Subgroups — C₅₃×#9705 #27217217

Order 13568.

538 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	C₅₃×#2868	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#332	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
6	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
7	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
10	C₂×#97×C₅₃	—	2	1	no	—
11	C₅₃×#384	—	2	2	no	—
12	C₅₃×#384	C₄	1	4	yes	—
13	C₄³×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#336	—	2	1	no	—
15	C₅₃×#336	—	2	2	no	—
16	C₅₃×#332	—	2	8	no	—
17	C₂²×Q₈×C₅₃	—	2	2	no	—
18	C₂²×C₈×C₅₃	—	2	2	no	—
19	C₂²×C₈×C₅₃	—	2	2	no	—
20	C₄⋊Q₈×C₅₃	—	2	4	no	—
21	C₄⋊Q₈×C₅₃	—	2	4	no	—
22	C₄⋊Q₈×C₅₃	—	4	4	no	—
23	C₄⋊Q₈×C₅₃	—	4	2	no	—
24	C₄⋊Q₈×C₅₃	—	4	2	no	—
25	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	D₄	1	2	yes	—
32	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	4	no	—
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
37	#97×C₅₃	—	4	2	no	—
38	C₂×Q₈×C₅₃	—	4	4	no	—
39	C₂×Q₈×C₅₃	—	4	2	no	—
40	C₂×Q₈×C₅₃	—	4	2	no	—
41	C₂×Q₈×C₅₃	—	4	4	no	—
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	—	2	4	no	—
49	C₂×C₈×C₅₃	—	4	2	no	—
50	C₂×C₈×C₅₃	—	4	2	no	—
51	C₂×C₈×C₅₃	—	4	2	no	—
52	C₂×C₈×C₅₃	—	4	4	no	—
53	C₄⋊C₄×C₅₃	—	2	8	no	—
54	C₄⋊C₄×C₅₃	—	4	8	no	—
55	C₄⋊C₄×C₅₃	—	4	8	no	—
56	C₄⋊C₄×C₅₃	—	4	2	no	—
57	C₄⋊C₄×C₅₃	—	4	2	no	—
58	C₄²×C₅₃	—	2	4	no	—
59	C₄²×C₅₃	—	2	2	no	—
60	C₄²×C₅₃	—	2	4	no	—
61	C₄²×C₅₃	D₈	1	2	yes	—
62	C₄²×C₅₃	SD₁₆	1	2	yes	—
63	C₄²×C₅₃	C₂×D₄	1	2	yes	—
64	C₄²×C₅₃	#41	1	2	yes	—
65	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	Q₈×C₅₃	—	8	4	no	—
67	Q₈×C₅₃	—	8	4	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	—	4	4	no	—
74	C₂×C₄×C₅₃	—	4	4	no	—
75	C₂×C₄×C₅₃	—	4	4	no	—
76	C₂×C₄×C₅₃	#101	1	4	yes	—
77	C₂×C₄×C₅₃	#102	1	4	yes	—
78	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₅₃	#121	1	2	yes	—
80	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
85	C₂×C₄×C₅₃	#137	1	4	yes	—
86	C₈×C₅₃	—	4	4	no	—
87	C₈×C₅₃	—	8	2	no	—
88	#9705	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₅₃	#390	1	1	yes	characteristic
90	C₂²×C₅₃	C₂×#101	1	2	yes	—
91	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
92	C₂²×C₅₃	#418	1	1	yes	characteristic
93	C₂²×C₅₃	#419	1	2	yes	—
94	C₄×C₅₃	—	2	4	no	—
95	C₄×C₅₃	—	2	4	no	—
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	—	8	4	no	—
98	C₄×C₅₃	#459	1	2	yes	—
99	C₄×C₅₃	#460	1	2	yes	—
100	C₄×C₅₃	#491	1	2	yes	—
101	C₄×C₅₃	#497	1	2	yes	—
102	#2868	C₂×C₅₃	1	1	yes	characteristic
103	#1831	C₂×C₅₃	1	1	yes	characteristic
104	#1767	C₂×C₅₃	1	1	yes	characteristic
105	C₂×#332	C₂×C₅₃	1	4	yes	—
106	C₂×C₅₃	#1546	1	2	yes	—
107	C₂×C₅₃	#1556	1	2	yes	—
108	C₂×C₅₃	#1960	1	1	yes	characteristic
109	C₂×C₅₃	#1966	1	1	yes	characteristic
110	C₂×C₅₃	#1982	1	1	yes	characteristic
111	C₂×C₄⋊Q₈	—	2	1	no	—
112	C₂×C₄⋊Q₈	—	2	1	no	—
113	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
114	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
115	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
116	C₂×#97	—	2	1	no	—
117	#384	—	2	2	no	—
118	#384	C₄×C₅₃	1	4	yes	—
119	C₄³	C₂²×C₅₃	1	1	yes	characteristic
120	#336	—	2	2	no	—
121	#336	—	2	1	no	—
122	#332	—	2	8	no	—
123	C₅₃	#9705	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²×Q₈	—	2	2	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	4	no	—
128	C₄⋊Q₈	—	2	4	no	—
129	C₄⋊Q₈	—	4	4	no	—
130	C₄⋊Q₈	—	4	2	no	—
131	C₄⋊Q₈	—	4	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	4	no	—
134	C₂×C₄⋊C₄	—	2	4	no	—
135	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
136	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
137	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
138	C₂×C₄²	D₄×C₅₃	1	2	yes	—
139	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
140	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
141	C₄⋊C₈	—	2	4	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
144	#97	—	4	2	no	—
145	C₂×Q₈	—	4	4	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	4	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
151	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
152	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	4	4	no	—
160	C₄⋊C₄	—	2	8	no	—
161	C₄⋊C₄	—	4	8	no	—
162	C₄⋊C₄	—	4	8	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	4	no	—
168	C₄²	D₈×C₅₃	1	2	yes	—
169	C₄²	SD₁₆×C₅₃	1	2	yes	—
170	C₄²	C₂×D₄×C₅₃	1	2	yes	—
171	C₄²	#41×C₅₃	1	2	yes	—
172	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
173	Q₈	—	8	4	no	—
174	Q₈	—	8	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	#101×C₅₃	1	4	yes	—
184	C₂×C₄	#102×C₅₃	1	4	yes	—
185	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
186	C₂×C₄	#121×C₅₃	1	2	yes	—
187	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
188	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
189	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
190	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
191	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
192	C₂×C₄	#137×C₅₃	1	4	yes	—
193	C₈	—	4	4	no	—
194	C₈	—	8	2	no	—
195	C₂²	C₅₃×#390	1	1	yes	characteristic
196	C₂²	C₂×#101×C₅₃	1	2	yes	—
197	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
198	C₂²	C₅₃×#418	1	1	yes	characteristic
199	C₂²	C₅₃×#419	1	2	yes	—
200	C₄	—	2	2	no	—
201	C₄	—	2	4	no	—
202	C₄	—	2	4	no	—
203	C₄	—	8	4	no	—
204	C₄	C₅₃×#459	1	2	yes	—
205	C₄	C₅₃×#460	1	2	yes	—
206	C₄	C₅₃×#491	1	2	yes	—
207	C₄	C₅₃×#497	1	2	yes	—
208	C₂	C₅₃×#1546	1	2	yes	—
209	C₂	C₅₃×#1556	1	2	yes	—
210	C₂	C₅₃×#1960	1	1	yes	characteristic
211	C₂	C₅₃×#1966	1	1	yes	characteristic
212	C₂	C₅₃×#1982	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
88. #9705	C₅₃	direct (N × H)
123. C₅₃	#9705	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2868	2
C₅₃×#1831	2
C₅₃×#1767	2
C₂×C₅₃×#332	2
#9705	53

Subgroups — C53×#9705 #27217217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9705 #27217217