Subgroups — #27300505

Order 13568.

370 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810441	C₂	1	4	yes	maximal
3	#20274401	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×#13133128	C₄	1	1	yes	characteristic
7	C₂×#13133127	C₄	1	2	yes	—
8	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
9	#16810469	C₄	1	1	yes	characteristic
10	C₄⋊#11719908	—	2	2	no	—
11	#16810441	—	2	8	no	—
12	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×C₄⋊C₁₆	—	2	2	no	—
15	C₂²×#88322	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
17	C₂×#11719908	—	2	4	no	—
18	C₂×#11719908	—	2	2	no	—
19	#13133128	—	2	2	no	—
20	#13133127	—	2	4	no	—
21	C₈×Dic₅₃	D₄	1	4	yes	—
22	C₈×Dic₅₃	Q₈	1	4	yes	—
23	#13133124	C₈	1	2	yes	—
24	#13133124	C₈	1	2	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	—	2	2	no	—
29	C₂×C₁₆×C₅₃	—	2	4	no	—
30	C₄×C₈×C₅₃	D₄	1	2	yes	—
31	C₄×C₈×C₅₃	Q₈	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×Dic₅₃	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
39	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
42	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₄²×C₅₃	M₁₆	1	2	yes	—
44	C₄×Dic₅₃	C₄⋊C₄	1	4	yes	—
45	C₄×Dic₅₃	M₁₆	1	4	yes	—
46	C₂×#88322	—	2	2	no	—
47	C₂×#88322	—	2	2	no	—
48	C₂×#88322	C₄⋊C₄	1	4	yes	—
49	C₁₆×C₅₃	—	4	2	no	—
50	#11719908	—	4	2	no	—
51	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
60	C₂×Dic₅₃	—	2	4	no	—
61	C₂×Dic₅₃	—	2	4	no	—
62	C₂×Dic₅₃	C₄⋊C₈	1	8	yes	—
63	C₈×C₅₃	—	2	2	no	—
64	C₈×C₅₃	#127	1	1	yes	characteristic
65	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
66	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
67	#88322	—	2	4	no	—
68	#12899	—	53	1	no	maximal, Sylow
69	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
70	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₂²×C₅₃	#504	1	2	yes	—
72	C₂²×C₅₃	#504	1	2	yes	—
73	C₄×C₅₃	—	2	2	no	—
74	C₄×C₅₃	—	2	2	no	—
75	C₄×C₅₃	#384	1	1	yes	characteristic
76	C₄×C₅₃	#423	1	2	yes	—
77	C₄×C₅₃	#423	1	1	yes	characteristic
78	Dic₅₃	—	2	8	no	—
79	C₂×C₄⋊C₁₆	—	53	4	no	—
80	C₂×C₄⋊C₁₆	—	53	1	no	—
81	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
82	#1726	—	53	1	no	—
83	C₂×C₅₃	#1842	1	1	yes	characteristic
84	C₂×C₅₃	#2151	1	4	yes	—
85	C₂×C₅₃	#2151	1	1	yes	characteristic
86	C₂×C₅₃	#2151	1	1	yes	characteristic
87	C₂²×C₁₆	—	53	2	no	—
88	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
89	C₂×#97	—	53	2	no	—
90	C₂×#97	—	53	1	no	—
91	C₂×C₄×C₈	—	53	2	no	—
92	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
93	#376	—	53	1	no	—
94	C₄⋊C₁₆	—	2	2	no	—
95	C₄⋊C₁₆	—	106	8	no	—
96	C₄⋊C₁₆	—	106	2	no	—
97	C₅₃	#12899	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₈	—	53	2	no	—
99	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
100	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
101	C₂×C₄²	—	53	2	no	—
102	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
103	C₂×C₁₆	—	2	4	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	106	4	no	—
106	C₂×C₁₆	—	106	2	no	—
107	#97	—	106	4	no	—
108	#97	—	106	2	no	—
109	C₄×C₈	—	53	4	no	—
110	C₄×C₈	—	53	4	no	—
111	C₄×C₈	Dic₁₀₆	1	2	yes	—
112	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
113	#95	—	53	2	no	—
114	#95	—	53	2	no	—
115	C₂²×C₄	—	53	4	no	—
116	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
117	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
118	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	53	4	no	—
121	C₂×C₈	—	106	2	no	—
122	C₂×C₈	—	106	2	no	—
123	C₂×C₈	#11719919	1	2	yes	—
124	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
125	C₂×C₈	D₄×D₅₃	1	2	yes	—
126	C₂×C₈	Q₈×D₅₃	1	2	yes	—
127	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
128	C₄²	—	53	4	no	—
129	C₄²	—	53	4	no	—
130	C₄²	#11719912	1	2	yes	—
131	C₄²	#11719919	1	2	yes	—
132	C₁₆	—	4	2	no	—
133	C₁₆	—	212	2	no	—
134	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	53	8	no	—
138	C₂×C₄	—	106	4	no	—
139	C₂×C₄	—	106	4	no	—
140	C₂×C₄	#13133146	1	2	yes	—
141	C₂×C₄	#13133146	1	2	yes	—
142	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
143	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
144	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
145	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	106	4	no	—
148	C₈	#13133213	1	1	yes	characteristic
149	C₈	#13133215	1	1	yes	characteristic
150	C₈	#13133216	1	2	yes	—
151	C₂²	C₄⋊C₈×D₅₃	1	2	yes	—
152	C₂²	C₂×#13133146	1	1	yes	characteristic
153	C₂²	#16811039	1	2	yes	—
154	C₂²	#16811041	1	2	yes	—
155	C₄	—	2	2	no	—
156	C₄	—	2	2	no	—
157	C₄	—	106	8	no	—
158	C₄	#16810492	1	1	yes	characteristic
159	C₄	#16810645	1	2	yes	—
160	C₄	#16810646	1	1	yes	characteristic
161	C₂	#20274866	1	1	yes	characteristic
162	C₂	#20276472	1	4	yes	—
163	C₂	#20276473	1	1	yes	characteristic
164	C₂	#20276475	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂²×C₈
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#12899
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
97. C₅₃	#12899	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719908	2
C₂×#16810441	2
#20274401	2
C₂×C₅₃×C₄⋊C₁₆	2
#12899	53