Subgroups — #28328612

Order 13568.

462 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	#20291139	C₂	1	8	yes	maximal, complemented
2	D₅₃⋊M₆₄	C₂	1	4	yes	maximal, complemented
3	C₂×#16811555	C₂	1	2	yes	maximal, complemented
4	C₂×#16811041	C₂	1	1	yes	maximal, characteristic
5	#16811556	C₂²	1	8	yes	complemented
6	#16811555	C₂²	1	4	yes	—
7	#16811555	C₂²	1	4	yes	complemented
8	D₅₃⋊C₃₂	C₂²	1	8	yes	complemented
9	C₂×#13133267	C₂²	1	4	yes	complemented
10	C₂×#13133223	C₄	1	1	yes	characteristic
11	#16811041	C₄	1	4	yes	—
12	#16811041	C₂²	1	4	yes	—
13	C₂×#13133156	C₄	1	2	yes	—
14	C₂²×#11719908	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
16	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
17	#13133267	C₂³	1	8	yes	—
18	C₂×#11719945	C₈	1	1	yes	characteristic
19	#13133223	C₈	1	4	yes	—
20	#13133223	C₂×C₄	1	4	yes	—
21	C₂×#11719917	C₈	1	1	yes	characteristic
22	C₂×#11719912	C₈	1	2	yes	—
23	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
24	#13133156	C₂×C₄	1	8	yes	—
25	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
26	C₂×#11719908	C₂×C₄	1	2	yes	—
27	C₂×#11719908	C₂³	1	1	yes	characteristic
28	C₂×#11719908	C₂³	1	2	yes	—
29	C₁₆×D₅₃	C₂×C₄	1	4	yes	—
30	C₁₆×D₅₃	C₂³	1	4	yes	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₂²⋊D₅₃	C₁₆	1	2	yes	—
36	#11719945	C₁₆	1	4	yes	—
37	#11719945	C₂×C₈	1	4	yes	—
38	C₂×D₂₁₂	C₁₆	1	1	yes	characteristic
39	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
40	C₂×Dic₁₀₆	C₁₆	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
45	#11719917	C₂×C₈	1	4	yes	—
46	C₂×#88322	C₂×C₈	1	2	yes	—
47	#11719912	C₂×C₈	1	8	yes	—
48	C₈×D₅₃	C₂×C₈	1	4	yes	—
49	C₈×D₅₃	C₂²×C₄	1	4	yes	—
50	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
51	#11719908	C₂²×C₄	1	1	yes	characteristic
52	#11719908	C₂²×C₄	1	2	yes	—
53	#11719908	C₂⁴	1	1	yes	characteristic
54	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
55	C₂²×D₅₃	C₂×C₁₆	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
58	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
60	C₂²⋊D₅₃	C₂×C₁₆	1	8	yes	—
61	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
62	D₂₁₂	C₂×C₁₆	1	4	yes	—
63	C₄×D₅₃	C₂×C₁₆	1	4	yes	—
64	C₄×D₅₃	C₂²×C₈	1	4	yes	—
65	Dic₁₀₆	C₂×C₁₆	1	4	yes	—
66	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
67	C₈×C₅₃	C₂²×C₈	1	2	yes	—
68	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
69	#88322	C₂²×C₈	1	4	yes	—
70	#33974	—	53	1	no	maximal, Sylow
71	C₂²×C₅₃	—	4	1	no	—
72	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
73	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
74	C₂×D₅₃	—	4	2	no	—
75	C₂×D₅₃	C₂²×C₁₆	1	4	yes	—
76	C₄×C₅₃	C₂²×C₁₆	1	2	yes	—
77	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
78	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
79	Dic₅₃	C₂²×C₁₆	1	4	yes	—
80	C₂×#504	—	53	1	no	—
81	#2258	—	53	8	no	—
82	C₂×M₆₄	—	53	2	no	—
83	C₂×M₆₄	—	53	4	no	—
84	C₂×C₅₃	—	2	1	no	—
85	C₂×C₅₃	—	2	2	no	—
86	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
87	D₅₃	—	2	4	no	—
88	C₂×#131	—	53	1	no	—
89	#504	—	53	4	no	—
90	#504	—	53	4	no	—
91	C₂×M₃₂	—	53	2	no	—
92	C₂×M₃₂	#6157	1	1	yes	characteristic
93	C₂²×C₁₆	—	53	1	no	—
94	C₂²×C₁₆	—	53	2	no	—
95	M₆₄	—	53	4	no	—
96	M₆₄	—	53	4	no	—
97	M₆₄	—	53	8	no	—
98	C₂×C₃₂	—	53	4	no	—
99	C₂×C₃₂	—	53	8	no	—
100	C₅₃	#33974	1	1	yes	characteristic, complemented, Sylow
101	C₂×#41	—	53	1	no	—
102	#131	—	53	4	no	—
103	#131	—	53	4	no	—
104	C₂×M₁₆	—	53	1	no	—
105	C₂×M₁₆	—	53	2	no	—
106	C₂²×C₈	—	53	2	no	—
107	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
108	M₃₂	—	53	8	no	—
109	M₃₂	C₂×#6157	1	4	yes	—
110	C₂×C₁₆	—	53	1	no	—
111	C₂×C₁₆	—	53	2	no	—
112	C₂×C₁₆	—	53	2	no	—
113	C₂×C₁₆	—	53	1	no	—
114	C₂×C₁₆	—	53	4	no	—
115	C₂×C₁₆	—	53	4	no	—
116	C₂×C₁₆	C₂×#6157	1	2	yes	—
117	C₃₂	—	53	8	no	—
118	#41	—	53	4	no	—
119	#41	—	53	4	no	—
120	C₂×Q₈	—	53	1	no	—
121	C₂×D₄	—	53	2	no	—
122	C₂×D₄	—	53	1	no	—
123	C₂²×C₄	—	53	2	no	—
124	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
125	M₁₆	—	53	4	no	—
126	M₁₆	—	53	8	no	—
127	C₂×C₈	—	53	2	no	—
128	C₂×C₈	—	53	4	no	—
129	C₂×C₈	—	53	4	no	—
130	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
131	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
132	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
133	C₂×C₈	C₂²×#6157	1	2	yes	—
134	C₁₆	—	53	1	no	—
135	C₁₆	—	53	1	no	—
136	C₁₆	—	53	2	no	—
137	C₁₆	C₂²×#6157	1	4	yes	—
138	C₂³	—	53	2	no	—
139	C₂³	D₅₃⋊C₁₆	1	1	yes	characteristic
140	Q₈	—	53	4	no	—
141	D₄	—	53	8	no	—
142	D₄	—	53	4	no	—
143	C₂×C₄	—	53	2	no	—
144	C₂×C₄	—	53	4	no	—
145	C₂×C₄	—	53	4	no	—
146	C₂×C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
147	C₂×C₄	D₅₃⋊C₁₆	1	2	yes	—
148	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
149	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
150	C₈	—	53	4	no	—
151	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
152	C₈	C₂×D₅₃⋊C₈	1	2	yes	—
153	C₈	C₂³×#6157	1	1	yes	characteristic
154	C₂²	—	4	1	no	—
155	C₂²	—	53	4	no	—
156	C₂²	—	212	2	no	—
157	C₂²	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
158	C₂²	C₂×D₅₃⋊C₁₆	1	2	yes	—
159	C₄	—	53	4	no	—
160	C₄	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
161	C₄	C₂×D₅₃⋊C₁₆	1	2	yes	—
162	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	2	no	—
165	C₂	—	106	4	no	—
166	C₂	C₂²×D₅₃⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×#6157
commutator, socle	C₂×C₅₃	C₂³×C₁₆
Fitting	C₂×M₃₂×C₅₃	C₄
prime radical	C₂×#11719945	C₈
cosocle	#11719908	C₂⁴
hypercenter	C₂×M₃₂	#6157
hypocenter	C₅₃	#33974
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
1. #20291139	C₂	semidirect
2. D₅₃⋊M₆₄	C₂	semidirect
3. C₂×#16811555	C₂	semidirect
5. #16811556	C₂²	semidirect
7. #16811555	C₂²	semidirect
8. D₅₃⋊C₃₂	C₂²	semidirect
9. C₂×#13133267	C₂²	semidirect
100. 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
D₅₃⋊M₆₄	2
C₂×#16811555	2
C₂×#16811041	2
#33974	53