Subgroups — C₂×#20276608 #27537527

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811066	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810746	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810731	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276608	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810446	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810440	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16810434	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₃×#347	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133239	C₂²	1	1	yes	characteristic
10	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
11	C₂×#13133223	—	2	1	no	—
12	C₂×C₄×Dic₁₀₆	—	2	1	no	—
13	#16811066	C₄	1	4	yes	complemented
14	#16811066	C₂²	1	4	yes	—
15	C₂×Dic₄₂₄	C₄	1	2	yes	complemented
16	C₂×#13133162	C₄	1	2	yes	complemented
17	#16810746	—	2	2	no	—
18	#16810746	C₂²	1	4	yes	—
19	#16810731	—	2	2	no	—
20	#16810731	C₂²	1	4	yes	complemented
21	C₂×#13133153	C₂²	1	1	yes	characteristic
22	C₂×#13133149	C₂²	1	1	yes	characteristic
23	C₂×#13133145	—	2	1	no	—
24	C₂×#13133142	—	2	1	no	—
25	#16810446	C₂²	1	4	yes	complemented
26	#16810440	C₂²	1	4	yes	complemented
27	#16810434	C₂²	1	4	yes	—
28	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#347	C₂²	1	4	yes	complemented
32	C₂×#11719945	—	2	1	no	—
33	C₂²×Dic₁₀₆	—	2	1	no	—
34	#13133239	—	2	1	no	—
35	#13133239	—	2	1	no	—
36	#13133239	C₂×C₄	1	2	yes	—
37	#13133239	C₂³	1	2	yes	—
38	C₂×Dic₂₁₂	—	2	1	no	—
39	C₂×Dic₂₁₂	—	2	1	no	—
40	C₂×Dic₂₁₂	—	2	1	no	—
41	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
42	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
43	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
44	C₂×Dic₂₁₂	C₂³	1	2	yes	—
45	C₂×#11719913	—	2	1	no	—
46	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
47	#13133223	—	2	4	no	—
48	#13133223	—	4	2	no	—
49	C₂×#11719917	—	2	1	no	—
50	C₂×#11719912	—	4	1	no	—
51	C₂×C₈×D₅₃	—	4	1	no	—
52	C₄×Dic₁₀₆	—	2	4	no	—
53	C₄×Dic₁₀₆	—	4	2	no	—
54	C₂×C₄⋊Dic₅₃	—	2	1	no	—
55	C₂×#11719919	—	4	1	no	—
56	C₂×C₄×Dic₅₃	—	4	1	no	—
57	Dic₄₂₄	C₂×C₄	1	8	yes	complemented
58	#13133162	C₂×C₄	1	8	yes	complemented
59	#13133153	—	2	1	no	—
60	#13133153	C₂³	1	2	yes	—
61	#13133149	—	2	1	no	—
62	#13133149	C₂³	1	2	yes	—
63	#13133145	—	2	4	no	—
64	#13133142	—	2	4	no	—
65	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
68	M₃₂×C₅₃	—	2	1	no	—
69	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
70	M₃₂×C₅₃	C₂³	1	2	yes	—
71	M₃₂×C₅₃	C₂³	1	2	yes	—
72	C₂×C₁₆×C₅₃	—	2	1	no	—
73	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
74	#97×C₅₃	D₄	1	2	yes	—
75	#97×C₅₃	C₂³	1	2	yes	—
76	C₂²×C₄×C₅₃	—	2	1	no	—
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₅₃	—	4	1	no	—
79	C₂²×Dic₅₃	—	4	1	no	—
80	C₂²×Dic₅₃	—	4	1	no	—
81	#11719945	—	2	2	no	—
82	#11719945	—	2	2	no	—
83	#11719945	—	4	1	no	—
84	#11719945	—	4	1	no	—
85	C₂×D₂₁₂	—	2	1	no	—
86	C₂×C₄×D₅₃	—	4	1	no	—
87	C₂×Dic₁₀₆	—	2	1	no	—
88	C₂×Dic₁₀₆	—	2	2	no	—
89	C₂×Dic₁₀₆	—	2	2	no	—
90	C₂×Dic₁₀₆	—	2	2	no	—
91	C₂×Dic₁₀₆	—	4	1	no	—
92	C₂×Dic₁₀₆	—	4	1	no	—
93	C₂×Dic₁₀₆	—	4	1	no	—
94	C₂×C₈×C₅₃	—	2	1	no	—
95	C₂×C₈×C₅₃	—	2	1	no	—
96	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
101	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
103	C₄²×C₅₃	—	2	1	no	—
104	C₄²×C₅₃	C₂×D₄	1	2	yes	—
105	C₄⋊Dic₅₃	—	2	2	no	—
106	C₄⋊Dic₅₃	—	4	1	no	—
107	#11719919	—	4	4	no	—
108	C₄×Dic₅₃	—	4	4	no	—
109	#11719917	—	2	2	no	—
110	#11719917	—	4	1	no	—
111	C₂×#88322	—	4	1	no	—
112	Dic₂₁₂	—	2	2	no	—
113	Dic₂₁₂	—	2	2	no	—
114	Dic₂₁₂	—	2	1	no	—
115	Dic₂₁₂	—	2	2	no	—
116	Dic₂₁₂	C₂²×C₄	1	4	yes	—
117	Dic₂₁₂	C₂²×C₄	1	2	yes	—
118	D₄₂₄	—	2	1	no	—
119	D₄₂₄	C₂²×C₄	1	2	yes	—
120	#11719913	—	2	2	no	—
121	#11719913	—	2	2	no	—
122	#11719912	—	4	4	no	—
123	C₈×D₅₃	—	4	4	no	—
124	C₁₆×C₅₃	—	2	2	no	—
125	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
126	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂²×D₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	4	2	no	—
130	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
136	C₂²⋊D₅₃	—	4	2	no	—
137	C₂²⋊D₅₃	—	4	2	no	—
138	C₂×Dic₅₃	—	4	2	no	—
139	C₂×Dic₅₃	—	4	2	no	—
140	C₂×Dic₅₃	—	4	2	no	—
141	C₂×Dic₅₃	—	4	1	no	—
142	C₂×Dic₅₃	—	4	2	no	—
143	C₂×Dic₅₃	—	8	2	no	—
144	D₂₁₂	—	2	2	no	—
145	D₂₁₂	—	4	1	no	—
146	C₄×D₅₃	—	4	2	no	—
147	C₄×D₅₃	—	4	2	no	—
148	Dic₁₀₆	—	2	4	no	—
149	Dic₁₀₆	—	2	2	no	—
150	Dic₁₀₆	—	4	2	no	—
151	Dic₁₀₆	—	4	2	no	—
152	Dic₁₀₆	—	4	2	no	—
153	Dic₁₀₆	—	4	1	no	—
154	C₈×C₅₃	—	2	2	no	—
155	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
156	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
157	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
158	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
159	#88322	—	4	2	no	—
160	C₂×#2180	—	53	1	no	maximal, Sylow
161	C₂²×C₅₃	—	2	2	no	—
162	C₂²×C₅₃	#442	1	1	yes	characteristic
163	C₂²×C₅₃	#442	1	1	yes	characteristic
164	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	C₂×D₅₃	—	4	2	no	—
166	C₂×D₅₃	—	8	2	no	—
167	C₄×C₅₃	—	4	2	no	—
168	C₄×C₅₃	#442	1	1	yes	characteristic
169	C₄×C₅₃	#442	1	1	yes	characteristic
170	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
171	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	Dic₅₃	—	4	4	no	—
173	Dic₅₃	—	4	2	no	—
174	Dic₅₃	—	8	2	no	—
175	C₂×#510	—	53	1	no	—
176	C₂×#444	—	53	1	no	—
177	C₂×#441	—	53	1	no	—
178	#2180	—	53	8	no	—
179	C₂×#365	—	53	1	no	—
180	C₂×#362	—	53	1	no	—
181	C₂×#360	—	53	1	no	—
182	C₂×#347	D₅₃	1	1	yes	characteristic, complemented
183	C₂×C₅₃	—	2	1	no	—
184	C₂×C₅₃	—	2	1	no	—
185	C₂×C₅₃	#2180	1	2	yes	complemented
186	C₂×C₅₃	C₂×#442	1	1	yes	characteristic
187	D₅₃	—	8	2	no	—
188	C₂×#135	—	53	1	no	—
189	C₂²×Q₁₆	—	53	1	no	—
190	C₂×#131	—	106	1	no	—
191	C₂×C₄×Q₈	—	106	1	no	—
192	#510	—	53	4	no	—
193	#510	—	53	4	no	—
194	C₂×Q₃₂	—	53	2	no	—
195	C₂×SD₃₂	—	53	2	no	—
196	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
197	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
198	#444	—	53	4	no	—
199	#444	—	106	2	no	—
200	#441	—	53	4	no	—
201	#441	—	106	2	no	—
202	C₂×#108	—	53	1	no	—
203	C₂×#106	—	53	1	no	—
204	C₂×#104	—	106	1	no	—
205	C₂×#103	—	106	1	no	—
206	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
207	#365	—	53	4	no	—
208	#362	—	53	4	no	—
209	#360	—	53	4	no	—
210	#347	C₂×D₅₃	1	4	yes	complemented
211	C₅₃	C₂×#2180	1	1	yes	characteristic, complemented, Sylow
212	C₂×#41	—	106	1	no	—
213	C₂²×Q₈	—	106	1	no	—
214	#135	—	53	2	no	—
215	#135	—	53	2	no	—
216	#135	—	106	1	no	—
217	#135	—	106	1	no	—
218	C₂×Q₁₆	—	53	2	no	—
219	C₂×Q₁₆	—	53	2	no	—
220	C₂×Q₁₆	—	53	1	no	—
221	C₂×Q₁₆	—	53	2	no	—
222	C₂×Q₁₆	—	106	1	no	—
223	C₂×Q₁₆	—	106	1	no	—
224	C₂×Q₁₆	—	106	1	no	—
225	C₂×SD₁₆	—	106	1	no	—
226	C₂×D₈	—	53	1	no	—
227	#131	—	106	4	no	—
228	#131	—	212	2	no	—
229	C₂×M₁₆	—	106	1	no	—
230	C₂×M₁₆	—	212	1	no	—
231	C₂²×C₈	—	212	1	no	—
232	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
234	C₄×Q₈	—	106	4	no	—
235	C₄×Q₈	—	212	2	no	—
236	C₂×C₄⋊C₄	—	106	1	no	—
237	C₂×C₄⋊C₄	—	212	1	no	—
238	C₂×C₄²	—	212	1	no	—
239	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
240	Q₃₂	—	53	8	no	—
241	SD₃₂	—	53	8	no	—
242	M₃₂	—	2	1	no	—
243	M₃₂	C₄×D₅₃	1	2	yes	—
244	M₃₂	C₂²×D₅₃	1	2	yes	—
245	M₃₂	C₂²×D₅₃	1	2	yes	—
246	C₂×C₁₆	—	2	1	no	—
247	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
248	#108	—	53	2	no	—
249	#108	—	106	1	no	—
250	#106	—	53	2	no	—
251	#106	—	106	1	no	—
252	#104	—	106	4	no	—
253	#103	—	106	4	no	—
254	#97	D₂₁₂	1	2	yes	—
255	#97	C₂²×D₅₃	1	2	yes	—
256	#41	—	106	2	no	—
257	#41	—	106	2	no	—
258	#41	—	212	1	no	—
259	#41	—	212	1	no	—
260	C₂×Q₈	—	106	1	no	—
261	C₂×Q₈	—	106	2	no	—
262	C₂×Q₈	—	106	2	no	—
263	C₂×Q₈	—	106	2	no	—
264	C₂×Q₈	—	212	1	no	—
265	C₂×Q₈	—	212	1	no	—
266	C₂×Q₈	—	212	1	no	—
267	C₂×D₄	—	106	1	no	—
268	C₂×D₄	—	212	1	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	212	1	no	—
271	C₂²×C₄	—	212	1	no	—
272	C₂²×C₄	—	212	1	no	—
273	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
274	Q₁₆	—	53	4	no	—
275	Q₁₆	—	53	2	no	—
276	Q₁₆	—	106	2	no	—
277	Q₁₆	—	106	2	no	—
278	Q₁₆	—	106	1	no	—
279	Q₁₆	—	106	2	no	—
280	SD₁₆	—	106	2	no	—
281	SD₁₆	—	106	2	no	—
282	D₈	—	53	2	no	—
283	D₈	—	106	1	no	—
284	M₁₆	—	106	2	no	—
285	M₁₆	—	212	4	no	—
286	M₁₆	—	212	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	212	4	no	—
290	C₂×C₈	—	212	1	no	—
291	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
293	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
294	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
295	C₂×C₈	#11719945	1	1	yes	characteristic
296	C₂×C₈	#11719945	1	1	yes	characteristic
297	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
298	C₄⋊C₄	—	106	2	no	—
299	C₄⋊C₄	—	212	4	no	—
300	C₄⋊C₄	—	212	1	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	212	4	no	—
303	C₄²	C₂×D₂₁₂	1	2	yes	—
304	C₁₆	—	2	2	no	—
305	C₁₆	C₂×C₄×D₅₃	1	4	yes	—
306	C₂³	—	212	1	no	—
307	C₂³	#13133245	1	1	yes	characteristic
308	Q₈	—	106	2	no	—
309	Q₈	—	106	4	no	—
310	Q₈	—	212	2	no	—
311	Q₈	—	212	2	no	—
312	Q₈	—	212	2	no	—
313	Q₈	—	212	1	no	—
314	D₄	—	106	2	no	—
315	D₄	—	212	2	no	—
316	D₄	—	212	2	no	—
317	D₄	—	212	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	424	2	no	—
321	C₂×C₄	—	212	2	no	—
322	C₂×C₄	—	212	2	no	—
323	C₂×C₄	—	212	2	no	—
324	C₂×C₄	—	212	1	no	—
325	C₂×C₄	—	212	2	no	—
326	C₂×C₄	—	212	2	no	—
327	C₂×C₄	—	212	2	no	—
328	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
329	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
330	C₂×C₄	#13133245	1	1	yes	characteristic
331	C₂×C₄	#13133245	1	1	yes	characteristic
332	C₂×C₄	#13133245	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	212	2	no	—
336	C₈	C₄×D₂₁₂	1	1	yes	characteristic
337	C₈	C₄×D₂₁₂	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
339	C₈	C₂×#11719945	1	1	yes	characteristic
340	C₂²	—	2	2	no	—
341	C₂²	—	424	2	no	—
342	C₂²	—	212	2	no	—
343	C₂²	#16810736	1	1	yes	characteristic
344	C₂²	#16810736	1	1	yes	characteristic
345	C₂²	C₂×#13133245	1	1	yes	characteristic
346	C₄	—	4	2	no	—
347	C₄	—	424	2	no	—
348	C₄	—	212	2	no	—
349	C₄	—	212	4	no	—
350	C₄	#16810736	1	1	yes	characteristic
351	C₄	#16810736	1	1	yes	characteristic
352	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
353	C₄	C₂×#13133245	1	1	yes	characteristic
354	C₂	—	2	1	no	—
355	C₂	—	2	1	no	—
356	C₂	—	424	2	no	—
357	C₂	#20276608	1	2	yes	direct product factor
358	C₂	C₂×#16810736	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810736
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#347	C₂
socle	C₂²×C₅₃	#442
prime radical	C₂×#13133239	C₂²
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#347	D₅₃
hypocenter	C₅₃	C₂×#2180
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
3. C₂×#16810731	C₂	semidirect
4. #20276608	C₂	direct (N × H)
5. C₂×#16810446	C₂	semidirect
6. C₂×#16810440	C₂	semidirect
8. C₂×C₅₃×#347	C₂	semidirect
13. #16811066	C₄	semidirect
15. C₂×Dic₄₂₄	C₄	semidirect
16. C₂×#13133162	C₄	semidirect
20. #16810731	C₂²	semidirect
25. #16810446	C₂²	semidirect
26. #16810440	C₂²	semidirect
31. C₅₃×#347	C₂²	semidirect
57. Dic₄₂₄	C₂×C₄	semidirect
58. #13133162	C₂×C₄	semidirect
182. C₂×#347	D₅₃	semidirect
185. C₂×C₅₃	#2180	semidirect
210. #347	C₂×D₅₃	semidirect
211. C₅₃	C₂×#2180	semidirect
357. C₂	#20276608	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×#16811066	2
C₂×#16810746	2
C₂×#16810731	2
#20276608	2
C₂×#16810446	2
C₂×#16810440	2
C₂×#16810434	2
C₂×C₅₃×#347	2
C₂×#2180	53

Subgroups — C2×#20276608 #27537527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276608 #27537527