Subgroups — C₂×#16810740 #20283605

Order 6784.

472 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133224	C₂	1	2	yes	maximal, complemented
2	#16810740	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133153	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13133144	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
7	#13133224	—	2	4	no	—
8	#13133224	C₂²	1	8	yes	complemented
9	C₂×#11719917	—	2	2	no	—
10	C₂×#11719917	C₂²	1	2	yes	—
11	C₂²×#88322	—	2	2	no	—
12	C₂²×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	#13133153	C₂²	1	4	yes	complemented
15	#13133144	C₂²	1	8	yes	complemented
16	C₈×Dic₅₃	C₂²	1	4	yes	complemented
17	C₂×#41×C₅₃	C₂²	1	2	yes	—
18	#135×C₅₃	C₄	1	4	yes	complemented
19	#135×C₅₃	C₂²	1	4	yes	—
20	C₂×Q₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
21	C₂×SD₁₆×C₅₃	C₄	1	2	yes	complemented
22	C₂×D₈×C₅₃	C₄	1	1	yes	characteristic, complemented
23	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	#41×C₅₃	—	2	2	no	—
25	#41×C₅₃	—	2	2	no	—
26	#41×C₅₃	C₂×C₄	1	4	yes	—
27	#41×C₅₃	C₂³	1	4	yes	—
28	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×D₄×C₅₃	—	2	2	no	—
30	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₅₃	—	2	1	no	—
34	Q₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
35	SD₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
36	D₈×C₅₃	C₂×C₄	1	4	yes	complemented
37	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₅₃	C₂³	1	2	yes	—
41	C₄×Dic₅₃	—	2	1	no	—
42	C₄×Dic₅₃	C₂³	1	2	yes	—
43	#11719917	—	2	2	no	—
44	#11719917	—	2	8	no	—
45	#11719917	C₂³	1	4	yes	—
46	C₂×#88322	—	2	2	no	—
47	C₂×#88322	—	2	8	no	—
48	C₂×#88322	D₄	1	1	yes	characteristic
49	C₂×#88322	D₄	1	1	yes	characteristic
50	C₂×#88322	D₄	1	2	yes	—
51	C₂×#88322	C₂³	1	2	yes	—
52	C₂³×C₅₃	—	2	2	no	—
53	C₂³×C₅₃	#41	1	1	yes	characteristic
54	Q₈×C₅₃	—	2	2	no	—
55	Q₈×C₅₃	C₂²×C₄	1	4	yes	—
56	D₄×C₅₃	—	2	4	no	—
57	D₄×C₅₃	—	2	4	no	—
58	D₄×C₅₃	—	2	2	no	—
59	D₄×C₅₃	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	4	no	—
62	C₂×C₄×C₅₃	—	2	4	no	—
63	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₂×Dic₅₃	—	2	4	no	—
70	C₂×Dic₅₃	—	2	2	no	—
71	C₈×C₅₃	C₂²×C₄	1	2	yes	—
72	C₈×C₅₃	C₂²×C₄	1	2	yes	—
73	#88322	—	2	4	no	—
74	#88322	C₂×D₄	1	2	yes	—
75	#88322	C₂×D₄	1	2	yes	—
76	C₂²×C₅₃	—	2	2	no	—
77	C₂²×C₅₃	—	2	4	no	—
78	C₂²×C₅₃	—	4	4	no	—
79	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
80	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
81	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
82	C₄×C₅₃	—	2	4	no	—
83	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
84	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
85	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
86	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
87	Dic₅₃	—	2	4	no	—
88	C₂×#443	—	53	1	no	maximal, Sylow
89	C₂×C₅₃	—	2	1	no	—
90	C₂×C₅₃	—	2	1	no	—
91	C₂×C₅₃	—	4	4	no	—
92	C₂×C₅₃	#443	1	2	yes	complemented
93	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
94	C₂×#135	D₅₃	1	1	yes	characteristic
95	C₂×#131	—	53	2	no	—
96	#443	—	53	8	no	—
97	C₂×#108	—	53	1	no	—
98	C₂×#104	—	53	2	no	—
99	C₂×C₄×C₈	—	53	1	no	—
100	C₅₃	C₂×#443	1	1	yes	characteristic, complemented, Sylow
101	C₂×#41	C₂×D₅₃	1	2	yes	—
102	#135	Dic₅₃	1	4	yes	complemented
103	#135	C₂×D₅₃	1	4	yes	—
104	C₂×Q₁₆	Dic₅₃	1	1	yes	characteristic, complemented
105	C₂×SD₁₆	Dic₅₃	1	2	yes	complemented
106	C₂×D₈	Dic₅₃	1	1	yes	characteristic, complemented
107	#131	—	53	8	no	—
108	#131	—	106	4	no	—
109	C₂×M₁₆	—	53	2	no	—
110	C₂×M₁₆	—	106	2	no	—
111	C₂²×C₈	—	53	1	no	—
112	C₂²×C₈	—	106	2	no	—
113	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
114	C₂×C₄²	—	53	1	no	—
115	#108	—	53	4	no	—
116	#104	—	53	8	no	—
117	C₄×C₈	—	53	4	no	—
118	#41	—	2	2	no	—
119	#41	—	2	2	no	—
120	#41	C₂×Dic₅₃	1	4	yes	—
121	#41	C₂²×D₅₃	1	4	yes	—
122	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
123	C₂×D₄	—	2	2	no	—
124	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	106	1	no	—
127	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
128	Q₁₆	C₂×Dic₅₃	1	4	yes	complemented
129	SD₁₆	C₂×Dic₅₃	1	8	yes	complemented
130	D₈	C₂×Dic₅₃	1	4	yes	complemented
131	M₁₆	—	53	4	no	—
132	M₁₆	—	106	8	no	—
133	M₁₆	—	106	2	no	—
134	C₂×C₈	—	53	2	no	—
135	C₂×C₈	—	53	1	no	—
136	C₂×C₈	—	53	1	no	—
137	C₂×C₈	—	53	2	no	—
138	C₂×C₈	—	106	8	no	—
139	C₂×C₈	—	106	2	no	—
140	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
142	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
143	C₂×C₈	C₂²×D₅₃	1	2	yes	—
144	C₄²	—	53	2	no	—
145	C₄²	—	106	1	no	—
146	C₂³	—	2	2	no	—
147	C₂³	#11719947	1	1	yes	characteristic
148	Q₈	—	2	2	no	—
149	Q₈	C₂²×Dic₅₃	1	4	yes	—
150	D₄	—	2	4	no	—
151	D₄	—	2	4	no	—
152	D₄	—	2	2	no	—
153	D₄	C₂²×Dic₅₃	1	4	yes	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	106	4	no	—
158	C₂×C₄	—	106	2	no	—
159	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
160	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
161	C₂×C₄	#11719947	1	1	yes	characteristic
162	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
163	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
164	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
165	C₈	—	53	2	no	—
166	C₈	—	53	2	no	—
167	C₈	—	106	4	no	—
168	C₈	C₂²×Dic₅₃	1	2	yes	—
169	C₈	C₂²×Dic₅₃	1	2	yes	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	4	no	—
172	C₂²	—	4	4	no	—
173	C₂²	D₄×Dic₅₃	1	1	yes	characteristic
174	C₂²	D₄×Dic₅₃	1	1	yes	characteristic
175	C₂²	C₂×#11719947	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	106	4	no	—
178	C₄	D₄×Dic₅₃	1	1	yes	characteristic
179	C₄	D₄×Dic₅₃	1	1	yes	characteristic
180	C₄	C₂³×Dic₅₃	1	1	yes	characteristic
181	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	—	4	4	no	—
185	C₂	#16810740	1	2	yes	direct product factor
186	C₂	C₂×D₄×Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#135×C₅₃	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	C₂×#135	D₅₃
hypocenter	C₅₃	C₂×#443
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#13133224	C₂	semidirect
2. #16810740	C₂	direct (N × H)
3. C₂×#13133153	C₂	semidirect
4. C₂×#13133144	C₂	semidirect
5. C₂×C₈×Dic₅₃	C₂	semidirect
8. #13133224	C₂²	semidirect
14. #13133153	C₂²	semidirect
15. #13133144	C₂²	semidirect
16. C₈×Dic₅₃	C₂²	semidirect
18. #135×C₅₃	C₄	semidirect
20. C₂×Q₁₆×C₅₃	C₄	semidirect
21. C₂×SD₁₆×C₅₃	C₄	semidirect
22. C₂×D₈×C₅₃	C₄	semidirect
34. Q₁₆×C₅₃	C₂×C₄	semidirect
35. SD₁₆×C₅₃	C₂×C₄	semidirect
36. D₈×C₅₃	C₂×C₄	semidirect
92. C₂×C₅₃	#443	semidirect
100. C₅₃	C₂×#443	semidirect
102. #135	Dic₅₃	semidirect
104. C₂×Q₁₆	Dic₅₃	semidirect
105. C₂×SD₁₆	Dic₅₃	semidirect
106. C₂×D₈	Dic₅₃	semidirect
128. Q₁₆	C₂×Dic₅₃	semidirect
129. SD₁₆	C₂×Dic₅₃	semidirect
130. D₈	C₂×Dic₅₃	semidirect
185. C₂	#16810740	direct (N × H)

Subgroups — C2×#16810740 #20283605

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810740 #20283605