Subgroups — C₅₃×#2627 #20271564

Order 6784.

508 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×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#400	C₂	1	2	yes	maximal, complemented
4	C₅₃×#398	C₂	1	4	yes	maximal, complemented
5	C₅₃×#395	C₂	1	2	yes	maximal, complemented
6	C₅₃×#385	C₂	1	2	yes	maximal
7	C₅₃×#381	C₂	1	2	yes	maximal
8	C₅₃×#380	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²⋊Q₈×C₅₃	—	2	4	no	—
14	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
23	#95×C₅₃	C₂²	1	2	yes	—
24	#95×C₅₃	C₂²	1	4	yes	—
25	#95×C₅₃	C₂²	1	4	yes	—
26	#95×C₅₃	C₂²	1	4	yes	—
27	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×Q₈×C₅₃	—	2	2	no	—
29	C₂×Q₈×C₅₃	—	4	2	no	—
30	C₂²×C₄×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	—	2	2	no	—
33	C₂²×C₄×C₅₃	—	4	2	no	—
34	C₂²×C₄×C₅₃	—	4	2	no	—
35	C₂²×C₄×C₅₃	D₄	1	4	yes	—
36	C₂²×C₄×C₅₃	Q₈	1	4	yes	complemented
37	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
39	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
44	C₄⋊C₄×C₅₃	—	2	4	no	—
45	C₄⋊C₄×C₅₃	—	2	4	no	—
46	C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₄⋊C₄×C₅₃	—	4	4	no	—
50	C₂²⋊C₄×C₅₃	—	2	4	no	—
51	C₂²⋊C₄×C₅₃	—	2	4	no	—
52	C₂³×C₅₃	—	2	2	no	—
53	C₂³×C₅₃	—	2	2	no	—
54	C₂³×C₅₃	C₂×D₄	1	2	yes	—
55	C₂³×C₅₃	C₂×Q₈	1	2	yes	—
56	C₂³×C₅₃	#41	1	2	yes	—
57	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
58	Q₈×C₅₃	—	4	4	no	—
59	C₂×C₄×C₅₃	—	2	8	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	4	no	—
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	—	2	4	no	—
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	2	4	no	—
72	C₂×C₄×C₅₃	—	2	4	no	—
73	C₂×C₄×C₅₃	—	4	4	no	—
74	C₂×C₄×C₅₃	—	4	4	no	—
75	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
78	C₂²×C₅₃	—	2	2	no	—
79	C₂²×C₅₃	—	2	2	no	—
80	C₂²×C₅₃	—	2	4	no	—
81	C₂²×C₅₃	—	2	4	no	—
82	C₂²×C₅₃	C₂²⋊Q₈	1	4	yes	—
83	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
84	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
85	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
86	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
87	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
88	C₄×C₅₃	—	2	4	no	—
89	C₄×C₅₃	—	2	2	no	—
90	C₄×C₅₃	—	2	2	no	—
91	C₄×C₅₃	—	4	2	no	—
92	C₄×C₅₃	—	4	2	no	—
93	C₄×C₅₃	—	4	2	no	—
94	C₄×C₅₃	—	4	4	no	—
95	#2627	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×C₅₃	—	2	4	no	—
97	C₂×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₅₃	#534	1	1	yes	characteristic
99	C₂×C₅₃	#536	1	1	yes	characteristic
100	C₂×C₅₃	#539	1	2	yes	—
101	C₂×C₅₃	#544	1	2	yes	—
102	C₂×C₂²⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
103	C₂²×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
104	#400	C₂×C₅₃	1	2	yes	complemented
105	#398	C₂×C₅₃	1	4	yes	complemented
106	#395	C₂×C₅₃	1	2	yes	complemented
107	#385	C₂×C₅₃	1	2	yes	—
108	#381	C₂×C₅₃	1	2	yes	—
109	#380	C₂×C₅₃	1	1	yes	characteristic
110	C₅₃	#2627	1	1	yes	characteristic, direct product factor, Sylow
111	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
112	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
113	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
114	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
115	C₂²⋊Q₈	—	2	4	no	—
116	C₂×C₄⋊C₄	—	2	4	no	—
117	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
118	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
119	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
120	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
122	C₂×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
123	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
124	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
125	#95	C₂²×C₅₃	1	2	yes	—
126	#95	C₂²×C₅₃	1	4	yes	—
127	#95	C₂²×C₅₃	1	4	yes	—
128	#95	C₂²×C₅₃	1	4	yes	—
129	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	4	2	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	4	2	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	D₄×C₅₃	1	4	yes	—
138	C₂²×C₄	Q₈×C₅₃	1	4	yes	complemented
139	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
140	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
141	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
142	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
143	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
144	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
145	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₂²⋊C₄	—	2	4	no	—
153	C₂²⋊C₄	—	2	4	no	—
154	C₂³	—	2	2	no	—
155	C₂³	—	2	2	no	—
156	C₂³	C₂×D₄×C₅₃	1	2	yes	—
157	C₂³	C₂×Q₈×C₅₃	1	2	yes	—
158	C₂³	#41×C₅₃	1	2	yes	—
159	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
160	Q₈	—	4	4	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
178	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
179	C₂×C₄	C₂×Q₈×C₅₃	1	4	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	2	4	no	—
183	C₂²	—	2	4	no	—
184	C₂²	C₂²⋊Q₈×C₅₃	1	4	yes	—
185	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
186	C₂²	C₂²×Q₈×C₅₃	1	1	yes	characteristic
187	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
188	C₂²	H(5,𝔽₂)×C₅₃	1	2	yes	—
189	C₂²	2₋¹⁺⁴×C₅₃	1	2	yes	—
190	C₄	—	2	4	no	—
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	4	2	no	—
194	C₄	—	4	2	no	—
195	C₄	—	4	2	no	—
196	C₄	—	4	4	no	—
197	C₂	—	2	4	no	—
198	C₂	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
199	C₂	C₅₃×#534	1	1	yes	characteristic
200	C₂	C₅₃×#536	1	1	yes	characteristic
201	C₂	C₅₃×#539	1	2	yes	—
202	C₂	C₅₃×#544	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂⁴×C₅₃	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#400	C₂	semidirect
4. C₅₃×#398	C₂	semidirect
5. C₅₃×#395	C₂	semidirect
36. C₂²×C₄×C₅₃	Q₈	semidirect
95. #2627	C₅₃	direct (N × H)
104. #400	C₂×C₅₃	semidirect
105. #398	C₂×C₅₃	semidirect
106. #395	C₂×C₅₃	semidirect
110. C₅₃	#2627	direct (N × H)
138. C₂²×C₄	Q₈×C₅₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂²⋊Q₈×C₅₃	2
C₂²×C₄⋊C₄×C₅₃	2
C₅₃×#400	2
C₅₃×#398	2
C₅₃×#395	2
C₅₃×#385	2
C₅₃×#381	2
C₅₃×#380	2
#2627	53

Subgroups — C53×#2627 #20271564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2627 #20271564