Subgroups — C₂×#20288591 #28292452

Order 13568.

8622 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20288591	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16811358	C₂	1	4	yes	maximal, complemented
3	C₂×#16811294	C₂	1	4	yes	maximal, complemented
4	C₂×#16811269	C₂	1	8	yes	maximal, complemented
5	C₂×#16811228	C₂	1	4	yes	maximal, complemented
6	C₂×#16811187	C₂	1	2	yes	maximal, complemented
7	C₂×#16811180	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16811173	C₂	1	4	yes	maximal, complemented
9	C₂²×#13133214	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16811097	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#11719945	C₂²	1	8	yes	complemented
13	C₂³×D₂₁₂	C₂²	1	2	yes	—
14	C₂³×D₂₁₂	C₂²	1	4	yes	—
15	#16811358	C₂²	1	64	yes	complemented
16	#16811294	C₂²	1	64	yes	complemented
17	#16811269	C₂²	1	128	yes	complemented
18	#16811228	C₂²	1	64	yes	complemented
19	#16811187	C₂²	1	32	yes	complemented
20	#16811180	C₂²	1	16	yes	complemented
21	#16811173	C₂²	1	64	yes	complemented
22	C₂×#13133217	C₂²	1	2	yes	complemented
23	C₂×#13133214	C₂²	1	4	yes	—
24	C₂×#13133214	C₂²	1	2	yes	—
25	C₂×#13133214	C₂²	1	2	yes	—
26	C₂×#13133214	C₂²	1	16	yes	complemented
27	C₂×#13133213	C₂²	1	4	yes	complemented
28	C₂×#13133209	C₂²	1	2	yes	complemented
29	C₂×#13133206	C₂²	1	4	yes	complemented
30	C₂×#13133205	C₂²	1	8	yes	complemented
31	C₂×#13133202	C₂²	1	16	yes	complemented
32	C₂×#13133196	C₂²	1	4	yes	—
33	C₂×#13133191	C₂²	1	8	yes	complemented
34	C₂×#13133191	C₂²	1	8	yes	complemented
35	C₂×#13133190	C₂²	1	16	yes	complemented
36	C₂×#13133185	C₂²	1	8	yes	—
37	#16811097	C₂²	1	32	yes	complemented
38	C₂×C₄×Dic₁₀₆	C₂²	1	4	yes	complemented
39	C₂×C₄×D₂₁₂	C₂²	1	8	yes	complemented
40	C₂×C₄×D₂₁₂	C₂²	1	8	yes	complemented
41	C₂×#13133179	C₂²	1	8	yes	complemented
42	C₂×#13133176	C₂²	1	2	yes	—
43	C₂×C₂²⋊C₄×D₅₃	C₂²	1	8	yes	—
44	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
45	C₄×#41×C₅₃	C₂²	1	16	yes	complemented
46	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
47	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
48	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
49	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
50	C₂×#117×C₅₃	C₂²	1	2	yes	complemented
51	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
52	C₂²×C₄²×C₅₃	C₂²	1	2	yes	complemented
53	C₂⁴×D₅₃	C₂³	1	4	yes	—
54	C₂×#11719945	—	4	32	no	—
55	C₂×#11719945	C₂³	1	64	yes	complemented
56	C₂²×Dic₁₀₆	C₂³	1	4	yes	—
57	C₂²×D₂₁₂	—	2	16	no	—
58	C₂²×D₂₁₂	—	4	16	no	—
59	C₂²×D₂₁₂	C₂³	1	4	yes	—
60	C₂²×D₂₁₂	C₂³	1	4	yes	—
61	C₂²×D₂₁₂	C₂³	1	8	yes	—
62	C₂²×D₂₁₂	C₂³	1	16	yes	—
63	C₂²×D₂₁₂	C₂³	1	32	yes	—
64	C₂²×D₂₁₂	C₂³	1	8	yes	—
65	C₂²×C₂²⋊D₅₃	C₂³	1	16	yes	—
66	C₂²×C₄×D₅₃	C₂³	1	16	yes	—
67	#13133217	C₂³	1	16	yes	complemented
68	#13133214	C₂³	1	32	yes	—
69	#13133214	C₂³	1	16	yes	—
70	#13133214	C₂³	1	16	yes	—
71	#13133213	C₂³	1	32	yes	complemented
72	#13133209	C₂³	1	16	yes	complemented
73	#13133206	C₂³	1	32	yes	complemented
74	#13133205	C₂³	1	64	yes	complemented
75	#13133202	C₂³	1	128	yes	complemented
76	#13133196	C₂³	1	32	yes	—
77	#13133191	C₂³	1	64	yes	complemented
78	#13133191	C₂³	1	64	yes	complemented
79	#13133190	C₂³	1	128	yes	complemented
80	#13133185	C₂³	1	64	yes	—
81	C₄×Dic₁₀₆	C₂³	1	32	yes	complemented
82	C₄×D₂₁₂	C₂³	1	64	yes	complemented
83	C₄×D₂₁₂	C₂³	1	64	yes	complemented
84	#13133179	C₂³	1	64	yes	complemented
85	#13133176	C₂³	1	16	yes	—
86	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
87	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
88	C₂×#11719919	C₂³	1	8	yes	—
89	C₂×C₂²⋊Dic₅₃	C₂³	1	4	yes	—
90	C₂×#11719921	C₂³	1	16	yes	—
91	C₂×#11719921	C₂³	1	16	yes	—
92	C₂²⋊C₄×D₅₃	C₂³	1	64	yes	—
93	C₂×C₄×Dic₅₃	C₂³	1	4	yes	—
94	C₂×#41×C₅₃	—	2	8	no	—
95	C₂×#41×C₅₃	C₂³	1	8	yes	—
96	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
97	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
98	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
99	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
100	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
101	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
102	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
103	C₄×Q₈×C₅₃	C₂³	1	16	yes	complemented
104	C₄×D₄×C₅₃	C₂³	1	32	yes	—
105	C₄×D₄×C₅₃	C₂³	1	16	yes	complemented
106	#117×C₅₃	C₂³	1	8	yes	—
107	#117×C₅₃	C₂³	1	16	yes	complemented
108	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
112	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
115	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
116	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
117	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
118	C₂×C₄²×C₅₃	C₂³	1	16	yes	complemented
119	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
120	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
121	C₂³×D₅₃	—	2	16	no	—
122	C₂³×D₅₃	—	4	16	no	—
123	C₂³×D₅₃	C₂⁴	1	8	yes	—
124	C₂³×D₅₃	C₂⁴	1	16	yes	—
125	#41×C₅₃	—	2	32	no	—
126	C₂×Q₈×C₅₃	—	2	4	no	—
127	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
128	C₂×D₄×C₅₃	—	2	8	no	—
129	C₂×D₄×C₅₃	—	2	4	no	—
130	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
131	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
132	C₂²×C₄×C₅₃	—	2	8	no	—
133	C₂²×C₄×C₅₃	—	2	8	no	—
134	C₂²×C₄×C₅₃	—	2	16	no	—
135	C₂²×C₄×C₅₃	#41	1	8	yes	complemented
136	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
137	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
138	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
139	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
140	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
141	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
142	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
143	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
144	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
145	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
146	C₂×C₂²⋊D₅₃	—	4	32	no	—
147	C₂×C₂²⋊D₅₃	C₂⁴	1	64	yes	—
148	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
149	#11719945	—	4	128	no	—
150	C₂×D₂₁₂	—	2	16	no	—
151	C₂×D₂₁₂	—	2	16	no	—
152	C₂×D₂₁₂	—	2	64	no	—
153	C₂×D₂₁₂	—	2	32	no	—
154	C₂×D₂₁₂	—	4	64	no	—
155	C₂×D₂₁₂	—	4	16	no	—
156	C₂×D₂₁₂	C₂⁴	1	16	yes	—
157	C₂×D₂₁₂	C₂⁴	1	16	yes	—
158	C₂×D₂₁₂	C₂⁴	1	32	yes	—
159	C₂×D₂₁₂	C₂⁴	1	32	yes	—
160	C₂×C₄×D₅₃	—	4	32	no	—
161	C₂×C₄×D₅₃	C₂⁴	1	64	yes	—
162	C₂×Dic₁₀₆	—	2	16	no	—
163	C₂×Dic₁₀₆	C₂⁴	1	16	yes	—
164	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
165	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
166	C₂²⋊C₄×C₅₃	C₂⁴	1	8	yes	—
167	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
168	C₄²×C₅₃	C₂⁴	1	4	yes	—
169	C₄²×C₅₃	C₂⁴	1	4	yes	—
170	C₄²×C₅₃	C₂⁴	1	8	yes	—
171	C₄²×C₅₃	C₂⁴	1	8	yes	—
172	C₄²×C₅₃	C₂⁴	1	16	yes	—
173	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
174	#11719921	C₂⁴	1	64	yes	—
175	#11719921	C₂⁴	1	64	yes	—
176	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
177	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
178	#11719919	C₂⁴	1	32	yes	—
179	C₄×Dic₅₃	C₂⁴	1	16	yes	—
180	C₂³×C₅₃	—	2	4	no	—
181	C₂³×C₅₃	—	2	4	no	—
182	C₂³×C₅₃	—	2	8	no	—
183	C₂³×C₅₃	C₂×#41	1	4	yes	—
184	C₂³×C₅₃	C₂⁵	1	4	yes	—
185	C₂³×C₅₃	C₂⁵	1	2	yes	—
186	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
187	C₂²×D₅₃	—	2	16	no	—
188	C₂²×D₅₃	—	2	32	no	—
189	C₂²×D₅₃	—	2	32	no	—
190	C₂²×D₅₃	—	4	32	no	—
191	C₂²×D₅₃	—	8	32	no	—
192	C₂²×D₅₃	C₂⁵	1	16	yes	—
193	Q₈×C₅₃	—	2	8	no	—
194	D₄×C₅₃	—	2	16	no	—
195	D₄×C₅₃	—	2	8	no	—
196	C₂×C₄×C₅₃	—	2	4	no	—
197	C₂×C₄×C₅₃	—	2	8	no	—
198	C₂×C₄×C₅₃	—	2	4	no	—
199	C₂×C₄×C₅₃	—	2	16	no	—
200	C₂×C₄×C₅₃	—	2	16	no	—
201	C₂×C₄×C₅₃	—	2	32	no	—
202	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
203	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
204	C₂×C₄×C₅₃	C₂×#41	1	16	yes	complemented
205	C₂×C₄×C₅₃	H(5,𝔽₂)	1	8	yes	—
206	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
207	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
208	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
209	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
210	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
211	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
212	C₂²⋊D₅₃	—	4	64	no	—
213	C₂×Dic₅₃	—	2	16	no	—
214	C₂×Dic₅₃	C₂⁵	1	16	yes	—
215	D₂₁₂	—	2	32	no	—
216	D₂₁₂	—	2	32	no	—
217	D₂₁₂	—	2	64	no	—
218	D₂₁₂	—	4	32	no	—
219	C₄×D₅₃	—	4	64	no	—
220	Dic₁₀₆	—	2	32	no	—
221	C₂×#3508	—	53	1	no	maximal, Sylow
222	C₂²×C₅₃	—	2	4	no	—
223	C₂²×C₅₃	—	2	8	no	—
224	C₂²×C₅₃	—	2	4	no	—
225	C₂²×C₅₃	—	2	8	no	—
226	C₂²×C₅₃	—	4	8	no	—
227	C₂²×C₅₃	C₂²×#41	1	4	yes	—
228	C₂²×C₅₃	C₂²×#41	1	2	yes	—
229	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
230	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
231	C₂×D₅₃	—	2	16	no	—
232	C₂×D₅₃	—	2	32	no	—
233	C₂×D₅₃	—	4	32	no	—
234	C₂×D₅₃	—	8	32	no	—
235	C₄×C₅₃	—	2	4	no	—
236	C₄×C₅₃	—	2	8	no	—
237	C₄×C₅₃	—	2	4	no	—
238	C₄×C₅₃	C₂²×#41	1	4	yes	—
239	C₄×C₅₃	C₂²×#41	1	4	yes	—
240	C₄×C₅₃	C₂×H(5,𝔽₂)	1	8	yes	—
241	Dic₅₃	—	2	16	no	—
242	#3508	—	53	32	no	—
243	C₂×#559	—	53	4	no	—
244	C₂×#550	—	53	4	no	—
245	C₂×#546	—	53	8	no	—
246	C₂×#540	—	53	4	no	—
247	C₂×#535	—	53	2	no	—
248	C₂×#533	—	53	1	no	—
249	C₂×#532	—	53	4	no	—
250	C₂²×#127	—	53	1	no	—
251	C₂×#518	—	53	2	no	—
252	C₂×C₄×#41	D₅₃	1	1	yes	characteristic, complemented
253	C₂×C₅₃	—	2	4	no	—
254	C₂×C₅₃	—	4	4	no	—
255	C₂×C₅₃	#3508	1	4	yes	complemented
256	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
257	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	2	yes	—
258	D₅₃	—	4	16	no	—
259	C₂²×#41	—	53	8	no	—
260	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
261	C₂³×D₄	—	53	2	no	—
262	C₂³×D₄	—	53	4	no	—
263	#559	—	53	64	no	—
264	#550	—	53	64	no	—
265	#546	—	53	128	no	—
266	#540	—	53	64	no	—
267	#535	—	53	32	no	—
268	#533	—	53	16	no	—
269	#532	—	53	64	no	—
270	C₂×C₄⋊Q₈	—	53	2	no	—
271	C₂×#127	—	53	4	no	—
272	C₂×#127	—	53	2	no	—
273	C₂×#127	—	53	2	no	—
274	C₂×#127	—	53	4	no	—
275	C₂×#127	—	53	16	no	—
276	C₂×#125	—	53	2	no	—
277	C₂×#124	—	53	4	no	—
278	C₂×#124	—	53	8	no	—
279	C₂×#123	—	53	16	no	—
280	C₂×C₂²⋊Q₈	—	53	4	no	—
281	C₂×#121	—	53	8	no	—
282	C₂×#121	—	53	8	no	—
283	C₂×#121	—	53	16	no	—
284	C₂×#120	—	53	8	no	—
285	#518	—	53	32	no	—
286	C₄×#41	C₂×D₅₃	1	16	yes	complemented
287	C₂×C₄×Q₈	—	53	4	no	—
288	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	complemented
289	C₂×C₄×D₄	—	53	8	no	—
290	C₂×C₄×D₄	—	53	8	no	—
291	C₂×C₄×D₄	—	53	8	no	—
292	C₂×C₄×D₄	C₂×D₅₃	1	4	yes	—
293	C₂×C₄×D₄	C₂×D₅₃	1	2	yes	complemented
294	C₂×#117	—	53	2	no	—
295	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
296	C₂×#117	C₂×D₅₃	1	2	yes	complemented
297	C₂²×C₂²⋊C₄	—	53	8	no	—
298	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
299	C₂²×C₄²	C₂×D₅₃	1	2	yes	complemented
300	C₅₃	C₂×#3508	1	1	yes	characteristic, complemented, Sylow
301	C₂⁵	—	53	4	no	—
302	C₂×#41	—	2	8	no	—
303	C₂×#41	—	53	64	no	—
304	C₂×#41	—	212	32	no	—
305	C₂×#41	C₂²×D₅₃	1	8	yes	—
306	C₂²×Q₈	—	53	4	no	—
307	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
308	C₂²×D₄	—	53	16	no	—
309	C₂²×D₄	—	53	4	no	—
310	C₂²×D₄	—	53	4	no	—
311	C₂²×D₄	—	53	8	no	—
312	C₂²×D₄	—	53	16	no	—
313	C₂²×D₄	—	53	8	no	—
314	C₂²×D₄	—	53	32	no	—
315	C₂²×D₄	—	106	16	no	—
316	C₂²×D₄	—	212	16	no	—
317	C₂²×D₄	C₂²×D₅₃	1	2	yes	—
318	C₂²×D₄	C₂²×D₅₃	1	1	yes	characteristic
319	C₂³×C₄	—	53	16	no	—
320	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
321	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
322	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
323	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
324	C₄⋊Q₈	—	53	16	no	—
325	#127	—	53	32	no	—
326	#127	—	53	16	no	—
327	#127	—	53	16	no	—
328	#127	—	53	32	no	—
329	#125	—	53	16	no	—
330	#124	—	53	32	no	—
331	#124	—	53	64	no	—
332	#123	—	53	128	no	—
333	C₂²⋊Q₈	—	53	32	no	—
334	#121	—	53	64	no	—
335	#121	—	53	64	no	—
336	#121	—	53	128	no	—
337	#120	—	53	64	no	—
338	C₄×Q₈	—	53	32	no	—
339	C₄×Q₈	C₂²×D₅₃	1	16	yes	complemented
340	C₄×D₄	—	53	64	no	—
341	C₄×D₄	—	53	64	no	—
342	C₄×D₄	—	53	64	no	—
343	C₄×D₄	C₂²×D₅₃	1	32	yes	—
344	C₄×D₄	C₂²×D₅₃	1	16	yes	complemented
345	#117	—	53	16	no	—
346	#117	C₂²×D₅₃	1	8	yes	—
347	#117	C₂²×D₅₃	1	16	yes	complemented
348	C₂×C₄⋊C₄	—	53	4	no	—
349	C₂×C₄⋊C₄	—	53	4	no	—
350	C₂×C₄⋊C₄	—	53	8	no	—
351	C₂×C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
352	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
353	C₂×C₂²⋊C₄	—	53	4	no	—
354	C₂×C₂²⋊C₄	—	53	16	no	—
355	C₂×C₂²⋊C₄	—	53	16	no	—
356	C₂×C₂²⋊C₄	—	53	64	no	—
357	C₂×C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
358	C₂×C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
359	C₂×C₄²	—	53	4	no	—
360	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
361	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
362	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
363	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
364	C₂×C₄²	C₂²×D₅₃	1	8	yes	—
365	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
366	C₂×C₄²	C₂²×D₅₃	1	16	yes	complemented
367	C₂⁴	—	53	8	no	—
368	C₂⁴	—	53	16	no	—
369	C₂⁴	—	106	16	no	—
370	C₂⁴	—	212	16	no	—
371	C₂⁴	C₂³×D₅₃	1	2	yes	—
372	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
373	#41	—	2	32	no	—
374	#41	—	212	128	no	—
375	C₂×Q₈	—	2	4	no	—
376	C₂×Q₈	—	53	16	no	—
377	C₂×Q₈	—	106	16	no	—
378	C₂×Q₈	C₂³×D₅₃	1	4	yes	—
379	C₂×D₄	—	2	8	no	—
380	C₂×D₄	—	2	4	no	—
381	C₂×D₄	—	53	64	no	—
382	C₂×D₄	—	53	16	no	—
383	C₂×D₄	—	53	16	no	—
384	C₂×D₄	—	53	32	no	—
385	C₂×D₄	—	53	32	no	—
386	C₂×D₄	—	106	16	no	—
387	C₂×D₄	—	106	16	no	—
388	C₂×D₄	—	106	64	no	—
389	C₂×D₄	—	106	32	no	—
390	C₂×D₄	—	212	64	no	—
391	C₂×D₄	—	212	16	no	—
392	C₂×D₄	—	212	32	no	—
393	C₂×D₄	C₂³×D₅₃	1	8	yes	—
394	C₂×D₄	C₂³×D₅₃	1	4	yes	—
395	C₂²×C₄	—	2	8	no	—
396	C₂²×C₄	—	2	8	no	—
397	C₂²×C₄	—	2	16	no	—
398	C₂²×C₄	—	53	8	no	—
399	C₂²×C₄	—	53	64	no	—
400	C₂²×C₄	—	212	32	no	—
401	C₂²×C₄	#11719945	1	8	yes	complemented
402	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
403	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
404	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
405	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
406	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
407	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
408	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
409	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
410	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
411	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
412	C₄⋊C₄	—	53	16	no	—
413	C₄⋊C₄	—	53	16	no	—
414	C₄⋊C₄	—	53	32	no	—
415	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
416	C₄⋊C₄	C₂³×D₅₃	1	16	yes	—
417	C₂²⋊C₄	—	53	16	no	—
418	C₂²⋊C₄	—	53	64	no	—
419	C₂²⋊C₄	—	53	64	no	—
420	C₂²⋊C₄	C₂³×D₅₃	1	8	yes	—
421	C₂²⋊C₄	C₂³×D₅₃	1	16	yes	—
422	C₄²	—	53	16	no	—
423	C₄²	C₂³×D₅₃	1	4	yes	—
424	C₄²	C₂³×D₅₃	1	4	yes	—
425	C₄²	C₂³×D₅₃	1	8	yes	—
426	C₄²	C₂³×D₅₃	1	8	yes	—
427	C₄²	C₂³×D₅₃	1	16	yes	—
428	C₂³	—	2	4	no	—
429	C₂³	—	2	4	no	—
430	C₂³	—	2	8	no	—
431	C₂³	—	53	16	no	—
432	C₂³	—	106	32	no	—
433	C₂³	—	106	16	no	—
434	C₂³	—	106	32	no	—
435	C₂³	—	424	32	no	—
436	C₂³	—	212	32	no	—
437	C₂³	C₂×#11719945	1	4	yes	—
438	C₂³	C₂⁴×D₅₃	1	4	yes	—
439	C₂³	C₂⁴×D₅₃	1	2	yes	—
440	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
441	Q₈	—	2	8	no	—
442	Q₈	—	106	32	no	—
443	D₄	—	2	16	no	—
444	D₄	—	2	8	no	—
445	D₄	—	106	32	no	—
446	D₄	—	106	32	no	—
447	D₄	—	106	64	no	—
448	D₄	—	212	32	no	—
449	D₄	—	212	64	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	8	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	16	no	—
454	C₂×C₄	—	2	16	no	—
455	C₂×C₄	—	2	32	no	—
456	C₂×C₄	—	53	16	no	—
457	C₂×C₄	—	106	16	no	—
458	C₂×C₄	—	212	64	no	—
459	C₂×C₄	C₂×#11719945	1	4	yes	—
460	C₂×C₄	C₂×#11719945	1	4	yes	—
461	C₂×C₄	C₂×#11719945	1	16	yes	complemented
462	C₂×C₄	#13133263	1	8	yes	—
463	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
464	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
465	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
466	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
467	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
468	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
469	C₂²	—	2	8	no	—
470	C₂²	—	2	4	no	—
471	C₂²	—	2	4	no	—
472	C₂²	—	2	8	no	—
473	C₂²	—	4	8	no	—
474	C₂²	—	106	16	no	—
475	C₂²	—	106	32	no	—
476	C₂²	—	424	32	no	—
477	C₂²	—	212	32	no	—
478	C₂²	C₂²×#11719945	1	4	yes	—
479	C₂²	C₂²×#11719945	1	2	yes	—
480	C₂²	C₂×#13133263	1	4	yes	—
481	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
482	C₄	—	2	4	no	—
483	C₄	—	2	8	no	—
484	C₄	—	2	4	no	—
485	C₄	—	106	16	no	—
486	C₄	C₂²×#11719945	1	4	yes	—
487	C₄	C₂²×#11719945	1	4	yes	—
488	C₄	C₂×#13133263	1	8	yes	—
489	C₂	—	2	4	no	—
490	C₂	—	4	4	no	—
491	C₂	—	212	16	no	—
492	C₂	#20288591	1	4	yes	direct product factor
493	C₂	C₂³×#11719945	1	1	yes	characteristic
494	C₂	C₂²×#13133263	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×C₄×#41×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×C₄×#41	D₅₃
hypocenter	C₅₃	C₂×#3508
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41

Complements

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

N	H ≅ G/N	type
1. #20288591	C₂	direct (N × H)
2. C₂×#16811358	C₂	semidirect
3. C₂×#16811294	C₂	semidirect
4. C₂×#16811269	C₂	semidirect
5. C₂×#16811228	C₂	semidirect
6. C₂×#16811187	C₂	semidirect
7. C₂×#16811180	C₂	semidirect
8. C₂×#16811173	C₂	semidirect
9. C₂²×#13133214	C₂	semidirect
10. C₂×#16811097	C₂	semidirect
11. C₂×C₄×#41×C₅₃	C₂	semidirect
12. C₂²×#11719945	C₂²	semidirect
15. #16811358	C₂²	semidirect
16. #16811294	C₂²	semidirect
17. #16811269	C₂²	semidirect
18. #16811228	C₂²	semidirect
19. #16811187	C₂²	semidirect
20. #16811180	C₂²	semidirect
21. #16811173	C₂²	semidirect
22. C₂×#13133217	C₂²	semidirect
26. C₂×#13133214	C₂²	semidirect
27. C₂×#13133213	C₂²	semidirect
28. C₂×#13133209	C₂²	semidirect
29. C₂×#13133206	C₂²	semidirect
30. C₂×#13133205	C₂²	semidirect
31. C₂×#13133202	C₂²	semidirect
33. C₂×#13133191	C₂²	semidirect
34. C₂×#13133191	C₂²	semidirect
35. C₂×#13133190	C₂²	semidirect
37. #16811097	C₂²	semidirect
38. C₂×C₄×Dic₁₀₆	C₂²	semidirect
39. C₂×C₄×D₂₁₂	C₂²	semidirect
40. C₂×C₄×D₂₁₂	C₂²	semidirect
41. C₂×#13133179	C₂²	semidirect
45. C₄×#41×C₅₃	C₂²	semidirect
46. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
48. C₂×C₄×D₄×C₅₃	C₂²	semidirect
50. C₂×#117×C₅₃	C₂²	semidirect
52. C₂²×C₄²×C₅₃	C₂²	semidirect
55. C₂×#11719945	C₂³	semidirect
67. #13133217	C₂³	semidirect
71. #13133213	C₂³	semidirect
72. #13133209	C₂³	semidirect
73. #13133206	C₂³	semidirect
74. #13133205	C₂³	semidirect
75. #13133202	C₂³	semidirect
77. #13133191	C₂³	semidirect
78. #13133191	C₂³	semidirect
79. #13133190	C₂³	semidirect
81. C₄×Dic₁₀₆	C₂³	semidirect
82. C₄×D₂₁₂	C₂³	semidirect
83. C₄×D₂₁₂	C₂³	semidirect
84. #13133179	C₂³	semidirect
103. C₄×Q₈×C₅₃	C₂³	semidirect
105. C₄×D₄×C₅₃	C₂³	semidirect
107. #117×C₅₃	C₂³	semidirect
118. C₂×C₄²×C₅₃	C₂³	semidirect
135. C₂²×C₄×C₅₃	#41	semidirect
204. C₂×C₄×C₅₃	C₂×#41	semidirect
252. C₂×C₄×#41	D₅₃	semidirect
255. C₂×C₅₃	#3508	semidirect
286. C₄×#41	C₂×D₅₃	semidirect
288. C₂×C₄×Q₈	C₂×D₅₃	semidirect
293. C₂×C₄×D₄	C₂×D₅₃	semidirect
296. C₂×#117	C₂×D₅₃	semidirect
299. C₂²×C₄²	C₂×D₅₃	semidirect
300. C₅₃	C₂×#3508	semidirect
339. C₄×Q₈	C₂²×D₅₃	semidirect
344. C₄×D₄	C₂²×D₅₃	semidirect
347. #117	C₂²×D₅₃	semidirect
366. C₂×C₄²	C₂²×D₅₃	semidirect
401. C₂²×C₄	#11719945	semidirect
461. C₂×C₄	C₂×#11719945	semidirect
492. C₂	#20288591	direct (N × H)

Maximal subgroups

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

subgroup	index
#20288591	2
C₂×#16811358	2
C₂×#16811294	2
C₂×#16811269	2
C₂×#16811228	2
C₂×#16811187	2
C₂×#16811180	2
C₂×#16811173	2
C₂²×#13133214	2
C₂×#16811097	2
C₂×C₄×#41×C₅₃	2
C₂×#3508	53

Subgroups — C2×#20288591 #28292452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288591 #28292452