Subgroups — #27537754

Order 13568.

460 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276624	C₂	1	4	yes	maximal, complemented
2	Q₈×#11719908	C₂	1	4	yes	maximal, complemented
3	#20276522	C₂	1	4	yes	maximal
4	#20276514	C₂	1	2	yes	maximal
5	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11719908	C₂²	1	2	yes	—
7	C₄⋊#11719908	C₂²	1	2	yes	—
8	C₄⋊#11719908	C₂²	1	8	yes	—
9	C₂²⋊#11719908	C₂²	1	4	yes	—
10	#16810399	C₂²	1	4	yes	—
11	C₄×#11719908	C₂²	1	4	yes	—
12	C₄×#11719908	C₂²	1	4	yes	—
13	C₅₃×#533	C₄	1	1	yes	characteristic
14	C₅₃×#446	C₄	1	4	yes	—
15	C₈×Q₈×C₅₃	C₂²	1	4	yes	—
16	C₅₃×#424	C₄	1	2	yes	—
17	C₅₃×#423	C₄	1	1	yes	characteristic
18	C₅₃×#405	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719908	C₂³	1	4	yes	—
21	C₂×#11719908	C₂³	1	4	yes	—
22	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₄⋊Q₈×C₅₃	C₈	1	2	yes	—
26	#125×C₅₃	C₈	1	2	yes	—
27	C₂²⋊Q₈×C₅₃	C₈	1	4	yes	—
28	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
29	#117×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
32	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
33	#97×C₅₃	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₃	C₂³	1	2	yes	—
36	C₄×C₈×C₅₃	C₂³	1	4	yes	—
37	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
38	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	M₁₆×C₅₃	—	4	2	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	4	1	no	—
43	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
46	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
47	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
48	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
49	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
50	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
51	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
52	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
54	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
55	#11719908	—	2	4	no	—
56	#11719908	—	2	4	no	—
57	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
58	Q₈×C₅₃	—	2	8	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	4	no	—
62	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
63	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
65	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
66	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
67	C₈×C₅₃	—	2	1	no	—
68	C₈×C₅₃	—	2	2	no	—
69	C₈×C₅₃	—	2	4	no	—
70	C₈×C₅₃	2₋¹⁺⁴	1	2	yes	—
71	#33500	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	—	2	2	no	—
73	C₂²×C₅₃	—	2	1	no	—
74	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	—	2	8	no	—
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	#504	1	4	yes	—
79	C₄×C₅₃	#519	1	1	yes	characteristic
80	C₄×C₅₃	#568	1	1	yes	characteristic
81	#2960	D₅₃	1	1	yes	characteristic
82	#2183	—	53	4	no	—
83	Q₈×C₁₆	—	53	4	no	—
84	#2164	—	53	4	no	—
85	#2162	—	53	2	no	—
86	C₂×C₅₃	—	4	1	no	—
87	C₂×C₅₃	#2982	1	1	yes	characteristic
88	C₂×C₅₃	C₂×#504	1	2	yes	—
89	#533	Dic₅₃	1	1	yes	characteristic
90	#446	Dic₅₃	1	4	yes	—
91	C₈×Q₈	C₂×D₅₃	1	4	yes	—
92	#424	Dic₅₃	1	2	yes	—
93	#423	Dic₅₃	1	1	yes	characteristic
94	#405	C₂×D₅₃	1	2	yes	—
95	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
96	C₄⋊C₁₆	—	53	2	no	—
97	C₄⋊C₁₆	—	53	2	no	—
98	C₄⋊C₁₆	—	53	8	no	—
99	C₂²⋊C₁₆	—	53	4	no	—
100	#346	—	53	4	no	—
101	C₄×C₁₆	—	53	4	no	—
102	C₄×C₁₆	—	53	4	no	—
103	C₅₃	#33500	1	1	yes	characteristic, complemented, Sylow
104	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
105	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
106	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
107	C₄⋊Q₈	#88322	1	2	yes	—
108	#125	#88322	1	2	yes	—
109	C₂²⋊Q₈	#88322	1	4	yes	—
110	C₄×Q₈	C₂×Dic₅₃	1	4	yes	—
111	#117	C₂×Dic₅₃	1	2	yes	—
112	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
113	C₂×C₁₆	—	53	4	no	—
114	C₂×C₁₆	—	53	4	no	—
115	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
116	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
117	#97	C₂×Dic₅₃	1	4	yes	—
118	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
119	C₄×C₈	C₂²×D₅₃	1	2	yes	—
120	C₄×C₈	C₂²×D₅₃	1	4	yes	—
121	C₂×Q₈	C₂×#88322	1	4	yes	—
122	C₂²×C₄	C₂×#88322	1	2	yes	—
123	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
128	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
129	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
130	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
131	C₄⋊C₄	C₂×#88322	1	4	yes	—
132	C₄⋊C₄	C₂×#88322	1	8	yes	—
133	C₄⋊C₄	C₂×#88322	1	4	yes	—
134	C₂²⋊C₄	C₂×#88322	1	4	yes	—
135	C₄²	C₂×#88322	1	1	yes	characteristic
136	C₄²	C₂×#88322	1	1	yes	characteristic
137	C₄²	C₂²×Dic₅₃	1	2	yes	—
138	C₄²	C₂²×Dic₅₃	1	4	yes	—
139	C₁₆	—	106	4	no	—
140	C₁₆	—	106	4	no	—
141	C₂³	C₂²×#88322	1	1	yes	characteristic
142	Q₈	—	2	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	C₂²×#88322	1	2	yes	—
147	C₂×C₄	C₂²×#88322	1	8	yes	—
148	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
149	C₂×C₄	C₂²×#88322	1	2	yes	—
150	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	2	2	no	—
153	C₈	—	2	4	no	—
154	C₈	#13133265	1	2	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	1	no	—
157	C₂²	C₂³×#88322	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	8	no	—
160	C₄	—	2	1	no	—
161	C₄	#16811040	1	4	yes	—
162	C₄	#16811101	1	1	yes	characteristic
163	C₄	#16811390	1	1	yes	characteristic
164	C₂	—	4	1	no	—
165	C₂	#20283796	1	1	yes	characteristic
166	C₂	C₂×#16811040	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2960	C₂
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#33500
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #20276624	C₂	semidirect
2. Q₈×#11719908	C₂	semidirect
103. C₅₃	#33500	semidirect

Maximal subgroups

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

subgroup	index
#20276624	2
Q₈×#11719908	2
#20276522	2
#20276514	2
C₅₃×#2960	2
#33500	53