Subgroups — #27270919

Order 13568.

306 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719908	C₄	1	2	yes	—
5	C₂×C₄×#88322	C₄	1	2	yes	—
6	C₄×#11719908	—	2	1	no	—
7	C₄×#11719908	—	2	1	no	—
8	C₈×#88322	—	2	1	no	—
9	C₈×#88322	—	2	1	no	—
10	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₅₃	—	2	1	no	—
13	C₄×C₁₆×C₅₃	—	2	1	no	—
14	C₂²×#88322	C₈	1	4	yes	—
15	C₂×#11719908	—	2	2	no	—
16	C₂×#11719908	—	2	2	no	—
17	C₂×#11719908	C₈	1	4	yes	complemented
18	C₄×#88322	—	2	2	no	—
19	C₄×#88322	—	2	2	no	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₅₃	—	2	2	no	—
24	C₂×C₁₆×C₅₃	—	2	2	no	—
25	C₂×C₁₆×C₅₃	C₈	1	4	yes	complemented
26	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
39	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
40	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₅₃	M₁₆	1	2	yes	—
43	C₂×#88322	—	2	4	no	—
44	C₂×#88322	—	2	4	no	—
45	C₂×#88322	C₁₆	1	8	yes	complemented
46	C₁₆×C₅₃	—	2	4	no	—
47	#11719908	—	2	4	no	—
48	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
58	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
59	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
60	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
61	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
62	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
63	#88322	—	2	8	no	—
64	#7136	—	53	1	no	maximal, Sylow
65	C₂²×C₅₃	C₈²	1	1	yes	characteristic
66	C₂²×C₅₃	#322	1	1	yes	characteristic
67	C₂²×C₅₃	#336	1	1	yes	characteristic
68	C₂²×C₅₃	C₄×C₁₆	1	2	yes	—
69	C₂²×C₅₃	#346	1	2	yes	—
70	C₄×C₅₃	#322	1	1	yes	characteristic
71	C₄×C₅₃	#322	1	1	yes	characteristic
72	C₄×C₅₃	#336	1	1	yes	characteristic
73	C₄×C₅₃	#336	1	1	yes	characteristic
74	C₄×C₅₃	C₂²⋊C₁₆	1	4	yes	—
75	C₄×C₅₃	C₄⋊C₁₆	1	4	yes	—
76	C₂×C₄×C₁₆	—	53	1	no	—
77	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
78	C₂×C₈²	—	53	1	no	—
79	C₂×C₅₃	#1273	1	1	yes	characteristic
80	C₂×C₅₃	C₈×C₁₆	1	1	yes	characteristic
81	C₂×C₅₃	#1311	1	1	yes	characteristic
82	C₂×C₅₃	#1312	1	2	yes	—
83	C₂×C₅₃	#1374	1	2	yes	—
84	C₂²×C₁₆	—	53	2	no	—
85	C₂²×C₁₆	Dic₅₃	1	2	yes	—
86	C₂×C₄×C₈	—	53	2	no	—
87	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
88	C₄×C₁₆	—	2	1	no	—
89	C₄×C₁₆	—	2	1	no	—
90	C₄×C₁₆	—	106	1	no	—
91	C₄×C₁₆	—	106	1	no	—
92	C₈²	—	106	1	no	—
93	C₈²	—	106	1	no	—
94	C₅₃	#7136	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₈	—	53	4	no	—
96	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
97	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
98	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	2	2	no	—
101	C₂×C₁₆	—	53	4	no	—
102	C₂×C₁₆	—	106	2	no	—
103	C₂×C₁₆	—	106	2	no	—
104	C₂×C₁₆	#88322	1	4	yes	complemented
105	C₄×C₈	—	106	2	no	—
106	C₄×C₈	—	106	2	no	—
107	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
108	C₄×C₈	D₂₁₂	1	1	yes	characteristic
109	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
110	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
111	C₂²×C₄	C₈×D₅₃	1	2	yes	—
112	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
113	C₂×C₈	—	53	8	no	—
114	C₂×C₈	—	106	4	no	—
115	C₂×C₈	—	106	4	no	—
116	C₂×C₈	C₈×D₅₃	1	2	yes	—
117	C₂×C₈	#11719912	1	2	yes	—
118	C₂×C₈	C₂×#88322	1	2	yes	—
119	C₂×C₈	#11719917	1	2	yes	—
120	C₂×C₈	#11719919	1	1	yes	characteristic
121	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
122	C₂×C₈	#11719921	1	1	yes	characteristic
123	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
124	C₄²	#11719912	1	2	yes	—
125	C₄²	#11719919	1	1	yes	characteristic
126	C₄²	#11719921	1	1	yes	characteristic
127	C₁₆	—	2	4	no	—
128	C₁₆	—	106	4	no	—
129	C₂³	C₈×Dic₅₃	1	1	yes	characteristic
130	C₂×C₄	#13133124	1	1	yes	characteristic
131	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
132	C₂×C₄	C₄×#88322	1	1	yes	characteristic
133	C₂×C₄	#13133127	1	1	yes	characteristic
134	C₂×C₄	#13133127	1	1	yes	characteristic
135	C₂×C₄	#13133128	1	1	yes	characteristic
136	C₂×C₄	#13133129	1	2	yes	—
137	C₂×C₄	#13133146	1	2	yes	—
138	C₂×C₄	C₁₆×D₅₃	1	4	yes	—
139	C₂×C₄	#13133156	1	4	yes	—
140	C₈	—	106	8	no	—
141	C₈	#13133129	1	2	yes	—
142	C₈	C₂²⋊#88322	1	2	yes	—
143	C₈	#13133146	1	2	yes	—
144	C₈	C₄⋊#88322	1	2	yes	—
145	C₂²	C₈×#88322	1	1	yes	characteristic
146	C₂²	#16810345	1	1	yes	characteristic
147	C₂²	#16810377	1	1	yes	characteristic
148	C₂²	C₁₆×Dic₅₃	1	2	yes	—
149	C₂²	#16810398	1	2	yes	—
150	C₄	#16810344	1	1	yes	characteristic
151	C₄	#16810345	1	1	yes	characteristic
152	C₄	#16810377	1	1	yes	characteristic
153	C₄	#16810378	1	1	yes	characteristic
154	C₄	#16810402	1	4	yes	—
155	C₄	#16810441	1	4	yes	—
156	C₂	#20272542	1	1	yes	characteristic
157	C₂	C₁₆×#88322	1	1	yes	characteristic
158	C₂	#20272631	1	1	yes	characteristic
159	C₂	#20272634	1	2	yes	—
160	C₂	#20272792	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₈×C₁₆
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#7136
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
17. C₂×#11719908	C₈	semidirect
25. C₂×C₁₆×C₅₃	C₈	semidirect
45. C₂×#88322	C₁₆	semidirect
94. C₅₃	#7136	semidirect
104. C₂×C₁₆	#88322	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11719908	2
C₂×C₈×#88322	2
C₂×C₄×C₁₆×C₅₃	2
#7136	53