Subgroups — C₂×#20276499 #27537121

Order 13568.

694 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810661	C₂	1	2	yes	maximal
2	#20276499	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810451	C₂	1	2	yes	maximal, complemented
4	C₂×#16810449	C₂	1	2	yes	maximal, complemented
5	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133176	—	2	2	no	—
7	#16810661	—	2	4	no	—
8	#16810661	C₂²	1	8	yes	—
9	C₂×#13133151	C₂²	1	4	yes	—
10	C₂×#13133149	—	2	2	no	—
11	#16810451	C₂²	1	8	yes	complemented
12	#16810449	C₂²	1	8	yes	complemented
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₅₃	C₄	1	8	yes	complemented
15	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₅₃	C₂²	1	4	yes	—
18	#13133176	—	2	8	no	—
19	#13133176	—	4	4	no	—
20	C₂×C₄⋊Dic₅₃	—	2	4	no	—
21	C₂×C₂²⋊Dic₅₃	—	4	2	no	—
22	C₂×C₄×Dic₅₃	—	4	2	no	—
23	#13133151	—	2	4	no	—
24	#13133151	C₂³	1	8	yes	—
25	#13133149	—	2	8	no	—
26	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	D₄	1	2	yes	—
30	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
33	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
34	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₁₆×C₅₃	C₂×C₄	1	16	yes	complemented
36	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
37	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
38	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₅₃	D₈	1	2	yes	—
41	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
45	C₂²×Dic₅₃	—	4	4	no	—
46	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
47	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
48	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
50	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
54	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
55	C₂²⋊Dic₅₃	—	4	8	no	—
56	C₄⋊Dic₅₃	—	2	8	no	—
57	C₄⋊Dic₅₃	—	4	4	no	—
58	C₄×Dic₅₃	—	4	8	no	—
59	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
60	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
61	C₂³×C₅₃	—	2	4	no	—
62	C₂³×C₅₃	#107	1	4	yes	—
63	C₂³×C₅₃	C₂×D₈	1	2	yes	—
64	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#107	1	8	yes	—
66	C₂×C₄×C₅₃	#107	1	4	yes	—
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
68	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
72	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
73	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
74	C₂×Dic₅₃	—	4	8	no	—
75	C₂×Dic₅₃	—	8	8	no	—
76	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
77	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
78	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
79	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
80	C₂×#2158	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	—	2	4	no	—
82	C₂²×C₅₃	—	2	8	no	—
83	C₂²×C₅₃	C₂×#107	1	4	yes	—
84	C₂²×C₅₃	C₂×#107	1	2	yes	—
85	C₂²×C₅₃	#508	1	4	yes	—
86	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
87	C₄×C₅₃	C₂×#107	1	4	yes	—
88	C₄×C₅₃	C₂×#107	1	2	yes	—
89	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
91	Dic₅₃	—	8	8	no	—
92	C₂³×C₁₆	D₅₃	1	1	yes	characteristic
93	C₂×#427	—	53	2	no	—
94	#2158	—	53	8	no	—
95	C₂×#367	—	53	2	no	—
96	C₂×#366	—	53	2	no	—
97	C₂×C₅₃	—	2	4	no	—
98	C₂×C₅₃	#2158	1	4	yes	complemented
99	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
100	C₂×C₅₃	C₂×#508	1	2	yes	—
101	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
102	C₂×#117	—	106	2	no	—
103	C₂²×C₁₆	Dic₅₃	1	8	yes	complemented
104	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
105	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
106	C₂²×C₁₆	C₂×D₅₃	1	4	yes	—
107	#427	—	53	8	no	—
108	#427	—	106	4	no	—
109	C₂×#107	—	53	4	no	—
110	C₂×#106	—	106	2	no	—
111	#367	—	53	8	no	—
112	#366	—	53	8	no	—
113	C₅₃	C₂×#2158	1	1	yes	characteristic, complemented, Sylow
114	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
115	C₂²×C₈	Dic₁₀₆	1	4	yes	—
116	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
117	C₂²×C₈	D₂₁₂	1	2	yes	—
118	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
119	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
121	#117	—	106	8	no	—
122	#117	—	212	4	no	—
123	C₂×C₄⋊C₄	—	106	4	no	—
124	C₂×C₂²⋊C₄	—	212	2	no	—
125	C₂×C₄²	—	212	2	no	—
126	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
127	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
128	C₂×C₁₆	C₂×Dic₅₃	1	16	yes	complemented
129	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
130	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
131	#107	—	53	8	no	—
132	#107	—	106	4	no	—
133	#106	—	106	8	no	—
134	C₂⁴	D₄₂₄	1	1	yes	characteristic
135	C₂²×C₄	—	212	4	no	—
136	C₂²×C₄	D₄₂₄	1	2	yes	—
137	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
138	C₂²×C₄	Dic₂₁₂	1	4	yes	—
139	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
140	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
142	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
143	C₂×C₈	C₄⋊Dic₅₃	1	8	yes	—
144	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
145	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
146	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
147	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
148	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
149	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
150	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
151	C₄⋊C₄	—	106	8	no	—
152	C₄⋊C₄	—	212	4	no	—
153	C₂²⋊C₄	—	212	8	no	—
154	C₄²	—	212	8	no	—
155	C₁₆	C₂²×Dic₅₃	1	4	yes	—
156	C₁₆	C₂²×Dic₅₃	1	4	yes	—
157	C₂³	—	2	4	no	—
158	C₂³	#13133151	1	4	yes	—
159	C₂³	C₂×D₄₂₄	1	2	yes	—
160	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
161	C₂×C₄	—	424	8	no	—
162	C₂×C₄	—	212	8	no	—
163	C₂×C₄	#13133151	1	8	yes	—
164	C₂×C₄	#13133151	1	4	yes	—
165	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
167	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
168	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
169	C₂×C₄	C₂×Dic₂₁₂	1	4	yes	—
170	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
171	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
172	C₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
173	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
174	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
175	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
176	C₂²	—	2	8	no	—
177	C₂²	—	2	4	no	—
178	C₂²	C₂×#13133151	1	4	yes	—
179	C₂²	C₂×#13133151	1	2	yes	—
180	C₂²	#16811055	1	4	yes	—
181	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
182	C₄	—	424	8	no	—
183	C₄	C₂×#13133151	1	2	yes	—
184	C₄	C₂×#13133151	1	4	yes	—
185	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
186	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
187	C₂	—	2	4	no	—
188	C₂	#20276499	1	4	yes	direct product factor
189	C₂	C₂²×#13133151	1	1	yes	characteristic
190	C₂	C₂×#16811055	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄₂₄
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂³×C₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂³×C₁₆	D₅₃
hypocenter	C₅₃	C₂×#2158
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20276499	C₂	direct (N × H)
3. C₂×#16810451	C₂	semidirect
4. C₂×#16810449	C₂	semidirect
11. #16810451	C₂²	semidirect
12. #16810449	C₂²	semidirect
14. C₂²×C₁₆×C₅₃	C₄	semidirect
35. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
98. C₂×C₅₃	#2158	semidirect
103. C₂²×C₁₆	Dic₅₃	semidirect
113. C₅₃	C₂×#2158	semidirect
128. C₂×C₁₆	C₂×Dic₅₃	semidirect
188. C₂	#20276499	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×#16810661	2
#20276499	2
C₂×#16810451	2
C₂×#16810449	2
C₂³×C₁₆×C₅₃	2
C₂×#2158	53

Subgroups — C2×#20276499 #27537121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276499 #27537121