Subgroups — C₂×#16810683 #20283340

Order 6784.

412 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133174	C₂	1	1	yes	maximal, characteristic
2	#16810683	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13133129	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133129	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133127	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	C₂²×#88322	C₂²	1	1	yes	characteristic
12	#13133174	C₄	1	4	yes	—
13	#13133174	C₂²	1	4	yes	—
14	C₂×#11719919	C₄	1	1	yes	characteristic
15	C₂×#11719919	C₄	1	1	yes	characteristic
16	C₂×#11719921	C₄	1	1	yes	characteristic
17	C₂×#11719921	C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊#88322	C₂²	1	4	yes	complemented
20	#13133129	C₂²	1	4	yes	—
21	#13133129	C₂²	1	4	yes	—
22	#13133127	C₂²	1	4	yes	complemented
23	C₈×Dic₅₃	C₂²	1	4	yes	complemented
24	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₅₃	C₂²	1	4	yes	complemented
28	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₅₃	—	2	2	no	—
36	C₂×C₄×D₅₃	—	2	2	no	—
37	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₅₃	C₂³	1	2	yes	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	C₂³	1	2	yes	—
42	C₂×C₈×C₅₃	C₂³	1	2	yes	—
43	C₄²×C₅₃	C₂×C₄	1	2	yes	—
44	C₄²×C₅₃	C₂³	1	2	yes	—
45	#11719921	C₂×C₄	1	4	yes	—
46	#11719921	C₂×C₄	1	4	yes	—
47	#11719919	C₂×C₄	1	4	yes	—
48	#11719919	C₂×C₄	1	4	yes	—
49	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
50	C₄×Dic₅₃	C₂³	1	2	yes	—
51	C₂×#88322	—	2	2	no	—
52	C₂×#88322	—	2	2	no	—
53	C₂×#88322	C₂³	1	2	yes	—
54	C₂×#88322	C₂³	1	2	yes	—
55	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×D₅₃	—	2	2	no	—
57	C₂²×D₅₃	—	2	2	no	—
58	C₂²×D₅₃	—	2	2	no	—
59	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#41	1	2	yes	—
66	C₂×C₄×C₅₃	#41	1	2	yes	—
67	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₅₃	—	2	2	no	—
69	C₂×Dic₅₃	—	2	2	no	—
70	C₂×Dic₅₃	—	2	2	no	—
71	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
72	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
73	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
74	C₄×D₅₃	—	2	4	no	—
75	C₄×D₅₃	—	2	2	no	—
76	C₄×D₅₃	—	2	2	no	—
77	C₈×C₅₃	—	2	2	no	—
78	C₈×C₅₃	—	2	2	no	—
79	#88322	—	2	2	no	—
80	#88322	—	2	2	no	—
81	C₂²×C₅₃	#117	1	2	yes	—
82	C₂²×C₅₃	#131	1	2	yes	—
83	C₂²×C₅₃	#131	1	2	yes	—
84	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
85	C₂×D₅₃	—	2	2	no	—
86	C₂×D₅₃	—	2	2	no	—
87	C₂×D₅₃	—	2	2	no	—
88	C₂×D₅₃	—	4	4	no	—
89	C₄×C₅₃	—	2	2	no	—
90	C₄×C₅₃	—	2	2	no	—
91	C₄×C₅₃	#117	1	2	yes	—
92	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
93	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
94	Dic₅₃	—	2	2	no	—
95	Dic₅₃	—	2	2	no	—
96	Dic₅₃	—	2	2	no	—
97	C₂×#433	—	53	1	no	maximal, Sylow
98	C₂×C₅₃	#433	1	4	yes	complemented
99	C₂×C₅₃	C₂×#117	1	1	yes	characteristic
100	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
101	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
102	D₅₃	—	4	2	no	—
103	C₂×#117	—	53	1	no	—
104	#433	—	53	8	no	—
105	C₂×C₄⋊C₈	—	53	1	no	—
106	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic, complemented
107	C₂×C₂²⋊C₈	—	53	1	no	—
108	C₂×C₂²⋊C₈	—	53	1	no	—
109	C₂×#97	—	53	1	no	—
110	C₂×C₄×C₈	—	53	1	no	—
111	C₅₃	C₂×#433	1	1	yes	characteristic, complemented, Sylow
112	C₂³×C₄	—	53	1	no	—
113	C₂²×C₈	—	53	1	no	—
114	C₂²×C₈	—	53	1	no	—
115	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
116	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
117	#117	—	53	4	no	—
118	#117	—	53	4	no	—
119	C₂×C₄⋊C₄	—	53	1	no	—
120	C₂×C₄⋊C₄	—	53	1	no	—
121	C₂×C₂²⋊C₄	—	53	1	no	—
122	C₂×C₂²⋊C₄	—	53	1	no	—
123	C₂×C₄²	—	53	1	no	—
124	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
125	C₄⋊C₈	—	53	4	no	—
126	C₄⋊C₈	C₂×D₅₃	1	4	yes	complemented
127	C₂²⋊C₈	—	53	4	no	—
128	C₂²⋊C₈	—	53	4	no	—
129	#97	—	53	4	no	—
130	C₄×C₈	—	53	4	no	—
131	C₂⁴	—	53	1	no	—
132	C₂²×C₄	—	53	1	no	—
133	C₂²×C₄	—	53	1	no	—
134	C₂²×C₄	—	53	1	no	—
135	C₂²×C₄	—	53	2	no	—
136	C₂²×C₄	—	53	2	no	—
137	C₂²×C₄	—	106	2	no	—
138	C₂²×C₄	—	106	2	no	—
139	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
140	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
141	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	53	2	no	—
145	C₂×C₈	—	53	2	no	—
146	C₂×C₈	—	106	2	no	—
147	C₂×C₈	—	106	2	no	—
148	C₂×C₈	C₂²×D₅₃	1	2	yes	—
149	C₂×C₈	C₂²×D₅₃	1	2	yes	—
150	C₄⋊C₄	—	53	4	no	—
151	C₄⋊C₄	—	53	4	no	—
152	C₂²⋊C₄	—	53	4	no	—
153	C₂²⋊C₄	—	53	4	no	—
154	C₄²	—	53	2	no	—
155	C₄²	—	53	2	no	—
156	C₄²	C₄×D₅₃	1	2	yes	—
157	C₄²	C₂²×D₅₃	1	2	yes	—
158	C₂³	—	53	2	no	—
159	C₂³	—	106	2	no	—
160	C₂³	—	106	2	no	—
161	C₂³	—	106	2	no	—
162	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	53	2	no	—
166	C₂×C₄	—	53	2	no	—
167	C₂×C₄	—	53	2	no	—
168	C₂×C₄	—	106	2	no	—
169	C₂×C₄	—	106	2	no	—
170	C₂×C₄	—	106	2	no	—
171	C₂×C₄	—	106	4	no	—
172	C₂×C₄	—	106	2	no	—
173	C₂×C₄	—	106	2	no	—
174	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
175	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
176	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
177	C₂×C₄	#11719947	1	2	yes	—
178	C₂×C₄	#11719949	1	2	yes	—
179	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₈	—	106	2	no	—
183	C₈	—	106	2	no	—
184	C₂²	—	106	2	no	—
185	C₂²	—	106	2	no	—
186	C₂²	—	106	2	no	—
187	C₂²	—	212	4	no	—
188	C₂²	#13133175	1	2	yes	—
189	C₂²	#13133223	1	2	yes	—
190	C₂²	#13133225	1	2	yes	—
191	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	106	2	no	—
195	C₄	—	106	2	no	—
196	C₄	—	106	2	no	—
197	C₄	#13133175	1	2	yes	—
198	C₄	C₂×#11719947	1	1	yes	characteristic
199	C₄	C₂×#11719949	1	1	yes	characteristic
200	C₂	—	212	2	no	—
201	C₂	#16810683	1	4	yes	direct product factor
202	C₂	C₂×#13133175	1	1	yes	characteristic
203	C₂	C₂×#13133223	1	1	yes	characteristic
204	C₂	C₂×#13133225	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	C₂×#433
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #16810683	C₂	direct (N × H)
3. C₂×C₄⋊#88322	C₂	semidirect
6. C₂×#13133127	C₂	semidirect
7. C₂×C₈×Dic₅₃	C₂	semidirect
8. C₂×C₄⋊C₈×C₅₃	C₂	semidirect
19. C₄⋊#88322	C₂²	semidirect
22. #13133127	C₂²	semidirect
23. C₈×Dic₅₃	C₂²	semidirect
27. C₄⋊C₈×C₅₃	C₂²	semidirect
98. C₂×C₅₃	#433	semidirect
106. C₂×C₄⋊C₈	D₅₃	semidirect
111. C₅₃	C₂×#433	semidirect
126. C₄⋊C₈	C₂×D₅₃	semidirect
201. C₂	#16810683	direct (N × H)

Subgroups — C2×#16810683 #20283340

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810683 #20283340