Subgroups — C₂²×#16811187 #28290847

Order 13568.

14414 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16811187	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133213	C₂	1	2	yes	maximal, complemented
4	C₂²×#13133206	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133191	C₂	1	4	yes	maximal, complemented
6	C₂²×#13133185	C₂	1	4	yes	maximal, complemented
7	C₂²×#13133176	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₅₃	C₂²	1	2	yes	—
10	C₂³×D₂₁₂	—	2	16	no	—
11	C₂³×D₂₁₂	C₂²	1	2	yes	—
12	C₂³×D₂₁₂	C₂²	1	2	yes	—
13	C₂³×D₂₁₂	C₂²	1	24	yes	complemented
14	C₂³×C₂²⋊D₅₃	C₂²	1	8	yes	complemented
15	#16811187	C₂²	1	256	yes	direct product factor
16	C₂×#13133213	C₂²	1	48	yes	complemented
17	C₂×#13133206	C₂²	1	48	yes	complemented
18	C₂×#13133191	C₂²	1	96	yes	complemented
19	C₂×#13133185	C₂²	1	96	yes	complemented
20	C₂×#13133176	C₂²	1	24	yes	complemented
21	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	complemented
22	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	—
23	C₂²×#11719921	C₂²	1	8	yes	complemented
24	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	complemented
25	C₂²×#41×C₅₃	—	2	16	no	—
26	C₂²×#41×C₅₃	C₂²	1	24	yes	complemented
27	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
28	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
29	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
30	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
31	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	complemented
32	C₂⁴×D₅₃	—	2	16	no	—
33	C₂⁴×D₅₃	—	4	16	no	—
34	C₂⁴×D₅₃	C₂³	1	4	yes	—
35	C₂⁴×D₅₃	C₂³	1	24	yes	—
36	C₂²×D₂₁₂	—	2	16	no	—
37	C₂²×D₂₁₂	—	2	16	no	—
38	C₂²×D₂₁₂	—	2	192	no	—
39	C₂²×D₂₁₂	C₂³	1	24	yes	—
40	C₂²×D₂₁₂	C₂³	1	24	yes	—
41	C₂²×D₂₁₂	C₂³	1	64	yes	complemented
42	C₂²×C₂²⋊D₅₃	—	4	32	no	—
43	C₂²×C₂²⋊D₅₃	C₂³	1	96	yes	complemented
44	C₂³×Dic₅₃	C₂³	1	4	yes	—
45	#13133213	C₂³	1	128	yes	complemented
46	#13133206	C₂³	1	128	yes	complemented
47	#13133191	C₂³	1	256	yes	complemented
48	#13133185	C₂³	1	256	yes	complemented
49	#13133176	C₂³	1	64	yes	complemented
50	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	complemented
51	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	—
52	C₂×#11719921	C₂³	1	96	yes	complemented
53	C₂×C₄×Dic₅₃	C₂³	1	24	yes	complemented
54	C₂⁵×C₅₃	C₂³	1	2	yes	—
55	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×#41×C₅₃	—	2	192	no	—
57	C₂×#41×C₅₃	C₂³	1	64	yes	complemented
58	C₂²×Q₈×C₅₃	—	2	8	no	—
59	C₂²×Q₈×C₅₃	C₂³	1	12	yes	complemented
60	C₂²×D₄×C₅₃	—	2	16	no	—
61	C₂²×D₄×C₅₃	—	2	8	no	—
62	C₂²×D₄×C₅₃	C₂³	1	24	yes	—
63	C₂²×D₄×C₅₃	C₂³	1	12	yes	complemented
64	C₂³×C₄×C₅₃	—	2	16	no	—
65	C₂³×C₄×C₅₃	D₄	1	16	yes	complemented
66	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
67	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
70	C₂³×C₄×C₅₃	C₂³	1	24	yes	complemented
71	C₂⁴×C₅₃	—	2	8	no	—
72	C₂⁴×C₅₃	—	2	8	no	—
73	C₂⁴×C₅₃	—	2	16	no	—
74	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
75	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
76	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
77	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
78	C₂³×D₅₃	—	2	96	no	—
79	C₂³×D₅₃	—	2	16	no	—
80	C₂³×D₅₃	—	2	32	no	—
81	C₂³×D₅₃	—	4	96	no	—
82	C₂³×D₅₃	—	4	128	no	—
83	C₂³×D₅₃	C₂⁴	1	24	yes	—
84	C₂³×D₅₃	C₂⁴	1	32	yes	—
85	#41×C₅₃	—	2	256	no	—
86	C₂×Q₈×C₅₃	—	2	48	no	—
87	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	complemented
88	C₂×D₄×C₅₃	—	2	96	no	—
89	C₂×D₄×C₅₃	—	2	48	no	—
90	C₂×D₄×C₅₃	C₂⁴	1	32	yes	—
91	C₂×D₄×C₅₃	C₂⁴	1	16	yes	complemented
92	C₂²×C₄×C₅₃	—	2	8	no	—
93	C₂²×C₄×C₅₃	—	2	96	no	—
94	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
95	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
96	C₂²×C₄×C₅₃	C₂×D₄	1	96	yes	complemented
97	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
98	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
99	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
100	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
101	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	complemented
102	C₂×C₂²⋊D₅₃	—	4	192	no	—
103	C₂×C₂²⋊D₅₃	C₂⁴	1	128	yes	complemented
104	C₂²×Dic₅₃	—	2	16	no	—
105	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
106	C₂×D₂₁₂	—	2	96	no	—
107	C₂×D₂₁₂	—	2	96	no	—
108	C₂×D₂₁₂	—	2	256	no	—
109	C₂×D₂₁₂	C₂⁴	1	32	yes	—
110	C₂×D₂₁₂	C₂⁴	1	32	yes	—
111	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	—
112	#11719921	C₂⁴	1	128	yes	complemented
113	C₄⋊Dic₅₃	C₂⁴	1	32	yes	complemented
114	C₄×Dic₅₃	C₂⁴	1	32	yes	complemented
115	C₂³×C₅₃	—	2	32	no	—
116	C₂³×C₅₃	—	2	24	no	—
117	C₂³×C₅₃	—	2	24	no	—
118	C₂³×C₅₃	—	2	48	no	—
119	C₂³×C₅₃	—	4	32	no	—
120	C₂³×C₅₃	C₂²×D₄	1	24	yes	—
121	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
122	C₂³×C₅₃	H(5,𝔽₂)	1	8	yes	—
123	C₂³×C₅₃	C₂⁵	1	8	yes	—
124	C₂³×C₅₃	C₂⁵	1	4	yes	—
125	C₂³×C₅₃	C₂⁵	1	3	yes	—
126	C₂²×D₅₃	—	2	64	no	—
127	C₂²×D₅₃	—	2	48	no	—
128	C₂²×D₅₃	—	2	96	no	—
129	C₂²×D₅₃	—	4	64	no	—
130	C₂²×D₅₃	—	4	384	no	—
131	C₂²×D₅₃	—	4	64	no	—
132	C₂²×D₅₃	C₂⁵	1	16	yes	—
133	Q₈×C₅₃	—	2	32	no	—
134	D₄×C₅₃	—	2	64	no	—
135	D₄×C₅₃	—	2	32	no	—
136	C₂×C₄×C₅₃	—	2	24	no	—
137	C₂×C₄×C₅₃	—	2	64	no	—
138	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
139	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
140	C₂×C₄×C₅₃	C₂²×D₄	1	64	yes	complemented
141	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
142	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
143	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
144	C₂²⋊D₅₃	—	4	128	no	—
145	C₂×Dic₅₃	—	2	48	no	—
146	C₂×Dic₅₃	C₂⁵	1	16	yes	—
147	D₂₁₂	—	2	64	no	—
148	D₂₁₂	—	2	64	no	—
149	C₂²×#535	—	53	1	no	maximal, Sylow
150	C₂²×C₅₃	—	2	8	no	—
151	C₂²×C₅₃	—	2	48	no	—
152	C₂²×C₅₃	—	2	8	no	—
153	C₂²×C₅₃	—	2	16	no	—
154	C₂²×C₅₃	—	4	48	no	—
155	C₂²×C₅₃	#535	1	16	yes	complemented
156	C₂²×C₅₃	C₂³×D₄	1	8	yes	—
157	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
158	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	12	yes	—
159	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
160	C₂×D₅₃	—	2	16	no	—
161	C₂×D₅₃	—	2	32	no	—
162	C₂×D₅₃	—	4	128	no	—
163	C₂×D₅₃	—	4	96	no	—
164	C₄×C₅₃	—	2	8	no	—
165	C₄×C₅₃	C₂³×D₄	1	8	yes	—
166	C₄×C₅₃	C₂³×D₄	1	8	yes	—
167	Dic₅₃	—	2	16	no	—
168	C₂³×#41	D₅₃	1	1	yes	characteristic, complemented
169	C₂⁴×D₄	—	53	1	no	—
170	C₂×#535	—	53	48	no	—
171	C₂²×#127	—	53	2	no	—
172	C₂²×#124	—	53	2	no	—
173	C₂²×#121	—	53	4	no	—
174	C₂²×#120	—	53	4	no	—
175	C₂²×#117	—	53	1	no	—
176	C₂×C₅₃	—	2	8	no	—
177	C₂×C₅₃	—	4	8	no	—
178	C₂×C₅₃	C₂×#535	1	12	yes	complemented
179	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
180	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	2	yes	—
181	D₅₃	—	4	16	no	—
182	C₂⁶	—	53	2	no	—
183	C₂²×#41	—	2	16	no	—
184	C₂²×#41	C₂×D₅₃	1	24	yes	complemented
185	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
186	C₂³×D₄	—	53	8	no	—
187	C₂³×D₄	—	53	2	no	—
188	C₂³×D₄	—	53	2	no	—
189	C₂³×D₄	—	53	24	no	—
190	C₂³×D₄	—	106	16	no	—
191	C₂³×D₄	C₂×D₅₃	1	2	yes	—
192	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
193	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
194	C₂⁴×C₄	C₂×D₅₃	1	2	yes	complemented
195	#535	—	53	256	no	—
196	C₂×#127	—	53	48	no	—
197	C₂×#124	—	53	48	no	—
198	C₂×#121	—	53	96	no	—
199	C₂×#120	—	53	96	no	—
200	C₂×#117	—	53	24	no	—
201	C₂²×C₄⋊C₄	—	53	2	no	—
202	C₂²×C₂²⋊C₄	—	53	2	no	—
203	C₂²×C₂²⋊C₄	—	53	8	no	—
204	C₂²×C₄²	—	53	2	no	—
205	C₅₃	C₂²×#535	1	1	yes	characteristic, complemented, Sylow
206	C₂⁵	—	53	4	no	—
207	C₂⁵	—	53	24	no	—
208	C₂⁵	—	106	16	no	—
209	C₂⁵	—	212	16	no	—
210	C₂⁵	C₂²×D₅₃	1	2	yes	—
211	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
212	C₂×#41	—	2	192	no	—
213	C₂×#41	C₂²×D₅₃	1	64	yes	complemented
214	C₂²×Q₈	—	2	8	no	—
215	C₂²×Q₈	C₂²×D₅₃	1	12	yes	complemented
216	C₂²×D₄	—	2	16	no	—
217	C₂²×D₄	—	2	8	no	—
218	C₂²×D₄	—	53	96	no	—
219	C₂²×D₄	—	53	24	no	—
220	C₂²×D₄	—	53	24	no	—
221	C₂²×D₄	—	53	64	no	—
222	C₂²×D₄	—	106	16	no	—
223	C₂²×D₄	—	106	16	no	—
224	C₂²×D₄	—	106	192	no	—
225	C₂²×D₄	—	212	32	no	—
226	C₂²×D₄	C₂²×D₅₃	1	24	yes	—
227	C₂²×D₄	C₂²×D₅₃	1	12	yes	complemented
228	C₂³×C₄	—	2	16	no	—
229	C₂³×C₄	—	53	4	no	—
230	C₂³×C₄	C₂²⋊D₅₃	1	16	yes	complemented
231	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
232	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
233	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
234	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
235	C₂³×C₄	C₂²×D₅₃	1	24	yes	complemented
236	#127	—	53	128	no	—
237	#124	—	53	128	no	—
238	#121	—	53	256	no	—
239	#120	—	53	256	no	—
240	#117	—	53	64	no	—
241	C₂×C₄⋊C₄	—	53	24	no	—
242	C₂×C₂²⋊C₄	—	53	24	no	—
243	C₂×C₂²⋊C₄	—	53	96	no	—
244	C₂×C₄²	—	53	24	no	—
245	C₂⁴	—	2	8	no	—
246	C₂⁴	—	2	8	no	—
247	C₂⁴	—	2	16	no	—
248	C₂⁴	—	53	24	no	—
249	C₂⁴	—	53	32	no	—
250	C₂⁴	—	106	96	no	—
251	C₂⁴	—	106	16	no	—
252	C₂⁴	—	106	32	no	—
253	C₂⁴	—	212	96	no	—
254	C₂⁴	—	212	128	no	—
255	C₂⁴	C₂×C₂²⋊D₅₃	1	8	yes	—
256	C₂⁴	C₂³×D₅₃	1	12	yes	—
257	C₂⁴	C₂³×D₅₃	1	6	yes	—
258	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
259	#41	—	2	256	no	—
260	C₂×Q₈	—	2	48	no	—
261	C₂×Q₈	C₂³×D₅₃	1	16	yes	complemented
262	C₂×D₄	—	2	96	no	—
263	C₂×D₄	—	2	48	no	—
264	C₂×D₄	—	53	128	no	—
265	C₂×D₄	—	53	32	no	—
266	C₂×D₄	—	53	32	no	—
267	C₂×D₄	—	106	96	no	—
268	C₂×D₄	—	106	96	no	—
269	C₂×D₄	—	106	256	no	—
270	C₂×D₄	—	212	192	no	—
271	C₂×D₄	C₂³×D₅₃	1	32	yes	—
272	C₂×D₄	C₂³×D₅₃	1	16	yes	complemented
273	C₂²×C₄	—	2	8	no	—
274	C₂²×C₄	—	2	96	no	—
275	C₂²×C₄	—	53	24	no	—
276	C₂²×C₄	—	106	16	no	—
277	C₂²×C₄	C₂×C₂²⋊D₅₃	1	8	yes	—
278	C₂²×C₄	C₂×C₂²⋊D₅₃	1	8	yes	—
279	C₂²×C₄	C₂×C₂²⋊D₅₃	1	96	yes	complemented
280	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
281	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
282	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
283	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
284	C₂²×C₄	C₂³×D₅₃	1	32	yes	complemented
285	C₄⋊C₄	—	53	32	no	—
286	C₂²⋊C₄	—	53	32	no	—
287	C₂²⋊C₄	—	53	128	no	—
288	C₄²	—	53	32	no	—
289	C₂³	—	2	32	no	—
290	C₂³	—	2	24	no	—
291	C₂³	—	2	24	no	—
292	C₂³	—	2	48	no	—
293	C₂³	—	4	32	no	—
294	C₂³	—	53	16	no	—
295	C₂³	—	106	64	no	—
296	C₂³	—	106	48	no	—
297	C₂³	—	106	96	no	—
298	C₂³	—	212	64	no	—
299	C₂³	—	212	384	no	—
300	C₂³	—	212	64	no	—
301	C₂³	C₂²×C₂²⋊D₅₃	1	24	yes	—
302	C₂³	C₂²×C₂²⋊D₅₃	1	4	yes	—
303	C₂³	#13133263	1	8	yes	—
304	C₂³	C₂⁴×D₅₃	1	8	yes	—
305	C₂³	C₂⁴×D₅₃	1	4	yes	—
306	C₂³	C₂⁴×D₅₃	1	3	yes	—
307	Q₈	—	2	32	no	—
308	D₄	—	2	64	no	—
309	D₄	—	2	32	no	—
310	D₄	—	106	64	no	—
311	D₄	—	106	64	no	—
312	D₄	—	212	128	no	—
313	C₂×C₄	—	2	24	no	—
314	C₂×C₄	—	2	64	no	—
315	C₂×C₄	—	53	16	no	—
316	C₂×C₄	—	106	48	no	—
317	C₂×C₄	C₂²×C₂²⋊D₅₃	1	24	yes	—
318	C₂×C₄	C₂²×C₂²⋊D₅₃	1	24	yes	—
319	C₂×C₄	C₂²×C₂²⋊D₅₃	1	64	yes	complemented
320	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
321	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
322	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
323	C₂²	—	2	48	no	—
324	C₂²	—	2	8	no	—
325	C₂²	—	2	8	no	—
326	C₂²	—	2	16	no	—
327	C₂²	—	4	48	no	—
328	C₂²	—	106	16	no	—
329	C₂²	—	106	32	no	—
330	C₂²	—	212	128	no	—
331	C₂²	—	212	96	no	—
332	C₂²	#16811187	1	16	yes	direct product factor
333	C₂²	C₂³×C₂²⋊D₅₃	1	8	yes	—
334	C₂²	C₂³×C₂²⋊D₅₃	1	6	yes	—
335	C₂²	C₂×#13133263	1	12	yes	—
336	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
337	C₄	—	2	8	no	—
338	C₄	—	106	16	no	—
339	C₄	C₂³×C₂²⋊D₅₃	1	8	yes	—
340	C₄	C₂³×C₂²⋊D₅₃	1	8	yes	—
341	C₂	—	2	8	no	—
342	C₂	—	4	8	no	—
343	C₂	—	212	16	no	—
344	C₂	C₂×#16811187	1	12	yes	direct product factor
345	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic
346	C₂	C₂²×#13133263	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×#41×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂³×#41	D₅₃
hypocenter	C₅₃	C₂²×#535
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₂₁₂	C₂	semidirect
2. C₂×#16811187	C₂	direct (N × H)
3. C₂²×#13133213	C₂	semidirect
4. C₂²×#13133206	C₂	semidirect
5. C₂²×#13133191	C₂	semidirect
6. C₂²×#13133185	C₂	semidirect
7. C₂²×#13133176	C₂	semidirect
8. C₂³×#41×C₅₃	C₂	semidirect
13. C₂³×D₂₁₂	C₂²	semidirect
14. C₂³×C₂²⋊D₅₃	C₂²	semidirect
15. #16811187	C₂²	direct (N × H)
16. C₂×#13133213	C₂²	semidirect
17. C₂×#13133206	C₂²	semidirect
18. C₂×#13133191	C₂²	semidirect
19. C₂×#13133185	C₂²	semidirect
20. C₂×#13133176	C₂²	semidirect
21. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
23. C₂²×#11719921	C₂²	semidirect
24. C₂²×C₄×Dic₅₃	C₂²	semidirect
26. C₂²×#41×C₅₃	C₂²	semidirect
27. C₂³×Q₈×C₅₃	C₂²	semidirect
29. C₂³×D₄×C₅₃	C₂²	semidirect
31. C₂⁴×C₄×C₅₃	C₂²	semidirect
41. C₂²×D₂₁₂	C₂³	semidirect
43. C₂²×C₂²⋊D₅₃	C₂³	semidirect
45. #13133213	C₂³	semidirect
46. #13133206	C₂³	semidirect
47. #13133191	C₂³	semidirect
48. #13133185	C₂³	semidirect
49. #13133176	C₂³	semidirect
50. C₂×C₄⋊Dic₅₃	C₂³	semidirect
52. C₂×#11719921	C₂³	semidirect
53. C₂×C₄×Dic₅₃	C₂³	semidirect
57. C₂×#41×C₅₃	C₂³	semidirect
59. C₂²×Q₈×C₅₃	C₂³	semidirect
63. C₂²×D₄×C₅₃	C₂³	semidirect
65. C₂³×C₄×C₅₃	D₄	semidirect
70. C₂³×C₄×C₅₃	C₂³	semidirect
87. C₂×Q₈×C₅₃	C₂⁴	semidirect
91. C₂×D₄×C₅₃	C₂⁴	semidirect
96. C₂²×C₄×C₅₃	C₂×D₄	semidirect
101. C₂²×C₄×C₅₃	C₂⁴	semidirect
103. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
112. #11719921	C₂⁴	semidirect
113. C₄⋊Dic₅₃	C₂⁴	semidirect
114. C₄×Dic₅₃	C₂⁴	semidirect
140. C₂×C₄×C₅₃	C₂²×D₄	semidirect
155. C₂²×C₅₃	#535	semidirect
168. C₂³×#41	D₅₃	semidirect
178. C₂×C₅₃	C₂×#535	semidirect
184. C₂²×#41	C₂×D₅₃	semidirect
185. C₂³×Q₈	C₂×D₅₃	semidirect
192. C₂³×D₄	C₂×D₅₃	semidirect
194. C₂⁴×C₄	C₂×D₅₃	semidirect
205. C₅₃	C₂²×#535	semidirect
213. C₂×#41	C₂²×D₅₃	semidirect
215. C₂²×Q₈	C₂²×D₅₃	semidirect
227. C₂²×D₄	C₂²×D₅₃	semidirect
230. C₂³×C₄	C₂²⋊D₅₃	semidirect
235. C₂³×C₄	C₂²×D₅₃	semidirect
261. C₂×Q₈	C₂³×D₅₃	semidirect
272. C₂×D₄	C₂³×D₅₃	semidirect
279. C₂²×C₄	C₂×C₂²⋊D₅₃	semidirect
284. C₂²×C₄	C₂³×D₅₃	semidirect
319. C₂×C₄	C₂²×C₂²⋊D₅₃	semidirect
332. C₂²	#16811187	direct (N × H)
344. C₂	C₂×#16811187	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₂₁₂	2
C₂×#16811187	2
C₂²×#13133213	2
C₂²×#13133206	2
C₂²×#13133191	2
C₂²×#13133185	2
C₂²×#13133176	2
C₂³×#41×C₅₃	2
C₂²×#535	53

Subgroups — C22×#16811187 #28290847

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811187 #28290847