Subgroups — C₂×#20291106 #28328344

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811682	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16811681	C₂	1	1	yes	maximal, characteristic, complemented
3	#20291106	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16811546	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16811522	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16811521	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16811511	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16810737	C₂	1	1	yes	maximal, characteristic
9	#16811682	—	2	2	no	—
10	#16811682	C₂²	1	4	yes	complemented
11	#16811681	—	2	2	no	—
12	#16811681	C₂²	1	4	yes	complemented
13	C₂×#13133292	—	2	1	no	—
14	C₂×#13133291	—	2	1	no	—
15	C₂×#13133291	C₂²	1	1	yes	characteristic
16	C₂×#13133291	C₂²	1	1	yes	characteristic
17	C₂×D₅₃⋊C₁₆	—	2	1	no	—
18	C₂²×#11719910	—	2	1	no	—
19	C₂²×#11719910	C₂²	1	1	yes	characteristic
20	C₂²×#11719910	C₂²	1	1	yes	characteristic
21	#16811546	C₂²	1	4	yes	complemented
22	#16811522	C₂²	1	4	yes	complemented
23	#16811521	C₂²	1	4	yes	complemented
24	#16811511	C₂²	1	4	yes	complemented
25	C₂×#13133237	C₄	1	1	yes	characteristic
26	C₂×#13133225	C₂²	1	1	yes	characteristic
27	C₂×#13133224	C₂²	1	1	yes	characteristic
28	#16810737	C₄	1	4	yes	—
29	#16810737	C₂²	1	4	yes	—
30	C₂×#13133152	C₄	1	1	yes	characteristic
31	C₂×#13133145	C₄	1	1	yes	characteristic
32	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
33	C₂×#104×C₅₃	C₄	1	1	yes	characteristic
34	#13133292	—	2	4	no	—
35	#13133291	—	2	1	no	—
36	#13133291	—	2	1	no	—
37	#13133291	—	2	4	no	—
38	#13133291	C₂³	1	2	yes	—
39	#13133291	C₂³	1	2	yes	—
40	D₅₃⋊C₁₆	—	2	4	no	—
41	C₂×#11719910	—	2	1	no	—
42	C₂×#11719910	—	2	1	no	—
43	C₂×#11719910	—	2	1	no	—
44	C₂×#11719910	—	2	1	no	—
45	C₂×#11719910	—	2	4	no	—
46	C₂×#11719910	D₄	1	1	yes	characteristic
47	C₂×#11719910	D₄	1	1	yes	characteristic
48	C₂×#11719910	D₄	1	2	yes	—
49	C₂×#11719910	C₂³	1	2	yes	—
50	C₂×#11719910	C₂³	1	2	yes	—
51	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
52	#13133237	C₈	1	4	yes	—
53	#13133237	C₂×C₄	1	4	yes	—
54	C₂×#11719925	C₈	1	1	yes	characteristic
55	C₂×#11719923	C₈	1	1	yes	characteristic
56	C₂×#11719924	C₈	1	1	yes	characteristic
57	C₂×#11719922	C₈	1	1	yes	characteristic
58	#13133225	—	2	1	no	—
59	#13133225	—	2	1	no	—
60	#13133225	C₂×C₄	1	2	yes	—
61	#13133225	C₂³	1	2	yes	—
62	#13133224	—	2	1	no	—
63	#13133224	—	2	1	no	—
64	#13133224	C₂×C₄	1	2	yes	—
65	#13133224	C₂³	1	2	yes	—
66	C₂×#11719917	—	2	1	no	—
67	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
68	C₂×#11719912	—	2	1	no	—
69	C₂²×#88322	—	2	1	no	—
70	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
71	C₂²×#88322	C₂³	1	1	yes	characteristic
72	C₂×C₈×D₅₃	—	2	1	no	—
73	#13133152	C₂×C₄	1	4	yes	—
74	#13133145	C₂×C₄	1	4	yes	—
75	C₄×#88322	C₂×C₄	1	2	yes	—
76	C₄×#88322	C₂³	1	2	yes	—
77	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
80	#104×C₅₃	C₂×C₄	1	4	yes	—
81	#41×C₅₃	—	2	1	no	—
82	#41×C₅₃	—	2	1	no	—
83	#41×C₅₃	C₂×C₈	1	2	yes	—
84	#41×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
86	C₂×D₄×C₅₃	—	2	1	no	—
87	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	—	2	1	no	—
89	C₂²×C₄×C₅₃	—	2	1	no	—
90	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₂²⋊D₅₃	—	2	1	no	—
92	#11719945	—	2	1	no	—
93	#11719945	—	2	1	no	—
94	#11719945	C₂×C₈	1	2	yes	—
95	#11719945	C₂²×C₄	1	2	yes	—
96	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
97	C₂×C₄×D₅₃	—	2	1	no	—
98	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
99	M₁₆×C₅₃	—	2	1	no	—
100	M₁₆×C₅₃	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₅₃	—	2	1	no	—
102	C₄²×C₅₃	—	2	1	no	—
103	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
104	#11719925	C₂×C₈	1	4	yes	—
105	#11719924	C₂×C₈	1	4	yes	—
106	#11719923	C₂×C₈	1	4	yes	—
107	#11719922	C₂×C₈	1	4	yes	—
108	#11719917	—	2	1	no	—
109	#11719917	—	2	2	no	—
110	#11719917	—	2	2	no	—
111	#11719917	C₂²×C₄	1	2	yes	—
112	C₂×#88322	—	2	1	no	—
113	C₂×#88322	—	2	2	no	—
114	C₂×#88322	—	2	2	no	—
115	C₂×#88322	C₂×C₈	1	1	yes	characteristic
116	C₂×#88322	C₂×C₈	1	1	yes	characteristic
117	C₂×#88322	C₂×C₈	1	2	yes	—
118	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
119	C₂×#88322	C₂²×C₄	1	2	yes	—
120	C₂×#88322	C₂×D₄	1	1	yes	characteristic
121	C₂×#88322	C₂×D₄	1	1	yes	characteristic
122	C₂×#88322	#41	1	1	yes	characteristic
123	C₂×#88322	#41	1	1	yes	characteristic
124	C₂×#88322	C₂⁴	1	1	yes	characteristic
125	#11719912	—	2	2	no	—
126	#11719912	—	2	2	no	—
127	C₈×D₅₃	—	2	2	no	—
128	C₈×D₅₃	—	2	2	no	—
129	#11719910	—	2	2	no	—
130	#11719910	—	2	2	no	—
131	#11719910	—	2	2	no	—
132	#11719910	C₂×D₄	1	2	yes	—
133	#11719910	C₂×D₄	1	2	yes	—
134	C₂³×C₅₃	—	2	1	no	—
135	C₂³×C₅₃	#131	1	1	yes	characteristic
136	C₂²×D₅₃	—	2	1	no	—
137	Q₈×C₅₃	—	2	1	no	—
138	Q₈×C₅₃	C₂²×C₈	1	2	yes	—
139	D₄×C₅₃	—	2	2	no	—
140	D₄×C₅₃	—	2	2	no	—
141	D₄×C₅₃	—	2	1	no	—
142	D₄×C₅₃	C₂²×C₈	1	2	yes	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	4	2	no	—
148	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
152	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
154	C₂²⋊D₅₃	—	2	2	no	—
155	C₂²⋊D₅₃	—	2	2	no	—
156	C₂×Dic₅₃	—	2	1	no	—
157	D₂₁₂	—	2	1	no	—
158	D₂₁₂	C₂²×C₈	1	2	yes	—
159	C₄×D₅₃	—	2	2	no	—
160	C₄×D₅₃	—	2	2	no	—
161	Dic₁₀₆	—	2	1	no	—
162	Dic₁₀₆	C₂²×C₈	1	2	yes	—
163	C₈×C₅₃	—	2	2	no	—
164	#88322	—	2	2	no	—
165	#88322	C₄×D₄	1	1	yes	characteristic
166	#88322	C₄×D₄	1	1	yes	characteristic
167	#88322	C₂²×C₈	1	2	yes	—
168	#88322	C₂²×C₈	1	2	yes	—
169	#88322	C₂²×D₄	1	1	yes	characteristic
170	#88322	C₂×#41	1	1	yes	characteristic
171	C₂×#2171	—	53	1	no	maximal, Sylow
172	C₂²×C₅₃	—	2	2	no	—
173	C₂²×C₅₃	—	2	2	no	—
174	C₂²×C₅₃	—	4	2	no	—
175	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
176	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
177	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
178	C₂×D₅₃	—	2	2	no	—
179	C₂×D₅₃	—	4	2	no	—
180	C₄×C₅₃	—	2	2	no	—
181	C₄×C₅₃	—	4	2	no	—
182	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
183	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
184	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
185	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
186	Dic₅₃	—	2	2	no	—
187	C₂×#504	—	53	1	no	—
188	C₂×#504	—	53	1	no	—
189	C₂×#443	—	53	1	no	—
190	#2171	—	53	8	no	—
191	C₂×#364	—	53	1	no	—
192	C₂×#350	—	53	1	no	—
193	C₂×#350	—	53	1	no	—
194	C₂×#346	—	53	1	no	—
195	C₂×C₅₃	—	2	1	no	—
196	C₂×C₅₃	—	2	1	no	—
197	C₂×C₅₃	—	4	2	no	—
198	C₂×C₅₃	#2171	1	2	yes	complemented
199	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
200	D₅₃	—	4	2	no	—
201	C₂×#135	—	53	1	no	—
202	C₂×#131	—	53	1	no	—
203	C₂×#131	—	53	1	no	—
204	#504	—	53	4	no	—
205	#504	—	53	4	no	—
206	#504	—	106	2	no	—
207	#504	—	106	2	no	—
208	C₂×M₃₂	—	53	1	no	—
209	C₂×M₃₂	—	53	1	no	—
210	C₂×M₃₂	—	106	1	no	—
211	C₂×M₃₂	—	106	1	no	—
212	C₂²×C₁₆	—	53	1	no	—
213	C₂²×C₁₆	—	53	1	no	—
214	C₂²×C₁₆	—	106	1	no	—
215	C₂²×C₁₆	—	106	1	no	—
216	#443	—	53	4	no	—
217	#443	—	53	4	no	—
218	C₂×#108	—	53	1	no	—
219	C₂×#104	—	53	1	no	—
220	C₂×#104	#6157	1	1	yes	characteristic
221	C₂×C₄×C₈	—	53	1	no	—
222	#364	—	53	4	no	—
223	#350	—	53	4	no	—
224	#350	—	53	4	no	—
225	#346	—	53	4	no	—
226	C₅₃	C₂×#2171	1	1	yes	characteristic, complemented, Sylow
227	C₂×#41	—	53	1	no	—
228	C₂×#41	C₂×#6157	1	1	yes	characteristic
229	#135	—	53	4	no	—
230	#135	—	53	4	no	—
231	C₂×Q₁₆	—	53	1	no	—
232	C₂×SD₁₆	—	53	1	no	—
233	C₂×SD₁₆	—	53	1	no	—
234	C₂×D₈	—	53	1	no	—
235	#131	—	53	2	no	—
236	#131	—	53	2	no	—
237	#131	—	53	2	no	—
238	#131	—	53	2	no	—
239	#131	—	106	1	no	—
240	#131	—	106	1	no	—
241	#131	—	106	1	no	—
242	#131	—	106	1	no	—
243	C₂×M₁₆	—	53	1	no	—
244	C₂×M₁₆	—	106	1	no	—
245	C₂×M₁₆	—	106	1	no	—
246	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
247	C₂²×C₈	—	53	1	no	—
248	C₂²×C₈	—	53	1	no	—
249	C₂²×C₈	—	106	1	no	—
250	C₂²×C₈	—	106	1	no	—
251	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
252	M₃₂	—	53	2	no	—
253	M₃₂	—	53	2	no	—
254	M₃₂	—	106	4	no	—
255	M₃₂	—	106	4	no	—
256	M₃₂	—	106	1	no	—
257	M₃₂	—	106	1	no	—
258	C₂×C₁₆	—	53	2	no	—
259	C₂×C₁₆	—	53	1	no	—
260	C₂×C₁₆	—	53	1	no	—
261	C₂×C₁₆	—	53	2	no	—
262	C₂×C₁₆	—	53	2	no	—
263	C₂×C₁₆	—	106	4	no	—
264	C₂×C₁₆	—	106	4	no	—
265	C₂×C₁₆	—	106	1	no	—
266	C₂×C₁₆	—	106	1	no	—
267	C₂×C₁₆	—	106	1	no	—
268	C₂×C₁₆	—	106	1	no	—
269	#108	—	53	4	no	—
270	#104	—	53	4	no	—
271	#104	C₂×#6157	1	4	yes	—
272	C₄×C₈	—	53	2	no	—
273	C₄×C₈	—	53	2	no	—
274	#41	—	2	1	no	—
275	#41	—	2	1	no	—
276	#41	—	53	2	no	—
277	#41	—	53	2	no	—
278	#41	—	106	1	no	—
279	#41	—	106	1	no	—
280	#41	D₅₃⋊C₈	1	2	yes	—
281	#41	C₂²×#6157	1	2	yes	—
282	C₂×Q₈	—	53	1	no	—
283	C₂×Q₈	D₅₃⋊C₈	1	1	yes	characteristic
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	53	1	no	—
286	C₂×D₄	—	106	1	no	—
287	C₂×D₄	D₅₃⋊C₈	1	1	yes	characteristic
288	C₂²×C₄	—	2	1	no	—
289	C₂²×C₄	—	2	1	no	—
290	C₂²×C₄	—	106	1	no	—
291	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
292	Q₁₆	—	53	4	no	—
293	SD₁₆	—	53	4	no	—
294	SD₁₆	—	53	4	no	—
295	D₈	—	53	4	no	—
296	M₁₆	—	2	1	no	—
297	M₁₆	—	53	2	no	—
298	M₁₆	—	106	2	no	—
299	M₁₆	—	106	2	no	—
300	M₁₆	—	106	2	no	—
301	M₁₆	—	106	2	no	—
302	M₁₆	—	106	1	no	—
303	M₁₆	C₂²×#6157	1	2	yes	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	53	1	no	—
306	C₂×C₈	—	53	1	no	—
307	C₂×C₈	—	53	1	no	—
308	C₂×C₈	—	53	1	no	—
309	C₂×C₈	—	53	1	no	—
310	C₂×C₈	—	53	1	no	—
311	C₂×C₈	—	53	2	no	—
312	C₂×C₈	—	53	2	no	—
313	C₂×C₈	—	53	1	no	—
314	C₂×C₈	—	53	1	no	—
315	C₂×C₈	—	106	2	no	—
316	C₂×C₈	—	106	2	no	—
317	C₂×C₈	—	106	2	no	—
318	C₂×C₈	—	106	2	no	—
319	C₂×C₈	—	106	1	no	—
320	C₄²	—	2	1	no	—
321	C₄²	C₂²×#6157	1	2	yes	—
322	C₁₆	—	53	2	no	—
323	C₁₆	—	53	2	no	—
324	C₁₆	—	106	2	no	—
325	C₁₆	—	106	2	no	—
326	C₁₆	—	106	2	no	—
327	C₂³	—	2	1	no	—
328	C₂³	—	106	1	no	—
329	C₂³	#13133304	1	1	yes	characteristic
330	Q₈	—	2	1	no	—
331	Q₈	—	53	2	no	—
332	Q₈	—	106	1	no	—
333	Q₈	C₂×D₅₃⋊C₈	1	2	yes	—
334	D₄	—	2	2	no	—
335	D₄	—	2	2	no	—
336	D₄	—	2	1	no	—
337	D₄	—	53	2	no	—
338	D₄	—	106	2	no	—
339	D₄	—	106	2	no	—
340	D₄	—	106	1	no	—
341	D₄	C₂×D₅₃⋊C₈	1	2	yes	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	106	1	no	—
348	C₂×C₄	—	106	2	no	—
349	C₂×C₄	—	106	2	no	—
350	C₂×C₄	D₄×#6157	1	1	yes	characteristic
351	C₂×C₄	D₄×#6157	1	1	yes	characteristic
352	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
353	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
354	C₂×C₄	#13133304	1	1	yes	characteristic
355	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
356	C₈	—	2	2	no	—
357	C₈	—	53	1	no	—
358	C₈	—	53	1	no	—
359	C₈	—	53	1	no	—
360	C₈	—	53	1	no	—
361	C₈	—	53	2	no	—
362	C₈	—	53	2	no	—
363	C₈	—	106	2	no	—
364	C₂²	—	2	2	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	106	2	no	—
368	C₂²	—	212	2	no	—
369	C₂²	#16811650	1	1	yes	characteristic
370	C₂²	#16811650	1	1	yes	characteristic
371	C₂²	C₂×#13133304	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	4	2	no	—
374	C₄	—	106	2	no	—
375	C₄	#16811650	1	1	yes	characteristic
376	C₄	#16811650	1	1	yes	characteristic
377	C₄	C₂×D₄×#6157	1	1	yes	characteristic
378	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	—	4	2	no	—
382	C₂	—	212	2	no	—
383	C₂	#20291106	1	2	yes	direct product factor
384	C₂	C₂×#16811650	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×#104×C₅₃	C₄
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×#13133237	C₄
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×#104	#6157
hypocenter	C₅₃	C₂×#2171
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#104

Complements

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

N	H ≅ G/N	type
1. C₂×#16811682	C₂	semidirect
2. C₂×#16811681	C₂	semidirect
3. #20291106	C₂	direct (N × H)
4. C₂×#16811546	C₂	semidirect
5. C₂×#16811522	C₂	semidirect
6. C₂×#16811521	C₂	semidirect
7. C₂×#16811511	C₂	semidirect
10. #16811682	C₂²	semidirect
12. #16811681	C₂²	semidirect
21. #16811546	C₂²	semidirect
22. #16811522	C₂²	semidirect
23. #16811521	C₂²	semidirect
24. #16811511	C₂²	semidirect
198. C₂×C₅₃	#2171	semidirect
226. C₅₃	C₂×#2171	semidirect
383. C₂	#20291106	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×#16811682	2
C₂×#16811681	2
#20291106	2
C₂×#16811546	2
C₂×#16811522	2
C₂×#16811521	2
C₂×#16811511	2
C₂×#16810737	2
C₂×#2171	53

Subgroups — C2×#20291106 #28328344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20291106 #28328344