Subgroups — C₂×#20276844 #27538660

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133161	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16811041	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810899	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16810890	C₂	1	1	yes	maximal, characteristic
5	#20276844	C₂	1	8	yes	maximal, direct product factor
6	C₂×#16810452	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16810440	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133250	—	2	1	no	—
10	C₂×#13133243	—	2	1	no	—
11	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
12	C₂×#13133223	C₂²	1	1	yes	characteristic
13	C₂×#13133161	—	2	1	no	—
14	C₂×#13133161	—	2	1	no	—
15	C₂×#13133161	—	2	2	no	—
16	C₂×#13133161	C₂²	1	4	yes	complemented
17	#16811041	—	2	2	no	—
18	#16811041	C₂²	1	4	yes	complemented
19	C₂×#13133156	—	2	1	no	—
20	C₂²×#11719908	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₅₃	—	2	1	no	—
22	#16810899	—	2	2	no	—
23	#16810899	C₂²	1	4	yes	complemented
24	#16810890	—	2	2	no	—
25	#16810890	C₂²	1	4	yes	—
26	C₂×#13133153	C₂²	1	1	yes	characteristic
27	C₂×#13133135	—	2	1	no	—
28	C₂×#13133134	—	2	1	no	—
29	#16810452	C₂²	1	4	yes	complemented
30	#16810440	C₂²	1	4	yes	complemented
31	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#361	C₂²	1	4	yes	complemented
35	C₂×#11719945	D₄	1	1	yes	characteristic
36	C₂²×Dic₁₀₆	—	2	1	no	—
37	#13133250	—	2	4	no	—
38	#13133250	—	4	2	no	—
39	#13133243	—	2	4	no	—
40	#13133243	—	4	2	no	—
41	C₂×Dic₂₁₂	—	2	1	no	—
42	C₂×Dic₂₁₂	—	2	1	no	—
43	C₂×Dic₂₁₂	—	2	1	no	—
44	C₂×Dic₂₁₂	—	2	1	no	—
45	C₂×Dic₂₁₂	—	2	1	no	—
46	C₂×Dic₂₁₂	—	4	1	no	—
47	C₂×Dic₂₁₂	C₂³	1	2	yes	—
48	C₂×#11719913	—	4	1	no	—
49	C₂×#11719923	—	4	1	no	—
50	C₂×#11719922	—	4	1	no	—
51	#13133223	—	2	1	no	—
52	#13133223	—	2	1	no	—
53	#13133223	D₄	1	2	yes	—
54	#13133223	C₂³	1	2	yes	—
55	C₂×#11719917	—	2	1	no	—
56	C₂×#11719917	D₄	1	1	yes	characteristic
57	C₂×#11719912	—	2	1	no	—
58	C₂×C₈×D₅₃	—	2	1	no	—
59	#13133161	—	2	4	no	—
60	#13133161	—	2	4	no	—
61	#13133156	—	2	4	no	—
62	C₂×#11719908	D₄	1	1	yes	characteristic
63	C₂×#11719908	D₄	1	1	yes	characteristic
64	C₂×#11719908	D₄	1	2	yes	—
65	C₂×#11719908	C₂³	1	2	yes	—
66	C₁₆×D₅₃	—	2	4	no	—
67	#13133153	—	2	1	no	—
68	#13133153	C₂³	1	2	yes	—
69	#13133135	—	2	4	no	—
70	#13133134	—	2	4	no	—
71	C₂²×D₄×C₅₃	—	2	1	no	—
72	C₂×D₈×C₅₃	—	2	1	no	—
73	C₂×D₈×C₅₃	—	2	1	no	—
74	C₂×D₈×C₅₃	—	2	1	no	—
75	C₂×D₈×C₅₃	—	2	1	no	—
76	C₂×D₈×C₅₃	—	2	1	no	—
77	C₂×D₈×C₅₃	C₂³	1	2	yes	—
78	C₂×M₁₆×C₅₃	—	2	1	no	—
79	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
80	M₃₂×C₅₃	—	2	1	no	—
81	M₃₂×C₅₃	C₂³	1	2	yes	—
82	C₂×C₁₆×C₅₃	—	2	1	no	—
83	#108×C₅₃	—	2	1	no	—
84	#108×C₅₃	C₂³	1	2	yes	—
85	C₂⁴×C₅₃	—	4	1	no	—
86	C₂×D₄×C₅₃	—	2	2	no	—
87	C₂×D₄×C₅₃	—	4	1	no	—
88	C₂×D₄×C₅₃	—	4	1	no	—
89	C₂×D₄×C₅₃	—	4	1	no	—
90	C₂×D₄×C₅₃	—	4	1	no	—
91	C₂×D₄×C₅₃	—	4	1	no	—
92	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₂²⋊D₅₃	—	2	1	no	—
94	C₂²×Dic₅₃	—	4	1	no	—
95	#11719945	—	2	1	no	—
96	#11719945	—	2	1	no	—
97	#11719945	D₈	1	2	yes	—
98	#11719945	C₂×D₄	1	2	yes	—
99	C₂×D₂₁₂	D₈	1	1	yes	characteristic
100	C₂×C₄×D₅₃	—	2	1	no	—
101	C₂×Dic₁₀₆	—	2	2	no	—
102	C₂×Dic₁₀₆	—	4	1	no	—
103	C₂×Dic₁₀₆	—	4	1	no	—
104	C₂×Dic₁₀₆	—	4	1	no	—
105	C₂×Dic₁₀₆	—	4	1	no	—
106	C₂×Dic₁₀₆	—	4	1	no	—
107	C₂×Dic₁₀₆	D₈	1	1	yes	characteristic
108	D₈×C₅₃	—	2	2	no	—
109	D₈×C₅₃	—	2	2	no	—
110	D₈×C₅₃	—	4	1	no	—
111	D₈×C₅₃	—	4	1	no	—
112	M₁₆×C₅₃	—	2	2	no	—
113	M₁₆×C₅₃	—	4	1	no	—
114	C₂×C₈×C₅₃	—	4	1	no	—
115	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
119	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
120	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
121	#11719923	—	4	4	no	—
122	#11719922	—	4	4	no	—
123	#11719917	—	2	1	no	—
124	#11719917	—	2	2	no	—
125	#11719917	—	4	1	no	—
126	#11719917	C₂×D₄	1	2	yes	—
127	C₂×#88322	—	2	1	no	—
128	C₂×#88322	—	4	1	no	—
129	Dic₂₁₂	—	2	2	no	—
130	Dic₂₁₂	—	2	2	no	—
131	Dic₂₁₂	—	4	4	no	—
132	Dic₂₁₂	—	4	1	no	—
133	Dic₂₁₂	—	4	1	no	—
134	#11719913	—	4	4	no	—
135	#11719912	—	2	2	no	—
136	#11719912	—	2	2	no	—
137	C₈×D₅₃	—	2	2	no	—
138	C₈×D₅₃	—	2	2	no	—
139	C₁₆×C₅₃	—	2	2	no	—
140	#11719908	C₂×D₄	1	2	yes	—
141	#11719908	C₂×D₄	1	2	yes	—
142	C₂³×C₅₃	—	4	2	no	—
143	C₂³×C₅₃	—	8	1	no	—
144	C₂³×C₅₃	—	8	1	no	—
145	C₂³×C₅₃	—	16	2	no	—
146	C₂³×C₅₃	#135	1	1	yes	characteristic
147	C₂²×D₅₃	—	2	1	no	—
148	D₄×C₅₃	—	4	2	no	—
149	D₄×C₅₃	—	4	2	no	—
150	D₄×C₅₃	—	8	1	no	—
151	D₄×C₅₃	—	8	1	no	—
152	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
153	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
158	C₂²⋊D₅₃	—	2	2	no	—
159	C₂²⋊D₅₃	—	2	2	no	—
160	C₂×Dic₅₃	—	2	1	no	—
161	C₂×Dic₅₃	—	4	2	no	—
162	C₂×Dic₅₃	—	8	1	no	—
163	C₂×Dic₅₃	—	8	1	no	—
164	D₂₁₂	—	2	1	no	—
165	D₂₁₂	C₂×D₈	1	2	yes	—
166	C₄×D₅₃	—	2	2	no	—
167	C₄×D₅₃	—	2	2	no	—
168	Dic₁₀₆	—	2	1	no	—
169	Dic₁₀₆	—	4	2	no	—
170	Dic₁₀₆	—	4	2	no	—
171	Dic₁₀₆	—	8	1	no	—
172	Dic₁₀₆	—	8	1	no	—
173	Dic₁₀₆	C₂×D₈	1	2	yes	—
174	C₈×C₅₃	—	4	2	no	—
175	C₈×C₅₃	#121	1	1	yes	characteristic
176	C₈×C₅₃	#121	1	1	yes	characteristic
177	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
178	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
179	#88322	—	2	2	no	—
180	#88322	—	4	2	no	—
181	C₂×#2223	—	53	1	no	maximal, Sylow
182	C₂²×C₅₃	—	2	2	no	—
183	C₂²×C₅₃	—	8	2	no	—
184	C₂²×C₅₃	—	16	2	no	—
185	C₂²×C₅₃	—	16	2	no	—
186	C₂²×C₅₃	—	16	2	no	—
187	C₂²×C₅₃	#466	1	1	yes	characteristic
188	C₂²×C₅₃	#466	1	1	yes	characteristic
189	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
190	C₂×D₅₃	—	2	2	no	—
191	C₂×D₅₃	—	4	2	no	—
192	C₄×C₅₃	#466	1	1	yes	characteristic
193	C₄×C₅₃	#466	1	1	yes	characteristic
194	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
195	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
196	Dic₅₃	—	2	2	no	—
197	Dic₅₃	—	8	2	no	—
198	C₂²×SD₃₂	—	53	1	no	—
199	C₂×#504	—	53	1	no	—
200	C₂×#473	—	53	1	no	—
201	C₂×#472	—	53	1	no	—
202	#2223	—	53	8	no	—
203	C₂×#368	—	53	1	no	—
204	C₂×#362	—	53	1	no	—
205	C₂×#361	D₅₃	1	1	yes	characteristic, complemented
206	C₂×C₅₃	—	2	1	no	—
207	C₂×C₅₃	—	2	1	no	—
208	C₂×C₅₃	—	16	2	no	—
209	C₂×C₅₃	#2223	1	2	yes	complemented
210	C₂×C₅₃	C₂×#466	1	1	yes	characteristic
211	D₅₃	—	4	2	no	—
212	C₂×#137	—	106	1	no	—
213	C₂×AGL(1,ℤ/8)	—	106	1	no	—
214	C₂²×Q₁₆	—	53	1	no	—
215	C₂²×D₈	C₂×D₅₃	1	1	yes	characteristic
216	C₂×#131	—	53	1	no	—
217	C₂×SD₃₂	—	53	4	no	—
218	C₂×SD₃₂	—	106	1	no	—
219	C₂×SD₃₂	—	106	1	no	—
220	C₂×SD₃₂	—	106	2	no	—
221	#504	—	53	4	no	—
222	#504	—	106	2	no	—
223	C₂×M₃₂	—	106	1	no	—
224	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
225	C₂²×C₁₆	—	53	1	no	—
226	C₂²×C₁₆	—	106	1	no	—
227	#473	—	53	4	no	—
228	#473	—	106	2	no	—
229	#472	—	53	4	no	—
230	#472	—	106	2	no	—
231	C₂×#108	—	53	1	no	—
232	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
233	C₂×#101	—	106	1	no	—
234	C₂×#100	—	106	1	no	—
235	#368	—	53	4	no	—
236	#362	—	53	4	no	—
237	#361	C₂×D₅₃	1	4	yes	complemented
238	C₅₃	C₂×#2223	1	1	yes	characteristic, complemented, Sylow
239	C₂×#41	—	53	1	no	—
240	C₂²×Q₈	—	106	1	no	—
241	C₂²×D₄	—	2	1	no	—
242	#137	—	106	4	no	—
243	#137	—	212	2	no	—
244	AGL(1,ℤ/8)	—	106	4	no	—
245	AGL(1,ℤ/8)	—	212	2	no	—
246	C₂×Q₁₆	—	53	2	no	—
247	C₂×Q₁₆	—	106	1	no	—
248	C₂×Q₁₆	—	106	1	no	—
249	C₂×Q₁₆	—	106	1	no	—
250	C₂×Q₁₆	—	106	1	no	—
251	C₂×Q₁₆	—	106	1	no	—
252	C₂×Q₁₆	—	212	1	no	—
253	C₂×SD₁₆	—	212	1	no	—
254	C₂×SD₁₆	—	212	1	no	—
255	C₂×D₈	—	2	1	no	—
256	C₂×D₈	—	2	1	no	—
257	C₂×D₈	—	2	1	no	—
258	C₂×D₈	—	2	1	no	—
259	C₂×D₈	—	2	1	no	—
260	C₂×D₈	—	212	1	no	—
261	C₂×D₈	C₂²×D₅₃	1	2	yes	—
262	#131	—	53	2	no	—
263	#131	—	53	2	no	—
264	#131	—	106	1	no	—
265	#131	—	106	1	no	—
266	C₂×M₁₆	—	2	1	no	—
267	C₂×M₁₆	—	53	1	no	—
268	C₂×M₁₆	—	106	1	no	—
269	C₂×M₁₆	—	106	1	no	—
270	C₂²×C₈	—	106	1	no	—
271	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
272	SD₃₂	—	106	4	no	—
273	SD₃₂	—	106	4	no	—
274	M₃₂	—	2	1	no	—
275	M₃₂	—	106	4	no	—
276	M₃₂	C₂²×D₅₃	1	2	yes	—
277	C₂×C₁₆	—	2	1	no	—
278	C₂×C₁₆	—	53	1	no	—
279	C₂×C₁₆	—	53	1	no	—
280	C₂×C₁₆	—	53	2	no	—
281	C₂×C₁₆	—	53	2	no	—
282	C₂×C₁₆	—	106	4	no	—
283	#108	—	2	1	no	—
284	#108	—	53	2	no	—
285	#108	—	106	1	no	—
286	#108	C₂²×D₅₃	1	2	yes	—
287	#101	—	106	4	no	—
288	#100	—	106	4	no	—
289	C₂⁴	—	4	1	no	—
290	#41	—	53	2	no	—
291	#41	—	53	2	no	—
292	#41	—	106	1	no	—
293	#41	—	106	1	no	—
294	C₂×Q₈	—	53	1	no	—
295	C₂×Q₈	—	106	2	no	—
296	C₂×Q₈	—	212	1	no	—
297	C₂×Q₈	—	212	1	no	—
298	C₂×Q₈	—	212	1	no	—
299	C₂×Q₈	—	212	1	no	—
300	C₂×Q₈	—	212	1	no	—
301	C₂×D₄	—	2	2	no	—
302	C₂×D₄	—	4	1	no	—
303	C₂×D₄	—	4	1	no	—
304	C₂×D₄	—	4	1	no	—
305	C₂×D₄	—	4	1	no	—
306	C₂×D₄	—	4	1	no	—
307	C₂×D₄	—	53	1	no	—
308	C₂×D₄	—	106	1	no	—
309	C₂²×C₄	—	106	1	no	—
310	C₂²×C₄	—	212	1	no	—
311	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
312	Q₁₆	—	106	2	no	—
313	Q₁₆	—	106	2	no	—
314	Q₁₆	—	212	4	no	—
315	Q₁₆	—	212	1	no	—
316	Q₁₆	—	212	1	no	—
317	SD₁₆	—	212	4	no	—
318	SD₁₆	—	212	4	no	—
319	D₈	—	2	2	no	—
320	D₈	—	2	2	no	—
321	D₈	—	4	1	no	—
322	D₈	—	4	1	no	—
323	D₈	—	212	4	no	—
324	M₁₆	—	2	2	no	—
325	M₁₆	—	4	1	no	—
326	M₁₆	—	53	2	no	—
327	M₁₆	—	106	1	no	—
328	M₁₆	—	106	2	no	—
329	M₁₆	—	106	2	no	—
330	M₁₆	—	106	2	no	—
331	M₁₆	—	212	1	no	—
332	C₂×C₈	—	4	1	no	—
333	C₂×C₈	—	106	1	no	—
334	C₂×C₈	—	106	2	no	—
335	C₂×C₈	—	106	2	no	—
336	C₂×C₈	—	212	1	no	—
337	C₂×C₈	#11719945	1	1	yes	characteristic
338	C₂×C₈	#11719945	1	1	yes	characteristic
339	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
340	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
341	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
342	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
343	C₁₆	—	2	2	no	—
344	C₁₆	—	53	2	no	—
345	C₁₆	—	53	2	no	—
346	C₂³	—	4	2	no	—
347	C₂³	—	8	1	no	—
348	C₂³	—	8	1	no	—
349	C₂³	—	16	2	no	—
350	C₂³	—	106	1	no	—
351	C₂³	#13133238	1	1	yes	characteristic
352	Q₈	—	53	2	no	—
353	Q₈	—	106	1	no	—
354	Q₈	—	424	1	no	—
355	Q₈	—	424	1	no	—
356	Q₈	—	212	2	no	—
357	Q₈	—	212	2	no	—
358	D₄	—	4	2	no	—
359	D₄	—	4	2	no	—
360	D₄	—	8	1	no	—
361	D₄	—	8	1	no	—
362	D₄	—	53	2	no	—
363	D₄	—	106	2	no	—
364	D₄	—	106	2	no	—
365	D₄	—	106	1	no	—
366	C₂×C₄	—	106	1	no	—
367	C₂×C₄	—	106	2	no	—
368	C₂×C₄	—	106	2	no	—
369	C₂×C₄	—	424	1	no	—
370	C₂×C₄	—	424	1	no	—
371	C₂×C₄	—	212	2	no	—
372	C₂×C₄	#13133190	1	1	yes	characteristic
373	C₂×C₄	#13133190	1	1	yes	characteristic
374	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
375	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
376	C₂×C₄	#13133238	1	1	yes	characteristic
377	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
378	C₈	—	4	2	no	—
379	C₈	—	106	2	no	—
380	C₈	—	212	2	no	—
381	C₈	#13133190	1	1	yes	characteristic
382	C₈	#13133190	1	1	yes	characteristic
383	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
384	C₈	C₂×#11719945	1	1	yes	characteristic
385	C₂²	—	2	2	no	—
386	C₂²	—	8	2	no	—
387	C₂²	—	16	2	no	—
388	C₂²	—	16	2	no	—
389	C₂²	—	16	2	no	—
390	C₂²	—	106	2	no	—
391	C₂²	—	212	2	no	—
392	C₂²	#16810858	1	1	yes	characteristic
393	C₂²	#16810858	1	1	yes	characteristic
394	C₂²	C₂×#13133238	1	1	yes	characteristic
395	C₄	—	106	2	no	—
396	C₄	—	424	2	no	—
397	C₄	#16810858	1	1	yes	characteristic
398	C₄	#16810858	1	1	yes	characteristic
399	C₄	C₂×#13133190	1	1	yes	characteristic
400	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	16	2	no	—
404	C₂	—	212	2	no	—
405	C₂	#20276844	1	2	yes	direct product factor
406	C₂	C₂×#16810858	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810858
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#361	C₂
socle	C₂²×C₅₃	#466
prime radical	C₂×#16810890	C₂
hypercenter	C₂×#361	D₅₃
hypocenter	C₅₃	C₂×#2223
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#361

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133161	C₂	semidirect
2. C₂×#16811041	C₂	semidirect
3. C₂×#16810899	C₂	semidirect
5. #20276844	C₂	direct (N × H)
6. C₂×#16810452	C₂	semidirect
7. C₂×#16810440	C₂	semidirect
8. C₂×C₅₃×#361	C₂	semidirect
16. C₂×#13133161	C₂²	semidirect
18. #16811041	C₂²	semidirect
23. #16810899	C₂²	semidirect
29. #16810452	C₂²	semidirect
30. #16810440	C₂²	semidirect
34. C₅₃×#361	C₂²	semidirect
205. C₂×#361	D₅₃	semidirect
209. C₂×C₅₃	#2223	semidirect
237. #361	C₂×D₅₃	semidirect
238. C₅₃	C₂×#2223	semidirect
405. C₂	#20276844	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×#13133161	2
C₂×#16811041	2
C₂×#16810899	2
C₂×#16810890	2
#20276844	2
C₂×#16810452	2
C₂×#16810440	2
C₂×C₅₃×#361	2
C₂×#2223	53

Subgroups — C2×#20276844 #27538660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276844 #27538660