Subgroups — #27299471

Order 13568.

308 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276488	C₂	1	2	yes	maximal
2	#20274594	C₂	1	2	yes	maximal
3	#20272819	C₂	1	1	yes	maximal, characteristic
4	#20272817	C₂	1	1	yes	maximal, characteristic
5	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133153	C₂²	1	2	yes	—
7	#16810661	C₂²	1	2	yes	—
8	#16810578	—	2	2	no	—
9	#16810446	C₄	1	8	yes	complemented
10	#16810383	—	2	2	no	—
11	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#346	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₅₃	C₄	1	2	yes	complemented
16	C₂×#11719917	—	2	2	no	—
17	#13133176	—	2	2	no	—
18	#13133153	—	2	2	no	—
19	#13133153	C₂×C₄	1	4	yes	—
20	#13133151	—	2	2	no	—
21	#13133149	C₂×C₄	1	4	yes	—
22	#13133127	—	4	2	no	—
23	C₈×Dic₅₃	—	4	2	no	—
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
27	M₃₂×C₅₃	—	2	2	no	—
28	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	—	2	1	no	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
40	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
41	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
43	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₅₃	#41	1	2	yes	—
45	C₄²×C₅₃	—	2	1	no	—
46	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²⋊Dic₅₃	—	4	2	no	—
49	C₄⋊Dic₅₃	—	2	4	no	—
50	C₄×Dic₅₃	—	4	2	no	—
51	#11719917	—	2	4	no	—
52	#11719917	—	4	2	no	—
53	C₂×#88322	—	4	2	no	—
54	C₁₆×C₅₃	—	2	2	no	—
55	C₁₆×C₅₃	C₄²	1	4	yes	—
56	C₂³×C₅₃	#135	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	4	2	no	—
60	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
63	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
64	C₂×Dic₅₃	—	4	4	no	—
65	C₈×C₅₃	#106	1	2	yes	—
66	C₈×C₅₃	#106	1	1	yes	characteristic
67	C₈×C₅₃	#106	1	1	yes	characteristic
68	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₅₃	#117	1	1	yes	characteristic
70	C₈×C₅₃	C₄×Q₈	1	2	yes	—
71	#88322	—	4	4	no	—
72	#12668	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	—	4	1	no	—
74	C₂²×C₅₃	#427	1	1	yes	characteristic
75	C₂²×C₅₃	C₄×SD₁₆	1	2	yes	—
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	4	1	no	—
78	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
79	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
80	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
81	Dic₅₃	—	8	4	no	—
82	#2155	—	53	2	no	—
83	C₄×M₃₂	D₅₃	1	1	yes	characteristic
84	#1772	—	53	2	no	—
85	#1385	—	53	1	no	—
86	#1384	—	53	1	no	—
87	C₂×C₅₃	—	2	2	no	—
88	C₂×C₅₃	—	2	1	no	—
89	C₂×C₅₃	C₄×#106	1	1	yes	characteristic
90	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
91	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
92	C₂×#108	—	53	2	no	—
93	#427	—	53	2	no	—
94	#405	—	106	2	no	—
95	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
96	#365	—	53	8	no	—
97	#346	Dic₅₃	1	2	yes	complemented
98	C₄×C₁₆	Dic₅₃	1	2	yes	complemented
99	#339	—	106	2	no	—
100	C₅₃	#12668	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	106	2	no	—
102	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
104	#117	—	106	2	no	—
105	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	C₄×D₅₃	1	4	yes	—
108	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
109	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
110	#108	—	53	4	no	—
111	#108	—	106	2	no	—
112	#107	—	106	2	no	—
113	#106	—	53	4	no	—
114	#97	—	212	2	no	—
115	C₄×C₈	—	212	2	no	—
116	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
117	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
118	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
119	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
122	M₁₆	—	106	4	no	—
123	M₁₆	—	212	2	no	—
124	C₂×C₈	—	212	2	no	—
125	C₂×C₈	#11719913	1	2	yes	—
126	C₂×C₈	#11719913	1	2	yes	—
127	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
128	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
129	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
131	C₂×C₈	#11719945	1	2	yes	—
132	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
133	C₄⋊C₄	—	106	4	no	—
134	C₂²⋊C₄	—	212	2	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	212	2	no	—
137	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
138	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
139	C₁₆	—	2	2	no	—
140	C₁₆	C₄×Dic₅₃	1	4	yes	—
141	C₂³	#13133239	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	212	4	no	—
146	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
147	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
148	C₂×C₄	C₂×#11719913	1	2	yes	—
149	C₂×C₄	#13133239	1	1	yes	characteristic
150	C₈	—	212	4	no	—
151	C₈	#13133149	1	1	yes	characteristic
152	C₈	#13133149	1	2	yes	—
153	C₈	#13133149	1	1	yes	characteristic
154	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
155	C₈	#13133176	1	1	yes	characteristic
156	C₈	C₄×Dic₁₀₆	1	2	yes	—
157	C₂²	—	4	1	no	—
158	C₂²	#16810661	1	1	yes	characteristic
159	C₂²	C₄×#11719913	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	1	no	—
162	C₄	—	424	4	no	—
163	C₄	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
164	C₄	C₂×#13133149	1	1	yes	characteristic
165	C₄	C₄×#11719913	1	2	yes	—
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	C₄×#13133149	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133149
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂×C₅₃	C₄×#106
prime radical	C₂³×C₅₃	#135
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#12668
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
9. #16810446	C₄	semidirect
14. C₅₃×#346	C₄	semidirect
15. C₄×C₁₆×C₅₃	C₄	semidirect
97. #346	Dic₅₃	semidirect
98. C₄×C₁₆	Dic₅₃	semidirect
100. C₅₃	#12668	semidirect

Maximal subgroups

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

subgroup	index
#20276488	2
#20274594	2
#20272819	2
#20272817	2
C₄×M₃₂×C₅₃	2
#12668	53