Subgroups — C₂×#20291683 #28330793

Order 13568.

2074 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291683	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16811682	C₂	1	6	yes	maximal, complemented
3	C₂×#16811681	C₂	1	2	yes	maximal, complemented
4	C₂²×#13133292	C₂	1	3	yes	maximal, complemented
5	C₂×D₅₃⋊M₃₂	C₂	1	3	yes	maximal, complemented
6	C₂×#131×D₅₃	C₂	1	1	yes	maximal, characteristic
7	#16811682	C₂²	1	48	yes	complemented
8	#16811681	C₂²	1	16	yes	complemented
9	C₂×#13133292	C₂²	1	3	yes	—
10	C₂×#13133292	C₂²	1	1	yes	characteristic
11	C₂×#13133292	C₂²	1	6	yes	complemented
12	C₂×#13133292	C₂²	1	24	yes	complemented
13	D₅₃⋊M₃₂	C₂²	1	24	yes	complemented
14	C₂×#13133291	C₂²	1	6	yes	complemented
15	C₂×D₅₃⋊C₁₆	C₂²	1	6	yes	complemented
16	C₂²×#11719910	C₂²	1	6	yes	complemented
17	C₂×#41×D₅₃	C₄	1	1	yes	characteristic
18	C₂×#13133225	C₂²	1	3	yes	—
19	C₂×#13133224	C₂²	1	1	yes	characteristic
20	C₂×#13133223	C₄	1	3	yes	—
21	#131×D₅₃	C₄	1	8	yes	—
22	#131×D₅₃	C₂²	1	8	yes	—
23	C₂×M₁₆×D₅₃	C₄	1	3	yes	—
24	C₂²×C₈×D₅₃	C₂²	1	3	yes	—
25	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
26	#13133292	C₂³	1	12	yes	—
27	#13133292	C₂³	1	4	yes	—
28	#13133292	C₂³	1	24	yes	complemented
29	#13133291	C₂³	1	24	yes	complemented
30	D₅₃⋊C₁₆	C₂³	1	24	yes	complemented
31	C₂×#11719910	C₂³	1	2	yes	—
32	C₂×#11719910	C₂³	1	6	yes	—
33	C₂×#11719910	C₂³	1	24	yes	complemented
34	C₂×#11719949	C₈	1	1	yes	characteristic
35	C₂×#11719945	C₂×C₄	1	3	yes	—
36	C₂×#11719947	C₈	1	3	yes	—
37	#41×D₅₃	C₈	1	8	yes	—
38	#41×D₅₃	C₂×C₄	1	8	yes	—
39	C₂×Q₈×D₅₃	C₈	1	1	yes	characteristic
40	C₂×D₄×D₅₃	C₈	1	3	yes	—
41	C₂²×C₄×D₅₃	C₂×C₄	1	3	yes	—
42	#13133225	C₂×C₄	1	12	yes	—
43	#13133225	C₂³	1	12	yes	—
44	#13133224	C₂×C₄	1	4	yes	—
45	#13133224	C₂³	1	4	yes	—
46	#13133223	C₂×C₄	1	24	yes	—
47	C₂×#11719917	C₂×C₄	1	3	yes	—
48	C₂×#11719912	C₂×C₄	1	6	yes	—
49	M₁₆×D₅₃	C₂×C₄	1	24	yes	—
50	C₂²×#88322	C₂³	1	3	yes	—
51	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
52	C₂×C₈×D₅₃	C₂×C₄	1	12	yes	—
53	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
54	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×D₅₃	C₂³	1	12	yes	—
56	C₂×C₈×D₅₃	C₂³	1	3	yes	—
57	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
58	#131×C₅₃	C₂×C₄	1	8	yes	—
59	C₂×M₁₆×C₅₃	C₂×C₄	1	3	yes	—
60	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
61	C₂³×D₅₃	C₂×C₈	1	3	yes	—
62	#41×C₅₃	C₂×C₈	1	4	yes	—
63	#41×C₅₃	C₂²×C₄	1	4	yes	—
64	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
65	C₂×D₄×C₅₃	C₂×C₈	1	3	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
67	C₂×C₂²⋊D₅₃	C₂×C₈	1	6	yes	—
68	C₂²×Dic₅₃	C₂×C₈	1	3	yes	—
69	#11719949	C₂×C₈	1	8	yes	—
70	Q₈×D₅₃	C₂×C₈	1	8	yes	—
71	#11719947	C₂×C₈	1	24	yes	—
72	D₄×D₅₃	C₂×C₈	1	24	yes	—
73	#11719945	C₂×C₈	1	12	yes	—
74	#11719945	C₂²×C₄	1	12	yes	—
75	C₂×D₂₁₂	C₂×C₈	1	3	yes	—
76	C₂×C₄×D₅₃	C₂×C₈	1	3	yes	—
77	C₂×C₄×D₅₃	C₂×C₈	1	12	yes	—
78	C₂×C₄×D₅₃	C₂×C₈	1	3	yes	—
79	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×D₅₃	C₂²×C₄	1	12	yes	—
81	C₂×C₄×D₅₃	C₂²×C₄	1	3	yes	—
82	C₂×Dic₁₀₆	C₂×C₈	1	3	yes	—
83	M₁₆×C₅₃	C₂²×C₄	1	12	yes	—
84	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
86	C₂×C₈×C₅₃	C₂²×C₄	1	12	yes	—
87	#11719917	C₂²×C₄	1	12	yes	—
88	C₂×#88322	C₂²×C₄	1	3	yes	—
89	C₂×#88322	C₂²×C₄	1	6	yes	—
90	C₂×#88322	C₂⁴	1	1	yes	characteristic
91	C₂×#88322	C₂⁴	1	6	yes	—
92	#11719912	C₂²×C₄	1	24	yes	—
93	C₈×D₅₃	C₂²×C₄	1	2	yes	—
94	C₈×D₅₃	C₂²×C₄	1	12	yes	—
95	C₈×D₅₃	C₂²×C₄	1	12	yes	—
96	C₈×D₅₃	C₂²×C₄	1	6	yes	—
97	C₈×D₅₃	C₂⁴	1	2	yes	—
98	C₈×D₅₃	C₂⁴	1	6	yes	—
99	#11719910	C₂⁴	1	4	yes	—
100	#11719910	C₂⁴	1	12	yes	—
101	C₂³×C₅₃	C₂²×C₈	1	3	yes	—
102	C₂²×D₅₃	—	4	6	no	—
103	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
104	C₂²×D₅₃	C₂²×C₈	1	3	yes	—
105	C₂²×D₅₃	C₂²×C₈	1	12	yes	—
106	Q₈×C₅₃	C₂²×C₈	1	4	yes	—
107	D₄×C₅₃	C₂²×C₈	1	12	yes	—
108	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
109	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
110	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
111	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
112	C₂²⋊D₅₃	C₂²×C₈	1	24	yes	—
113	C₂×Dic₅₃	C₂²×C₈	1	3	yes	—
114	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×Dic₅₃	C₂²×C₈	1	12	yes	—
116	D₂₁₂	C₂²×C₈	1	12	yes	—
117	C₄×D₅₃	C₂²×C₈	1	2	yes	—
118	C₄×D₅₃	C₂²×C₈	1	6	yes	—
119	C₄×D₅₃	C₂²×C₈	1	12	yes	—
120	C₄×D₅₃	C₂²×C₈	1	12	yes	—
121	C₄×D₅₃	C₂³×C₄	1	2	yes	—
122	C₄×D₅₃	C₂³×C₄	1	6	yes	—
123	Dic₁₀₆	C₂²×C₈	1	12	yes	—
124	C₈×C₅₃	C₂³×C₄	1	2	yes	—
125	C₈×C₅₃	C₂³×C₄	1	6	yes	—
126	#88322	C₂³×C₄	1	1	yes	characteristic
127	#88322	C₂³×C₄	1	6	yes	—
128	#88322	C₂⁵	1	1	yes	characteristic
129	C₂×#3407	—	53	1	no	maximal, Sylow
130	C₂²×C₅₃	—	2	6	no	—
131	C₂²×C₅₃	C₂³×C₈	1	6	yes	—
132	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
133	C₂×D₅₃	—	2	2	no	—
134	C₂×D₅₃	—	2	6	no	—
135	C₂×D₅₃	—	4	12	no	—
136	C₂×D₅₃	C₂³×C₈	1	2	yes	—
137	C₂×D₅₃	C₂³×C₈	1	6	yes	—
138	C₄×C₅₃	C₂³×C₈	1	6	yes	—
139	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
140	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
141	Dic₅₃	C₂³×C₈	1	6	yes	—
142	Dic₅₃	C₂³×C₈	1	2	yes	—
143	C₂²×#131	—	53	1	no	—
144	#3407	—	53	16	no	—
145	C₂×#504	—	53	2	no	—
146	C₂×#504	—	53	6	no	—
147	C₂²×M₃₂	—	53	3	no	—
148	C₂²×M₃₂	—	53	3	no	—
149	C₂×C₅₃	—	2	6	no	—
150	C₂×C₅₃	#3407	1	2	yes	complemented
151	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
152	D₅₃	—	2	2	no	—
153	D₅₃	—	2	6	no	—
154	C₂²×#41	—	53	1	no	—
155	C₂×#131	—	53	1	no	—
156	C₂×#131	—	53	8	no	—
157	C₂×#131	—	53	8	no	—
158	C₂×#131	—	53	3	no	—
159	C₂×#131	—	53	3	no	—
160	C₂×#131	#6157	1	1	yes	characteristic
161	C₂²×M₁₆	—	53	3	no	—
162	C₂³×C₈	—	53	3	no	—
163	#504	—	53	16	no	—
164	#504	—	53	48	no	—
165	C₂×M₃₂	—	53	3	no	—
166	C₂×M₃₂	—	53	1	no	—
167	C₂×M₃₂	—	53	6	no	—
168	C₂×M₃₂	—	53	6	no	—
169	C₂×M₃₂	—	53	24	no	—
170	C₂×M₃₂	—	53	24	no	—
171	C₂²×C₁₆	—	53	6	no	—
172	C₂²×C₁₆	—	53	6	no	—
173	C₅₃	C₂×#3407	1	1	yes	characteristic, complemented, Sylow
174	C₂×#41	—	53	8	no	—
175	C₂×#41	—	53	8	no	—
176	C₂×#41	—	53	1	no	—
177	C₂×#41	—	53	3	no	—
178	C₂×#41	—	53	3	no	—
179	C₂×#41	C₂×#6157	1	1	yes	characteristic
180	C₂²×Q₈	—	53	1	no	—
181	C₂²×D₄	—	53	3	no	—
182	C₂³×C₄	—	53	3	no	—
183	#131	—	53	4	no	—
184	#131	—	53	4	no	—
185	#131	—	53	12	no	—
186	#131	—	53	12	no	—
187	#131	—	53	24	no	—
188	#131	C₂×#6157	1	8	yes	—
189	C₂×M₁₆	—	53	3	no	—
190	C₂×M₁₆	—	53	24	no	—
191	C₂×M₁₆	—	53	6	no	—
192	C₂×M₁₆	C₂×#6157	1	3	yes	—
193	C₂²×C₈	—	53	3	no	—
194	C₂²×C₈	—	53	1	no	—
195	C₂²×C₈	—	53	3	no	—
196	C₂²×C₈	—	53	12	no	—
197	C₂²×C₈	—	53	3	no	—
198	C₂²×C₈	—	53	3	no	—
199	C₂²×C₈	—	53	12	no	—
200	C₂²×C₈	C₂×#6157	1	3	yes	—
201	M₃₂	—	53	24	no	—
202	M₃₂	—	53	12	no	—
203	M₃₂	—	53	4	no	—
204	M₃₂	—	53	24	no	—
205	C₂×C₁₆	—	53	2	no	—
206	C₂×C₁₆	—	53	6	no	—
207	C₂×C₁₆	—	53	24	no	—
208	C₂×C₁₆	—	53	24	no	—
209	C₂⁴	—	53	3	no	—
210	#41	—	53	8	no	—
211	#41	—	53	24	no	—
212	#41	—	53	12	no	—
213	#41	—	53	12	no	—
214	#41	D₅₃⋊C₈	1	4	yes	—
215	#41	C₂²×#6157	1	4	yes	—
216	C₂×Q₈	—	53	8	no	—
217	C₂×Q₈	—	53	3	no	—
218	C₂×Q₈	D₅₃⋊C₈	1	1	yes	characteristic
219	C₂×D₄	—	53	6	no	—
220	C₂×D₄	—	53	24	no	—
221	C₂×D₄	—	53	3	no	—
222	C₂×D₄	D₅₃⋊C₈	1	3	yes	—
223	C₂²×C₄	—	53	3	no	—
224	C₂²×C₄	—	53	3	no	—
225	C₂²×C₄	—	53	3	no	—
226	C₂²×C₄	—	53	12	no	—
227	C₂²×C₄	—	53	12	no	—
228	C₂²×C₄	—	53	1	no	—
229	C₂²×C₄	—	53	3	no	—
230	C₂²×C₄	C₂²×#6157	1	3	yes	—
231	M₁₆	—	53	12	no	—
232	M₁₆	—	53	24	no	—
233	M₁₆	C₂²×#6157	1	12	yes	—
234	C₂×C₈	—	53	6	no	—
235	C₂×C₈	—	53	6	no	—
236	C₂×C₈	—	53	1	no	—
237	C₂×C₈	—	53	3	no	—
238	C₂×C₈	—	53	6	no	—
239	C₂×C₈	—	53	2	no	—
240	C₂×C₈	—	53	2	no	—
241	C₂×C₈	—	53	12	no	—
242	C₂×C₈	—	53	6	no	—
243	C₂×C₈	—	53	12	no	—
244	C₂×C₈	C₂²×#6157	1	12	yes	—
245	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
246	C₂×C₈	C₂²×#6157	1	3	yes	—
247	C₁₆	—	53	4	no	—
248	C₁₆	—	53	12	no	—
249	C₂³	—	53	1	no	—
250	C₂³	—	53	3	no	—
251	C₂³	—	53	12	no	—
252	C₂³	—	212	6	no	—
253	C₂³	C₂×D₅₃⋊C₈	1	3	yes	—
254	Q₈	—	53	12	no	—
255	Q₈	C₂×D₅₃⋊C₈	1	4	yes	—
256	D₄	—	53	24	no	—
257	D₄	—	53	12	no	—
258	D₄	C₂×D₅₃⋊C₈	1	12	yes	—
259	C₂×C₄	—	53	1	no	—
260	C₂×C₄	—	53	3	no	—
261	C₂×C₄	—	53	12	no	—
262	C₂×C₄	—	53	2	no	—
263	C₂×C₄	—	53	12	no	—
264	C₂×C₄	—	53	12	no	—
265	C₂×C₄	—	53	6	no	—
266	C₂×C₄	—	53	2	no	—
267	C₂×C₄	—	53	6	no	—
268	C₂×C₄	C₂×D₅₃⋊C₈	1	3	yes	—
269	C₂×C₄	C₂×D₅₃⋊C₈	1	6	yes	—
270	C₂×C₄	C₂³×#6157	1	6	yes	—
271	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
272	C₈	—	53	1	no	—
273	C₈	—	53	6	no	—
274	C₈	—	53	1	no	—
275	C₈	C₂³×#6157	1	2	yes	—
276	C₈	C₂³×#6157	1	6	yes	—
277	C₂²	—	2	6	no	—
278	C₂²	—	53	2	no	—
279	C₂²	—	53	6	no	—
280	C₂²	—	106	2	no	—
281	C₂²	—	106	6	no	—
282	C₂²	—	212	12	no	—
283	C₂²	C₂²×D₅₃⋊C₈	1	6	yes	—
284	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
285	C₄	—	53	2	no	—
286	C₄	—	53	6	no	—
287	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
288	C₄	C₂²×D₅₃⋊C₈	1	6	yes	—
289	C₄	C₂⁴×#6157	1	1	yes	characteristic
290	C₂	—	2	6	no	—
291	C₂	—	106	2	no	—
292	C₂	—	106	6	no	—
293	C₂	#20291683	1	2	yes	direct product factor
294	C₂	C₂³×D₅₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×#6157
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₄	C₂⁴×#6157
Fitting	C₂×#131×C₅₃	C₄
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₂×#41×D₅₃	C₄
cosocle	#88322	C₂⁵
hypercenter	C₂×#131	#6157
hypocenter	C₅₃	C₂×#3407
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. #20291683	C₂	direct (N × H)
2. C₂×#16811682	C₂	semidirect
3. C₂×#16811681	C₂	semidirect
4. C₂²×#13133292	C₂	semidirect
5. C₂×D₅₃⋊M₃₂	C₂	semidirect
7. #16811682	C₂²	semidirect
8. #16811681	C₂²	semidirect
11. C₂×#13133292	C₂²	semidirect
12. C₂×#13133292	C₂²	semidirect
13. D₅₃⋊M₃₂	C₂²	semidirect
14. C₂×#13133291	C₂²	semidirect
15. C₂×D₅₃⋊C₁₆	C₂²	semidirect
16. C₂²×#11719910	C₂²	semidirect
28. #13133292	C₂³	semidirect
29. #13133291	C₂³	semidirect
30. D₅₃⋊C₁₆	C₂³	semidirect
33. C₂×#11719910	C₂³	semidirect
150. C₂×C₅₃	#3407	semidirect
173. C₅₃	C₂×#3407	semidirect
293. C₂	#20291683	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20291683	2
C₂×#16811682	2
C₂×#16811681	2
C₂²×#13133292	2
C₂×D₅₃⋊M₃₂	2
C₂×#131×D₅₃	2
C₂×#3407	53

Subgroups — C2×#20291683 #28330793

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20291683 #28330793