Subgroups — #27274517

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274461	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
3	#20272542	C₂	1	4	yes	maximal
4	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#88322	C₄	1	4	yes	—
6	C₂×C₄×#88322	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	2	yes	—
8	#16810378	C₄	1	4	yes	—
9	C₂×#97×C₅₃	C₄	1	2	yes	—
10	C₂×#97×C₅₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
12	C₄³×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×#88322	C₂×C₄	1	4	yes	—
14	C₂²×#88322	C₂×C₄	1	4	yes	—
15	C₄×#88322	—	2	4	no	—
16	C₄×#88322	—	2	4	no	—
17	C₄×#88322	—	2	8	no	—
18	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	#97×C₅₃	—	2	4	no	—
25	#97×C₅₃	C₈	1	8	yes	complemented
26	C₄×C₈×C₅₃	—	2	4	no	—
27	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
28	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
29	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	—	2	4	no	—
33	C₂×C₈×C₅₃	—	2	4	no	—
34	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	—
35	C₄²×C₅₃	—	2	4	no	—
36	C₄²×C₅₃	—	2	4	no	—
37	C₄²×C₅₃	C₂×C₈	1	4	yes	—
38	C₄²×C₅₃	M₁₆	1	4	yes	—
39	C₄²×C₅₃	#41	1	4	yes	—
40	C₂×#88322	—	2	4	no	—
41	C₂×#88322	—	2	4	no	—
42	C₂×#88322	—	2	4	no	—
43	C₂×#88322	—	2	8	no	—
44	C₂×#88322	M₁₆	1	8	yes	—
45	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
51	C₂×C₄×C₅₃	#97	1	4	yes	—
52	C₂×C₄×C₅₃	#117	1	2	yes	—
53	C₂×C₄×C₅₃	#117	1	2	yes	—
54	C₂×C₄×C₅₃	#117	1	2	yes	—
55	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
57	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
58	C₂×C₄×C₅₃	#131	1	4	yes	—
59	C₈×C₅₃	—	2	8	no	—
60	#88322	—	2	8	no	—
61	#88322	—	4	4	no	—
62	#8132	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	#376	1	1	yes	characteristic
64	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
65	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
66	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
67	C₂²×C₅₃	#405	1	2	yes	—
68	C₄×C₅₃	—	2	4	no	—
69	C₄×C₅₃	—	2	4	no	—
70	C₄×C₅₃	#431	1	4	yes	—
71	C₄×C₅₃	#432	1	4	yes	—
72	C₄×C₅₃	#433	1	4	yes	—
73	#1744	—	53	1	no	—
74	C₄×#97	D₅₃	1	1	yes	characteristic
75	C₄²×C₈	—	53	1	no	—
76	#1273	—	53	4	no	—
77	C₂×C₅₃	C₈×M₁₆	1	2	yes	—
78	C₂×C₅₃	#1451	1	2	yes	—
79	C₂×C₅₃	#1744	1	1	yes	characteristic
80	C₂×C₅₃	#1745	1	1	yes	characteristic
81	C₂×C₅₃	#1746	1	1	yes	characteristic
82	C₂×#97	Dic₅₃	1	2	yes	—
83	C₂×#97	Dic₅₃	1	2	yes	—
84	C₂×C₄×C₈	—	53	4	no	—
85	C₂×C₄×C₈	—	53	2	no	—
86	C₂×C₄×C₈	—	53	2	no	—
87	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
88	C₄³	C₂×D₅₃	1	1	yes	characteristic
89	#336	—	53	4	no	—
90	C₅₃	#8132	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₈	—	53	4	no	—
92	C₂²×C₈	—	53	4	no	—
93	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
94	C₂×C₄²	C₄×D₅₃	1	2	yes	—
95	C₂×C₄²	C₄×D₅₃	1	2	yes	—
96	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
97	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
98	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
99	#97	—	2	4	no	—
100	#97	#88322	1	8	yes	complemented
101	C₄×C₈	—	2	4	no	—
102	C₄×C₈	—	106	8	no	—
103	C₄×C₈	—	106	4	no	—
104	C₄×C₈	—	106	4	no	—
105	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
106	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
107	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
108	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
109	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	53	8	no	—
113	C₂×C₈	—	106	4	no	—
114	C₂×C₈	—	106	4	no	—
115	C₂×C₈	—	106	4	no	—
116	C₂×C₈	—	106	8	no	—
117	C₂×C₈	C₂×#88322	1	8	yes	—
118	C₄²	—	2	4	no	—
119	C₄²	—	2	4	no	—
120	C₄²	C₂×#88322	1	4	yes	—
121	C₄²	#11719917	1	4	yes	—
122	C₄²	#11719945	1	4	yes	—
123	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₄×#88322	1	4	yes	—
129	C₂×C₄	#13133128	1	4	yes	—
130	C₂×C₄	#13133174	1	2	yes	—
131	C₂×C₄	#13133174	1	2	yes	—
132	C₂×C₄	#13133176	1	2	yes	—
133	C₂×C₄	C₂²×#88322	1	2	yes	—
134	C₂×C₄	M₁₆×D₅₃	1	4	yes	—
135	C₂×C₄	C₂×#11719917	1	2	yes	—
136	C₂×C₄	#13133225	1	4	yes	—
137	C₈	—	2	8	no	—
138	C₈	—	106	8	no	—
139	C₈	—	212	4	no	—
140	C₂²	#16810469	1	1	yes	characteristic
141	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
142	C₂²	C₂×#13133128	1	1	yes	characteristic
143	C₂²	M₁₆×Dic₅₃	1	2	yes	—
144	C₂²	#16810581	1	2	yes	—
145	C₄	—	2	4	no	—
146	C₄	—	2	4	no	—
147	C₄	#16810676	1	4	yes	—
148	C₄	#16810678	1	4	yes	—
149	C₄	#16810685	1	4	yes	—
150	C₂	M₁₆×#88322	1	2	yes	—
151	C₂	#20273002	1	2	yes	—
152	C₂	#20274461	1	1	yes	characteristic
153	C₂	#20274463	1	1	yes	characteristic
154	C₂	#20274468	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₄×#97×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#97	D₅₃
hypocenter	C₅₃	#8132
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
25. #97×C₅₃	C₈	semidirect
90. C₅₃	#8132	semidirect
100. #97	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20274461	2
C₄²×#88322	2
#20272542	2
C₄×#97×C₅₃	2
#8132	53