Subgroups — C₂²×#16811363 #28290961

Order 13568.

9606 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811363	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13133213	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13133212	C₂	1	2	yes	maximal, complemented
4	C₂²×#13133199	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133192	C₂	1	2	yes	maximal, complemented
6	C₂²×#13133191	C₂	1	2	yes	maximal, complemented
7	C₂²×#13133190	C₂	1	2	yes	maximal, complemented
8	C₂²×#13133187	C₂	1	2	yes	maximal, complemented
9	C₂²×#13133185	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#123×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
12	C₂³×D₂₁₂	C₂²	1	2	yes	—
13	C₂³×D₂₁₂	C₂²	1	2	yes	—
14	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
15	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
16	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
17	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
18	C₂³×C₄×D₅₃	C₂²	1	2	yes	complemented
19	#16811363	C₂²	1	256	yes	direct product factor
20	C₂×#13133213	C₂²	1	24	yes	complemented
21	C₂×#13133212	C₂²	1	48	yes	complemented
22	C₂×#13133199	C₂²	1	48	yes	complemented
23	C₂×#13133192	C₂²	1	48	yes	complemented
24	C₂×#13133191	C₂²	1	48	yes	complemented
25	C₂×#13133190	C₂²	1	48	yes	complemented
26	C₂×#13133187	C₂²	1	48	yes	complemented
27	C₂×#13133185	C₂²	1	24	yes	complemented
28	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	complemented
29	C₂²×#11719919	C₂²	1	2	yes	complemented
30	C₂²×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
31	C₂²×#11719921	C₂²	1	2	yes	—
32	C₂²×#11719921	C₂²	1	2	yes	—
33	C₂²×#11719921	C₂²	1	2	yes	—
34	C₂²×#11719921	C₂²	1	2	yes	complemented
35	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic, 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	—	4	8	no	—
43	C₂⁴×D₅₃	—	4	4	no	—
44	C₂⁴×D₅₃	C₂³	1	2	yes	—
45	C₂⁴×D₅₃	C₂³	1	2	yes	—
46	C₂⁴×D₅₃	C₂³	1	12	yes	—
47	C₂²×D₂₁₂	—	4	8	no	—
48	C₂²×D₂₁₂	—	4	8	no	—
49	C₂²×D₂₁₂	—	4	4	no	—
50	C₂²×D₂₁₂	C₂³	1	24	yes	—
51	C₂²×D₂₁₂	C₂³	1	24	yes	—
52	C₂²×D₂₁₂	C₂³	1	12	yes	complemented
53	C₂²×C₂²⋊D₅₃	—	4	8	no	—
54	C₂²×C₂²⋊D₅₃	—	4	8	no	—
55	C₂²×C₂²⋊D₅₃	—	4	8	no	—
56	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
57	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
58	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
59	C₂³×Dic₅₃	C₂³	1	2	yes	—
60	C₂³×Dic₅₃	C₂³	1	2	yes	—
61	C₂²×C₄×D₅₃	—	4	8	no	—
62	C₂²×C₄×D₅₃	C₂³	1	24	yes	complemented
63	#13133213	C₂³	1	64	yes	complemented
64	#13133212	C₂³	1	128	yes	complemented
65	#13133199	C₂³	1	128	yes	complemented
66	#13133192	C₂³	1	128	yes	complemented
67	#13133191	C₂³	1	128	yes	complemented
68	#13133190	C₂³	1	128	yes	complemented
69	#13133187	C₂³	1	128	yes	complemented
70	#13133185	C₂³	1	64	yes	complemented
71	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	complemented
72	C₂×#11719919	C₂³	1	24	yes	complemented
73	C₂×C₂²⋊Dic₅₃	C₂³	1	12	yes	—
74	C₂×#11719921	C₂³	1	24	yes	—
75	C₂×#11719921	C₂³	1	24	yes	—
76	C₂×#11719921	C₂³	1	24	yes	—
77	C₂×#11719921	C₂³	1	24	yes	complemented
78	C₂×C₄×Dic₅₃	C₂³	1	12	yes	complemented
79	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
80	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
81	C₂²×D₄×C₅₃	—	4	4	no	—
82	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
83	C₂³×C₄×C₅₃	—	4	4	no	—
84	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
85	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
86	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
87	C₂³×C₄×C₅₃	C₂³	1	12	yes	complemented
88	#123×C₅₃	C₂³	1	64	yes	complemented
89	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	complemented
90	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
91	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	complemented
92	C₂⁴×C₅₃	—	2	4	no	—
93	C₂⁴×C₅₃	—	2	4	no	—
94	C₂⁴×C₅₃	—	2	8	no	—
95	C₂⁴×C₅₃	—	2	8	no	—
96	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
97	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
98	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
99	C₂³×D₅₃	—	2	8	no	—
100	C₂³×D₅₃	—	2	8	no	—
101	C₂³×D₅₃	—	2	8	no	—
102	C₂³×D₅₃	—	2	8	no	—
103	C₂³×D₅₃	—	2	8	no	—
104	C₂³×D₅₃	—	2	8	no	—
105	C₂³×D₅₃	—	4	24	no	—
106	C₂³×D₅₃	—	4	48	no	—
107	C₂³×D₅₃	—	4	64	no	—
108	C₂³×D₅₃	C₂⁴	1	12	yes	—
109	C₂³×D₅₃	C₂⁴	1	12	yes	—
110	C₂³×D₅₃	C₂⁴	1	16	yes	—
111	C₂×D₄×C₅₃	—	4	24	no	—
112	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
113	C₂²×C₄×C₅₃	—	2	8	no	—
114	C₂²×C₄×C₅₃	—	2	4	no	—
115	C₂²×C₄×C₅₃	—	2	8	no	—
116	C₂²×C₄×C₅₃	—	4	24	no	—
117	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
118	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
119	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
120	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	complemented
121	C₂×C₂²⋊D₅₃	—	4	48	no	—
122	C₂×C₂²⋊D₅₃	—	4	48	no	—
123	C₂×C₂²⋊D₅₃	—	4	48	no	—
124	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
125	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
126	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
127	C₂²×Dic₅₃	—	2	8	no	—
128	C₂²×Dic₅₃	—	2	8	no	—
129	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
130	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
131	C₂×D₂₁₂	—	4	48	no	—
132	C₂×D₂₁₂	—	4	48	no	—
133	C₂×D₂₁₂	—	4	24	no	—
134	C₂×D₂₁₂	C₂⁴	1	32	yes	—
135	C₂×D₂₁₂	C₂⁴	1	32	yes	—
136	C₂×D₂₁₂	C₂⁴	1	16	yes	complemented
137	C₂×C₄×D₅₃	—	4	48	no	—
138	C₂×C₄×D₅₃	C₂⁴	1	32	yes	complemented
139	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
140	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
141	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
142	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
143	#11719921	C₂⁴	1	32	yes	—
144	#11719921	C₂⁴	1	32	yes	—
145	#11719921	C₂⁴	1	32	yes	—
146	#11719921	C₂⁴	1	32	yes	complemented
147	C₄⋊Dic₅₃	C₂⁴	1	32	yes	complemented
148	#11719919	C₂⁴	1	32	yes	complemented
149	C₄×Dic₅₃	C₂⁴	1	16	yes	complemented
150	C₂³×C₅₃	—	2	24	no	—
151	C₂³×C₅₃	—	2	24	no	—
152	C₂³×C₅₃	—	2	12	no	—
153	C₂³×C₅₃	—	2	12	no	—
154	C₂³×C₅₃	—	4	16	no	—
155	C₂³×C₅₃	—	4	16	no	—
156	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
157	C₂³×C₅₃	H(5,𝔽₂)	1	8	yes	—
158	C₂³×C₅₃	C₂⁵	1	4	yes	—
159	C₂³×C₅₃	C₂⁵	1	4	yes	—
160	C₂³×C₅₃	C₂⁵	1	3	yes	—
161	C₂²×D₅₃	—	2	24	no	—
162	C₂²×D₅₃	—	2	24	no	—
163	C₂²×D₅₃	—	2	24	no	—
164	C₂²×D₅₃	—	2	24	no	—
165	C₂²×D₅₃	—	2	24	no	—
166	C₂²×D₅₃	—	2	24	no	—
167	C₂²×D₅₃	—	4	16	no	—
168	C₂²×D₅₃	—	4	32	no	—
169	C₂²×D₅₃	—	4	192	no	—
170	C₂²×D₅₃	—	4	32	no	—
171	C₂²×D₅₃	—	4	32	no	—
172	C₂²×D₅₃	C₂⁵	1	8	yes	—
173	C₂²×D₅₃	C₂⁵	1	8	yes	—
174	D₄×C₅₃	—	4	16	no	—
175	C₂×C₄×C₅₃	—	2	24	no	—
176	C₂×C₄×C₅₃	—	2	24	no	—
177	C₂×C₄×C₅₃	—	2	12	no	—
178	C₂×C₄×C₅₃	—	4	16	no	—
179	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
180	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
181	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
182	C₂²⋊D₅₃	—	4	32	no	—
183	C₂²⋊D₅₃	—	4	32	no	—
184	C₂²⋊D₅₃	—	4	32	no	—
185	C₂×Dic₅₃	—	2	24	no	—
186	C₂×Dic₅₃	—	2	24	no	—
187	C₂×Dic₅₃	C₂⁵	1	8	yes	—
188	C₂×Dic₅₃	C₂⁵	1	8	yes	—
189	D₂₁₂	—	4	32	no	—
190	D₂₁₂	—	4	32	no	—
191	D₂₁₂	—	4	16	no	—
192	C₄×D₅₃	—	4	32	no	—
193	C₂²×#560	—	53	1	no	maximal, Sylow
194	C₂²×C₅₃	—	2	8	no	—
195	C₂²×C₅₃	—	2	8	no	—
196	C₂²×C₅₃	—	2	4	no	—
197	C₂²×C₅₃	—	2	4	no	—
198	C₂²×C₅₃	—	4	24	no	—
199	C₂²×C₅₃	—	4	24	no	—
200	C₂²×C₅₃	#560	1	16	yes	complemented
201	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
202	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	12	yes	—
203	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
204	C₂×D₅₃	—	2	8	no	—
205	C₂×D₅₃	—	2	8	no	—
206	C₂×D₅₃	—	2	8	no	—
207	C₂×D₅₃	—	2	8	no	—
208	C₂×D₅₃	—	2	8	no	—
209	C₂×D₅₃	—	2	8	no	—
210	C₂×D₅₃	—	4	64	no	—
211	C₂×D₅₃	—	4	48	no	—
212	C₂×D₅₃	—	4	48	no	—
213	C₄×C₅₃	—	2	8	no	—
214	C₄×C₅₃	—	2	8	no	—
215	C₄×C₅₃	—	2	4	no	—
216	Dic₅₃	—	2	8	no	—
217	Dic₅₃	—	2	8	no	—
218	C₂×#560	—	53	48	no	—
219	C₂²×#127	—	53	1	no	—
220	C₂²×#126	—	53	2	no	—
221	C₂²×#123	—	53	2	no	—
222	C₂²×#123	D₅₃	1	1	yes	characteristic, complemented
223	C₂²×#121	—	53	2	no	—
224	C₂²×#121	—	53	2	no	—
225	C₂²×#121	—	53	2	no	—
226	C₂²×#120	—	53	1	no	—
227	C₂²×#120	—	53	2	no	—
228	C₂×C₅₃	—	4	4	no	—
229	C₂×C₅₃	—	4	4	no	—
230	C₂×C₅₃	C₂×#560	1	12	yes	complemented
231	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
232	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	2	yes	—
233	D₅₃	—	4	8	no	—
234	D₅₃	—	4	8	no	—
235	C₂⁶	—	53	1	no	—
236	C₂³×D₄	—	53	2	no	—
237	C₂³×D₄	—	53	2	no	—
238	C₂³×D₄	—	53	2	no	—
239	C₂³×D₄	—	53	2	no	—
240	C₂³×D₄	—	53	2	no	—
241	C₂³×D₄	—	53	1	no	—
242	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
243	C₂⁴×C₄	—	53	2	no	—
244	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
245	#560	—	53	256	no	—
246	C₂×#127	—	53	24	no	—
247	C₂×#126	—	53	48	no	—
248	C₂×#123	—	53	48	no	—
249	C₂×#123	C₂×D₅₃	1	24	yes	complemented
250	C₂×#121	—	53	48	no	—
251	C₂×#121	—	53	48	no	—
252	C₂×#121	—	53	48	no	—
253	C₂×#120	—	53	24	no	—
254	C₂×#120	—	53	48	no	—
255	C₂²×C₄⋊C₄	—	53	2	no	—
256	C₂²×C₄⋊C₄	—	53	2	no	—
257	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	complemented
258	C₂²×C₂²⋊C₄	—	53	1	no	—
259	C₂²×C₂²⋊C₄	—	53	2	no	—
260	C₂²×C₂²⋊C₄	—	53	2	no	—
261	C₂²×C₂²⋊C₄	—	53	2	no	—
262	C₂²×C₂²⋊C₄	—	53	2	no	—
263	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
264	C₂²×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
265	C₂²×C₄²	—	53	1	no	—
266	C₅₃	C₂²×#560	1	1	yes	characteristic, complemented, Sylow
267	C₂⁵	—	53	2	no	—
268	C₂⁵	—	53	2	no	—
269	C₂⁵	—	53	12	no	—
270	C₂⁵	—	212	4	no	—
271	C₂⁵	—	212	8	no	—
272	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
273	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
274	C₂²×D₄	—	4	4	no	—
275	C₂²×D₄	—	53	24	no	—
276	C₂²×D₄	—	53	24	no	—
277	C₂²×D₄	—	53	24	no	—
278	C₂²×D₄	—	53	24	no	—
279	C₂²×D₄	—	53	24	no	—
280	C₂²×D₄	—	53	12	no	—
281	C₂²×D₄	—	212	8	no	—
282	C₂²×D₄	—	212	4	no	—
283	C₂²×D₄	—	212	8	no	—
284	C₂²×D₄	—	212	8	no	—
285	C₂²×D₄	—	212	8	no	—
286	C₂²×D₄	—	212	8	no	—
287	C₂²×D₄	C₂²×D₅₃	1	12	yes	—
288	C₂³×C₄	—	4	4	no	—
289	C₂³×C₄	—	53	2	no	—
290	C₂³×C₄	—	53	2	no	—
291	C₂³×C₄	—	53	24	no	—
292	C₂³×C₄	—	212	8	no	—
293	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
294	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
295	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
296	C₂³×C₄	C₂²×D₅₃	1	12	yes	complemented
297	#127	—	53	64	no	—
298	#126	—	53	128	no	—
299	#123	—	53	128	no	—
300	#123	C₂²×D₅₃	1	64	yes	complemented
301	#121	—	53	128	no	—
302	#121	—	53	128	no	—
303	#121	—	53	128	no	—
304	#120	—	53	64	no	—
305	#120	—	53	128	no	—
306	C₂×C₄⋊C₄	—	53	24	no	—
307	C₂×C₄⋊C₄	—	53	24	no	—
308	C₂×C₄⋊C₄	C₂²×D₅₃	1	24	yes	complemented
309	C₂×C₂²⋊C₄	—	53	12	no	—
310	C₂×C₂²⋊C₄	—	53	24	no	—
311	C₂×C₂²⋊C₄	—	53	24	no	—
312	C₂×C₂²⋊C₄	—	53	24	no	—
313	C₂×C₂²⋊C₄	—	53	24	no	—
314	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	—
315	C₂×C₂²⋊C₄	C₂²×D₅₃	1	12	yes	complemented
316	C₂×C₄²	—	53	12	no	—
317	C₂⁴	—	2	4	no	—
318	C₂⁴	—	2	4	no	—
319	C₂⁴	—	2	8	no	—
320	C₂⁴	—	2	8	no	—
321	C₂⁴	—	53	12	no	—
322	C₂⁴	—	53	12	no	—
323	C₂⁴	—	53	16	no	—
324	C₂⁴	—	106	8	no	—
325	C₂⁴	—	106	8	no	—
326	C₂⁴	—	106	8	no	—
327	C₂⁴	—	106	8	no	—
328	C₂⁴	—	106	8	no	—
329	C₂⁴	—	106	8	no	—
330	C₂⁴	—	212	24	no	—
331	C₂⁴	—	212	48	no	—
332	C₂⁴	—	212	64	no	—
333	C₂⁴	C₂³×D₅₃	1	6	yes	—
334	C₂⁴	C₂³×D₅₃	1	6	yes	—
335	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
336	C₂×D₄	—	4	24	no	—
337	C₂×D₄	—	53	32	no	—
338	C₂×D₄	—	53	32	no	—
339	C₂×D₄	—	53	32	no	—
340	C₂×D₄	—	53	32	no	—
341	C₂×D₄	—	53	32	no	—
342	C₂×D₄	—	53	16	no	—
343	C₂×D₄	—	212	48	no	—
344	C₂×D₄	—	212	24	no	—
345	C₂×D₄	—	212	48	no	—
346	C₂×D₄	—	212	48	no	—
347	C₂×D₄	—	212	48	no	—
348	C₂×D₄	—	212	48	no	—
349	C₂×D₄	C₂³×D₅₃	1	16	yes	—
350	C₂²×C₄	—	2	8	no	—
351	C₂²×C₄	—	2	4	no	—
352	C₂²×C₄	—	2	8	no	—
353	C₂²×C₄	—	4	24	no	—
354	C₂²×C₄	—	53	12	no	—
355	C₂²×C₄	—	53	12	no	—
356	C₂²×C₄	—	53	32	no	—
357	C₂²×C₄	—	106	8	no	—
358	C₂²×C₄	—	106	8	no	—
359	C₂²×C₄	—	212	48	no	—
360	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
361	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
362	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
363	C₂²×C₄	C₂³×D₅₃	1	16	yes	complemented
364	C₄⋊C₄	—	53	32	no	—
365	C₄⋊C₄	—	53	32	no	—
366	C₄⋊C₄	C₂³×D₅₃	1	32	yes	complemented
367	C₂²⋊C₄	—	53	16	no	—
368	C₂²⋊C₄	—	53	32	no	—
369	C₂²⋊C₄	—	53	32	no	—
370	C₂²⋊C₄	—	53	32	no	—
371	C₂²⋊C₄	—	53	32	no	—
372	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	—
373	C₂²⋊C₄	C₂³×D₅₃	1	16	yes	complemented
374	C₄²	—	53	16	no	—
375	C₂³	—	2	12	no	—
376	C₂³	—	2	12	no	—
377	C₂³	—	2	24	no	—
378	C₂³	—	2	24	no	—
379	C₂³	—	4	16	no	—
380	C₂³	—	4	16	no	—
381	C₂³	—	53	8	no	—
382	C₂³	—	53	8	no	—
383	C₂³	—	106	24	no	—
384	C₂³	—	106	24	no	—
385	C₂³	—	106	24	no	—
386	C₂³	—	106	24	no	—
387	C₂³	—	106	24	no	—
388	C₂³	—	106	24	no	—
389	C₂³	—	212	16	no	—
390	C₂³	—	212	32	no	—
391	C₂³	—	212	192	no	—
392	C₂³	—	212	32	no	—
393	C₂³	—	212	32	no	—
394	C₂³	#13133262	1	4	yes	—
395	C₂³	#13133263	1	8	yes	—
396	C₂³	C₂⁴×D₅₃	1	4	yes	—
397	C₂³	C₂⁴×D₅₃	1	4	yes	—
398	C₂³	C₂⁴×D₅₃	1	3	yes	—
399	D₄	—	4	16	no	—
400	D₄	—	212	32	no	—
401	D₄	—	212	16	no	—
402	D₄	—	212	32	no	—
403	D₄	—	212	32	no	—
404	D₄	—	212	32	no	—
405	D₄	—	212	32	no	—
406	C₂×C₄	—	2	24	no	—
407	C₂×C₄	—	2	12	no	—
408	C₂×C₄	—	2	24	no	—
409	C₂×C₄	—	4	16	no	—
410	C₂×C₄	—	53	8	no	—
411	C₂×C₄	—	53	8	no	—
412	C₂×C₄	—	106	24	no	—
413	C₂×C₄	—	106	24	no	—
414	C₂×C₄	—	212	32	no	—
415	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
416	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
417	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
418	C₂²	—	2	4	no	—
419	C₂²	—	2	4	no	—
420	C₂²	—	2	8	no	—
421	C₂²	—	2	8	no	—
422	C₂²	—	4	24	no	—
423	C₂²	—	4	24	no	—
424	C₂²	—	106	8	no	—
425	C₂²	—	106	8	no	—
426	C₂²	—	106	8	no	—
427	C₂²	—	106	8	no	—
428	C₂²	—	106	8	no	—
429	C₂²	—	106	8	no	—
430	C₂²	—	212	64	no	—
431	C₂²	—	212	48	no	—
432	C₂²	—	212	48	no	—
433	C₂²	#16811363	1	16	yes	direct product factor
434	C₂²	C₂×#13133262	1	6	yes	—
435	C₂²	C₂×#13133263	1	12	yes	—
436	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
437	C₄	—	2	8	no	—
438	C₄	—	2	4	no	—
439	C₄	—	2	8	no	—
440	C₄	—	106	8	no	—
441	C₄	—	106	8	no	—
442	C₂	—	4	4	no	—
443	C₂	—	4	4	no	—
444	C₂	—	212	8	no	—
445	C₂	—	212	8	no	—
446	C₂	C₂×#16811363	1	12	yes	direct product factor
447	C₂	C₂²×#13133262	1	1	yes	characteristic
448	C₂	C₂²×#13133263	1	2	yes	—

	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
hypercenter	C₂²×#123	D₅₃
hypocenter	C₅₃	C₂²×#560
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×#16811363	C₂	direct (N × H)
2. C₂²×#13133213	C₂	semidirect
3. C₂²×#13133212	C₂	semidirect
4. C₂²×#13133199	C₂	semidirect
5. C₂²×#13133192	C₂	semidirect
6. C₂²×#13133191	C₂	semidirect
7. C₂²×#13133190	C₂	semidirect
8. C₂²×#13133187	C₂	semidirect
9. C₂²×#13133185	C₂	semidirect
10. C₂²×#123×C₅₃	C₂	semidirect
14. C₂³×D₂₁₂	C₂²	semidirect
17. C₂³×C₂²⋊D₅₃	C₂²	semidirect
18. C₂³×C₄×D₅₃	C₂²	semidirect
19. #16811363	C₂²	direct (N × H)
20. C₂×#13133213	C₂²	semidirect
21. C₂×#13133212	C₂²	semidirect
22. C₂×#13133199	C₂²	semidirect
23. C₂×#13133192	C₂²	semidirect
24. C₂×#13133191	C₂²	semidirect
25. C₂×#13133190	C₂²	semidirect
26. C₂×#13133187	C₂²	semidirect
27. C₂×#13133185	C₂²	semidirect
28. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
29. C₂²×#11719919	C₂²	semidirect
34. C₂²×#11719921	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
52. C₂²×D₂₁₂	C₂³	semidirect
58. C₂²×C₂²⋊D₅₃	C₂³	semidirect
62. C₂²×C₄×D₅₃	C₂³	semidirect
63. #13133213	C₂³	semidirect
64. #13133212	C₂³	semidirect
65. #13133199	C₂³	semidirect
66. #13133192	C₂³	semidirect
67. #13133191	C₂³	semidirect
68. #13133190	C₂³	semidirect
69. #13133187	C₂³	semidirect
70. #13133185	C₂³	semidirect
71. C₂×C₄⋊Dic₅₃	C₂³	semidirect
72. C₂×#11719919	C₂³	semidirect
77. C₂×#11719921	C₂³	semidirect
78. C₂×C₄×Dic₅₃	C₂³	semidirect
87. C₂³×C₄×C₅₃	C₂³	semidirect
88. #123×C₅₃	C₂³	semidirect
89. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
91. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
120. C₂²×C₄×C₅₃	C₂⁴	semidirect
126. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
136. C₂×D₂₁₂	C₂⁴	semidirect
138. C₂×C₄×D₅₃	C₂⁴	semidirect
139. C₄⋊C₄×C₅₃	C₂⁴	semidirect
141. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
146. #11719921	C₂⁴	semidirect
147. C₄⋊Dic₅₃	C₂⁴	semidirect
148. #11719919	C₂⁴	semidirect
149. C₄×Dic₅₃	C₂⁴	semidirect
200. C₂²×C₅₃	#560	semidirect
222. C₂²×#123	D₅₃	semidirect
230. C₂×C₅₃	C₂×#560	semidirect
244. C₂⁴×C₄	C₂×D₅₃	semidirect
249. C₂×#123	C₂×D₅₃	semidirect
257. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
264. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
266. C₅₃	C₂²×#560	semidirect
296. C₂³×C₄	C₂²×D₅₃	semidirect
300. #123	C₂²×D₅₃	semidirect
308. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
315. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
363. C₂²×C₄	C₂³×D₅₃	semidirect
366. C₄⋊C₄	C₂³×D₅₃	semidirect
373. C₂²⋊C₄	C₂³×D₅₃	semidirect
433. C₂²	#16811363	direct (N × H)
446. C₂	C₂×#16811363	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×#16811363	2
C₂²×#13133213	2
C₂²×#13133212	2
C₂²×#13133199	2
C₂²×#13133192	2
C₂²×#13133191	2
C₂²×#13133190	2
C₂²×#13133187	2
C₂²×#13133185	2
C₂²×#123×C₅₃	2
C₂²×#560	53

Subgroups — C22×#16811363 #28290961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811363 #28290961