Subgroups — C₂×#20272807 #27298969

Order 13568.

694 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133151	C₂	1	2	yes	maximal
2	#20272807	C₂	1	4	yes	maximal, direct product factor
3	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₅₃	—	2	2	no	—
5	C₂×#13133151	—	2	4	no	—
6	C₂×#13133151	—	2	2	no	—
7	C₂×#13133151	C₄	1	8	yes	complemented
8	C₂×#13133151	C₂²	1	4	yes	—
9	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₅₃	—	2	4	no	—
11	C₂²×C₁₆×C₅₃	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
13	C₂³×Dic₅₃	—	4	2	no	—
14	C₂×C₄⋊Dic₅₃	—	2	8	no	—
15	C₂×C₄⋊Dic₅₃	—	2	4	no	—
16	C₂×C₄⋊Dic₅₃	—	4	2	no	—
17	C₂×C₄⋊Dic₅₃	—	4	4	no	—
18	#13133151	—	2	8	no	—
19	#13133151	—	2	4	no	—
20	#13133151	C₂×C₄	1	8	yes	complemented
21	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₅₃	D₄	1	4	yes	—
25	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	—	2	8	no	—
29	C₂×C₁₆×C₅₃	—	2	4	no	—
30	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
31	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
33	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
36	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₅₃	—	4	8	no	—
39	C₂²×Dic₅₃	—	4	4	no	—
40	C₂²×Dic₅₃	—	8	4	no	—
41	C₂²×Dic₅₃	—	8	4	no	—
42	C₂×C₈×C₅₃	C₄²	1	4	yes	—
43	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
44	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
45	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
47	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
50	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₄⋊Dic₅₃	—	2	8	no	—
52	C₄⋊Dic₅₃	—	4	8	no	—
53	C₄⋊Dic₅₃	—	4	4	no	—
54	C₁₆×C₅₃	—	2	4	no	—
55	C₂³×C₅₃	#102	1	4	yes	—
56	C₂³×C₅₃	#107	1	2	yes	—
57	C₂³×C₅₃	D₁₆	1	2	yes	—
58	C₂³×C₅₃	SD₃₂	1	4	yes	—
59	C₂³×C₅₃	Q₃₂	1	2	yes	—
60	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#95	1	4	yes	—
62	C₂×C₄×C₅₃	#102	1	4	yes	—
63	C₂×C₄×C₅₃	#103	1	8	yes	—
64	C₂×C₄×C₅₃	#106	1	4	yes	—
65	C₂×C₄×C₅₃	#107	1	2	yes	—
66	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
69	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×Dic₅₃	—	4	8	no	—
71	C₂×Dic₅₃	—	8	16	no	—
72	C₂×Dic₅₃	—	8	4	no	—
73	C₂×Dic₅₃	—	8	4	no	—
74	C₈×C₅₃	#95	1	4	yes	—
75	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
76	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×#1380	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	#340	1	4	yes	—
80	C₂²×C₅₃	#357	1	8	yes	—
81	C₂²×C₅₃	#358	1	8	yes	—
82	C₂²×C₅₃	#366	1	4	yes	—
83	C₂²×C₅₃	#367	1	4	yes	—
84	C₂²×C₅₃	C₂×#102	1	2	yes	—
85	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
86	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
87	C₂²×C₅₃	C₂×SD₃₂	1	2	yes	—
88	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
89	C₄×C₅₃	#340	1	4	yes	—
90	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
91	C₄×C₅₃	C₂×#103	1	2	yes	—
92	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
93	Dic₅₃	—	8	8	no	—
94	C₂³×C₁₆	D₅₃	1	1	yes	characteristic
95	C₂²×#107	—	53	2	no	—
96	#1380	—	53	4	no	—
97	C₂×C₅₃	#1380	1	8	yes	complemented
98	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
99	C₂×C₅₃	C₂×#357	1	2	yes	—
100	C₂×C₅₃	C₂×#358	1	2	yes	—
101	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
102	C₂×C₅₃	C₂×#367	1	1	yes	characteristic
103	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
104	C₂²×C₄⋊C₄	—	106	2	no	—
105	C₂²×C₁₆	—	2	4	no	—
106	C₂²×C₁₆	Dic₅₃	1	4	yes	complemented
107	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
108	C₂×#107	—	53	8	no	—
109	C₂×#107	—	53	4	no	—
110	C₂×#107	—	106	2	no	—
111	C₂×#107	—	106	4	no	—
112	C₅₃	C₂×#1380	1	1	yes	characteristic, complemented, Sylow
113	C₂³×C₄	—	212	2	no	—
114	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
115	C₂²×C₈	Dic₁₀₆	1	2	yes	—
116	C₂²×C₈	C₄×D₅₃	1	4	yes	—
117	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
118	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
119	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
120	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
121	C₂×C₄⋊C₄	—	106	4	no	—
122	C₂×C₄⋊C₄	—	106	8	no	—
123	C₂×C₄⋊C₄	—	212	4	no	—
124	C₂×C₄⋊C₄	—	212	2	no	—
125	C₂×C₁₆	—	2	4	no	—
126	C₂×C₁₆	—	2	8	no	—
127	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	complemented
128	#107	—	53	8	no	—
129	#107	—	106	4	no	—
130	#107	—	106	8	no	—
131	C₂⁴	D₄₂₄	1	1	yes	characteristic
132	C₂²×C₄	—	424	4	no	—
133	C₂²×C₄	—	424	4	no	—
134	C₂²×C₄	—	212	8	no	—
135	C₂²×C₄	—	212	4	no	—
136	C₂²×C₄	#11719913	1	4	yes	—
137	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
138	C₂²×C₄	Dic₂₁₂	1	2	yes	—
139	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
140	C₂²×C₄	#11719921	1	4	yes	—
141	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
142	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
143	C₂×C₈	#11719919	1	8	yes	—
144	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
145	C₂×C₈	#11719921	1	4	yes	—
146	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
147	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
148	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
149	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
150	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
151	C₄⋊C₄	—	106	8	no	—
152	C₄⋊C₄	—	212	8	no	—
153	C₄⋊C₄	—	212	4	no	—
154	C₁₆	—	2	4	no	—
155	C₂³	#13133139	1	4	yes	—
156	C₂³	#13133151	1	2	yes	—
157	C₂³	D₈₄₈	1	2	yes	—
158	C₂³	#13133162	1	4	yes	—
159	C₂³	Dic₄₂₄	1	2	yes	—
160	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
161	C₂×C₄	—	424	4	no	—
162	C₂×C₄	—	424	4	no	—
163	C₂×C₄	—	424	16	no	—
164	C₂×C₄	—	212	8	no	—
165	C₂×C₄	#13133124	1	4	yes	—
166	C₂×C₄	#13133139	1	4	yes	—
167	C₂×C₄	#13133142	1	8	yes	—
168	C₂×C₄	#13133149	1	4	yes	—
169	C₂×C₄	#13133151	1	2	yes	—
170	C₂×C₄	C₂×#11719921	1	2	yes	—
171	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
172	C₂×C₄	C₂×#11719913	1	2	yes	—
173	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
174	C₈	#13133124	1	4	yes	—
175	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
176	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
177	C₈	C₂×#11719919	1	2	yes	—
178	C₂²	#16810386	1	4	yes	—
179	C₂²	#16810425	1	8	yes	—
180	C₂²	#16810428	1	8	yes	—
181	C₂²	#16810449	1	4	yes	—
182	C₂²	#16810451	1	4	yes	—
183	C₂²	C₂×#13133139	1	2	yes	—
184	C₂²	C₂×#13133151	1	1	yes	characteristic
185	C₂²	C₂×D₈₄₈	1	1	yes	characteristic
186	C₂²	C₂×#13133162	1	2	yes	—
187	C₂²	C₂×Dic₄₂₄	1	1	yes	characteristic
188	C₄	—	424	8	no	—
189	C₄	#16810386	1	4	yes	—
190	C₄	C₂×#13133124	1	1	yes	characteristic
191	C₄	C₂×#13133142	1	2	yes	—
192	C₄	C₂×#13133149	1	1	yes	characteristic
193	C₂	#20272807	1	8	yes	direct product factor
194	C₂	C₂×#16810386	1	1	yes	characteristic
195	C₂	C₂×#16810425	1	2	yes	—
196	C₂	C₂×#16810428	1	2	yes	—
197	C₂	C₂×#16810449	1	1	yes	characteristic
198	C₂	C₂×#16810451	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄₂₄
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂³×C₁₆×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂³×C₁₆	D₅₃
hypocenter	C₅₃	C₂×#1380
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20272807	C₂	direct (N × H)
7. C₂×#13133151	C₄	semidirect
11. C₂²×C₁₆×C₅₃	C₄	semidirect
20. #13133151	C₂×C₄	semidirect
30. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
97. C₂×C₅₃	#1380	semidirect
106. C₂²×C₁₆	Dic₅₃	semidirect
112. C₅₃	C₂×#1380	semidirect
127. C₂×C₁₆	C₂×Dic₅₃	semidirect
193. C₂	#20272807	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×#13133151	2
#20272807	2
C₂³×C₁₆×C₅₃	2
C₂×#1380	53

Subgroups — C2×#20272807 #27298969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272807 #27298969