Subgroups — C₂×#20289773 #28310269

Order 13568.

694 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133286	C₂	1	2	yes	maximal
2	#20289773	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊#6157	—	2	2	no	—
5	C₂×#13133286	—	2	4	no	—
6	C₂×#13133286	—	2	2	no	—
7	C₂×#13133286	C₄	1	8	yes	complemented
8	C₂×#13133286	C₂²	1	4	yes	—
9	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₂²×#11719908	C₄	1	1	yes	characteristic, complemented
11	C₂×C₁₆×D₅₃	—	2	4	no	—
12	C₂×C₁₆×D₅₃	C₄	1	2	yes	complemented
13	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
14	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
15	C₂³×#6157	—	4	2	no	—
16	C₂×C₄⋊#6157	—	2	8	no	—
17	C₂×C₄⋊#6157	—	2	4	no	—
18	C₂×C₄⋊#6157	—	4	2	no	—
19	C₂×C₄⋊#6157	—	4	4	no	—
20	#13133286	—	2	8	no	—
21	#13133286	—	2	4	no	—
22	#13133286	C₂×C₄	1	8	yes	complemented
23	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
24	C₂²×#88322	Q₈	1	1	yes	characteristic
25	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
26	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₅₃	D₄	1	4	yes	—
28	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
29	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
30	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
31	C₂×#11719908	—	2	2	no	—
32	C₂×#11719908	C₂×C₄	1	2	yes	complemented
33	C₁₆×D₅₃	—	2	4	no	—
34	C₁₆×D₅₃	—	2	4	no	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆×C₅₃	—	2	2	no	—
37	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
38	C₂³×D₅₃	D₈	1	1	yes	characteristic
39	C₂²×#6157	—	4	8	no	—
40	C₂²×#6157	—	4	4	no	—
41	C₂²×#6157	—	8	4	no	—
42	C₂²×#6157	—	8	4	no	—
43	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	Q₁₆	1	1	yes	characteristic
45	C₂×C₄×D₅₃	C₂²⋊C₄	1	4	yes	—
46	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
48	C₂×C₄×D₅₃	SD₁₆	1	4	yes	—
49	C₂×C₄×D₅₃	Q₁₆	1	1	yes	characteristic
50	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
51	C₄⋊#6157	—	2	8	no	—
52	C₄⋊#6157	—	4	8	no	—
53	C₄⋊#6157	—	4	4	no	—
54	C₂×C₈×C₅₃	C₄²	1	2	yes	—
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
59	C₂×#88322	C₄⋊C₄	1	4	yes	—
60	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
61	C₈×D₅₃	C₄²	1	2	yes	—
62	C₈×D₅₃	C₂²⋊C₄	1	4	yes	—
63	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
64	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
65	C₈×D₅₃	C₂²×C₄	1	2	yes	—
66	C₈×D₅₃	C₂×D₄	1	2	yes	—
67	C₁₆×C₅₃	—	2	2	no	—
68	#11719908	—	2	2	no	—
69	C₂³×C₅₃	#107	1	1	yes	characteristic
70	C₂²×D₅₃	#102	1	4	yes	—
71	C₂²×D₅₃	#107	1	1	yes	characteristic
72	C₂²×D₅₃	D₁₆	1	2	yes	—
73	C₂²×D₅₃	SD₃₂	1	4	yes	—
74	C₂²×D₅₃	Q₃₂	1	2	yes	—
75	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
76	C₂×#6157	—	4	8	no	—
77	C₂×#6157	—	8	16	no	—
78	C₂×#6157	—	8	4	no	—
79	C₂×#6157	—	8	4	no	—
80	C₂×C₄×C₅₃	#95	1	2	yes	—
81	C₂×C₄×C₅₃	#106	1	2	yes	—
82	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×Dic₅₃	#103	1	4	yes	—
85	C₂×Dic₅₃	#107	1	1	yes	characteristic
86	C₂×Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
87	C₄×D₅₃	#95	1	2	yes	—
88	C₄×D₅₃	#102	1	4	yes	—
89	C₄×D₅₃	#103	1	4	yes	—
90	C₄×D₅₃	#106	1	2	yes	—
91	C₄×D₅₃	C₂×C₂²⋊C₄	1	2	yes	—
92	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
93	C₈×C₅₃	#95	1	2	yes	—
94	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
95	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	#88322	#95	1	2	yes	—
97	#88322	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×#1380	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	#340	1	2	yes	—
100	C₂²×C₅₃	#366	1	2	yes	—
101	C₂²×C₅₃	#367	1	2	yes	—
102	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
103	C₂×D₅₃	#340	1	2	yes	—
104	C₂×D₅₃	#357	1	8	yes	—
105	C₂×D₅₃	#358	1	8	yes	—
106	C₂×D₅₃	#366	1	2	yes	—
107	C₂×D₅₃	#367	1	2	yes	—
108	C₂×D₅₃	C₂×#102	1	2	yes	—
109	C₂×D₅₃	C₂×D₁₆	1	1	yes	characteristic
110	C₂×D₅₃	C₂×SD₃₂	1	2	yes	—
111	C₂×D₅₃	C₂×Q₃₂	1	1	yes	characteristic
112	#6157	—	8	8	no	—
113	C₄×C₅₃	#340	1	2	yes	—
114	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
115	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
116	Dic₅₃	#340	1	2	yes	—
117	Dic₅₃	C₂×#103	1	2	yes	—
118	C₂³×C₁₆	—	53	1	no	—
119	C₂²×#107	—	53	2	no	—
120	#1380	—	53	4	no	—
121	C₂×C₅₃	#1380	1	4	yes	complemented
122	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
123	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
124	C₂×C₅₃	C₂×#367	1	1	yes	characteristic
125	D₅₃	#1380	1	4	yes	complemented
126	D₅₃	C₂×#357	1	2	yes	—
127	D₅₃	C₂×#358	1	2	yes	—
128	C₂³×C₈	—	53	1	no	—
129	C₂²×C₄⋊C₄	—	106	2	no	—
130	C₂²×C₁₆	—	53	1	no	—
131	C₂²×C₁₆	—	53	2	no	—
132	C₂²×C₁₆	—	53	2	no	—
133	C₂²×C₁₆	—	106	4	no	—
134	C₂²×C₁₆	#6157	1	1	yes	characteristic, complemented
135	C₂×#107	—	53	4	no	—
136	C₂×#107	—	53	8	no	—
137	C₂×#107	—	106	2	no	—
138	C₂×#107	—	106	4	no	—
139	C₅₃	C₂×#1380	1	1	yes	characteristic, complemented, Sylow
140	C₂³×C₄	—	53	1	no	—
141	C₂³×C₄	—	212	2	no	—
142	C₂²×C₈	—	53	1	no	—
143	C₂²×C₈	—	53	4	no	—
144	C₂²×C₈	—	53	1	no	—
145	C₂²×C₈	—	53	1	no	—
146	C₂²×C₈	—	53	4	no	—
147	C₂²×C₈	—	53	1	no	—
148	C₂²×C₈	—	53	1	no	—
149	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
150	C₂×C₄⋊C₄	—	106	8	no	—
151	C₂×C₄⋊C₄	—	106	4	no	—
152	C₂×C₄⋊C₄	—	212	4	no	—
153	C₂×C₄⋊C₄	—	212	2	no	—
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	53	2	no	—
156	C₂×C₁₆	—	106	4	no	—
157	C₂×C₁₆	—	106	4	no	—
158	C₂×C₁₆	—	106	2	no	—
159	C₂×C₁₆	C₂×#6157	1	2	yes	complemented
160	#107	—	53	8	no	—
161	#107	—	106	8	no	—
162	#107	—	106	4	no	—
163	C₂⁴	—	53	1	no	—
164	C₂²×C₄	—	53	1	no	—
165	C₂²×C₄	—	53	4	no	—
166	C₂²×C₄	—	53	4	no	—
167	C₂²×C₄	—	53	1	no	—
168	C₂²×C₄	—	53	1	no	—
169	C₂²×C₄	—	53	1	no	—
170	C₂²×C₄	—	53	1	no	—
171	C₂²×C₄	—	424	4	no	—
172	C₂²×C₄	—	424	4	no	—
173	C₂²×C₄	—	212	4	no	—
174	C₂²×C₄	—	212	8	no	—
175	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
176	C₂×C₈	—	53	4	no	—
177	C₂×C₈	—	53	1	no	—
178	C₂×C₈	—	53	1	no	—
179	C₂×C₈	—	53	4	no	—
180	C₂×C₈	—	53	4	no	—
181	C₂×C₈	—	53	2	no	—
182	C₂×C₈	—	53	2	no	—
183	C₂×C₈	—	53	2	no	—
184	C₂×C₈	—	53	2	no	—
185	C₂×C₈	C₄×#6157	1	2	yes	—
186	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
187	C₂×C₈	C₂²⋊#6157	1	2	yes	—
188	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
189	C₄⋊C₄	—	106	8	no	—
190	C₄⋊C₄	—	212	4	no	—
191	C₄⋊C₄	—	212	8	no	—
192	C₁₆	—	2	2	no	—
193	C₁₆	—	106	2	no	—
194	C₂³	—	53	2	no	—
195	C₂³	—	53	2	no	—
196	C₂³	—	53	4	no	—
197	C₂³	—	53	1	no	—
198	C₂³	—	53	1	no	—
199	C₂³	—	53	4	no	—
200	C₂³	#13133286	1	1	yes	characteristic
201	C₂×C₄	—	53	1	no	—
202	C₂×C₄	—	53	1	no	—
203	C₂×C₄	—	53	4	no	—
204	C₂×C₄	—	53	2	no	—
205	C₂×C₄	—	53	2	no	—
206	C₂×C₄	—	53	2	no	—
207	C₂×C₄	—	53	2	no	—
208	C₂×C₄	—	53	4	no	—
209	C₂×C₄	—	53	4	no	—
210	C₂×C₄	—	424	16	no	—
211	C₂×C₄	—	424	4	no	—
212	C₂×C₄	—	424	4	no	—
213	C₂×C₄	—	212	8	no	—
214	C₂×C₄	(C₂×#6157)⋊C₄	1	2	yes	—
215	C₂×C₄	#13133285	1	2	yes	—
216	C₂×C₄	#13133286	1	1	yes	characteristic
217	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
218	C₈	—	53	2	no	—
219	C₈	—	53	2	no	—
220	C₈	(C₂×#6157)⋊C₄	1	2	yes	—
221	C₈	C₂×C₄×#6157	1	1	yes	characteristic
222	C₈	C₂×C₂²⋊#6157	1	1	yes	characteristic
223	C₂²	—	53	2	no	—
224	C₂²	—	53	2	no	—
225	C₂²	—	53	8	no	—
226	C₂²	—	53	8	no	—
227	C₂²	—	53	1	no	—
228	C₂²	—	53	2	no	—
229	C₂²	—	53	1	no	—
230	C₂²	—	53	2	no	—
231	C₂²	—	53	2	no	—
232	C₂²	#16811497	1	2	yes	—
233	C₂²	#16811549	1	2	yes	—
234	C₂²	#16811550	1	2	yes	—
235	C₂²	C₂×#13133286	1	1	yes	characteristic
236	C₄	—	53	2	no	—
237	C₄	—	53	2	no	—
238	C₄	—	424	8	no	—
239	C₄	#16811497	1	2	yes	—
240	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
241	C₄	C₂×#13133285	1	1	yes	characteristic
242	C₂	—	53	2	no	—
243	C₂	—	53	2	no	—
244	C₂	—	53	4	no	—
245	C₂	#20289773	1	4	yes	direct product factor
246	C₂	C₂×#16811497	1	1	yes	characteristic
247	C₂	C₂×#16811549	1	1	yes	characteristic
248	C₂	C₂×#16811550	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133286
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂³×C₅₃	#107
prime radical	C₂³×D₅₃	D₈
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	C₂×#1380
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20289773	C₂	direct (N × H)
7. C₂×#13133286	C₄	semidirect
10. C₂²×#11719908	C₄	semidirect
12. C₂×C₁₆×D₅₃	C₄	semidirect
14. C₂²×C₁₆×C₅₃	C₄	semidirect
22. #13133286	C₂×C₄	semidirect
32. C₂×#11719908	C₂×C₄	semidirect
37. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
121. C₂×C₅₃	#1380	semidirect
125. D₅₃	#1380	semidirect
134. C₂²×C₁₆	#6157	semidirect
139. C₅₃	C₂×#1380	semidirect
159. C₂×C₁₆	C₂×#6157	semidirect
245. C₂	#20289773	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×#13133286	2
#20289773	2
C₂²×C₁₆×D₅₃	2
C₂×#1380	53

Subgroups — C2×#20289773 #28310269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20289773 #28310269