Subgroups — C₅₃×#12025 #27219537

Order 13568.

224 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1372	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1371	C₂	1	2	yes	maximal, complemented
6	C₂×M₃₂×C₅₃	C₄	1	2	yes	—
7	C₅₃×#432	—	2	1	no	—
8	C₅₃×#423	—	2	1	no	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#346	C₄	1	1	yes	characteristic
11	C₅₃×#346	C₄	1	1	yes	characteristic
12	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#335	—	2	1	no	—
14	C₅₃×#335	C₂²	1	2	yes	—
15	C₅₃×#335	C₂²	1	1	yes	characteristic
16	C₅₃×#335	C₂²	1	1	yes	characteristic
17	C₅₃×#334	—	2	1	no	—
18	C₂×M₁₆×C₅₃	—	4	1	no	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
22	M₃₂×C₅₃	—	2	4	no	—
23	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
25	C₄⋊C₈×C₅₃	—	2	1	no	—
26	C₄⋊C₈×C₅₃	—	2	1	no	—
27	C₄⋊C₈×C₅₃	—	2	2	no	—
28	C₄⋊C₈×C₅₃	—	4	1	no	—
29	C₂²⋊C₈×C₅₃	—	4	1	no	—
30	#97×C₅₃	—	4	1	no	—
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
38	M₁₆×C₅₃	—	8	2	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	—	4	1	no	—
42	C₂×C₈×C₅₃	—	4	1	no	—
43	C₂×C₈×C₅₃	—	4	2	no	—
44	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₅₃	D₈	1	1	yes	characteristic
50	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
51	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₁₆×C₅₃	—	2	2	no	—
53	C₁₆×C₅₃	M₁₆	1	4	yes	complemented
54	C₂³×C₅₃	#107	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	4	1	no	—
60	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₈×C₅₃	—	2	1	no	—
67	C₈×C₅₃	—	2	1	no	—
68	C₈×C₅₃	—	8	1	no	—
69	C₈×C₅₃	—	8	2	no	—
70	C₈×C₅₃	—	8	1	no	—
71	C₈×C₅₃	#108	1	2	yes	—
72	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
73	#12025	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂²×C₅₃	—	2	1	no	—
75	C₂²×C₅₃	—	2	1	no	—
76	C₂²×C₅₃	—	2	1	no	—
77	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
78	C₄×C₅₃	—	2	1	no	—
79	C₄×C₅₃	—	2	1	no	—
80	C₄×C₅₃	—	2	1	no	—
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	#423	1	1	yes	characteristic
83	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
84	C₄×C₅₃	#509	1	1	yes	characteristic
85	C₄×C₅₃	#510	1	1	yes	characteristic
86	C₄×M₃₂	C₂×C₅₃	1	1	yes	characteristic
87	#1568	C₂×C₅₃	1	1	yes	characteristic
88	#1567	C₂×C₅₃	1	1	yes	characteristic
89	#1372	C₂×C₅₃	1	2	yes	complemented
90	#1371	C₂×C₅₃	1	2	yes	complemented
91	C₂×C₅₃	—	4	1	no	—
92	C₂×C₅₃	#1567	1	1	yes	characteristic
93	C₂×C₅₃	#2159	1	1	yes	characteristic
94	C₂×C₅₃	#2161	1	1	yes	characteristic
95	C₂×M₃₂	C₄×C₅₃	1	2	yes	—
96	#432	—	2	1	no	—
97	#423	—	2	1	no	—
98	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
99	#346	C₄×C₅₃	1	1	yes	characteristic
100	#346	C₄×C₅₃	1	1	yes	characteristic
101	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
102	#335	—	2	1	no	—
103	#335	C₂²×C₅₃	1	2	yes	—
104	#335	C₂²×C₅₃	1	1	yes	characteristic
105	#335	C₂²×C₅₃	1	1	yes	characteristic
106	#334	—	2	1	no	—
107	C₅₃	#12025	1	1	yes	characteristic, direct product factor, Sylow
108	C₂×M₁₆	—	4	1	no	—
109	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
110	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
111	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
112	M₃₂	—	2	4	no	—
113	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
114	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
115	C₄⋊C₈	—	2	1	no	—
116	C₄⋊C₈	—	2	1	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	4	1	no	—
119	C₂²⋊C₈	—	4	1	no	—
120	#97	—	4	1	no	—
121	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
122	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
123	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
124	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
125	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
126	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
127	C₂²×C₄	Q₁₆×C₅₃	1	1	yes	characteristic
128	M₁₆	—	8	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
135	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
136	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
137	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
138	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
139	C₄²	D₈×C₅₃	1	1	yes	characteristic
140	C₄²	Q₁₆×C₅₃	1	1	yes	characteristic
141	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
142	C₁₆	—	2	2	no	—
143	C₁₆	M₁₆×C₅₃	1	4	yes	complemented
144	C₂³	#107×C₅₃	1	1	yes	characteristic
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
151	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
152	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
154	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
155	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	8	1	no	—
159	C₈	—	8	2	no	—
160	C₈	—	8	1	no	—
161	C₈	#108×C₅₃	1	2	yes	—
162	C₈	C₂×M₁₆×C₅₃	1	2	yes	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₂×#107×C₅₃	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	C₅₃×#423	1	1	yes	characteristic
172	C₄	C₂×#108×C₅₃	1	1	yes	characteristic
173	C₄	C₅₃×#509	1	1	yes	characteristic
174	C₄	C₅₃×#510	1	1	yes	characteristic
175	C₂	—	4	1	no	—
176	C₂	C₅₃×#1567	1	1	yes	characteristic
177	C₂	C₅₃×#2159	1	1	yes	characteristic
178	C₂	C₅₃×#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1372	C₂	semidirect
5. C₅₃×#1371	C₂	semidirect
53. C₁₆×C₅₃	M₁₆	semidirect
73. #12025	C₅₃	direct (N × H)
89. #1372	C₂×C₅₃	semidirect
90. #1371	C₂×C₅₃	semidirect
107. C₅₃	#12025	direct (N × H)
143. C₁₆	M₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₃₂×C₅₃	2
C₅₃×#1568	2
C₅₃×#1567	2
C₅₃×#1372	2
C₅₃×#1371	2
#12025	53

Subgroups — C53×#12025 #27219537

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12025 #27219537