Subgroups — C₂×C₅₃×#1709 #27227109

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#486	C₂	1	4	yes	maximal, complemented
4	C₅₃×#1709	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₅₃	—	2	2	no	—
7	C₂²×D₈×C₅₃	—	2	2	no	—
8	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
9	C₂×#127×C₅₃	C₂²	1	2	yes	—
10	C₅₃×#494	—	2	2	no	—
11	C₅₃×#494	C₂²	1	4	yes	complemented
12	C₅₃×#493	—	2	2	no	—
13	C₅₃×#493	C₂²	1	4	yes	—
14	C₅₃×#486	—	2	8	no	—
15	C₅₃×#486	C₂²	1	16	yes	complemented
16	C₂×#102×C₅₃	—	2	8	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₈²×C₅₃	C₂²	1	4	yes	complemented
20	C₂²×Q₈×C₅₃	—	2	2	no	—
21	C₂²×D₄×C₅₃	—	2	2	no	—
22	C₂²×D₄×C₅₃	—	2	4	no	—
23	C₂×Q₁₆×C₅₃	—	2	8	no	—
24	C₂×Q₁₆×C₅₃	—	4	8	no	—
25	C₂×D₈×C₅₃	—	2	8	no	—
26	C₂×D₈×C₅₃	—	4	8	no	—
27	C₂²×C₈×C₅₃	D₄	1	4	yes	—
28	C₂²×C₈×C₅₃	D₄	1	2	yes	—
29	C₄⋊Q₈×C₅₃	—	2	2	no	—
30	C₄⋊Q₈×C₅₃	—	2	4	no	—
31	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
32	#127×C₅₃	—	2	4	no	—
33	#127×C₅₃	—	2	2	no	—
34	#127×C₅₃	C₂³	1	4	yes	—
35	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	#102×C₅₃	—	2	32	no	—
38	C₄×C₈×C₅₃	D₄	1	4	yes	—
39	C₄×C₈×C₅₃	D₄	1	2	yes	—
40	C₄×C₈×C₅₃	C₂³	1	4	yes	—
41	C₄×C₈×C₅₃	C₂³	1	2	yes	—
42	C₂⁴×C₅₃	—	4	2	no	—
43	C₂×Q₈×C₅₃	—	2	4	no	—
44	C₂×Q₈×C₅₃	—	4	2	no	—
45	C₂×Q₈×C₅₃	—	4	8	no	—
46	C₂×D₄×C₅₃	—	2	8	no	—
47	C₂×D₄×C₅₃	—	2	4	no	—
48	C₂×D₄×C₅₃	—	4	2	no	—
49	C₂×D₄×C₅₃	—	4	4	no	—
50	C₂×D₄×C₅₃	—	4	8	no	—
51	C₂×D₄×C₅₃	—	4	16	no	—
52	C₂²×C₄×C₅₃	—	4	2	no	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	Q₁₆×C₅₃	—	4	16	no	—
56	D₈×C₅₃	—	4	16	no	—
57	C₂×C₈×C₅₃	—	2	8	no	—
58	C₂×C₈×C₅₃	D₈	1	8	yes	—
59	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	—
60	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
61	C₄⋊C₄×C₅₃	—	2	8	no	—
62	C₄⋊C₄×C₅₃	—	4	4	no	—
63	C₄²×C₅₃	C₂×D₄	1	2	yes	—
64	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
66	C₂³×C₅₃	—	4	4	no	—
67	C₂³×C₅₃	—	8	4	no	—
68	C₂³×C₅₃	—	8	8	no	—
69	C₂³×C₅₃	#127	1	1	yes	characteristic
70	Q₈×C₅₃	—	4	8	no	—
71	Q₈×C₅₃	—	8	4	no	—
72	D₄×C₅₃	—	4	8	no	—
73	D₄×C₅₃	—	4	16	no	—
74	D₄×C₅₃	—	8	4	no	—
75	D₄×C₅₃	—	8	8	no	—
76	C₂×C₄×C₅₃	—	4	4	no	—
77	C₂×C₄×C₅₃	—	8	4	no	—
78	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#127	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
81	C₂×C₄×C₅₃	#135	1	8	yes	—
82	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
84	C₈×C₅₃	—	2	8	no	—
85	C₈×C₅₃	C₂×D₈	1	8	yes	—
86	C₂×#1709	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₅₃	—	8	4	no	—
88	C₂²×C₅₃	—	8	8	no	—
89	C₂²×C₅₃	—	16	8	no	—
90	C₂²×C₅₃	#493	1	2	yes	—
91	C₂²×C₅₃	#495	1	4	yes	—
92	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
93	C₄×C₅₃	—	8	4	no	—
94	C₄×C₅₃	#493	1	2	yes	—
95	C₄×C₅₃	#495	1	4	yes	—
96	C₄×C₅₃	C₂²×D₈	1	2	yes	—
97	C₄×C₅₃	C₂×#135	1	4	yes	—
98	C₂×#494	C₂×C₅₃	1	1	yes	characteristic, complemented
99	C₂×#493	C₂×C₅₃	1	1	yes	characteristic
100	C₂×#486	C₂×C₅₃	1	4	yes	complemented
101	#1709	C₂×C₅₃	1	8	yes	direct product factor
102	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic, complemented
103	C₂×C₅₃	—	16	4	no	—
104	C₂×C₅₃	#1709	1	4	yes	direct product factor
105	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
106	C₂×C₅₃	C₂×#495	1	2	yes	—
107	C₂²×Q₁₆	—	2	2	no	—
108	C₂²×D₈	—	2	2	no	—
109	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
110	C₂×#127	C₂²×C₅₃	1	2	yes	—
111	#494	—	2	2	no	—
112	#494	C₂²×C₅₃	1	4	yes	complemented
113	#493	—	2	2	no	—
114	#493	C₂²×C₅₃	1	4	yes	—
115	#486	—	2	8	no	—
116	#486	C₂²×C₅₃	1	16	yes	complemented
117	C₂×#102	—	2	8	no	—
118	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
119	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
120	C₈²	C₂²×C₅₃	1	4	yes	complemented
121	C₅₃	C₂×#1709	1	1	yes	characteristic, direct product factor, Sylow
122	C₂²×Q₈	—	2	2	no	—
123	C₂²×D₄	—	2	2	no	—
124	C₂²×D₄	—	2	4	no	—
125	C₂×Q₁₆	—	2	8	no	—
126	C₂×Q₁₆	—	4	8	no	—
127	C₂×D₈	—	2	8	no	—
128	C₂×D₈	—	4	8	no	—
129	C₂²×C₈	D₄×C₅₃	1	4	yes	—
130	C₂²×C₈	D₄×C₅₃	1	2	yes	—
131	C₄⋊Q₈	—	2	4	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
134	#127	—	2	4	no	—
135	#127	—	2	2	no	—
136	#127	C₂³×C₅₃	1	4	yes	—
137	C₂×C₄⋊C₄	—	2	4	no	—
138	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
139	#102	—	2	32	no	—
140	C₄×C₈	D₄×C₅₃	1	4	yes	—
141	C₄×C₈	D₄×C₅₃	1	2	yes	—
142	C₄×C₈	C₂³×C₅₃	1	4	yes	—
143	C₄×C₈	C₂³×C₅₃	1	2	yes	—
144	C₂⁴	—	4	2	no	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	4	8	no	—
148	C₂×D₄	—	2	8	no	—
149	C₂×D₄	—	2	4	no	—
150	C₂×D₄	—	4	2	no	—
151	C₂×D₄	—	4	4	no	—
152	C₂×D₄	—	4	8	no	—
153	C₂×D₄	—	4	16	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
157	Q₁₆	—	4	16	no	—
158	D₈	—	4	16	no	—
159	C₂×C₈	—	2	8	no	—
160	C₂×C₈	D₈×C₅₃	1	8	yes	—
161	C₂×C₈	C₂×D₄×C₅₃	1	8	yes	—
162	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
163	C₄⋊C₄	—	2	8	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄²	C₂×D₄×C₅₃	1	2	yes	—
166	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
167	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
168	C₂³	—	4	4	no	—
169	C₂³	—	8	4	no	—
170	C₂³	—	8	8	no	—
171	C₂³	#127×C₅₃	1	1	yes	characteristic
172	Q₈	—	4	8	no	—
173	Q₈	—	8	4	no	—
174	D₄	—	4	8	no	—
175	D₄	—	4	16	no	—
176	D₄	—	8	4	no	—
177	D₄	—	8	8	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	4	no	—
180	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
181	C₂×C₄	#127×C₅₃	1	2	yes	—
182	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
183	C₂×C₄	#135×C₅₃	1	8	yes	—
184	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
185	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
186	C₈	—	2	8	no	—
187	C₈	C₂×D₈×C₅₃	1	8	yes	—
188	C₂²	—	8	4	no	—
189	C₂²	—	8	8	no	—
190	C₂²	—	16	8	no	—
191	C₂²	C₅₃×#493	1	2	yes	—
192	C₂²	C₅₃×#495	1	4	yes	—
193	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
194	C₄	—	8	4	no	—
195	C₄	C₅₃×#493	1	2	yes	—
196	C₄	C₅₃×#495	1	4	yes	—
197	C₄	C₂²×D₈×C₅₃	1	2	yes	—
198	C₄	C₂×#135×C₅₃	1	4	yes	—
199	C₂	—	16	4	no	—
200	C₂	C₅₃×#1709	1	4	yes	direct product factor
201	C₂	C₂×C₅₃×#493	1	1	yes	characteristic
202	C₂	C₂×C₅₃×#495	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂×C₅₃×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#494	C₂	semidirect
3. C₂×C₅₃×#486	C₂	semidirect
4. C₅₃×#1709	C₂	direct (N × H)
5. C₂×C₈²×C₅₃	C₂	semidirect
11. C₅₃×#494	C₂²	semidirect
15. C₅₃×#486	C₂²	semidirect
19. C₈²×C₅₃	C₂²	semidirect
86. C₂×#1709	C₅₃	direct (N × H)
98. C₂×#494	C₂×C₅₃	semidirect
100. C₂×#486	C₂×C₅₃	semidirect
101. #1709	C₂×C₅₃	direct (N × H)
102. C₂×C₈²	C₂×C₅₃	semidirect
104. C₂×C₅₃	#1709	direct (N × H)
112. #494	C₂²×C₅₃	semidirect
116. #486	C₂²×C₅₃	semidirect
120. C₈²	C₂²×C₅₃	semidirect
121. C₅₃	C₂×#1709	direct (N × H)
200. C₂	C₅₃×#1709	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×C₅₃×#494	2
C₂×C₅₃×#493	2
C₂×C₅₃×#486	2
C₅₃×#1709	2
C₂×C₈²×C₅₃	2
C₂×#1709	53

Subgroups — C2×C53×#1709 #27227109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1709 #27227109