Subgroups — #27299486

Order 13568.

310 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	C₂×#16810453	C₂	1	1	yes	maximal, characteristic
2	#20276499	C₂	1	1	yes	maximal, characteristic
3	#20274594	C₂	1	2	yes	maximal
4	#20272811	C₂	1	2	yes	maximal
5	C₂×C₅₃×#346	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	#16810453	C₄	1	4	yes	complemented
10	#16810451	C₄	1	2	yes	complemented
11	#16810449	C₄	1	2	yes	complemented
12	#16810383	—	2	2	no	—
13	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#346	C₄	1	4	yes	complemented
17	C₂×#11719917	—	2	2	no	—
18	#13133176	—	2	2	no	—
19	#13133153	—	2	2	no	—
20	#13133153	C₂×C₄	1	4	yes	—
21	#13133151	C₂×C₄	1	4	yes	—
22	#13133149	—	2	2	no	—
23	#13133127	—	4	2	no	—
24	C₈×Dic₅₃	—	4	2	no	—
25	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	—	2	1	no	—
29	C₂×C₁₆×C₅₃	—	2	1	no	—
30	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₅₃	Q₈	1	2	yes	—
37	C₂²×C₄×C₅₃	—	2	1	no	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₅₃	D₈	1	2	yes	—
41	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
47	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
48	C₄²×C₅₃	—	2	1	no	—
49	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
50	C₂²⋊Dic₅₃	—	4	2	no	—
51	C₄⋊Dic₅₃	—	2	4	no	—
52	C₄×Dic₅₃	—	4	2	no	—
53	#11719917	—	2	4	no	—
54	#11719917	—	4	2	no	—
55	C₂×#88322	—	4	2	no	—
56	C₁₆×C₅₃	—	2	2	no	—
57	C₁₆×C₅₃	C₄²	1	2	yes	—
58	C₁₆×C₅₃	C₄²	1	2	yes	—
59	C₂³×C₅₃	#135	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	4	2	no	—
62	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
68	C₂×Dic₅₃	—	4	4	no	—
69	C₈×C₅₃	#107	1	4	yes	—
70	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
71	C₈×C₅₃	#117	1	1	yes	characteristic
72	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
73	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
74	#88322	—	4	4	no	—
75	#12671	—	53	1	no	maximal, Sylow
76	C₂²×C₅₃	—	2	2	no	—
77	C₂²×C₅₃	#427	1	1	yes	characteristic
78	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
79	C₂²×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
80	C₄×C₅₃	—	4	2	no	—
81	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
83	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
84	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
85	Dic₅₃	—	8	4	no	—
86	C₂×#368	—	53	1	no	—
87	#2158	—	53	1	no	—
88	C₂×#346	D₅₃	1	1	yes	characteristic
89	#1772	—	53	2	no	—
90	#1382	—	53	2	no	—
91	C₂×C₅₃	—	2	1	no	—
92	C₂×C₅₃	—	2	1	no	—
93	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
94	C₂×C₅₃	#2179	1	2	yes	—
95	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
96	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
97	C₂×#108	—	53	2	no	—
98	#427	—	53	2	no	—
99	#405	—	106	2	no	—
100	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
101	#368	—	53	4	no	—
102	#367	—	53	2	no	—
103	#366	—	53	2	no	—
104	#346	Dic₅₃	1	4	yes	complemented
105	#339	—	106	2	no	—
106	C₅₃	#12671	1	1	yes	characteristic, complemented, Sylow
107	C₂×M₁₆	—	106	2	no	—
108	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
109	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
110	#117	—	106	2	no	—
111	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
112	C₂×C₁₆	—	2	1	no	—
113	C₂×C₁₆	—	2	1	no	—
114	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
115	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
116	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
117	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
118	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
119	#108	—	53	4	no	—
120	#108	—	106	2	no	—
121	#107	—	53	4	no	—
122	#106	—	106	2	no	—
123	#97	—	212	2	no	—
124	C₄×C₈	—	212	2	no	—
125	C₄×C₈	Dic₁₀₆	1	2	yes	—
126	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
129	M₁₆	—	106	4	no	—
130	M₁₆	—	212	2	no	—
131	C₂×C₈	—	212	2	no	—
132	C₂×C₈	D₄₂₄	1	2	yes	—
133	C₂×C₈	Dic₂₁₂	1	2	yes	—
134	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
135	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
137	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
138	C₂×C₈	#11719945	1	1	yes	characteristic
139	C₂×C₈	#11719945	1	1	yes	characteristic
140	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
141	C₄⋊C₄	—	106	4	no	—
142	C₂²⋊C₄	—	212	2	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	212	2	no	—
145	C₄²	C₄⋊Dic₅₃	1	2	yes	—
146	C₁₆	—	2	2	no	—
147	C₁₆	C₄×Dic₅₃	1	2	yes	—
148	C₁₆	C₄×Dic₅₃	1	2	yes	—
149	C₂³	#13133239	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	212	4	no	—
153	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
154	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
155	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
156	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
157	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
158	C₂×C₄	#13133239	1	1	yes	characteristic
159	C₈	—	212	4	no	—
160	C₈	#13133151	1	4	yes	—
161	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
162	C₈	#13133176	1	1	yes	characteristic
163	C₈	C₄×Dic₁₀₆	1	1	yes	characteristic
164	C₈	C₄×Dic₁₀₆	1	1	yes	characteristic
165	C₂²	—	2	2	no	—
166	C₂²	#16810661	1	1	yes	characteristic
167	C₂²	C₄×D₄₂₄	1	1	yes	characteristic
168	C₂²	C₄×Dic₂₁₂	1	1	yes	characteristic
169	C₄	—	4	2	no	—
170	C₄	—	424	4	no	—
171	C₄	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
172	C₄	C₂×#13133151	1	1	yes	characteristic
173	C₄	C₄×D₄₂₄	1	1	yes	characteristic
174	C₄	C₄×Dic₂₁₂	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	C₄×#13133151	1	1	yes	characteristic
178	C₂	#20276602	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle	C₂²×C₅₃	C₄×D₈
prime radical	C₂³×C₅₃	#135
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#12671
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
9. #16810453	C₄	semidirect
10. #16810451	C₄	semidirect
11. #16810449	C₄	semidirect
16. C₅₃×#346	C₄	semidirect
104. #346	Dic₅₃	semidirect
106. C₅₃	#12671	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810453	2
#20276499	2
#20274594	2
#20272811	2
C₂×C₅₃×#346	2
#12671	53