Subgroups — C₂×#20290930 #28327918

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 362 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	C₂²×#13133271	C₂	1	1	yes	maximal, characteristic
3	#20290930	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	C₂×M₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×D₅₃⋊C₈	C₄	1	1	yes	characteristic
8	C₂²×C₄×#6157	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	C₂×#13133271	C₄	1	4	yes	—
15	C₂×#13133271	C₄	1	1	yes	characteristic
16	C₂×#13133271	C₄	1	1	yes	characteristic
17	C₂×#13133271	C₂²	1	1	yes	characteristic
18	C₂×#13133271	C₂²	1	1	yes	characteristic
19	C₂×#13133271	C₂²	1	4	yes	—
20	#16811514	C₂²	1	8	yes	complemented
21	#16811513	C₂²	1	8	yes	complemented
22	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
23	C₂×#13133157	C₄	1	1	yes	characteristic, complemented
24	C₂×#13133156	C₂²	1	2	yes	—
25	M₃₂×D₅₃	C₄	1	4	yes	complemented
26	M₃₂×D₅₃	C₂²	1	4	yes	—
27	C₂×C₁₆×D₅₃	C₄	1	2	yes	complemented
28	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic, complemented
29	C₂³×#6157	—	2	1	no	—
30	C₂×D₅₃⋊C₈	—	2	2	no	—
31	C₂×D₅₃⋊C₈	—	2	1	no	—
32	C₂×D₅₃⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×D₅₃⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×D₅₃⋊C₈	C₂×C₄	1	4	yes	—
35	C₂²×#88324	—	2	1	no	—
36	C₂×C₄×#6157	—	2	2	no	—
37	C₂×C₄×#6157	—	2	1	no	—
38	C₂×C₄×#6157	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×#6157	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×#6157	C₂×C₄	1	4	yes	—
41	#13133292	C₂×C₄	1	4	yes	complemented
42	#13133292	C₂³	1	4	yes	—
43	#13133291	C₂×C₄	1	8	yes	complemented
44	D₅₃⋊C₁₆	C₂×C₄	1	8	yes	complemented
45	C₂×#11719910	C₂×C₄	1	4	yes	complemented
46	#13133271	C₂×C₄	1	2	yes	—
47	#13133271	C₂×C₄	1	2	yes	—
48	#13133271	C₂×C₄	1	4	yes	—
49	#13133271	C₂×C₄	1	4	yes	—
50	#13133271	C₂³	1	2	yes	—
51	#13133271	C₂³	1	2	yes	—
52	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
56	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
58	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₈×D₅₃	C₂³	1	2	yes	—
61	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
62	#13133157	C₂×C₄	1	4	yes	complemented
63	#13133156	C₂×C₄	1	4	yes	complemented
64	#13133156	C₂³	1	4	yes	—
65	C₂×#11719908	C₂×C₄	1	2	yes	complemented
66	C₁₆×D₅₃	C₂×C₄	1	8	yes	complemented
67	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
68	M₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
69	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
70	C₂³×D₅₃	M₁₆	1	1	yes	characteristic
71	C₂²×#6157	—	2	1	no	—
72	C₂²×#6157	—	2	1	no	—
73	C₂²×#6157	—	2	4	no	—
74	C₂²×#6157	—	4	4	no	—
75	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
76	C₂²×Dic₅₃	M₁₆	1	1	yes	characteristic
77	C₂×C₄×D₅₃	C₄²	1	2	yes	—
78	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
79	C₂×C₄×D₅₃	M₁₆	1	1	yes	characteristic
80	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
81	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
82	C₂×C₄×D₅₃	M₁₆	1	1	yes	characteristic
83	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×#88324	—	2	4	no	—
86	C₂×#88324	—	2	1	no	—
87	C₂×#88324	—	2	1	no	—
88	C₄×#6157	—	2	1	no	—
89	C₄×#6157	—	2	1	no	—
90	C₄×#6157	—	2	4	no	—
91	C₄×#6157	C₂²×C₄	1	2	yes	—
92	C₄×#6157	C₂²×C₄	1	2	yes	—
93	D₅₃⋊C₈	—	2	1	no	—
94	D₅₃⋊C₈	—	2	1	no	—
95	D₅₃⋊C₈	—	2	4	no	—
96	D₅₃⋊C₈	C₂²×C₄	1	2	yes	—
97	D₅₃⋊C₈	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₅₃	C₄²	1	2	yes	—
99	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
100	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
102	C₂×#88322	C₄²	1	2	yes	—
103	C₂×#88322	C₄²	1	1	yes	characteristic
104	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
105	C₂×#88322	C₂²×C₄	1	2	yes	—
106	C₈×D₅₃	C₄²	1	2	yes	—
107	C₈×D₅₃	C₄²	1	2	yes	—
108	C₈×D₅₃	C₄²	1	2	yes	—
109	C₈×D₅₃	C₂²×C₄	1	2	yes	—
110	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₈×D₅₃	C₂²×C₄	1	2	yes	—
112	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₈×D₅₃	C₂²×C₄	1	2	yes	—
114	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₈×D₅₃	C₂⁴	1	1	yes	characteristic
116	#11719910	C₂²×C₄	1	8	yes	complemented
117	C₁₆×C₅₃	C₂²×C₄	1	4	yes	complemented
118	#11719908	C₂²×C₄	1	4	yes	complemented
119	C₂³×C₅₃	#97	1	1	yes	characteristic
120	C₂²×D₅₃	—	2	4	no	—
121	C₂²×D₅₃	#97	1	2	yes	—
122	C₂²×D₅₃	#97	1	1	yes	characteristic
123	C₂²×D₅₃	C₂×M₁₆	1	1	yes	characteristic
124	C₂²×D₅₃	C₂×M₁₆	1	2	yes	—
125	C₂×#6157	—	2	2	no	—
126	C₂×#6157	—	2	2	no	—
127	C₂×#6157	—	4	2	no	—
128	C₂×#6157	—	4	2	no	—
129	C₂×#6157	—	4	8	no	—
130	C₂×C₄×C₅₃	#97	1	2	yes	—
131	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
134	C₂×Dic₅₃	#97	1	2	yes	—
135	C₂×Dic₅₃	#97	1	1	yes	characteristic
136	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
137	C₂×Dic₅₃	C₂×M₁₆	1	1	yes	characteristic
138	C₄×D₅₃	#97	1	2	yes	—
139	C₄×D₅₃	#97	1	2	yes	—
140	C₄×D₅₃	#97	1	2	yes	—
141	C₄×D₅₃	C₂×C₄²	1	1	yes	characteristic
142	C₄×D₅₃	C₂×C₄²	1	2	yes	—
143	C₄×D₅₃	C₂×M₁₆	1	1	yes	characteristic
144	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
145	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
146	C₄×D₅₃	C₂×M₁₆	1	1	yes	characteristic
147	C₄×D₅₃	C₂³×C₄	1	1	yes	characteristic
148	#88324	—	2	2	no	—
149	#88324	—	2	2	no	—
150	C₈×C₅₃	C₂×C₄²	1	2	yes	—
151	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
152	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
153	#88322	C₂×C₄²	1	2	yes	—
154	#88322	C₂×C₄²	1	1	yes	characteristic
155	#88322	C₂³×C₄	1	1	yes	characteristic
156	C₂²×#347	—	53	1	no	maximal, Sylow
157	C₂²×C₅₃	—	2	2	no	—
158	C₂²×C₅₃	C₂×#97	1	2	yes	—
159	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
160	C₂×D₅₃	—	2	2	no	—
161	C₂×D₅₃	—	2	4	no	—
162	C₂×D₅₃	—	2	4	no	—
163	C₂×D₅₃	#347	1	4	yes	complemented
164	C₂×D₅₃	C₂×#97	1	1	yes	characteristic
165	C₂×D₅₃	C₂×#97	1	2	yes	—
166	C₂×D₅₃	C₂²×M₁₆	1	1	yes	characteristic
167	#6157	—	4	2	no	—
168	#6157	—	4	2	no	—
169	C₄×C₅₃	C₂×#97	1	2	yes	—
170	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
171	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
172	Dic₅₃	C₂×#97	1	2	yes	—
173	Dic₅₃	C₂×#97	1	1	yes	characteristic
174	Dic₅₃	C₂²×M₁₆	1	1	yes	characteristic
175	C₂²×M₃₂	—	53	1	no	—
176	C₂²×M₃₂	—	53	1	no	—
177	C₂²×#97	—	53	1	no	—
178	C₂×#347	—	53	2	no	—
179	C₂×#347	—	53	2	no	—
180	C₂×#347	—	53	8	no	—
181	C₂×C₅₃	—	2	2	no	—
182	C₂×C₅₃	C₂×#347	1	2	yes	complemented
183	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
184	D₅₃	—	2	1	no	—
185	D₅₃	—	2	1	no	—
186	D₅₃	C₂×#347	1	4	yes	complemented
187	C₂³×C₈	—	53	1	no	—
188	C₂³×C₈	—	53	1	no	—
189	C₂²×C₄²	—	53	1	no	—
190	C₂×M₃₂	—	53	4	no	—
191	C₂×M₃₂	—	53	2	no	—
192	C₂×M₃₂	—	53	2	no	—
193	C₂×M₃₂	—	53	4	no	—
194	C₂×M₃₂	—	53	1	no	—
195	C₂×M₃₂	—	53	4	no	—
196	C₂×M₃₂	—	53	4	no	—
197	C₂×M₃₂	—	53	2	no	—
198	C₂×M₃₂	#6157	1	1	yes	characteristic, complemented
199	C₂²×C₁₆	—	53	2	no	—
200	C₂²×C₁₆	—	53	2	no	—
201	C₂×#97	—	53	4	no	—
202	C₂×#97	—	53	1	no	—
203	C₂×#97	—	53	1	no	—
204	C₂×#97	—	53	1	no	—
205	C₂×#97	—	53	1	no	—
206	C₂×#97	—	53	4	no	—
207	#347	—	53	8	no	—
208	#347	—	53	8	no	—
209	C₅₃	C₂²×#347	1	1	yes	characteristic, complemented, Sylow
210	C₂³×C₄	—	53	1	no	—
211	C₂³×C₄	—	106	1	no	—
212	C₂²×C₈	—	53	1	no	—
213	C₂²×C₈	—	53	2	no	—
214	C₂²×C₈	—	53	1	no	—
215	C₂²×C₈	—	53	1	no	—
216	C₂²×C₈	—	53	2	no	—
217	C₂²×C₈	—	53	2	no	—
218	C₂²×C₈	—	53	1	no	—
219	C₂²×C₈	—	53	1	no	—
220	C₂²×C₈	—	53	2	no	—
221	C₂²×C₈	—	53	4	no	—
222	C₂²×C₈	—	53	1	no	—
223	C₂²×C₈	—	53	1	no	—
224	C₂²×C₈	—	106	1	no	—
225	C₂²×C₈	—	106	1	no	—
226	C₂²×C₈	—	106	2	no	—
227	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
228	C₂×C₄²	—	53	4	no	—
229	C₂×C₄²	—	53	1	no	—
230	C₂×C₄²	—	53	1	no	—
231	C₂×C₄²	—	106	1	no	—
232	C₂×C₄²	—	106	2	no	—
233	M₃₂	—	53	4	no	—
234	M₃₂	—	53	8	no	—
235	M₃₂	—	53	4	no	—
236	M₃₂	—	53	4	no	—
237	M₃₂	—	53	4	no	—
238	M₃₂	—	53	4	no	—
239	M₃₂	C₂×#6157	1	4	yes	complemented
240	C₂×C₁₆	—	53	4	no	—
241	C₂×C₁₆	—	53	8	no	—
242	C₂×C₁₆	—	53	2	no	—
243	C₂×C₁₆	—	53	8	no	—
244	C₂×C₁₆	C₂×#6157	1	2	yes	complemented
245	#97	—	53	2	no	—
246	#97	—	53	2	no	—
247	#97	—	53	2	no	—
248	#97	—	53	2	no	—
249	#97	—	53	4	no	—
250	#97	—	53	4	no	—
251	C₂⁴	—	53	1	no	—
252	C₂²×C₄	—	53	1	no	—
253	C₂²×C₄	—	53	2	no	—
254	C₂²×C₄	—	53	1	no	—
255	C₂²×C₄	—	53	1	no	—
256	C₂²×C₄	—	53	2	no	—
257	C₂²×C₄	—	53	2	no	—
258	C₂²×C₄	—	53	2	no	—
259	C₂²×C₄	—	53	1	no	—
260	C₂²×C₄	—	53	1	no	—
261	C₂²×C₄	—	106	4	no	—
262	C₂²×C₄	—	106	1	no	—
263	C₂²×C₄	—	106	1	no	—
264	C₂²×C₄	—	212	4	no	—
265	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
266	C₂×C₈	—	53	2	no	—
267	C₂×C₈	—	53	2	no	—
268	C₂×C₈	—	53	1	no	—
269	C₂×C₈	—	53	1	no	—
270	C₂×C₈	—	53	2	no	—
271	C₂×C₈	—	53	1	no	—
272	C₂×C₈	—	53	2	no	—
273	C₂×C₈	—	53	1	no	—
274	C₂×C₈	—	53	2	no	—
275	C₂×C₈	—	53	2	no	—
276	C₂×C₈	—	53	2	no	—
277	C₂×C₈	—	53	1	no	—
278	C₂×C₈	—	53	2	no	—
279	C₂×C₈	—	53	1	no	—
280	C₂×C₈	—	53	2	no	—
281	C₂×C₈	—	53	2	no	—
282	C₂×C₈	—	106	4	no	—
283	C₂×C₈	—	106	1	no	—
284	C₂×C₈	—	106	1	no	—
285	C₂×C₈	—	106	4	no	—
286	C₂×C₈	—	106	1	no	—
287	C₂×C₈	—	106	1	no	—
288	C₂×C₈	C₄×#6157	1	2	yes	—
289	C₂×C₈	C₄×#6157	1	1	yes	characteristic
290	C₂×C₈	C₂²×#6157	1	2	yes	—
291	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
292	C₄²	—	53	2	no	—
293	C₄²	—	53	2	no	—
294	C₄²	—	106	4	no	—
295	C₄²	—	106	1	no	—
296	C₄²	—	106	1	no	—
297	C₁₆	—	53	8	no	—
298	C₁₆	—	53	4	no	—
299	C₁₆	C₂²×#6157	1	4	yes	complemented
300	C₂³	—	53	1	no	—
301	C₂³	—	53	2	no	—
302	C₂³	—	53	2	no	—
303	C₂³	—	53	1	no	—
304	C₂³	—	106	4	no	—
305	C₂³	#13133271	1	1	yes	characteristic
306	C₂×C₄	—	53	2	no	—
307	C₂×C₄	—	53	1	no	—
308	C₂×C₄	—	53	2	no	—
309	C₂×C₄	—	53	1	no	—
310	C₂×C₄	—	53	2	no	—
311	C₂×C₄	—	53	2	no	—
312	C₂×C₄	—	53	2	no	—
313	C₂×C₄	—	53	2	no	—
314	C₂×C₄	—	53	1	no	—
315	C₂×C₄	—	53	2	no	—
316	C₂×C₄	—	53	1	no	—
317	C₂×C₄	—	53	2	no	—
318	C₂×C₄	—	53	1	no	—
319	C₂×C₄	—	53	1	no	—
320	C₂×C₄	—	106	2	no	—
321	C₂×C₄	—	106	2	no	—
322	C₂×C₄	—	212	8	no	—
323	C₂×C₄	—	212	2	no	—
324	C₂×C₄	—	212	2	no	—
325	C₂×C₄	#13133271	1	1	yes	characteristic
326	C₂×C₄	#13133271	1	2	yes	—
327	C₂×C₄	C₂×C₄×#6157	1	2	yes	—
328	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
329	C₈	—	53	2	no	—
330	C₈	—	53	1	no	—
331	C₈	—	53	1	no	—
332	C₈	—	106	2	no	—
333	C₈	—	106	2	no	—
334	C₈	C₂×C₄×#6157	1	2	yes	—
335	C₈	C₂×C₄×#6157	1	1	yes	characteristic
336	C₈	C₂³×#6157	1	1	yes	characteristic
337	C₂²	—	2	2	no	—
338	C₂²	—	53	4	no	—
339	C₂²	—	53	2	no	—
340	C₂²	—	53	1	no	—
341	C₂²	—	53	1	no	—
342	C₂²	—	106	4	no	—
343	C₂²	—	106	4	no	—
344	C₂²	—	106	2	no	—
345	C₂²	C₂×#13133271	1	2	yes	—
346	C₂²	C₂×#13133271	1	1	yes	characteristic
347	C₄	—	53	1	no	—
348	C₄	—	53	1	no	—
349	C₄	—	53	2	no	—
350	C₄	—	212	2	no	—
351	C₄	—	212	2	no	—
352	C₄	C₂×#13133271	1	2	yes	—
353	C₄	C₂×#13133271	1	1	yes	characteristic
354	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
355	C₂	—	2	2	no	—
356	C₂	—	53	4	no	—
357	C₂	—	106	1	no	—
358	C₂	—	106	1	no	—
359	C₂	#20290930	1	2	yes	direct product factor
360	C₂	C₂²×#13133271	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#6157
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₈	C₂³×#6157
Fitting	C₂×M₃₂×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₈×D₅₃	C₂⁴
hypercenter	C₂×M₃₂	#6157
hypocenter	C₅₃	C₂²×#347
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20290930	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
20. #16811514	C₂²	semidirect
21. #16811513	C₂²	semidirect
23. C₂×#13133157	C₄	semidirect
25. M₃₂×D₅₃	C₄	semidirect
27. C₂×C₁₆×D₅₃	C₄	semidirect
28. C₂×M₃₂×C₅₃	C₄	semidirect
41. #13133292	C₂×C₄	semidirect
43. #13133291	C₂×C₄	semidirect
44. D₅₃⋊C₁₆	C₂×C₄	semidirect
45. C₂×#11719910	C₂×C₄	semidirect
62. #13133157	C₂×C₄	semidirect
63. #13133156	C₂×C₄	semidirect
65. C₂×#11719908	C₂×C₄	semidirect
66. C₁₆×D₅₃	C₂×C₄	semidirect
68. M₃₂×C₅₃	C₂×C₄	semidirect
69. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
116. #11719910	C₂²×C₄	semidirect
117. C₁₆×C₅₃	C₂²×C₄	semidirect
118. #11719908	C₂²×C₄	semidirect
163. C₂×D₅₃	#347	semidirect
182. C₂×C₅₃	C₂×#347	semidirect
186. D₅₃	C₂×#347	semidirect
198. C₂×M₃₂	#6157	semidirect
209. C₅₃	C₂²×#347	semidirect
239. M₃₂	C₂×#6157	semidirect
244. C₂×C₁₆	C₂×#6157	semidirect
299. C₁₆	C₂²×#6157	semidirect
359. C₂	#20290930	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
C₂²×#13133271	2
#20290930	2
C₂×#16811514	2
C₂×#16811513	2
C₂×M₃₂×D₅₃	2
C₂²×#347	53

Subgroups — C2×#20290930 #28327918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20290930 #28327918