Subgroups — #27293076

Order 13568.

330 conjugacy classes of subgroups.

The subgroup lattice has 144 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	#20272998	C₂	1	1	yes	maximal, characteristic
3	#20272631	C₂	1	2	yes	maximal, complemented
4	C₁₆×#88322	C₂	1	2	yes	maximal, complemented
5	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₄	1	2	yes	—
7	#16810676	C₄	1	2	yes	—
8	C₄×#11719908	C₄	1	2	yes	—
9	C₄×#11719908	C₂²	1	1	yes	characteristic
10	C₄×#11719908	C₂²	1	1	yes	characteristic
11	#16810344	C₄	1	2	yes	—
12	C₈×#88322	C₂²	1	2	yes	—
13	C₂×M₃₂×C₅₃	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#346	C₄	1	2	yes	—
16	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂×#11719908	C₈	1	8	yes	complemented
18	C₂×#11719908	C₂×C₄	1	2	yes	—
19	C₂×#11719908	C₂×C₄	1	2	yes	—
20	C₄⋊#88322	C₈	1	4	yes	—
21	C₂²⋊#88322	C₈	1	4	yes	—
22	C₄×#88322	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₅₃	C₈	1	8	yes	complemented
27	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
37	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
38	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₅₃	C₄²	1	1	yes	characteristic
41	C₄²×C₅₃	C₂×C₈	1	2	yes	—
42	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×#88322	C₂×C₈	1	8	yes	—
44	C₁₆×C₅₃	C₂×C₈	1	8	yes	—
45	#11719908	C₂×C₈	1	4	yes	—
46	#11719908	C₂×C₈	1	4	yes	—
47	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	—	2	4	no	—
49	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
50	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
53	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
55	C₈×C₅₃	C₄×C₈	1	2	yes	—
56	C₈×C₅₃	C₄×C₈	1	2	yes	—
57	C₈×C₅₃	C₂²×C₈	1	2	yes	—
58	C₈×C₅₃	C₂²×C₈	1	2	yes	—
59	#88322	—	2	8	no	—
60	#11399	—	53	1	no	maximal, Sylow
61	C₂²×C₅₃	—	2	2	no	—
62	C₂²×C₅₃	C₈²	1	2	yes	—
63	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
64	C₄×C₅₃	—	2	2	no	—
65	C₄×C₅₃	C₈²	1	2	yes	—
66	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
67	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
68	C₄×C₅₃	#504	1	4	yes	—
69	C₄×M₃₂	D₅₃	1	1	yes	characteristic
70	C₂×C₄×C₁₆	—	53	1	no	—
71	#1450	—	53	1	no	—
72	#1311	—	53	2	no	—
73	C₈×C₁₆	—	53	2	no	—
74	C₂×C₅₃	—	2	2	no	—
75	C₂×C₅₃	C₂×C₈²	1	1	yes	characteristic
76	C₂×C₅₃	#2108	1	2	yes	—
77	C₂×M₃₂	Dic₅₃	1	2	yes	—
78	C₂²×C₁₆	—	53	2	no	—
79	#431	—	53	2	no	—
80	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
81	#346	Dic₅₃	1	2	yes	—
82	C₄×C₁₆	—	53	2	no	—
83	C₄×C₁₆	—	53	1	no	—
84	C₄×C₁₆	—	53	1	no	—
85	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
86	#322	—	53	2	no	—
87	C₈²	—	53	2	no	—
88	C₅₃	#11399	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
90	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
91	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
92	M₃₂	#88322	1	8	yes	complemented
93	C₂×C₁₆	—	53	2	no	—
94	C₂×C₁₆	—	53	2	no	—
95	C₂×C₁₆	—	53	8	no	—
96	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
97	C₄⋊C₈	—	53	4	no	—
98	C₂²⋊C₈	—	53	4	no	—
99	C₄×C₈	—	53	4	no	—
100	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
101	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
102	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
103	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
104	C₂²×C₄	C₈×D₅₃	1	2	yes	—
105	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
106	C₂×C₈	—	53	8	no	—
107	C₂×C₈	C₈×D₅₃	1	4	yes	—
108	C₂×C₈	C₂×#88322	1	4	yes	—
109	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
110	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
111	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
112	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
113	C₄²	C₈×D₅₃	1	2	yes	—
114	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
115	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
116	C₁₆	—	53	4	no	—
117	C₁₆	—	53	4	no	—
118	C₁₆	C₂×#88322	1	8	yes	—
119	C₂³	C₈×Dic₅₃	1	1	yes	characteristic
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
122	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
123	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
124	C₂×C₄	C₄×#88322	1	2	yes	—
125	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
126	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
127	C₈	—	106	8	no	—
128	C₈	C₈×Dic₅₃	1	2	yes	—
129	C₈	C₄×#88322	1	2	yes	—
130	C₈	C₂×C₈×D₅₃	1	2	yes	—
131	C₈	C₂²×#88322	1	2	yes	—
132	C₂²	—	2	2	no	—
133	C₂²	C₈×#88322	1	2	yes	—
134	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	C₈×#88322	1	2	yes	—
137	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
138	C₄	C₂×C₄×#88322	1	1	yes	characteristic
139	C₄	#16811041	1	4	yes	—
140	C₂	—	2	2	no	—
141	C₂	C₂×C₈×#88322	1	1	yes	characteristic
142	C₂	#20276314	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₅₃
commutator	C₂×C₅₃	C₂×C₈²
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#11399
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20272631	C₂	semidirect
4. C₁₆×#88322	C₂	semidirect
17. C₂×#11719908	C₈	semidirect
26. M₃₂×C₅₃	C₈	semidirect
88. C₅₃	#11399	semidirect
92. M₃₂	#88322	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
#20272998	2
#20272631	2
C₁₆×#88322	2
C₄×M₃₂×C₅₃	2
#11399	53