Subgroups — #27300549

Order 13568.

266 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276499	C₂	1	1	yes	maximal, characteristic
2	#20276489	C₂	1	2	yes	maximal
3	#20276311	C₂	1	1	yes	maximal, characteristic
4	#20274840	C₂	1	2	yes	maximal
5	C₂×C₅₃×#368	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	1	yes	characteristic
7	#16810661	C₂²	1	2	yes	—
8	#16810578	C₂²	1	1	yes	characteristic
9	#16810451	—	2	1	no	—
10	#16810449	—	2	1	no	—
11	#16810447	—	2	4	no	—
12	#16810398	C₄	1	2	yes	complemented
13	C₁₆×Dic₅₃	C₄	1	2	yes	complemented
14	#16810395	—	2	2	no	—
15	#16810380	—	2	2	no	—
16	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#108×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#368	—	2	1	no	—
19	C₅₃×#368	—	2	1	no	—
20	C₂×#11719917	D₄	1	1	yes	characteristic
21	#13133176	—	2	2	no	—
22	#13133176	D₄	1	1	yes	characteristic
23	#13133157	D₄	1	2	yes	—
24	#13133157	Q₈	1	2	yes	—
25	C₂×#11719908	C₂×C₄	1	2	yes	—
26	#13133151	—	2	2	no	—
27	#13133149	—	2	2	no	—
28	#13133127	Q₈	1	2	yes	—
29	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
37	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
38	#108×C₅₃	—	2	2	no	—
39	#108×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	M₁₆×C₅₃	—	4	2	no	—
42	M₁₆×C₅₃	—	4	2	no	—
43	C₂×C₈×C₅₃	—	4	2	no	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×Q₈	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	C₂²⋊Dic₅₃	—	2	1	no	—
51	C₂²⋊Dic₅₃	—	4	2	no	—
52	C₄⋊Dic₅₃	—	2	1	no	—
53	C₄⋊Dic₅₃	—	4	2	no	—
54	C₄×Dic₅₃	—	4	2	no	—
55	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
56	#11719917	—	2	1	no	—
57	#11719917	—	2	1	no	—
58	C₂×#88322	C₄⋊C₄	1	2	yes	—
59	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
61	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
62	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
63	#11719908	C₄⋊C₄	1	4	yes	—
64	C₂³×C₅₃	#135	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
71	C₂×Dic₅₃	—	2	2	no	—
72	C₂×Dic₅₃	—	8	2	no	—
73	C₈×C₅₃	—	8	2	no	—
74	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
78	#88322	—	2	2	no	—
79	#12911	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	—	4	1	no	—
81	C₂²×C₅₃	#427	1	1	yes	characteristic
82	C₂²×C₅₃	#495	1	1	yes	characteristic
83	C₂²×C₅₃	#499	1	1	yes	characteristic
84	C₄×C₅₃	#384	1	1	yes	characteristic
85	C₄×C₅₃	#427	1	1	yes	characteristic
86	C₄×C₅₃	#495	1	1	yes	characteristic
87	C₄×C₅₃	#499	1	1	yes	characteristic
88	Dic₅₃	—	4	2	no	—
89	Dic₅₃	—	16	2	no	—
90	C₂×#368	D₅₃	1	1	yes	characteristic
91	#2158	—	53	1	no	—
92	#2155	—	53	2	no	—
93	#2108	—	53	1	no	—
94	#1833	—	53	2	no	—
95	C₂×C₅₃	—	2	1	no	—
96	C₂×C₅₃	—	2	1	no	—
97	C₂×C₅₃	—	2	1	no	—
98	C₂×C₅₃	#1846	1	1	yes	characteristic
99	C₂×M₃₂	—	53	1	no	—
100	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
101	C₂×#108	C₂×D₅₃	1	2	yes	—
102	#427	—	53	2	no	—
103	#405	—	53	1	no	—
104	#368	—	2	1	no	—
105	#368	—	2	1	no	—
106	#367	—	106	1	no	—
107	#366	—	106	1	no	—
108	#365	—	106	4	no	—
109	#346	—	53	2	no	—
110	C₄×C₁₆	—	53	2	no	—
111	#344	—	106	2	no	—
112	H(3,ℤ/4)	—	106	2	no	—
113	C₅₃	#12911	1	1	yes	characteristic, complemented, Sylow
114	C₂×M₁₆	—	2	2	no	—
115	C₂×M₁₆	—	53	1	no	—
116	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
117	#117	—	53	1	no	—
118	#117	—	106	2	no	—
119	M₃₂	—	53	2	no	—
120	M₃₂	—	53	2	no	—
121	C₂×C₁₆	—	53	2	no	—
122	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
123	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
124	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
125	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
126	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
127	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
128	#108	—	2	2	no	—
129	#108	—	2	2	no	—
130	#107	—	106	2	no	—
131	#106	—	106	2	no	—
132	#97	—	53	2	no	—
133	C₄×C₈	—	53	2	no	—
134	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
135	M₁₆	—	4	2	no	—
136	M₁₆	—	4	2	no	—
137	M₁₆	—	106	1	no	—
138	M₁₆	—	106	1	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	53	2	no	—
141	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
142	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
143	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
145	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
146	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
147	C₄⋊C₄	—	106	1	no	—
148	C₄⋊C₄	—	212	2	no	—
149	C₂²⋊C₄	—	106	1	no	—
150	C₂²⋊C₄	—	212	2	no	—
151	C₄²	—	53	2	no	—
152	C₄²	—	212	2	no	—
153	C₁₆	—	53	4	no	—
154	C₁₆	#11719919	1	1	yes	characteristic
155	C₁₆	#11719919	1	1	yes	characteristic
156	C₁₆	#11719919	1	1	yes	characteristic
157	C₁₆	#11719919	1	1	yes	characteristic
158	C₂³	#13133240	1	1	yes	characteristic
159	C₂×C₄	—	106	2	no	—
160	C₂×C₄	—	424	2	no	—
161	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
162	C₂×C₄	#13133213	1	1	yes	characteristic
163	C₂×C₄	#13133216	1	1	yes	characteristic
164	C₂×C₄	#13133236	1	1	yes	characteristic
165	C₂×C₄	#13133236	1	1	yes	characteristic
166	C₂×C₄	#13133240	1	1	yes	characteristic
167	C₈	—	8	2	no	—
168	C₈	—	106	2	no	—
169	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
170	C₈	C₂×#11719919	1	1	yes	characteristic
171	C₈	#13133215	1	1	yes	characteristic
172	C₈	#13133216	1	1	yes	characteristic
173	C₂²	—	4	1	no	—
174	C₂²	#16810660	1	1	yes	characteristic
175	C₂²	#16811008	1	1	yes	characteristic
176	C₂²	#16811024	1	1	yes	characteristic
177	C₄	—	848	2	no	—
178	C₄	—	212	2	no	—
179	C₄	#16810492	1	1	yes	characteristic
180	C₄	#16810660	1	1	yes	characteristic
181	C₄	#16811006	1	1	yes	characteristic
182	C₄	#16811024	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#20274878	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16810660
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#368	C₂
socle	C₂×C₅₃	#1846
prime radical	C₂³×C₅₃	#135
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#368	D₅₃
hypocenter	C₅₃	#12911
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
12. #16810398	C₄	semidirect
13. C₁₆×Dic₅₃	C₄	semidirect
113. C₅₃	#12911	semidirect

Maximal subgroups

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

subgroup	index
#20276499	2
#20276489	2
#20276311	2
#20274840	2
C₂×C₅₃×#368	2
#12911	53