Subgroups — #27537136

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283315	C₂	1	2	yes	maximal
2	#20276503	C₂	1	6	yes	maximal, complemented
3	#20276499	C₂	1	3	yes	maximal, complemented
4	C₂×#16810449	C₂	1	3	yes	maximal, complemented
5	C₂×C₅₃×#504	C₂	1	1	yes	maximal, characteristic
6	#16811106	—	2	2	no	—
7	#16810665	—	2	6	no	—
8	#16810661	C₂²	1	6	yes	—
9	C₂×#13133151	C₂²	1	6	yes	—
10	#16810451	C₂²	1	6	yes	—
11	#16810449	C₂²	1	3	yes	—
12	#16810449	C₂²	1	6	yes	—
13	#16810449	C₂²	1	1	yes	characteristic
14	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#504	C₄	1	8	yes	complemented
16	C₂×M₃₂×C₅₃	C₂²	1	3	yes	—
17	C₂²×C₁₆×C₅₃	C₂²	1	3	yes	—
18	Q₈×Dic₅₃	—	4	2	no	—
19	D₄×Dic₅₃	—	4	6	no	—
20	#13133176	—	2	6	no	—
21	C₂×C₄⋊Dic₅₃	—	2	6	no	—
22	#13133151	—	2	6	no	—
23	#13133151	C₂³	1	6	yes	—
24	#13133151	C₂³	1	2	yes	—
25	#13133149	—	2	6	no	—
26	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
27	#131×C₅₃	C₂×C₄	1	4	yes	—
28	#131×C₅₃	Q₈	1	4	yes	—
29	C₂×M₁₆×C₅₃	D₄	1	3	yes	—
30	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
31	M₃₂×C₅₃	C₂×C₄	1	12	yes	complemented
32	C₂×C₁₆×C₅₃	C₂×C₄	1	12	yes	complemented
33	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₁₆×C₅₃	C₂³	1	3	yes	—
35	#41×C₅₃	C₄⋊C₄	1	4	yes	—
36	#41×C₅₃	Q₁₆	1	4	yes	—
37	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
38	C₂×D₄×C₅₃	D₈	1	3	yes	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
40	C₂²×Dic₅₃	—	4	6	no	—
41	M₁₆×C₅₃	C₄⋊C₄	1	12	yes	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
43	C₂×C₈×C₅₃	C₂×D₄	1	3	yes	—
44	C₂×C₈×C₅₃	C₂×Q₈	1	6	yes	—
45	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₂²⋊Dic₅₃	—	4	6	no	—
47	C₄⋊Dic₅₃	—	2	2	no	—
48	C₄⋊Dic₅₃	—	2	6	no	—
49	C₄⋊Dic₅₃	—	4	6	no	—
50	C₄×Dic₅₃	—	4	6	no	—
51	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
52	C₁₆×C₅₃	C₂²×C₄	1	6	yes	—
53	C₂³×C₅₃	C₂×D₈	1	3	yes	—
54	Q₈×C₅₃	#107	1	4	yes	—
55	D₄×C₅₃	#107	1	12	yes	—
56	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
57	C₂×C₄×C₅₃	C₂×D₈	1	3	yes	—
58	C₂×C₄×C₅₃	C₂×Q₁₆	1	6	yes	—
59	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
60	C₂×Dic₅₃	—	4	6	no	—
61	C₂×Dic₅₃	—	4	2	no	—
62	C₂×Dic₅₃	—	16	6	no	—
63	C₈×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
64	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
65	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
66	#33410	—	53	1	no	maximal, Sylow
67	C₂²×C₅₃	—	2	3	no	—
68	C₂²×C₅₃	—	2	3	no	—
69	C₂²×C₅₃	C₂×#107	1	6	yes	—
70	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
71	C₄×C₅₃	C₂×#107	1	6	yes	—
72	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
73	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
74	Dic₅₃	—	8	6	no	—
75	Dic₅₃	—	8	2	no	—
76	C₂×#504	D₅₃	1	1	yes	characteristic
77	#2913	—	53	2	no	—
78	#2159	—	53	6	no	—
79	#2158	—	53	3	no	—
80	C₂×#366	—	53	3	no	—
81	C₂×C₅₃	—	2	6	no	—
82	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
83	C₂×C₅₃	#3415	1	1	yes	characteristic
84	C₂×C₅₃	#3417	1	1	yes	characteristic
85	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
86	#520	—	106	2	no	—
87	#504	Dic₅₃	1	8	yes	complemented
88	C₂×M₃₂	C₂×D₅₃	1	3	yes	—
89	C₂²×C₁₆	C₂×D₅₃	1	3	yes	—
90	#428	—	106	6	no	—
91	#427	—	53	6	no	—
92	C₂×#107	—	53	6	no	—
93	#367	—	53	6	no	—
94	#366	—	53	3	no	—
95	#366	—	53	6	no	—
96	#366	—	53	1	no	—
97	C₅₃	#33410	1	1	yes	characteristic, complemented, Sylow
98	C₂×#41	D₂₁₂	1	1	yes	characteristic
99	#131	Dic₁₀₆	1	4	yes	—
100	#131	C₂×Dic₅₃	1	4	yes	—
101	C₂×M₁₆	D₂₁₂	1	3	yes	—
102	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
103	C₄×Q₈	—	212	2	no	—
104	C₄×D₄	—	212	6	no	—
105	#117	—	106	6	no	—
106	C₂×C₄⋊C₄	—	106	6	no	—
107	M₃₂	C₂×Dic₅₃	1	12	yes	complemented
108	C₂×C₁₆	C₂×Dic₅₃	1	12	yes	complemented
109	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
110	C₂×C₁₆	C₂²×D₅₃	1	3	yes	—
111	#107	—	53	6	no	—
112	#107	—	53	2	no	—
113	#107	—	106	6	no	—
114	#106	—	106	6	no	—
115	#41	Dic₂₁₂	1	4	yes	—
116	#41	C₄⋊Dic₅₃	1	4	yes	—
117	C₂×Q₈	D₄₂₄	1	1	yes	characteristic
118	C₂×D₄	D₄₂₄	1	3	yes	—
119	C₂²×C₄	—	212	6	no	—
120	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
121	M₁₆	C₄⋊Dic₅₃	1	12	yes	—
122	C₂×C₈	C₂×Dic₁₀₆	1	6	yes	—
123	C₂×C₈	C₂×D₂₁₂	1	3	yes	—
124	C₂×C₈	C₂²×Dic₅₃	1	6	yes	—
125	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
126	C₄⋊C₄	—	106	2	no	—
127	C₄⋊C₄	—	106	6	no	—
128	C₄⋊C₄	—	212	6	no	—
129	C₂²⋊C₄	—	212	6	no	—
130	C₄²	—	212	6	no	—
131	C₁₆	C₂²×Dic₅₃	1	2	yes	—
132	C₁₆	C₂²×Dic₅₃	1	6	yes	—
133	C₂³	C₂×D₄₂₄	1	3	yes	—
134	Q₈	#13133151	1	4	yes	—
135	D₄	#13133151	1	12	yes	—
136	C₂×C₄	—	848	6	no	—
137	C₂×C₄	—	212	2	no	—
138	C₂×C₄	—	212	6	no	—
139	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
140	C₂×C₄	C₂×D₄₂₄	1	3	yes	—
141	C₂×C₄	C₂×Dic₂₁₂	1	6	yes	—
142	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
143	C₈	C₂×C₄⋊Dic₅₃	1	6	yes	—
144	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
145	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
146	C₂²	—	2	3	no	—
147	C₂²	—	2	3	no	—
148	C₂²	C₂×#13133151	1	6	yes	—
149	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
150	C₄	—	424	6	no	—
151	C₄	—	424	2	no	—
152	C₄	C₂×#13133151	1	6	yes	—
153	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
154	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
155	C₂	—	2	6	no	—
156	C₂	C₂²×#13133151	1	1	yes	characteristic
157	C₂	#20287856	1	1	yes	characteristic
158	C₂	#20287868	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₂₄
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#504	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#41×C₅₃	D₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#504	D₅₃
hypocenter	C₅₃	#33410
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
2. #20276503	C₂	semidirect
3. #20276499	C₂	semidirect
4. C₂×#16810449	C₂	semidirect
15. C₅₃×#504	C₄	semidirect
31. M₃₂×C₅₃	C₂×C₄	semidirect
32. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
87. #504	Dic₅₃	semidirect
97. C₅₃	#33410	semidirect
107. M₃₂	C₂×Dic₅₃	semidirect
108. C₂×C₁₆	C₂×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20283315	2
#20276503	2
#20276499	2
C₂×#16810449	2
C₂×C₅₃×#504	2
#33410	53