Subgroups — C₂×C₅₃×#1365 #27219450

Order 13568.

350 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1365	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
6	C₂²×C₁₆×C₅₃	—	2	1	no	—
7	C₅₃×#500	—	2	1	no	—
8	C₅₃×#500	—	2	1	no	—
9	C₅₃×#500	C₄	1	2	yes	—
10	C₅₃×#500	C₂²	1	2	yes	—
11	C₂×#107×C₅₃	—	2	1	no	—
12	C₂×#107×C₅₃	C₄	1	2	yes	—
13	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×C₄⋊C₁₆	—	2	1	no	—
16	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₅₃×#335	—	2	1	no	—
18	C₅₃×#335	C₂²	1	2	yes	—
19	C₂²×Q₈×C₅₃	—	4	1	no	—
20	C₂²×C₈×C₅₃	—	4	1	no	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₄⋊Q₈×C₅₃	—	2	2	no	—
24	C₄⋊Q₈×C₅₃	—	2	2	no	—
25	C₄⋊Q₈×C₅₃	—	4	1	no	—
26	C₄⋊Q₈×C₅₃	—	4	1	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	—	2	2	no	—
31	C₂×C₁₆×C₅₃	—	4	2	no	—
32	#107×C₅₃	—	2	2	no	—
33	#107×C₅₃	—	2	2	no	—
34	#107×C₅₃	—	2	2	no	—
35	#107×C₅₃	C₂×C₄	1	4	yes	—
36	C₄⋊C₈×C₅₃	—	2	2	no	—
37	C₄⋊C₈×C₅₃	—	4	1	no	—
38	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×Q₈×C₅₃	—	4	2	no	—
43	C₂×Q₈×C₅₃	—	4	2	no	—
44	C₂×Q₈×C₅₃	—	8	4	no	—
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	—	4	2	no	—
51	C₂×C₈×C₅₃	—	8	2	no	—
52	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
55	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄⋊C₄×C₅₃	—	2	4	no	—
59	C₄⋊C₄×C₅₃	—	4	2	no	—
60	C₄⋊C₄×C₅₃	—	4	2	no	—
61	C₄⋊C₄×C₅₃	—	4	2	no	—
62	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₅₃	D₈	1	1	yes	characteristic
64	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
65	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₁₆×C₅₃	—	4	2	no	—
67	C₂³×C₅₃	#102	1	1	yes	characteristic
68	Q₈×C₅₃	—	8	4	no	—
69	Q₈×C₅₃	—	8	4	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	4	4	no	—
72	C₂×C₄×C₅₃	—	8	4	no	—
73	C₂×C₄×C₅₃	#101	1	2	yes	—
74	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#103	1	2	yes	—
78	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
79	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
83	C₈×C₅₃	—	2	1	no	—
84	C₈×C₅₃	—	2	1	no	—
85	C₈×C₅₃	—	8	2	no	—
86	C₈×C₅₃	#103	1	2	yes	—
87	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×#1365	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂²×C₅₃	#332	1	2	yes	—
91	C₂²×C₅₃	#358	1	2	yes	—
92	C₂²×C₅₃	#361	1	2	yes	—
93	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	—	8	4	no	—
97	C₄×C₅₃	#332	1	2	yes	—
98	C₄×C₅₃	#358	1	2	yes	—
99	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
100	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
101	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
102	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
103	C₂×#500	C₂×C₅₃	1	1	yes	characteristic
104	C₂×C₄⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic
105	C₂×#335	C₂×C₅₃	1	1	yes	characteristic
106	#1365	C₂×C₅₃	1	4	yes	direct product factor
107	C₂×C₅₃	#1365	1	4	yes	direct product factor
108	C₂×C₅₃	C₂×#332	1	1	yes	characteristic
109	C₂×C₅₃	C₂×#358	1	1	yes	characteristic
110	C₂×C₅₃	C₂×#361	1	1	yes	characteristic
111	C₂×C₄⋊Q₈	—	2	1	no	—
112	C₂²×C₁₆	—	2	1	no	—
113	#500	—	2	1	no	—
114	#500	—	2	1	no	—
115	#500	C₄×C₅₃	1	2	yes	—
116	#500	C₂²×C₅₃	1	2	yes	—
117	C₂×#107	—	2	1	no	—
118	C₂×#107	C₄×C₅₃	1	2	yes	—
119	C₂×C₄⋊C₈	—	2	1	no	—
120	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
121	C₄⋊C₁₆	—	2	1	no	—
122	C₄⋊C₁₆	C₂²×C₅₃	1	2	yes	—
123	#335	—	2	1	no	—
124	#335	C₂²×C₅₃	1	2	yes	—
125	C₅₃	C₂×#1365	1	1	yes	characteristic, direct product factor, Sylow
126	C₂²×Q₈	—	4	1	no	—
127	C₂²×C₈	—	4	1	no	—
128	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
129	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	4	1	no	—
133	C₄⋊Q₈	—	4	1	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	4	1	no	—
136	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	4	2	no	—
139	#107	—	2	2	no	—
140	#107	—	2	2	no	—
141	#107	—	2	2	no	—
142	#107	C₂×C₄×C₅₃	1	4	yes	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	4	1	no	—
145	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
146	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
147	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
148	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
149	C₂×Q₈	—	4	2	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×Q₈	—	8	4	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
154	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
155	C₂²×C₄	SD₁₆×C₅₃	1	1	yes	characteristic
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	8	2	no	—
159	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
160	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
161	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
162	C₂×C₈	Q₁₆×C₅₃	1	2	yes	—
163	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
164	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
170	C₄²	D₈×C₅₃	1	1	yes	characteristic
171	C₄²	SD₁₆×C₅₃	1	1	yes	characteristic
172	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
173	C₁₆	—	4	2	no	—
174	C₂³	#102×C₅₃	1	1	yes	characteristic
175	Q₈	—	8	4	no	—
176	Q₈	—	8	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	4	no	—
180	C₂×C₄	#101×C₅₃	1	2	yes	—
181	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
182	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
183	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
184	C₂×C₄	#103×C₅₃	1	2	yes	—
185	C₂×C₄	SD₃₂×C₅₃	1	2	yes	—
186	C₂×C₄	Q₃₂×C₅₃	1	2	yes	—
187	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
188	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
189	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	8	2	no	—
193	C₈	#103×C₅₃	1	2	yes	—
194	C₈	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
195	C₈	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
196	C₂²	C₅₃×#332	1	2	yes	—
197	C₂²	C₅₃×#358	1	2	yes	—
198	C₂²	C₅₃×#361	1	2	yes	—
199	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	8	4	no	—
203	C₄	C₅₃×#332	1	2	yes	—
204	C₄	C₅₃×#358	1	2	yes	—
205	C₄	C₂×#101×C₅₃	1	1	yes	characteristic
206	C₄	C₂×#103×C₅₃	1	1	yes	characteristic
207	C₄	C₂×SD₃₂×C₅₃	1	1	yes	characteristic
208	C₄	C₂×Q₃₂×C₅₃	1	1	yes	characteristic
209	C₂	C₅₃×#1365	1	4	yes	direct product factor
210	C₂	C₂×C₅₃×#332	1	1	yes	characteristic
211	C₂	C₂×C₅₃×#358	1	1	yes	characteristic
212	C₂	C₂×C₅₃×#361	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#102
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1365	C₂	direct (N × H)
89. C₂×#1365	C₅₃	direct (N × H)
106. #1365	C₂×C₅₃	direct (N × H)
107. C₂×C₅₃	#1365	direct (N × H)
125. C₅₃	C₂×#1365	direct (N × H)
209. C₂	C₅₃×#1365	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×#500	2
C₂×C₅₃×C₄⋊C₁₆	2
C₂×C₅₃×#335	2
C₅₃×#1365	2
C₂×#1365	53

Subgroups — C2×C53×#1365 #27219450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1365 #27219450