Subgroups — C₂²×#16811301 #28291072

Order 13568.

9526 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811301	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13133210	C₂	1	2	yes	maximal, complemented
3	C₂²×#13133204	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#13133199	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133195	C₂	1	2	yes	maximal, complemented
6	C₂²×#13133185	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄×C₂²⋊D₅₃	C₂	1	2	yes	maximal, complemented
8	C₂²×#13133175	C₂	1	2	yes	maximal, complemented
9	C₂²×C₂²⋊C₄×D₅₃	C₂	1	2	yes	maximal, complemented
10	C₂²×#123×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
12	C₂³×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
13	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
14	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
15	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
16	#16811301	C₂²	1	256	yes	direct product factor
17	C₂×#13133210	C₂²	1	48	yes	complemented
18	C₂×#13133204	C₂²	1	24	yes	complemented
19	C₂×#13133199	C₂²	1	48	yes	complemented
20	C₂×#13133195	C₂²	1	48	yes	complemented
21	C₂×#13133185	C₂²	1	24	yes	—
22	C₂×C₄×C₂²⋊D₅₃	C₂²	1	48	yes	complemented
23	C₂×#13133175	C₂²	1	48	yes	complemented
24	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	complemented
25	C₂²×#11719919	C₂²	1	2	yes	complemented
26	C₂²×#11719919	C₂²	1	2	yes	complemented
27	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	—
28	C₂²×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	complemented
30	C₂²×#11719921	C₂²	1	2	yes	—
31	C₂²×#11719921	C₂²	1	2	yes	—
32	C₂²×#11719921	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₄×D₅₃	C₂²	1	48	yes	complemented
34	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
35	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	complemented
36	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
38	C₂×#123×C₅₃	C₂²	1	24	yes	complemented
39	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
40	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
41	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
42	C₂⁴×D₅₃	—	2	16	no	—
43	C₂⁴×D₅₃	—	4	4	no	—
44	C₂⁴×D₅₃	C₂³	1	12	yes	—
45	C₂⁴×D₅₃	C₂³	1	2	yes	—
46	C₂²×Dic₁₀₆	—	4	4	no	—
47	C₂²×Dic₁₀₆	C₂³	1	12	yes	complemented
48	C₂²×C₂²⋊D₅₃	—	2	16	no	—
49	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
50	C₂³×Dic₅₃	C₂³	1	2	yes	—
51	C₂³×Dic₅₃	C₂³	1	2	yes	—
52	C₂³×Dic₅₃	C₂³	1	2	yes	—
53	C₂²×C₄×D₅₃	—	2	16	no	—
54	C₂²×C₄×D₅₃	—	4	8	no	—
55	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
56	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
57	#13133210	C₂³	1	128	yes	complemented
58	#13133204	C₂³	1	64	yes	complemented
59	#13133199	C₂³	1	128	yes	complemented
60	#13133195	C₂³	1	128	yes	complemented
61	#13133185	C₂³	1	64	yes	—
62	C₄×C₂²⋊D₅₃	C₂³	1	128	yes	complemented
63	#13133175	C₂³	1	128	yes	complemented
64	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	complemented
65	C₂×#11719919	C₂³	1	24	yes	complemented
66	C₂×#11719919	C₂³	1	24	yes	complemented
67	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	—
68	C₂×C₂²⋊Dic₅₃	C₂³	1	12	yes	—
69	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	complemented
70	C₂×#11719921	C₂³	1	24	yes	—
71	C₂×#11719921	C₂³	1	24	yes	—
72	C₂×#11719921	C₂³	1	24	yes	—
73	C₂²⋊C₄×D₅₃	C₂³	1	128	yes	complemented
74	C₂×C₄×Dic₅₃	C₂³	1	12	yes	complemented
75	C₂×C₄×Dic₅₃	C₂³	1	24	yes	complemented
76	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
77	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
78	C₂²×D₄×C₅₃	—	4	4	no	—
79	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
80	C₂³×C₄×C₅₃	—	4	4	no	—
81	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
82	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
83	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₅₃	C₂³	1	12	yes	complemented
85	#123×C₅₃	C₂³	1	64	yes	complemented
86	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	complemented
87	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
88	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	complemented
89	C₂⁴×C₅₃	—	2	8	no	—
90	C₂⁴×C₅₃	—	2	8	no	—
91	C₂⁴×C₅₃	—	2	4	no	—
92	C₂⁴×C₅₃	—	2	4	no	—
93	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
94	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
95	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
96	C₂³×D₅₃	—	2	8	no	—
97	C₂³×D₅₃	—	2	8	no	—
98	C₂³×D₅₃	—	2	96	no	—
99	C₂³×D₅₃	—	4	24	no	—
100	C₂³×D₅₃	—	4	64	no	—
101	C₂³×D₅₃	#41	1	16	yes	—
102	C₂³×D₅₃	C₂⁴	1	16	yes	—
103	C₂³×D₅₃	C₂⁴	1	12	yes	—
104	C₂×D₄×C₅₃	—	4	24	no	—
105	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
106	C₂²×C₄×C₅₃	—	2	8	no	—
107	C₂²×C₄×C₅₃	—	2	8	no	—
108	C₂²×C₄×C₅₃	—	2	4	no	—
109	C₂²×C₄×C₅₃	—	4	24	no	—
110	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
111	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
112	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
113	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	complemented
114	C₂×C₂²⋊D₅₃	—	2	96	no	—
115	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
116	C₂²×Dic₅₃	—	2	8	no	—
117	C₂²×Dic₅₃	—	2	8	no	—
118	C₂²×Dic₅₃	—	2	8	no	—
119	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
120	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
121	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
122	C₂×C₄×D₅₃	—	2	96	no	—
123	C₂×C₄×D₅₃	—	4	48	no	—
124	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
125	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
126	C₂×Dic₁₀₆	—	4	24	no	—
127	C₂×Dic₁₀₆	C₂⁴	1	16	yes	complemented
128	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
129	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
130	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
131	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	—
132	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
133	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	complemented
134	#11719921	C₂⁴	1	32	yes	—
135	#11719921	C₂⁴	1	32	yes	—
136	#11719921	C₂⁴	1	32	yes	—
137	C₄⋊Dic₅₃	C₂⁴	1	32	yes	complemented
138	#11719919	C₂⁴	1	32	yes	complemented
139	#11719919	C₂⁴	1	32	yes	complemented
140	C₄×Dic₅₃	C₂⁴	1	16	yes	complemented
141	C₄×Dic₅₃	C₂⁴	1	32	yes	complemented
142	C₂³×C₅₃	—	2	24	no	—
143	C₂³×C₅₃	—	2	24	no	—
144	C₂³×C₅₃	—	2	12	no	—
145	C₂³×C₅₃	—	2	12	no	—
146	C₂³×C₅₃	—	4	16	no	—
147	C₂³×C₅₃	—	4	16	no	—
148	C₂³×C₅₃	C₂×#41	1	8	yes	—
149	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
150	C₂³×C₅₃	C₂⁵	1	4	yes	—
151	C₂³×C₅₃	C₂⁵	1	4	yes	—
152	C₂³×C₅₃	C₂⁵	1	3	yes	—
153	C₂²×D₅₃	—	2	64	no	—
154	C₂²×D₅₃	—	2	24	no	—
155	C₂²×D₅₃	—	2	24	no	—
156	C₂²×D₅₃	—	2	64	no	—
157	C₂²×D₅₃	—	4	16	no	—
158	C₂²×D₅₃	—	4	192	no	—
159	C₂²×D₅₃	C₂×#41	1	48	yes	—
160	C₂²×D₅₃	C₂⁵	1	8	yes	—
161	D₄×C₅₃	—	4	16	no	—
162	C₂×C₄×C₅₃	—	2	24	no	—
163	C₂×C₄×C₅₃	—	2	12	no	—
164	C₂×C₄×C₅₃	—	2	24	no	—
165	C₂×C₄×C₅₃	—	4	16	no	—
166	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
167	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
168	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
169	C₂²⋊D₅₃	—	2	64	no	—
170	C₂×Dic₅₃	—	2	24	no	—
171	C₂×Dic₅₃	—	2	24	no	—
172	C₂×Dic₅₃	—	2	24	no	—
173	C₂×Dic₅₃	C₂⁵	1	8	yes	—
174	C₂×Dic₅₃	C₂⁵	1	8	yes	—
175	C₂×Dic₅₃	C₂⁵	1	8	yes	—
176	C₄×D₅₃	—	2	64	no	—
177	C₄×D₅₃	—	4	32	no	—
178	Dic₁₀₆	—	4	16	no	—
179	C₂²×#551	—	53	1	no	maximal, Sylow
180	C₂²×C₅₃	—	2	8	no	—
181	C₂²×C₅₃	—	2	8	no	—
182	C₂²×C₅₃	—	2	4	no	—
183	C₂²×C₅₃	—	2	4	no	—
184	C₂²×C₅₃	—	4	24	no	—
185	C₂²×C₅₃	—	4	24	no	—
186	C₂²×C₅₃	#551	1	16	yes	complemented
187	C₂²×C₅₃	C₂²×#41	1	12	yes	—
188	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
189	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
190	C₂×D₅₃	—	2	8	no	—
191	C₂×D₅₃	—	2	8	no	—
192	C₂×D₅₃	—	2	96	no	—
193	C₂×D₅₃	—	4	64	no	—
194	C₂×D₅₃	C₂²×#41	1	16	yes	—
195	C₄×C₅₃	—	2	8	no	—
196	C₄×C₅₃	—	2	4	no	—
197	C₄×C₅₃	—	2	8	no	—
198	Dic₅₃	—	2	8	no	—
199	Dic₅₃	—	2	8	no	—
200	Dic₅₃	—	2	8	no	—
201	C₂×#551	—	53	48	no	—
202	C₂²×#126	—	53	2	no	—
203	C₂²×#124	—	53	1	no	—
204	C₂²×#123	—	53	2	no	—
205	C₂²×#123	D₅₃	1	1	yes	characteristic, complemented
206	C₂²×C₂²⋊Q₈	—	53	2	no	—
207	C₂²×#120	—	53	1	no	—
208	C₂²×C₄×D₄	—	53	2	no	—
209	C₂²×#117	—	53	2	no	—
210	C₂³×C₂²⋊C₄	—	53	2	no	—
211	C₂×C₅₃	—	4	4	no	—
212	C₂×C₅₃	—	4	4	no	—
213	C₂×C₅₃	C₂×#551	1	12	yes	complemented
214	C₂×C₅₃	C₂³×#41	1	2	yes	—
215	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
216	D₅₃	—	2	16	no	—
217	C₂⁶	—	53	1	no	—
218	C₂³×Q₈	—	53	1	no	—
219	C₂³×D₄	—	53	2	no	—
220	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
221	C₂⁴×C₄	—	53	2	no	—
222	C₂⁴×C₄	—	53	2	no	—
223	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
224	#551	—	53	256	no	—
225	C₂×#126	—	53	48	no	—
226	C₂×#124	—	53	24	no	—
227	C₂×#123	—	53	48	no	—
228	C₂×#123	C₂×D₅₃	1	24	yes	complemented
229	C₂×C₂²⋊Q₈	—	53	48	no	—
230	C₂×#120	—	53	24	no	—
231	C₂×C₄×D₄	—	53	48	no	—
232	C₂×#117	—	53	48	no	—
233	C₂²×C₄⋊C₄	—	53	2	no	—
234	C₂²×C₄⋊C₄	—	53	2	no	—
235	C₂²×C₄⋊C₄	—	53	2	no	—
236	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	complemented
237	C₂²×C₂²⋊C₄	—	53	2	no	—
238	C₂²×C₂²⋊C₄	—	53	1	no	—
239	C₂²×C₂²⋊C₄	—	53	2	no	—
240	C₂²×C₂²⋊C₄	—	53	2	no	—
241	C₂²×C₂²⋊C₄	—	53	2	no	—
242	C₂²×C₂²⋊C₄	—	53	2	no	—
243	C₂²×C₂²⋊C₄	—	53	48	no	—
244	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
245	C₂²×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
246	C₂²×C₄²	—	53	1	no	—
247	C₂²×C₄²	—	53	2	no	—
248	C₅₃	C₂²×#551	1	1	yes	characteristic, complemented, Sylow
249	C₂⁵	—	53	2	no	—
250	C₂⁵	—	53	12	no	—
251	C₂⁵	—	106	16	no	—
252	C₂⁵	—	212	4	no	—
253	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
254	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
255	C₂²×Q₈	—	53	12	no	—
256	C₂²×Q₈	—	212	4	no	—
257	C₂²×D₄	—	4	4	no	—
258	C₂²×D₄	—	53	24	no	—
259	C₂²×D₄	—	106	16	no	—
260	C₂²×D₄	C₂²×D₅₃	1	12	yes	—
261	C₂³×C₄	—	4	4	no	—
262	C₂³×C₄	—	53	2	no	—
263	C₂³×C₄	—	53	2	no	—
264	C₂³×C₄	—	53	2	no	—
265	C₂³×C₄	—	53	24	no	—
266	C₂³×C₄	—	53	24	no	—
267	C₂³×C₄	—	106	16	no	—
268	C₂³×C₄	—	212	8	no	—
269	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
270	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
271	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
272	C₂³×C₄	C₂²×D₅₃	1	12	yes	complemented
273	#126	—	53	128	no	—
274	#124	—	53	64	no	—
275	#123	—	53	128	no	—
276	#123	C₂²×D₅₃	1	64	yes	complemented
277	C₂²⋊Q₈	—	53	128	no	—
278	#120	—	53	64	no	—
279	C₄×D₄	—	53	128	no	—
280	#117	—	53	128	no	—
281	C₂×C₄⋊C₄	—	53	24	no	—
282	C₂×C₄⋊C₄	—	53	24	no	—
283	C₂×C₄⋊C₄	—	53	24	no	—
284	C₂×C₄⋊C₄	C₂²×D₅₃	1	24	yes	complemented
285	C₂×C₂²⋊C₄	—	53	24	no	—
286	C₂×C₂²⋊C₄	—	53	24	no	—
287	C₂×C₂²⋊C₄	—	53	12	no	—
288	C₂×C₂²⋊C₄	—	53	24	no	—
289	C₂×C₂²⋊C₄	—	53	24	no	—
290	C₂×C₂²⋊C₄	—	53	24	no	—
291	C₂×C₂²⋊C₄	—	53	128	no	—
292	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	—
293	C₂×C₂²⋊C₄	C₂²×D₅₃	1	12	yes	complemented
294	C₂×C₄²	—	53	12	no	—
295	C₂×C₄²	—	53	24	no	—
296	C₂⁴	—	2	8	no	—
297	C₂⁴	—	2	8	no	—
298	C₂⁴	—	2	4	no	—
299	C₂⁴	—	2	4	no	—
300	C₂⁴	—	53	16	no	—
301	C₂⁴	—	53	12	no	—
302	C₂⁴	—	53	16	no	—
303	C₂⁴	—	106	96	no	—
304	C₂⁴	—	106	8	no	—
305	C₂⁴	—	106	8	no	—
306	C₂⁴	—	212	24	no	—
307	C₂⁴	—	212	64	no	—
308	C₂⁴	C₂³×D₅₃	1	6	yes	—
309	C₂⁴	C₂³×D₅₃	1	6	yes	—
310	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
311	C₂×Q₈	—	53	16	no	—
312	C₂×Q₈	—	212	24	no	—
313	C₂×D₄	—	4	24	no	—
314	C₂×D₄	—	53	32	no	—
315	C₂×D₄	—	106	96	no	—
316	C₂×D₄	C₂³×D₅₃	1	16	yes	—
317	C₂²×C₄	—	2	8	no	—
318	C₂²×C₄	—	2	8	no	—
319	C₂²×C₄	—	2	4	no	—
320	C₂²×C₄	—	4	24	no	—
321	C₂²×C₄	—	53	12	no	—
322	C₂²×C₄	—	53	12	no	—
323	C₂²×C₄	—	53	12	no	—
324	C₂²×C₄	—	53	32	no	—
325	C₂²×C₄	—	53	32	no	—
326	C₂²×C₄	—	106	96	no	—
327	C₂²×C₄	—	106	8	no	—
328	C₂²×C₄	—	106	8	no	—
329	C₂²×C₄	—	106	8	no	—
330	C₂²×C₄	—	212	48	no	—
331	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
332	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
333	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
334	C₂²×C₄	C₂³×D₅₃	1	16	yes	complemented
335	C₄⋊C₄	—	53	32	no	—
336	C₄⋊C₄	—	53	32	no	—
337	C₄⋊C₄	—	53	32	no	—
338	C₄⋊C₄	C₂³×D₅₃	1	32	yes	complemented
339	C₂²⋊C₄	—	53	32	no	—
340	C₂²⋊C₄	—	53	32	no	—
341	C₂²⋊C₄	—	53	16	no	—
342	C₂²⋊C₄	—	53	32	no	—
343	C₂²⋊C₄	—	53	32	no	—
344	C₂²⋊C₄	—	53	32	no	—
345	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	—
346	C₂²⋊C₄	C₂³×D₅₃	1	16	yes	complemented
347	C₄²	—	53	16	no	—
348	C₄²	—	53	32	no	—
349	C₂³	—	2	24	no	—
350	C₂³	—	2	24	no	—
351	C₂³	—	2	12	no	—
352	C₂³	—	2	12	no	—
353	C₂³	—	4	16	no	—
354	C₂³	—	4	16	no	—
355	C₂³	—	53	48	no	—
356	C₂³	—	53	8	no	—
357	C₂³	—	106	64	no	—
358	C₂³	—	106	64	no	—
359	C₂³	—	106	24	no	—
360	C₂³	—	106	24	no	—
361	C₂³	—	212	16	no	—
362	C₂³	—	212	192	no	—
363	C₂³	#41×D₅₃	1	8	yes	—
364	C₂³	#13133262	1	4	yes	—
365	C₂³	C₂⁴×D₅₃	1	4	yes	—
366	C₂³	C₂⁴×D₅₃	1	4	yes	—
367	C₂³	C₂⁴×D₅₃	1	3	yes	—
368	Q₈	—	212	16	no	—
369	D₄	—	4	16	no	—
370	D₄	—	106	64	no	—
371	C₂×C₄	—	2	24	no	—
372	C₂×C₄	—	2	24	no	—
373	C₂×C₄	—	2	12	no	—
374	C₂×C₄	—	4	16	no	—
375	C₂×C₄	—	53	8	no	—
376	C₂×C₄	—	53	8	no	—
377	C₂×C₄	—	53	8	no	—
378	C₂×C₄	—	106	64	no	—
379	C₂×C₄	—	106	24	no	—
380	C₂×C₄	—	106	24	no	—
381	C₂×C₄	—	106	24	no	—
382	C₂×C₄	—	212	32	no	—
383	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
384	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
385	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
386	C₂²	—	2	8	no	—
387	C₂²	—	2	8	no	—
388	C₂²	—	2	4	no	—
389	C₂²	—	2	4	no	—
390	C₂²	—	4	24	no	—
391	C₂²	—	4	24	no	—
392	C₂²	—	53	16	no	—
393	C₂²	—	106	96	no	—
394	C₂²	—	106	8	no	—
395	C₂²	—	106	8	no	—
396	C₂²	—	212	64	no	—
397	C₂²	#16811301	1	16	yes	direct product factor
398	C₂²	C₂×#41×D₅₃	1	12	yes	—
399	C₂²	C₂×#13133262	1	6	yes	—
400	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
401	C₄	—	2	8	no	—
402	C₄	—	2	8	no	—
403	C₄	—	2	4	no	—
404	C₄	—	106	8	no	—
405	C₄	—	106	8	no	—
406	C₄	—	106	8	no	—
407	C₂	—	4	4	no	—
408	C₂	—	4	4	no	—
409	C₂	—	106	16	no	—
410	C₂	C₂×#16811301	1	12	yes	direct product factor
411	C₂	C₂²×#41×D₅₃	1	2	yes	—
412	C₂	C₂²×#13133262	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×#123×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂²×#13133185	C₂
hypercenter	C₂²×#123	D₅₃
hypocenter	C₅₃	C₂²×#551
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#123

Complements

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

N	H ≅ G/N	type
1. C₂×#16811301	C₂	direct (N × H)
2. C₂²×#13133210	C₂	semidirect
3. C₂²×#13133204	C₂	semidirect
4. C₂²×#13133199	C₂	semidirect
5. C₂²×#13133195	C₂	semidirect
7. C₂²×C₄×C₂²⋊D₅₃	C₂	semidirect
8. C₂²×#13133175	C₂	semidirect
9. C₂²×C₂²⋊C₄×D₅₃	C₂	semidirect
10. C₂²×#123×C₅₃	C₂	semidirect
12. C₂³×Dic₁₀₆	C₂²	semidirect
16. #16811301	C₂²	direct (N × H)
17. C₂×#13133210	C₂²	semidirect
18. C₂×#13133204	C₂²	semidirect
19. C₂×#13133199	C₂²	semidirect
20. C₂×#13133195	C₂²	semidirect
22. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
23. C₂×#13133175	C₂²	semidirect
24. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
25. C₂²×#11719919	C₂²	semidirect
26. C₂²×#11719919	C₂²	semidirect
29. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
33. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
34. C₂²×C₄×Dic₅₃	C₂²	semidirect
35. C₂²×C₄×Dic₅₃	C₂²	semidirect
37. C₂⁴×C₄×C₅₃	C₂²	semidirect
38. C₂×#123×C₅₃	C₂²	semidirect
39. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
41. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
47. C₂²×Dic₁₀₆	C₂³	semidirect
57. #13133210	C₂³	semidirect
58. #13133204	C₂³	semidirect
59. #13133199	C₂³	semidirect
60. #13133195	C₂³	semidirect
62. C₄×C₂²⋊D₅₃	C₂³	semidirect
63. #13133175	C₂³	semidirect
64. C₂×C₄⋊Dic₅₃	C₂³	semidirect
65. C₂×#11719919	C₂³	semidirect
66. C₂×#11719919	C₂³	semidirect
69. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
73. C₂²⋊C₄×D₅₃	C₂³	semidirect
74. C₂×C₄×Dic₅₃	C₂³	semidirect
75. C₂×C₄×Dic₅₃	C₂³	semidirect
84. C₂³×C₄×C₅₃	C₂³	semidirect
85. #123×C₅₃	C₂³	semidirect
86. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
88. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
113. C₂²×C₄×C₅₃	C₂⁴	semidirect
127. C₂×Dic₁₀₆	C₂⁴	semidirect
128. C₄⋊C₄×C₅₃	C₂⁴	semidirect
130. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
133. C₂²⋊Dic₅₃	C₂⁴	semidirect
137. C₄⋊Dic₅₃	C₂⁴	semidirect
138. #11719919	C₂⁴	semidirect
139. #11719919	C₂⁴	semidirect
140. C₄×Dic₅₃	C₂⁴	semidirect
141. C₄×Dic₅₃	C₂⁴	semidirect
186. C₂²×C₅₃	#551	semidirect
205. C₂²×#123	D₅₃	semidirect
213. C₂×C₅₃	C₂×#551	semidirect
223. C₂⁴×C₄	C₂×D₅₃	semidirect
228. C₂×#123	C₂×D₅₃	semidirect
236. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
245. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
248. C₅₃	C₂²×#551	semidirect
272. C₂³×C₄	C₂²×D₅₃	semidirect
276. #123	C₂²×D₅₃	semidirect
284. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
293. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
334. C₂²×C₄	C₂³×D₅₃	semidirect
338. C₄⋊C₄	C₂³×D₅₃	semidirect
346. C₂²⋊C₄	C₂³×D₅₃	semidirect
397. C₂²	#16811301	direct (N × H)
410. C₂	C₂×#16811301	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×#16811301	2
C₂²×#13133210	2
C₂²×#13133204	2
C₂²×#13133199	2
C₂²×#13133195	2
C₂²×#13133185	2
C₂²×C₄×C₂²⋊D₅₃	2
C₂²×#13133175	2
C₂²×C₂²⋊C₄×D₅₃	2
C₂²×#123×C₅₃	2
C₂²×#551	53

Subgroups — C22×#16811301 #28291072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811301 #28291072