Subgroups — C₂×#20290934 #28327922

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₃⋊M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811586	C₂	1	1	yes	maximal, characteristic
3	#20290934	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16811514	C₂	1	2	yes	maximal, complemented
5	C₂×#16811513	C₂	1	2	yes	maximal, complemented
6	C₂²×#13133156	C₂	1	1	yes	maximal, characteristic
7	C₂×D₅₃⋊M₁₆	C₄	1	1	yes	characteristic
8	C₂×#13133296	C₄	1	1	yes	characteristic
9	C₂×#13133292	C₂²	1	2	yes	—
10	D₅₃⋊M₃₂	C₄	1	4	yes	complemented
11	D₅₃⋊M₃₂	C₂²	1	4	yes	—
12	C₂×#13133291	C₄	1	2	yes	complemented
13	C₂×D₅₃⋊C₁₆	C₄	1	2	yes	complemented
14	#16811586	C₄	1	4	yes	—
15	#16811586	C₂²	1	4	yes	—
16	C₂×#13133271	C₂²	1	2	yes	—
17	C₂×C₈×#6157	C₄	1	2	yes	—
18	#16811514	C₂²	1	8	yes	complemented
19	#16811513	C₂²	1	8	yes	complemented
20	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
21	C₂×#13133156	C₄	1	1	yes	characteristic, complemented
22	C₂×#13133156	C₄	1	1	yes	characteristic, complemented
23	C₂×#13133156	C₄	1	4	yes	complemented
24	C₂×#13133156	C₂²	1	1	yes	characteristic
25	C₂×#13133156	C₂²	1	1	yes	characteristic
26	C₂×#13133156	C₂²	1	4	yes	—
27	C₂²×#11719908	C₄	1	1	yes	characteristic, complemented
28	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
29	C₂×#11719936	—	2	1	no	—
30	D₅₃⋊M₁₆	—	2	2	no	—
31	D₅₃⋊M₁₆	C₂×C₄	1	4	yes	—
32	C₂×#11719940	—	2	1	no	—
33	C₂×D₅₃⋊C₈	C₂×C₄	1	2	yes	—
34	#13133296	—	2	2	no	—
35	#13133296	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊#6157	—	2	1	no	—
37	C₂×C₂²⋊#6157	—	2	1	no	—
38	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
39	#13133292	C₂×C₄	1	4	yes	—
40	#13133292	C₂³	1	4	yes	—
41	#13133291	C₂×C₄	1	8	yes	complemented
42	D₅₃⋊C₁₆	C₂×C₄	1	8	yes	complemented
43	C₂×#11719910	C₂×C₄	1	4	yes	—
44	#13133271	C₂×C₄	1	4	yes	—
45	#13133271	C₂³	1	4	yes	—
46	C₈×#6157	C₂×C₄	1	8	yes	—
47	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
55	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₈×D₅₃	C₂³	1	2	yes	—
57	#13133156	C₂×C₄	1	2	yes	—
58	#13133156	C₂×C₄	1	2	yes	—
59	#13133156	C₂×C₄	1	4	yes	complemented
60	#13133156	C₂×C₄	1	4	yes	complemented
61	#13133156	C₂³	1	2	yes	—
62	#13133156	C₂³	1	2	yes	—
63	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
64	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
65	C₂×#11719908	C₂×C₄	1	4	yes	complemented
66	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
70	C₂³×D₅₃	M₁₆	1	1	yes	characteristic
71	C₂²×#6157	—	2	2	no	—
72	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
73	C₂²×Dic₅₃	M₁₆	1	1	yes	characteristic
74	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
75	C₂×C₄×D₅₃	C₄²	1	2	yes	—
76	C₂×C₄×D₅₃	M₁₆	1	1	yes	characteristic
77	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
78	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
79	C₂×C₄×D₅₃	M₁₆	1	1	yes	characteristic
80	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
82	C₂²⋊#6157	—	2	4	no	—
83	#11719940	—	2	4	no	—
84	C₂×#88324	—	2	2	no	—
85	C₄⋊#6157	—	2	4	no	—
86	C₄×#6157	—	2	2	no	—
87	C₄×#6157	C₂²×C₄	1	4	yes	—
88	#11719936	—	2	4	no	—
89	D₅₃⋊C₈	—	2	2	no	—
90	D₅₃⋊C₈	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₅₃	C₄²	1	2	yes	—
92	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂×#88322	C₄²	1	1	yes	characteristic
96	C₂×#88322	C₄²	1	2	yes	—
97	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
98	C₂×#88322	C₂²×C₄	1	2	yes	—
99	C₈×D₅₃	C₄²	1	2	yes	—
100	C₈×D₅₃	C₄²	1	2	yes	—
101	C₈×D₅₃	C₄²	1	2	yes	—
102	C₈×D₅₃	C₂²×C₄	1	2	yes	—
103	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₈×D₅₃	C₂²×C₄	1	2	yes	—
106	C₈×D₅₃	C₂²×C₄	1	2	yes	—
107	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₈×D₅₃	C₂⁴	1	1	yes	characteristic
109	#11719910	C₂²×C₄	1	8	yes	—
110	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
111	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
112	#11719908	C₂²×C₄	1	2	yes	—
113	#11719908	C₂²×C₄	1	2	yes	—
114	C₂³×C₅₃	#97	1	1	yes	characteristic
115	C₂²×D₅₃	—	4	2	no	—
116	C₂²×D₅₃	#97	1	1	yes	characteristic
117	C₂²×D₅₃	#97	1	2	yes	—
118	C₂²×D₅₃	C₂×M₁₆	1	1	yes	characteristic
119	C₂²×D₅₃	C₂×M₁₆	1	2	yes	—
120	C₂×#6157	—	2	4	no	—
121	C₂×#6157	—	4	4	no	—
122	C₂×C₄×C₅₃	#97	1	2	yes	—
123	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
126	C₂×Dic₅₃	#97	1	2	yes	—
127	C₂×Dic₅₃	#97	1	1	yes	characteristic
128	C₂×Dic₅₃	C₂×M₁₆	1	1	yes	characteristic
129	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
130	C₄×D₅₃	#97	1	2	yes	—
131	C₄×D₅₃	#97	1	2	yes	—
132	C₄×D₅₃	#97	1	2	yes	—
133	C₄×D₅₃	C₂×C₄²	1	1	yes	characteristic
134	C₄×D₅₃	C₂×C₄²	1	2	yes	—
135	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
136	C₄×D₅₃	C₂×M₁₆	1	1	yes	characteristic
137	C₄×D₅₃	C₂×M₁₆	1	1	yes	characteristic
138	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
139	C₄×D₅₃	C₂³×C₄	1	1	yes	characteristic
140	#88324	—	2	4	no	—
141	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
142	C₈×C₅₃	C₂×C₄²	1	2	yes	—
143	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
144	#88322	C₂×C₄²	1	2	yes	—
145	#88322	C₂×C₄²	1	1	yes	characteristic
146	#88322	C₂³×C₄	1	1	yes	characteristic
147	C₂×#2110	—	53	1	no	maximal, Sylow
148	C₂²×C₅₃	—	2	2	no	—
149	C₂²×C₅₃	C₂×#97	1	2	yes	—
150	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
151	C₂×D₅₃	—	2	2	no	—
152	C₂×D₅₃	—	2	2	no	—
153	C₂×D₅₃	—	4	2	no	—
154	C₂×D₅₃	—	4	2	no	—
155	C₂×D₅₃	C₂×#97	1	2	yes	—
156	C₂×D₅₃	C₂×#97	1	1	yes	characteristic
157	C₂×D₅₃	C₂²×M₁₆	1	1	yes	characteristic
158	#6157	—	4	4	no	—
159	C₄×C₅₃	C₂×#97	1	2	yes	—
160	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
161	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
162	Dic₅₃	C₂×#97	1	1	yes	characteristic
163	Dic₅₃	C₂×#97	1	2	yes	—
164	Dic₅₃	C₂²×M₁₆	1	1	yes	characteristic
165	C₂²×M₃₂	—	53	1	no	—
166	C₂²×M₃₂	—	53	1	no	—
167	C₂×#405	—	53	1	no	—
168	#2110	—	53	8	no	—
169	C₂×#347	—	53	2	no	—
170	C₂×#347	—	53	2	no	—
171	C₂×C₅₃	—	2	2	no	—
172	C₂×C₅₃	#2110	1	2	yes	complemented
173	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
174	D₅₃	—	2	1	no	—
175	D₅₃	—	2	1	no	—
176	D₅₃	—	2	2	no	—
177	C₂²×M₁₆	—	53	1	no	—
178	C₂³×C₈	—	53	1	no	—
179	C₂×#117	—	53	1	no	—
180	C₂×M₃₂	—	53	2	no	—
181	C₂×M₃₂	—	53	4	no	—
182	C₂×M₃₂	—	53	2	no	—
183	C₂×M₃₂	—	53	4	no	—
184	C₂×M₃₂	—	53	1	no	—
185	C₂×M₃₂	—	53	1	no	—
186	C₂×M₃₂	—	53	1	no	—
187	C₂×M₃₂	—	53	1	no	—
188	C₂×M₃₂	—	53	4	no	—
189	C₂×M₃₂	—	53	4	no	—
190	C₂²×C₁₆	—	53	2	no	—
191	C₂²×C₁₆	—	53	1	no	—
192	C₂²×C₁₆	#6157	1	1	yes	characteristic, complemented
193	#405	—	53	4	no	—
194	#405	—	53	4	no	—
195	C₂×#97	—	53	2	no	—
196	C₂×C₄×C₈	—	53	2	no	—
197	#347	—	53	8	no	—
198	#347	—	53	8	no	—
199	C₅₃	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
200	C₂³×C₄	—	53	1	no	—
201	C₂×M₁₆	—	53	4	no	—
202	C₂×M₁₆	—	106	1	no	—
203	C₂×M₁₆	—	106	2	no	—
204	C₂×M₁₆	—	106	1	no	—
205	C₂²×C₈	—	53	1	no	—
206	C₂²×C₈	—	53	2	no	—
207	C₂²×C₈	—	53	2	no	—
208	C₂²×C₈	—	53	1	no	—
209	C₂²×C₈	—	53	1	no	—
210	C₂²×C₈	—	53	1	no	—
211	C₂²×C₈	—	53	2	no	—
212	C₂²×C₈	—	53	1	no	—
213	C₂²×C₈	—	53	2	no	—
214	C₂²×C₈	—	53	2	no	—
215	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
216	#117	—	53	4	no	—
217	#117	—	106	2	no	—
218	C₂×C₄⋊C₄	—	106	1	no	—
219	C₂×C₂²⋊C₄	—	106	1	no	—
220	C₂×C₄²	—	53	2	no	—
221	M₃₂	—	53	4	no	—
222	M₃₂	—	53	8	no	—
223	M₃₂	—	53	4	no	—
224	M₃₂	—	53	2	no	—
225	M₃₂	—	53	2	no	—
226	M₃₂	—	53	2	no	—
227	M₃₂	—	53	2	no	—
228	M₃₂	—	53	4	no	—
229	M₃₂	—	53	4	no	—
230	C₂×C₁₆	—	53	4	no	—
231	C₂×C₁₆	—	53	8	no	—
232	C₂×C₁₆	—	53	1	no	—
233	C₂×C₁₆	—	53	1	no	—
234	C₂×C₁₆	—	53	4	no	—
235	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
236	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
237	C₂×C₁₆	C₂×#6157	1	4	yes	complemented
238	#97	—	53	4	no	—
239	#97	—	53	4	no	—
240	C₄×C₈	—	53	8	no	—
241	C₂⁴	—	53	1	no	—
242	C₂²×C₄	—	53	1	no	—
243	C₂²×C₄	—	53	1	no	—
244	C₂²×C₄	—	53	2	no	—
245	C₂²×C₄	—	53	1	no	—
246	C₂²×C₄	—	53	2	no	—
247	C₂²×C₄	—	53	2	no	—
248	C₂²×C₄	—	53	1	no	—
249	C₂²×C₄	—	53	1	no	—
250	C₂²×C₄	—	53	2	no	—
251	C₂²×C₄	—	106	2	no	—
252	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
253	M₁₆	—	106	4	no	—
254	M₁₆	—	106	4	no	—
255	C₂×C₈	—	53	1	no	—
256	C₂×C₈	—	53	2	no	—
257	C₂×C₈	—	53	2	no	—
258	C₂×C₈	—	53	1	no	—
259	C₂×C₈	—	53	2	no	—
260	C₂×C₈	—	53	1	no	—
261	C₂×C₈	—	53	2	no	—
262	C₂×C₈	—	53	1	no	—
263	C₂×C₈	—	53	2	no	—
264	C₂×C₈	—	53	2	no	—
265	C₂×C₈	—	53	2	no	—
266	C₂×C₈	—	53	1	no	—
267	C₂×C₈	—	53	1	no	—
268	C₂×C₈	—	53	2	no	—
269	C₂×C₈	—	53	4	no	—
270	C₂×C₈	—	106	2	no	—
271	C₂×C₈	—	106	2	no	—
272	C₂×C₈	C₄×#6157	1	2	yes	—
273	C₂×C₈	C₄×#6157	1	1	yes	characteristic
274	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
275	C₂×C₈	C₂²×#6157	1	2	yes	—
276	C₄⋊C₄	—	106	4	no	—
277	C₂²⋊C₄	—	106	4	no	—
278	C₄²	—	53	4	no	—
279	C₄²	—	106	2	no	—
280	C₁₆	—	53	8	no	—
281	C₁₆	—	53	2	no	—
282	C₁₆	—	53	2	no	—
283	C₁₆	C₂²×#6157	1	2	yes	—
284	C₁₆	C₂²×#6157	1	2	yes	—
285	C₂³	—	53	1	no	—
286	C₂³	—	53	1	no	—
287	C₂³	—	53	2	no	—
288	C₂³	—	53	2	no	—
289	C₂³	—	212	2	no	—
290	C₂³	#13133271	1	1	yes	characteristic
291	C₂×C₄	—	53	2	no	—
292	C₂×C₄	—	53	1	no	—
293	C₂×C₄	—	53	1	no	—
294	C₂×C₄	—	53	2	no	—
295	C₂×C₄	—	53	2	no	—
296	C₂×C₄	—	53	2	no	—
297	C₂×C₄	—	53	2	no	—
298	C₂×C₄	—	53	1	no	—
299	C₂×C₄	—	53	2	no	—
300	C₂×C₄	—	53	1	no	—
301	C₂×C₄	—	53	2	no	—
302	C₂×C₄	—	53	2	no	—
303	C₂×C₄	—	53	1	no	—
304	C₂×C₄	—	53	1	no	—
305	C₂×C₄	—	106	4	no	—
306	C₂×C₄	—	212	4	no	—
307	C₂×C₄	#13133271	1	2	yes	—
308	C₂×C₄	#13133271	1	1	yes	characteristic
309	C₂×C₄	C₂×C₄×#6157	1	2	yes	—
310	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
311	C₈	—	53	1	no	—
312	C₈	—	53	2	no	—
313	C₈	—	53	1	no	—
314	C₈	—	106	4	no	—
315	C₈	C₂×C₄×#6157	1	1	yes	characteristic
316	C₈	C₂×C₄×#6157	1	2	yes	—
317	C₈	C₂³×#6157	1	1	yes	characteristic
318	C₂²	—	2	2	no	—
319	C₂²	—	53	1	no	—
320	C₂²	—	53	1	no	—
321	C₂²	—	53	2	no	—
322	C₂²	—	106	2	no	—
323	C₂²	—	106	2	no	—
324	C₂²	—	212	2	no	—
325	C₂²	—	212	2	no	—
326	C₂²	C₂×#13133271	1	2	yes	—
327	C₂²	C₂×#13133271	1	1	yes	characteristic
328	C₄	—	53	1	no	—
329	C₄	—	53	1	no	—
330	C₄	—	53	2	no	—
331	C₄	—	212	4	no	—
332	C₄	C₂×#13133271	1	1	yes	characteristic
333	C₄	C₂×#13133271	1	2	yes	—
334	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
335	C₂	—	2	2	no	—
336	C₂	—	106	1	no	—
337	C₂	—	106	1	no	—
338	C₂	—	106	2	no	—
339	C₂	#20290934	1	2	yes	direct product factor
340	C₂	C₂²×#13133271	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×#6157
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₈	C₂³×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₈×D₅₃	C₂⁴
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	C₂×#2110
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #20290934	C₂	direct (N × H)
4. C₂×#16811514	C₂	semidirect
5. C₂×#16811513	C₂	semidirect
10. D₅₃⋊M₃₂	C₄	semidirect
12. C₂×#13133291	C₄	semidirect
13. C₂×D₅₃⋊C₁₆	C₄	semidirect
18. #16811514	C₂²	semidirect
19. #16811513	C₂²	semidirect
21. C₂×#13133156	C₄	semidirect
22. C₂×#13133156	C₄	semidirect
23. C₂×#13133156	C₄	semidirect
27. C₂²×#11719908	C₄	semidirect
28. C₂²×C₁₆×C₅₃	C₄	semidirect
41. #13133291	C₂×C₄	semidirect
42. D₅₃⋊C₁₆	C₂×C₄	semidirect
59. #13133156	C₂×C₄	semidirect
60. #13133156	C₂×C₄	semidirect
65. C₂×#11719908	C₂×C₄	semidirect
69. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
172. C₂×C₅₃	#2110	semidirect
192. C₂²×C₁₆	#6157	semidirect
199. C₅₃	C₂×#2110	semidirect
237. C₂×C₁₆	C₂×#6157	semidirect
339. C₂	#20290934	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×D₅₃⋊M₃₂	2
C₂×#16811586	2
#20290934	2
C₂×#16811514	2
C₂×#16811513	2
C₂²×#13133156	2
C₂×#2110	53

Subgroups — C2×#20290934 #28327922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20290934 #28327922