Subgroups — C₂×#20288351 #28291976

Order 13568.

14438 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133262	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#20288351	C₂	1	32	yes	maximal, direct product factor
4	C₂×#16811268	C₂	1	8	yes	maximal, complemented
5	C₂×D₄×D₂₁₂	C₂	1	8	yes	maximal, complemented
6	C₂×#16811192	C₂	1	2	yes	maximal, complemented
7	C₂×#16811185	C₂	1	4	yes	maximal, complemented
8	C₂×#16811135	C₂	1	2	yes	maximal, complemented
9	C₂²×#13133187	C₂	1	4	yes	maximal, complemented
10	C₂×C₅₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵×D₅₃	C₂²	1	2	yes	complemented
12	C₂×#13133262	—	2	16	no	—
13	C₂×#13133262	C₂²	1	16	yes	complemented
14	C₂²×#11719945	C₂²	1	2	yes	complemented
15	C₂²×#11719947	C₂²	1	4	yes	complemented
16	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
17	C₂³×D₂₁₂	C₂²	1	8	yes	complemented
18	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
19	C₂³×C₂²⋊D₅₃	C₂²	1	4	yes	complemented
20	C₂²×D₄×D₅₃	—	2	16	no	—
21	C₂²×D₄×D₅₃	C₂²	1	8	yes	complemented
22	C₂²×D₄×D₅₃	C₂²	1	16	yes	complemented
23	C₂²×D₄×D₅₃	C₂²	1	4	yes	complemented
24	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
25	#16811268	C₂²	1	128	yes	complemented
26	D₄×D₂₁₂	C₂²	1	128	yes	complemented
27	#16811192	C₂²	1	32	yes	complemented
28	#16811185	C₂²	1	64	yes	complemented
29	C₂×#13133214	C₂²	1	4	yes	complemented
30	C₂×#13133205	C₂²	1	4	yes	complemented
31	#16811135	C₂²	1	32	yes	complemented
32	C₂×#13133201	C₂²	1	4	yes	complemented
33	C₂×#13133201	C₂²	1	8	yes	complemented
34	C₂×#13133197	C₂²	1	4	yes	complemented
35	C₂×#13133192	C₂²	1	4	yes	complemented
36	C₂×#13133192	C₂²	1	8	yes	complemented
37	C₂×#13133191	C₂²	1	16	yes	complemented
38	C₂×#13133187	C₂²	1	4	yes	complemented
39	C₂×#13133187	C₂²	1	8	yes	complemented
40	C₂×#13133187	C₂²	1	16	yes	complemented
41	C₂×#13133187	C₂²	1	8	yes	complemented
42	C₂×#13133187	C₂²	1	64	yes	complemented
43	C₂×C₄×D₂₁₂	C₂²	1	8	yes	complemented
44	C₂²×#11719921	C₂²	1	8	yes	complemented
45	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
46	C₅₃×#518	C₂²	1	16	yes	complemented
47	C₂×C₄×D₄×C₅₃	C₂²	1	8	yes	complemented
48	C₂×#117×C₅₃	C₂²	1	2	yes	complemented
49	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
50	C₂⁴×D₅₃	—	2	16	no	—
51	C₂⁴×D₅₃	—	4	16	no	—
52	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
53	C₂⁴×D₅₃	C₂³	1	8	yes	—
54	C₂⁴×D₅₃	C₂³	1	2	yes	—
55	C₂⁴×D₅₃	C₂³	1	16	yes	complemented
56	C₂⁴×D₅₃	C₂³	1	4	yes	complemented
57	#13133262	—	2	128	no	—
58	C₂×#11719945	—	2	16	no	—
59	C₂×#11719945	C₂³	1	16	yes	complemented
60	C₂×#11719947	—	2	32	no	—
61	C₂×#11719947	C₂³	1	32	yes	complemented
62	C₂²×Dic₁₀₆	C₂³	1	2	yes	complemented
63	C₂²×D₂₁₂	—	2	8	no	—
64	C₂²×D₂₁₂	—	4	32	no	—
65	C₂²×D₂₁₂	C₂³	1	8	yes	—
66	C₂²×D₂₁₂	C₂³	1	4	yes	—
67	C₂²×D₂₁₂	C₂³	1	8	yes	complemented
68	C₂²×D₂₁₂	C₂³	1	2	yes	complemented
69	C₂²×D₂₁₂	C₂³	1	64	yes	complemented
70	C₂²×D₂₁₂	C₂³	1	16	yes	complemented
71	C₂²×C₂²⋊D₅₃	—	2	16	no	—
72	C₂²×C₂²⋊D₅₃	—	2	32	no	—
73	C₂²×C₂²⋊D₅₃	C₂³	1	4	yes	—
74	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
75	C₂²×C₂²⋊D₅₃	C₂³	1	16	yes	complemented
76	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	complemented
77	C₂²×C₂²⋊D₅₃	C₂³	1	32	yes	complemented
78	C₂×D₄×D₅₃	—	2	64	no	—
79	C₂×D₄×D₅₃	—	2	128	no	—
80	C₂×D₄×D₅₃	—	2	32	no	—
81	C₂×D₄×D₅₃	C₂³	1	64	yes	complemented
82	C₂×D₄×D₅₃	C₂³	1	32	yes	complemented
83	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
84	C₂³×Dic₅₃	C₂³	1	4	yes	complemented
85	C₂²×C₄×D₅₃	—	2	8	no	—
86	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
87	C₂²×C₄×D₅₃	C₂³	1	8	yes	complemented
88	C₂²×C₄×D₅₃	C₂³	1	4	yes	complemented
89	#13133214	C₂³	1	32	yes	complemented
90	#13133205	C₂³	1	32	yes	complemented
91	#13133201	C₂³	1	32	yes	complemented
92	#13133201	C₂³	1	64	yes	complemented
93	#13133197	C₂³	1	32	yes	complemented
94	#13133192	C₂³	1	32	yes	complemented
95	#13133192	C₂³	1	64	yes	complemented
96	#13133191	C₂³	1	128	yes	complemented
97	#13133187	C₂³	1	32	yes	complemented
98	#13133187	C₂³	1	128	yes	complemented
99	#13133187	C₂³	1	64	yes	complemented
100	#13133187	C₂³	1	64	yes	complemented
101	C₄×D₂₁₂	C₂³	1	64	yes	complemented
102	C₂×C₄⋊Dic₅₃	C₂³	1	8	yes	complemented
103	C₂×#11719921	C₂³	1	8	yes	—
104	C₂×#11719921	C₂³	1	16	yes	complemented
105	C₂×#11719921	C₂³	1	8	yes	complemented
106	C₂×#11719921	C₂³	1	64	yes	complemented
107	C₂⁵×C₅₃	C₂³	1	2	yes	—
108	C₂²×D₄×C₅₃	D₄	1	16	yes	complemented
109	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
110	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
111	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
112	C₂³×C₄×C₅₃	C₂³	1	8	yes	complemented
113	C₄×D₄×C₅₃	C₂³	1	64	yes	complemented
114	#117×C₅₃	C₂³	1	16	yes	complemented
115	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	complemented
116	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
117	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	complemented
118	C₂×C₂²⋊C₄×C₅₃	C₂³	1	32	yes	complemented
119	C₂×C₄²×C₅₃	C₂³	1	4	yes	complemented
120	C₂⁴×C₅₃	—	4	8	no	—
121	C₂⁴×C₅₃	C₂×D₄	1	16	yes	complemented
122	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
123	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
124	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
125	C₂³×D₅₃	—	2	4	no	—
126	C₂³×D₅₃	—	2	8	no	—
127	C₂³×D₅₃	—	2	32	no	—
128	C₂³×D₅₃	—	2	8	no	—
129	C₂³×D₅₃	—	2	64	no	—
130	C₂³×D₅₃	—	2	16	no	—
131	C₂³×D₅₃	—	2	32	no	—
132	C₂³×D₅₃	—	4	64	no	—
133	C₂³×D₅₃	—	4	16	no	—
134	C₂³×D₅₃	—	4	8	no	—
135	C₂³×D₅₃	—	4	16	no	—
136	C₂³×D₅₃	—	4	128	no	—
137	C₂³×D₅₃	C₂⁴	1	4	yes	—
138	C₂³×D₅₃	C₂⁴	1	4	yes	—
139	C₂³×D₅₃	C₂⁴	1	4	yes	—
140	C₂³×D₅₃	C₂⁴	1	32	yes	—
141	C₂³×D₅₃	C₂⁴	1	2	yes	—
142	C₂³×D₅₃	C₂⁴	1	8	yes	—
143	C₂³×D₅₃	C₂⁴	1	16	yes	complemented
144	C₂×D₄×C₅₃	C₂×D₄	1	32	yes	complemented
145	C₂×D₄×C₅₃	C₂×D₄	1	64	yes	complemented
146	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
147	C₂²×C₄×C₅₃	—	4	16	no	—
148	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	complemented
149	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
150	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	complemented
153	C₂×C₂²⋊D₅₃	—	2	32	no	—
154	C₂×C₂²⋊D₅₃	—	2	32	no	—
155	C₂×C₂²⋊D₅₃	—	2	16	no	—
156	C₂×C₂²⋊D₅₃	—	2	64	no	—
157	C₂×C₂²⋊D₅₃	—	2	128	no	—
158	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
159	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
160	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
161	C₂²×Dic₅₃	—	2	4	no	—
162	C₂²×Dic₅₃	—	2	16	no	—
163	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
164	C₂²×Dic₅₃	C₂⁴	1	2	yes	—
165	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
166	C₂²×Dic₅₃	C₂⁴	1	16	yes	complemented
167	#11719947	—	2	128	no	—
168	D₄×D₅₃	—	2	256	no	—
169	D₄×D₅₃	—	2	128	no	—
170	#11719945	—	2	64	no	—
171	C₂×D₂₁₂	—	2	32	no	—
172	C₂×D₂₁₂	—	2	16	no	—
173	C₂×D₂₁₂	—	2	8	no	—
174	C₂×D₂₁₂	—	2	32	no	—
175	C₂×D₂₁₂	—	4	128	no	—
176	C₂×D₂₁₂	—	4	32	no	—
177	C₂×D₂₁₂	C₂⁴	1	32	yes	—
178	C₂×D₂₁₂	C₂⁴	1	16	yes	—
179	C₂×D₂₁₂	C₂⁴	1	8	yes	complemented
180	C₂×D₂₁₂	C₂⁴	1	64	yes	complemented
181	C₂×C₄×D₅₃	—	2	16	no	—
182	C₂×C₄×D₅₃	—	2	16	no	—
183	C₂×C₄×D₅₃	—	2	32	no	—
184	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
185	C₂×C₄×D₅₃	C₂⁴	1	16	yes	complemented
186	C₂×Dic₁₀₆	—	2	8	no	—
187	C₂×Dic₁₀₆	C₂⁴	1	8	yes	complemented
188	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
189	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
190	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
191	C₄²×C₅₃	C₂⁴	1	16	yes	complemented
192	#11719921	C₂⁴	1	32	yes	—
193	#11719921	C₂⁴	1	64	yes	complemented
194	#11719921	C₂⁴	1	32	yes	complemented
195	C₄⋊Dic₅₃	C₂⁴	1	32	yes	complemented
196	C₂³×C₅₃	—	2	16	no	—
197	C₂³×C₅₃	—	2	4	no	—
198	C₂³×C₅₃	—	4	32	no	—
199	C₂³×C₅₃	—	4	16	no	—
200	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
201	C₂³×C₅₃	C₂²×D₄	1	16	yes	—
202	C₂³×C₅₃	C₂²×D₄	1	32	yes	complemented
203	C₂³×C₅₃	C₂⁵	1	2	yes	—
204	C₂³×C₅₃	C₂⁵	1	8	yes	—
205	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
206	C₂²×D₅₃	—	2	32	no	—
207	C₂²×D₅₃	—	2	8	no	—
208	C₂²×D₅₃	—	2	4	no	—
209	C₂²×D₅₃	—	2	4	no	—
210	C₂²×D₅₃	—	2	16	no	—
211	C₂²×D₅₃	—	2	8	no	—
212	C₂²×D₅₃	—	2	16	no	—
213	C₂²×D₅₃	—	2	8	no	—
214	C₂²×D₅₃	—	2	64	no	—
215	C₂²×D₅₃	—	2	16	no	—
216	C₂²×D₅₃	—	2	16	no	—
217	C₂²×D₅₃	—	2	32	no	—
218	C₂²×D₅₃	—	2	64	no	—
219	C₂²×D₅₃	—	4	32	no	—
220	C₂²×D₅₃	—	4	16	no	—
221	C₂²×D₅₃	—	4	32	no	—
222	C₂²×D₅₃	—	4	128	no	—
223	C₂²×D₅₃	—	4	256	no	—
224	C₂²×D₅₃	—	4	32	no	—
225	C₂²×D₅₃	—	8	32	no	—
226	C₂²×D₅₃	H(5,𝔽₂)	1	8	yes	complemented
227	C₂²×D₅₃	C₂⁵	1	8	yes	—
228	C₂²×D₅₃	C₂⁵	1	8	yes	—
229	C₂²×D₅₃	C₂⁵	1	4	yes	—
230	D₄×C₅₃	C₂²×D₄	1	64	yes	complemented
231	C₂×C₄×C₅₃	—	2	16	no	—
232	C₂×C₄×C₅₃	—	4	32	no	—
233	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
234	C₂×C₄×C₅₃	C₂²×D₄	1	16	yes	complemented
235	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
236	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
237	C₂²⋊D₅₃	—	2	32	no	—
238	C₂²⋊D₅₃	—	2	64	no	—
239	C₂²⋊D₅₃	—	2	64	no	—
240	C₂×Dic₅₃	—	2	8	no	—
241	C₂×Dic₅₃	—	2	8	no	—
242	C₂×Dic₅₃	—	2	4	no	—
243	C₂×Dic₅₃	—	2	32	no	—
244	C₂×Dic₅₃	C₂⁵	1	8	yes	—
245	C₂×Dic₅₃	C₂⁵	1	4	yes	—
246	D₂₁₂	—	2	64	no	—
247	D₂₁₂	—	2	32	no	—
248	D₂₁₂	—	2	16	no	—
249	D₂₁₂	—	4	64	no	—
250	C₄×D₅₃	—	2	32	no	—
251	C₄×D₅₃	—	2	32	no	—
252	Dic₁₀₆	—	2	16	no	—
253	C₂×#3484	—	53	1	no	maximal, Sylow
254	C₂²×C₅₃	—	2	16	no	—
255	C₂²×C₅₃	—	2	4	no	—
256	C₂²×C₅₃	—	2	8	no	—
257	C₂²×C₅₃	—	2	32	no	—
258	C₂²×C₅₃	—	4	32	no	—
259	C₂²×C₅₃	C₂³×D₄	1	4	yes	—
260	C₂²×C₅₃	C₂³×D₄	1	16	yes	—
261	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
262	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
263	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
264	C₂×D₅₃	—	2	32	no	—
265	C₂×D₅₃	—	2	4	no	—
266	C₂×D₅₃	—	2	4	no	—
267	C₂×D₅₃	—	2	16	no	—
268	C₂×D₅₃	—	2	8	no	—
269	C₂×D₅₃	—	2	8	no	—
270	C₂×D₅₃	—	2	16	no	—
271	C₂×D₅₃	—	2	16	no	—
272	C₂×D₅₃	—	4	128	no	—
273	C₂×D₅₃	—	4	32	no	—
274	C₂×D₅₃	—	4	16	no	—
275	C₂×D₅₃	—	4	16	no	—
276	C₂×D₅₃	—	8	32	no	—
277	C₂×D₅₃	C₂×H(5,𝔽₂)	1	8	yes	complemented
278	C₄×C₅₃	—	2	16	no	—
279	C₄×C₅₃	C₂³×D₄	1	8	yes	—
280	Dic₅₃	—	2	8	no	—
281	Dic₅₃	—	2	4	no	—
282	C₂²×H(5,𝔽₂)	—	53	1	no	—
283	C₂⁴×D₄	—	53	1	no	—
284	#3484	—	53	32	no	—
285	C₂×#546	—	53	8	no	—
286	C₂×D₄²	—	53	8	no	—
287	C₂×#535	—	53	2	no	—
288	C₂×#534	—	53	4	no	—
289	C₂×#525	—	53	2	no	—
290	C₂²×#120	—	53	4	no	—
291	C₂×#518	D₅₃	1	1	yes	characteristic, complemented
292	C₂×C₅₃	—	2	4	no	—
293	C₂×C₅₃	—	2	16	no	—
294	C₂×C₅₃	#3484	1	4	yes	complemented
295	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
296	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	2	yes	—
297	D₅₃	—	2	8	no	—
298	D₅₃	—	4	8	no	—
299	D₅₃	—	4	8	no	—
300	C₂⁶	—	53	2	no	—
301	C₂×H(5,𝔽₂)	—	53	16	no	—
302	C₂×H(5,𝔽₂)	—	106	16	no	—
303	C₂²×#41	—	53	4	no	—
304	C₂²×#41	—	53	2	no	—
305	C₂³×D₄	—	53	2	no	—
306	C₂³×D₄	—	53	4	no	—
307	C₂³×D₄	—	53	8	no	—
308	C₂³×D₄	—	53	16	no	—
309	C₂³×D₄	—	53	4	no	—
310	C₂³×D₄	—	53	1	no	—
311	C₂³×D₄	—	53	8	no	—
312	C₂³×D₄	—	106	16	no	—
313	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
314	C₂⁴×C₄	—	53	1	no	—
315	#546	—	53	128	no	—
316	D₄²	—	53	128	no	—
317	#535	—	53	32	no	—
318	#534	—	53	64	no	—
319	C₂×#127	—	53	4	no	—
320	C₂×#124	—	53	4	no	—
321	#525	—	53	32	no	—
322	C₂×#123	—	53	4	no	—
323	C₂×#123	—	53	8	no	—
324	C₂×C₂²⋊Q₈	—	53	4	no	—
325	C₂×#121	—	53	16	no	—
326	C₂×#121	—	53	4	no	—
327	C₂×#121	—	53	8	no	—
328	C₂×#120	—	53	4	no	—
329	C₂×#120	—	53	8	no	—
330	C₂×#120	—	53	16	no	—
331	C₂×#120	—	53	64	no	—
332	C₂×#120	—	53	8	no	—
333	#518	C₂×D₅₃	1	16	yes	complemented
334	C₂×C₄×D₄	—	53	8	no	—
335	C₂×C₄×D₄	C₂×D₅₃	1	8	yes	complemented
336	C₂×#117	C₂×D₅₃	1	2	yes	complemented
337	C₂²×C₂²⋊C₄	—	53	8	no	—
338	C₂²×C₂²⋊C₄	C₂×D₅₃	1	4	yes	complemented
339	C₅₃	C₂×#3484	1	1	yes	characteristic, complemented, Sylow
340	C₂⁵	—	53	1	no	—
341	C₂⁵	—	53	8	no	—
342	C₂⁵	—	53	2	no	—
343	C₂⁵	—	53	16	no	—
344	C₂⁵	—	53	4	no	—
345	C₂⁵	—	106	16	no	—
346	C₂⁵	—	212	16	no	—
347	C₂⁵	C₂²×D₅₃	1	2	yes	—
348	H(5,𝔽₂)	—	106	128	no	—
349	C₂×#41	—	53	32	no	—
350	C₂×#41	—	53	16	no	—
351	C₂×#41	—	106	32	no	—
352	C₂×#41	—	106	16	no	—
353	C₂²×Q₈	—	53	2	no	—
354	C₂²×D₄	—	53	4	no	—
355	C₂²×D₄	—	53	8	no	—
356	C₂²×D₄	—	53	32	no	—
357	C₂²×D₄	—	53	8	no	—
358	C₂²×D₄	—	53	16	no	—
359	C₂²×D₄	—	53	32	no	—
360	C₂²×D₄	—	53	64	no	—
361	C₂²×D₄	—	53	8	no	—
362	C₂²×D₄	—	53	4	no	—
363	C₂²×D₄	—	53	2	no	—
364	C₂²×D₄	—	53	8	no	—
365	C₂²×D₄	—	53	16	no	—
366	C₂²×D₄	—	53	64	no	—
367	C₂²×D₄	—	106	16	no	—
368	C₂²×D₄	—	106	32	no	—
369	C₂²×D₄	—	106	64	no	—
370	C₂²×D₄	—	106	128	no	—
371	C₂²×D₄	—	106	32	no	—
372	C₂²×D₄	—	106	8	no	—
373	C₂²×D₄	—	212	32	no	—
374	C₂²×D₄	D₂₁₂	1	16	yes	complemented
375	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
376	C₂²×D₄	C₂²×D₅₃	1	8	yes	—
377	C₂³×C₄	—	53	4	no	—
378	C₂³×C₄	—	53	1	no	—
379	C₂³×C₄	—	53	4	no	—
380	C₂³×C₄	—	53	4	no	—
381	C₂³×C₄	—	53	8	no	—
382	C₂³×C₄	—	106	8	no	—
383	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
384	C₂³×C₄	C₂²×D₅₃	1	8	yes	complemented
385	#127	—	53	32	no	—
386	#124	—	53	32	no	—
387	#123	—	53	32	no	—
388	#123	—	53	64	no	—
389	C₂²⋊Q₈	—	53	32	no	—
390	#121	—	53	128	no	—
391	#121	—	53	32	no	—
392	#121	—	53	64	no	—
393	#120	—	53	32	no	—
394	#120	—	53	64	no	—
395	#120	—	53	128	no	—
396	#120	—	53	64	no	—
397	C₄×D₄	—	53	64	no	—
398	C₄×D₄	C₂²×D₅₃	1	64	yes	complemented
399	#117	C₂²×D₅₃	1	16	yes	complemented
400	C₂×C₄⋊C₄	—	53	8	no	—
401	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	complemented
402	C₂×C₂²⋊C₄	—	53	8	no	—
403	C₂×C₂²⋊C₄	—	53	16	no	—
404	C₂×C₂²⋊C₄	—	53	8	no	—
405	C₂×C₂²⋊C₄	—	53	64	no	—
406	C₂×C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
407	C₂×C₂²⋊C₄	C₂²×D₅₃	1	32	yes	complemented
408	C₂×C₂²⋊C₄	C₂²×D₅₃	1	8	yes	complemented
409	C₂×C₄²	C₂²×D₅₃	1	4	yes	complemented
410	C₂⁴	—	4	8	no	—
411	C₂⁴	—	53	2	no	—
412	C₂⁴	—	53	4	no	—
413	C₂⁴	—	53	4	no	—
414	C₂⁴	—	53	4	no	—
415	C₂⁴	—	53	32	no	—
416	C₂⁴	—	53	8	no	—
417	C₂⁴	—	53	16	no	—
418	C₂⁴	—	106	8	no	—
419	C₂⁴	—	106	32	no	—
420	C₂⁴	—	106	4	no	—
421	C₂⁴	—	106	16	no	—
422	C₂⁴	—	106	32	no	—
423	C₂⁴	—	106	8	no	—
424	C₂⁴	—	106	64	no	—
425	C₂⁴	—	212	16	no	—
426	C₂⁴	—	212	16	no	—
427	C₂⁴	—	212	8	no	—
428	C₂⁴	—	212	64	no	—
429	C₂⁴	—	212	128	no	—
430	C₂⁴	C₂×D₂₁₂	1	16	yes	complemented
431	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
432	C₂⁴	C₂³×D₅₃	1	4	yes	—
433	C₂⁴	C₂³×D₅₃	1	8	yes	—
434	#41	—	106	128	no	—
435	#41	—	106	64	no	—
436	C₂×Q₈	—	53	8	no	—
437	C₂×Q₈	—	106	8	no	—
438	C₂×D₄	—	53	16	no	—
439	C₂×D₄	—	53	32	no	—
440	C₂×D₄	—	53	32	no	—
441	C₂×D₄	—	53	32	no	—
442	C₂×D₄	—	53	16	no	—
443	C₂×D₄	—	53	8	no	—
444	C₂×D₄	—	53	64	no	—
445	C₂×D₄	—	106	32	no	—
446	C₂×D₄	—	106	32	no	—
447	C₂×D₄	—	106	16	no	—
448	C₂×D₄	—	106	32	no	—
449	C₂×D₄	—	106	64	no	—
450	C₂×D₄	—	106	128	no	—
451	C₂×D₄	—	106	256	no	—
452	C₂×D₄	—	106	128	no	—
453	C₂×D₄	—	106	16	no	—
454	C₂×D₄	—	106	8	no	—
455	C₂×D₄	—	106	32	no	—
456	C₂×D₄	—	212	32	no	—
457	C₂×D₄	—	212	128	no	—
458	C₂×D₄	C₂×D₂₁₂	1	32	yes	complemented
459	C₂×D₄	C₂×D₂₁₂	1	64	yes	complemented
460	C₂×D₄	C₂³×D₅₃	1	16	yes	—
461	C₂²×C₄	—	4	16	no	—
462	C₂²×C₄	—	53	4	no	—
463	C₂²×C₄	—	53	16	no	—
464	C₂²×C₄	—	53	4	no	—
465	C₂²×C₄	—	53	2	no	—
466	C₂²×C₄	—	53	16	no	—
467	C₂²×C₄	—	53	16	no	—
468	C₂²×C₄	—	106	4	no	—
469	C₂²×C₄	—	106	16	no	—
470	C₂²×C₄	—	106	16	no	—
471	C₂²×C₄	—	106	16	no	—
472	C₂²×C₄	—	106	32	no	—
473	C₂²×C₄	C₂×D₂₁₂	1	8	yes	complemented
474	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
475	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
476	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
477	C₂²×C₄	C₂³×D₅₃	1	32	yes	complemented
478	C₄⋊C₄	—	53	32	no	—
479	C₄⋊C₄	C₂³×D₅₃	1	16	yes	complemented
480	C₂²⋊C₄	—	53	32	no	—
481	C₂²⋊C₄	—	53	64	no	—
482	C₂²⋊C₄	—	53	32	no	—
483	C₂²⋊C₄	C₂³×D₅₃	1	16	yes	—
484	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	complemented
485	C₄²	C₂³×D₅₃	1	16	yes	complemented
486	C₂³	—	2	16	no	—
487	C₂³	—	2	4	no	—
488	C₂³	—	4	32	no	—
489	C₂³	—	4	16	no	—
490	C₂³	—	53	8	no	—
491	C₂³	—	53	4	no	—
492	C₂³	—	53	8	no	—
493	C₂³	—	53	8	no	—
494	C₂³	—	106	16	no	—
495	C₂³	—	106	64	no	—
496	C₂³	—	106	32	no	—
497	C₂³	—	106	8	no	—
498	C₂³	—	106	4	no	—
499	C₂³	—	106	4	no	—
500	C₂³	—	106	32	no	—
501	C₂³	—	106	8	no	—
502	C₂³	—	106	64	no	—
503	C₂³	—	106	16	no	—
504	C₂³	—	106	8	no	—
505	C₂³	—	106	16	no	—
506	C₂³	—	106	16	no	—
507	C₂³	—	424	32	no	—
508	C₂³	—	212	32	no	—
509	C₂³	—	212	32	no	—
510	C₂³	—	212	16	no	—
511	C₂³	—	212	128	no	—
512	C₂³	—	212	32	no	—
513	C₂³	—	212	256	no	—
514	C₂³	C₂²×D₂₁₂	1	4	yes	—
515	C₂³	C₂²×D₂₁₂	1	16	yes	—
516	C₂³	C₂²×D₂₁₂	1	32	yes	complemented
517	C₂³	C₂⁴×D₅₃	1	2	yes	—
518	C₂³	C₂⁴×D₅₃	1	8	yes	—
519	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
520	Q₈	—	106	16	no	—
521	D₄	—	106	64	no	—
522	D₄	—	106	64	no	—
523	D₄	—	106	32	no	—
524	D₄	—	106	64	no	—
525	D₄	—	106	32	no	—
526	D₄	—	106	16	no	—
527	D₄	—	212	64	no	—
528	D₄	C₂²×D₂₁₂	1	64	yes	complemented
529	C₂×C₄	—	2	16	no	—
530	C₂×C₄	—	4	32	no	—
531	C₂×C₄	—	53	8	no	—
532	C₂×C₄	—	53	4	no	—
533	C₂×C₄	—	106	8	no	—
534	C₂×C₄	—	106	4	no	—
535	C₂×C₄	—	106	8	no	—
536	C₂×C₄	—	106	32	no	—
537	C₂×C₄	—	106	32	no	—
538	C₂×C₄	—	106	32	no	—
539	C₂×C₄	C₂²×D₂₁₂	1	8	yes	—
540	C₂×C₄	C₂²×D₂₁₂	1	16	yes	complemented
541	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
542	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
543	C₂²	—	2	8	no	—
544	C₂²	—	2	32	no	—
545	C₂²	—	2	16	no	—
546	C₂²	—	2	4	no	—
547	C₂²	—	4	32	no	—
548	C₂²	—	53	8	no	—
549	C₂²	—	106	32	no	—
550	C₂²	—	106	16	no	—
551	C₂²	—	106	4	no	—
552	C₂²	—	106	4	no	—
553	C₂²	—	106	8	no	—
554	C₂²	—	106	8	no	—
555	C₂²	—	106	16	no	—
556	C₂²	—	106	16	no	—
557	C₂²	—	424	32	no	—
558	C₂²	—	212	128	no	—
559	C₂²	—	212	32	no	—
560	C₂²	—	212	16	no	—
561	C₂²	—	212	16	no	—
562	C₂²	C₂³×D₂₁₂	1	4	yes	—
563	C₂²	C₂³×D₂₁₂	1	16	yes	—
564	C₂²	C₂³×D₂₁₂	1	2	yes	—
565	C₂²	H(5,𝔽₂)×D₅₃	1	4	yes	—
566	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
567	C₄	—	2	16	no	—
568	C₄	—	106	4	no	—
569	C₄	—	106	8	no	—
570	C₄	C₂³×D₂₁₂	1	8	yes	—
571	C₂	—	2	4	no	—
572	C₂	—	2	16	no	—
573	C₂	—	106	8	no	—
574	C₂	—	212	8	no	—
575	C₂	—	212	8	no	—
576	C₂	#20288351	1	4	yes	direct product factor
577	C₂	C₂⁴×D₂₁₂	1	1	yes	characteristic
578	C₂	C₂×H(5,𝔽₂)×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×C₅₃×#518	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×#518	D₅₃
hypocenter	C₅₃	C₂×#3484
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#518

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133262	C₂	semidirect
2. C₂³×D₄×D₅₃	C₂	semidirect
3. #20288351	C₂	direct (N × H)
4. C₂×#16811268	C₂	semidirect
5. C₂×D₄×D₂₁₂	C₂	semidirect
6. C₂×#16811192	C₂	semidirect
7. C₂×#16811185	C₂	semidirect
8. C₂×#16811135	C₂	semidirect
9. C₂²×#13133187	C₂	semidirect
10. C₂×C₅₃×#518	C₂	semidirect
11. C₂⁵×D₅₃	C₂²	semidirect
13. C₂×#13133262	C₂²	semidirect
14. C₂²×#11719945	C₂²	semidirect
15. C₂²×#11719947	C₂²	semidirect
16. C₂³×D₂₁₂	C₂²	semidirect
17. C₂³×D₂₁₂	C₂²	semidirect
18. C₂³×C₂²⋊D₅₃	C₂²	semidirect
19. C₂³×C₂²⋊D₅₃	C₂²	semidirect
21. C₂²×D₄×D₅₃	C₂²	semidirect
22. C₂²×D₄×D₅₃	C₂²	semidirect
23. C₂²×D₄×D₅₃	C₂²	semidirect
24. C₂³×C₄×D₅₃	C₂²	semidirect
25. #16811268	C₂²	semidirect
26. D₄×D₂₁₂	C₂²	semidirect
27. #16811192	C₂²	semidirect
28. #16811185	C₂²	semidirect
29. C₂×#13133214	C₂²	semidirect
30. C₂×#13133205	C₂²	semidirect
31. #16811135	C₂²	semidirect
32. C₂×#13133201	C₂²	semidirect
33. C₂×#13133201	C₂²	semidirect
34. C₂×#13133197	C₂²	semidirect
35. C₂×#13133192	C₂²	semidirect
36. C₂×#13133192	C₂²	semidirect
37. C₂×#13133191	C₂²	semidirect
38. C₂×#13133187	C₂²	semidirect
39. C₂×#13133187	C₂²	semidirect
40. C₂×#13133187	C₂²	semidirect
41. C₂×#13133187	C₂²	semidirect
42. C₂×#13133187	C₂²	semidirect
43. C₂×C₄×D₂₁₂	C₂²	semidirect
44. C₂²×#11719921	C₂²	semidirect
46. C₅₃×#518	C₂²	semidirect
47. C₂×C₄×D₄×C₅₃	C₂²	semidirect
48. C₂×#117×C₅₃	C₂²	semidirect
49. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
55. C₂⁴×D₅₃	C₂³	semidirect
56. C₂⁴×D₅₃	C₂³	semidirect
59. C₂×#11719945	C₂³	semidirect
61. C₂×#11719947	C₂³	semidirect
62. C₂²×Dic₁₀₆	C₂³	semidirect
67. C₂²×D₂₁₂	C₂³	semidirect
68. C₂²×D₂₁₂	C₂³	semidirect
69. C₂²×D₂₁₂	C₂³	semidirect
70. C₂²×D₂₁₂	C₂³	semidirect
75. C₂²×C₂²⋊D₅₃	C₂³	semidirect
76. C₂²×C₂²⋊D₅₃	C₂³	semidirect
77. C₂²×C₂²⋊D₅₃	C₂³	semidirect
81. C₂×D₄×D₅₃	C₂³	semidirect
82. C₂×D₄×D₅₃	C₂³	semidirect
84. C₂³×Dic₅₃	C₂³	semidirect
87. C₂²×C₄×D₅₃	C₂³	semidirect
88. C₂²×C₄×D₅₃	C₂³	semidirect
89. #13133214	C₂³	semidirect
90. #13133205	C₂³	semidirect
91. #13133201	C₂³	semidirect
92. #13133201	C₂³	semidirect
93. #13133197	C₂³	semidirect
94. #13133192	C₂³	semidirect
95. #13133192	C₂³	semidirect
96. #13133191	C₂³	semidirect
97. #13133187	C₂³	semidirect
98. #13133187	C₂³	semidirect
99. #13133187	C₂³	semidirect
100. #13133187	C₂³	semidirect
101. C₄×D₂₁₂	C₂³	semidirect
102. C₂×C₄⋊Dic₅₃	C₂³	semidirect
104. C₂×#11719921	C₂³	semidirect
105. C₂×#11719921	C₂³	semidirect
106. C₂×#11719921	C₂³	semidirect
108. C₂²×D₄×C₅₃	D₄	semidirect
112. C₂³×C₄×C₅₃	C₂³	semidirect
113. C₄×D₄×C₅₃	C₂³	semidirect
114. #117×C₅₃	C₂³	semidirect
115. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
117. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
118. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
119. C₂×C₄²×C₅₃	C₂³	semidirect
121. C₂⁴×C₅₃	C₂×D₄	semidirect
143. C₂³×D₅₃	C₂⁴	semidirect
144. C₂×D₄×C₅₃	C₂×D₄	semidirect
145. C₂×D₄×C₅₃	C₂×D₄	semidirect
148. C₂²×C₄×C₅₃	C₂×D₄	semidirect
152. C₂²×C₄×C₅₃	C₂⁴	semidirect
160. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
166. C₂²×Dic₅₃	C₂⁴	semidirect
179. C₂×D₂₁₂	C₂⁴	semidirect
180. C₂×D₂₁₂	C₂⁴	semidirect
185. C₂×C₄×D₅₃	C₂⁴	semidirect
187. C₂×Dic₁₀₆	C₂⁴	semidirect
188. C₄⋊C₄×C₅₃	C₂⁴	semidirect
190. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
191. C₄²×C₅₃	C₂⁴	semidirect
193. #11719921	C₂⁴	semidirect
194. #11719921	C₂⁴	semidirect
195. C₄⋊Dic₅₃	C₂⁴	semidirect
202. C₂³×C₅₃	C₂²×D₄	semidirect
226. C₂²×D₅₃	H(5,𝔽₂)	semidirect
230. D₄×C₅₃	C₂²×D₄	semidirect
234. C₂×C₄×C₅₃	C₂²×D₄	semidirect
277. C₂×D₅₃	C₂×H(5,𝔽₂)	semidirect
291. C₂×#518	D₅₃	semidirect
294. C₂×C₅₃	#3484	semidirect
333. #518	C₂×D₅₃	semidirect
335. C₂×C₄×D₄	C₂×D₅₃	semidirect
336. C₂×#117	C₂×D₅₃	semidirect
338. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
339. C₅₃	C₂×#3484	semidirect
374. C₂²×D₄	D₂₁₂	semidirect
384. C₂³×C₄	C₂²×D₅₃	semidirect
398. C₄×D₄	C₂²×D₅₃	semidirect
399. #117	C₂²×D₅₃	semidirect
401. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
407. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
408. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
409. C₂×C₄²	C₂²×D₅₃	semidirect
430. C₂⁴	C₂×D₂₁₂	semidirect
458. C₂×D₄	C₂×D₂₁₂	semidirect
459. C₂×D₄	C₂×D₂₁₂	semidirect
473. C₂²×C₄	C₂×D₂₁₂	semidirect
477. C₂²×C₄	C₂³×D₅₃	semidirect
479. C₄⋊C₄	C₂³×D₅₃	semidirect
484. C₂²⋊C₄	C₂³×D₅₃	semidirect
485. C₄²	C₂³×D₅₃	semidirect
516. C₂³	C₂²×D₂₁₂	semidirect
528. D₄	C₂²×D₂₁₂	semidirect
540. C₂×C₄	C₂²×D₂₁₂	semidirect
576. C₂	#20288351	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133262	2
C₂³×D₄×D₅₃	2
#20288351	2
C₂×#16811268	2
C₂×D₄×D₂₁₂	2
C₂×#16811192	2
C₂×#16811185	2
C₂×#16811135	2
C₂²×#13133187	2
C₂×C₅₃×#518	2
C₂×#3484	53

Subgroups — C2×#20288351 #28291976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288351 #28291976