Subgroups — C₂²×#16811091 #28290665

Order 13568.

12878 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719945	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811091	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133182	C₂	1	2	yes	maximal, complemented
4	C₂²×#13133179	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133177	C₂	1	4	yes	maximal, complemented
6	C₂²×#13133174	C₂	1	2	yes	maximal, complemented
7	C₂³×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄²×D₅₃	C₂	1	2	yes	maximal, complemented
9	C₂²×#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#11719945	C₄	1	32	yes	complemented
11	C₂²×#11719945	C₂²	1	24	yes	—
12	C₂³×Dic₁₀₆	C₂²	1	1	yes	characteristic
13	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
14	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
15	C₂⁴×Dic₅₃	C₂²	1	2	yes	—
16	C₂³×C₄×D₅₃	C₂²	1	4	yes	—
17	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
18	#16811091	C₂²	1	256	yes	direct product factor
19	C₂×#13133182	C₂²	1	48	yes	complemented
20	C₂×#13133179	C₂²	1	48	yes	complemented
21	C₂×#13133177	C₂²	1	96	yes	complemented
22	C₂×#13133174	C₂²	1	48	yes	complemented
23	C₂²×#11719919	C₂²	1	4	yes	—
24	C₂²×#11719921	C₂²	1	4	yes	—
25	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
26	C₂²×C₄×Dic₅₃	C₂²	1	4	yes	—
27	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	—
29	C₂²×C₄×Dic₅₃	C₂²	1	24	yes	complemented
30	C₂×C₄²×D₅₃	C₂²	1	48	yes	complemented
31	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#117×C₅₃	C₂²	1	24	yes	complemented
33	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
34	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
35	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
36	C₂⁴×D₅₃	C₂³	1	2	yes	—
37	C₂×#11719945	C₂×C₄	1	384	yes	complemented
38	C₂×#11719945	C₂³	1	64	yes	—
39	C₂²×Dic₁₀₆	C₂×C₄	1	16	yes	complemented
40	C₂²×Dic₁₀₆	C₂³	1	12	yes	—
41	C₂²×D₂₁₂	C₂×C₄	1	16	yes	complemented
42	C₂²×D₂₁₂	C₂³	1	12	yes	—
43	C₂²×C₂²⋊D₅₃	C₂×C₄	1	32	yes	complemented
44	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
45	C₂³×Dic₅₃	—	2	16	no	—
46	C₂³×Dic₅₃	C₂³	1	2	yes	—
47	C₂³×Dic₅₃	C₂³	1	24	yes	—
48	C₂³×Dic₅₃	C₂³	1	2	yes	—
49	C₂³×Dic₅₃	C₂³	1	2	yes	—
50	C₂²×C₄×D₅₃	—	2	32	no	—
51	C₂²×C₄×D₅₃	C₂×C₄	1	32	yes	complemented
52	C₂²×C₄×D₅₃	C₂³	1	48	yes	—
53	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
54	#13133182	C₂³	1	128	yes	complemented
55	#13133179	C₂³	1	128	yes	complemented
56	#13133177	C₂³	1	256	yes	complemented
57	#13133174	C₂³	1	128	yes	complemented
58	C₂×#11719919	C₂³	1	48	yes	—
59	C₂×#11719921	C₂³	1	48	yes	—
60	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
61	C₂×C₄×Dic₅₃	C₂³	1	48	yes	—
62	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
63	C₂×C₄×Dic₅₃	C₂³	1	24	yes	—
64	C₂×C₄×Dic₅₃	C₂³	1	64	yes	complemented
65	C₄²×D₅₃	C₂³	1	128	yes	complemented
66	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	complemented
68	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
69	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
72	#117×C₅₃	C₂³	1	64	yes	complemented
73	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
74	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
75	C₂×C₄²×C₅₃	C₂³	1	24	yes	—
76	C₂⁴×C₅₃	—	2	8	no	—
77	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	—
78	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
79	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₂³×D₅₃	—	2	16	no	—
81	C₂³×D₅₃	C₂²×C₄	1	16	yes	—
82	C₂³×D₅₃	C₂⁴	1	12	yes	—
83	C₂²×C₄×C₅₃	—	2	16	no	—
84	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
85	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
86	C₂²×C₄×C₅₃	C₂²×C₄	1	96	yes	complemented
87	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
88	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
89	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
90	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
91	C₂×C₂²⋊D₅₃	C₂²×C₄	1	192	yes	complemented
92	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
93	C₂²×Dic₅₃	—	2	8	no	—
94	C₂²×Dic₅₃	—	2	96	no	—
95	C₂²×Dic₅₃	C₂²×C₄	1	16	yes	—
96	C₂²×Dic₅₃	#41	1	16	yes	complemented
97	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
98	C₂²×Dic₅₃	C₂⁴	1	32	yes	—
99	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
100	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
101	#11719945	C₂²×C₄	1	512	yes	complemented
102	C₂×D₂₁₂	C₂²×C₄	1	96	yes	complemented
103	C₂×D₂₁₂	C₂⁴	1	16	yes	—
104	C₂×C₄×D₅₃	—	2	192	no	—
105	C₂×C₄×D₅₃	C₂²×C₄	1	192	yes	complemented
106	C₂×C₄×D₅₃	C₂⁴	1	64	yes	—
107	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
108	C₂×Dic₁₀₆	C₂²×C₄	1	96	yes	complemented
109	C₂×Dic₁₀₆	C₂⁴	1	16	yes	—
110	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	—
111	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
112	C₄²×C₅₃	C₂⁴	1	32	yes	—
113	#11719921	C₂⁴	1	64	yes	—
114	#11719919	C₂⁴	1	64	yes	—
115	C₄×Dic₅₃	C₂⁴	1	16	yes	—
116	C₄×Dic₅₃	C₂⁴	1	64	yes	—
117	C₄×Dic₅₃	C₂⁴	1	16	yes	—
118	C₄×Dic₅₃	C₂⁴	1	32	yes	—
119	C₂³×C₅₃	—	2	32	no	—
120	C₂³×C₅₃	—	2	24	no	—
121	C₂³×C₅₃	C₂³×C₄	1	24	yes	—
122	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
123	C₂³×C₅₃	C₂×#41	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	C₂³×C₄	1	48	yes	—
129	C₂²×D₅₃	C₂⁵	1	8	yes	—
130	C₂×C₄×C₅₃	—	2	48	no	—
131	C₂×C₄×C₅₃	C₂³×C₄	1	24	yes	—
132	C₂×C₄×C₅₃	C₂³×C₄	1	24	yes	—
133	C₂×C₄×C₅₃	C₂³×C₄	1	64	yes	complemented
134	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
135	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
136	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
137	C₂²⋊D₅₃	C₂³×C₄	1	128	yes	complemented
138	C₂×Dic₅₃	—	2	24	no	—
139	C₂×Dic₅₃	—	2	64	no	—
140	C₂×Dic₅₃	C₂³×C₄	1	48	yes	—
141	C₂×Dic₅₃	C₂×#41	1	48	yes	complemented
142	C₂×Dic₅₃	C₂⁵	1	8	yes	—
143	C₂×Dic₅₃	C₂⁵	1	8	yes	—
144	C₂×Dic₅₃	C₂⁵	1	8	yes	—
145	D₂₁₂	C₂³×C₄	1	64	yes	complemented
146	C₄×D₅₃	—	2	128	no	—
147	C₄×D₅₃	C₂³×C₄	1	128	yes	complemented
148	Dic₁₀₆	C₂³×C₄	1	64	yes	complemented
149	C₂²×C₄×#41	—	53	1	no	maximal, Sylow
150	C₂²×C₅₃	—	2	8	no	—
151	C₂²×C₅₃	—	2	48	no	—
152	C₂²×C₅₃	C₄×#41	1	16	yes	complemented
153	C₂²×C₅₃	C₂⁴×C₄	1	8	yes	—
154	C₂²×C₅₃	C₂⁴×C₄	1	6	yes	—
155	C₂²×C₅₃	C₂²×#41	1	12	yes	—
156	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
157	C₂×D₅₃	—	2	16	no	—
158	C₂×D₅₃	—	2	96	no	—
159	C₂×D₅₃	C₂⁴×C₄	1	16	yes	—
160	C₄×C₅₃	—	2	16	no	—
161	C₄×C₅₃	C₂⁴×C₄	1	8	yes	—
162	C₄×C₅₃	C₂⁴×C₄	1	8	yes	—
163	Dic₅₃	—	2	8	no	—
164	Dic₅₃	C₂⁴×C₄	1	16	yes	—
165	Dic₅₃	C₂²×#41	1	16	yes	complemented
166	C₂³×#41	—	53	1	no	—
167	C₂×C₄×#41	—	53	48	no	—
168	C₂²×C₄×Q₈	—	53	2	no	—
169	C₂²×C₄×D₄	—	53	2	no	—
170	C₂²×C₄×D₄	—	53	4	no	—
171	C₂²×#117	—	53	2	no	—
172	C₂²×#117	D₅₃	1	1	yes	characteristic, complemented
173	C₂³×C₄²	—	53	2	no	—
174	C₂³×C₄²	—	53	1	no	—
175	C₂×C₅₃	—	2	8	no	—
176	C₂×C₅₃	C₂×C₄×#41	1	12	yes	complemented
177	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
178	C₂×C₅₃	C₂³×#41	1	2	yes	—
179	D₅₃	—	2	16	no	—
180	C₂²×#41	—	53	32	no	—
181	C₂²×#41	—	53	24	no	—
182	C₂³×Q₈	—	53	1	no	—
183	C₂³×D₄	—	53	2	no	—
184	C₂³×D₄	—	53	1	no	—
185	C₂⁴×C₄	—	53	2	no	—
186	C₂⁴×C₄	—	53	2	no	—
187	C₂⁴×C₄	—	53	4	no	—
188	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
189	C₄×#41	—	53	256	no	—
190	C₂×C₄×Q₈	—	53	48	no	—
191	C₂×C₄×D₄	—	53	48	no	—
192	C₂×C₄×D₄	—	53	96	no	—
193	C₂×#117	—	53	48	no	—
194	C₂×#117	C₂×D₅₃	1	24	yes	complemented
195	C₂²×C₄⋊C₄	—	53	4	no	—
196	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
197	C₂²×C₂²⋊C₄	—	53	4	no	—
198	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
199	C₂²×C₄²	—	53	4	no	—
200	C₂²×C₄²	—	53	2	no	—
201	C₂²×C₄²	—	53	24	no	—
202	C₂²×C₄²	—	53	1	no	—
203	C₂²×C₄²	—	53	1	no	—
204	C₂²×C₄²	—	53	48	no	—
205	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
206	C₅₃	C₂²×C₄×#41	1	1	yes	characteristic, complemented, Sylow
207	C₂⁵	—	53	2	no	—
208	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
209	C₂×#41	—	53	384	no	—
210	C₂×#41	—	53	64	no	—
211	C₂²×Q₈	—	53	16	no	—
212	C₂²×Q₈	—	53	12	no	—
213	C₂²×D₄	—	53	32	no	—
214	C₂²×D₄	—	53	24	no	—
215	C₂²×D₄	—	53	16	no	—
216	C₂²×D₄	—	53	12	no	—
217	C₂³×C₄	—	53	2	no	—
218	C₂³×C₄	—	53	2	no	—
219	C₂³×C₄	—	53	32	no	—
220	C₂³×C₄	—	53	24	no	—
221	C₂³×C₄	—	53	2	no	—
222	C₂³×C₄	—	53	24	no	—
223	C₂³×C₄	—	53	48	no	—
224	C₂³×C₄	—	106	32	no	—
225	C₂³×C₄	—	106	16	no	—
226	C₂³×C₄	C₄×D₅₃	1	16	yes	complemented
227	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
228	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
229	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
230	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
231	C₄×Q₈	—	53	128	no	—
232	C₄×D₄	—	53	128	no	—
233	C₄×D₄	—	53	256	no	—
234	#117	—	53	128	no	—
235	#117	C₂²×D₅₃	1	64	yes	complemented
236	C₂×C₄⋊C₄	—	53	48	no	—
237	C₂×C₄⋊C₄	C₂²×D₅₃	1	24	yes	—
238	C₂×C₂²⋊C₄	—	53	48	no	—
239	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	—
240	C₂×C₄²	—	53	12	no	—
241	C₂×C₄²	—	53	48	no	—
242	C₂×C₄²	—	53	24	no	—
243	C₂×C₄²	—	53	64	no	—
244	C₂×C₄²	—	53	12	no	—
245	C₂×C₄²	—	53	128	no	—
246	C₂×C₄²	C₂²×D₅₃	1	24	yes	—
247	C₂⁴	—	2	8	no	—
248	C₂⁴	—	53	16	no	—
249	C₂⁴	—	53	12	no	—
250	C₂⁴	—	106	16	no	—
251	C₂⁴	C₂×C₄×D₅₃	1	8	yes	—
252	C₂⁴	C₂³×D₅₃	1	6	yes	—
253	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
254	#41	—	53	512	no	—
255	C₂×Q₈	—	53	96	no	—
256	C₂×Q₈	—	53	16	no	—
257	C₂×D₄	—	53	192	no	—
258	C₂×D₄	—	53	32	no	—
259	C₂×D₄	—	53	96	no	—
260	C₂×D₄	—	53	16	no	—
261	C₂²×C₄	—	2	16	no	—
262	C₂²×C₄	—	53	16	no	—
263	C₂²×C₄	—	53	16	no	—
264	C₂²×C₄	—	53	12	no	—
265	C₂²×C₄	—	53	12	no	—
266	C₂²×C₄	—	53	192	no	—
267	C₂²×C₄	—	53	32	no	—
268	C₂²×C₄	—	53	12	no	—
269	C₂²×C₄	—	53	32	no	—
270	C₂²×C₄	—	53	64	no	—
271	C₂²×C₄	—	106	192	no	—
272	C₂²×C₄	—	106	96	no	—
273	C₂²×C₄	—	106	8	no	—
274	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
275	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
276	C₂²×C₄	C₂×C₄×D₅₃	1	96	yes	complemented
277	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
278	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
279	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
280	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
281	C₄⋊C₄	—	53	64	no	—
282	C₄⋊C₄	C₂³×D₅₃	1	32	yes	—
283	C₂²⋊C₄	—	53	64	no	—
284	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	—
285	C₄²	—	53	16	no	—
286	C₄²	—	53	16	no	—
287	C₄²	—	53	64	no	—
288	C₄²	—	53	32	no	—
289	C₄²	C₂³×D₅₃	1	32	yes	—
290	C₂³	—	2	32	no	—
291	C₂³	—	2	24	no	—
292	C₂³	—	53	48	no	—
293	C₂³	—	53	8	no	—
294	C₂³	—	106	64	no	—
295	C₂³	—	106	48	no	—
296	C₂³	C₂²×C₄×D₅₃	1	24	yes	—
297	C₂³	C₂²×C₄×D₅₃	1	4	yes	—
298	C₂³	#41×D₅₃	1	8	yes	—
299	C₂³	C₂⁴×D₅₃	1	4	yes	—
300	C₂³	C₂⁴×D₅₃	1	3	yes	—
301	Q₈	—	53	64	no	—
302	D₄	—	53	128	no	—
303	D₄	—	53	64	no	—
304	C₂×C₄	—	2	48	no	—
305	C₂×C₄	—	53	48	no	—
306	C₂×C₄	—	53	48	no	—
307	C₂×C₄	—	53	8	no	—
308	C₂×C₄	—	53	8	no	—
309	C₂×C₄	—	53	128	no	—
310	C₂×C₄	—	53	8	no	—
311	C₂×C₄	—	106	128	no	—
312	C₂×C₄	—	106	64	no	—
313	C₂×C₄	—	106	24	no	—
314	C₂×C₄	C₂²×C₄×D₅₃	1	24	yes	—
315	C₂×C₄	C₂²×C₄×D₅₃	1	24	yes	—
316	C₂×C₄	C₂²×C₄×D₅₃	1	64	yes	complemented
317	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
318	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
319	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
320	C₂²	—	2	48	no	—
321	C₂²	—	2	8	no	—
322	C₂²	—	53	16	no	—
323	C₂²	—	106	96	no	—
324	C₂²	—	106	16	no	—
325	C₂²	#16811091	1	16	yes	direct product factor
326	C₂²	C₂³×C₄×D₅₃	1	8	yes	—
327	C₂²	C₂³×C₄×D₅₃	1	6	yes	—
328	C₂²	C₂×#41×D₅₃	1	12	yes	—
329	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
330	C₄	—	2	16	no	—
331	C₄	—	53	16	no	—
332	C₄	—	53	16	no	—
333	C₄	—	106	8	no	—
334	C₄	C₂³×C₄×D₅₃	1	8	yes	—
335	C₄	C₂³×C₄×D₅₃	1	8	yes	—
336	C₂	—	2	8	no	—
337	C₂	—	106	16	no	—
338	C₂	C₂×#16811091	1	12	yes	direct product factor
339	C₂	C₂⁴×C₄×D₅₃	1	1	yes	characteristic
340	C₂	C₂²×#41×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×#117×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂³×#11719945	C₂
cosocle	C₂²×C₅₃	C₂⁶
hypercenter	C₂²×#117	D₅₃
hypocenter	C₅₃	C₂²×C₄×#41
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
2. C₂×#16811091	C₂	direct (N × H)
3. C₂²×#13133182	C₂	semidirect
4. C₂²×#13133179	C₂	semidirect
5. C₂²×#13133177	C₂	semidirect
6. C₂²×#13133174	C₂	semidirect
7. C₂³×C₄×Dic₅₃	C₂	semidirect
8. C₂²×C₄²×D₅₃	C₂	semidirect
9. C₂²×#117×C₅₃	C₂	semidirect
10. C₂²×#11719945	C₄	semidirect
18. #16811091	C₂²	direct (N × H)
19. C₂×#13133182	C₂²	semidirect
20. C₂×#13133179	C₂²	semidirect
21. C₂×#13133177	C₂²	semidirect
22. C₂×#13133174	C₂²	semidirect
29. C₂²×C₄×Dic₅₃	C₂²	semidirect
30. C₂×C₄²×D₅₃	C₂²	semidirect
32. C₂×#117×C₅₃	C₂²	semidirect
37. C₂×#11719945	C₂×C₄	semidirect
39. C₂²×Dic₁₀₆	C₂×C₄	semidirect
41. C₂²×D₂₁₂	C₂×C₄	semidirect
43. C₂²×C₂²⋊D₅₃	C₂×C₄	semidirect
51. C₂²×C₄×D₅₃	C₂×C₄	semidirect
54. #13133182	C₂³	semidirect
55. #13133179	C₂³	semidirect
56. #13133177	C₂³	semidirect
57. #13133174	C₂³	semidirect
64. C₂×C₄×Dic₅₃	C₂³	semidirect
65. C₄²×D₅₃	C₂³	semidirect
67. C₂³×C₄×C₅₃	C₂×C₄	semidirect
72. #117×C₅₃	C₂³	semidirect
86. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
91. C₂×C₂²⋊D₅₃	C₂²×C₄	semidirect
96. C₂²×Dic₅₃	#41	semidirect
101. #11719945	C₂²×C₄	semidirect
102. C₂×D₂₁₂	C₂²×C₄	semidirect
105. C₂×C₄×D₅₃	C₂²×C₄	semidirect
108. C₂×Dic₁₀₆	C₂²×C₄	semidirect
133. C₂×C₄×C₅₃	C₂³×C₄	semidirect
137. C₂²⋊D₅₃	C₂³×C₄	semidirect
141. C₂×Dic₅₃	C₂×#41	semidirect
145. D₂₁₂	C₂³×C₄	semidirect
147. C₄×D₅₃	C₂³×C₄	semidirect
148. Dic₁₀₆	C₂³×C₄	semidirect
152. C₂²×C₅₃	C₄×#41	semidirect
165. Dic₅₃	C₂²×#41	semidirect
172. C₂²×#117	D₅₃	semidirect
176. C₂×C₅₃	C₂×C₄×#41	semidirect
194. C₂×#117	C₂×D₅₃	semidirect
206. C₅₃	C₂²×C₄×#41	semidirect
226. C₂³×C₄	C₄×D₅₃	semidirect
235. #117	C₂²×D₅₃	semidirect
276. C₂²×C₄	C₂×C₄×D₅₃	semidirect
316. C₂×C₄	C₂²×C₄×D₅₃	semidirect
325. C₂²	#16811091	direct (N × H)
338. C₂	C₂×#16811091	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11719945	2
C₂×#16811091	2
C₂²×#13133182	2
C₂²×#13133179	2
C₂²×#13133177	2
C₂²×#13133174	2
C₂³×C₄×Dic₅₃	2
C₂²×C₄²×D₅₃	2
C₂²×#117×C₅₃	2
C₂²×C₄×#41	53

Subgroups — C22×#16811091 #28290665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811091 #28290665