Subgroups — C₂×#20286515 #28260318

Order 13568.

2766 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811428	C₂	1	2	yes	maximal, complemented
2	C₂×#16811425	C₂	1	2	yes	maximal, complemented
3	#20286515	C₂	1	16	yes	maximal, direct product factor
4	C₂×#16811000	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810894	C₂	1	2	yes	maximal, complemented
6	C₂×#16810893	C₂	1	2	yes	maximal, complemented
7	C₂×#16810792	C₂	1	4	yes	maximal, complemented
8	C₂×#16810741	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133263	C₂²	1	4	yes	complemented
11	#16811428	—	2	8	no	—
12	#16811428	C₂²	1	16	yes	complemented
13	#16811425	—	2	8	no	—
14	#16811425	C₂²	1	16	yes	complemented
15	C₂×#13133245	—	2	4	no	—
16	C₂×#13133245	C₂²	1	4	yes	complemented
17	C₂×#13133244	—	2	4	no	—
18	C₂×#13133242	C₂²	1	4	yes	complemented
19	C₂×#13133240	—	2	2	no	—
20	C₂×#13133239	—	2	2	no	—
21	C₂×#13133239	C₂²	1	1	yes	characteristic
22	C₂²×D₄₂₄	—	2	2	no	—
23	C₂²×D₄₂₄	C₂²	1	2	yes	complemented
24	C₂²×D₄₂₄	C₂²	1	2	yes	complemented
25	C₂×D₈×D₅₃	—	2	2	no	—
26	C₂×#13133223	C₂²	1	2	yes	complemented
27	C₂×#13133214	C₂²	1	2	yes	complemented
28	#16811000	C₂²	1	8	yes	complemented
29	#16810894	C₂²	1	16	yes	complemented
30	#16810893	C₂²	1	16	yes	complemented
31	#16810792	C₂²	1	32	yes	complemented
32	#16810741	C₂²	1	8	yes	complemented
33	C₂×#13133153	C₂²	1	1	yes	characteristic, complemented
34	C₂×#13133145	C₂²	1	2	yes	complemented
35	C₂×#13133133	C₂²	1	4	yes	complemented
36	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×#131×C₅₃	C₂²	1	2	yes	complemented
38	C₅₃×#443	C₂²	1	8	yes	complemented
39	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic, complemented
40	C₂×#104×C₅₃	C₂²	1	2	yes	complemented
41	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
42	#13133263	—	2	8	no	—
43	#13133263	—	2	8	no	—
44	#13133263	—	4	4	no	—
45	#13133263	C₂³	1	16	yes	complemented
46	C₂×#11719949	—	2	4	no	—
47	C₂×#11719945	—	2	4	no	—
48	C₂×#11719945	C₂³	1	2	yes	—
49	C₂²×D₂₁₂	—	2	4	no	—
50	C₂²×D₂₁₂	—	4	2	no	—
51	C₂²×D₂₁₂	C₂³	1	4	yes	—
52	C₂×D₄×D₅₃	—	2	4	no	—
53	C₂×D₄×D₅₃	—	4	4	no	—
54	#13133245	—	2	32	no	—
55	#13133245	—	2	8	no	—
56	#13133245	C₂³	1	16	yes	complemented
57	#13133244	—	2	32	no	—
58	#13133242	—	2	8	no	—
59	#13133242	C₂³	1	16	yes	complemented
60	#13133240	—	2	16	no	—
61	#13133239	—	2	16	no	—
62	#13133239	D₄	1	4	yes	complemented
63	#13133239	C₂³	1	4	yes	—
64	C₂×Dic₂₁₂	D₄	1	1	yes	characteristic, complemented
65	C₂×#11719913	—	2	4	no	—
66	C₂×#11719913	D₄	1	2	yes	complemented
67	C₂×#11719924	—	2	4	no	—
68	C₂×D₄₂₄	—	2	2	no	—
69	C₂×D₄₂₄	—	2	2	no	—
70	C₂×D₄₂₄	—	2	2	no	—
71	C₂×D₄₂₄	—	2	2	no	—
72	C₂×D₄₂₄	—	2	4	no	—
73	C₂×D₄₂₄	—	2	4	no	—
74	C₂×D₄₂₄	—	2	16	no	—
75	C₂×D₄₂₄	—	2	4	no	—
76	C₂×D₄₂₄	D₄	1	1	yes	characteristic, complemented
77	C₂×D₄₂₄	C₂³	1	8	yes	complemented
78	C₂×D₄₂₄	C₂³	1	8	yes	complemented
79	C₂×#11719922	—	2	4	no	—
80	D₈×D₅₃	—	2	16	no	—
81	#13133223	—	2	4	no	—
82	#13133223	C₂³	1	8	yes	complemented
83	C₂×#11719917	C₂³	1	2	yes	—
84	C₂×#11719912	—	2	2	no	—
85	C₂×C₈×D₅₃	—	2	2	no	—
86	#13133214	—	4	2	no	—
87	#13133214	C₂³	1	8	yes	complemented
88	#13133153	C₂³	1	4	yes	complemented
89	#13133145	C₂³	1	8	yes	complemented
90	#13133133	C₂³	1	16	yes	complemented
91	C₂×#41×C₅₃	C₂³	1	2	yes	—
92	#135×C₅₃	D₄	1	4	yes	complemented
93	#135×C₅₃	C₂³	1	4	yes	—
94	C₂×Q₁₆×C₅₃	D₄	1	1	yes	characteristic, complemented
95	C₂×SD₁₆×C₅₃	D₄	1	2	yes	complemented
96	C₂×D₈×C₅₃	D₄	1	1	yes	characteristic, complemented
97	#131×C₅₃	—	2	4	no	—
98	#131×C₅₃	C₂³	1	8	yes	complemented
99	C₂×M₁₆×C₅₃	—	2	2	no	—
100	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
101	C₂²×C₈×C₅₃	—	2	2	no	—
102	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
103	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
105	#108×C₅₃	C₂³	1	4	yes	complemented
106	#104×C₅₃	C₂³	1	8	yes	complemented
107	C₄×C₈×C₅₃	C₂³	1	4	yes	complemented
108	C₂³×D₅₃	—	2	4	no	—
109	C₂³×D₅₃	—	4	4	no	—
110	#41×C₅₃	—	2	2	no	—
111	#41×C₅₃	—	2	2	no	—
112	#41×C₅₃	C₂×D₄	1	4	yes	—
113	#41×C₅₃	C₂⁴	1	4	yes	—
114	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
115	C₂×D₄×C₅₃	—	2	2	no	—
116	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₅₃	—	2	1	no	—
118	C₂²×C₄×C₅₃	—	2	2	no	—
119	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
120	C₂×C₂²⋊D₅₃	—	2	2	no	—
121	C₂×C₂²⋊D₅₃	—	4	4	no	—
122	#11719949	—	2	16	no	—
123	#11719949	—	4	8	no	—
124	D₄×D₅₃	—	2	16	no	—
125	D₄×D₅₃	—	4	8	no	—
126	D₄×D₅₃	—	4	16	no	—
127	D₄×D₅₃	—	4	16	no	—
128	#11719945	—	2	2	no	—
129	#11719945	—	2	2	no	—
130	#11719945	—	2	16	no	—
131	#11719945	—	4	8	no	—
132	#11719945	C₂×D₄	1	4	yes	—
133	#11719945	C₂⁴	1	4	yes	—
134	C₂×D₂₁₂	—	2	4	no	—
135	C₂×D₂₁₂	—	2	4	no	—
136	C₂×D₂₁₂	—	2	4	no	—
137	C₂×D₂₁₂	—	2	4	no	—
138	C₂×D₂₁₂	—	2	16	no	—
139	C₂×D₂₁₂	—	2	4	no	—
140	C₂×D₂₁₂	—	4	8	no	—
141	C₂×D₂₁₂	—	4	4	no	—
142	C₂×D₂₁₂	—	4	8	no	—
143	C₂×D₂₁₂	—	8	8	no	—
144	C₂×D₂₁₂	C₂×D₄	1	2	yes	—
145	C₂×D₂₁₂	C₂⁴	1	8	yes	—
146	C₂×C₄×D₅₃	—	2	2	no	—
147	C₂×C₄×D₅₃	—	4	4	no	—
148	C₂×Dic₁₀₆	C₂×D₄	1	2	yes	—
149	Q₁₆×C₅₃	C₂×D₄	1	4	yes	complemented
150	SD₁₆×C₅₃	C₂×D₄	1	8	yes	complemented
151	D₈×C₅₃	C₂×D₄	1	4	yes	complemented
152	M₁₆×C₅₃	—	2	2	no	—
153	M₁₆×C₅₃	—	2	8	no	—
154	M₁₆×C₅₃	C₂⁴	1	4	yes	—
155	C₂×C₈×C₅₃	—	2	2	no	—
156	C₂×C₈×C₅₃	—	2	8	no	—
157	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
158	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
159	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
160	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
161	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
162	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
163	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
164	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
165	C₄²×C₅₃	—	2	1	no	—
166	C₄²×C₅₃	C₂⁴	1	2	yes	—
167	#11719924	—	2	16	no	—
168	#11719922	—	2	16	no	—
169	#11719917	—	2	2	no	—
170	#11719917	C₂⁴	1	4	yes	—
171	C₂×#88322	—	2	2	no	—
172	Dic₂₁₂	C₂×D₄	1	4	yes	complemented
173	D₄₂₄	—	2	16	no	—
174	D₄₂₄	—	2	8	no	—
175	D₄₂₄	—	2	8	no	—
176	D₄₂₄	—	2	8	no	—
177	D₄₂₄	—	2	8	no	—
178	D₄₂₄	C₂×D₄	1	4	yes	complemented
179	#11719913	—	2	16	no	—
180	#11719913	C₂×D₄	1	8	yes	complemented
181	#11719912	—	2	8	no	—
182	C₈×D₅₃	—	2	8	no	—
183	C₂³×C₅₃	—	2	2	no	—
184	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
185	C₂²×D₅₃	—	2	2	no	—
186	C₂²×D₅₃	—	2	8	no	—
187	C₂²×D₅₃	—	4	16	no	—
188	C₂²×D₅₃	—	4	4	no	—
189	C₂²×D₅₃	—	4	4	no	—
190	C₂²×D₅₃	—	8	8	no	—
191	C₂²×D₅₃	—	16	8	no	—
192	Q₈×C₅₃	—	2	2	no	—
193	Q₈×C₅₃	C₂²×D₄	1	4	yes	—
194	D₄×C₅₃	—	2	4	no	—
195	D₄×C₅₃	—	2	4	no	—
196	D₄×C₅₃	—	2	2	no	—
197	D₄×C₅₃	C₂²×D₄	1	4	yes	—
198	C₂×C₄×C₅₃	—	2	4	no	—
199	C₂×C₄×C₅₃	—	2	2	no	—
200	C₂×C₄×C₅₃	—	2	2	no	—
201	C₂×C₄×C₅₃	—	2	4	no	—
202	C₂×C₄×C₅₃	—	2	4	no	—
203	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
208	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
209	C₂²⋊D₅₃	—	2	4	no	—
210	C₂²⋊D₅₃	—	2	4	no	—
211	C₂²⋊D₅₃	—	4	16	no	—
212	C₂×Dic₅₃	—	2	2	no	—
213	D₂₁₂	—	2	2	no	—
214	D₂₁₂	—	2	8	no	—
215	D₂₁₂	—	2	8	no	—
216	D₂₁₂	—	4	16	no	—
217	D₂₁₂	—	4	4	no	—
218	D₂₁₂	—	4	4	no	—
219	D₂₁₂	—	8	16	no	—
220	D₂₁₂	C₂²×D₄	1	4	yes	—
221	C₄×D₅₃	—	2	4	no	—
222	C₄×D₅₃	—	2	4	no	—
223	C₄×D₅₃	—	4	16	no	—
224	Dic₁₀₆	—	2	2	no	—
225	Dic₁₀₆	C₂²×D₄	1	4	yes	—
226	C₈×C₅₃	—	2	4	no	—
227	C₈×C₅₃	C₂²×D₄	1	2	yes	—
228	C₈×C₅₃	C₂²×D₄	1	2	yes	—
229	C₈×C₅₃	C₂²×D₄	1	2	yes	—
230	C₈×C₅₃	C₂²×D₄	1	2	yes	—
231	#88322	—	2	4	no	—
232	C₂×#3292	—	53	1	no	maximal, Sylow
233	C₂²×C₅₃	—	2	2	no	—
234	C₂²×C₅₃	—	2	4	no	—
235	C₂²×C₅₃	—	4	4	no	—
236	C₂²×C₅₃	D₄²	1	1	yes	characteristic
237	C₂²×C₅₃	D₄²	1	1	yes	characteristic
238	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
239	C₂×D₅₃	—	2	4	no	—
240	C₂×D₅₃	—	4	4	no	—
241	C₂×D₅₃	—	4	8	no	—
242	C₂×D₅₃	—	8	8	no	—
243	C₂×D₅₃	—	8	8	no	—
244	C₂×D₅₃	—	16	16	no	—
245	C₂×D₅₃	—	16	4	no	—
246	C₄×C₅₃	—	2	4	no	—
247	C₄×C₅₃	—	2	4	no	—
248	C₄×C₅₃	D₄²	1	1	yes	characteristic
249	C₄×C₅₃	D₄²	1	1	yes	characteristic
250	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
251	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
252	Dic₅₃	—	2	4	no	—
253	C₂×#576	—	53	2	no	—
254	C₂×#576	—	53	2	no	—
255	#3292	—	53	16	no	—
256	C₂×#493	—	53	1	no	—
257	C₂×#472	—	53	2	no	—
258	C₂×#472	—	53	2	no	—
259	C₂×#453	—	53	4	no	—
260	C₂×#443	—	53	1	no	—
261	C₂×#443	D₅₃	1	1	yes	characteristic, complemented
262	C₂×C₅₃	—	2	1	no	—
263	C₂×C₅₃	—	2	1	no	—
264	C₂×C₅₃	—	4	4	no	—
265	C₂×C₅₃	#3292	1	2	yes	complemented
266	C₂×C₅₃	C₂×D₄²	1	1	yes	characteristic
267	D₅₃	—	4	4	no	—
268	D₅₃	—	8	8	no	—
269	C₂×H(5,𝔽₂)	—	53	4	no	—
270	#576	—	53	16	no	—
271	#576	—	53	16	no	—
272	#576	—	106	8	no	—
273	#576	—	106	8	no	—
274	C₂×AGL(1,ℤ/8)	—	53	4	no	—
275	C₂×AGL(1,ℤ/8)	—	53	4	no	—
276	C₂×AGL(1,ℤ/8)	—	106	4	no	—
277	C₂×AGL(1,ℤ/8)	—	106	4	no	—
278	C₂×#135	—	53	1	no	—
279	C₂×#135	—	106	2	no	—
280	C₂×#135	—	106	2	no	—
281	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
282	C₂²×D₈	—	53	2	no	—
283	C₂²×D₈	—	53	2	no	—
284	C₂²×D₈	—	106	2	no	—
285	C₂²×D₈	—	106	2	no	—
286	C₂×#131	—	53	2	no	—
287	C₂×#131	C₂×D₅₃	1	2	yes	complemented
288	C₂×#127	—	53	2	no	—
289	#493	—	53	8	no	—
290	#472	—	53	16	no	—
291	#472	—	53	16	no	—
292	#453	—	53	32	no	—
293	#443	—	53	8	no	—
294	#443	C₂×D₅₃	1	8	yes	complemented
295	C₂×#108	—	53	1	no	—
296	C₂×#108	C₂×D₅₃	1	1	yes	characteristic, complemented
297	C₂×#104	—	53	2	no	—
298	C₂×#104	C₂×D₅₃	1	2	yes	complemented
299	C₂×#100	—	53	4	no	—
300	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
301	C₅₃	C₂×#3292	1	1	yes	characteristic, complemented, Sylow
302	H(5,𝔽₂)	—	53	16	no	—
303	H(5,𝔽₂)	—	106	8	no	—
304	H(5,𝔽₂)	—	106	8	no	—
305	H(5,𝔽₂)	—	212	4	no	—
306	C₂×#41	—	53	2	no	—
307	C₂×#41	—	106	4	no	—
308	C₂×#41	—	106	4	no	—
309	C₂×#41	C₂²×D₅₃	1	2	yes	—
310	C₂²×D₄	—	53	4	no	—
311	C₂²×D₄	—	106	4	no	—
312	C₂²×D₄	—	106	4	no	—
313	C₂²×D₄	—	212	2	no	—
314	C₂²×D₄	—	212	4	no	—
315	AGL(1,ℤ/8)	—	53	16	no	—
316	AGL(1,ℤ/8)	—	53	16	no	—
317	AGL(1,ℤ/8)	—	106	32	no	—
318	AGL(1,ℤ/8)	—	106	8	no	—
319	AGL(1,ℤ/8)	—	106	8	no	—
320	AGL(1,ℤ/8)	—	106	32	no	—
321	#135	—	53	4	no	—
322	#135	—	53	4	no	—
323	#135	—	106	16	no	—
324	#135	—	106	16	no	—
325	#135	D₂₁₂	1	4	yes	complemented
326	#135	C₂²×D₅₃	1	4	yes	—
327	C₂×Q₁₆	—	53	1	no	—
328	C₂×Q₁₆	D₂₁₂	1	1	yes	characteristic, complemented
329	C₂×SD₁₆	—	53	2	no	—
330	C₂×SD₁₆	—	106	4	no	—
331	C₂×SD₁₆	—	106	4	no	—
332	C₂×SD₁₆	D₂₁₂	1	2	yes	complemented
333	C₂×D₈	—	53	1	no	—
334	C₂×D₈	—	53	8	no	—
335	C₂×D₈	—	53	8	no	—
336	C₂×D₈	—	106	16	no	—
337	C₂×D₈	—	106	4	no	—
338	C₂×D₈	—	106	4	no	—
339	C₂×D₈	—	106	16	no	—
340	C₂×D₈	—	106	2	no	—
341	C₂×D₈	—	106	2	no	—
342	C₂×D₈	—	106	4	no	—
343	C₂×D₈	—	106	2	no	—
344	C₂×D₈	—	106	2	no	—
345	C₂×D₈	—	106	4	no	—
346	C₂×D₈	D₂₁₂	1	1	yes	characteristic, complemented
347	#131	—	2	4	no	—
348	#131	—	53	8	no	—
349	#131	—	106	4	no	—
350	#131	C₂²×D₅₃	1	8	yes	complemented
351	C₂×M₁₆	—	2	2	no	—
352	C₂×M₁₆	—	53	2	no	—
353	C₂×M₁₆	—	106	2	no	—
354	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
355	C₂²×C₈	—	2	2	no	—
356	C₂²×C₈	—	106	2	no	—
357	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
358	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
359	#127	—	53	8	no	—
360	#127	—	212	2	no	—
361	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
362	#108	—	53	4	no	—
363	#108	C₂²×D₅₃	1	4	yes	complemented
364	#104	—	53	8	no	—
365	#104	C₂²×D₅₃	1	8	yes	complemented
366	#100	—	53	16	no	—
367	C₄×C₈	C₂²×D₅₃	1	4	yes	complemented
368	C₂⁴	—	106	4	no	—
369	C₂⁴	—	212	4	no	—
370	#41	—	2	2	no	—
371	#41	—	2	2	no	—
372	#41	—	53	4	no	—
373	#41	—	53	4	no	—
374	#41	—	106	16	no	—
375	#41	—	106	2	no	—
376	#41	—	106	2	no	—
377	#41	—	106	16	no	—
378	#41	—	212	8	no	—
379	#41	—	212	8	no	—
380	#41	C₂×D₂₁₂	1	4	yes	—
381	#41	C₂³×D₅₃	1	4	yes	—
382	C₂×Q₈	—	53	2	no	—
383	C₂×Q₈	C₂×D₂₁₂	1	2	yes	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	—	53	2	no	—
386	C₂×D₄	—	53	8	no	—
387	C₂×D₄	—	106	16	no	—
388	C₂×D₄	—	106	2	no	—
389	C₂×D₄	—	106	16	no	—
390	C₂×D₄	—	106	4	no	—
391	C₂×D₄	—	106	4	no	—
392	C₂×D₄	—	106	4	no	—
393	C₂×D₄	—	106	4	no	—
394	C₂×D₄	—	106	4	no	—
395	C₂×D₄	—	424	8	no	—
396	C₂×D₄	—	212	8	no	—
397	C₂×D₄	—	212	8	no	—
398	C₂×D₄	—	212	4	no	—
399	C₂×D₄	—	212	8	no	—
400	C₂×D₄	—	212	4	no	—
401	C₂×D₄	—	212	16	no	—
402	C₂×D₄	—	212	16	no	—
403	C₂×D₄	C₂×D₂₁₂	1	2	yes	—
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	106	2	no	—
407	C₂²×C₄	—	212	4	no	—
408	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
409	Q₁₆	—	53	4	no	—
410	Q₁₆	C₂×D₂₁₂	1	4	yes	complemented
411	SD₁₆	—	53	8	no	—
412	SD₁₆	—	106	16	no	—
413	SD₁₆	—	106	16	no	—
414	SD₁₆	C₂×D₂₁₂	1	8	yes	complemented
415	D₈	—	53	4	no	—
416	D₈	—	106	16	no	—
417	D₈	—	106	16	no	—
418	D₈	—	106	8	no	—
419	D₈	—	106	8	no	—
420	D₈	—	106	8	no	—
421	D₈	—	106	8	no	—
422	D₈	C₂×D₂₁₂	1	4	yes	complemented
423	M₁₆	—	2	2	no	—
424	M₁₆	—	2	8	no	—
425	M₁₆	—	53	4	no	—
426	M₁₆	—	106	2	no	—
427	M₁₆	—	106	8	no	—
428	M₁₆	C₂³×D₅₃	1	4	yes	—
429	C₂×C₈	—	2	2	no	—
430	C₂×C₈	—	2	8	no	—
431	C₂×C₈	—	106	2	no	—
432	C₂×C₈	—	106	8	no	—
433	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
434	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
435	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
436	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
437	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
438	C₂×C₈	D₄×D₅₃	1	2	yes	—
439	C₂×C₈	C₂³×D₅₃	1	2	yes	—
440	C₂×C₈	C₂³×D₅₃	1	2	yes	—
441	C₄²	—	2	1	no	—
442	C₄²	C₂³×D₅₃	1	2	yes	—
443	C₂³	—	2	2	no	—
444	C₂³	—	848	8	no	—
445	C₂³	—	106	2	no	—
446	C₂³	—	106	8	no	—
447	C₂³	—	424	8	no	—
448	C₂³	—	212	16	no	—
449	C₂³	—	212	4	no	—
450	C₂³	—	212	4	no	—
451	C₂³	#13133263	1	1	yes	characteristic
452	Q₈	—	2	2	no	—
453	Q₈	—	53	4	no	—
454	Q₈	—	106	2	no	—
455	Q₈	C₂²×D₂₁₂	1	4	yes	—
456	D₄	—	2	4	no	—
457	D₄	—	2	4	no	—
458	D₄	—	2	2	no	—
459	D₄	—	53	4	no	—
460	D₄	—	106	4	no	—
461	D₄	—	106	4	no	—
462	D₄	—	106	8	no	—
463	D₄	—	106	8	no	—
464	D₄	—	106	2	no	—
465	D₄	—	424	16	no	—
466	D₄	—	212	16	no	—
467	D₄	—	212	16	no	—
468	D₄	—	212	4	no	—
469	D₄	—	212	4	no	—
470	D₄	C₂²×D₂₁₂	1	4	yes	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	106	2	no	—
477	C₂×C₄	—	106	4	no	—
478	C₂×C₄	—	106	4	no	—
479	C₂×C₄	—	212	16	no	—
480	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
481	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
482	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
483	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
484	C₂×C₄	#13133263	1	1	yes	characteristic
485	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
486	C₈	—	2	4	no	—
487	C₈	—	106	4	no	—
488	C₈	C₂×D₄×D₅₃	1	2	yes	—
489	C₈	C₂×D₄×D₅₃	1	2	yes	—
490	C₈	C₂²×D₂₁₂	1	2	yes	—
491	C₈	C₂²×D₂₁₂	1	2	yes	—
492	C₂²	—	2	2	no	—
493	C₂²	—	2	4	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	848	4	no	—
496	C₂²	—	848	16	no	—
497	C₂²	—	106	4	no	—
498	C₂²	—	424	8	no	—
499	C₂²	—	424	8	no	—
500	C₂²	—	212	8	no	—
501	C₂²	—	212	4	no	—
502	C₂²	D₄×D₂₁₂	1	1	yes	characteristic
503	C₂²	D₄×D₂₁₂	1	1	yes	characteristic
504	C₂²	C₂×#13133263	1	1	yes	characteristic
505	C₄	—	2	4	no	—
506	C₄	—	2	4	no	—
507	C₄	—	106	4	no	—
508	C₄	D₄×D₂₁₂	1	1	yes	characteristic
509	C₄	D₄×D₂₁₂	1	1	yes	characteristic
510	C₄	C₂²×D₄×D₅₃	1	1	yes	characteristic
511	C₄	C₂³×D₂₁₂	1	1	yes	characteristic
512	C₂	—	2	1	no	—
513	C₂	—	2	1	no	—
514	C₂	—	4	4	no	—
515	C₂	—	424	8	no	—
516	C₂	—	212	4	no	—
517	C₂	#20286515	1	2	yes	direct product factor
518	C₂	C₂×D₄×D₂₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₁₂
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂×C₅₃×#443	C₂
socle	C₂²×C₅₃	D₄²
hypercenter	C₂×#443	D₅₃
hypocenter	C₅₃	C₂×#3292
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#16811428	C₂	semidirect
2. C₂×#16811425	C₂	semidirect
3. #20286515	C₂	direct (N × H)
4. C₂×#16811000	C₂	semidirect
5. C₂×#16810894	C₂	semidirect
6. C₂×#16810893	C₂	semidirect
7. C₂×#16810792	C₂	semidirect
8. C₂×#16810741	C₂	semidirect
9. C₂×C₅₃×#443	C₂	semidirect
10. C₂×#13133263	C₂²	semidirect
12. #16811428	C₂²	semidirect
14. #16811425	C₂²	semidirect
16. C₂×#13133245	C₂²	semidirect
18. C₂×#13133242	C₂²	semidirect
23. C₂²×D₄₂₄	C₂²	semidirect
24. C₂²×D₄₂₄	C₂²	semidirect
26. C₂×#13133223	C₂²	semidirect
27. C₂×#13133214	C₂²	semidirect
28. #16811000	C₂²	semidirect
29. #16810894	C₂²	semidirect
30. #16810893	C₂²	semidirect
31. #16810792	C₂²	semidirect
32. #16810741	C₂²	semidirect
33. C₂×#13133153	C₂²	semidirect
34. C₂×#13133145	C₂²	semidirect
35. C₂×#13133133	C₂²	semidirect
37. C₂×#131×C₅₃	C₂²	semidirect
38. C₅₃×#443	C₂²	semidirect
39. C₂×#108×C₅₃	C₂²	semidirect
40. C₂×#104×C₅₃	C₂²	semidirect
41. C₂×C₄×C₈×C₅₃	C₂²	semidirect
45. #13133263	C₂³	semidirect
56. #13133245	C₂³	semidirect
59. #13133242	C₂³	semidirect
62. #13133239	D₄	semidirect
64. C₂×Dic₂₁₂	D₄	semidirect
66. C₂×#11719913	D₄	semidirect
76. C₂×D₄₂₄	D₄	semidirect
77. C₂×D₄₂₄	C₂³	semidirect
78. C₂×D₄₂₄	C₂³	semidirect
82. #13133223	C₂³	semidirect
87. #13133214	C₂³	semidirect
88. #13133153	C₂³	semidirect
89. #13133145	C₂³	semidirect
90. #13133133	C₂³	semidirect
92. #135×C₅₃	D₄	semidirect
94. C₂×Q₁₆×C₅₃	D₄	semidirect
95. C₂×SD₁₆×C₅₃	D₄	semidirect
96. C₂×D₈×C₅₃	D₄	semidirect
98. #131×C₅₃	C₂³	semidirect
105. #108×C₅₃	C₂³	semidirect
106. #104×C₅₃	C₂³	semidirect
107. C₄×C₈×C₅₃	C₂³	semidirect
149. Q₁₆×C₅₃	C₂×D₄	semidirect
150. SD₁₆×C₅₃	C₂×D₄	semidirect
151. D₈×C₅₃	C₂×D₄	semidirect
172. Dic₂₁₂	C₂×D₄	semidirect
178. D₄₂₄	C₂×D₄	semidirect
180. #11719913	C₂×D₄	semidirect
261. C₂×#443	D₅₃	semidirect
265. C₂×C₅₃	#3292	semidirect
287. C₂×#131	C₂×D₅₃	semidirect
294. #443	C₂×D₅₃	semidirect
296. C₂×#108	C₂×D₅₃	semidirect
298. C₂×#104	C₂×D₅₃	semidirect
300. C₂×C₄×C₈	C₂×D₅₃	semidirect
301. C₅₃	C₂×#3292	semidirect
325. #135	D₂₁₂	semidirect
328. C₂×Q₁₆	D₂₁₂	semidirect
332. C₂×SD₁₆	D₂₁₂	semidirect
346. C₂×D₈	D₂₁₂	semidirect
350. #131	C₂²×D₅₃	semidirect
363. #108	C₂²×D₅₃	semidirect
365. #104	C₂²×D₅₃	semidirect
367. C₄×C₈	C₂²×D₅₃	semidirect
410. Q₁₆	C₂×D₂₁₂	semidirect
414. SD₁₆	C₂×D₂₁₂	semidirect
422. D₈	C₂×D₂₁₂	semidirect
517. C₂	#20286515	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×#16811428	2
C₂×#16811425	2
#20286515	2
C₂×#16811000	2
C₂×#16810894	2
C₂×#16810893	2
C₂×#16810792	2
C₂×#16810741	2
C₂×C₅₃×#443	2
C₂×#3292	53

Subgroups — C2×#20286515 #28260318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20286515 #28260318