	4	2	no	—
160	C₄×C₅₃	—	4	6	no	—
161	C₄×C₅₃	#548	1	2	yes	—
162	C₄×C₅₃	C₂²×SD₁₆	1	12	yes	—
163	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
164	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
165	C₂×Q₈²	C₂×C₅₃	1	1	yes	characteristic, complemented
166	C₂×#550	C₂×C₅₃	1	1	yes	characteristic
167	#3362	C₂×C₅₃	1	16	yes	direct product factor
168	C₂×#486	C₂×C₅₃	1	3	yes	complemented
169	C₂×#476	C₂×C₅₃	1	3	yes	complemented
170	C₂×#460	C₂×C₅₃	1	3	yes	complemented
171	C₂×C₈×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
172	C₂×C₄×SD₁₆	C₂×C₅₃	1	3	yes	complemented
173	C₂×C₅₃	—	8	6	no	—
174	C₂×C₅₃	#3362	1	4	yes	direct product factor
175	C₂×C₅₃	C₂×#548	1	1	yes	characteristic
176	C₂×C₅₃	C₂³×SD₁₆	1	1	yes	characteristic
177	C₂×C₅₃	C₂×#576	1	1	yes	characteristic
178	C₂²×#41	—	2	1	no	—
179	C₂²×SD₁₆	C₂²×C₅₃	1	3	yes	—
180	Q₈²	—	2	4	no	—
181	Q₈²	C₂²×C₅₃	1	8	yes	complemented
182	#550	—	2	4	no	—
183	#550	C₂²×C₅₃	1	8	yes	—
184	C₂×C₄⋊Q₈	—	2	3	no	—
185	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	—
186	C₂×#127	C₂²×C₅₃	1	3	yes	—
187	C₂×#121	—	2	3	no	—
188	C₂×C₄×Q₈	—	2	1	no	—
189	C₂×C₄×Q₈	C₂²×C₅₃	1	3	yes	—
190	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
191	C₂×C₄×D₄	C₂²×C₅₃	1	3	yes	—
192	#486	C₂²×C₅₃	1	24	yes	complemented
193	#476	C₂²×C₅₃	1	24	yes	complemented
194	#460	C₂²×C₅₃	1	24	yes	complemented
195	C₈×Q₈	C₂²×C₅₃	1	8	yes	complemented
196	C₄×SD₁₆	C₂²×C₅₃	1	24	yes	complemented
197	C₂×#106	C₂²×C₅₃	1	3	yes	—
198	C₂×C₄⋊C₈	C₂²×C₅₃	1	3	yes	—
199	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
200	C₂×#102	C₂²×C₅₃	1	3	yes	—
201	C₂×#102	C₂²×C₅₃	1	6	yes	—
202	C₂×C₄×C₈	C₂²×C₅₃	1	3	yes	—
203	C₅₃	C₂×#3362	1	1	yes	characteristic, direct product factor, Sylow
204	C₂×#41	—	2	8	no	—
205	C₂×#41	—	4	8	no	—
206	C₂²×Q₈	—	2	3	no	—
207	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
208	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
209	C₂²×D₄	—	2	3	no	—
210	C₂²×D₄	—	2	3	no	—
211	C₂²×D₄	C₂³×C₅₃	1	3	yes	—
212	C₂³×C₄	—	2	3	no	—
213	C₂×SD₁₆	—	2	12	no	—
214	C₂×SD₁₆	C₂³×C₅₃	1	12	yes	—
215	C₂²×C₈	C₂³×C₅₃	1	3	yes	—
216	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
217	C₄⋊Q₈	—	2	24	no	—
218	C₄⋊Q₈	—	2	6	no	—
219	C₄⋊Q₈	C₂³×C₅₃	1	12	yes	—
220	#127	—	2	6	no	—
221	#127	C₂³×C₅₃	1	12	yes	—
222	#121	—	2	24	no	—
223	C₄×Q₈	—	2	8	no	—
224	C₄×Q₈	—	2	6	no	—
225	C₄×Q₈	D₄×C₅₃	1	4	yes	—
226	C₄×Q₈	C₂³×C₅₃	1	12	yes	—
227	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
228	C₄×D₄	—	2	6	no	—
229	C₄×D₄	C₂³×C₅₃	1	12	yes	—
230	C₂×C₄⋊C₄	—	2	3	no	—
231	C₂×C₄⋊C₄	—	2	3	no	—
232	C₂×C₄⋊C₄	D₄×C₅₃	1	3	yes	—
233	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
234	C₂×C₄⋊C₄	C₂³×C₅₃	1	3	yes	—
235	C₂×C₂²⋊C₄	—	2	3	no	—
236	C₂×C₄²	—	2	3	no	—
237	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
238	#106	C₂³×C₅₃	1	12	yes	—
239	C₄⋊C₈	C₂³×C₅₃	1	12	yes	—
240	#103	C₂³×C₅₃	1	4	yes	—
241	#102	C₂³×C₅₃	1	12	yes	—
242	#102	C₂³×C₅₃	1	24	yes	—
243	C₄×C₈	C₂³×C₅₃	1	12	yes	—
244	C₂⁴	—	2	3	no	—
245	#41	—	4	32	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	2	12	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	8	6	no	—
250	C₂×Q₈	SD₁₆×C₅₃	1	8	yes	complemented
251	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
252	C₂×Q₈	#41×C₅₃	1	4	yes	complemented
253	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
254	C₂×D₄	—	2	3	no	—
255	C₂×D₄	—	2	12	no	—
256	C₂×D₄	—	2	12	no	—
257	C₂×D₄	—	2	6	no	—
258	C₂×D₄	—	2	6	no	—
259	C₂×D₄	—	4	12	no	—
260	C₂×D₄	—	8	6	no	—
261	C₂×D₄	C₂⁴×C₅₃	1	6	yes	—
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	2	3	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	2	12	no	—
266	C₂²×C₄	—	4	12	no	—
267	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
268	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
269	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
270	SD₁₆	—	2	24	no	—
271	C₂×C₈	—	2	6	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
274	C₂×C₈	C₂⁴×C₅₃	1	6	yes	—
275	C₄⋊C₄	—	2	12	no	—
276	C₄⋊C₄	—	2	12	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	3	no	—
279	C₄⋊C₄	C₂×D₄×C₅₃	1	12	yes	—
280	C₄⋊C₄	C₂⁴×C₅₃	1	6	yes	—
281	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
282	C₂²⋊C₄	—	2	12	no	—
283	C₄²	—	2	12	no	—
284	C₄²	C₂×D₄×C₅₃	1	6	yes	—
285	C₄²	C₂⁴×C₅₃	1	6	yes	—
286	C₂³	—	2	6	no	—
287	C₂³	—	4	6	no	—
288	C₂³	—	4	6	no	—
289	C₂³	—	4	6	no	—
290	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
291	Q₈	—	2	4	no	—
292	Q₈	—	2	2	no	—
293	Q₈	—	8	12	no	—
294	Q₈	C₂×SD₁₆×C₅₃	1	16	yes	complemented
295	Q₈	C₂×#41×C₅₃	1	4	yes	complemented
296	D₄	—	2	6	no	—
297	D₄	—	2	6	no	—
298	D₄	—	4	6	no	—
299	D₄	—	4	24	no	—
300	D₄	—	8	12	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	6	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	6	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	6	no	—
308	C₂×C₄	—	4	24	no	—
309	C₂×C₄	C₂×SD₁₆×C₅₃	1	12	yes	—
310	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
311	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
313	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
314	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
315	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
316	C₈	—	2	6	no	—
317	C₈	—	2	2	no	—
318	C₂²	—	4	6	no	—
319	C₂²	—	4	6	no	—
320	C₂²	—	4	6	no	—
321	C₂²	—	8	12	no	—
322	C₂²	C₅₃×#548	1	2	yes	—
323	C₂²	C₂²×SD₁₆×C₅₃	1	2	yes	—
324	C₂²	C₅₃×#576	1	2	yes	—
325	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
326	C₄	—	2	4	no	—
327	C₄	—	2	6	no	—
328	C₄	—	4	2	no	—
329	C₄	—	4	6	no	—
330	C₄	C₅₃×#548	1	2	yes	—
331	C₄	C₂²×SD₁₆×C₅₃	1	12	yes	—
332	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
333	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
334	C₂	—	8	6	no	—
335	C₂	C₅₃×#3362	1	4	yes	direct product factor
336	C₂	C₂×C₅₃×#548	1	1	yes	characteristic
337	C₂	C₂³×SD₁₆×C₅₃	1	1	yes	characteristic
338	C₂	C₂×C₅₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₂×C₅₃×#550	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈²×C₅₃	C₂	semidirect
3. C₅₃×#3362	C₂	direct (N × H)
4. C₂×C₅₃×#486	C₂	semidirect
5. C₂×C₅₃×#476	C₂	semidirect
6. C₂×C₅₃×#460	C₂	semidirect
7. C₂×C₈×Q₈×C₅₃	C₂	semidirect
8. C₂×C₄×SD₁₆×C₅₃	C₂	semidirect
12. Q₈²×C₅₃	C₂²	semidirect
23. C₅₃×#486	C₂²	semidirect
24. C₅₃×#476	C₂²	semidirect
25. C₅₃×#460	C₂²	semidirect
26. C₈×Q₈×C₅₃	C₂²	semidirect
27. C₄×SD₁₆×C₅₃	C₂²	semidirect
80. C₂×Q₈×C₅₃	SD₁₆	semidirect
82. C₂×Q₈×C₅₃	#41	semidirect
124. Q₈×C₅₃	C₂×SD₁₆	semidirect
125. Q₈×C₅₃	C₂×#41	semidirect
148. C₂×#3362	C₅₃	direct (N × H)
165. C₂×Q₈²	C₂×C₅₃	semidirect
167. #3362	C₂×C₅₃	direct (N × H)
168. C₂×#486	C₂×C₅₃	semidirect
169. C₂×#476	C₂×C₅₃	semidirect
170. C₂×#460	C₂×C₅₃	semidirect
171. C₂×C₈×Q₈	C₂×C₅₃	semidirect
172. C₂×C₄×SD₁₆	C₂×C₅₃	semidirect
174. C₂×C₅₃	#3362	direct (N × H)
181. Q₈²	C₂²×C₅₃	semidirect
192. #486	C₂²×C₅₃	semidirect
193. #476	C₂²×C₅₃	semidirect
194. #460	C₂²×C₅₃	semidirect
195. C₈×Q₈	C₂²×C₅₃	semidirect
196. C₄×SD₁₆	C₂²×C₅₃	semidirect
203. C₅₃	C₂×#3362	direct (N × H)
250. C₂×Q₈	SD₁₆×C₅₃	semidirect
252. C₂×Q₈	#41×C₅₃	semidirect
294. Q₈	C₂×SD₁₆×C₅₃	semidirect
295. Q₈	C₂×#41×C₅₃	semidirect
335. C₂	C₅₃×#3362	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×Q₈²×C₅₃	2
C₂×C₅₃×#550	2
C₅₃×#3362	2
C₂×C₅₃×#486	2
C₂×C₅₃×#476	2
C₂×C₅₃×#460	2
C₂×C₈×Q₈×C₅₃	2
C₂×C₄×SD₁₆×C₅₃	2
C₂×#3362	53

Subgroups — C2×C53×#3362 #27268811

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3362 #27268811