Subgroups — C₅₃×#13172 #27220684

Order 13568.

274 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#368	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#367	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1952	C₂	1	2	yes	maximal
5	C₅₃×#1387	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#108×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#428	—	2	2	no	—
10	C₂×#107×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#405	C₂²	1	1	yes	characteristic
12	C₅₃×#368	C₄	1	4	yes	complemented
13	C₅₃×#367	—	2	1	no	—
14	C₅₃×#367	—	2	1	no	—
15	C₅₃×#346	—	2	1	no	—
16	C₄×C₁₆×C₅₃	—	2	1	no	—
17	C₅₃×#343	—	2	2	no	—
18	C₂×M₁₆×C₅₃	—	2	2	no	—
19	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	#117×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
23	M₃₂×C₅₃	—	2	1	no	—
24	M₃₂×C₅₃	Q₈	1	2	yes	—
25	C₂×C₁₆×C₅₃	—	2	1	no	—
26	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
30	#108×C₅₃	—	2	2	no	—
31	#108×C₅₃	C₂×C₄	1	4	yes	—
32	#107×C₅₃	—	2	2	no	—
33	#107×C₅₃	—	2	2	no	—
34	#107×C₅₃	—	4	2	no	—
35	#106×C₅₃	—	4	2	no	—
36	#97×C₅₃	—	2	1	no	—
37	C₄×C₈×C₅₃	—	2	1	no	—
38	C₂²×C₄×C₅₃	—	4	2	no	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	M₁₆×C₅₃	—	2	1	no	—
41	M₁₆×C₅₃	—	2	4	no	—
42	M₁₆×C₅₃	—	4	2	no	—
43	M₁₆×C₅₃	Q₁₆	1	2	yes	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	—	4	2	no	—
46	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
51	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
52	C₄⋊C₄×C₅₃	—	4	2	no	—
53	C₄⋊C₄×C₅₃	—	4	2	no	—
54	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
55	C₂²⋊C₄×C₅₃	—	2	1	no	—
56	C₄²×C₅₃	—	2	1	no	—
57	C₁₆×C₅₃	—	2	2	no	—
58	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
59	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
60	C₂³×C₅₃	#135	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	8	2	no	—
63	C₂×C₄×C₅₃	—	16	2	no	—
64	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
70	C₈×C₅₃	—	2	2	no	—
71	C₈×C₅₃	—	4	4	no	—
72	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
74	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₈×C₅₃	#125	1	1	yes	characteristic
76	#13172	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂²×C₅₃	—	2	2	no	—
78	C₂²×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
79	C₂²×C₅₃	#466	1	1	yes	characteristic
80	C₂²×C₅₃	#499	1	1	yes	characteristic
81	C₄×C₅₃	—	4	2	no	—
82	C₄×C₅₃	—	16	2	no	—
83	C₄×C₅₃	#389	1	1	yes	characteristic
84	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
85	C₄×C₅₃	#467	1	1	yes	characteristic
86	C₄×C₅₃	#500	1	1	yes	characteristic
87	C₂×#368	C₂×C₅₃	1	1	yes	characteristic
88	C₂×#367	C₂×C₅₃	1	1	yes	characteristic
89	#2108	C₂×C₅₃	1	1	yes	characteristic
90	#1952	C₂×C₅₃	1	2	yes	—
91	#1387	C₂×C₅₃	1	2	yes	—
92	C₂×C₅₃	—	2	1	no	—
93	C₂×C₅₃	—	2	1	no	—
94	C₂×C₅₃	#1942	1	1	yes	characteristic
95	C₂×C₅₃	#2223	1	2	yes	—
96	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
97	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
98	C₂×#108	C₂²×C₅₃	1	2	yes	—
99	#428	—	2	2	no	—
100	C₂×#107	C₂²×C₅₃	1	2	yes	—
101	#405	C₂²×C₅₃	1	1	yes	characteristic
102	#368	C₄×C₅₃	1	4	yes	complemented
103	#367	—	2	1	no	—
104	#367	—	2	1	no	—
105	#346	—	2	1	no	—
106	C₄×C₁₆	—	2	1	no	—
107	#343	—	2	2	no	—
108	C₅₃	#13172	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
111	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
112	#117	D₄×C₅₃	1	1	yes	characteristic
113	C₂×C₄⋊C₄	—	2	2	no	—
114	M₃₂	—	2	1	no	—
115	M₃₂	Q₈×C₅₃	1	2	yes	—
116	C₂×C₁₆	—	2	1	no	—
117	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
118	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
119	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
120	C₂×C₁₆	Q₈×C₅₃	1	2	yes	—
121	#108	—	2	2	no	—
122	#108	C₂×C₄×C₅₃	1	4	yes	—
123	#107	—	2	2	no	—
124	#107	—	2	2	no	—
125	#107	—	4	2	no	—
126	#106	—	4	2	no	—
127	#97	—	2	1	no	—
128	C₄×C₈	—	2	1	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
131	M₁₆	—	2	1	no	—
132	M₁₆	—	2	4	no	—
133	M₁₆	—	4	2	no	—
134	M₁₆	Q₁₆×C₅₃	1	2	yes	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
138	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
139	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
140	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
141	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
142	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	D₈×C₅₃	1	2	yes	—
146	C₂²⋊C₄	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	C₄⋊C₄×C₅₃	1	2	yes	—
150	C₁₆	C₄⋊C₄×C₅₃	1	2	yes	—
151	C₂³	#135×C₅₃	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	—	16	2	no	—
155	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
156	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
157	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
158	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
159	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
160	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	4	4	no	—
163	C₈	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
164	C₈	C₄×Q₈×C₅₃	1	1	yes	characteristic
165	C₈	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
166	C₈	#125×C₅₃	1	1	yes	characteristic
167	C₂²	—	2	2	no	—
168	C₂²	C₄×Q₁₆×C₅₃	1	1	yes	characteristic
169	C₂²	C₅₃×#466	1	1	yes	characteristic
170	C₂²	C₅₃×#499	1	1	yes	characteristic
171	C₄	—	4	2	no	—
172	C₄	—	16	2	no	—
173	C₄	C₅₃×#389	1	1	yes	characteristic
174	C₄	C₄×D₈×C₅₃	1	1	yes	characteristic
175	C₄	C₅₃×#467	1	1	yes	characteristic
176	C₄	C₅₃×#500	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	C₅₃×#1942	1	1	yes	characteristic
180	C₂	C₅₃×#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	#466
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂³×C₅₃	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
12. C₅₃×#368	C₄	semidirect
76. #13172	C₅₃	direct (N × H)
102. #368	C₄×C₅₃	semidirect
108. C₅₃	#13172	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×#368	2
C₂×C₅₃×#367	2
C₅₃×#2108	2
C₅₃×#1952	2
C₅₃×#1387	2
#13172	53

Subgroups — C53×#13172 #27220684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13172 #27220684