Subgroups — #27537094

Order 13568.

684 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287818	C₂	1	1	yes	maximal, characteristic
2	#131×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20276480	C₂	1	6	yes	maximal, complemented
4	C₂×#16810443	C₂	1	6	yes	maximal, complemented
5	C₅₃×#3407	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133224	C₄	1	1	yes	characteristic
7	#41×Dic₅₃	C₄	1	1	yes	characteristic
8	#16811040	—	2	4	no	—
9	C₂×#13133157	C₂²	1	6	yes	—
10	#16810581	C₄	1	3	yes	—
11	#16810578	C₂²	1	3	yes	—
12	M₁₆×Dic₅₃	C₄	1	3	yes	—
13	C₂×C₈×Dic₅₃	C₂²	1	3	yes	—
14	#16810443	C₂²	1	12	yes	—
15	#16810443	C₂²	1	4	yes	—
16	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#504	—	2	4	no	—
18	C₂×M₃₂×C₅₃	C₂²	1	6	yes	—
19	#13133224	C₈	1	8	yes	—
20	C₂×#11719917	C₂×C₄	1	3	yes	—
21	C₂²×#88322	C₂×C₄	1	3	yes	—
22	Q₈×Dic₅₃	—	2	1	no	—
23	D₄×Dic₅₃	—	2	3	no	—
24	#13133176	C₂×C₄	1	3	yes	—
25	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
26	#13133157	—	2	6	no	—
27	#13133157	C₂³	1	4	yes	—
28	C₂×#11719908	—	2	6	no	—
29	#13133127	C₂×C₄	1	6	yes	—
30	C₈×Dic₅₃	C₂×C₄	1	3	yes	—
31	C₈×Dic₅₃	C₂×C₄	1	3	yes	—
32	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
33	C₈×Dic₅₃	C₂³	1	3	yes	—
34	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
35	#131×C₅₃	D₄	1	4	yes	—
36	#131×C₅₃	Q₈	1	4	yes	—
37	C₂×M₁₆×C₅₃	C₂×C₄	1	3	yes	—
38	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
39	M₃₂×C₅₃	—	2	6	no	—
40	M₃₂×C₅₃	C₂³	1	4	yes	—
41	C₂×C₁₆×C₅₃	—	2	6	no	—
42	#41×C₅₃	C₄⋊C₄	1	4	yes	—
43	#41×C₅₃	C₂×C₈	1	4	yes	—
44	C₂×Q₈×C₅₃	M₁₆	1	1	yes	characteristic
45	C₂×D₄×C₅₃	M₁₆	1	3	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
47	C₂²×Dic₅₃	—	2	3	no	—
48	M₁₆×C₅₃	C₄⋊C₄	1	12	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₅₃	C₂×D₄	1	6	yes	—
51	C₂×C₈×C₅₃	C₂×Q₈	1	6	yes	—
52	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
53	C₂²⋊Dic₅₃	—	2	3	no	—
54	C₄⋊Dic₅₃	—	2	3	no	—
55	C₄×Dic₅₃	—	2	3	no	—
56	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
58	#11719917	C₂×C₈	1	12	yes	—
59	C₂×#88322	C₂×C₈	1	12	yes	—
60	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
61	C₂×#88322	C₂²×C₄	1	3	yes	—
62	C₁₆×C₅₃	—	2	4	no	—
63	#11719908	—	2	4	no	—
64	C₂³×C₅₃	C₂×M₁₆	1	3	yes	—
65	Q₈×C₅₃	C₄⋊C₈	1	4	yes	—
66	D₄×C₅₃	C₄⋊C₈	1	12	yes	—
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
68	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
69	C₂×C₄×C₅₃	C₂×M₁₆	1	3	yes	—
70	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
71	C₂×Dic₅₃	—	2	1	no	—
72	C₂×Dic₅₃	—	2	3	no	—
73	C₂×Dic₅₃	—	4	6	no	—
74	C₈×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
75	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
76	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
77	#88322	C₂²×C₈	1	2	yes	—
78	#88322	C₂²×C₈	1	6	yes	—
79	#33401	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	—	4	3	no	—
81	C₂²×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
82	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
83	C₄×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
84	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
85	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
86	Dic₅₃	—	2	2	no	—
87	Dic₅₃	—	4	3	no	—
88	#3407	—	53	1	no	—
89	#3407	D₅₃	1	1	yes	characteristic
90	C₄×#131	—	53	1	no	—
91	#2153	—	53	6	no	—
92	C₂×#364	—	53	6	no	—
93	C₂×C₅₃	—	2	1	no	—
94	C₂×C₅₃	—	2	6	no	—
95	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
96	C₂×#131	—	53	1	no	—
97	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
98	C₄×#41	—	53	1	no	—
99	#504	—	2	4	no	—
100	#504	—	106	4	no	—
101	C₂×M₃₂	—	53	6	no	—
102	C₂×M₃₂	C₂×D₅₃	1	6	yes	—
103	#405	—	53	3	no	—
104	#405	—	53	3	no	—
105	C₄×M₁₆	—	53	3	no	—
106	C₂×C₄×C₈	—	53	3	no	—
107	#364	—	53	4	no	—
108	#364	—	53	12	no	—
109	C₅₃	#33401	1	1	yes	characteristic, complemented, Sylow
110	C₂×#41	C₄×D₅₃	1	1	yes	characteristic
111	#131	—	53	8	no	—
112	#131	Dic₁₀₆	1	4	yes	—
113	#131	C₂²⋊D₅₃	1	4	yes	—
114	C₂×M₁₆	—	53	3	no	—
115	C₂×M₁₆	C₄×D₅₃	1	3	yes	—
116	C₂²×C₈	—	53	3	no	—
117	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
118	C₄×Q₈	—	106	1	no	—
119	C₄×D₄	—	106	3	no	—
120	#117	—	53	3	no	—
121	C₂×C₄²	—	53	3	no	—
122	M₃₂	—	2	6	no	—
123	M₃₂	—	53	4	no	—
124	M₃₂	—	106	6	no	—
125	M₃₂	C₂²×D₅₃	1	4	yes	—
126	C₂×C₁₆	—	2	6	no	—
127	C₂×C₁₆	—	106	6	no	—
128	#97	—	53	6	no	—
129	C₄×C₈	—	53	3	no	—
130	C₄×C₈	—	53	3	no	—
131	C₄×C₈	—	53	1	no	—
132	C₄×C₈	—	53	3	no	—
133	#41	C₈×D₅₃	1	4	yes	—
134	#41	#11719919	1	4	yes	—
135	C₂×Q₈	#11719912	1	1	yes	characteristic
136	C₂×D₄	#11719912	1	3	yes	—
137	C₂²×C₄	—	106	3	no	—
138	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
139	M₁₆	—	53	12	no	—
140	M₁₆	#11719919	1	12	yes	—
141	C₂×C₈	—	53	12	no	—
142	C₂×C₈	—	53	1	no	—
143	C₂×C₈	—	53	3	no	—
144	C₂×C₈	C₂×Dic₁₀₆	1	6	yes	—
145	C₂×C₈	C₂×C₄×D₅₃	1	3	yes	—
146	C₂×C₈	C₂×C₂²⋊D₅₃	1	6	yes	—
147	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
148	C₄⋊C₄	—	106	3	no	—
149	C₂²⋊C₄	—	106	3	no	—
150	C₄²	—	53	1	no	—
151	C₄²	—	53	3	no	—
152	C₄²	—	106	3	no	—
153	C₁₆	—	2	4	no	—
154	C₁₆	—	106	4	no	—
155	C₂³	C₂×#11719912	1	3	yes	—
156	Q₈	#13133146	1	4	yes	—
157	D₄	#13133146	1	12	yes	—
158	C₂×C₄	—	106	1	no	—
159	C₂×C₄	—	106	3	no	—
160	C₂×C₄	—	212	6	no	—
161	C₂×C₄	C₂×#11719919	1	6	yes	—
162	C₂×C₄	C₂×C₈×D₅₃	1	6	yes	—
163	C₂×C₄	C₂×#11719912	1	3	yes	—
164	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
165	C₈	—	53	2	no	—
166	C₈	—	53	6	no	—
167	C₈	C₂×#11719919	1	6	yes	—
168	C₈	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
169	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
170	C₂²	—	4	3	no	—
171	C₂²	C₂×#13133146	1	6	yes	—
172	C₂²	C₂²×#11719912	1	1	yes	characteristic
173	C₄	—	106	2	no	—
174	C₄	—	212	3	no	—
175	C₄	C₂×#13133146	1	6	yes	—
176	C₄	C₂²×#11719919	1	1	yes	characteristic
177	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	6	no	—
180	C₂	C₂²×#13133146	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3407	C₂
socle	C₂×C₅₃	C₂²×C₄⋊C₈
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#3407	D₅₃
hypocenter	C₅₃	#33401
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×#131 ◁ #3407

Complements

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

N	H ≅ G/N	type
3. #20276480	C₂	semidirect
4. C₂×#16810443	C₂	semidirect
109. C₅₃	#33401	semidirect

Maximal subgroups

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

subgroup	index
#20287818	2
#131×Dic₅₃	2
#20276480	2
C₂×#16810443	2
C₅₃×#3407	2
#33401	53