Subgroups — #28293106

Order 13568.

11770 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133262	C₂	1	2	yes	maximal, complemented
2	#20288479	C₂	1	16	yes	maximal, complemented
3	#20288360	C₂	1	16	yes	maximal, complemented
4	#20288351	C₂	1	16	yes	maximal, complemented
5	#20288048	C₂	1	4	yes	maximal, complemented
6	C₂×#16811185	C₂	1	8	yes	maximal, complemented
7	C₂×C₅₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133262	—	2	32	no	—
9	C₂×#13133262	C₂²	1	16	yes	complemented
10	C₂×#13133262	C₂²	1	16	yes	complemented
11	C₂²×#11719945	C₂²	1	4	yes	complemented
12	C₂²×#11719947	C₂²	1	8	yes	complemented
13	C₂³×C₂²⋊D₅₃	C₂²	1	8	yes	complemented
14	C₂²×D₄×D₅₃	C₂²	1	8	yes	complemented
15	#16811277	C₂²	1	64	yes	complemented
16	#16811268	C₂²	1	128	yes	complemented
17	D₄×D₂₁₂	C₂²	1	64	yes	complemented
18	#16811204	C₂²	1	16	yes	complemented
19	#16811199	C₂²	1	16	yes	complemented
20	#16811192	C₂²	1	16	yes	complemented
21	#16811185	C₂²	1	16	yes	complemented
22	#16811185	C₂²	1	64	yes	complemented
23	#16811185	C₂²	1	32	yes	complemented
24	#16811173	C₂²	1	16	yes	complemented
25	#16811135	C₂²	1	32	yes	complemented
26	C₂×#13133201	C₂²	1	32	yes	complemented
27	C₂×#13133191	C₂²	1	32	yes	complemented
28	C₂×#13133187	C₂²	1	32	yes	complemented
29	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#518	C₂²	1	16	yes	complemented
31	C₂×C₄×D₄×C₅₃	C₂²	1	8	yes	complemented
32	C₂×#117×C₅₃	C₂²	1	2	yes	complemented
33	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
34	C₂⁴×D₅₃	C₂³	1	8	yes	complemented
35	#13133262	—	2	128	no	—
36	#13133262	—	2	128	no	—
37	C₂×#11719945	—	2	32	no	—
38	C₂×#11719945	C₂³	1	16	yes	—
39	C₂×#11719945	C₂³	1	16	yes	complemented
40	C₂×#11719947	—	2	64	no	—
41	C₂×#11719947	C₂³	1	64	yes	complemented
42	C₂²×Dic₁₀₆	C₂³	1	4	yes	complemented
43	C₂²×D₂₁₂	C₂³	1	4	yes	complemented
44	C₂²×D₂₁₂	C₂³	1	32	yes	complemented
45	C₂²×C₂²⋊D₅₃	—	2	64	no	—
46	C₂²×C₂²⋊D₅₃	C₂³	1	32	yes	—
47	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
48	C₂²×C₂²⋊D₅₃	C₂³	1	32	yes	complemented
49	C₂²×C₂²⋊D₅₃	C₂³	1	16	yes	complemented
50	C₂×D₄×D₅₃	—	2	64	no	—
51	C₂×D₄×D₅₃	C₂³	1	64	yes	complemented
52	C₂³×Dic₅₃	C₂³	1	8	yes	complemented
53	C₂²×C₄×D₅₃	C₂³	1	8	yes	complemented
54	#13133217	C₂³	1	16	yes	complemented
55	#13133214	C₂³	1	16	yes	complemented
56	#13133205	C₂³	1	32	yes	complemented
57	#13133201	C₂³	1	128	yes	complemented
58	#13133201	C₂³	1	64	yes	complemented
59	#13133197	C₂³	1	64	yes	complemented
60	#13133192	C₂³	1	64	yes	complemented
61	#13133191	C₂³	1	128	yes	complemented
62	#13133191	C₂³	1	64	yes	complemented
63	#13133187	C₂³	1	128	yes	complemented
64	#13133187	C₂³	1	64	yes	complemented
65	C₄×D₂₁₂	C₂³	1	64	yes	complemented
66	C₂×C₄⋊Dic₅₃	C₂³	1	32	yes	complemented
67	C₂×#11719921	C₂³	1	64	yes	complemented
68	C₂⁵×C₅₃	C₂³	1	2	yes	—
69	C₂²×D₄×C₅₃	D₄	1	16	yes	complemented
70	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
71	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
72	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₅₃	C₂³	1	8	yes	complemented
74	C₄×D₄×C₅₃	C₂³	1	64	yes	complemented
75	#117×C₅₃	C₂³	1	16	yes	complemented
76	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	complemented
77	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
78	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	complemented
79	C₂×C₂²⋊C₄×C₅₃	C₂³	1	32	yes	complemented
80	C₂×C₄²×C₅₃	C₂³	1	4	yes	complemented
81	C₂⁴×C₅₃	—	8	4	no	—
82	C₂⁴×C₅₃	C₂×D₄	1	16	yes	complemented
83	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
84	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
85	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
86	C₂³×D₅₃	—	2	32	no	—
87	C₂³×D₅₃	—	8	16	no	—
88	C₂³×D₅₃	C₂⁴	1	32	yes	—
89	C₂³×D₅₃	C₂⁴	1	8	yes	—
90	C₂×D₄×C₅₃	C₂×D₄	1	64	yes	complemented
91	C₂×D₄×C₅₃	C₂×D₄	1	32	yes	complemented
92	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
93	C₂²×C₄×C₅₃	—	8	8	no	—
94	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	complemented
95	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
96	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	—
97	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
98	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
99	C₂×C₂²⋊D₅₃	—	2	64	no	—
100	C₂×C₂²⋊D₅₃	—	2	32	no	—
101	C₂×C₂²⋊D₅₃	—	2	128	no	—
102	C₂×C₂²⋊D₅₃	—	2	128	no	—
103	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
104	C₂×C₂²⋊D₅₃	C₂⁴	1	64	yes	—
105	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
106	C₂²×Dic₅₃	—	2	32	no	—
107	C₂²×Dic₅₃	C₂⁴	1	32	yes	—
108	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
109	#11719947	—	2	256	no	—
110	D₄×D₅₃	—	2	256	no	—
111	#11719945	—	2	64	no	—
112	#11719945	—	2	64	no	—
113	C₂×D₂₁₂	—	2	16	no	—
114	C₂×D₂₁₂	—	8	32	no	—
115	C₂×D₂₁₂	C₂⁴	1	16	yes	—
116	C₂×D₂₁₂	C₂⁴	1	64	yes	—
117	C₂×C₄×D₅₃	—	2	32	no	—
118	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
119	C₂×Dic₁₀₆	—	2	16	no	—
120	C₂×Dic₁₀₆	C₂⁴	1	16	yes	—
121	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
122	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
123	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
124	C₄²×C₅₃	C₂⁴	1	16	yes	—
125	#11719921	C₂⁴	1	128	yes	—
126	C₄⋊Dic₅₃	C₂⁴	1	64	yes	—
127	C₂³×C₅₃	—	4	8	no	—
128	C₂³×C₅₃	—	4	16	no	—
129	C₂³×C₅₃	—	4	2	no	—
130	C₂³×C₅₃	—	8	16	no	—
131	C₂³×C₅₃	C₂²×D₄	1	32	yes	—
132	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
133	C₂³×C₅₃	C₂²×D₄	1	16	yes	—
134	C₂³×C₅₃	C₂⁵	1	8	yes	—
135	C₂³×C₅₃	C₂⁵	1	2	yes	—
136	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
137	C₂²×D₅₃	—	2	16	no	—
138	C₂²×D₅₃	—	2	64	no	—
139	C₂²×D₅₃	—	4	16	no	—
140	C₂²×D₅₃	—	4	64	no	—
141	C₂²×D₅₃	—	8	64	no	—
142	C₂²×D₅₃	C₂⁵	1	16	yes	—
143	D₄×C₅₃	C₂²×D₄	1	64	yes	—
144	C₂×C₄×C₅₃	—	4	32	no	—
145	C₂×C₄×C₅₃	—	4	8	no	—
146	C₂×C₄×C₅₃	C₂²×D₄	1	16	yes	—
147	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
148	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
149	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
150	C₂²⋊D₅₃	—	2	32	no	—
151	C₂²⋊D₅₃	—	2	32	no	—
152	C₂²⋊D₅₃	—	2	128	no	—
153	C₂×Dic₅₃	—	2	16	no	—
154	C₂×Dic₅₃	—	2	64	no	—
155	C₂×Dic₅₃	C₂⁵	1	16	yes	—
156	D₂₁₂	—	2	32	no	—
157	D₂₁₂	—	4	64	no	—
158	C₄×D₅₃	—	2	64	no	—
159	Dic₁₀₆	—	2	32	no	—
160	#62842	—	53	1	no	maximal, Sylow
161	C₂²×C₅₃	—	2	2	no	—
162	C₂²×C₅₃	—	2	16	no	—
163	C₂²×C₅₃	—	2	4	no	—
164	C₂²×C₅₃	—	4	32	no	—
165	C₂²×C₅₃	—	4	4	no	—
166	C₂²×C₅₃	—	4	16	no	—
167	C₂²×C₅₃	C₂³×D₄	1	4	yes	—
168	C₂²×C₅₃	C₂³×D₄	1	16	yes	—
169	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
170	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
171	C₂×D₅₃	—	2	16	no	—
172	C₂×D₅₃	—	2	32	no	—
173	C₂×D₅₃	—	4	32	no	—
174	C₂×D₅₃	—	8	64	no	—
175	C₄×C₅₃	—	2	16	no	—
176	C₄×C₅₃	C₂³×D₄	1	8	yes	—
177	Dic₅₃	—	2	16	no	—
178	C₂²×H(5,𝔽₂)	—	53	2	no	—
179	#3496	—	53	16	no	—
180	#3485	—	53	16	no	—
181	#3484	—	53	16	no	—
182	#3449	—	53	4	no	—
183	C₂×#534	—	53	8	no	—
184	C₂×#518	D₅₃	1	1	yes	characteristic, complemented
185	C₂×C₅₃	—	2	4	no	—
186	C₂×C₅₃	—	2	16	no	—
187	C₂×C₅₃	—	2	2	no	—
188	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
189	C₂×C₅₃	H(7,𝔽₂)	1	2	yes	—
190	D₅₃	—	4	16	no	—
191	C₂×H(5,𝔽₂)	—	53	16	no	—
192	C₂×H(5,𝔽₂)	—	53	16	no	—
193	C₂×H(5,𝔽₂)	—	106	32	no	—
194	C₂²×#41	—	53	8	no	—
195	C₂²×#41	—	53	4	no	—
196	C₂³×D₄	—	53	8	no	—
197	C₂³×D₄	—	53	8	no	—
198	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
199	#547	—	53	64	no	—
200	#546	—	53	128	no	—
201	D₄²	—	53	64	no	—
202	#537	—	53	16	no	—
203	#536	—	53	16	no	—
204	#535	—	53	16	no	—
205	#534	—	53	16	no	—
206	#534	—	53	64	no	—
207	#534	—	53	32	no	—
208	#532	—	53	16	no	—
209	#525	—	53	32	no	—
210	C₂×#123	—	53	32	no	—
211	C₂×#121	—	53	32	no	—
212	C₂×#120	—	53	32	no	—
213	#518	C₂×D₅₃	1	16	yes	complemented
214	C₂×C₄×D₄	C₂×D₅₃	1	8	yes	complemented
215	C₂×#117	C₂×D₅₃	1	2	yes	complemented
216	C₂²×C₂²⋊C₄	C₂×D₅₃	1	4	yes	complemented
217	C₅₃	#62842	1	1	yes	characteristic, complemented, Sylow
218	C₂⁵	—	53	8	no	—
219	C₂⁵	C₂²×D₅₃	1	2	yes	—
220	H(5,𝔽₂)	—	106	128	no	—
221	H(5,𝔽₂)	—	106	128	no	—
222	C₂×#41	—	53	64	no	—
223	C₂×#41	—	53	16	no	—
224	C₂×#41	—	53	16	no	—
225	C₂×#41	—	106	64	no	—
226	C₂×#41	—	106	32	no	—
227	C₂²×Q₈	—	53	4	no	—
228	C₂²×D₄	—	53	32	no	—
229	C₂²×D₄	—	53	32	no	—
230	C₂²×D₄	—	53	8	no	—
231	C₂²×D₄	—	53	16	no	—
232	C₂²×D₄	—	53	64	no	—
233	C₂²×D₄	—	53	4	no	—
234	C₂²×D₄	—	53	32	no	—
235	C₂²×D₄	—	106	64	no	—
236	C₂²×D₄	—	106	64	no	—
237	C₂²×D₄	D₂₁₂	1	16	yes	complemented
238	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
239	C₂²×D₄	C₂²×D₅₃	1	8	yes	—
240	C₂³×C₄	—	53	8	no	—
241	C₂³×C₄	—	53	8	no	—
242	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
243	C₂³×C₄	C₂²×D₅₃	1	8	yes	complemented
244	C₄⋊Q₈	—	53	16	no	—
245	#127	—	53	16	no	—
246	#124	—	53	32	no	—
247	#123	—	53	64	no	—
248	#123	—	53	128	no	—
249	C₂²⋊Q₈	—	53	64	no	—
250	#121	—	53	64	no	—
251	#121	—	53	128	no	—
252	#121	—	53	64	no	—
253	#120	—	53	64	no	—
254	#120	—	53	128	no	—
255	C₄×D₄	—	53	64	no	—
256	C₄×D₄	C₂²×D₅₃	1	64	yes	complemented
257	#117	C₂²×D₅₃	1	16	yes	complemented
258	C₂×C₄⋊C₄	—	53	32	no	—
259	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	complemented
260	C₂×C₂²⋊C₄	—	53	64	no	—
261	C₂×C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
262	C₂×C₂²⋊C₄	C₂²×D₅₃	1	8	yes	complemented
263	C₂×C₂²⋊C₄	C₂²×D₅₃	1	32	yes	complemented
264	C₂×C₄²	C₂²×D₅₃	1	4	yes	complemented
265	C₂⁴	—	8	4	no	—
266	C₂⁴	—	53	32	no	—
267	C₂⁴	—	53	8	no	—
268	C₂⁴	—	106	32	no	—
269	C₂⁴	—	424	16	no	—
270	C₂⁴	C₂×D₂₁₂	1	16	yes	complemented
271	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
272	C₂⁴	C₂³×D₅₃	1	8	yes	—
273	C₂⁴	C₂³×D₅₃	1	4	yes	—
274	#41	—	106	256	no	—
275	#41	—	106	64	no	—
276	#41	—	106	64	no	—
277	C₂×Q₈	—	53	16	no	—
278	C₂×Q₈	—	106	16	no	—
279	C₂×D₄	—	53	16	no	—
280	C₂×D₄	—	53	64	no	—
281	C₂×D₄	—	53	16	no	—
282	C₂×D₄	—	53	16	no	—
283	C₂×D₄	—	53	64	no	—
284	C₂×D₄	—	106	64	no	—
285	C₂×D₄	—	106	32	no	—
286	C₂×D₄	—	106	128	no	—
287	C₂×D₄	—	106	128	no	—
288	C₂×D₄	—	106	256	no	—
289	C₂×D₄	—	106	16	no	—
290	C₂×D₄	—	424	32	no	—
291	C₂×D₄	C₂×D₂₁₂	1	64	yes	complemented
292	C₂×D₄	C₂×D₂₁₂	1	32	yes	complemented
293	C₂×D₄	C₂³×D₅₃	1	16	yes	—
294	C₂²×C₄	—	8	8	no	—
295	C₂²×C₄	—	53	32	no	—
296	C₂²×C₄	—	53	8	no	—
297	C₂²×C₄	—	53	32	no	—
298	C₂²×C₄	—	106	32	no	—
299	C₂²×C₄	—	106	32	no	—
300	C₂²×C₄	C₂×D₂₁₂	1	8	yes	complemented
301	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
302	C₂²×C₄	C₂³×D₅₃	1	32	yes	—
303	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
304	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
305	C₄⋊C₄	—	53	64	no	—
306	C₄⋊C₄	C₂³×D₅₃	1	16	yes	—
307	C₂²⋊C₄	—	53	128	no	—
308	C₂²⋊C₄	C₂³×D₅₃	1	16	yes	—
309	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	—
310	C₄²	C₂³×D₅₃	1	16	yes	—
311	C₂³	—	4	8	no	—
312	C₂³	—	4	2	no	—
313	C₂³	—	4	16	no	—
314	C₂³	—	8	16	no	—
315	C₂³	—	53	16	no	—
316	C₂³	—	106	64	no	—
317	C₂³	—	106	16	no	—
318	C₂³	—	424	64	no	—
319	C₂³	—	212	64	no	—
320	C₂³	—	212	16	no	—
321	C₂³	C₂²×D₂₁₂	1	32	yes	—
322	C₂³	C₂²×D₂₁₂	1	4	yes	—
323	C₂³	C₂²×D₂₁₂	1	16	yes	—
324	C₂³	C₂⁴×D₅₃	1	2	yes	—
325	C₂³	C₂⁴×D₅₃	1	8	yes	—
326	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
327	Q₈	—	106	32	no	—
328	D₄	—	106	32	no	—
329	D₄	—	106	128	no	—
330	D₄	—	106	32	no	—
331	D₄	—	106	32	no	—
332	D₄	—	212	64	no	—
333	D₄	C₂²×D₂₁₂	1	64	yes	—
334	C₂×C₄	—	4	8	no	—
335	C₂×C₄	—	4	32	no	—
336	C₂×C₄	—	53	16	no	—
337	C₂×C₄	—	106	16	no	—
338	C₂×C₄	—	106	64	no	—
339	C₂×C₄	—	106	64	no	—
340	C₂×C₄	C₂²×D₂₁₂	1	16	yes	—
341	C₂×C₄	C₂²×D₂₁₂	1	8	yes	—
342	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
343	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	16	no	—
346	C₂²	—	2	4	no	—
347	C₂²	—	4	4	no	—
348	C₂²	—	4	32	no	—
349	C₂²	—	4	16	no	—
350	C₂²	—	106	16	no	—
351	C₂²	—	106	32	no	—
352	C₂²	—	424	64	no	—
353	C₂²	—	212	32	no	—
354	C₂²	C₂³×D₂₁₂	1	4	yes	—
355	C₂²	C₂³×D₂₁₂	1	16	yes	—
356	C₂²	C₂³×D₂₁₂	1	2	yes	—
357	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
358	C₄	—	2	16	no	—
359	C₄	—	106	16	no	—
360	C₄	C₂³×D₂₁₂	1	8	yes	—
361	C₂	—	2	4	no	—
362	C₂	—	2	2	no	—
363	C₂	—	2	16	no	—
364	C₂	—	212	16	no	—
365	C₂	C₂⁴×D₂₁₂	1	1	yes	characteristic
366	C₂	#20289525	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁶
Fitting	C₂×C₅₃×#518	C₂
hypercenter	C₂×#518	D₅₃
hypocenter	C₅₃	#62842
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×#518

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133262	C₂	semidirect
2. #20288479	C₂	semidirect
3. #20288360	C₂	semidirect
4. #20288351	C₂	semidirect
5. #20288048	C₂	semidirect
6. C₂×#16811185	C₂	semidirect
7. C₂×C₅₃×#518	C₂	semidirect
9. C₂×#13133262	C₂²	semidirect
10. C₂×#13133262	C₂²	semidirect
11. C₂²×#11719945	C₂²	semidirect
12. C₂²×#11719947	C₂²	semidirect
13. C₂³×C₂²⋊D₅₃	C₂²	semidirect
14. C₂²×D₄×D₅₃	C₂²	semidirect
15. #16811277	C₂²	semidirect
16. #16811268	C₂²	semidirect
17. D₄×D₂₁₂	C₂²	semidirect
18. #16811204	C₂²	semidirect
19. #16811199	C₂²	semidirect
20. #16811192	C₂²	semidirect
21. #16811185	C₂²	semidirect
22. #16811185	C₂²	semidirect
23. #16811185	C₂²	semidirect
24. #16811173	C₂²	semidirect
25. #16811135	C₂²	semidirect
26. C₂×#13133201	C₂²	semidirect
27. C₂×#13133191	C₂²	semidirect
28. C₂×#13133187	C₂²	semidirect
30. C₅₃×#518	C₂²	semidirect
31. C₂×C₄×D₄×C₅₃	C₂²	semidirect
32. C₂×#117×C₅₃	C₂²	semidirect
33. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
34. C₂⁴×D₅₃	C₂³	semidirect
39. C₂×#11719945	C₂³	semidirect
41. C₂×#11719947	C₂³	semidirect
42. C₂²×Dic₁₀₆	C₂³	semidirect
43. C₂²×D₂₁₂	C₂³	semidirect
44. C₂²×D₂₁₂	C₂³	semidirect
48. C₂²×C₂²⋊D₅₃	C₂³	semidirect
49. C₂²×C₂²⋊D₅₃	C₂³	semidirect
51. C₂×D₄×D₅₃	C₂³	semidirect
52. C₂³×Dic₅₃	C₂³	semidirect
53. C₂²×C₄×D₅₃	C₂³	semidirect
54. #13133217	C₂³	semidirect
55. #13133214	C₂³	semidirect
56. #13133205	C₂³	semidirect
57. #13133201	C₂³	semidirect
58. #13133201	C₂³	semidirect
59. #13133197	C₂³	semidirect
60. #13133192	C₂³	semidirect
61. #13133191	C₂³	semidirect
62. #13133191	C₂³	semidirect
63. #13133187	C₂³	semidirect
64. #13133187	C₂³	semidirect
65. C₄×D₂₁₂	C₂³	semidirect
66. C₂×C₄⋊Dic₅₃	C₂³	semidirect
67. C₂×#11719921	C₂³	semidirect
69. C₂²×D₄×C₅₃	D₄	semidirect
73. C₂³×C₄×C₅₃	C₂³	semidirect
74. C₄×D₄×C₅₃	C₂³	semidirect
75. #117×C₅₃	C₂³	semidirect
76. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
78. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
79. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
80. C₂×C₄²×C₅₃	C₂³	semidirect
82. C₂⁴×C₅₃	C₂×D₄	semidirect
90. C₂×D₄×C₅₃	C₂×D₄	semidirect
91. C₂×D₄×C₅₃	C₂×D₄	semidirect
94. C₂²×C₄×C₅₃	C₂×D₄	semidirect
184. C₂×#518	D₅₃	semidirect
213. #518	C₂×D₅₃	semidirect
214. C₂×C₄×D₄	C₂×D₅₃	semidirect
215. C₂×#117	C₂×D₅₃	semidirect
216. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
217. C₅₃	#62842	semidirect
237. C₂²×D₄	D₂₁₂	semidirect
243. C₂³×C₄	C₂²×D₅₃	semidirect
256. C₄×D₄	C₂²×D₅₃	semidirect
257. #117	C₂²×D₅₃	semidirect
259. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
262. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
263. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
264. C₂×C₄²	C₂²×D₅₃	semidirect
270. C₂⁴	C₂×D₂₁₂	semidirect
291. C₂×D₄	C₂×D₂₁₂	semidirect
292. C₂×D₄	C₂×D₂₁₂	semidirect
300. C₂²×C₄	C₂×D₂₁₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133262	2
#20288479	2
#20288360	2
#20288351	2
#20288048	2
C₂×#16811185	2
C₂×C₅₃×#518	2
#62842	53