Subgroups — C₅₃×#9801 #27217313

Order 13568.

410 conjugacy classes of subgroups.

The subgroup lattice has 208 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	C₅₃×#1997	C₂	1	2	yes	maximal
3	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1549	C₂	1	2	yes	maximal
5	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
7	C₅₃×#500	—	2	1	no	—
8	C₅₃×#500	—	2	1	no	—
9	C₅₃×#500	—	2	2	no	—
10	C₅₃×#431	C₂²	1	1	yes	characteristic
11	C₂×#107×C₅₃	C₄	1	4	yes	—
12	C₅₃×#423	C₂²	1	2	yes	—
13	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#343	—	2	4	no	—
16	C₅₃×#335	—	2	2	no	—
17	C₅₃×#332	—	2	4	no	—
18	C₅₃×#322	—	2	1	no	—
19	C₈²×C₅₃	—	2	1	no	—
20	C₂²×Q₈×C₅₃	—	2	2	no	—
21	C₂×M₁₆×C₅₃	—	2	2	no	—
22	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₄⋊Q₈×C₅₃	—	2	2	no	—
25	C₄⋊Q₈×C₅₃	—	2	2	no	—
26	C₄⋊Q₈×C₅₃	—	2	2	no	—
27	C₄⋊Q₈×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	#107×C₅₃	—	2	4	no	—
32	#107×C₅₃	—	2	4	no	—
33	C₄⋊C₈×C₅₃	—	2	2	no	—
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
36	C₂²⋊C₈×C₅₃	—	2	1	no	—
37	#97×C₅₃	—	2	1	no	—
38	C₄×C₈×C₅₃	—	2	1	no	—
39	C₄×C₈×C₅₃	—	2	1	no	—
40	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	D₄	1	2	yes	—
43	C₂×Q₈×C₅₃	—	4	2	no	—
44	C₂×Q₈×C₅₃	—	4	2	no	—
45	C₂×Q₈×C₅₃	—	8	4	no	—
46	C₂²×C₄×C₅₃	—	2	4	no	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
50	M₁₆×C₅₃	—	2	2	no	—
51	M₁₆×C₅₃	—	4	4	no	—
52	M₁₆×C₅₃	Q₁₆	1	4	yes	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	2	4	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	4	2	no	—
57	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₄⋊C₄×C₅₃	—	2	4	no	—
60	C₄⋊C₄×C₅₃	—	2	4	no	—
61	C₄⋊C₄×C₅₃	—	4	2	no	—
62	C₄⋊C₄×C₅₃	—	4	2	no	—
63	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₅₃	#41	1	1	yes	characteristic
67	C₂³×C₅₃	C₄×D₄	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	—	2	2	no	—
72	C₂×C₄×C₅₃	—	4	4	no	—
73	C₂×C₄×C₅₃	—	8	4	no	—
74	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
81	C₂×C₄×C₅₃	#135	1	2	yes	—
82	C₈×C₅₃	—	2	2	no	—
83	C₈×C₅₃	—	2	4	no	—
84	C₈×C₅₃	—	4	2	no	—
85	C₈×C₅₃	—	8	2	no	—
86	C₈×C₅₃	#101	1	4	yes	—
87	#9801	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂²×C₅₃	—	2	1	no	—
89	C₂²×C₅₃	—	2	1	no	—
90	C₂²×C₅₃	#390	1	1	yes	characteristic
91	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
92	C₄×C₅₃	—	2	1	no	—
93	C₄×C₅₃	—	2	1	no	—
94	C₄×C₅₃	—	8	4	no	—
95	C₄×C₅₃	C₂×#101	1	2	yes	—
96	C₄×C₅₃	#467	1	2	yes	—
97	C₄×C₅₃	#472	1	2	yes	—
98	C₄×C₅₃	#494	1	1	yes	characteristic
99	C₄×C₅₃	#495	1	1	yes	characteristic
100	C₂×#500	C₂×C₅₃	1	1	yes	characteristic
101	#1997	C₂×C₅₃	1	2	yes	—
102	#1567	C₂×C₅₃	1	1	yes	characteristic
103	#1549	C₂×C₅₃	1	2	yes	—
104	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic
105	C₂×C₅₃	—	2	2	no	—
106	C₂×C₅₃	#1966	1	1	yes	characteristic
107	C₂×C₅₃	#1969	1	1	yes	characteristic
108	C₂×C₅₃	#1978	1	1	yes	characteristic
109	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
110	#500	—	2	1	no	—
111	#500	—	2	1	no	—
112	#500	—	2	2	no	—
113	#431	C₂²×C₅₃	1	1	yes	characteristic
114	C₂×#107	C₄×C₅₃	1	4	yes	—
115	#423	C₂²×C₅₃	1	2	yes	—
116	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
118	#343	—	2	4	no	—
119	#335	—	2	2	no	—
120	#332	—	2	4	no	—
121	#322	—	2	1	no	—
122	C₈²	—	2	1	no	—
123	C₅₃	#9801	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²×Q₈	—	2	2	no	—
125	C₂×M₁₆	—	2	2	no	—
126	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
127	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
134	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
135	#107	—	2	4	no	—
136	#107	—	2	4	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
140	C₂²⋊C₈	—	2	1	no	—
141	#97	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
145	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
146	C₄×C₈	D₄×C₅₃	1	2	yes	—
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	8	4	no	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
153	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
154	M₁₆	—	2	2	no	—
155	M₁₆	—	4	4	no	—
156	M₁₆	Q₁₆×C₅₃	1	4	yes	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
162	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
168	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
169	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
170	C₄²	#41×C₅₃	1	1	yes	characteristic
171	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
172	Q₈	—	8	4	no	—
173	Q₈	—	8	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	—	8	4	no	—
178	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
179	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
180	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
181	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
182	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
183	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
184	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
185	C₂×C₄	#135×C₅₃	1	2	yes	—
186	C₈	—	2	2	no	—
187	C₈	—	2	4	no	—
188	C₈	—	4	2	no	—
189	C₈	—	8	2	no	—
190	C₈	#101×C₅₃	1	4	yes	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	C₅₃×#390	1	1	yes	characteristic
194	C₂²	C₄×Q₁₆×C₅₃	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	8	4	no	—
198	C₄	C₂×#101×C₅₃	1	2	yes	—
199	C₄	C₅₃×#467	1	2	yes	—
200	C₄	C₅₃×#472	1	2	yes	—
201	C₄	C₅₃×#494	1	1	yes	characteristic
202	C₄	C₅₃×#495	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	C₅₃×#1966	1	1	yes	characteristic
205	C₂	C₅₃×#1969	1	1	yes	characteristic
206	C₂	C₅₃×#1978	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#390
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
87. #9801	C₅₃	direct (N × H)
123. C₅₃	#9801	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
C₅₃×#1997	2
C₅₃×#1567	2
C₅₃×#1549	2
C₈×M₁₆×C₅₃	2
#9801	53

Subgroups — C53×#9801 #27217313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9801 #27217313