Subgroups — #27539584

Order 13568.

490 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285229	C₂	1	2	yes	maximal
2	#20276997	C₂	1	2	yes	maximal, complemented
3	#20276994	C₂	1	4	yes	maximal, complemented
4	#20276990	C₂	1	2	yes	maximal, complemented
5	#20276499	C₂	1	4	yes	maximal
6	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	#16811180	—	2	2	no	—
8	#16811028	C₂²	1	4	yes	—
9	#16811025	—	2	2	no	—
10	#16811025	C₂²	1	4	yes	—
11	#16811021	—	2	2	no	—
12	#16810661	—	2	2	no	—
13	#16810661	C₂²	1	4	yes	—
14	#16810451	C₂²	1	8	yes	—
15	#16810449	C₂²	1	8	yes	—
16	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
19	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	#13133217	—	2	4	no	—
22	#13133209	—	2	4	no	—
23	#13133196	—	4	4	no	—
24	C₄×Dic₁₀₆	—	4	4	no	—
25	#13133176	—	2	4	no	—
26	#13133151	—	2	4	no	—
27	#13133151	C₂³	1	8	yes	—
28	#13133149	—	2	4	no	—
29	#13133149	—	2	4	no	—
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₅₃	Q₈	1	8	yes	complemented
34	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
35	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
36	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	D₈	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₁₀₆	—	4	4	no	—
43	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
44	C₂×C₈×C₅₃	C₂×D₄	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	4	yes	—
48	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
49	C₄²×C₅₃	D₈	1	2	yes	—
50	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²⋊Dic₅₃	—	4	4	no	—
53	C₄⋊Dic₅₃	—	2	8	no	—
54	C₄⋊Dic₅₃	—	4	4	no	—
55	#11719919	—	4	8	no	—
56	C₄×Dic₅₃	—	4	4	no	—
57	C₁₆×C₅₃	C₂×Q₈	1	4	yes	—
58	C₁₆×C₅₃	C₂×Q₈	1	4	yes	—
59	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	4	no	—
61	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
63	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
64	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
66	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
67	C₂×Dic₅₃	—	4	8	no	—
68	Dic₁₀₆	—	8	8	no	—
69	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
70	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
71	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
72	C₈×C₅₃	C₂²×Q₈	1	2	yes	—
73	#33726	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	—	2	2	no	—
75	C₂²×C₅₃	#500	1	2	yes	—
76	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	#500	1	2	yes	—
79	C₄×C₅₃	#508	1	4	yes	—
80	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
81	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
82	Dic₅₃	—	8	8	no	—
83	#3160	—	53	2	no	—
84	#2254	—	53	2	no	—
85	#2253	—	53	4	no	—
86	#2252	—	53	2	no	—
87	#2158	—	53	4	no	—
88	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
89	C₂×C₅₃	—	2	2	no	—
90	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
91	C₂×C₅₃	C₂×#508	1	2	yes	—
92	#533	—	106	2	no	—
93	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
94	#500	—	53	4	no	—
95	#499	—	53	4	no	—
96	#499	—	106	2	no	—
97	#498	—	106	2	no	—
98	#427	—	53	4	no	—
99	#427	—	106	2	no	—
100	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
101	#367	—	53	8	no	—
102	#366	—	53	8	no	—
103	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
104	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
105	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
106	C₅₃	#33726	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
109	C₄⋊Q₈	—	106	4	no	—
110	#125	—	106	4	no	—
111	C₂²⋊Q₈	—	212	4	no	—
112	C₄×Q₈	—	212	4	no	—
113	#117	—	106	4	no	—
114	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
115	C₂×C₁₆	Dic₁₀₆	1	8	yes	complemented
116	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
117	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
118	#107	—	53	8	no	—
119	#107	—	106	4	no	—
120	#106	—	106	4	no	—
121	#106	—	106	4	no	—
122	C₄×C₈	D₂₁₂	1	1	yes	characteristic
123	C₄×C₈	D₂₁₂	1	1	yes	characteristic
124	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
125	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
126	C₂×Q₈	—	212	4	no	—
127	C₂²×C₄	D₄₂₄	1	2	yes	—
128	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
129	C₂×C₈	Dic₂₁₂	1	4	yes	—
130	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
131	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
132	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
133	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
134	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
135	C₄⋊C₄	—	106	8	no	—
136	C₄⋊C₄	—	212	8	no	—
137	C₄⋊C₄	—	212	4	no	—
138	C₂²⋊C₄	—	212	4	no	—
139	C₄²	—	212	4	no	—
140	C₄²	D₄₂₄	1	2	yes	—
141	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
142	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
143	C₁₆	C₂×Dic₁₀₆	1	4	yes	—
144	C₁₆	C₂×Dic₁₀₆	1	4	yes	—
145	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
146	Q₈	—	424	8	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	212	8	no	—
149	C₂×C₄	#13133217	1	2	yes	—
150	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
151	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
152	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
153	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
154	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
155	C₈	#13133217	1	2	yes	—
156	C₈	C₂×Dic₂₁₂	1	2	yes	—
157	C₈	C₂×Dic₂₁₂	1	2	yes	—
158	C₈	C₂²×Dic₁₀₆	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	#16811028	1	2	yes	—
161	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	424	8	no	—
164	C₄	#16811028	1	2	yes	—
165	C₄	#16811055	1	4	yes	—
166	C₄	C₂×#13133217	1	1	yes	characteristic
167	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
168	C₂	—	2	2	no	—
169	C₂	C₂×#16811028	1	1	yes	characteristic
170	C₂	C₂×#16811055	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#33726
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20276997	C₂	semidirect
3. #20276994	C₂	semidirect
4. #20276990	C₂	semidirect
33. C₂×C₁₆×C₅₃	Q₈	semidirect
106. C₅₃	#33726	semidirect
115. C₂×C₁₆	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20285229	2
#20276997	2
#20276994	2
#20276990	2
#20276499	2
C₂×C₄×C₁₆×C₅₃	2
#33726	53