Subgroups — C₅₃×#3044 #20271981

Order 6784.

458 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#137×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#536	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#464	C₂	1	4	yes	maximal, complemented
6	C₅₃×#462	C₂	1	4	yes	maximal, complemented
7	C₅₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#419	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#416	C₂	1	1	yes	maximal, characteristic
10	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	#137×C₅₃	—	2	4	no	—
14	C₂×Q₁₆×C₅₃	—	2	4	no	—
15	C₂×Q₁₆×C₅₃	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂×SD₁₆×C₅₃	C₂²	1	2	yes	—
18	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
21	#124×C₅₃	C₂²	1	2	yes	—
22	#123×C₅₃	—	2	2	no	—
23	C₂²⋊Q₈×C₅₃	—	2	2	no	—
24	C₄×Q₈×C₅₃	—	2	2	no	—
25	C₄×Q₈×C₅₃	C₂²	1	4	yes	—
26	#117×C₅₃	—	2	2	no	—
27	#117×C₅₃	C₂²	1	1	yes	characteristic
28	#117×C₅₃	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
30	#103×C₅₃	C₂²	1	2	yes	—
31	#103×C₅₃	C₂²	1	4	yes	—
32	#102×C₅₃	C₂²	1	2	yes	—
33	#41×C₅₃	—	2	2	no	—
34	#41×C₅₃	—	2	2	no	—
35	C₂×Q₈×C₅₃	—	2	1	no	—
36	C₂×Q₈×C₅₃	—	2	1	no	—
37	C₂×Q₈×C₅₃	—	2	2	no	—
38	C₂×Q₈×C₅₃	—	2	2	no	—
39	C₂×Q₈×C₅₃	—	2	2	no	—
40	C₂×Q₈×C₅₃	D₄	1	4	yes	—
41	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
42	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
44	C₂×D₄×C₅₃	—	2	1	no	—
45	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	Q₁₆×C₅₃	—	2	8	no	—
51	Q₁₆×C₅₃	—	2	4	no	—
52	SD₁₆×C₅₃	—	2	4	no	—
53	M₁₆×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	C₂³	1	2	yes	—
56	C₂×C₈×C₅₃	C₂³	1	2	yes	—
57	C₄⋊C₄×C₅₃	—	2	4	no	—
58	C₄⋊C₄×C₅₃	—	2	4	no	—
59	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
60	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
61	C₂²⋊C₄×C₅₃	—	2	1	no	—
62	C₂²⋊C₄×C₅₃	—	2	4	no	—
63	C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
64	C₄²×C₅₃	—	2	4	no	—
65	C₄²×C₅₃	—	2	1	no	—
66	C₄²×C₅₃	C₂³	1	2	yes	—
67	C₂³×C₅₃	—	2	1	no	—
68	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₅₃	—	2	4	no	—
70	Q₈×C₅₃	—	2	2	no	—
71	Q₈×C₅₃	—	2	2	no	—
72	Q₈×C₅₃	—	2	1	no	—
73	Q₈×C₅₃	—	2	2	no	—
74	Q₈×C₅₃	—	2	1	no	—
75	Q₈×C₅₃	—	4	2	no	—
76	Q₈×C₅₃	C₂×D₄	1	4	yes	—
77	D₄×C₅₃	—	2	1	no	—
78	D₄×C₅₃	—	2	1	no	—
79	D₄×C₅₃	—	4	2	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
88	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₅₃	#41	1	2	yes	—
91	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
92	C₈×C₅₃	—	2	2	no	—
93	C₈×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	—	2	2	no	—
95	C₂²×C₅₃	—	4	1	no	—
96	C₂²×C₅₃	—	4	2	no	—
97	C₂²×C₅₃	#121	1	2	yes	—
98	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
99	C₄×C₅₃	—	2	4	no	—
100	C₄×C₅₃	—	2	4	no	—
101	C₄×C₅₃	—	4	1	no	—
102	C₄×C₅₃	—	4	2	no	—
103	C₄×C₅₃	—	4	2	no	—
104	C₄×C₅₃	#121	1	2	yes	—
105	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
107	#3044	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂×C₅₃	—	2	2	no	—
109	C₂×C₅₃	—	8	1	no	—
110	C₂×C₅₃	C₂×#121	1	1	yes	characteristic
111	C₂×C₅₃	#578	1	2	yes	—
112	C₂×#137	C₂×C₅₃	1	1	yes	characteristic
113	C₂²×Q₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
114	#536	C₂×C₅₃	1	1	yes	characteristic
115	#519	C₂×C₅₃	1	1	yes	characteristic, complemented
116	#464	C₂×C₅₃	1	4	yes	complemented
117	#462	C₂×C₅₃	1	4	yes	complemented
118	#424	C₂×C₅₃	1	1	yes	characteristic, complemented
119	#419	C₂×C₅₃	1	1	yes	characteristic, complemented
120	#416	C₂×C₅₃	1	1	yes	characteristic
121	C₅₃	#3044	1	1	yes	characteristic, direct product factor, Sylow
122	C₂×#41	C₂²×C₅₃	1	1	yes	characteristic
123	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
124	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
125	#137	—	2	4	no	—
126	C₂×Q₁₆	—	2	4	no	—
127	C₂×Q₁₆	C₂²×C₅₃	1	4	yes	—
128	C₂×Q₁₆	C₂²×C₅₃	1	2	yes	—
129	C₂×SD₁₆	C₂²×C₅₃	1	2	yes	—
130	C₂×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
131	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
132	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
133	#124	C₂²×C₅₃	1	2	yes	—
134	#123	—	2	2	no	—
135	C₂²⋊Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	2	no	—
137	C₄×Q₈	C₂²×C₅₃	1	4	yes	—
138	#117	—	2	2	no	—
139	#117	C₂²×C₅₃	1	1	yes	characteristic
140	#117	C₂²×C₅₃	1	1	yes	characteristic
141	C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
142	#103	C₂²×C₅₃	1	2	yes	—
143	#103	C₂²×C₅₃	1	4	yes	—
144	#102	C₂²×C₅₃	1	2	yes	—
145	#41	—	2	2	no	—
146	#41	—	2	2	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	2	1	no	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	—	2	2	no	—
151	C₂×Q₈	—	2	2	no	—
152	C₂×Q₈	D₄×C₅₃	1	4	yes	—
153	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
154	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
155	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
156	C₂×D₄	—	2	1	no	—
157	C₂×D₄	C₂³×C₅₃	1	1	yes	characteristic
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
162	Q₁₆	—	2	8	no	—
163	Q₁₆	—	2	4	no	—
164	SD₁₆	—	2	4	no	—
165	M₁₆	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	C₂³×C₅₃	1	2	yes	—
168	C₂×C₈	C₂³×C₅₃	1	2	yes	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	D₄×C₅₃	1	2	yes	—
172	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
173	C₂²⋊C₄	—	2	1	no	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
176	C₄²	—	2	4	no	—
177	C₄²	—	2	1	no	—
178	C₄²	C₂³×C₅₃	1	2	yes	—
179	C₂³	—	2	1	no	—
180	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
181	Q₈	—	2	4	no	—
182	Q₈	—	2	2	no	—
183	Q₈	—	2	2	no	—
184	Q₈	—	2	1	no	—
185	Q₈	—	2	2	no	—
186	Q₈	—	2	1	no	—
187	Q₈	—	4	2	no	—
188	Q₈	C₂×D₄×C₅₃	1	4	yes	—
189	D₄	—	2	1	no	—
190	D₄	—	2	1	no	—
191	D₄	—	4	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	C₂×D₄×C₅₃	1	4	yes	—
200	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
201	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
202	C₂×C₄	#41×C₅₃	1	2	yes	—
203	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	4	1	no	—
208	C₂²	—	4	2	no	—
209	C₂²	#121×C₅₃	1	2	yes	—
210	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	2	4	no	—
213	C₄	—	4	1	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	2	no	—
216	C₄	#121×C₅₃	1	2	yes	—
217	C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
218	C₄	C₂×#41×C₅₃	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	—	8	1	no	—
221	C₂	C₂×#121×C₅₃	1	1	yes	characteristic
222	C₂	C₅₃×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅₃
prime radical	C₂×#41×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ #117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×Q₁₆×C₅₃	C₂	semidirect
4. C₅₃×#519	C₂	semidirect
5. C₅₃×#464	C₂	semidirect
6. C₅₃×#462	C₂	semidirect
7. C₅₃×#424	C₂	semidirect
8. C₅₃×#419	C₂	semidirect
107. #3044	C₅₃	direct (N × H)
113. C₂²×Q₁₆	C₂×C₅₃	semidirect
115. #519	C₂×C₅₃	semidirect
116. #464	C₂×C₅₃	semidirect
117. #462	C₂×C₅₃	semidirect
118. #424	C₂×C₅₃	semidirect
119. #419	C₂×C₅₃	semidirect
121. C₅₃	#3044	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×#137×C₅₃	2
C₂²×Q₁₆×C₅₃	2
C₅₃×#536	2
C₅₃×#519	2
C₅₃×#464	2
C₅₃×#462	2
C₅₃×#424	2
C₅₃×#419	2
C₅₃×#416	2
#3044	53

Subgroups — C53×#3044 #20271981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3044 #20271981