Subgroups — #28328611

Order 13568.

462 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291139	C₂	1	6	yes	maximal, complemented
2	#20291138	C₂	1	2	yes	maximal, complemented
3	C₂×#16811556	C₂	1	3	yes	maximal, complemented
4	D₅₃⋊M₆₄	C₂	1	3	yes	maximal, complemented
5	#504×D₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811556	C₂²	1	3	yes	—
7	#16811556	C₂²	1	1	yes	characteristic
8	#16811556	C₂²	1	6	yes	complemented
9	#16811555	C₂²	1	6	yes	complemented
10	D₅₃⋊C₃₂	C₂²	1	6	yes	complemented
11	C₂×#13133267	C₂²	1	6	yes	complemented
12	#131×D₅₃	C₄	1	1	yes	characteristic
13	#16811041	C₂²	1	3	yes	—
14	#16811040	C₂²	1	1	yes	characteristic
15	#16811039	C₄	1	3	yes	—
16	M₃₂×D₅₃	C₄	1	3	yes	—
17	C₂×C₁₆×D₅₃	C₂²	1	3	yes	—
18	C₅₃×#504	C₄	1	1	yes	characteristic
19	#13133267	C₂³	1	2	yes	—
20	#13133267	C₂³	1	6	yes	—
21	#41×D₅₃	C₈	1	1	yes	characteristic
22	#13133225	C₈	1	3	yes	—
23	#13133224	C₈	1	1	yes	characteristic
24	#13133223	C₂×C₄	1	3	yes	—
25	M₁₆×D₅₃	C₈	1	3	yes	—
26	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
27	#13133157	C₂×C₄	1	3	yes	—
28	#13133156	C₂×C₄	1	6	yes	—
29	C₂×#11719908	C₂³	1	3	yes	—
30	C₁₆×D₅₃	C₂×C₄	1	3	yes	—
31	C₁₆×D₅₃	C₂×C₄	1	3	yes	—
32	C₁₆×D₅₃	C₂³	1	1	yes	characteristic
33	C₁₆×D₅₃	C₂³	1	3	yes	—
34	#131×C₅₃	C₂×C₄	1	1	yes	characteristic
35	M₃₂×C₅₃	C₂×C₄	1	3	yes	—
36	C₂×C₁₆×C₅₃	C₂×C₄	1	3	yes	—
37	#41×C₅₃	C₂×C₈	1	1	yes	characteristic
38	#11719949	C₁₆	1	1	yes	characteristic
39	Q₈×D₅₃	C₁₆	1	1	yes	characteristic
40	#11719947	C₁₆	1	3	yes	—
41	D₄×D₅₃	C₁₆	1	3	yes	—
42	#11719945	C₂×C₈	1	3	yes	—
43	C₂×C₄×D₅₃	C₂×C₈	1	3	yes	—
44	M₁₆×C₅₃	C₂×C₈	1	3	yes	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
46	#11719917	C₂×C₈	1	3	yes	—
47	C₂×#88322	C₂×C₈	1	3	yes	—
48	#11719912	C₂×C₈	1	6	yes	—
49	C₈×D₅₃	C₂×C₈	1	3	yes	—
50	C₈×D₅₃	C₂×C₈	1	3	yes	—
51	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₈×D₅₃	C₂²×C₄	1	3	yes	—
53	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₁₆×C₅₃	C₂²×C₄	1	3	yes	—
55	#11719908	C₂²×C₄	1	3	yes	—
56	#11719908	C₂⁴	1	1	yes	characteristic
57	C₂²×D₅₃	C₂×C₁₆	1	3	yes	—
58	Q₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
59	D₄×C₅₃	C₂×C₁₆	1	3	yes	—
60	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
61	C₂²⋊D₅₃	C₂×C₁₆	1	6	yes	—
62	C₂×Dic₅₃	C₂×C₁₆	1	3	yes	—
63	D₂₁₂	C₂×C₁₆	1	3	yes	—
64	C₄×D₅₃	C₂×C₁₆	1	3	yes	—
65	C₄×D₅₃	C₂×C₁₆	1	3	yes	—
66	C₄×D₅₃	C₂²×C₈	1	1	yes	characteristic
67	C₄×D₅₃	C₂²×C₈	1	3	yes	—
68	Dic₁₀₆	C₂×C₁₆	1	3	yes	—
69	C₈×C₅₃	C₂²×C₈	1	3	yes	—
70	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
71	#88322	C₂²×C₈	1	1	yes	characteristic
72	#88322	C₂²×C₈	1	3	yes	—
73	#33974	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	C₂²×C₁₆	1	3	yes	—
75	C₂×D₅₃	—	4	3	no	—
76	C₂×D₅₃	C₂²×C₁₆	1	1	yes	characteristic
77	C₂×D₅₃	C₂²×C₁₆	1	3	yes	—
78	C₄×C₅₃	C₂²×C₁₆	1	3	yes	—
79	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
80	Dic₅₃	C₂²×C₁₆	1	3	yes	—
81	Dic₅₃	C₂²×C₁₆	1	1	yes	characteristic
82	C₂×#504	—	53	1	no	—
83	#2258	—	53	2	no	—
84	#2258	—	53	6	no	—
85	C₂×M₆₄	—	53	3	no	—
86	C₂×M₆₄	—	53	3	no	—
87	C₂×C₅₃	—	2	3	no	—
88	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
89	D₅₃	—	2	1	no	—
90	D₅₃	—	2	3	no	—
91	C₂×#131	—	53	1	no	—
92	#504	—	53	1	no	—
93	#504	—	53	3	no	—
94	#504	—	53	3	no	—
95	#504	#6157	1	1	yes	characteristic
96	C₂×M₃₂	—	53	3	no	—
97	C₂²×C₁₆	—	53	3	no	—
98	M₆₄	—	53	6	no	—
99	M₆₄	—	53	3	no	—
100	M₆₄	—	53	1	no	—
101	M₆₄	—	53	6	no	—
102	C₂×C₃₂	—	53	6	no	—
103	C₂×C₃₂	—	53	6	no	—
104	C₅₃	#33974	1	1	yes	characteristic, complemented, Sylow
105	C₂×#41	—	53	1	no	—
106	#131	—	53	1	no	—
107	#131	—	53	3	no	—
108	#131	—	53	3	no	—
109	#131	C₂×#6157	1	1	yes	characteristic
110	C₂×M₁₆	—	53	3	no	—
111	C₂²×C₈	—	53	3	no	—
112	M₃₂	—	53	3	no	—
113	M₃₂	—	53	6	no	—
114	M₃₂	C₂×#6157	1	3	yes	—
115	C₂×C₁₆	—	53	3	no	—
116	C₂×C₁₆	—	53	3	no	—
117	C₂×C₁₆	—	53	3	no	—
118	C₂×C₁₆	—	53	3	no	—
119	C₂×C₁₆	—	53	1	no	—
120	C₂×C₁₆	C₂×#6157	1	3	yes	—
121	C₃₂	—	53	2	no	—
122	C₃₂	—	53	6	no	—
123	#41	—	53	1	no	—
124	#41	—	53	3	no	—
125	#41	—	53	3	no	—
126	#41	D₅₃⋊C₈	1	1	yes	characteristic
127	C₂×Q₈	—	53	1	no	—
128	C₂×D₄	—	53	3	no	—
129	C₂²×C₄	—	53	3	no	—
130	M₁₆	—	53	3	no	—
131	M₁₆	—	53	6	no	—
132	M₁₆	D₅₃⋊C₈	1	3	yes	—
133	C₂×C₈	—	53	3	no	—
134	C₂×C₈	—	53	3	no	—
135	C₂×C₈	—	53	3	no	—
136	C₂×C₈	—	53	1	no	—
137	C₂×C₈	—	53	3	no	—
138	C₂×C₈	C₂²×#6157	1	3	yes	—
139	C₁₆	—	53	1	no	—
140	C₁₆	—	53	3	no	—
141	C₁₆	C₂²×#6157	1	1	yes	characteristic
142	C₁₆	C₂²×#6157	1	3	yes	—
143	C₂³	—	53	3	no	—
144	Q₈	—	53	3	no	—
145	Q₈	D₅₃⋊C₁₆	1	1	yes	characteristic
146	D₄	—	53	6	no	—
147	D₄	—	53	3	no	—
148	D₄	D₅₃⋊C₁₆	1	3	yes	—
149	C₂×C₄	—	53	3	no	—
150	C₂×C₄	—	53	3	no	—
151	C₂×C₄	—	53	3	no	—
152	C₂×C₄	—	53	1	no	—
153	C₂×C₄	—	53	3	no	—
154	C₂×C₄	C₂×D₅₃⋊C₈	1	3	yes	—
155	C₈	—	53	1	no	—
156	C₈	—	53	3	no	—
157	C₈	C₂×D₅₃⋊C₈	1	3	yes	—
158	C₈	C₂³×#6157	1	1	yes	characteristic
159	C₂²	—	53	1	no	—
160	C₂²	—	53	3	no	—
161	C₂²	—	212	3	no	—
162	C₂²	C₂×D₅₃⋊C₁₆	1	3	yes	—
163	C₄	—	53	1	no	—
164	C₄	—	53	3	no	—
165	C₄	C₂×D₅₃⋊C₁₆	1	3	yes	—
166	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
167	C₂	—	2	3	no	—
168	C₂	—	106	1	no	—
169	C₂	—	106	3	no	—
170	C₂	C₂²×D₅₃⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×#6157
commutator, socle	C₂×C₅₃	C₂³×C₁₆
Fitting	C₅₃×#504	C₄
prime radical	#41×D₅₃	C₈
cosocle	#11719908	C₂⁴
hypercenter	#504	#6157
hypocenter	C₅₃	#33974
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ #504

Complements

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

N	H ≅ G/N	type
1. #20291139	C₂	semidirect
2. #20291138	C₂	semidirect
3. C₂×#16811556	C₂	semidirect
4. D₅₃⋊M₆₄	C₂	semidirect
8. #16811556	C₂²	semidirect
9. #16811555	C₂²	semidirect
10. D₅₃⋊C₃₂	C₂²	semidirect
11. C₂×#13133267	C₂²	semidirect
104. C₅₃	#33974	semidirect

Maximal subgroups

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

subgroup	index
#20291139	2
#20291138	2
C₂×#16811556	2
D₅₃⋊M₆₄	2
#504×D₅₃	2
#33974	53