Subgroups — C₂×#20289495 #28297933

Order 13568.

11138 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811459	C₂	1	1	yes	maximal, characteristic, complemented
2	#20289495	C₂	1	32	yes	maximal, direct product factor
3	C₂×#16811434	C₂	1	8	yes	maximal, complemented
4	C₂×#16811426	C₂	1	8	yes	maximal, complemented
5	C₂×#16811421	C₂	1	6	yes	maximal, complemented
6	C₂²×#13133243	C₂	1	6	yes	maximal, complemented
7	C₂×#16811390	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×H(5,𝔽₂)×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811459	—	2	8	no	—
10	#16811459	C₂²	1	16	yes	complemented
11	C₂×#13133265	—	2	1	no	—
12	C₂×#13133264	C₂²	1	4	yes	complemented
13	C₂×#13133263	C₂²	1	4	yes	complemented
14	C₂×#13133262	—	2	3	no	—
15	C₂²×#11719945	C₂²	1	6	yes	complemented
16	C₂×#41×D₅₃	—	2	4	no	—
17	#16811434	C₂²	1	128	yes	complemented
18	#16811426	C₂²	1	128	yes	complemented
19	#16811421	C₂²	1	96	yes	complemented
20	C₂×#13133248	C₂²	1	4	yes	complemented
21	C₂×#13133247	C₂²	1	12	yes	complemented
22	C₂×#13133243	C₂²	1	12	yes	complemented
23	C₂×#13133243	C₂²	1	24	yes	complemented
24	C₂×#13133243	C₂²	1	96	yes	complemented
25	C₂×#13133242	C₂²	1	12	yes	complemented
26	C₂×#13133237	C₂²	1	24	yes	complemented
27	C₂²×#11719923	C₂²	1	12	yes	complemented
28	C₂²×#11719922	C₂²	1	12	yes	complemented
29	#16811390	C₂²	1	16	yes	complemented
30	C₂×#13133224	C₂²	1	8	yes	complemented
31	C₂²×#11719917	C₂²	1	6	yes	complemented
32	C₂×H(5,𝔽₂)×C₅₃	—	2	4	no	—
33	C₂×H(5,𝔽₂)×C₅₃	—	2	8	no	—
34	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	8	yes	complemented
35	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	16	yes	complemented
36	C₂²×#41×C₅₃	—	2	1	no	—
37	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
38	C₂²×#41×C₅₃	C₂²	1	3	yes	complemented
39	C₂³×D₄×C₅₃	—	2	3	no	—
40	C₂³×D₄×C₅₃	C₂²	1	3	yes	complemented
41	#13133265	—	2	16	no	—
42	#13133264	—	2	16	no	—
43	#13133264	C₂³	1	32	yes	complemented
44	#13133263	—	2	16	no	—
45	#13133263	C₂³	1	32	yes	complemented
46	#13133262	—	2	48	no	—
47	C₂×#11719949	—	2	4	no	—
48	C₂×#11719945	—	2	6	no	—
49	C₂×#11719945	—	2	6	no	—
50	C₂×#11719945	—	2	24	no	—
51	C₂×#11719945	C₂³	1	4	yes	—
52	C₂×#11719945	C₂³	1	12	yes	complemented
53	C₂×#11719945	C₂³	1	48	yes	complemented
54	C₂×#11719947	—	2	12	no	—
55	#41×D₅₃	—	2	64	no	—
56	C₂²×Dic₁₀₆	C₂³	1	6	yes	complemented
57	C₂×Q₈×D₅₃	—	2	4	no	—
58	C₂²×D₂₁₂	C₂³	1	6	yes	complemented
59	C₂²×C₂²⋊D₅₃	—	2	6	no	—
60	C₂×D₄×D₅₃	—	2	12	no	—
61	C₂²×C₄×D₅₃	—	2	6	no	—
62	#13133248	C₂³	1	32	yes	complemented
63	#13133247	C₂³	1	96	yes	complemented
64	#13133243	C₂³	1	96	yes	complemented
65	#13133243	C₂³	1	192	yes	complemented
66	#13133242	C₂³	1	96	yes	complemented
67	#13133237	C₂³	1	192	yes	complemented
68	C₂×#11719925	C₂³	1	8	yes	complemented
69	C₂×#11719923	C₂³	1	24	yes	complemented
70	C₂×#11719923	C₂³	1	96	yes	complemented
71	C₂×#11719924	C₂³	1	8	yes	complemented
72	C₂×#11719922	C₂³	1	24	yes	complemented
73	C₂×#11719922	C₂³	1	96	yes	complemented
74	#13133224	C₂³	1	64	yes	complemented
75	C₂×#11719917	C₂³	1	4	yes	—
76	C₂×#11719917	C₂³	1	12	yes	complemented
77	C₂×#11719917	C₂³	1	48	yes	complemented
78	C₂²×#88322	C₂³	1	12	yes	complemented
79	C₂⁵×C₅₃	—	2	3	no	—
80	H(5,𝔽₂)×C₅₃	—	2	32	no	—
81	H(5,𝔽₂)×C₅₃	—	2	64	no	—
82	H(5,𝔽₂)×C₅₃	C₂³	1	64	yes	complemented
83	C₂×#41×C₅₃	—	2	8	no	—
84	C₂×#41×C₅₃	—	2	16	no	—
85	C₂×#41×C₅₃	—	2	6	no	—
86	C₂×#41×C₅₃	—	2	12	no	—
87	C₂×#41×C₅₃	D₄	1	4	yes	—
88	C₂×#41×C₅₃	D₄	1	8	yes	—
89	C₂×#41×C₅₃	C₂³	1	12	yes	—
90	C₂×#41×C₅₃	C₂³	1	4	yes	—
91	C₂×#41×C₅₃	C₂³	1	8	yes	—
92	C₂×#41×C₅₃	C₂³	1	24	yes	complemented
93	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
94	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₂²×D₄×C₅₃	—	2	3	no	—
96	C₂²×D₄×C₅₃	—	2	24	no	—
97	C₂²×D₄×C₅₃	—	2	48	no	—
98	C₂²×D₄×C₅₃	—	2	3	no	—
99	C₂²×D₄×C₅₃	—	2	6	no	—
100	C₂²×D₄×C₅₃	—	2	12	no	—
101	C₂²×D₄×C₅₃	D₄	1	3	yes	—
102	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
103	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
104	C₂²×D₄×C₅₃	C₂³	1	24	yes	complemented
105	C₂³×C₄×C₅₃	—	2	3	no	—
106	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
107	C₂⁴×C₅₃	—	2	3	no	—
108	C₂⁴×C₅₃	—	2	12	no	—
109	C₂⁴×C₅₃	—	2	24	no	—
110	C₂⁴×C₅₃	—	8	12	no	—
111	C₂⁴×C₅₃	C₂×D₄	1	3	yes	—
112	C₂³×D₅₃	—	2	6	no	—
113	#41×C₅₃	—	2	64	no	—
114	#41×C₅₃	—	2	24	no	—
115	#41×C₅₃	—	2	48	no	—
116	#41×C₅₃	C₂×D₄	1	32	yes	—
117	#41×C₅₃	C₂×D₄	1	16	yes	—
118	#41×C₅₃	C₂⁴	1	48	yes	—
119	#41×C₅₃	C₂⁴	1	16	yes	—
120	C₂×Q₈×C₅₃	—	2	1	no	—
121	C₂×Q₈×C₅₃	—	2	2	no	—
122	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
123	C₂×Q₈×C₅₃	C₂×D₄	1	8	yes	—
124	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
125	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
126	C₂×D₄×C₅₃	—	2	12	no	—
127	C₂×D₄×C₅₃	—	2	24	no	—
128	C₂×D₄×C₅₃	—	2	192	no	—
129	C₂×D₄×C₅₃	—	2	12	no	—
130	C₂×D₄×C₅₃	—	2	3	no	—
131	C₂×D₄×C₅₃	—	2	6	no	—
132	C₂×D₄×C₅₃	—	2	24	no	—
133	C₂×D₄×C₅₃	—	2	24	no	—
134	C₂×D₄×C₅₃	—	2	48	no	—
135	C₂×D₄×C₅₃	C₂×D₄	1	12	yes	—
136	C₂×D₄×C₅₃	C₂×D₄	1	24	yes	—
137	C₂×D₄×C₅₃	C₂⁴	1	6	yes	—
138	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
139	C₂×D₄×C₅₃	C₂⁴	1	48	yes	—
140	C₂²×C₄×C₅₃	—	2	1	no	—
141	C₂²×C₄×C₅₃	—	2	12	no	—
142	C₂²×C₄×C₅₃	—	2	24	no	—
143	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
144	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
145	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
146	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
147	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
148	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
149	C₂×C₂²⋊D₅₃	—	2	12	no	—
150	C₂×C₂²⋊D₅₃	—	2	4	no	—
151	C₂×C₂²⋊D₅₃	—	2	48	no	—
152	C₂²×Dic₅₃	—	2	6	no	—
153	#11719949	—	2	32	no	—
154	Q₈×D₅₃	—	2	32	no	—
155	#11719947	—	2	96	no	—
156	D₄×D₅₃	—	2	96	no	—
157	#11719945	—	2	48	no	—
158	#11719945	—	2	8	no	—
159	#11719945	—	2	48	no	—
160	#11719945	—	2	24	no	—
161	#11719945	C₂⁴	1	16	yes	—
162	#11719945	C₂⁴	1	48	yes	complemented
163	C₂×D₂₁₂	—	2	6	no	—
164	C₂×D₂₁₂	—	2	12	no	—
165	C₂×D₂₁₂	C₂⁴	1	4	yes	—
166	C₂×D₂₁₂	C₂⁴	1	24	yes	complemented
167	C₂×C₄×D₅₃	—	2	4	no	—
168	C₂×C₄×D₅₃	—	2	12	no	—
169	C₂×C₄×D₅₃	—	2	48	no	—
170	C₂×Dic₁₀₆	—	2	6	no	—
171	C₂×Dic₁₀₆	—	2	12	no	—
172	C₂×Dic₁₀₆	C₂⁴	1	4	yes	—
173	C₂×Dic₁₀₆	C₂⁴	1	24	yes	complemented
174	#11719925	C₂⁴	1	32	yes	complemented
175	#11719924	C₂⁴	1	32	yes	complemented
176	#11719923	C₂⁴	1	96	yes	complemented
177	#11719922	C₂⁴	1	96	yes	complemented
178	#11719917	C₂⁴	1	16	yes	—
179	#11719917	C₂⁴	1	48	yes	complemented
180	C₂×#88322	C₂⁴	1	8	yes	—
181	C₂×#88322	C₂⁴	1	48	yes	complemented
182	C₂³×C₅₃	—	2	12	no	—
183	C₂³×C₅₃	—	2	6	no	—
184	C₂³×C₅₃	—	2	48	no	—
185	C₂³×C₅₃	—	4	6	no	—
186	C₂³×C₅₃	—	4	12	no	—
187	C₂³×C₅₃	—	8	24	no	—
188	C₂³×C₅₃	—	8	48	no	—
189	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
190	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
191	C₂³×C₅₃	C₂²×D₄	1	12	yes	—
192	C₂²×D₅₃	—	2	4	no	—
193	C₂²×D₅₃	—	2	24	no	—
194	C₂²×D₅₃	—	8	12	no	—
195	Q₈×C₅₃	—	2	2	no	—
196	Q₈×C₅₃	—	2	4	no	—
197	Q₈×C₅₃	C₂²×D₄	1	16	yes	—
198	Q₈×C₅₃	C₂⁵	1	4	yes	—
199	D₄×C₅₃	—	2	48	no	—
200	D₄×C₅₃	—	2	6	no	—
201	D₄×C₅₃	—	2	48	no	—
202	D₄×C₅₃	—	2	12	no	—
203	D₄×C₅₃	C₂²×D₄	1	48	yes	—
204	D₄×C₅₃	C₂⁵	1	12	yes	—
205	C₂×C₄×C₅₃	—	2	4	no	—
206	C₂×C₄×C₅₃	—	2	2	no	—
207	C₂×C₄×C₅₃	—	2	48	no	—
208	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
209	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
210	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
212	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
213	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
215	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
216	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
217	C₂²⋊D₅₃	—	2	16	no	—
218	C₂²⋊D₅₃	—	2	48	no	—
219	C₂×Dic₅₃	—	2	4	no	—
220	C₂×Dic₅₃	—	2	24	no	—
221	D₂₁₂	—	2	4	no	—
222	D₂₁₂	—	2	24	no	—
223	D₂₁₂	C₂⁵	1	8	yes	—
224	C₄×D₅₃	—	2	16	no	—
225	C₄×D₅₃	—	2	48	no	—
226	Dic₁₀₆	—	2	4	no	—
227	Dic₁₀₆	—	2	24	no	—
228	Dic₁₀₆	C₂⁵	1	8	yes	—
229	#88322	C₂⁵	1	16	yes	—
230	C₂×#3585	—	53	1	no	maximal, Sylow
231	C₂²×C₅₃	—	2	12	no	—
232	C₂²×C₅₃	—	2	2	no	—
233	C₂²×C₅₃	—	2	12	no	—
234	C₂²×C₅₃	—	4	12	no	—
235	C₂²×C₅₃	—	4	24	no	—
236	C₂²×C₅₃	—	4	12	no	—
237	C₂²×C₅₃	—	8	48	no	—
238	C₂²×C₅₃	C₂³×D₄	1	12	yes	—
239	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
240	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
241	C₂×D₅₃	—	2	8	no	—
242	C₂×D₅₃	—	4	8	no	—
243	C₂×D₅₃	—	8	24	no	—
244	C₄×C₅₃	—	2	4	no	—
245	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
246	C₄×C₅₃	C₂³×D₄	1	12	yes	—
247	C₄×C₅₃	C₂³×D₄	1	2	yes	—
248	C₄×C₅₃	C₂⁶	1	1	yes	characteristic
249	Dic₅₃	—	2	8	no	—
250	C₂×#585	—	53	1	no	—
251	C₂²×H(5,𝔽₂)	D₅₃	1	1	yes	characteristic, complemented
252	#3585	—	53	32	no	—
253	C₂×#577	—	53	8	no	—
254	C₂×#576	—	53	8	no	—
255	C₂×#575	—	53	6	no	—
256	C₂²×AGL(1,ℤ/8)	—	53	6	no	—
257	C₂×#568	—	53	1	no	—
258	C₂×C₅₃	—	2	12	no	—
259	C₂×C₅₃	—	2	2	no	—
260	C₂×C₅₃	—	4	12	no	—
261	C₂×C₅₃	#3585	1	2	yes	complemented
262	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
263	D₅₃	—	4	8	no	—
264	#585	—	53	16	no	—
265	#585	—	106	8	no	—
266	C₂×2₋¹⁺⁴	—	53	4	no	—
267	C₂×2₋¹⁺⁴	—	106	1	no	—
268	C₂×H(5,𝔽₂)	—	2	4	no	—
269	C₂×H(5,𝔽₂)	—	2	8	no	—
270	C₂×H(5,𝔽₂)	—	53	4	no	—
271	C₂×H(5,𝔽₂)	—	106	3	no	—
272	C₂×H(5,𝔽₂)	C₂×D₅₃	1	8	yes	complemented
273	C₂×H(5,𝔽₂)	C₂×D₅₃	1	16	yes	complemented
274	C₂²×#41	—	2	1	no	—
275	C₂²×#41	—	53	6	no	—
276	C₂²×#41	—	106	4	no	—
277	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
278	C₂²×#41	C₂×D₅₃	1	3	yes	complemented
279	C₂³×D₄	—	2	3	no	—
280	C₂³×D₄	C₂×D₅₃	1	3	yes	complemented
281	#577	—	53	128	no	—
282	#576	—	53	128	no	—
283	#575	—	53	96	no	—
284	C₂×#137	—	53	12	no	—
285	C₂×#137	—	53	4	no	—
286	C₂×AGL(1,ℤ/8)	—	53	12	no	—
287	C₂×AGL(1,ℤ/8)	—	53	12	no	—
288	C₂×AGL(1,ℤ/8)	—	53	24	no	—
289	C₂×AGL(1,ℤ/8)	—	53	96	no	—
290	C₂×#135	—	53	24	no	—
291	C₂²×SD₁₆	—	53	12	no	—
292	C₂²×D₈	—	53	12	no	—
293	#568	—	53	16	no	—
294	C₂×#131	—	53	8	no	—
295	C₂²×M₁₆	—	53	6	no	—
296	C₅₃	C₂×#3585	1	1	yes	characteristic, complemented, Sylow
297	C₂⁵	—	2	3	no	—
298	2₋¹⁺⁴	—	53	32	no	—
299	2₋¹⁺⁴	—	106	16	no	—
300	2₋¹⁺⁴	—	106	16	no	—
301	H(5,𝔽₂)	—	2	32	no	—
302	H(5,𝔽₂)	—	2	64	no	—
303	H(5,𝔽₂)	—	53	32	no	—
304	H(5,𝔽₂)	—	106	16	no	—
305	H(5,𝔽₂)	—	106	48	no	—
306	H(5,𝔽₂)	C₂²×D₅₃	1	64	yes	complemented
307	C₂×#41	—	2	8	no	—
308	C₂×#41	—	2	16	no	—
309	C₂×#41	—	2	6	no	—
310	C₂×#41	—	2	12	no	—
311	C₂×#41	—	53	12	no	—
312	C₂×#41	—	53	4	no	—
313	C₂×#41	—	53	48	no	—
314	C₂×#41	—	106	64	no	—
315	C₂×#41	—	106	4	no	—
316	C₂×#41	—	106	12	no	—
317	C₂×#41	—	106	6	no	—
318	C₂×#41	—	106	6	no	—
319	C₂×#41	—	106	24	no	—
320	C₂×#41	C₂²⋊D₅₃	1	4	yes	—
321	C₂×#41	C₂²⋊D₅₃	1	8	yes	—
322	C₂×#41	C₂²×D₅₃	1	12	yes	—
323	C₂×#41	C₂²×D₅₃	1	4	yes	—
324	C₂×#41	C₂²×D₅₃	1	8	yes	—
325	C₂×#41	C₂²×D₅₃	1	24	yes	complemented
326	C₂²×Q₈	—	53	6	no	—
327	C₂²×Q₈	—	106	4	no	—
328	C₂²×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
329	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
330	C₂²×D₄	—	2	3	no	—
331	C₂²×D₄	—	2	24	no	—
332	C₂²×D₄	—	2	48	no	—
333	C₂²×D₄	—	2	3	no	—
334	C₂²×D₄	—	2	6	no	—
335	C₂²×D₄	—	2	12	no	—
336	C₂²×D₄	—	53	6	no	—
337	C₂²×D₄	—	106	6	no	—
338	C₂²×D₄	—	106	12	no	—
339	C₂²×D₄	C₂²⋊D₅₃	1	3	yes	—
340	C₂²×D₄	C₂²×D₅₃	1	12	yes	—
341	C₂²×D₄	C₂²×D₅₃	1	3	yes	—
342	C₂²×D₄	C₂²×D₅₃	1	24	yes	complemented
343	C₂³×C₄	—	2	3	no	—
344	C₂³×C₄	—	106	6	no	—
345	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
346	#137	—	53	96	no	—
347	#137	—	53	32	no	—
348	AGL(1,ℤ/8)	—	53	96	no	—
349	AGL(1,ℤ/8)	—	53	96	no	—
350	AGL(1,ℤ/8)	—	53	192	no	—
351	#135	—	53	192	no	—
352	C₂×Q₁₆	—	53	8	no	—
353	C₂×SD₁₆	—	53	8	no	—
354	C₂×SD₁₆	—	53	24	no	—
355	C₂×SD₁₆	—	53	96	no	—
356	C₂×D₈	—	53	24	no	—
357	C₂×D₈	—	53	96	no	—
358	#131	—	53	64	no	—
359	C₂×M₁₆	—	53	4	no	—
360	C₂×M₁₆	—	53	12	no	—
361	C₂×M₁₆	—	53	48	no	—
362	C₂²×C₈	—	53	12	no	—
363	C₂⁴	—	2	3	no	—
364	C₂⁴	—	2	12	no	—
365	C₂⁴	—	2	24	no	—
366	C₂⁴	—	8	12	no	—
367	C₂⁴	—	106	6	no	—
368	C₂⁴	C₂×C₂²⋊D₅₃	1	3	yes	—
369	#41	—	2	64	no	—
370	#41	—	2	24	no	—
371	#41	—	2	48	no	—
372	#41	—	53	16	no	—
373	#41	—	53	48	no	—
374	#41	—	106	32	no	—
375	#41	—	106	96	no	—
376	#41	—	106	48	no	—
377	#41	—	106	8	no	—
378	#41	—	106	48	no	—
379	#41	—	106	24	no	—
380	#41	C₂×C₂²⋊D₅₃	1	32	yes	—
381	#41	C₂×C₂²⋊D₅₃	1	16	yes	—
382	#41	C₂³×D₅₃	1	48	yes	—
383	#41	C₂³×D₅₃	1	16	yes	—
384	C₂×Q₈	—	2	2	no	—
385	C₂×Q₈	—	2	1	no	—
386	C₂×Q₈	—	53	4	no	—
387	C₂×Q₈	—	53	24	no	—
388	C₂×Q₈	—	106	32	no	—
389	C₂×Q₈	—	106	6	no	—
390	C₂×Q₈	—	106	12	no	—
391	C₂×Q₈	C₂×C₂²⋊D₅₃	1	4	yes	—
392	C₂×Q₈	C₂×C₂²⋊D₅₃	1	8	yes	—
393	C₂×Q₈	C₂³×D₅₃	1	2	yes	—
394	C₂×Q₈	C₂³×D₅₃	1	4	yes	—
395	C₂×D₄	—	2	12	no	—
396	C₂×D₄	—	2	24	no	—
397	C₂×D₄	—	2	192	no	—
398	C₂×D₄	—	2	6	no	—
399	C₂×D₄	—	2	12	no	—
400	C₂×D₄	—	2	3	no	—
401	C₂×D₄	—	2	24	no	—
402	C₂×D₄	—	2	24	no	—
403	C₂×D₄	—	2	48	no	—
404	C₂×D₄	—	53	4	no	—
405	C₂×D₄	—	53	24	no	—
406	C₂×D₄	—	106	4	no	—
407	C₂×D₄	—	106	12	no	—
408	C₂×D₄	—	106	48	no	—
409	C₂×D₄	—	106	96	no	—
410	C₂×D₄	—	106	6	no	—
411	C₂×D₄	—	106	12	no	—
412	C₂×D₄	C₂×C₂²⋊D₅₃	1	12	yes	—
413	C₂×D₄	C₂×C₂²⋊D₅₃	1	24	yes	—
414	C₂×D₄	C₂³×D₅₃	1	6	yes	—
415	C₂×D₄	C₂³×D₅₃	1	48	yes	—
416	C₂×D₄	C₂³×D₅₃	1	12	yes	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	12	no	—
419	C₂²×C₄	—	2	24	no	—
420	C₂²×C₄	—	106	6	no	—
421	C₂²×C₄	—	106	4	no	—
422	C₂²×C₄	—	106	12	no	—
423	C₂²×C₄	—	106	48	no	—
424	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
425	C₂²×C₄	C₂×C₂²⋊D₅₃	1	6	yes	—
426	C₂²×C₄	C₂×C₂²⋊D₅₃	1	12	yes	—
427	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
428	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
429	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
430	Q₁₆	—	53	32	no	—
431	SD₁₆	—	53	32	no	—
432	SD₁₆	—	53	96	no	—
433	D₈	—	53	96	no	—
434	M₁₆	—	53	16	no	—
435	M₁₆	—	53	48	no	—
436	C₂×C₈	—	53	8	no	—
437	C₂×C₈	—	53	48	no	—
438	C₂³	—	2	12	no	—
439	C₂³	—	2	6	no	—
440	C₂³	—	2	48	no	—
441	C₂³	—	4	6	no	—
442	C₂³	—	4	12	no	—
443	C₂³	—	8	24	no	—
444	C₂³	—	8	48	no	—
445	C₂³	—	106	24	no	—
446	C₂³	—	106	4	no	—
447	C₂³	—	424	12	no	—
448	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
449	C₂³	C₂²×C₂²⋊D₅₃	1	6	yes	—
450	C₂³	C₂²×C₂²⋊D₅₃	1	12	yes	—
451	Q₈	—	2	4	no	—
452	Q₈	—	2	2	no	—
453	Q₈	—	53	8	no	—
454	Q₈	—	106	24	no	—
455	Q₈	—	106	4	no	—
456	Q₈	C₂²×C₂²⋊D₅₃	1	16	yes	—
457	Q₈	C₂⁴×D₅₃	1	4	yes	—
458	D₄	—	2	48	no	—
459	D₄	—	2	48	no	—
460	D₄	—	2	12	no	—
461	D₄	—	2	6	no	—
462	D₄	—	53	8	no	—
463	D₄	—	106	16	no	—
464	D₄	—	106	48	no	—
465	D₄	—	106	24	no	—
466	D₄	—	106	4	no	—
467	D₄	C₂²×C₂²⋊D₅₃	1	48	yes	—
468	D₄	C₂⁴×D₅₃	1	12	yes	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	48	no	—
472	C₂×C₄	—	106	24	no	—
473	C₂×C₄	—	106	4	no	—
474	C₂×C₄	—	106	16	no	—
475	C₂×C₄	—	106	48	no	—
476	C₂×C₄	C₂²×C₂²⋊D₅₃	1	12	yes	—
477	C₂×C₄	C₂²×C₂²⋊D₅₃	1	24	yes	—
478	C₂×C₄	C₂²×C₂²⋊D₅₃	1	2	yes	—
479	C₂×C₄	C₂²×C₂²⋊D₅₃	1	12	yes	—
480	C₂×C₄	C₂²×C₂²⋊D₅₃	1	6	yes	—
481	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
482	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
483	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
484	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
485	C₈	—	53	16	no	—
486	C₂²	—	2	2	no	—
487	C₂²	—	2	12	no	—
488	C₂²	—	2	12	no	—
489	C₂²	—	4	12	no	—
490	C₂²	—	4	24	no	—
491	C₂²	—	4	12	no	—
492	C₂²	—	8	48	no	—
493	C₂²	—	106	8	no	—
494	C₂²	—	424	24	no	—
495	C₂²	—	212	8	no	—
496	C₂²	C₂³×C₂²⋊D₅₃	1	2	yes	—
497	C₂²	C₂³×C₂²⋊D₅₃	1	12	yes	—
498	C₂²	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
499	C₄	—	2	4	no	—
500	C₄	—	106	8	no	—
501	C₄	C₂³×C₂²⋊D₅₃	1	12	yes	—
502	C₄	C₂³×C₂²⋊D₅₃	1	2	yes	—
503	C₄	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
504	C₄	C₂⁵×D₅₃	1	1	yes	characteristic
505	C₂	—	2	2	no	—
506	C₂	—	2	12	no	—
507	C₂	—	4	12	no	—
508	C₂	—	212	8	no	—
509	C₂	#20289495	1	2	yes	direct product factor
510	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×C₂²⋊D₅₃
commutator, cosocle	C₄×C₅₃	C₂⁶
Frattini	C₄	C₂⁵×D₅₃
Fitting	C₂²×H(5,𝔽₂)×C₅₃	C₂
socle	C₂²×C₅₃	C₂³×D₄
hypercenter	C₂²×H(5,𝔽₂)	D₅₃
hypocenter	C₅₃	C₂×#3585
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×#16811459	C₂	semidirect
2. #20289495	C₂	direct (N × H)
3. C₂×#16811434	C₂	semidirect
4. C₂×#16811426	C₂	semidirect
5. C₂×#16811421	C₂	semidirect
6. C₂²×#13133243	C₂	semidirect
7. C₂×#16811390	C₂	semidirect
8. C₂²×H(5,𝔽₂)×C₅₃	C₂	semidirect
10. #16811459	C₂²	semidirect
12. C₂×#13133264	C₂²	semidirect
13. C₂×#13133263	C₂²	semidirect
15. C₂²×#11719945	C₂²	semidirect
17. #16811434	C₂²	semidirect
18. #16811426	C₂²	semidirect
19. #16811421	C₂²	semidirect
20. C₂×#13133248	C₂²	semidirect
21. C₂×#13133247	C₂²	semidirect
22. C₂×#13133243	C₂²	semidirect
23. C₂×#13133243	C₂²	semidirect
24. C₂×#13133243	C₂²	semidirect
25. C₂×#13133242	C₂²	semidirect
26. C₂×#13133237	C₂²	semidirect
27. C₂²×#11719923	C₂²	semidirect
28. C₂²×#11719922	C₂²	semidirect
29. #16811390	C₂²	semidirect
30. C₂×#13133224	C₂²	semidirect
31. C₂²×#11719917	C₂²	semidirect
34. C₂×H(5,𝔽₂)×C₅₃	C₂²	semidirect
35. C₂×H(5,𝔽₂)×C₅₃	C₂²	semidirect
38. C₂²×#41×C₅₃	C₂²	semidirect
40. C₂³×D₄×C₅₃	C₂²	semidirect
43. #13133264	C₂³	semidirect
45. #13133263	C₂³	semidirect
52. C₂×#11719945	C₂³	semidirect
53. C₂×#11719945	C₂³	semidirect
56. C₂²×Dic₁₀₆	C₂³	semidirect
58. C₂²×D₂₁₂	C₂³	semidirect
62. #13133248	C₂³	semidirect
63. #13133247	C₂³	semidirect
64. #13133243	C₂³	semidirect
65. #13133243	C₂³	semidirect
66. #13133242	C₂³	semidirect
67. #13133237	C₂³	semidirect
68. C₂×#11719925	C₂³	semidirect
69. C₂×#11719923	C₂³	semidirect
70. C₂×#11719923	C₂³	semidirect
71. C₂×#11719924	C₂³	semidirect
72. C₂×#11719922	C₂³	semidirect
73. C₂×#11719922	C₂³	semidirect
74. #13133224	C₂³	semidirect
76. C₂×#11719917	C₂³	semidirect
77. C₂×#11719917	C₂³	semidirect
78. C₂²×#88322	C₂³	semidirect
82. H(5,𝔽₂)×C₅₃	C₂³	semidirect
92. C₂×#41×C₅₃	C₂³	semidirect
104. C₂²×D₄×C₅₃	C₂³	semidirect
162. #11719945	C₂⁴	semidirect
166. C₂×D₂₁₂	C₂⁴	semidirect
173. C₂×Dic₁₀₆	C₂⁴	semidirect
174. #11719925	C₂⁴	semidirect
175. #11719924	C₂⁴	semidirect
176. #11719923	C₂⁴	semidirect
177. #11719922	C₂⁴	semidirect
179. #11719917	C₂⁴	semidirect
181. C₂×#88322	C₂⁴	semidirect
251. C₂²×H(5,𝔽₂)	D₅₃	semidirect
261. C₂×C₅₃	#3585	semidirect
272. C₂×H(5,𝔽₂)	C₂×D₅₃	semidirect
273. C₂×H(5,𝔽₂)	C₂×D₅₃	semidirect
278. C₂²×#41	C₂×D₅₃	semidirect
280. C₂³×D₄	C₂×D₅₃	semidirect
296. C₅₃	C₂×#3585	semidirect
306. H(5,𝔽₂)	C₂²×D₅₃	semidirect
325. C₂×#41	C₂²×D₅₃	semidirect
342. C₂²×D₄	C₂²×D₅₃	semidirect
509. C₂	#20289495	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×#16811459	2
#20289495	2
C₂×#16811434	2
C₂×#16811426	2
C₂×#16811421	2
C₂²×#13133243	2
C₂×#16811390	2
C₂²×H(5,𝔽₂)×C₅₃	2
C₂×#3585	53

Subgroups — C2×#20289495 #28297933

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20289495 #28297933