Subgroups — C₂×#20290925 #28327898

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133292	C₂	1	1	yes	maximal, characteristic
2	C₂²×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#20290925	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16811512	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16811511	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16811509	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×#11719910	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16810580	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133292	C₄	1	8	yes	complemented
10	C₂×#13133292	C₂²	1	4	yes	—
11	C₂×D₅₃⋊C₁₆	C₄	1	8	yes	complemented
12	C₂×D₅₃⋊C₁₆	C₂²	1	4	yes	—
13	C₂²×#11719910	C₂²	1	1	yes	characteristic
14	C₂²×#11719910	C₂²	1	1	yes	characteristic
15	C₂²×#11719910	C₂²	1	1	yes	characteristic
16	C₂²×#11719910	C₂²	1	1	yes	characteristic
17	#16811512	C₂²	1	4	yes	complemented
18	#16811511	C₂²	1	4	yes	complemented
19	#16811509	C₂²	1	4	yes	complemented
20	C₄×#11719910	C₂²	1	4	yes	complemented
21	C₂²×#11719912	C₄	1	1	yes	characteristic
22	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
23	C₂×#13133174	C₄	1	1	yes	characteristic
24	#16810580	C₄	1	4	yes	—
25	#16810580	C₂²	1	4	yes	—
26	C₂×#13133127	C₄	1	1	yes	characteristic
27	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
29	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
30	#13133292	C₂×C₄	1	16	yes	complemented
31	D₅₃⋊C₁₆	C₂×C₄	1	16	yes	complemented
32	C₂×#11719910	C₂×C₄	1	4	yes	—
33	C₂×#11719910	C₂×C₄	1	4	yes	—
34	C₂×#11719910	C₂×C₄	1	4	yes	—
35	C₂×#11719910	C₂×C₄	1	4	yes	—
36	C₂×#11719910	C₂³	1	2	yes	—
37	C₂×#11719910	C₂³	1	2	yes	—
38	C₂×#11719910	C₂³	1	2	yes	—
39	C₂×#11719910	C₂³	1	2	yes	—
40	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×#11719912	C₈	1	8	yes	—
42	C₂×#11719912	C₂×C₄	1	4	yes	—
43	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
44	C₂²×#88322	C₂³	1	1	yes	characteristic
45	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
46	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
47	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₅₃	C₂³	1	2	yes	—
49	#13133174	C₈	1	4	yes	—
50	#13133174	C₂×C₄	1	4	yes	—
51	C₂×#11719919	C₈	1	2	yes	—
52	C₂×#11719921	C₈	1	2	yes	—
53	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
54	#13133127	C₂×C₄	1	4	yes	—
55	C₄×#88322	C₂×C₄	1	2	yes	—
56	C₄×#88322	C₂³	1	2	yes	—
57	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
58	C₈×Dic₅₃	C₂³	1	2	yes	—
59	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
62	#97×C₅₃	C₂×C₄	1	4	yes	—
63	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
67	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
68	C₂×C₄×D₅₃	C₄²	1	4	yes	—
69	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
70	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
71	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₄²	1	4	yes	—
73	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₄²×C₅₃	C₂×C₈	1	2	yes	—
77	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
78	#11719921	C₂×C₈	1	8	yes	—
79	#11719919	C₂×C₈	1	8	yes	—
80	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
81	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
82	C₂×#88322	C₂×C₈	1	4	yes	—
83	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
84	C₂×#88322	C₂²×C₄	1	2	yes	—
85	C₂×#88322	C₂²×C₄	1	2	yes	—
86	C₂×#88322	C₂²×C₄	1	2	yes	—
87	C₂×#88322	C₂⁴	1	1	yes	characteristic
88	#11719912	C₂×C₈	1	16	yes	—
89	C₈×D₅₃	C₄²	1	8	yes	—
90	C₈×D₅₃	C₂²×C₄	1	4	yes	—
91	C₈×D₅₃	C₂²×C₄	1	4	yes	—
92	#11719910	C₂²×C₄	1	4	yes	—
93	#11719910	C₂²×C₄	1	4	yes	—
94	#11719910	C₂²×C₄	1	4	yes	—
95	#11719910	C₂²×C₄	1	4	yes	—
96	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
97	C₂²×D₅₃	—	2	4	no	—
98	C₂²×D₅₃	C₄×C₈	1	4	yes	—
99	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
103	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×Dic₅₃	—	2	4	no	—
107	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
108	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
109	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
110	C₄×D₅₃	C₄×C₈	1	8	yes	—
111	C₄×D₅₃	C₂×C₄²	1	4	yes	—
112	C₄×D₅₃	C₂²×C₈	1	4	yes	—
113	C₈×C₅₃	C₂×C₄²	1	4	yes	—
114	C₈×C₅₃	C₂²×C₈	1	4	yes	—
115	#88322	C₂×C₄²	1	2	yes	—
116	#88322	C₂²×C₈	1	4	yes	—
117	#88322	C₂³×C₄	1	1	yes	characteristic
118	#88322	C₂³×C₄	1	1	yes	characteristic
119	C₂×#2108	—	53	1	no	maximal, Sylow
120	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
121	C₂²×C₅₃	#504	1	4	yes	—
122	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
123	C₂×D₅₃	—	2	8	no	—
124	C₂×D₅₃	—	2	4	no	—
125	C₂×D₅₃	C₂×C₄×C₈	1	4	yes	—
126	C₄×C₅₃	—	2	4	no	—
127	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
128	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
129	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
130	Dic₅₃	—	2	4	no	—
131	Dic₅₃	C₂×C₄×C₈	1	4	yes	—
132	C₂²×M₃₂	—	53	1	no	—
133	C₂³×C₁₆	—	53	1	no	—
134	C₂×#405	—	53	1	no	—
135	#2108	—	53	8	no	—
136	C₂×#346	—	53	1	no	—
137	C₂×#346	—	53	1	no	—
138	C₂×C₄×C₁₆	—	53	1	no	—
139	C₂×C₄×C₁₆	—	53	1	no	—
140	C₂×C₅₃	#2108	1	4	yes	complemented
141	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
142	C₂×C₅₃	C₂×#504	1	2	yes	—
143	D₅₃	—	2	4	no	—
144	C₂²×M₁₆	—	53	1	no	—
145	C₂³×C₈	—	53	1	no	—
146	C₂×#117	—	53	1	no	—
147	C₂×M₃₂	—	53	4	no	—
148	C₂×M₃₂	—	53	8	no	—
149	C₂²×C₁₆	—	53	1	no	—
150	C₂²×C₁₆	—	53	1	no	—
151	C₂²×C₁₆	—	53	1	no	—
152	C₂²×C₁₆	—	53	1	no	—
153	C₂²×C₁₆	—	53	4	no	—
154	C₂²×C₁₆	—	53	8	no	—
155	#405	—	53	4	no	—
156	#405	—	53	4	no	—
157	C₂×#97	—	53	1	no	—
158	C₂×#97	#6157	1	1	yes	characteristic
159	C₂×C₄×C₈	—	53	1	no	—
160	C₂×C₄×C₈	—	53	1	no	—
161	#346	—	53	4	no	—
162	#346	—	53	4	no	—
163	C₄×C₁₆	—	53	4	no	—
164	C₄×C₁₆	—	53	4	no	—
165	C₅₃	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	—	53	1	no	—
167	C₂×M₁₆	—	53	4	no	—
168	C₂×M₁₆	—	53	8	no	—
169	C₂²×C₈	—	53	1	no	—
170	C₂²×C₈	—	53	1	no	—
171	C₂²×C₈	—	53	4	no	—
172	C₂²×C₈	—	53	4	no	—
173	C₂²×C₈	—	53	2	no	—
174	C₂²×C₈	—	53	2	no	—
175	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
176	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
177	#117	—	53	4	no	—
178	#117	—	53	4	no	—
179	C₂×C₄⋊C₄	—	53	2	no	—
180	C₂×C₂²⋊C₄	—	53	2	no	—
181	C₂×C₄²	—	53	1	no	—
182	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
183	M₃₂	—	53	16	no	—
184	C₂×C₁₆	—	53	2	no	—
185	C₂×C₁₆	—	53	2	no	—
186	C₂×C₁₆	—	53	2	no	—
187	C₂×C₁₆	—	53	2	no	—
188	C₂×C₁₆	—	53	4	no	—
189	C₂×C₁₆	—	53	4	no	—
190	C₂×C₁₆	—	53	4	no	—
191	C₂×C₁₆	—	53	4	no	—
192	C₂×C₁₆	—	53	16	no	—
193	#97	—	53	4	no	—
194	#97	C₂×#6157	1	4	yes	—
195	C₄×C₈	—	53	2	no	—
196	C₄×C₈	—	53	2	no	—
197	C₄×C₈	—	53	2	no	—
198	C₄×C₈	—	53	2	no	—
199	C₂⁴	—	53	1	no	—
200	C₂²×C₄	—	53	1	no	—
201	C₂²×C₄	—	53	4	no	—
202	C₂²×C₄	—	53	2	no	—
203	C₂²×C₄	—	53	4	no	—
204	C₂²×C₄	—	53	2	no	—
205	C₂²×C₄	—	53	2	no	—
206	C₂²×C₄	D₅₃⋊C₈	1	2	yes	—
207	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
208	M₁₆	—	53	16	no	—
209	C₂×C₈	—	53	2	no	—
210	C₂×C₈	—	53	2	no	—
211	C₂×C₈	—	53	4	no	—
212	C₂×C₈	—	53	1	no	—
213	C₂×C₈	—	53	1	no	—
214	C₂×C₈	—	53	2	no	—
215	C₂×C₈	—	53	4	no	—
216	C₂×C₈	—	53	4	no	—
217	C₂×C₈	—	53	8	no	—
218	C₂×C₈	D₅₃⋊C₈	1	4	yes	—
219	C₂×C₈	C₄×#6157	1	4	yes	—
220	C₂×C₈	C₂²×#6157	1	2	yes	—
221	C₂×C₈	C₂²×#6157	1	2	yes	—
222	C₄⋊C₄	—	53	8	no	—
223	C₂²⋊C₄	—	53	8	no	—
224	C₄²	—	53	2	no	—
225	C₄²	—	53	2	no	—
226	C₄²	D₅₃⋊C₈	1	2	yes	—
227	C₄²	C₂²×#6157	1	2	yes	—
228	C₁₆	—	53	4	no	—
229	C₁₆	—	53	4	no	—
230	C₁₆	—	53	4	no	—
231	C₁₆	—	53	4	no	—
232	C₂³	—	53	4	no	—
233	C₂³	—	53	2	no	—
234	C₂³	—	106	4	no	—
235	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	53	4	no	—
238	C₂×C₄	—	53	2	no	—
239	C₂×C₄	—	53	8	no	—
240	C₂×C₄	—	53	4	no	—
241	C₂×C₄	—	53	4	no	—
242	C₂×C₄	—	53	4	no	—
243	C₂×C₄	—	106	4	no	—
244	C₂×C₄	C₂×C₄×#6157	1	2	yes	—
245	C₂×C₄	C₂×D₅₃⋊C₈	1	4	yes	—
246	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
247	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
248	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
249	C₈	—	53	2	no	—
250	C₈	—	53	1	no	—
251	C₈	—	53	1	no	—
252	C₈	—	53	4	no	—
253	C₈	C₂×C₄×#6157	1	4	yes	—
254	C₈	C₂×D₅₃⋊C₈	1	4	yes	—
255	C₂²	—	53	4	no	—
256	C₂²	—	106	4	no	—
257	C₂²	—	106	8	no	—
258	C₂²	C₄×D₅₃⋊C₈	1	2	yes	—
259	C₂²	#16811682	1	4	yes	—
260	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
261	C₄	—	2	4	no	—
262	C₄	—	53	4	no	—
263	C₄	—	106	4	no	—
264	C₄	C₄×D₅₃⋊C₈	1	2	yes	—
265	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
266	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
267	C₂	—	106	4	no	—
268	C₂	#20290925	1	4	yes	direct product factor
269	C₂	C₂×C₄×D₅₃⋊C₈	1	1	yes	characteristic
270	C₂	C₂×#16811682	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×#97×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×#97	#6157
hypocenter	C₅₃	C₂×#2108
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
3. #20290925	C₂	direct (N × H)
4. C₂×#16811512	C₂	semidirect
5. C₂×#16811511	C₂	semidirect
6. C₂×#16811509	C₂	semidirect
7. C₂×C₄×#11719910	C₂	semidirect
9. C₂×#13133292	C₄	semidirect
11. C₂×D₅₃⋊C₁₆	C₄	semidirect
17. #16811512	C₂²	semidirect
18. #16811511	C₂²	semidirect
19. #16811509	C₂²	semidirect
20. C₄×#11719910	C₂²	semidirect
30. #13133292	C₂×C₄	semidirect
31. D₅₃⋊C₁₆	C₂×C₄	semidirect
140. C₂×C₅₃	#2108	semidirect
165. C₅₃	C₂×#2108	semidirect
268. C₂	#20290925	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×#13133292	2
C₂²×D₅₃⋊C₁₆	2
#20290925	2
C₂×#16811512	2
C₂×#16811511	2
C₂×#16811509	2
C₂×C₄×#11719910	2
C₂×#16810580	2
C₂×#2108	53

Subgroups — C2×#20290925 #28327898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20290925 #28327898