Subgroups — C₂×#20285229 #28253697

Order 13568.

1942 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811180	C₂	1	2	yes	maximal
2	#20285229	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16811028	C₂	1	2	yes	maximal, complemented
4	C₂×#16811025	C₂	1	4	yes	maximal, complemented
5	C₂×#16811021	C₂	1	2	yes	maximal, complemented
6	C₂×#16810661	C₂	1	4	yes	maximal
7	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
8	#16811180	—	2	8	no	—
9	#16811180	C₂²	1	16	yes	—
10	C₂×#13133217	C₂²	1	4	yes	—
11	C₂×#13133209	C₂²	1	4	yes	—
12	C₂×#13133196	—	2	4	no	—
13	C₂×C₄×Dic₁₀₆	—	2	4	no	—
14	C₂×#13133176	C₂²	1	4	yes	—
15	#16811028	C₂²	1	16	yes	complemented
16	#16811025	C₂²	1	32	yes	complemented
17	#16811021	C₂²	1	16	yes	complemented
18	#16810661	C₂²	1	32	yes	—
19	C₂×#13133151	C₂²	1	8	yes	—
20	C₂×#13133149	C₂²	1	8	yes	—
21	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
22	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	8	yes	—
27	C₂²×Dic₁₀₆	—	2	4	no	—
28	#13133217	—	2	8	no	—
29	#13133217	C₂³	1	16	yes	—
30	#13133209	—	2	8	no	—
31	#13133209	C₂³	1	16	yes	—
32	#13133196	—	2	32	no	—
33	C₄×Dic₁₀₆	—	2	32	no	—
34	#13133176	—	2	8	no	—
35	#13133176	C₂³	1	16	yes	—
36	C₂×C₄⋊Dic₅₃	—	2	4	no	—
37	C₂×C₄⋊Dic₅₃	C₂³	1	8	yes	—
38	C₂×#11719919	—	2	8	no	—
39	C₂×C₂²⋊Dic₅₃	—	2	4	no	—
40	C₂×C₄×Dic₅₃	—	2	4	no	—
41	#13133151	C₂³	1	32	yes	—
42	#13133149	C₂³	1	32	yes	—
43	C₂³×C₄×C₅₃	D₄	1	2	yes	—
44	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	Q₈	1	16	yes	complemented
46	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	8	yes	—
49	C₂×C₄²×C₅₃	D₄	1	2	yes	—
50	C₂×C₄²×C₅₃	D₄	1	4	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
54	C₄×C₈×C₅₃	C₂³	1	4	yes	—
55	C₄×C₈×C₅₃	C₂³	1	8	yes	—
56	C₄×C₈×C₅₃	C₂³	1	4	yes	—
57	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	—	2	8	no	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
65	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
67	C₂²×Dic₅₃	—	2	8	no	—
68	C₂×Dic₁₀₆	—	2	16	no	—
69	C₂×Dic₁₀₆	—	4	16	no	—
70	C₂×C₈×C₅₃	C₂×Q₈	1	8	yes	—
71	C₂×C₈×C₅₃	C₂×Q₈	1	8	yes	—
72	C₂×C₈×C₅₃	C₂×Q₈	1	32	yes	complemented
73	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
74	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
75	C₄²×C₅₃	C₂×D₄	1	8	yes	—
76	C₄²×C₅₃	C₂×D₄	1	2	yes	—
77	C₄²×C₅₃	C₂×D₄	1	2	yes	—
78	C₄²×C₅₃	C₂⁴	1	2	yes	—
79	C₄²×C₅₃	C₂⁴	1	2	yes	—
80	C₂²⋊Dic₅₃	—	2	16	no	—
81	C₄⋊Dic₅₃	—	2	16	no	—
82	C₄⋊Dic₅₃	—	2	8	no	—
83	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
84	#11719919	—	2	32	no	—
85	C₄×Dic₅₃	—	2	16	no	—
86	C₂³×C₅₃	—	2	4	no	—
87	C₂³×C₅₃	C₄⋊Q₈	1	4	yes	—
88	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
89	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	2	16	no	—
92	C₂×C₄×C₅₃	C₄⋊Q₈	1	8	yes	—
93	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
94	C₂×C₄×C₅₃	#135	1	8	yes	—
95	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
96	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
97	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
99	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
100	C₂×C₄×C₅₃	C₂²×Q₈	1	8	yes	—
101	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
102	C₂×Dic₅₃	—	2	16	no	—
103	C₂×Dic₅₃	—	4	16	no	—
104	Dic₁₀₆	—	4	32	no	—
105	C₈×C₅₃	C₂²×Q₈	1	8	yes	—
106	C₈×C₅₃	C₂²×Q₈	1	8	yes	—
107	C₂×#3160	—	53	1	no	maximal, Sylow
108	C₂²×C₅₃	—	2	8	no	—
109	C₂²×C₅₃	—	2	4	no	—
110	C₂²×C₅₃	C₂×C₄⋊Q₈	1	4	yes	—
111	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
112	C₂²×C₅₃	C₂×#135	1	4	yes	—
113	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
114	C₄×C₅₃	—	2	4	no	—
115	C₄×C₅₃	C₂×C₄⋊Q₈	1	4	yes	—
116	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
117	C₄×C₅₃	C₂×#135	1	8	yes	—
118	C₄×C₅₃	C₂³×Q₈	1	2	yes	—
119	Dic₅₃	—	4	16	no	—
120	C₂×#533	—	53	2	no	—
121	#3160	—	53	16	no	—
122	C₂×#500	—	53	2	no	—
123	C₂×#499	—	53	4	no	—
124	C₂×#498	—	53	2	no	—
125	C₂×#427	—	53	4	no	—
126	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
127	C₂×C₅₃	—	2	4	no	—
128	C₂×C₅₃	#3160	1	4	yes	complemented
129	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
130	C₂×C₅₃	C₂²×#135	1	2	yes	—
131	C₂³×C₈	C₂×D₅₃	1	2	yes	—
132	#533	—	53	16	no	—
133	#533	—	106	8	no	—
134	C₂×C₄⋊Q₈	—	53	4	no	—
135	C₂×#125	—	53	4	no	—
136	C₂×C₂²⋊Q₈	—	106	4	no	—
137	C₂×C₄×Q₈	—	106	4	no	—
138	C₂×#117	—	53	4	no	—
139	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
140	#500	—	53	16	no	—
141	#499	—	53	32	no	—
142	#498	—	53	16	no	—
143	#427	—	53	32	no	—
144	C₂×#107	—	53	8	no	—
145	C₂×#106	—	53	8	no	—
146	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
148	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
149	C₂×C₄×C₈	C₂×D₅₃	1	8	yes	—
150	C₅₃	C₂×#3160	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₈	—	106	4	no	—
152	C₂³×C₄	D₂₁₂	1	2	yes	—
153	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
154	C₂²×C₈	Dic₁₀₆	1	16	yes	complemented
155	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
156	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
157	C₂²×C₈	C₂²×D₅₃	1	8	yes	—
158	C₄⋊Q₈	—	53	16	no	—
159	C₄⋊Q₈	—	106	8	no	—
160	#125	—	53	16	no	—
161	#125	—	106	8	no	—
162	C₂²⋊Q₈	—	106	32	no	—
163	C₄×Q₈	—	106	32	no	—
164	#117	—	53	16	no	—
165	#117	—	106	8	no	—
166	C₂×C₄⋊C₄	—	53	8	no	—
167	C₂×C₄⋊C₄	—	106	8	no	—
168	C₂×C₄⋊C₄	—	106	4	no	—
169	C₂×C₂²⋊C₄	—	106	4	no	—
170	C₂×C₄²	—	106	4	no	—
171	C₂×C₄²	D₂₁₂	1	4	yes	—
172	C₂×C₄²	D₂₁₂	1	2	yes	—
173	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
176	#107	—	53	32	no	—
177	#106	—	53	32	no	—
178	C₄×C₈	C₂²×D₅₃	1	4	yes	—
179	C₄×C₈	C₂²×D₅₃	1	8	yes	—
180	C₄×C₈	C₂²×D₅₃	1	4	yes	—
181	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
182	C₂×Q₈	—	106	16	no	—
183	C₂×Q₈	—	212	16	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	106	8	no	—
186	C₂²×C₄	C₂×Dic₁₀₆	1	8	yes	—
187	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
188	C₂²×C₄	C₂×D₂₁₂	1	8	yes	—
189	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
190	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
191	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
192	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
193	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
194	C₂×C₈	C₂×Dic₁₀₆	1	8	yes	—
195	C₂×C₈	C₂×Dic₁₀₆	1	8	yes	—
196	C₂×C₈	C₂×Dic₁₀₆	1	32	yes	complemented
197	C₂×C₈	C₂³×D₅₃	1	4	yes	—
198	C₂×C₈	C₂³×D₅₃	1	4	yes	—
199	C₄⋊C₄	—	53	16	no	—
200	C₄⋊C₄	—	106	8	no	—
201	C₄⋊C₄	—	106	32	no	—
202	C₄⋊C₄	—	106	16	no	—
203	C₂²⋊C₄	—	106	16	no	—
204	C₄²	—	106	16	no	—
205	C₄²	C₂×D₂₁₂	1	8	yes	—
206	C₄²	C₂×D₂₁₂	1	2	yes	—
207	C₄²	C₂×D₂₁₂	1	2	yes	—
208	C₄²	C₂³×D₅₃	1	2	yes	—
209	C₄²	C₂³×D₅₃	1	2	yes	—
210	C₂³	—	2	4	no	—
211	C₂³	#13133217	1	4	yes	—
212	C₂³	C₂²×D₂₁₂	1	2	yes	—
213	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
214	Q₈	—	212	32	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	16	no	—
217	C₂×C₄	—	106	16	no	—
218	C₂×C₄	—	212	16	no	—
219	C₂×C₄	#13133217	1	8	yes	—
220	C₂×C₄	#13133217	1	4	yes	—
221	C₂×C₄	#13133239	1	8	yes	—
222	C₂×C₄	C₂²×D₂₁₂	1	4	yes	—
223	C₂×C₄	C₂²×D₂₁₂	1	4	yes	—
224	C₂×C₄	C₂²×D₂₁₂	1	2	yes	—
225	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
226	C₂×C₄	C₂²×Dic₁₀₆	1	4	yes	—
227	C₂×C₄	C₂²×Dic₁₀₆	1	8	yes	—
228	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
229	C₈	C₂²×Dic₁₀₆	1	8	yes	—
230	C₈	C₂²×Dic₁₀₆	1	8	yes	—
231	C₂²	—	2	8	no	—
232	C₂²	—	2	4	no	—
233	C₂²	C₂×#13133217	1	4	yes	—
234	C₂²	C₂×#13133217	1	2	yes	—
235	C₂²	C₂×#13133239	1	4	yes	—
236	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
237	C₄	—	2	4	no	—
238	C₄	—	212	16	no	—
239	C₄	C₂×#13133217	1	2	yes	—
240	C₄	C₂×#13133217	1	4	yes	—
241	C₄	C₂×#13133239	1	8	yes	—
242	C₄	C₂³×Dic₁₀₆	1	2	yes	—
243	C₂	—	2	4	no	—
244	C₂	#20285229	1	4	yes	direct product factor
245	C₂	C₂²×#13133217	1	1	yes	characteristic
246	C₂	C₂²×#13133239	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂⁴×C₅₃	C₂×D₄
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×#3160
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20285229	C₂	direct (N × H)
3. C₂×#16811028	C₂	semidirect
4. C₂×#16811025	C₂	semidirect
5. C₂×#16811021	C₂	semidirect
15. #16811028	C₂²	semidirect
16. #16811025	C₂²	semidirect
17. #16811021	C₂²	semidirect
45. C₂²×C₈×C₅₃	Q₈	semidirect
72. C₂×C₈×C₅₃	C₂×Q₈	semidirect
128. C₂×C₅₃	#3160	semidirect
150. C₅₃	C₂×#3160	semidirect
154. C₂²×C₈	Dic₁₀₆	semidirect
196. C₂×C₈	C₂×Dic₁₀₆	semidirect
244. C₂	#20285229	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×#16811180	2
#20285229	2
C₂×#16811028	2
C₂×#16811025	2
C₂×#16811021	2
C₂×#16810661	2
C₂²×C₄×C₈×C₅₃	2
C₂×#3160	53

Subgroups — C2×#20285229 #28253697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20285229 #28253697