Subgroups — C₂×#20288346 #28291972

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	C₂³×D₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#20288346	C₂	1	32	yes	maximal, direct product factor
3	C₂×#16811261	C₂	1	8	yes	maximal, complemented
4	C₂×D₄×C₂²⋊D₅₃	C₂	1	8	yes	maximal, complemented
5	C₂×#16811187	C₂	1	2	yes	maximal, complemented
6	C₂×#16811182	C₂	1	4	yes	maximal, complemented
7	C₂×#16811130	C₂	1	2	yes	maximal, complemented
8	C₂²×#13133185	C₂	1	4	yes	maximal, complemented
9	C₂×#16811098	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×H(5,𝔽₂)×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵×D₅₃	C₂²	1	2	yes	complemented
12	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
13	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
14	C₂³×C₂²⋊D₅₃	C₂²	1	8	yes	complemented
15	C₂²×D₄×D₅₃	—	2	16	no	—
16	C₂²×D₄×D₅₃	C₂²	1	8	yes	complemented
17	C₂²×D₄×D₅₃	C₂²	1	16	yes	complemented
18	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
19	#16811261	C₂²	1	128	yes	complemented
20	D₄×C₂²⋊D₅₃	C₂²	1	128	yes	complemented
21	#16811187	C₂²	1	32	yes	complemented
22	#16811182	C₂²	1	64	yes	complemented
23	C₂×#13133213	C₂²	1	4	yes	complemented
24	C₂×#13133206	C₂²	1	4	yes	complemented
25	#16811130	C₂²	1	32	yes	complemented
26	C₂×#13133200	C₂²	1	8	yes	complemented
27	C₂×#13133198	C₂²	1	4	yes	complemented
28	C₂×#13133193	C₂²	1	4	yes	complemented
29	C₂×#13133191	C₂²	1	8	yes	complemented
30	C₂×#13133189	C₂²	1	16	yes	complemented
31	C₂×#13133188	C₂²	1	4	yes	complemented
32	C₂×#13133186	C₂²	1	8	yes	complemented
33	C₂×#13133185	C₂²	1	4	yes	complemented
34	C₂×#13133185	C₂²	1	8	yes	complemented
35	C₂×#13133185	C₂²	1	16	yes	complemented
36	C₂×#13133185	C₂²	1	64	yes	complemented
37	#16811098	C₂²	1	16	yes	complemented
38	C₂×D₄×Dic₅₃	C₂²	1	8	yes	complemented
39	C₂×C₄×C₂²⋊D₅₃	C₂²	1	8	yes	complemented
40	C₂×#13133176	C₂²	1	2	yes	complemented
41	C₂²×C₂²⋊Dic₅₃	C₂²	1	4	yes	complemented
42	C₂²×#11719921	C₂²	1	8	yes	complemented
43	C₂×H(5,𝔽₂)×C₅₃	—	2	16	no	—
44	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	16	yes	complemented
45	C₂²×#41×C₅₃	C₂²	1	2	yes	complemented
46	C₂²×#41×C₅₃	C₂²	1	4	yes	complemented
47	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
48	C₂³×D₄×C₅₃	C₂²	1	4	yes	complemented
49	C₂³×D₄×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	C₂²×D₂₁₂	—	2	8	no	—
57	C₂²×D₂₁₂	C₂³	1	4	yes	—
58	C₂²×D₂₁₂	C₂³	1	8	yes	complemented
59	C₂²×C₂²⋊D₅₃	—	2	16	no	—
60	C₂²×C₂²⋊D₅₃	—	4	32	no	—
61	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
62	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
63	C₂²×C₂²⋊D₅₃	C₂³	1	16	yes	complemented
64	C₂²×C₂²⋊D₅₃	C₂³	1	16	yes	complemented
65	C₂²×C₂²⋊D₅₃	C₂³	1	64	yes	complemented
66	C₂×D₄×D₅₃	—	2	64	no	—
67	C₂×D₄×D₅₃	—	2	128	no	—
68	C₂×D₄×D₅₃	C₂³	1	64	yes	complemented
69	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
70	C₂³×Dic₅₃	C₂³	1	8	yes	complemented
71	C₂²×C₄×D₅₃	—	2	8	no	—
72	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
73	C₂²×C₄×D₅₃	C₂³	1	8	yes	complemented
74	#13133213	C₂³	1	32	yes	complemented
75	#13133206	C₂³	1	32	yes	complemented
76	#13133200	C₂³	1	64	yes	complemented
77	#13133198	C₂³	1	32	yes	complemented
78	#13133193	C₂³	1	32	yes	complemented
79	#13133191	C₂³	1	64	yes	complemented
80	#13133189	C₂³	1	128	yes	complemented
81	#13133188	C₂³	1	32	yes	complemented
82	#13133186	C₂³	1	64	yes	complemented
83	#13133185	C₂³	1	32	yes	complemented
84	#13133185	C₂³	1	128	yes	complemented
85	#13133185	C₂³	1	64	yes	complemented
86	D₄×Dic₅₃	C₂³	1	64	yes	complemented
87	C₄×C₂²⋊D₅₃	C₂³	1	64	yes	complemented
88	#13133176	C₂³	1	16	yes	complemented
89	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	complemented
90	C₂×#11719919	C₂³	1	8	yes	complemented
91	C₂×C₂²⋊Dic₅₃	C₂³	1	4	yes	—
92	C₂×C₂²⋊Dic₅₃	C₂³	1	8	yes	complemented
93	C₂×C₂²⋊Dic₅₃	C₂³	1	8	yes	complemented
94	C₂×C₂²⋊Dic₅₃	C₂³	1	32	yes	complemented
95	C₂×#11719921	C₂³	1	8	yes	—
96	C₂×#11719921	C₂³	1	16	yes	complemented
97	C₂×#11719921	C₂³	1	64	yes	complemented
98	C₂×C₄×Dic₅₃	C₂³	1	4	yes	complemented
99	C₂⁵×C₅₃	C₂³	1	2	yes	—
100	C₂⁵×C₅₃	C₂³	1	4	yes	complemented
101	H(5,𝔽₂)×C₅₃	—	2	128	no	—
102	C₂×#41×C₅₃	—	2	32	no	—
103	C₂×#41×C₅₃	—	2	16	no	—
104	C₂×#41×C₅₃	C₂³	1	16	yes	complemented
105	C₂×#41×C₅₃	C₂³	1	32	yes	complemented
106	C₂²×Q₈×C₅₃	C₂³	1	2	yes	complemented
107	C₂²×D₄×C₅₃	—	2	32	no	—
108	C₂²×D₄×C₅₃	—	2	32	no	—
109	C₂²×D₄×C₅₃	D₄	1	16	yes	complemented
110	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
111	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
112	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
113	C₂²×D₄×C₅₃	C₂³	1	8	yes	complemented
114	C₂²×D₄×C₅₃	C₂³	1	2	yes	complemented
115	C₂²×D₄×C₅₃	C₂³	1	32	yes	complemented
116	C₂²×D₄×C₅₃	C₂³	1	32	yes	complemented
117	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
118	C₂³×C₄×C₅₃	C₂³	1	4	yes	complemented
119	C₂³×C₄×C₅₃	C₂³	1	4	yes	complemented
120	C₂⁴×C₅₃	—	2	16	no	—
121	C₂⁴×C₅₃	—	4	8	no	—
122	C₂⁴×C₅₃	—	4	16	no	—
123	C₂⁴×C₅₃	C₂×D₄	1	16	yes	complemented
124	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
125	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
126	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
127	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
128	C₂⁴×C₅₃	C₂⁴	1	16	yes	complemented
129	C₂³×D₅₃	—	2	4	no	—
130	C₂³×D₅₃	—	2	8	no	—
131	C₂³×D₅₃	—	2	32	no	—
132	C₂³×D₅₃	—	2	8	no	—
133	C₂³×D₅₃	—	2	64	no	—
134	C₂³×D₅₃	—	2	32	no	—
135	C₂³×D₅₃	—	4	64	no	—
136	C₂³×D₅₃	—	4	16	no	—
137	C₂³×D₅₃	—	4	8	no	—
138	C₂³×D₅₃	—	4	128	no	—
139	C₂³×D₅₃	C₂⁴	1	4	yes	—
140	C₂³×D₅₃	C₂⁴	1	4	yes	—
141	C₂³×D₅₃	C₂⁴	1	32	yes	—
142	C₂³×D₅₃	C₂⁴	1	2	yes	—
143	C₂³×D₅₃	C₂⁴	1	8	yes	—
144	#41×C₅₃	—	2	128	no	—
145	#41×C₅₃	—	2	64	no	—
146	C₂×Q₈×C₅₃	—	2	8	no	—
147	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	complemented
148	C₂×D₄×C₅₃	—	2	16	no	—
149	C₂×D₄×C₅₃	—	2	32	no	—
150	C₂×D₄×C₅₃	—	2	128	no	—
151	C₂×D₄×C₅₃	—	2	128	no	—
152	C₂×D₄×C₅₃	—	2	8	no	—
153	C₂×D₄×C₅₃	C₂×D₄	1	32	yes	complemented
154	C₂×D₄×C₅₃	C₂×D₄	1	64	yes	complemented
155	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
156	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
157	C₂×D₄×C₅₃	C₂⁴	1	32	yes	complemented
158	C₂×D₄×C₅₃	C₂⁴	1	8	yes	complemented
159	C₂²×C₄×C₅₃	—	2	16	no	—
160	C₂²×C₄×C₅₃	—	2	16	no	—
161	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	complemented
162	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	complemented
166	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	complemented
167	C₂×C₂²⋊D₅₃	—	2	32	no	—
168	C₂×C₂²⋊D₅₃	—	2	64	no	—
169	C₂×C₂²⋊D₅₃	—	2	32	no	—
170	C₂×C₂²⋊D₅₃	—	4	32	no	—
171	C₂×C₂²⋊D₅₃	—	4	128	no	—
172	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
173	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
174	C₂×C₂²⋊D₅₃	C₂⁴	1	64	yes	complemented
175	C₂²×Dic₅₃	—	2	4	no	—
176	C₂²×Dic₅₃	—	4	16	no	—
177	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
178	C₂²×Dic₅₃	C₂⁴	1	2	yes	—
179	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
180	C₂²×Dic₅₃	C₂⁴	1	32	yes	complemented
181	D₄×D₅₃	—	2	256	no	—
182	C₂×D₂₁₂	—	2	16	no	—
183	C₂×D₂₁₂	—	2	32	no	—
184	C₂×D₂₁₂	C₂⁴	1	16	yes	—
185	C₂×C₄×D₅₃	—	2	16	no	—
186	C₂×C₄×D₅₃	—	2	32	no	—
187	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
188	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
189	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	complemented
190	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	complemented
191	#11719921	C₂⁴	1	32	yes	—
192	#11719921	C₂⁴	1	64	yes	complemented
193	C₄⋊Dic₅₃	C₂⁴	1	16	yes	complemented
194	#11719919	C₂⁴	1	32	yes	complemented
195	C₄×Dic₅₃	C₂⁴	1	16	yes	complemented
196	C₂³×C₅₃	—	2	16	no	—
197	C₂³×C₅₃	—	2	4	no	—
198	C₂³×C₅₃	—	2	8	no	—
199	C₂³×C₅₃	—	2	16	no	—
200	C₂³×C₅₃	—	2	32	no	—
201	C₂³×C₅₃	—	4	32	no	—
202	C₂³×C₅₃	—	4	16	no	—
203	C₂³×C₅₃	—	4	32	no	—
204	C₂³×C₅₃	—	8	32	no	—
205	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
206	C₂³×C₅₃	C₂²×D₄	1	16	yes	—
207	C₂³×C₅₃	C₂²×D₄	1	32	yes	complemented
208	C₂³×C₅₃	C₂⁵	1	8	yes	—
209	C₂³×C₅₃	C₂⁵	1	2	yes	—
210	C₂³×C₅₃	C₂⁵	1	8	yes	—
211	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
212	C₂²×D₅₃	—	2	32	no	—
213	C₂²×D₅₃	—	2	8	no	—
214	C₂²×D₅₃	—	2	4	no	—
215	C₂²×D₅₃	—	2	4	no	—
216	C₂²×D₅₃	—	2	16	no	—
217	C₂²×D₅₃	—	2	16	no	—
218	C₂²×D₅₃	—	2	8	no	—
219	C₂²×D₅₃	—	2	64	no	—
220	C₂²×D₅₃	—	2	16	no	—
221	C₂²×D₅₃	—	2	64	no	—
222	C₂²×D₅₃	—	4	32	no	—
223	C₂²×D₅₃	—	4	16	no	—
224	C₂²×D₅₃	—	4	128	no	—
225	C₂²×D₅₃	—	4	256	no	—
226	C₂²×D₅₃	—	4	32	no	—
227	C₂²×D₅₃	H(5,𝔽₂)	1	8	yes	complemented
228	C₂²×D₅₃	C₂⁵	1	8	yes	—
229	C₂²×D₅₃	C₂⁵	1	4	yes	—
230	Q₈×C₅₃	—	2	16	no	—
231	D₄×C₅₃	—	2	32	no	—
232	D₄×C₅₃	—	2	64	no	—
233	D₄×C₅₃	—	2	16	no	—
234	D₄×C₅₃	C₂²×D₄	1	64	yes	complemented
235	C₂×C₄×C₅₃	—	2	8	no	—
236	C₂×C₄×C₅₃	—	2	32	no	—
237	C₂×C₄×C₅₃	—	2	32	no	—
238	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
239	C₂×C₄×C₅₃	C₂²×D₄	1	16	yes	complemented
240	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
241	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
242	C₂²⋊D₅₃	—	2	64	no	—
243	C₂²⋊D₅₃	—	2	64	no	—
244	C₂²⋊D₅₃	—	4	64	no	—
245	C₂×Dic₅₃	—	2	8	no	—
246	C₂×Dic₅₃	—	2	4	no	—
247	C₂×Dic₅₃	—	2	16	no	—
248	C₂×Dic₅₃	—	4	32	no	—
249	C₂×Dic₅₃	C₂⁵	1	4	yes	—
250	C₂×Dic₅₃	C₂⁵	1	16	yes	—
251	D₂₁₂	—	2	32	no	—
252	C₄×D₅₃	—	2	32	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	—	2	8	no	—
259	C₂²×C₅₃	—	2	16	no	—
260	C₂²×C₅₃	—	4	32	no	—
261	C₂²×C₅₃	—	4	16	no	—
262	C₂²×C₅₃	—	8	32	no	—
263	C₂²×C₅₃	C₂³×D₄	1	4	yes	—
264	C₂²×C₅₃	C₂³×D₄	1	16	yes	—
265	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
266	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
267	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
268	C₂×D₅₃	—	2	32	no	—
269	C₂×D₅₃	—	2	4	no	—
270	C₂×D₅₃	—	2	4	no	—
271	C₂×D₅₃	—	2	16	no	—
272	C₂×D₅₃	—	2	8	no	—
273	C₂×D₅₃	—	2	16	no	—
274	C₂×D₅₃	—	4	128	no	—
275	C₂×D₅₃	—	4	32	no	—
276	C₂×D₅₃	—	4	16	no	—
277	C₂×D₅₃	C₂×H(5,𝔽₂)	1	8	yes	complemented
278	C₄×C₅₃	—	2	8	no	—
279	C₄×C₅₃	C₂³×D₄	1	8	yes	—
280	Dic₅₃	—	2	4	no	—
281	Dic₅₃	—	2	16	no	—
282	C₂²×H(5,𝔽₂)	D₅₃	1	1	yes	characteristic, complemented
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	—	53	1	no	—
292	C₂×C₅₃	—	2	4	no	—
293	C₂×C₅₃	—	2	16	no	—
294	C₂×C₅₃	—	4	8	no	—
295	C₂×C₅₃	#3484	1	4	yes	complemented
296	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
297	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	2	yes	—
298	D₅₃	—	2	8	no	—
299	D₅₃	—	4	8	no	—
300	C₂⁶	—	53	2	no	—
301	C₂×H(5,𝔽₂)	—	2	16	no	—
302	C₂×H(5,𝔽₂)	C₂×D₅₃	1	16	yes	complemented
303	C₂²×#41	C₂×D₅₃	1	2	yes	complemented
304	C₂²×#41	C₂×D₅₃	1	4	yes	complemented
305	C₂³×D₄	—	53	2	no	—
306	C₂³×D₄	—	53	8	no	—
307	C₂³×D₄	—	53	8	no	—
308	C₂³×D₄	—	53	16	no	—
309	C₂³×D₄	—	53	1	no	—
310	C₂³×D₄	—	106	16	no	—
311	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
312	C₂³×D₄	C₂×D₅₃	1	4	yes	complemented
313	C₂³×D₄	C₂×D₅₃	1	4	yes	complemented
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	4	no	—
326	C₂×#121	—	53	16	no	—
327	C₂×#121	—	53	8	no	—
328	C₂×#120	—	53	8	no	—
329	C₂×#120	—	53	4	no	—
330	C₂×#120	—	53	8	no	—
331	C₂×#120	—	53	16	no	—
332	C₂×#120	—	53	64	no	—
333	#518	—	53	16	no	—
334	C₂×C₄×D₄	—	53	8	no	—
335	C₂×C₄×D₄	—	53	8	no	—
336	C₂×#117	—	53	2	no	—
337	C₂²×C₂²⋊C₄	—	53	4	no	—
338	C₂²×C₂²⋊C₄	—	53	8	no	—
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	—	106	16	no	—
345	C₂⁵	—	212	16	no	—
346	C₂⁵	C₂²×D₅₃	1	2	yes	—
347	C₂⁵	C₂²×D₅₃	1	4	yes	complemented
348	H(5,𝔽₂)	—	2	128	no	—
349	C₂×#41	—	2	32	no	—
350	C₂×#41	—	2	16	no	—
351	C₂×#41	C₂²×D₅₃	1	16	yes	complemented
352	C₂×#41	C₂²×D₅₃	1	32	yes	complemented
353	C₂²×Q₈	C₂²×D₅₃	1	2	yes	complemented
354	C₂²×D₄	—	2	32	no	—
355	C₂²×D₄	—	2	32	no	—
356	C₂²×D₄	—	53	8	no	—
357	C₂²×D₄	—	53	8	no	—
358	C₂²×D₄	—	53	16	no	—
359	C₂²×D₄	—	53	16	no	—
360	C₂²×D₄	—	53	64	no	—
361	C₂²×D₄	—	53	64	no	—
362	C₂²×D₄	—	53	4	no	—
363	C₂²×D₄	—	53	8	no	—
364	C₂²×D₄	—	106	16	no	—
365	C₂²×D₄	—	106	64	no	—
366	C₂²×D₄	—	106	128	no	—
367	C₂²×D₄	—	106	8	no	—
368	C₂²×D₄	—	212	32	no	—
369	C₂²×D₄	C₂²⋊D₅₃	1	16	yes	complemented
370	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
371	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
372	C₂²×D₄	C₂²×D₅₃	1	8	yes	—
373	C₂²×D₄	C₂²×D₅₃	1	8	yes	complemented
374	C₂²×D₄	C₂²×D₅₃	1	2	yes	complemented
375	C₂²×D₄	C₂²×D₅₃	1	32	yes	complemented
376	C₂²×D₄	C₂²×D₅₃	1	32	yes	complemented
377	C₂³×C₄	—	53	1	no	—
378	C₂³×C₄	—	53	8	no	—
379	C₂³×C₄	—	53	4	no	—
380	C₂³×C₄	—	53	8	no	—
381	C₂³×C₄	—	106	8	no	—
382	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
383	C₂³×C₄	C₂²×D₅₃	1	4	yes	complemented
384	C₂³×C₄	C₂²×D₅₃	1	4	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	32	no	—
391	#121	—	53	128	no	—
392	#121	—	53	64	no	—
393	#120	—	53	64	no	—
394	#120	—	53	32	no	—
395	#120	—	53	64	no	—
396	#120	—	53	128	no	—
397	C₄×D₄	—	53	64	no	—
398	C₄×D₄	—	53	64	no	—
399	#117	—	53	16	no	—
400	C₂×C₄⋊C₄	—	53	4	no	—
401	C₂×C₄⋊C₄	—	53	8	no	—
402	C₂×C₂²⋊C₄	—	53	4	no	—
403	C₂×C₂²⋊C₄	—	53	8	no	—
404	C₂×C₂²⋊C₄	—	53	32	no	—
405	C₂×C₂²⋊C₄	—	53	8	no	—
406	C₂×C₂²⋊C₄	—	53	8	no	—
407	C₂×C₂²⋊C₄	—	53	16	no	—
408	C₂×C₂²⋊C₄	—	53	64	no	—
409	C₂×C₄²	—	53	4	no	—
410	C₂⁴	—	2	16	no	—
411	C₂⁴	—	4	8	no	—
412	C₂⁴	—	4	16	no	—
413	C₂⁴	—	53	2	no	—
414	C₂⁴	—	53	4	no	—
415	C₂⁴	—	53	4	no	—
416	C₂⁴	—	53	32	no	—
417	C₂⁴	—	53	8	no	—
418	C₂⁴	—	106	8	no	—
419	C₂⁴	—	106	32	no	—
420	C₂⁴	—	106	4	no	—
421	C₂⁴	—	106	32	no	—
422	C₂⁴	—	106	8	no	—
423	C₂⁴	—	106	64	no	—
424	C₂⁴	—	212	16	no	—
425	C₂⁴	—	212	8	no	—
426	C₂⁴	—	212	64	no	—
427	C₂⁴	—	212	128	no	—
428	C₂⁴	C₂×C₂²⋊D₅₃	1	16	yes	complemented
429	C₂⁴	C₂³×D₅₃	1	4	yes	—
430	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
431	C₂⁴	C₂³×D₅₃	1	4	yes	—
432	C₂⁴	C₂³×D₅₃	1	8	yes	—
433	C₂⁴	C₂³×D₅₃	1	16	yes	complemented
434	#41	—	2	128	no	—
435	#41	—	2	64	no	—
436	C₂×Q₈	—	2	8	no	—
437	C₂×Q₈	C₂³×D₅₃	1	8	yes	complemented
438	C₂×D₄	—	2	16	no	—
439	C₂×D₄	—	2	32	no	—
440	C₂×D₄	—	2	128	no	—
441	C₂×D₄	—	2	128	no	—
442	C₂×D₄	—	2	8	no	—
443	C₂×D₄	—	53	32	no	—
444	C₂×D₄	—	53	32	no	—
445	C₂×D₄	—	53	64	no	—
446	C₂×D₄	—	53	16	no	—
447	C₂×D₄	—	106	32	no	—
448	C₂×D₄	—	106	32	no	—
449	C₂×D₄	—	106	64	no	—
450	C₂×D₄	—	106	256	no	—
451	C₂×D₄	—	106	16	no	—
452	C₂×D₄	—	106	32	no	—
453	C₂×D₄	—	212	32	no	—
454	C₂×D₄	—	212	128	no	—
455	C₂×D₄	C₂×C₂²⋊D₅₃	1	32	yes	complemented
456	C₂×D₄	C₂×C₂²⋊D₅₃	1	64	yes	complemented
457	C₂×D₄	C₂³×D₅₃	1	16	yes	—
458	C₂×D₄	C₂³×D₅₃	1	16	yes	—
459	C₂×D₄	C₂³×D₅₃	1	32	yes	complemented
460	C₂×D₄	C₂³×D₅₃	1	8	yes	complemented
461	C₂²×C₄	—	2	16	no	—
462	C₂²×C₄	—	2	16	no	—
463	C₂²×C₄	—	53	4	no	—
464	C₂²×C₄	—	53	8	no	—
465	C₂²×C₄	—	53	2	no	—
466	C₂²×C₄	—	53	32	no	—
467	C₂²×C₄	—	53	16	no	—
468	C₂²×C₄	—	106	4	no	—
469	C₂²×C₄	—	106	16	no	—
470	C₂²×C₄	—	106	32	no	—
471	C₂²×C₄	—	212	16	no	—
472	C₂²×C₄	C₂×C₂²⋊D₅₃	1	8	yes	complemented
473	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
474	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
475	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
476	C₂²×C₄	C₂³×D₅₃	1	16	yes	complemented
477	C₂²×C₄	C₂³×D₅₃	1	16	yes	complemented
478	C₄⋊C₄	—	53	16	no	—
479	C₄⋊C₄	—	53	32	no	—
480	C₂²⋊C₄	—	53	16	no	—
481	C₂²⋊C₄	—	53	32	no	—
482	C₂²⋊C₄	—	53	32	no	—
483	C₂²⋊C₄	—	53	32	no	—
484	C₂²⋊C₄	—	53	64	no	—
485	C₄²	—	53	16	no	—
486	C₂³	—	2	16	no	—
487	C₂³	—	2	4	no	—
488	C₂³	—	2	8	no	—
489	C₂³	—	2	16	no	—
490	C₂³	—	2	32	no	—
491	C₂³	—	4	32	no	—
492	C₂³	—	4	16	no	—
493	C₂³	—	4	32	no	—
494	C₂³	—	8	32	no	—
495	C₂³	—	53	8	no	—
496	C₂³	—	53	4	no	—
497	C₂³	—	53	8	no	—
498	C₂³	—	106	16	no	—
499	C₂³	—	106	64	no	—
500	C₂³	—	106	32	no	—
501	C₂³	—	106	8	no	—
502	C₂³	—	106	4	no	—
503	C₂³	—	106	4	no	—
504	C₂³	—	106	64	no	—
505	C₂³	—	106	16	no	—
506	C₂³	—	106	8	no	—
507	C₂³	—	106	16	no	—
508	C₂³	—	212	32	no	—
509	C₂³	—	212	16	no	—
510	C₂³	—	212	128	no	—
511	C₂³	—	212	32	no	—
512	C₂³	—	212	256	no	—
513	C₂³	C₂²×C₂²⋊D₅₃	1	16	yes	—
514	C₂³	C₂²×C₂²⋊D₅₃	1	4	yes	—
515	C₂³	C₂²×C₂²⋊D₅₃	1	32	yes	complemented
516	C₂³	C₂⁴×D₅₃	1	8	yes	—
517	C₂³	C₂⁴×D₅₃	1	2	yes	—
518	C₂³	C₂⁴×D₅₃	1	8	yes	—
519	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
520	Q₈	—	2	16	no	—
521	D₄	—	2	32	no	—
522	D₄	—	2	64	no	—
523	D₄	—	2	16	no	—
524	D₄	—	106	64	no	—
525	D₄	—	106	64	no	—
526	D₄	—	106	32	no	—
527	D₄	—	212	64	no	—
528	D₄	C₂²×C₂²⋊D₅₃	1	64	yes	complemented
529	C₂×C₄	—	2	8	no	—
530	C₂×C₄	—	2	32	no	—
531	C₂×C₄	—	2	32	no	—
532	C₂×C₄	—	53	16	no	—
533	C₂×C₄	—	53	4	no	—
534	C₂×C₄	—	106	8	no	—
535	C₂×C₄	—	106	4	no	—
536	C₂×C₄	—	106	16	no	—
537	C₂×C₄	—	106	32	no	—
538	C₂×C₄	—	212	32	no	—
539	C₂×C₄	C₂²×C₂²⋊D₅₃	1	8	yes	—
540	C₂×C₄	C₂²×C₂²⋊D₅₃	1	16	yes	complemented
541	C₂×C₄	C₂⁴×D₅₃	1	8	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	—	2	8	no	—
548	C₂²	—	2	16	no	—
549	C₂²	—	4	32	no	—
550	C₂²	—	4	16	no	—
551	C₂²	—	8	32	no	—
552	C₂²	—	53	8	no	—
553	C₂²	—	106	32	no	—
554	C₂²	—	106	16	no	—
555	C₂²	—	106	4	no	—
556	C₂²	—	106	4	no	—
557	C₂²	—	106	8	no	—
558	C₂²	—	106	16	no	—
559	C₂²	—	212	128	no	—
560	C₂²	—	212	32	no	—
561	C₂²	—	212	16	no	—
562	C₂²	C₂³×C₂²⋊D₅₃	1	16	yes	—
563	C₂²	C₂³×C₂²⋊D₅₃	1	4	yes	—
564	C₂²	C₂³×C₂²⋊D₅₃	1	2	yes	—
565	C₂²	H(5,𝔽₂)×D₅₃	1	4	yes	—
566	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
567	C₄	—	2	8	no	—
568	C₄	—	106	4	no	—
569	C₄	—	106	16	no	—
570	C₄	C₂³×C₂²⋊D₅₃	1	8	yes	—
571	C₂	—	2	4	no	—
572	C₂	—	2	16	no	—
573	C₂	—	4	8	no	—
574	C₂	—	106	8	no	—
575	C₂	—	212	8	no	—
576	C₂	#20288346	1	4	yes	direct product factor
577	C₂	C₂⁴×C₂²⋊D₅₃	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	C₂²×H(5,𝔽₂)×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂²×H(5,𝔽₂)	D₅₃
hypocenter	C₅₃	C₂×#3484
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×D₅₃	C₂	semidirect
2. #20288346	C₂	direct (N × H)
3. C₂×#16811261	C₂	semidirect
4. C₂×D₄×C₂²⋊D₅₃	C₂	semidirect
5. C₂×#16811187	C₂	semidirect
6. C₂×#16811182	C₂	semidirect
7. C₂×#16811130	C₂	semidirect
8. C₂²×#13133185	C₂	semidirect
9. C₂×#16811098	C₂	semidirect
10. C₂²×H(5,𝔽₂)×C₅₃	C₂	semidirect
11. C₂⁵×D₅₃	C₂²	semidirect
12. C₂³×D₂₁₂	C₂²	semidirect
13. C₂³×C₂²⋊D₅₃	C₂²	semidirect
14. C₂³×C₂²⋊D₅₃	C₂²	semidirect
16. C₂²×D₄×D₅₃	C₂²	semidirect
17. C₂²×D₄×D₅₃	C₂²	semidirect
18. C₂³×C₄×D₅₃	C₂²	semidirect
19. #16811261	C₂²	semidirect
20. D₄×C₂²⋊D₅₃	C₂²	semidirect
21. #16811187	C₂²	semidirect
22. #16811182	C₂²	semidirect
23. C₂×#13133213	C₂²	semidirect
24. C₂×#13133206	C₂²	semidirect
25. #16811130	C₂²	semidirect
26. C₂×#13133200	C₂²	semidirect
27. C₂×#13133198	C₂²	semidirect
28. C₂×#13133193	C₂²	semidirect
29. C₂×#13133191	C₂²	semidirect
30. C₂×#13133189	C₂²	semidirect
31. C₂×#13133188	C₂²	semidirect
32. C₂×#13133186	C₂²	semidirect
33. C₂×#13133185	C₂²	semidirect
34. C₂×#13133185	C₂²	semidirect
35. C₂×#13133185	C₂²	semidirect
36. C₂×#13133185	C₂²	semidirect
37. #16811098	C₂²	semidirect
38. C₂×D₄×Dic₅₃	C₂²	semidirect
39. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
40. C₂×#13133176	C₂²	semidirect
41. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
42. C₂²×#11719921	C₂²	semidirect
44. C₂×H(5,𝔽₂)×C₅₃	C₂²	semidirect
45. C₂²×#41×C₅₃	C₂²	semidirect
46. C₂²×#41×C₅₃	C₂²	semidirect
48. C₂³×D₄×C₅₃	C₂²	semidirect
49. C₂³×D₄×C₅₃	C₂²	semidirect
55. C₂⁴×D₅₃	C₂³	semidirect
58. C₂²×D₂₁₂	C₂³	semidirect
63. C₂²×C₂²⋊D₅₃	C₂³	semidirect
64. C₂²×C₂²⋊D₅₃	C₂³	semidirect
65. C₂²×C₂²⋊D₅₃	C₂³	semidirect
68. C₂×D₄×D₅₃	C₂³	semidirect
70. C₂³×Dic₅₃	C₂³	semidirect
73. C₂²×C₄×D₅₃	C₂³	semidirect
74. #13133213	C₂³	semidirect
75. #13133206	C₂³	semidirect
76. #13133200	C₂³	semidirect
77. #13133198	C₂³	semidirect
78. #13133193	C₂³	semidirect
79. #13133191	C₂³	semidirect
80. #13133189	C₂³	semidirect
81. #13133188	C₂³	semidirect
82. #13133186	C₂³	semidirect
83. #13133185	C₂³	semidirect
84. #13133185	C₂³	semidirect
85. #13133185	C₂³	semidirect
86. D₄×Dic₅₃	C₂³	semidirect
87. C₄×C₂²⋊D₅₃	C₂³	semidirect
88. #13133176	C₂³	semidirect
89. C₂×C₄⋊Dic₅₃	C₂³	semidirect
90. C₂×#11719919	C₂³	semidirect
92. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
93. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
94. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
96. C₂×#11719921	C₂³	semidirect
97. C₂×#11719921	C₂³	semidirect
98. C₂×C₄×Dic₅₃	C₂³	semidirect
100. C₂⁵×C₅₃	C₂³	semidirect
104. C₂×#41×C₅₃	C₂³	semidirect
105. C₂×#41×C₅₃	C₂³	semidirect
106. C₂²×Q₈×C₅₃	C₂³	semidirect
109. C₂²×D₄×C₅₃	D₄	semidirect
113. C₂²×D₄×C₅₃	C₂³	semidirect
114. C₂²×D₄×C₅₃	C₂³	semidirect
115. C₂²×D₄×C₅₃	C₂³	semidirect
116. C₂²×D₄×C₅₃	C₂³	semidirect
118. C₂³×C₄×C₅₃	C₂³	semidirect
119. C₂³×C₄×C₅₃	C₂³	semidirect
123. C₂⁴×C₅₃	C₂×D₄	semidirect
128. C₂⁴×C₅₃	C₂⁴	semidirect
147. C₂×Q₈×C₅₃	C₂⁴	semidirect
153. C₂×D₄×C₅₃	C₂×D₄	semidirect
154. C₂×D₄×C₅₃	C₂×D₄	semidirect
157. C₂×D₄×C₅₃	C₂⁴	semidirect
158. C₂×D₄×C₅₃	C₂⁴	semidirect
161. C₂²×C₄×C₅₃	C₂×D₄	semidirect
165. C₂²×C₄×C₅₃	C₂⁴	semidirect
166. C₂²×C₄×C₅₃	C₂⁴	semidirect
174. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
180. C₂²×Dic₅₃	C₂⁴	semidirect
189. C₂²⋊Dic₅₃	C₂⁴	semidirect
190. C₂²⋊Dic₅₃	C₂⁴	semidirect
192. #11719921	C₂⁴	semidirect
193. C₄⋊Dic₅₃	C₂⁴	semidirect
194. #11719919	C₂⁴	semidirect
195. C₄×Dic₅₃	C₂⁴	semidirect
207. C₂³×C₅₃	C₂²×D₄	semidirect
227. C₂²×D₅₃	H(5,𝔽₂)	semidirect
234. D₄×C₅₃	C₂²×D₄	semidirect
239. C₂×C₄×C₅₃	C₂²×D₄	semidirect
277. C₂×D₅₃	C₂×H(5,𝔽₂)	semidirect
282. C₂²×H(5,𝔽₂)	D₅₃	semidirect
295. C₂×C₅₃	#3484	semidirect
302. C₂×H(5,𝔽₂)	C₂×D₅₃	semidirect
303. C₂²×#41	C₂×D₅₃	semidirect
304. C₂²×#41	C₂×D₅₃	semidirect
312. C₂³×D₄	C₂×D₅₃	semidirect
313. C₂³×D₄	C₂×D₅₃	semidirect
339. C₅₃	C₂×#3484	semidirect
347. C₂⁵	C₂²×D₅₃	semidirect
351. C₂×#41	C₂²×D₅₃	semidirect
352. C₂×#41	C₂²×D₅₃	semidirect
353. C₂²×Q₈	C₂²×D₅₃	semidirect
369. C₂²×D₄	C₂²⋊D₅₃	semidirect
373. C₂²×D₄	C₂²×D₅₃	semidirect
374. C₂²×D₄	C₂²×D₅₃	semidirect
375. C₂²×D₄	C₂²×D₅₃	semidirect
376. C₂²×D₄	C₂²×D₅₃	semidirect
383. C₂³×C₄	C₂²×D₅₃	semidirect
384. C₂³×C₄	C₂²×D₅₃	semidirect
428. C₂⁴	C₂×C₂²⋊D₅₃	semidirect
433. C₂⁴	C₂³×D₅₃	semidirect
437. C₂×Q₈	C₂³×D₅₃	semidirect
455. C₂×D₄	C₂×C₂²⋊D₅₃	semidirect
456. C₂×D₄	C₂×C₂²⋊D₅₃	semidirect
459. C₂×D₄	C₂³×D₅₃	semidirect
460. C₂×D₄	C₂³×D₅₃	semidirect
472. C₂²×C₄	C₂×C₂²⋊D₅₃	semidirect
476. C₂²×C₄	C₂³×D₅₃	semidirect
477. C₂²×C₄	C₂³×D₅₃	semidirect
515. C₂³	C₂²×C₂²⋊D₅₃	semidirect
528. D₄	C₂²×C₂²⋊D₅₃	semidirect
540. C₂×C₄	C₂²×C₂²⋊D₅₃	semidirect
576. C₂	#20288346	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×D₄×D₅₃	2
#20288346	2
C₂×#16811261	2
C₂×D₄×C₂²⋊D₅₃	2
C₂×#16811187	2
C₂×#16811182	2
C₂×#16811130	2
C₂²×#13133185	2
C₂×#16811098	2
C₂²×H(5,𝔽₂)×C₅₃	2
C₂×#3484	53

Subgroups — C2×#20288346 #28291972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288346 #28291972