Subgroups — C₂×C₅₃×#538 #20272387

Order 6784.

1336 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	C₅₃×#538	C₂	1	16	yes	maximal, direct product factor
2	C₂×#126×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₂×#124×C₅₃	C₂	1	2	yes	maximal, complemented
4	C₂×#123×C₅₃	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#121×C₅₃	C₂	1	2	yes	maximal, complemented
7	C₂×#121×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#120×C₅₃	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×D₄×C₅₃	C₂	1	2	yes	maximal, complemented
10	C₂²×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₄×C₅₃	C₂²	1	2	yes	—
14	C₂²×D₄×C₅₃	C₂²	1	2	yes	—
15	C₂²×D₄×C₅₃	C₂²	1	2	yes	—
16	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
17	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂³×C₄×C₅₃	C₂²	1	2	yes	complemented
20	#126×C₅₃	C₂²	1	16	yes	complemented
21	#124×C₅₃	C₂²	1	16	yes	complemented
22	#123×C₅₃	C₂²	1	16	yes	complemented
23	C₂²⋊Q₈×C₅₃	C₂²	1	8	yes	complemented
24	#121×C₅₃	C₂²	1	8	yes	complemented
25	#121×C₅₃	C₂²	1	16	yes	complemented
26	#120×C₅₃	C₂²	1	16	yes	complemented
27	C₄×D₄×C₅₃	C₂²	1	16	yes	complemented
28	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
29	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
30	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
34	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
35	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
37	C₂×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	complemented
38	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
39	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
40	C₂×C₄²×C₅₃	C₂²	1	2	yes	complemented
41	C₂⁴×C₅₃	—	2	4	no	—
42	C₂⁴×C₅₃	—	4	2	no	—
43	C₂⁴×C₅₃	—	4	2	no	—
44	C₂⁴×C₅₃	C₂³	1	2	yes	—
45	C₂⁴×C₅₃	C₂³	1	2	yes	—
46	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
47	C₂⁴×C₅₃	C₂³	1	4	yes	—
48	C₂×Q₈×C₅₃	—	4	2	no	—
49	C₂×Q₈×C₅₃	C₂³	1	4	yes	complemented
50	C₂×D₄×C₅₃	—	2	8	no	—
51	C₂×D₄×C₅₃	—	4	4	no	—
52	C₂×D₄×C₅₃	—	4	4	no	—
53	C₂×D₄×C₅₃	—	4	2	no	—
54	C₂×D₄×C₅₃	C₂³	1	8	yes	—
55	C₂×D₄×C₅₃	C₂³	1	8	yes	—
56	C₂×D₄×C₅₃	C₂³	1	8	yes	—
57	C₂×D₄×C₅₃	C₂³	1	4	yes	complemented
58	C₂²×C₄×C₅₃	—	4	2	no	—
59	C₂²×C₄×C₅₃	—	4	2	no	—
60	C₂²×C₄×C₅₃	—	4	4	no	—
61	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
68	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
69	C₂²×C₄×C₅₃	C₂³	1	8	yes	complemented
70	C₄⋊C₄×C₅₃	C₂³	1	8	yes	complemented
71	C₄⋊C₄×C₅₃	C₂³	1	8	yes	complemented
72	C₄⋊C₄×C₅₃	C₂³	1	4	yes	complemented
73	C₄⋊C₄×C₅₃	C₂³	1	4	yes	complemented
74	C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
75	C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
76	C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
77	C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
78	C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
79	C₂²⋊C₄×C₅₃	C₂³	1	8	yes	complemented
80	C₂²⋊C₄×C₅₃	C₂³	1	8	yes	complemented
81	C₄²×C₅₃	C₂³	1	8	yes	complemented
82	C₂³×C₅₃	—	2	2	no	—
83	C₂³×C₅₃	—	2	2	no	—
84	C₂³×C₅₃	—	2	4	no	—
85	C₂³×C₅₃	—	2	8	no	—
86	C₂³×C₅₃	—	2	4	no	—
87	C₂³×C₅₃	—	2	4	no	—
88	C₂³×C₅₃	—	2	4	no	—
89	C₂³×C₅₃	—	2	4	no	—
90	C₂³×C₅₃	—	2	4	no	—
91	C₂³×C₅₃	—	4	4	no	—
92	C₂³×C₅₃	—	4	4	no	—
93	C₂³×C₅₃	—	4	16	no	—
94	C₂³×C₅₃	#41	1	4	yes	—
95	C₂³×C₅₃	C₂⁴	1	4	yes	—
96	C₂³×C₅₃	C₂⁴	1	4	yes	—
97	C₂³×C₅₃	C₂⁴	1	2	yes	—
98	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
99	Q₈×C₅₃	—	4	4	no	—
100	D₄×C₅₃	—	2	16	no	—
101	D₄×C₅₃	—	4	8	no	—
102	D₄×C₅₃	—	4	8	no	—
103	D₄×C₅₃	—	4	4	no	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	4	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	—	4	4	no	—
112	C₂×C₄×C₅₃	—	4	8	no	—
113	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
114	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
115	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
116	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
117	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
118	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
119	C₂²×C₅₃	—	2	8	no	—
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	—	2	4	no	—
123	C₂²×C₅₃	—	2	4	no	—
124	C₂²×C₅₃	—	2	4	no	—
125	C₂²×C₅₃	—	2	4	no	—
126	C₂²×C₅₃	—	2	4	no	—
127	C₂²×C₅₃	—	2	4	no	—
128	C₂²×C₅₃	—	4	16	no	—
129	C₂²×C₅₃	—	4	8	no	—
130	C₂²×C₅₃	—	4	8	no	—
131	C₂²×C₅₃	C₂×#41	1	4	yes	—
132	C₂²×C₅₃	C₂×#41	1	2	yes	—
133	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
134	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
135	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
136	C₄×C₅₃	—	2	4	no	—
137	C₄×C₅₃	—	2	4	no	—
138	C₄×C₅₃	—	2	2	no	—
139	C₄×C₅₃	—	2	4	no	—
140	C₄×C₅₃	—	2	2	no	—
141	C₄×C₅₃	—	2	4	no	—
142	C₂×#538	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂×C₅₃	—	2	4	no	—
144	C₂×C₅₃	—	4	4	no	—
145	C₂×C₅₃	—	4	4	no	—
146	C₂×C₅₃	#538	1	4	yes	direct product factor
147	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
148	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
149	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
150	#538	C₂×C₅₃	1	16	yes	direct product factor
151	C₂×#126	C₂×C₅₃	1	2	yes	complemented
152	C₂×#124	C₂×C₅₃	1	2	yes	complemented
153	C₂×#123	C₂×C₅₃	1	2	yes	complemented
154	C₂×C₂²⋊Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
155	C₂×#121	C₂×C₅₃	1	2	yes	complemented
156	C₂×#121	C₂×C₅₃	1	1	yes	characteristic, complemented
157	C₂×#120	C₂×C₅₃	1	2	yes	complemented
158	C₂×C₄×D₄	C₂×C₅₃	1	2	yes	complemented
159	C₂²×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
160	C₅₃	C₂×#538	1	1	yes	characteristic, direct product factor, Sylow
161	C₂⁵	C₂²×C₅₃	1	1	yes	characteristic
162	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
163	C₂²×D₄	C₂²×C₅₃	1	2	yes	—
164	C₂²×D₄	C₂²×C₅₃	1	2	yes	—
165	C₂²×D₄	C₂²×C₅₃	1	2	yes	—
166	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
167	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
168	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
169	C₂³×C₄	C₂²×C₅₃	1	2	yes	complemented
170	#126	C₂²×C₅₃	1	16	yes	complemented
171	#124	C₂²×C₅₃	1	16	yes	complemented
172	#123	C₂²×C₅₃	1	16	yes	complemented
173	C₂²⋊Q₈	C₂²×C₅₃	1	8	yes	complemented
174	#121	C₂²×C₅₃	1	16	yes	complemented
175	#121	C₂²×C₅₃	1	8	yes	complemented
176	#120	C₂²×C₅₃	1	16	yes	complemented
177	C₄×D₄	C₂²×C₅₃	1	16	yes	complemented
178	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
179	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
180	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
181	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
182	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
183	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
184	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
185	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
186	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
187	C₂×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	complemented
188	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	complemented
189	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	complemented
190	C₂×C₄²	C₂²×C₅₃	1	2	yes	complemented
191	C₂⁴	—	2	4	no	—
192	C₂⁴	—	4	2	no	—
193	C₂⁴	—	4	2	no	—
194	C₂⁴	C₂³×C₅₃	1	2	yes	—
195	C₂⁴	C₂³×C₅₃	1	2	yes	—
196	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
197	C₂⁴	C₂³×C₅₃	1	4	yes	—
198	C₂×Q₈	—	4	2	no	—
199	C₂×Q₈	C₂³×C₅₃	1	4	yes	complemented
200	C₂×D₄	—	2	8	no	—
201	C₂×D₄	—	4	4	no	—
202	C₂×D₄	—	4	4	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	C₂³×C₅₃	1	8	yes	—
205	C₂×D₄	C₂³×C₅₃	1	8	yes	—
206	C₂×D₄	C₂³×C₅₃	1	8	yes	—
207	C₂×D₄	C₂³×C₅₃	1	4	yes	complemented
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
212	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
213	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
214	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
215	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
216	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
217	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
218	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
219	C₂²×C₄	C₂³×C₅₃	1	8	yes	complemented
220	C₄⋊C₄	C₂³×C₅₃	1	8	yes	complemented
221	C₄⋊C₄	C₂³×C₅₃	1	8	yes	complemented
222	C₄⋊C₄	C₂³×C₅₃	1	4	yes	complemented
223	C₄⋊C₄	C₂³×C₅₃	1	4	yes	complemented
224	C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
225	C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
226	C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
227	C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
228	C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
229	C₂²⋊C₄	C₂³×C₅₃	1	8	yes	complemented
230	C₂²⋊C₄	C₂³×C₅₃	1	8	yes	complemented
231	C₄²	C₂³×C₅₃	1	8	yes	complemented
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	4	no	—
235	C₂³	—	2	8	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	2	4	no	—
238	C₂³	—	2	4	no	—
239	C₂³	—	2	4	no	—
240	C₂³	—	2	4	no	—
241	C₂³	—	4	4	no	—
242	C₂³	—	4	4	no	—
243	C₂³	—	4	16	no	—
244	C₂³	#41×C₅₃	1	4	yes	—
245	C₂³	C₂⁴×C₅₃	1	4	yes	—
246	C₂³	C₂⁴×C₅₃	1	4	yes	—
247	C₂³	C₂⁴×C₅₃	1	2	yes	—
248	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
249	Q₈	—	4	4	no	—
250	D₄	—	2	16	no	—
251	D₄	—	4	8	no	—
252	D₄	—	4	8	no	—
253	D₄	—	4	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	8	no	—
263	C₂×C₄	C₂⁴×C₅₃	1	4	yes	—
264	C₂×C₄	C₂⁴×C₅₃	1	4	yes	—
265	C₂×C₄	C₂⁴×C₅₃	1	4	yes	—
266	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
267	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
268	C₂×C₄	C₂⁴×C₅₃	1	4	yes	—
269	C₂²	—	2	8	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	4	16	no	—
279	C₂²	—	4	8	no	—
280	C₂²	—	4	8	no	—
281	C₂²	C₂×#41×C₅₃	1	4	yes	—
282	C₂²	C₂×#41×C₅₃	1	2	yes	—
283	C₂²	H(5,𝔽₂)×C₅₃	1	2	yes	—
284	C₂²	H(5,𝔽₂)×C₅₃	1	2	yes	—
285	C₂²	C₂⁵×C₅₃	1	1	yes	characteristic
286	C₄	—	2	4	no	—
287	C₄	—	2	4	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	4	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	4	no	—
292	C₂	—	2	4	no	—
293	C₂	—	4	4	no	—
294	C₂	—	4	4	no	—
295	C₂	C₅₃×#538	1	4	yes	direct product factor
296	C₂	C₂²×#41×C₅₃	1	1	yes	characteristic
297	C₂	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
298	C₂	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₅₃×#538	C₂	direct (N × H)
2. C₂×#126×C₅₃	C₂	semidirect
3. C₂×#124×C₅₃	C₂	semidirect
4. C₂×#123×C₅₃	C₂	semidirect
5. C₂×C₂²⋊Q₈×C₅₃	C₂	semidirect
6. C₂×#121×C₅₃	C₂	semidirect
7. C₂×#121×C₅₃	C₂	semidirect
8. C₂×#120×C₅₃	C₂	semidirect
9. C₂×C₄×D₄×C₅₃	C₂	semidirect
10. C₂²×C₂²⋊C₄×C₅₃	C₂	semidirect
12. C₂²×Q₈×C₅₃	C₂²	semidirect
16. C₂²×D₄×C₅₃	C₂²	semidirect
19. C₂³×C₄×C₅₃	C₂²	semidirect
20. #126×C₅₃	C₂²	semidirect
21. #124×C₅₃	C₂²	semidirect
22. #123×C₅₃	C₂²	semidirect
23. C₂²⋊Q₈×C₅₃	C₂²	semidirect
24. #121×C₅₃	C₂²	semidirect
25. #121×C₅₃	C₂²	semidirect
26. #120×C₅₃	C₂²	semidirect
27. C₄×D₄×C₅₃	C₂²	semidirect
28. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
29. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
30. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
31. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
37. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
38. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
39. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
40. C₂×C₄²×C₅₃	C₂²	semidirect
49. C₂×Q₈×C₅₃	C₂³	semidirect
57. C₂×D₄×C₅₃	C₂³	semidirect
69. C₂²×C₄×C₅₃	C₂³	semidirect
70. C₄⋊C₄×C₅₃	C₂³	semidirect
71. C₄⋊C₄×C₅₃	C₂³	semidirect
72. C₄⋊C₄×C₅₃	C₂³	semidirect
73. C₄⋊C₄×C₅₃	C₂³	semidirect
79. C₂²⋊C₄×C₅₃	C₂³	semidirect
80. C₂²⋊C₄×C₅₃	C₂³	semidirect
81. C₄²×C₅₃	C₂³	semidirect
142. C₂×#538	C₅₃	direct (N × H)
146. C₂×C₅₃	#538	direct (N × H)
150. #538	C₂×C₅₃	direct (N × H)
151. C₂×#126	C₂×C₅₃	semidirect
152. C₂×#124	C₂×C₅₃	semidirect
153. C₂×#123	C₂×C₅₃	semidirect
154. C₂×C₂²⋊Q₈	C₂×C₅₃	semidirect
155. C₂×#121	C₂×C₅₃	semidirect
156. C₂×#121	C₂×C₅₃	semidirect
157. C₂×#120	C₂×C₅₃	semidirect
158. C₂×C₄×D₄	C₂×C₅₃	semidirect
159. C₂²×C₂²⋊C₄	C₂×C₅₃	semidirect
160. C₅₃	C₂×#538	direct (N × H)
162. C₂²×Q₈	C₂²×C₅₃	semidirect
166. C₂²×D₄	C₂²×C₅₃	semidirect
169. C₂³×C₄	C₂²×C₅₃	semidirect
170. #126	C₂²×C₅₃	semidirect
171. #124	C₂²×C₅₃	semidirect
172. #123	C₂²×C₅₃	semidirect
173. C₂²⋊Q₈	C₂²×C₅₃	semidirect
174. #121	C₂²×C₅₃	semidirect
175. #121	C₂²×C₅₃	semidirect
176. #120	C₂²×C₅₃	semidirect
177. C₄×D₄	C₂²×C₅₃	semidirect
178. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
179. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
180. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
181. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
187. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
188. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
189. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
190. C₂×C₄²	C₂²×C₅₃	semidirect
199. C₂×Q₈	C₂³×C₅₃	semidirect
207. C₂×D₄	C₂³×C₅₃	semidirect
219. C₂²×C₄	C₂³×C₅₃	semidirect
220. C₄⋊C₄	C₂³×C₅₃	semidirect
221. C₄⋊C₄	C₂³×C₅₃	semidirect
222. C₄⋊C₄	C₂³×C₅₃	semidirect
223. C₄⋊C₄	C₂³×C₅₃	semidirect
229. C₂²⋊C₄	C₂³×C₅₃	semidirect
230. C₂²⋊C₄	C₂³×C₅₃	semidirect
231. C₄²	C₂³×C₅₃	semidirect
295. C₂	C₅₃×#538	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×#538	2
C₂×#126×C₅₃	2
C₂×#124×C₅₃	2
C₂×#123×C₅₃	2
C₂×C₂²⋊Q₈×C₅₃	2
C₂×#121×C₅₃	2
C₂×#120×C₅₃	2
C₂×C₄×D₄×C₅₃	2
C₂²×C₂²⋊C₄×C₅₃	2
C₂×#538	53

Subgroups — C2×C53×#538 #20272387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#538 #20272387