Subgroups — #27537096

Order 13568.

684 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	#20287818	C₂	1	2	yes	maximal
2	#20276481	C₂	1	6	yes	maximal, complemented
3	C₂×#16810444	C₂	1	6	yes	maximal, complemented
4	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#16811040	—	2	8	no	—
6	C₂×#13133157	C₂²	1	12	yes	—
7	#16810444	C₂²	1	12	yes	—
8	#16810444	C₂²	1	4	yes	—
9	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
10	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
11	C₄×#41×C₅₃	C₄	1	1	yes	characteristic
12	C₅₃×#405	C₄	1	3	yes	—
13	C₅₃×#405	C₂²	1	3	yes	—
14	C₄×M₁₆×C₅₃	C₄	1	3	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
16	#13133157	—	2	12	no	—
17	#13133157	C₂³	1	8	yes	—
18	C₂×#11719908	—	2	12	no	—
19	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
20	#131×C₅₃	C₈	1	8	yes	—
21	#131×C₅₃	D₄	1	4	yes	—
22	#131×C₅₃	Q₈	1	4	yes	—
23	C₂×M₁₆×C₅₃	C₂×C₄	1	3	yes	—
24	C₂×M₁₆×C₅₃	C₂×C₄	1	3	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
26	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
27	C₄×Q₈×C₅₃	—	2	1	no	—
28	C₄×D₄×C₅₃	—	2	3	no	—
29	#117×C₅₃	C₂×C₄	1	3	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
31	#97×C₅₃	C₂×C₄	1	6	yes	—
32	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
33	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
34	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₄×C₈×C₅₃	C₂³	1	3	yes	—
36	#41×C₅₃	C₄⋊C₄	1	4	yes	—
37	#41×C₅₃	C₂×C₈	1	4	yes	—
38	C₂×Q₈×C₅₃	M₁₆	1	1	yes	characteristic
39	C₂×D₄×C₅₃	M₁₆	1	3	yes	—
40	C₂²×C₄×C₅₃	—	2	3	no	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
42	M₁₆×C₅₃	C₄⋊C₄	1	12	yes	—
43	M₁₆×C₅₃	C₂×C₈	1	12	yes	—
44	C₂×C₈×C₅₃	C₂×C₈	1	12	yes	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
46	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₅₃	C₂×D₄	1	6	yes	—
49	C₂×C₈×C₅₃	C₂×Q₈	1	6	yes	—
50	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
51	C₄⋊C₄×C₅₃	—	2	3	no	—
52	C₂²⋊C₄×C₅₃	—	2	3	no	—
53	C₄²×C₅₃	—	2	3	no	—
54	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
56	#11719908	—	2	8	no	—
57	C₂³×C₅₃	C₂×M₁₆	1	3	yes	—
58	Q₈×C₅₃	C₄⋊C₈	1	4	yes	—
59	D₄×C₅₃	C₄⋊C₈	1	12	yes	—
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	3	no	—
62	C₂×C₄×C₅₃	—	4	6	no	—
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
64	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
65	C₂×C₄×C₅₃	C₂×M₁₆	1	3	yes	—
66	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
67	C₈×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
68	C₈×C₅₃	C₂²×C₈	1	2	yes	—
69	C₈×C₅₃	C₂²×C₈	1	6	yes	—
70	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
71	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
72	#33401	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	—	4	3	no	—
74	C₂²×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
75	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	4	3	no	—
78	C₄×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
79	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
81	#3407	—	53	2	no	—
82	C₄×#131	D₅₃	1	1	yes	characteristic
83	#2153	—	53	6	no	—
84	C₂×#364	—	53	6	no	—
85	C₂×C₅₃	—	2	1	no	—
86	C₂×C₅₃	—	2	6	no	—
87	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
88	C₂×#131	Dic₅₃	1	1	yes	characteristic
89	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
90	C₄×#41	Dic₅₃	1	1	yes	characteristic
91	#504	—	106	8	no	—
92	C₂×M₃₂	—	53	12	no	—
93	#405	Dic₅₃	1	3	yes	—
94	#405	C₂×D₅₃	1	3	yes	—
95	C₄×M₁₆	Dic₅₃	1	3	yes	—
96	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
97	#364	—	53	4	no	—
98	#364	—	53	12	no	—
99	C₅₃	#33401	1	1	yes	characteristic, complemented, Sylow
100	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
101	#131	#88322	1	8	yes	—
102	#131	Dic₁₀₆	1	4	yes	—
103	#131	D₂₁₂	1	4	yes	—
104	C₂×M₁₆	C₂×Dic₅₃	1	3	yes	—
105	C₂×M₁₆	C₂×Dic₅₃	1	3	yes	—
106	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
107	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
108	C₄×Q₈	—	2	1	no	—
109	C₄×D₄	—	2	3	no	—
110	#117	C₂×Dic₅₃	1	3	yes	—
111	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
112	M₃₂	—	53	8	no	—
113	M₃₂	—	106	12	no	—
114	C₂×C₁₆	—	106	12	no	—
115	#97	C₂×Dic₅₃	1	6	yes	—
116	C₄×C₈	C₂×Dic₅₃	1	3	yes	—
117	C₄×C₈	C₂×Dic₅₃	1	3	yes	—
118	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
119	C₄×C₈	C₂²×D₅₃	1	3	yes	—
120	#41	C₂×#88322	1	4	yes	—
121	#41	C₄⋊Dic₅₃	1	4	yes	—
122	C₂×Q₈	#11719917	1	1	yes	characteristic
123	C₂×D₄	#11719917	1	3	yes	—
124	C₂²×C₄	—	2	3	no	—
125	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
126	M₁₆	C₂×#88322	1	12	yes	—
127	M₁₆	C₄⋊Dic₅₃	1	12	yes	—
128	C₂×C₈	C₂×#88322	1	12	yes	—
129	C₂×C₈	C₂×Dic₁₀₆	1	6	yes	—
130	C₂×C₈	C₂×D₂₁₂	1	6	yes	—
131	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
132	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
133	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
134	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
135	C₄⋊C₄	—	2	3	no	—
136	C₂²⋊C₄	—	2	3	no	—
137	C₄²	—	2	3	no	—
138	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
139	C₄²	C₂²×Dic₅₃	1	3	yes	—
140	C₁₆	—	106	8	no	—
141	C₂³	C₂×#11719917	1	3	yes	—
142	Q₈	C₄⋊#88322	1	4	yes	—
143	D₄	C₄⋊#88322	1	12	yes	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	3	no	—
146	C₂×C₄	—	4	6	no	—
147	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
148	C₂×C₄	C₂²×#88322	1	6	yes	—
149	C₂×C₄	C₂×#11719917	1	3	yes	—
150	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
151	C₈	C₂×C₄⋊Dic₅₃	1	6	yes	—
152	C₈	C₂²×#88322	1	2	yes	—
153	C₈	C₂²×#88322	1	6	yes	—
154	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
155	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
156	C₂²	—	4	3	no	—
157	C₂²	C₂×C₄⋊#88322	1	6	yes	—
158	C₂²	C₂²×#11719917	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	4	3	no	—
161	C₄	C₂×C₄⋊#88322	1	6	yes	—
162	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
163	C₄	C₂³×#88322	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	6	no	—
166	C₂	C₂²×C₄⋊#88322	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×D₂₁₂
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×#131×C₅₃	C₂
socle	C₂×C₅₃	C₂²×C₄⋊C₈
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	#33401
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×#131 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #20276481	C₂	semidirect
3. C₂×#16810444	C₂	semidirect
99. 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
#20276481	2
C₂×#16810444	2
C₄×#131×C₅₃	2
#33401	53