Subgroups — #27300541

Order 13568.

314 conjugacy classes of subgroups.

The subgroup lattice has 146 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	4	yes	maximal
2	#20274887	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133153	—	2	2	no	—
5	C₂×#13133153	C₂²	1	4	yes	—
6	#16810647	—	2	2	no	—
7	#16810453	—	2	8	no	—
8	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
11	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
12	C₂×#11719917	—	2	4	no	—
13	#13133153	—	2	8	no	—
14	#13133153	—	4	4	no	—
15	C₄⋊#88322	—	4	4	no	—
16	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
20	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
21	C₂×C₁₆×C₅₃	Q₈	1	4	yes	—
22	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
27	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
28	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	D₈	1	2	yes	—
31	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
32	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
34	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
35	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₅₃	D₈	1	2	yes	—
39	#11719917	—	4	4	no	—
40	#11719917	—	4	4	no	—
41	C₂×#88322	—	4	4	no	—
42	C₁₆×C₅₃	C₄⋊C₄	1	8	yes	—
43	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	2	4	no	—
45	C₂×C₄×C₅₃	#107	1	2	yes	—
46	C₂×C₄×C₅₃	#107	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
53	C₈×C₅₃	#107	1	2	yes	—
54	C₈×C₅₃	#107	1	2	yes	—
55	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
56	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
57	#88322	—	8	4	no	—
58	#12909	—	53	1	no	maximal, Sylow
59	C₂²×C₅₃	—	2	2	no	—
60	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
61	C₂²×C₅₃	#494	1	1	yes	characteristic
62	C₂²×C₅₃	#500	1	1	yes	characteristic
63	C₄×C₅₃	—	2	2	no	—
64	C₄×C₅₃	#368	1	4	yes	complemented
65	C₄×C₅₃	#384	1	1	yes	characteristic
66	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
67	C₄×C₅₃	#493	1	1	yes	characteristic
68	C₄×C₅₃	#500	1	1	yes	characteristic
69	C₂×#368	—	53	4	no	—
70	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
71	#1848	—	53	2	no	—
72	C₂×C₅₃	—	2	1	no	—
73	C₂×C₅₃	—	2	1	no	—
74	C₂×C₅₃	#1845	1	1	yes	characteristic
75	C₂×C₅₃	C₂×#368	1	2	yes	—
76	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
77	C₂×#108	—	53	4	no	—
78	C₂×#108	—	106	2	no	—
79	#423	—	106	2	no	—
80	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
81	#368	—	106	8	no	—
82	C₄×C₁₆	Dic₅₃	1	2	yes	—
83	C₄×C₁₆	Dic₅₃	1	2	yes	—
84	C₅₃	#12909	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₁₆	—	106	4	no	—
86	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
87	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
88	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
89	C₂×C₁₆	Dic₁₀₆	1	4	yes	—
90	C₂×C₁₆	D₂₁₂	1	4	yes	—
91	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
92	#108	—	106	8	no	—
93	#108	—	212	4	no	—
94	C₄⋊C₈	—	212	4	no	—
95	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
96	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
97	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
98	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
99	C₂²×C₄	Dic₂₁₂	1	2	yes	—
100	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
101	M₁₆	—	212	4	no	—
102	M₁₆	—	212	4	no	—
103	C₂×C₈	—	212	4	no	—
104	C₂×C₈	D₄₂₄	1	2	yes	—
105	C₂×C₈	Dic₂₁₂	1	2	yes	—
106	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
107	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
108	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
109	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
110	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
111	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
112	C₄²	D₄₂₄	1	2	yes	—
113	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
114	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
115	C₁₆	C₄⋊Dic₅₃	1	8	yes	—
116	C₂³	C₂×Dic₂₁₂	1	1	yes	characteristic
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	#13133151	1	2	yes	—
119	C₂×C₄	#13133151	1	2	yes	—
120	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
121	C₂×C₄	#13133214	1	1	yes	characteristic
122	C₂×C₄	#13133217	1	1	yes	characteristic
123	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
124	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
125	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
126	C₈	—	424	4	no	—
127	C₈	#13133151	1	2	yes	—
128	C₈	#13133151	1	2	yes	—
129	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
130	C₈	#13133217	1	2	yes	—
131	C₂²	—	2	2	no	—
132	C₂²	C₂×#13133151	1	1	yes	characteristic
133	C₂²	#16811003	1	1	yes	characteristic
134	C₂²	#16811028	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	#16810453	1	4	yes	complemented
137	C₄	#16810493	1	1	yes	characteristic
138	C₄	C₂×#13133151	1	1	yes	characteristic
139	C₄	#16811000	1	1	yes	characteristic
140	C₄	#16811028	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	#20274875	1	1	yes	characteristic
144	C₂	C₂×#16810453	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#12909
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
64. C₄×C₅₃	#368	semidirect
84. C₅₃	#12909	semidirect
136. C₄	#16810453	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
#20274887	2
C₂×C₄×C₁₆×C₅₃	2
#12909	53