Subgroups — C₂×#20290927 #28327900

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811586	C₂	1	1	yes	maximal, characteristic
3	#20290927	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16811510	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×#6157	C₂	1	2	yes	maximal, complemented
6	C₂×M₃₂×D₅₃	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×D₅₃⋊C₁₆	C₄	1	2	yes	complemented
10	C₂×D₅₃⋊C₁₆	C₄	1	4	yes	complemented
11	C₂×D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×D₅₃⋊C₁₆	C₂²	1	4	yes	—
14	C₂²×#11719910	C₄	1	2	yes	complemented
15	#16811586	C₄	1	4	yes	—
16	#16811586	C₂²	1	4	yes	—
17	C₂×#13133271	C₄	1	2	yes	—
18	C₂×C₈×#6157	C₂²	1	2	yes	—
19	#16811510	C₂²	1	8	yes	complemented
20	C₁₆×#6157	C₂²	1	8	yes	complemented
21	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
22	C₂×#13133157	C₄	1	1	yes	characteristic, complemented
23	C₂×#13133156	C₄	1	2	yes	complemented
24	M₃₂×D₅₃	C₄	1	4	yes	complemented
25	M₃₂×D₅₃	C₂²	1	4	yes	—
26	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
27	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic, complemented
28	C₂×#11719936	C₈	1	2	yes	—
29	D₅₃⋊M₁₆	C₈	1	4	yes	—
30	D₅₃⋊M₁₆	C₂×C₄	1	4	yes	—
31	C₂×#11719940	C₈	1	2	yes	—
32	C₂×D₅₃⋊C₈	C₂×C₄	1	2	yes	—
33	#13133296	C₈	1	4	yes	—
34	#13133296	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊#6157	C₈	1	2	yes	—
36	C₂×C₂²⋊#6157	C₈	1	2	yes	—
37	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
38	D₅₃⋊C₁₆	C₂×C₄	1	2	yes	—
39	D₅₃⋊C₁₆	C₂×C₄	1	2	yes	—
40	D₅₃⋊C₁₆	C₂×C₄	1	8	yes	complemented
41	D₅₃⋊C₁₆	C₂³	1	2	yes	—
42	D₅₃⋊C₁₆	C₂³	1	2	yes	—
43	C₂×#11719910	C₂×C₄	1	2	yes	—
44	C₂×#11719910	C₂×C₄	1	2	yes	—
45	C₂×#11719910	C₂×C₄	1	8	yes	complemented
46	#13133271	C₂×C₄	1	8	yes	—
47	C₈×#6157	C₂×C₄	1	4	yes	—
48	C₈×#6157	C₂³	1	4	yes	—
49	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
54	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
57	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₈×D₅₃	C₂³	1	2	yes	—
59	#13133157	C₂×C₄	1	4	yes	complemented
60	#13133156	C₂×C₄	1	8	yes	complemented
61	C₂×#11719908	C₂×C₄	1	2	yes	—
62	C₁₆×D₅₃	C₂×C₄	1	4	yes	—
63	C₁₆×D₅₃	C₂³	1	4	yes	—
64	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	M₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
66	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
67	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×#6157	C₂×C₈	1	4	yes	—
69	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
70	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₅₃	C₄²	1	2	yes	—
72	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
73	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
74	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
75	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
76	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
79	C₂²⋊#6157	C₂×C₈	1	8	yes	—
80	#11719940	C₂×C₈	1	8	yes	—
81	C₂×#88324	C₂×C₈	1	4	yes	—
82	C₄⋊#6157	C₂×C₈	1	8	yes	—
83	C₄×#6157	C₂×C₈	1	4	yes	—
84	C₄×#6157	C₂²×C₄	1	4	yes	—
85	#11719936	C₂×C₈	1	8	yes	—
86	D₅₃⋊C₈	C₂×C₈	1	4	yes	—
87	D₅₃⋊C₈	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₄²	1	2	yes	—
90	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×#88322	C₄²	1	2	yes	—
93	C₂×#88322	C₄²	1	1	yes	characteristic
94	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
95	C₂×#88322	C₂²×C₄	1	2	yes	—
96	C₈×D₅₃	C₄²	1	2	yes	—
97	C₈×D₅₃	C₄²	1	2	yes	—
98	C₈×D₅₃	C₄²	1	2	yes	—
99	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₈×D₅₃	C₂²×C₄	1	2	yes	—
101	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₈×D₅₃	C₂²×C₄	1	2	yes	—
103	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₈×D₅₃	C₂²×C₄	1	2	yes	—
105	C₈×D₅₃	C₂⁴	1	1	yes	characteristic
106	#11719910	C₂²×C₄	1	4	yes	—
107	#11719910	C₂²×C₄	1	4	yes	—
108	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
109	#11719908	C₂²×C₄	1	4	yes	—
110	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
111	C₂²×D₅₃	—	2	4	no	—
112	C₂²×D₅₃	C₄×C₈	1	1	yes	characteristic
113	C₂²×D₅₃	C₄×C₈	1	2	yes	—
114	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
115	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×#6157	—	2	8	no	—
117	C₂×#6157	C₂²×C₈	1	8	yes	—
118	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
119	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
122	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
123	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
124	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
126	C₄×D₅₃	C₄×C₈	1	2	yes	—
127	C₄×D₅₃	C₄×C₈	1	2	yes	—
128	C₄×D₅₃	C₄×C₈	1	2	yes	—
129	C₄×D₅₃	C₂×C₄²	1	2	yes	—
130	C₄×D₅₃	C₂×C₄²	1	1	yes	characteristic
131	C₄×D₅₃	C₂²×C₈	1	2	yes	—
132	C₄×D₅₃	C₂²×C₈	1	1	yes	characteristic
133	C₄×D₅₃	C₂²×C₈	1	1	yes	characteristic
134	C₄×D₅₃	C₂²×C₈	1	2	yes	—
135	C₄×D₅₃	C₂³×C₄	1	1	yes	characteristic
136	#88324	C₂²×C₈	1	8	yes	—
137	C₈×C₅₃	C₂×C₄²	1	2	yes	—
138	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
139	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
140	#88322	C₂×C₄²	1	1	yes	characteristic
141	#88322	C₂×C₄²	1	2	yes	—
142	#88322	C₂³×C₄	1	1	yes	characteristic
143	C₂×#2108	—	53	1	no	maximal, Sylow
144	C₂²×C₅₃	—	2	2	no	—
145	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
146	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
147	C₂×D₅₃	—	2	2	no	—
148	C₂×D₅₃	—	2	4	no	—
149	C₂×D₅₃	—	2	4	no	—
150	C₂×D₅₃	C₂×C₄×C₈	1	2	yes	—
151	C₂×D₅₃	C₂×C₄×C₈	1	1	yes	characteristic
152	C₂×D₅₃	#504	1	4	yes	—
153	C₂×D₅₃	C₂³×C₈	1	1	yes	characteristic
154	#6157	—	2	8	no	—
155	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
156	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
157	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
158	Dic₅₃	C₂×C₄×C₈	1	1	yes	characteristic
159	Dic₅₃	C₂×C₄×C₈	1	2	yes	—
160	Dic₅₃	C₂³×C₈	1	1	yes	characteristic
161	C₂²×M₃₂	—	53	1	no	—
162	C₂³×C₁₆	—	53	1	no	—
163	C₂×#405	—	53	1	no	—
164	#2108	—	53	8	no	—
165	C₂×#346	—	53	2	no	—
166	C₂×C₄×C₁₆	—	53	2	no	—
167	C₂×C₅₃	—	2	2	no	—
168	C₂×C₅₃	#2108	1	2	yes	complemented
169	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
170	D₅₃	—	2	2	no	—
171	D₅₃	#2108	1	2	yes	complemented
172	D₅₃	C₂×#504	1	2	yes	—
173	C₂²×M₁₆	—	53	1	no	—
174	C₂³×C₈	—	53	1	no	—
175	C₂×#117	—	53	1	no	—
176	C₂×M₃₂	—	53	1	no	—
177	C₂×M₃₂	—	53	4	no	—
178	C₂×M₃₂	—	53	4	no	—
179	C₂×M₃₂	—	53	2	no	—
180	C₂×M₃₂	#6157	1	1	yes	characteristic, complemented
181	C₂²×C₁₆	—	53	1	no	—
182	C₂²×C₁₆	—	53	1	no	—
183	C₂²×C₁₆	—	53	4	no	—
184	C₂²×C₁₆	—	53	2	no	—
185	C₂²×C₁₆	—	53	2	no	—
186	C₂²×C₁₆	—	53	4	no	—
187	C₂²×C₁₆	—	53	2	no	—
188	#405	—	53	4	no	—
189	#405	—	53	4	no	—
190	C₂×#97	—	53	2	no	—
191	C₂×C₄×C₈	—	53	2	no	—
192	#346	—	53	8	no	—
193	C₄×C₁₆	—	53	8	no	—
194	C₅₃	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₄	—	53	1	no	—
196	C₂×M₁₆	—	53	4	no	—
197	C₂×M₁₆	—	53	2	no	—
198	C₂×M₁₆	—	53	4	no	—
199	C₂×M₁₆	—	53	2	no	—
200	C₂²×C₈	—	53	2	no	—
201	C₂²×C₈	—	53	1	no	—
202	C₂²×C₈	—	53	1	no	—
203	C₂²×C₈	—	53	2	no	—
204	C₂²×C₈	—	53	2	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	C₂×#6157	1	1	yes	characteristic
211	#117	—	53	4	no	—
212	#117	—	53	4	no	—
213	C₂×C₄⋊C₄	—	53	2	no	—
214	C₂×C₂²⋊C₄	—	53	2	no	—
215	C₂×C₄²	—	53	2	no	—
216	M₃₂	—	53	4	no	—
217	M₃₂	—	53	8	no	—
218	M₃₂	C₂×#6157	1	4	yes	complemented
219	C₂×C₁₆	—	53	2	no	—
220	C₂×C₁₆	—	53	2	no	—
221	C₂×C₁₆	—	53	2	no	—
222	C₂×C₁₆	—	53	2	no	—
223	C₂×C₁₆	—	53	8	no	—
224	C₂×C₁₆	—	53	2	no	—
225	C₂×C₁₆	—	53	2	no	—
226	C₂×C₁₆	—	53	8	no	—
227	C₂×C₁₆	—	53	2	no	—
228	C₂×C₁₆	—	53	4	no	—
229	C₂×C₁₆	—	53	4	no	—
230	C₂×C₁₆	C₂×#6157	1	2	yes	—
231	#97	—	53	8	no	—
232	C₄×C₈	—	53	4	no	—
233	C₄×C₈	—	53	4	no	—
234	C₂⁴	—	53	1	no	—
235	C₂²×C₄	—	53	1	no	—
236	C₂²×C₄	—	53	1	no	—
237	C₂²×C₄	—	53	2	no	—
238	C₂²×C₄	—	53	1	no	—
239	C₂²×C₄	—	53	2	no	—
240	C₂²×C₄	—	53	2	no	—
241	C₂²×C₄	—	53	1	no	—
242	C₂²×C₄	—	53	1	no	—
243	C₂²×C₄	—	53	2	no	—
244	C₂²×C₄	—	53	4	no	—
245	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
246	M₁₆	—	53	8	no	—
247	M₁₆	—	53	8	no	—
248	C₂×C₈	—	53	4	no	—
249	C₂×C₈	—	53	4	no	—
250	C₂×C₈	—	53	4	no	—
251	C₂×C₈	—	53	2	no	—
252	C₂×C₈	—	53	1	no	—
253	C₂×C₈	—	53	2	no	—
254	C₂×C₈	—	53	1	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	2	no	—
261	C₂×C₈	—	53	2	no	—
262	C₂×C₈	—	53	1	no	—
263	C₂×C₈	—	53	1	no	—
264	C₂×C₈	—	53	2	no	—
265	C₂×C₈	C₄×#6157	1	1	yes	characteristic
266	C₂×C₈	C₄×#6157	1	2	yes	—
267	C₂×C₈	C₂²×#6157	1	2	yes	—
268	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
269	C₄⋊C₄	—	53	8	no	—
270	C₂²⋊C₄	—	53	8	no	—
271	C₄²	—	53	4	no	—
272	C₄²	—	53	4	no	—
273	C₁₆	—	53	4	no	—
274	C₁₆	—	53	4	no	—
275	C₁₆	—	53	4	no	—
276	C₁₆	C₂²×#6157	1	4	yes	—
277	C₂³	—	53	1	no	—
278	C₂³	—	53	2	no	—
279	C₂³	—	53	1	no	—
280	C₂³	—	53	2	no	—
281	C₂³	—	106	4	no	—
282	C₂³	C₈×#6157	1	1	yes	characteristic
283	C₂×C₄	—	53	2	no	—
284	C₂×C₄	—	53	1	no	—
285	C₂×C₄	—	53	1	no	—
286	C₂×C₄	—	53	2	no	—
287	C₂×C₄	—	53	2	no	—
288	C₂×C₄	—	53	2	no	—
289	C₂×C₄	—	53	2	no	—
290	C₂×C₄	—	53	1	no	—
291	C₂×C₄	—	53	2	no	—
292	C₂×C₄	—	53	1	no	—
293	C₂×C₄	—	53	2	no	—
294	C₂×C₄	—	53	1	no	—
295	C₂×C₄	—	53	2	no	—
296	C₂×C₄	—	53	1	no	—
297	C₂×C₄	—	53	8	no	—
298	C₂×C₄	—	106	8	no	—
299	C₂×C₄	C₈×#6157	1	2	yes	—
300	C₂×C₄	C₈×#6157	1	1	yes	characteristic
301	C₂×C₄	C₂×C₄×#6157	1	2	yes	—
302	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
303	C₈	—	53	8	no	—
304	C₈	—	53	1	no	—
305	C₈	—	53	2	no	—
306	C₈	—	53	1	no	—
307	C₈	C₂×C₄×#6157	1	2	yes	—
308	C₈	C₂×C₄×#6157	1	1	yes	characteristic
309	C₈	C₂³×#6157	1	1	yes	characteristic
310	C₂²	—	2	2	no	—
311	C₂²	—	53	4	no	—
312	C₂²	—	53	1	no	—
313	C₂²	—	53	1	no	—
314	C₂²	—	53	2	no	—
315	C₂²	—	106	4	no	—
316	C₂²	—	106	4	no	—
317	C₂²	—	106	2	no	—
318	C₂²	C₂×C₈×#6157	1	2	yes	—
319	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
320	C₄	—	53	1	no	—
321	C₄	—	53	1	no	—
322	C₄	—	53	2	no	—
323	C₄	—	106	8	no	—
324	C₄	C₂×C₈×#6157	1	1	yes	characteristic
325	C₄	C₂×C₈×#6157	1	2	yes	—
326	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	—	53	2	no	—
329	C₂	—	53	2	no	—
330	C₂	—	106	2	no	—
331	C₂	#20290927	1	2	yes	direct product factor
332	C₂	C₂²×C₈×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×#6157
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₈	C₂³×#6157
Fitting	C₂×M₃₂×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₈×D₅₃	C₂⁴
hypercenter	C₂×M₃₂	#6157
hypocenter	C₅₃	C₂×#2108
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20290927	C₂	direct (N × H)
4. C₂×#16811510	C₂	semidirect
5. C₂×C₁₆×#6157	C₂	semidirect
9. C₂×D₅₃⋊C₁₆	C₄	semidirect
10. C₂×D₅₃⋊C₁₆	C₄	semidirect
14. C₂²×#11719910	C₄	semidirect
19. #16811510	C₂²	semidirect
20. C₁₆×#6157	C₂²	semidirect
22. C₂×#13133157	C₄	semidirect
23. C₂×#13133156	C₄	semidirect
24. M₃₂×D₅₃	C₄	semidirect
27. C₂×M₃₂×C₅₃	C₄	semidirect
40. D₅₃⋊C₁₆	C₂×C₄	semidirect
45. C₂×#11719910	C₂×C₄	semidirect
59. #13133157	C₂×C₄	semidirect
60. #13133156	C₂×C₄	semidirect
65. M₃₂×C₅₃	C₂×C₄	semidirect
168. C₂×C₅₃	#2108	semidirect
171. D₅₃	#2108	semidirect
180. C₂×M₃₂	#6157	semidirect
194. C₅₃	C₂×#2108	semidirect
218. M₃₂	C₂×#6157	semidirect
331. C₂	#20290927	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×D₅₃⋊C₁₆	2
C₂×#16811586	2
#20290927	2
C₂×#16811510	2
C₂×C₁₆×#6157	2
C₂×M₃₂×D₅₃	2
C₂×#2108	53

Subgroups — C2×#20290927 #28327900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20290927 #28327900