Subgroups — C₂×#20288872 #28291215

Order 13568.

9542 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20288872	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16811362	C₂	1	3	yes	maximal, complemented
3	C₂×#16811303	C₂	1	3	yes	maximal, complemented
4	C₂×#16811266	C₂	1	6	yes	maximal, complemented
5	C₂×#16811255	C₂	1	3	yes	maximal, complemented
6	C₂×#16811207	C₂	1	6	yes	maximal, complemented
7	C₂×#16811191	C₂	1	3	yes	maximal, complemented
8	C₂×#16811183	C₂	1	3	yes	maximal, complemented
9	C₂×#120×D₅₃	C₂	1	3	yes	maximal, complemented
10	C₂×C₅₃×H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic, complemented
12	C₂²×#11719949	C₂²	1	3	yes	complemented
13	C₂³×C₂²⋊D₅₃	C₂²	1	3	yes	complemented
14	C₂²×D₄×D₅₃	C₂²	1	3	yes	complemented
15	C₂²×D₄×D₅₃	C₂²	1	6	yes	complemented
16	#16811362	C₂²	1	48	yes	complemented
17	#16811303	C₂²	1	48	yes	complemented
18	#16811266	C₂²	1	96	yes	complemented
19	#16811255	C₂²	1	48	yes	complemented
20	#16811207	C₂²	1	96	yes	complemented
21	#16811191	C₂²	1	48	yes	complemented
22	#16811183	C₂²	1	48	yes	complemented
23	C₂×#13133214	C₂²	1	1	yes	characteristic, complemented
24	C₂×#13133213	C₂²	1	3	yes	complemented
25	C₂×#13133211	C₂²	1	2	yes	complemented
26	C₂×#13133210	C₂²	1	6	yes	complemented
27	C₂×#13133206	C₂²	1	3	yes	complemented
28	C₂×#13133201	C₂²	1	3	yes	complemented
29	C₂×#13133199	C₂²	1	6	yes	complemented
30	C₂×#13133199	C₂²	1	6	yes	complemented
31	C₂×#13133198	C₂²	1	3	yes	complemented
32	C₂×#13133193	C₂²	1	6	yes	complemented
33	C₂×#13133190	C₂²	1	6	yes	complemented
34	C₂×#13133190	C₂²	1	6	yes	complemented
35	C₂×#13133190	C₂²	1	6	yes	complemented
36	C₂×#13133189	C₂²	1	6	yes	complemented
37	C₂×#13133188	C₂²	1	6	yes	complemented
38	C₂×#13133187	C₂²	1	6	yes	complemented
39	C₂×#13133187	C₂²	1	3	yes	complemented
40	C₂×#13133187	C₂²	1	6	yes	complemented
41	C₂×#13133186	C₂²	1	3	yes	complemented
42	C₂×#13133186	C₂²	1	1	yes	characteristic, complemented
43	C₂×#13133185	C₂²	1	3	yes	—
44	C₂×#13133185	C₂²	1	6	yes	complemented
45	#120×D₅₃	C₂²	1	48	yes	complemented
46	C₂×C₄×D₂₁₂	C₂²	1	6	yes	complemented
47	C₂×#13133177	C₂²	1	6	yes	complemented
48	C₂×#13133174	C₂²	1	3	yes	complemented
49	C₂×C₂²⋊C₄×D₅₃	C₂²	1	3	yes	complemented
50	C₂×C₂²⋊C₄×D₅₃	C₂²	1	6	yes	complemented
51	C₂×C₂²⋊C₄×D₅₃	C₂²	1	3	yes	complemented
52	C₅₃×H(3,𝔽₄)	C₂²	1	16	yes	complemented
53	C₂×#124×C₅₃	C₂²	1	3	yes	complemented
54	C₂×#124×C₅₃	C₂²	1	6	yes	complemented
55	C₂×#120×C₅₃	C₂²	1	3	yes	complemented
56	C₂×#120×C₅₃	C₂²	1	3	yes	complemented
57	C₂⁴×D₅₃	—	4	12	no	—
58	C₂⁴×D₅₃	C₂³	1	3	yes	—
59	C₂⁴×D₅₃	C₂³	1	3	yes	—
60	C₂⁴×D₅₃	C₂³	1	3	yes	—
61	C₂⁴×D₅₃	C₂³	1	8	yes	complemented
62	C₂×#11719949	—	4	12	no	—
63	C₂×#11719949	C₂³	1	24	yes	complemented
64	C₂²×D₂₁₂	C₂³	1	6	yes	—
65	C₂²×D₂₁₂	C₂³	1	3	yes	—
66	C₂²×D₂₁₂	C₂³	1	3	yes	complemented
67	C₂²×C₂²⋊D₅₃	—	4	12	no	—
68	C₂²×C₂²⋊D₅₃	C₂³	1	6	yes	—
69	C₂²×C₂²⋊D₅₃	C₂³	1	3	yes	—
70	C₂²×C₂²⋊D₅₃	C₂³	1	3	yes	—
71	C₂²×C₂²⋊D₅₃	C₂³	1	6	yes	—
72	C₂²×C₂²⋊D₅₃	C₂³	1	6	yes	complemented
73	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
74	C₂×D₄×D₅₃	—	4	12	no	—
75	C₂×D₄×D₅₃	—	4	24	no	—
76	C₂×D₄×D₅₃	C₂³	1	48	yes	complemented
77	C₂×D₄×D₅₃	C₂³	1	24	yes	complemented
78	C₂³×Dic₅₃	C₂³	1	3	yes	complemented
79	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
80	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
81	C₂²×C₄×D₅₃	C₂³	1	6	yes	complemented
82	#13133214	C₂³	1	8	yes	complemented
83	#13133213	C₂³	1	24	yes	complemented
84	#13133211	C₂³	1	16	yes	complemented
85	#13133210	C₂³	1	48	yes	complemented
86	#13133206	C₂³	1	24	yes	complemented
87	#13133201	C₂³	1	24	yes	complemented
88	#13133199	C₂³	1	48	yes	complemented
89	#13133199	C₂³	1	48	yes	complemented
90	#13133198	C₂³	1	24	yes	complemented
91	#13133193	C₂³	1	48	yes	complemented
92	#13133190	C₂³	1	48	yes	complemented
93	#13133190	C₂³	1	48	yes	complemented
94	#13133190	C₂³	1	48	yes	complemented
95	#13133189	C₂³	1	48	yes	complemented
96	#13133188	C₂³	1	48	yes	complemented
97	#13133187	C₂³	1	24	yes	complemented
98	#13133187	C₂³	1	48	yes	complemented
99	#13133187	C₂³	1	48	yes	complemented
100	#13133186	C₂³	1	24	yes	complemented
101	#13133186	C₂³	1	8	yes	complemented
102	#13133185	C₂³	1	24	yes	—
103	#13133185	C₂³	1	48	yes	complemented
104	C₄×D₂₁₂	C₂³	1	48	yes	complemented
105	#13133177	C₂³	1	48	yes	complemented
106	#13133174	C₂³	1	24	yes	complemented
107	C₂×C₄⋊Dic₅₃	C₂³	1	6	yes	complemented
108	C₂×#11719919	C₂³	1	6	yes	complemented
109	C₂×#11719919	C₂³	1	6	yes	complemented
110	C₂×C₂²⋊Dic₅₃	C₂³	1	3	yes	—
111	C₂×C₂²⋊Dic₅₃	C₂³	1	6	yes	—
112	C₂×C₂²⋊Dic₅₃	C₂³	1	3	yes	complemented
113	C₂×#11719921	C₂³	1	6	yes	—
114	C₂×#11719921	C₂³	1	6	yes	—
115	C₂×#11719921	C₂³	1	6	yes	—
116	C₂×#11719921	C₂³	1	6	yes	complemented
117	C₂²⋊C₄×D₅₃	C₂³	1	48	yes	complemented
118	C₂²⋊C₄×D₅₃	C₂³	1	24	yes	complemented
119	C₂²⋊C₄×D₅₃	C₂³	1	24	yes	complemented
120	C₂×C₄×Dic₅₃	C₂³	1	3	yes	complemented
121	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
122	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic, complemented
123	C₂²×Q₈×C₅₃	C₂³	1	3	yes	complemented
124	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
125	C₂²×D₄×C₅₃	C₂³	1	6	yes	—
126	#124×C₅₃	C₂³	1	48	yes	complemented
127	#124×C₅₃	C₂³	1	24	yes	complemented
128	#120×C₅₃	C₂³	1	24	yes	complemented
129	#120×C₅₃	C₂³	1	24	yes	complemented
130	C₂×C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
131	C₂×C₂²⋊C₄×C₅₃	C₂³	1	3	yes	—
132	C₂×C₂²⋊C₄×C₅₃	C₂³	1	3	yes	complemented
133	C₂×C₂²⋊C₄×C₅₃	C₂³	1	6	yes	complemented
134	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic, complemented
135	C₂×C₄²×C₅₃	C₂³	1	2	yes	complemented
136	C₂⁴×C₅₃	—	4	6	no	—
137	C₂⁴×C₅₃	—	4	6	no	—
138	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
139	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
140	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
141	C₂⁴×C₅₃	C₂⁴	1	4	yes	complemented
142	C₂³×D₅₃	—	2	24	no	—
143	C₂³×D₅₃	—	2	12	no	—
144	C₂³×D₅₃	—	2	12	no	—
145	C₂³×D₅₃	—	4	48	no	—
146	C₂³×D₅₃	—	4	6	no	—
147	C₂³×D₅₃	—	4	12	no	—
148	C₂³×D₅₃	—	4	12	no	—
149	C₂³×D₅₃	—	4	64	no	—
150	C₂³×D₅₃	C₂⁴	1	1	yes	characteristic
151	C₂³×D₅₃	C₂⁴	1	3	yes	—
152	C₂³×D₅₃	C₂⁴	1	12	yes	—
153	C₂³×D₅₃	C₂⁴	1	3	yes	—
154	C₂³×D₅₃	C₂⁴	1	12	yes	—
155	C₂³×D₅₃	C₂⁴	1	12	yes	—
156	C₂×Q₈×C₅₃	—	4	6	no	—
157	C₂×Q₈×C₅₃	C₂⁴	1	12	yes	complemented
158	C₂×D₄×C₅₃	—	4	6	no	—
159	C₂×D₄×C₅₃	—	4	12	no	—
160	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
161	C₂×D₄×C₅₃	C₂⁴	1	24	yes	—
162	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
163	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
164	C₂×C₂²⋊D₅₃	—	2	24	no	—
165	C₂×C₂²⋊D₅₃	—	4	6	no	—
166	C₂×C₂²⋊D₅₃	—	4	6	no	—
167	C₂×C₂²⋊D₅₃	—	4	12	no	—
168	C₂×C₂²⋊D₅₃	—	4	12	no	—
169	C₂×C₂²⋊D₅₃	—	4	48	no	—
170	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
171	C₂×C₂²⋊D₅₃	C₂⁴	1	12	yes	—
172	C₂×C₂²⋊D₅₃	C₂⁴	1	12	yes	—
173	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
174	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	complemented
175	C₂²×Dic₅₃	—	4	6	no	—
176	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
177	C₂²×Dic₅₃	C₂⁴	1	3	yes	—
178	C₂²×Dic₅₃	C₂⁴	1	12	yes	complemented
179	#11719949	—	4	48	no	—
180	D₄×D₅₃	—	4	48	no	—
181	D₄×D₅₃	—	4	96	no	—
182	C₂×D₂₁₂	—	2	24	no	—
183	C₂×D₂₁₂	—	4	6	no	—
184	C₂×D₂₁₂	—	4	6	no	—
185	C₂×D₂₁₂	C₂⁴	1	24	yes	—
186	C₂×D₂₁₂	C₂⁴	1	12	yes	—
187	C₂×D₂₁₂	C₂⁴	1	12	yes	complemented
188	C₂×C₄×D₅₃	—	2	12	no	—
189	C₂×C₄×D₅₃	—	4	12	no	—
190	C₂×C₄×D₅₃	—	4	12	no	—
191	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
192	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
193	C₂×C₄×D₅₃	C₂⁴	1	24	yes	complemented
194	C₂²⋊C₄×C₅₃	C₂⁴	1	24	yes	—
195	C₂²⋊C₄×C₅₃	C₂⁴	1	12	yes	—
196	C₂²⋊C₄×C₅₃	C₂⁴	1	12	yes	complemented
197	C₂²⋊C₄×C₅₃	C₂⁴	1	24	yes	complemented
198	C₄²×C₅₃	C₂⁴	1	4	yes	complemented
199	C₄²×C₅₃	C₂⁴	1	8	yes	complemented
200	C₂²⋊Dic₅₃	C₂⁴	1	12	yes	—
201	C₂²⋊Dic₅₃	C₂⁴	1	24	yes	—
202	C₂²⋊Dic₅₃	C₂⁴	1	12	yes	complemented
203	#11719921	C₂⁴	1	24	yes	—
204	#11719921	C₂⁴	1	24	yes	—
205	#11719921	C₂⁴	1	24	yes	—
206	#11719921	C₂⁴	1	24	yes	complemented
207	C₄⋊Dic₅₃	C₂⁴	1	24	yes	complemented
208	#11719919	C₂⁴	1	24	yes	complemented
209	#11719919	C₂⁴	1	24	yes	complemented
210	C₄×Dic₅₃	C₂⁴	1	12	yes	complemented
211	C₂³×C₅₃	—	2	6	no	—
212	C₂³×C₅₃	—	2	12	no	—
213	C₂³×C₅₃	—	2	6	no	—
214	C₂³×C₅₃	—	2	12	no	—
215	C₂³×C₅₃	—	4	12	no	—
216	C₂³×C₅₃	—	4	12	no	—
217	C₂³×C₅₃	—	4	16	no	—
218	C₂³×C₅₃	—	8	8	no	—
219	C₂³×C₅₃	C₂⁵	1	6	yes	—
220	C₂³×C₅₃	C₂⁵	1	6	yes	—
221	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
222	C₂²×D₅₃	—	2	12	no	—
223	C₂²×D₅₃	—	2	12	no	—
224	C₂²×D₅₃	—	2	6	no	—
225	C₂²×D₅₃	—	2	6	no	—
226	C₂²×D₅₃	—	2	48	no	—
227	C₂²×D₅₃	—	2	24	no	—
228	C₂²×D₅₃	—	2	24	no	—
229	C₂²×D₅₃	—	4	12	no	—
230	C₂²×D₅₃	—	4	24	no	—
231	C₂²×D₅₃	—	4	48	no	—
232	C₂²×D₅₃	—	4	24	no	—
233	C₂²×D₅₃	—	4	48	no	—
234	C₂²×D₅₃	—	4	128	no	—
235	C₂²×D₅₃	—	8	24	no	—
236	C₂²×D₅₃	H(5,𝔽₂)	1	12	yes	—
237	C₂²×D₅₃	C₂⁵	1	6	yes	—
238	C₂²×D₅₃	C₂⁵	1	2	yes	—
239	C₂²×D₅₃	C₂⁵	1	6	yes	—
240	Q₈×C₅₃	—	4	12	no	—
241	D₄×C₅₃	—	4	12	no	—
242	D₄×C₅₃	—	4	24	no	—
243	C₂×C₄×C₅₃	—	2	6	no	—
244	C₂×C₄×C₅₃	—	2	12	no	—
245	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
246	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
247	C₂²⋊D₅₃	—	2	48	no	—
248	C₂²⋊D₅₃	—	4	12	no	—
249	C₂²⋊D₅₃	—	4	12	no	—
250	C₂²⋊D₅₃	—	4	24	no	—
251	C₂²⋊D₅₃	—	4	24	no	—
252	C₂×Dic₅₃	—	2	6	no	—
253	C₂×Dic₅₃	—	2	12	no	—
254	C₂×Dic₅₃	—	4	12	no	—
255	C₂×Dic₅₃	C₂⁵	1	6	yes	—
256	C₂×Dic₅₃	C₂⁵	1	12	yes	—
257	D₂₁₂	—	2	48	no	—
258	D₂₁₂	—	4	12	no	—
259	D₂₁₂	—	4	12	no	—
260	C₄×D₅₃	—	2	24	no	—
261	C₄×D₅₃	—	4	24	no	—
262	C₄×D₅₃	—	4	24	no	—
263	C₂×#3532	—	53	1	no	maximal, Sylow
264	C₂²×C₅₃	—	2	6	no	—
265	C₂²×C₅₃	—	2	12	no	—
266	C₂²×C₅₃	—	2	6	no	—
267	C₂²×C₅₃	—	2	12	no	—
268	C₂²×C₅₃	—	4	16	no	—
269	C₂²×C₅₃	—	4	12	no	—
270	C₂²×C₅₃	—	4	12	no	—
271	C₂²×C₅₃	—	8	8	no	—
272	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
273	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
274	C₂×D₅₃	—	2	12	no	—
275	C₂×D₅₃	—	2	24	no	—
276	C₂×D₅₃	—	2	12	no	—
277	C₂×D₅₃	—	2	6	no	—
278	C₂×D₅₃	—	2	6	no	—
279	C₂×D₅₃	—	4	48	no	—
280	C₂×D₅₃	—	4	48	no	—
281	C₂×D₅₃	—	4	4	no	—
282	C₂×D₅₃	—	4	12	no	—
283	C₂×D₅₃	—	8	24	no	—
284	C₂×D₅₃	C₂×H(5,𝔽₂)	1	12	yes	—
285	C₄×C₅₃	—	2	6	no	—
286	C₄×C₅₃	—	2	12	no	—
287	Dic₅₃	—	2	6	no	—
288	Dic₅₃	—	2	12	no	—
289	#3532	—	53	32	no	—
290	C₂×H(3,𝔽₄)	D₅₃	1	1	yes	characteristic, complemented
291	C₂×#560	—	53	3	no	—
292	C₂×#551	—	53	3	no	—
293	C₂×#546	—	53	6	no	—
294	C₂×D₄²	—	53	3	no	—
295	C₂×#538	—	53	6	no	—
296	C₂×#535	—	53	3	no	—
297	C₂×#534	—	53	3	no	—
298	C₂²×#120	—	53	3	no	—
299	C₂×C₅₃	—	4	6	no	—
300	C₂×C₅₃	—	4	6	no	—
301	C₂×C₅₃	#3532	1	4	yes	complemented
302	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	3	yes	—
303	D₅₃	—	2	12	no	—
304	D₅₃	—	4	2	no	—
305	D₅₃	—	4	6	no	—
306	C₂⁶	—	53	1	no	—
307	C₂²×#41	—	53	3	no	—
308	C₂³×D₄	—	53	3	no	—
309	C₂³×D₄	—	53	3	no	—
310	C₂³×D₄	—	53	6	no	—
311	H(3,𝔽₄)	C₂×D₅₃	1	16	yes	complemented
312	#560	—	53	48	no	—
313	#551	—	53	48	no	—
314	#546	—	53	96	no	—
315	D₄²	—	53	48	no	—
316	#538	—	53	96	no	—
317	#535	—	53	48	no	—
318	#534	—	53	48	no	—
319	C₂×#127	—	53	3	no	—
320	C₂×#127	—	53	1	no	—
321	C₂×#126	—	53	6	no	—
322	C₂×#126	—	53	2	no	—
323	C₂×#124	—	53	3	no	—
324	C₂×#124	C₂×D₅₃	1	3	yes	complemented
325	C₂×#124	C₂×D₅₃	1	6	yes	complemented
326	C₂×#123	—	53	3	no	—
327	C₂×#123	—	53	6	no	—
328	C₂×#123	—	53	6	no	—
329	C₂×#123	—	53	3	no	—
330	C₂×C₂²⋊Q₈	—	53	6	no	—
331	C₂×#121	—	53	6	no	—
332	C₂×#121	—	53	6	no	—
333	C₂×#121	—	53	6	no	—
334	C₂×#121	—	53	6	no	—
335	C₂×#121	—	53	6	no	—
336	C₂×#120	—	53	3	no	—
337	C₂×#120	—	53	1	no	—
338	C₂×#120	—	53	3	no	—
339	C₂×#120	—	53	6	no	—
340	C₂×#120	—	53	48	no	—
341	C₂×#120	—	53	6	no	—
342	C₂×#120	—	53	3	no	—
343	C₂×#120	—	53	6	no	—
344	C₂×#120	C₂×D₅₃	1	3	yes	complemented
345	C₂×#120	C₂×D₅₃	1	3	yes	complemented
346	C₂×C₄×D₄	—	53	6	no	—
347	C₂×C₄×D₄	—	53	6	no	—
348	C₂×#117	—	53	3	no	—
349	C₂²×C₂²⋊C₄	—	53	3	no	—
350	C₂²×C₂²⋊C₄	—	53	6	no	—
351	C₂²×C₂²⋊C₄	—	53	3	no	—
352	C₅₃	C₂×#3532	1	1	yes	characteristic, complemented, Sylow
353	C₂⁵	—	53	3	no	—
354	C₂⁵	—	53	3	no	—
355	C₂⁵	—	53	3	no	—
356	C₂⁵	—	53	8	no	—
357	C₂⁵	—	212	12	no	—
358	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
359	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic, complemented
360	C₂×#41	—	53	24	no	—
361	C₂×#41	—	212	12	no	—
362	C₂²×Q₈	C₂²×D₅₃	1	3	yes	complemented
363	C₂²×D₄	—	53	6	no	—
364	C₂²×D₄	—	53	3	no	—
365	C₂²×D₄	—	53	3	no	—
366	C₂²×D₄	—	53	6	no	—
367	C₂²×D₄	—	53	6	no	—
368	C₂²×D₄	—	53	24	no	—
369	C₂²×D₄	—	53	24	no	—
370	C₂²×D₄	—	53	48	no	—
371	C₂²×D₄	—	53	6	no	—
372	C₂²×D₄	—	53	3	no	—
373	C₂²×D₄	—	53	3	no	—
374	C₂²×D₄	—	212	12	no	—
375	C₂²×D₄	—	212	24	no	—
376	C₂²×D₄	—	212	12	no	—
377	C₂²×D₄	C₂²×D₅₃	1	6	yes	—
378	C₂²×D₄	C₂²×D₅₃	1	3	yes	—
379	C₂³×C₄	—	53	3	no	—
380	C₂³×C₄	—	53	3	no	—
381	C₂³×C₄	—	53	6	no	—
382	C₂³×C₄	—	53	6	no	—
383	#127	—	53	24	no	—
384	#127	—	53	8	no	—
385	#126	—	53	48	no	—
386	#126	—	53	16	no	—
387	#124	—	53	24	no	—
388	#124	C₂²×D₅₃	1	24	yes	complemented
389	#124	C₂²×D₅₃	1	48	yes	complemented
390	#123	—	53	24	no	—
391	#123	—	53	48	no	—
392	#123	—	53	24	no	—
393	#123	—	53	48	no	—
394	C₂²⋊Q₈	—	53	48	no	—
395	#121	—	53	48	no	—
396	#121	—	53	48	no	—
397	#121	—	53	48	no	—
398	#121	—	53	48	no	—
399	#121	—	53	48	no	—
400	#120	—	53	24	no	—
401	#120	—	53	8	no	—
402	#120	—	53	48	no	—
403	#120	—	53	24	no	—
404	#120	—	53	48	no	—
405	#120	—	53	24	no	—
406	#120	—	53	48	no	—
407	#120	C₂²×D₅₃	1	24	yes	complemented
408	#120	C₂²×D₅₃	1	24	yes	complemented
409	C₄×D₄	—	53	48	no	—
410	C₄×D₄	—	53	48	no	—
411	#117	—	53	24	no	—
412	C₂×C₄⋊C₄	—	53	6	no	—
413	C₂×C₄⋊C₄	—	53	6	no	—
414	C₂×C₄⋊C₄	—	53	6	no	—
415	C₂×C₂²⋊C₄	—	53	3	no	—
416	C₂×C₂²⋊C₄	—	53	6	no	—
417	C₂×C₂²⋊C₄	—	53	3	no	—
418	C₂×C₂²⋊C₄	—	53	6	no	—
419	C₂×C₂²⋊C₄	—	53	6	no	—
420	C₂×C₂²⋊C₄	—	53	6	no	—
421	C₂×C₂²⋊C₄	—	53	6	no	—
422	C₂×C₂²⋊C₄	—	53	24	no	—
423	C₂×C₂²⋊C₄	—	53	48	no	—
424	C₂×C₂²⋊C₄	—	53	24	no	—
425	C₂×C₂²⋊C₄	C₂²×D₅₃	1	6	yes	—
426	C₂×C₂²⋊C₄	C₂²×D₅₃	1	3	yes	—
427	C₂×C₂²⋊C₄	C₂²×D₅₃	1	3	yes	complemented
428	C₂×C₂²⋊C₄	C₂²×D₅₃	1	6	yes	complemented
429	C₂×C₄²	—	53	3	no	—
430	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic, complemented
431	C₂×C₄²	C₂²×D₅₃	1	2	yes	complemented
432	C₂⁴	—	4	6	no	—
433	C₂⁴	—	4	6	no	—
434	C₂⁴	—	53	3	no	—
435	C₂⁴	—	53	1	no	—
436	C₂⁴	—	53	3	no	—
437	C₂⁴	—	53	12	no	—
438	C₂⁴	—	53	12	no	—
439	C₂⁴	—	53	12	no	—
440	C₂⁴	—	106	12	no	—
441	C₂⁴	—	106	24	no	—
442	C₂⁴	—	106	12	no	—
443	C₂⁴	—	212	12	no	—
444	C₂⁴	—	212	6	no	—
445	C₂⁴	—	212	12	no	—
446	C₂⁴	—	212	48	no	—
447	C₂⁴	—	212	64	no	—
448	C₂⁴	C₂³×D₅₃	1	3	yes	—
449	C₂⁴	C₂³×D₅₃	1	3	yes	—
450	C₂⁴	C₂³×D₅₃	1	4	yes	—
451	C₂⁴	C₂³×D₅₃	1	4	yes	complemented
452	#41	—	212	48	no	—
453	C₂×Q₈	—	4	6	no	—
454	C₂×Q₈	C₂³×D₅₃	1	12	yes	complemented
455	C₂×D₄	—	4	6	no	—
456	C₂×D₄	—	4	12	no	—
457	C₂×D₄	—	53	24	no	—
458	C₂×D₄	—	53	12	no	—
459	C₂×D₄	—	53	12	no	—
460	C₂×D₄	—	53	24	no	—
461	C₂×D₄	—	53	24	no	—
462	C₂×D₄	—	53	24	no	—
463	C₂×D₄	—	53	12	no	—
464	C₂×D₄	—	53	12	no	—
465	C₂×D₄	—	106	24	no	—
466	C₂×D₄	—	106	24	no	—
467	C₂×D₄	—	212	6	no	—
468	C₂×D₄	—	212	6	no	—
469	C₂×D₄	—	212	48	no	—
470	C₂×D₄	—	212	6	no	—
471	C₂×D₄	—	212	6	no	—
472	C₂×D₄	—	212	12	no	—
473	C₂×D₄	—	212	12	no	—
474	C₂×D₄	—	212	96	no	—
475	C₂×D₄	—	212	48	no	—
476	C₂×D₄	C₂³×D₅₃	1	24	yes	—
477	C₂×D₄	C₂³×D₅₃	1	12	yes	—
478	C₂²×C₄	—	53	6	no	—
479	C₂²×C₄	—	53	3	no	—
480	C₂²×C₄	—	53	12	no	—
481	C₂²×C₄	—	53	12	no	—
482	C₂²×C₄	—	53	24	no	—
483	C₂²×C₄	—	53	24	no	—
484	C₂²×C₄	—	106	12	no	—
485	C₂²×C₄	—	212	12	no	—
486	C₂²×C₄	—	212	12	no	—
487	C₂²×C₄	—	212	6	no	—
488	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
489	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
490	C₄⋊C₄	—	53	24	no	—
491	C₄⋊C₄	—	53	24	no	—
492	C₄⋊C₄	—	53	24	no	—
493	C₂²⋊C₄	—	53	12	no	—
494	C₂²⋊C₄	—	53	24	no	—
495	C₂²⋊C₄	—	53	12	no	—
496	C₂²⋊C₄	—	53	24	no	—
497	C₂²⋊C₄	—	53	24	no	—
498	C₂²⋊C₄	—	53	24	no	—
499	C₂²⋊C₄	—	53	24	no	—
500	C₂²⋊C₄	C₂³×D₅₃	1	24	yes	—
501	C₂²⋊C₄	C₂³×D₅₃	1	12	yes	—
502	C₂²⋊C₄	C₂³×D₅₃	1	12	yes	complemented
503	C₂²⋊C₄	C₂³×D₅₃	1	24	yes	complemented
504	C₄²	—	53	12	no	—
505	C₄²	C₂³×D₅₃	1	4	yes	complemented
506	C₄²	C₂³×D₅₃	1	8	yes	complemented
507	C₂³	—	2	6	no	—
508	C₂³	—	2	12	no	—
509	C₂³	—	2	6	no	—
510	C₂³	—	2	12	no	—
511	C₂³	—	4	12	no	—
512	C₂³	—	4	12	no	—
513	C₂³	—	4	16	no	—
514	C₂³	—	8	8	no	—
515	C₂³	—	53	12	no	—
516	C₂³	—	53	6	no	—
517	C₂³	—	53	2	no	—
518	C₂³	—	53	6	no	—
519	C₂³	—	106	24	no	—
520	C₂³	—	106	48	no	—
521	C₂³	—	106	24	no	—
522	C₂³	—	106	12	no	—
523	C₂³	—	106	12	no	—
524	C₂³	—	106	6	no	—
525	C₂³	—	106	6	no	—
526	C₂³	—	424	24	no	—
527	C₂³	—	212	24	no	—
528	C₂³	—	212	12	no	—
529	C₂³	—	212	24	no	—
530	C₂³	—	212	48	no	—
531	C₂³	—	212	48	no	—
532	C₂³	—	212	128	no	—
533	C₂³	C₂⁴×D₅₃	1	6	yes	—
534	C₂³	C₂⁴×D₅₃	1	6	yes	—
535	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
536	Q₈	—	4	12	no	—
537	D₄	—	4	12	no	—
538	D₄	—	4	24	no	—
539	D₄	—	106	48	no	—
540	D₄	—	106	48	no	—
541	D₄	—	212	12	no	—
542	D₄	—	212	12	no	—
543	D₄	—	212	12	no	—
544	D₄	—	212	12	no	—
545	D₄	—	212	24	no	—
546	D₄	—	212	24	no	—
547	C₂×C₄	—	2	6	no	—
548	C₂×C₄	—	2	12	no	—
549	C₂×C₄	—	53	12	no	—
550	C₂×C₄	—	53	6	no	—
551	C₂×C₄	—	106	24	no	—
552	C₂×C₄	—	106	6	no	—
553	C₂×C₄	—	106	12	no	—
554	C₂×C₄	—	212	24	no	—
555	C₂×C₄	—	212	24	no	—
556	C₂×C₄	—	212	12	no	—
557	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
558	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
559	C₂²	—	2	6	no	—
560	C₂²	—	2	12	no	—
561	C₂²	—	2	6	no	—
562	C₂²	—	2	12	no	—
563	C₂²	—	4	16	no	—
564	C₂²	—	4	12	no	—
565	C₂²	—	4	12	no	—
566	C₂²	—	8	8	no	—
567	C₂²	—	53	12	no	—
568	C₂²	—	106	24	no	—
569	C₂²	—	106	12	no	—
570	C₂²	—	106	12	no	—
571	C₂²	—	106	6	no	—
572	C₂²	—	106	6	no	—
573	C₂²	—	424	24	no	—
574	C₂²	—	212	48	no	—
575	C₂²	—	212	48	no	—
576	C₂²	—	212	12	no	—
577	C₂²	—	212	4	no	—
578	C₂²	H(5,𝔽₂)×D₅₃	1	6	yes	—
579	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
580	C₄	—	2	6	no	—
581	C₄	—	2	12	no	—
582	C₄	—	106	6	no	—
583	C₄	—	106	12	no	—
584	C₂	—	4	6	no	—
585	C₂	—	4	6	no	—
586	C₂	—	106	12	no	—
587	C₂	—	212	6	no	—
588	C₂	—	212	2	no	—
589	C₂	#20288872	1	4	yes	direct product factor
590	C₂	C₂×H(5,𝔽₂)×D₅₃	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×C₅₃×H(3,𝔽₄)	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×H(3,𝔽₄)	D₅₃
hypocenter	C₅₃	C₂×#3532
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×H(3,𝔽₄)

Complements

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

N	H ≅ G/N	type
1. #20288872	C₂	direct (N × H)
2. C₂×#16811362	C₂	semidirect
3. C₂×#16811303	C₂	semidirect
4. C₂×#16811266	C₂	semidirect
5. C₂×#16811255	C₂	semidirect
6. C₂×#16811207	C₂	semidirect
7. C₂×#16811191	C₂	semidirect
8. C₂×#16811183	C₂	semidirect
9. C₂×#120×D₅₃	C₂	semidirect
10. C₂×C₅₃×H(3,𝔽₄)	C₂	semidirect
11. C₂⁵×D₅₃	C₂²	semidirect
12. C₂²×#11719949	C₂²	semidirect
13. C₂³×C₂²⋊D₅₃	C₂²	semidirect
14. C₂²×D₄×D₅₃	C₂²	semidirect
15. C₂²×D₄×D₅₃	C₂²	semidirect
16. #16811362	C₂²	semidirect
17. #16811303	C₂²	semidirect
18. #16811266	C₂²	semidirect
19. #16811255	C₂²	semidirect
20. #16811207	C₂²	semidirect
21. #16811191	C₂²	semidirect
22. #16811183	C₂²	semidirect
23. C₂×#13133214	C₂²	semidirect
24. C₂×#13133213	C₂²	semidirect
25. C₂×#13133211	C₂²	semidirect
26. C₂×#13133210	C₂²	semidirect
27. C₂×#13133206	C₂²	semidirect
28. C₂×#13133201	C₂²	semidirect
29. C₂×#13133199	C₂²	semidirect
30. C₂×#13133199	C₂²	semidirect
31. C₂×#13133198	C₂²	semidirect
32. C₂×#13133193	C₂²	semidirect
33. C₂×#13133190	C₂²	semidirect
34. C₂×#13133190	C₂²	semidirect
35. C₂×#13133190	C₂²	semidirect
36. C₂×#13133189	C₂²	semidirect
37. C₂×#13133188	C₂²	semidirect
38. C₂×#13133187	C₂²	semidirect
39. C₂×#13133187	C₂²	semidirect
40. C₂×#13133187	C₂²	semidirect
41. C₂×#13133186	C₂²	semidirect
42. C₂×#13133186	C₂²	semidirect
44. C₂×#13133185	C₂²	semidirect
45. #120×D₅₃	C₂²	semidirect
46. C₂×C₄×D₂₁₂	C₂²	semidirect
47. C₂×#13133177	C₂²	semidirect
48. C₂×#13133174	C₂²	semidirect
49. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
50. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
51. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
52. C₅₃×H(3,𝔽₄)	C₂²	semidirect
53. C₂×#124×C₅₃	C₂²	semidirect
54. C₂×#124×C₅₃	C₂²	semidirect
55. C₂×#120×C₅₃	C₂²	semidirect
56. C₂×#120×C₅₃	C₂²	semidirect
61. C₂⁴×D₅₃	C₂³	semidirect
63. C₂×#11719949	C₂³	semidirect
66. C₂²×D₂₁₂	C₂³	semidirect
72. C₂²×C₂²⋊D₅₃	C₂³	semidirect
73. C₂²×C₂²⋊D₅₃	C₂³	semidirect
76. C₂×D₄×D₅₃	C₂³	semidirect
77. C₂×D₄×D₅₃	C₂³	semidirect
78. C₂³×Dic₅₃	C₂³	semidirect
81. C₂²×C₄×D₅₃	C₂³	semidirect
82. #13133214	C₂³	semidirect
83. #13133213	C₂³	semidirect
84. #13133211	C₂³	semidirect
85. #13133210	C₂³	semidirect
86. #13133206	C₂³	semidirect
87. #13133201	C₂³	semidirect
88. #13133199	C₂³	semidirect
89. #13133199	C₂³	semidirect
90. #13133198	C₂³	semidirect
91. #13133193	C₂³	semidirect
92. #13133190	C₂³	semidirect
93. #13133190	C₂³	semidirect
94. #13133190	C₂³	semidirect
95. #13133189	C₂³	semidirect
96. #13133188	C₂³	semidirect
97. #13133187	C₂³	semidirect
98. #13133187	C₂³	semidirect
99. #13133187	C₂³	semidirect
100. #13133186	C₂³	semidirect
101. #13133186	C₂³	semidirect
103. #13133185	C₂³	semidirect
104. C₄×D₂₁₂	C₂³	semidirect
105. #13133177	C₂³	semidirect
106. #13133174	C₂³	semidirect
107. C₂×C₄⋊Dic₅₃	C₂³	semidirect
108. C₂×#11719919	C₂³	semidirect
109. C₂×#11719919	C₂³	semidirect
112. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
116. C₂×#11719921	C₂³	semidirect
117. C₂²⋊C₄×D₅₃	C₂³	semidirect
118. C₂²⋊C₄×D₅₃	C₂³	semidirect
119. C₂²⋊C₄×D₅₃	C₂³	semidirect
120. C₂×C₄×Dic₅₃	C₂³	semidirect
122. C₂⁵×C₅₃	C₂³	semidirect
123. C₂²×Q₈×C₅₃	C₂³	semidirect
126. #124×C₅₃	C₂³	semidirect
127. #124×C₅₃	C₂³	semidirect
128. #120×C₅₃	C₂³	semidirect
129. #120×C₅₃	C₂³	semidirect
132. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
133. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
134. C₂×C₄²×C₅₃	C₂³	semidirect
135. C₂×C₄²×C₅₃	C₂³	semidirect
141. C₂⁴×C₅₃	C₂⁴	semidirect
157. C₂×Q₈×C₅₃	C₂⁴	semidirect
174. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
178. C₂²×Dic₅₃	C₂⁴	semidirect
187. C₂×D₂₁₂	C₂⁴	semidirect
193. C₂×C₄×D₅₃	C₂⁴	semidirect
196. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
197. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
198. C₄²×C₅₃	C₂⁴	semidirect
199. C₄²×C₅₃	C₂⁴	semidirect
202. C₂²⋊Dic₅₃	C₂⁴	semidirect
206. #11719921	C₂⁴	semidirect
207. C₄⋊Dic₅₃	C₂⁴	semidirect
208. #11719919	C₂⁴	semidirect
209. #11719919	C₂⁴	semidirect
210. C₄×Dic₅₃	C₂⁴	semidirect
290. C₂×H(3,𝔽₄)	D₅₃	semidirect
301. C₂×C₅₃	#3532	semidirect
311. H(3,𝔽₄)	C₂×D₅₃	semidirect
324. C₂×#124	C₂×D₅₃	semidirect
325. C₂×#124	C₂×D₅₃	semidirect
344. C₂×#120	C₂×D₅₃	semidirect
345. C₂×#120	C₂×D₅₃	semidirect
352. C₅₃	C₂×#3532	semidirect
359. C₂⁵	C₂²×D₅₃	semidirect
362. C₂²×Q₈	C₂²×D₅₃	semidirect
388. #124	C₂²×D₅₃	semidirect
389. #124	C₂²×D₅₃	semidirect
407. #120	C₂²×D₅₃	semidirect
408. #120	C₂²×D₅₃	semidirect
427. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
428. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
430. C₂×C₄²	C₂²×D₅₃	semidirect
431. C₂×C₄²	C₂²×D₅₃	semidirect
451. C₂⁴	C₂³×D₅₃	semidirect
454. C₂×Q₈	C₂³×D₅₃	semidirect
502. C₂²⋊C₄	C₂³×D₅₃	semidirect
503. C₂²⋊C₄	C₂³×D₅₃	semidirect
505. C₄²	C₂³×D₅₃	semidirect
506. C₄²	C₂³×D₅₃	semidirect
589. C₂	#20288872	direct (N × H)

Maximal subgroups

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

subgroup	index
#20288872	2
C₂×#16811362	2
C₂×#16811303	2
C₂×#16811266	2
C₂×#16811255	2
C₂×#16811207	2
C₂×#16811191	2
C₂×#16811183	2
C₂×#120×D₅₃	2
C₂×C₅₃×H(3,𝔽₄)	2
C₂×#3532	53

Subgroups — C2×#20288872 #28291215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288872 #28291215