Subgroups — #27277856

Order 13568.

382 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274867	C₂	1	1	yes	maximal, characteristic
2	#20274832	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810376	C₂	1	2	yes	maximal
4	C₂×#16810374	C₂	1	2	yes	maximal
5	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	2	no	—
7	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
10	#16810378	—	2	2	no	—
11	#16810376	—	2	4	no	—
12	#16810374	—	2	4	no	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
14	C₄³×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#336	C₄	1	2	yes	—
16	C₅₃×#336	C₄	1	2	yes	—
17	C₂²×#88322	—	2	1	no	—
18	C₂²×#88322	—	2	1	no	—
19	C₂²×#88322	—	2	2	no	—
20	C₄⋊#88322	—	2	2	no	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	4	4	no	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	D₄	1	2	yes	—
29	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₅₃	D₄	1	4	yes	—
32	C₄×C₈×C₅₃	Q₈	1	4	yes	—
33	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₅₃	—	2	4	no	—
39	C₂×C₈×C₅₃	—	2	4	no	—
40	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
41	C₄²×C₅₃	—	2	4	no	—
42	C₄²×C₅₃	—	2	2	no	—
43	C₄²×C₅₃	—	2	2	no	—
44	C₄²×C₅₃	—	2	2	no	—
45	C₄²×C₅₃	—	2	2	no	—
46	C₄²×C₅₃	C₂×D₄	1	2	yes	—
47	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
48	C₂×#88322	—	4	2	no	—
49	C₂×#88322	—	4	1	no	—
50	C₂×#88322	—	4	1	no	—
51	C₂×#88322	—	4	4	no	—
52	C₂×#88322	—	4	2	no	—
53	C₂×#88322	—	4	2	no	—
54	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	#108	1	4	yes	—
62	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
63	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
65	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#131	1	2	yes	—
67	C₂×C₄×C₅₃	#131	1	2	yes	—
68	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×C₄×C₅₃	#137	1	2	yes	—
70	C₈×C₅₃	—	2	8	no	—
71	#88322	—	8	2	no	—
72	#88322	—	8	1	no	—
73	#88322	—	8	1	no	—
74	#8812	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	#384	1	1	yes	characteristic
76	C₂²×C₅₃	#424	1	1	yes	characteristic
77	C₂²×C₅₃	#424	1	1	yes	characteristic
78	C₂²×C₅₃	#428	1	2	yes	—
79	C₂²×C₅₃	C₂×#108	1	2	yes	—
80	C₄×C₅₃	—	2	4	no	—
81	C₄×C₅₃	—	2	2	no	—
82	C₄×C₅₃	—	4	2	no	—
83	C₄×C₅₃	#424	1	4	yes	—
84	C₄×C₅₃	#496	1	1	yes	characteristic
85	C₄×C₅₃	#497	1	1	yes	characteristic
86	C₄×C₅₃	#501	1	2	yes	—
87	#1842	—	53	1	no	—
88	#1831	—	53	1	no	—
89	#1744	D₅₃	1	1	yes	characteristic
90	C₂×#335	—	53	2	no	—
91	C₂×#334	—	53	2	no	—
92	C₂×C₅₃	#1573	1	2	yes	—
93	C₂×C₅₃	#1574	1	2	yes	—
94	C₂×C₅₃	#1843	1	1	yes	characteristic
95	C₂×C₅₃	#1847	1	1	yes	characteristic
96	C₂×C₅₃	#1848	1	1	yes	characteristic
97	C₂×C₄⋊C₈	—	53	1	no	—
98	C₂×C₄⋊C₈	—	53	2	no	—
99	C₂×C₄⋊C₈	—	53	1	no	—
100	C₂×C₄⋊C₈	—	106	2	no	—
101	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
102	C₄³	C₂×D₅₃	1	1	yes	characteristic
103	#336	—	106	2	no	—
104	#336	Dic₅₃	1	2	yes	—
105	#336	Dic₅₃	1	2	yes	—
106	#335	—	106	4	no	—
107	#334	—	106	4	no	—
108	C₅₃	#8812	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	106	1	no	—
110	C₂²×C₈	—	106	1	no	—
111	C₂²×C₈	—	106	2	no	—
112	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
113	C₂×C₄²	Dic₁₀₆	1	2	yes	—
114	C₂×C₄²	D₂₁₂	1	2	yes	—
115	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
116	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
117	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
118	C₄⋊C₈	—	106	2	no	—
119	C₄⋊C₈	—	106	2	no	—
120	C₄⋊C₈	—	106	2	no	—
121	C₄⋊C₈	—	106	2	no	—
122	C₄⋊C₈	—	212	4	no	—
123	C₄×C₈	Dic₁₀₆	1	4	yes	—
124	C₄×C₈	D₂₁₂	1	4	yes	—
125	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
126	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
127	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
128	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
129	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	—	212	2	no	—
133	C₂×C₈	—	212	1	no	—
134	C₂×C₈	—	212	1	no	—
135	C₂×C₈	—	212	4	no	—
136	C₂×C₈	—	212	2	no	—
137	C₂×C₈	—	212	2	no	—
138	C₂×C₈	C₄⋊Dic₅₃	1	8	yes	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	C₂×Dic₁₀₆	1	2	yes	—
145	C₄²	C₂×D₂₁₂	1	2	yes	—
146	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	#13133153	1	4	yes	—
154	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
155	C₂×C₄	#13133214	1	1	yes	characteristic
156	C₂×C₄	#13133217	1	2	yes	—
157	C₂×C₄	#13133217	1	1	yes	characteristic
158	C₂×C₄	#13133224	1	2	yes	—
159	C₂×C₄	#13133224	1	2	yes	—
160	C₂×C₄	#13133245	1	2	yes	—
161	C₂×C₄	#13133250	1	2	yes	—
162	C₈	—	2	8	no	—
163	C₈	—	424	2	no	—
164	C₈	—	424	1	no	—
165	C₈	—	424	1	no	—
166	C₂²	#16810493	1	1	yes	characteristic
167	C₂²	#16810650	1	1	yes	characteristic
168	C₂²	#16810650	1	1	yes	characteristic
169	C₂²	#16810665	1	2	yes	—
170	C₂²	C₂×#13133153	1	2	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	2	2	no	—
173	C₄	—	4	2	no	—
174	C₄	#16810650	1	4	yes	—
175	C₄	#16811012	1	1	yes	characteristic
176	C₄	#16811017	1	1	yes	characteristic
177	C₄	#16811032	1	2	yes	—
178	C₂	#20273500	1	2	yes	—
179	C₂	#20273504	1	2	yes	—
180	C₂	#20274869	1	1	yes	characteristic
181	C₂	#20274883	1	1	yes	characteristic
182	C₂	#20274887	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1744	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1744	D₅₃
hypocenter	C₅₃	#8812
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
108. C₅₃	#8812	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
#20274832	2
C₂×#16810376	2
C₂×#16810374	2
C₅₃×#1744	2
#8812	53