Subgroups — #27300540

Order 13568.

314 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810452	C₂	1	4	yes	maximal
2	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
3	#20274887	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×#13133153	—	2	1	no	—
7	C₂×#13133153	C₂²	1	2	yes	—
8	#16810647	—	2	1	no	—
9	#16810452	—	2	8	no	—
10	C₄×#11719908	C₄	1	2	yes	—
11	C₄×#11719908	C₄	1	2	yes	—
12	C₂×#108×C₅₃	—	2	1	no	—
13	C₂×#108×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#423	—	2	1	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#11719917	—	2	2	no	—
17	C₂×#11719908	C₂×C₄	1	4	yes	—
18	C₂×#11719908	D₄	1	4	yes	—
19	C₂×#11719908	Q₈	1	4	yes	—
20	#13133153	—	2	4	no	—
21	#13133153	—	4	2	no	—
22	C₄⋊#88322	—	4	2	no	—
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
27	#108×C₅₃	—	2	4	no	—
28	#108×C₅₃	—	4	2	no	—
29	C₄⋊C₈×C₅₃	—	4	2	no	—
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	M₁₆×C₅₃	—	4	2	no	—
37	M₁₆×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	D₈	1	2	yes	—
42	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
43	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
46	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₅₃	D₈	1	2	yes	—
50	#11719917	—	4	2	no	—
51	#11719917	—	4	2	no	—
52	C₂×#88322	—	4	2	no	—
53	#11719908	C₄⋊C₄	1	8	yes	—
54	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	4	no	—
56	C₂×C₄×C₅₃	#107	1	2	yes	—
57	C₂×C₄×C₅₃	#107	1	2	yes	—
58	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₈×C₅₃	—	8	2	no	—
65	C₈×C₅₃	#107	1	2	yes	—
66	C₈×C₅₃	#107	1	2	yes	—
67	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
68	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
69	#88322	—	8	2	no	—
70	#12909	—	53	1	no	maximal, Sylow
71	C₂²×C₅₃	—	2	2	no	—
72	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
73	C₂²×C₅₃	#494	1	1	yes	characteristic
74	C₂²×C₅₃	#500	1	1	yes	characteristic
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	#368	1	4	yes	complemented
77	C₄×C₅₃	#384	1	1	yes	characteristic
78	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
79	C₄×C₅₃	#493	1	1	yes	characteristic
80	C₄×C₅₃	#500	1	1	yes	characteristic
81	C₂×#368	—	53	4	no	—
82	C₂×C₄×C₁₆	—	53	1	no	—
83	#1848	—	53	1	no	—
84	#1848	D₅₃	1	1	yes	characteristic
85	C₂×C₅₃	—	2	1	no	—
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	#1845	1	1	yes	characteristic
88	C₂×C₅₃	C₂×#368	1	2	yes	—
89	C₂²×C₁₆	—	53	2	no	—
90	C₂×#108	—	2	1	no	—
91	C₂×#108	—	53	2	no	—
92	C₂×#108	—	106	1	no	—
93	C₂×#108	C₂×D₅₃	1	2	yes	—
94	#423	—	2	1	no	—
95	#423	—	106	1	no	—
96	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
97	#368	—	106	8	no	—
98	C₄×C₁₆	—	53	2	no	—
99	C₄×C₁₆	—	53	2	no	—
100	C₅₃	#12909	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	2	2	no	—
102	C₂×M₁₆	—	106	2	no	—
103	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
104	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
105	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
106	C₂×C₁₆	—	53	4	no	—
107	C₂×C₁₆	—	53	4	no	—
108	C₂×C₁₆	—	53	4	no	—
109	#108	—	2	4	no	—
110	#108	—	4	2	no	—
111	#108	—	106	4	no	—
112	#108	—	212	2	no	—
113	C₄⋊C₈	—	4	2	no	—
114	C₄⋊C₈	—	212	2	no	—
115	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
116	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
117	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
118	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
119	C₂²×C₄	#11719925	1	2	yes	—
120	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
121	M₁₆	—	4	2	no	—
122	M₁₆	—	4	2	no	—
123	M₁₆	—	212	2	no	—
124	M₁₆	—	212	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	212	2	no	—
127	C₂×C₈	#11719919	1	1	yes	characteristic
128	C₂×C₈	#11719919	1	1	yes	characteristic
129	C₂×C₈	#11719922	1	2	yes	—
130	C₂×C₈	#11719925	1	2	yes	—
131	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
133	C₂×C₈	D₄×D₅₃	1	2	yes	—
134	C₂×C₈	Q₈×D₅₃	1	2	yes	—
135	C₄²	#11719919	1	1	yes	characteristic
136	C₄²	#11719919	1	1	yes	characteristic
137	C₄²	#11719922	1	2	yes	—
138	C₁₆	—	53	8	no	—
139	C₂³	C₂×#11719925	1	1	yes	characteristic
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	#13133150	1	2	yes	—
142	C₂×C₄	#13133150	1	2	yes	—
143	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
144	C₂×C₄	#13133213	1	1	yes	characteristic
145	C₂×C₄	#13133215	1	1	yes	characteristic
146	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
147	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
148	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
149	C₈	—	8	2	no	—
150	C₈	—	424	2	no	—
151	C₈	#13133150	1	2	yes	—
152	C₈	#13133150	1	2	yes	—
153	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
154	C₈	#13133216	1	2	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	C₂×#13133150	1	1	yes	characteristic
157	C₂²	#16811002	1	1	yes	characteristic
158	C₂²	#16811026	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	#16810452	1	4	yes	complemented
161	C₄	#16810492	1	1	yes	characteristic
162	C₄	C₂×#13133150	1	1	yes	characteristic
163	C₄	#16810999	1	1	yes	characteristic
164	C₄	#16811026	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#20274874	1	1	yes	characteristic
168	C₂	C₂×#16810452	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1848	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1848	D₅₃
hypocenter	C₅₃	#12909
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
76. C₄×C₅₃	#368	semidirect
100. C₅₃	#12909	semidirect
160. C₄	#16810452	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810452	2
C₂×C₄×#11719908	2
#20274887	2
C₅₃×#1848	2
#12909	53