Subgroups — #27277861

Order 13568.

454 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274874	C₂	1	2	yes	maximal
2	#20274872	C₂	1	1	yes	maximal, characteristic
3	#20274842	C₂	1	2	yes	maximal
4	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133150	—	2	4	no	—
7	C₂×#13133149	—	2	2	no	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	#16810493	C₂²	1	2	yes	—
11	#16810385	—	2	4	no	—
12	C₈×#88322	C₄	1	4	yes	complemented
13	C₂×#106×C₅₃	—	2	2	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#384	C₂²	1	2	yes	—
16	C₂²×#88322	D₄	1	2	yes	—
17	C₂²×#88322	D₄	1	2	yes	—
18	C₂×C₄⋊Dic₅₃	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	#13133150	—	4	8	no	—
22	#13133149	—	4	4	no	—
23	C₄×#88322	C₂×C₄	1	2	yes	—
24	C₄×#88322	C₂×C₄	1	2	yes	—
25	C₄×#88322	Q₈	1	2	yes	—
26	C₄×#88322	Q₈	1	2	yes	—
27	C₂²×C₈×C₅₃	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	#106×C₅₃	—	4	4	no	—
33	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₃	Q₈	1	2	yes	—
35	C₂²×C₄×C₅₃	—	4	2	no	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₅₃	—	4	2	no	—
40	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
41	C₂×C₈×C₅₃	D₈	1	4	yes	—
42	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
43	C₄⋊C₄×C₅₃	—	4	4	no	—
44	C₄⋊C₄×C₅₃	—	4	4	no	—
45	C₄⋊C₄×C₅₃	—	4	4	no	—
46	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₄⋊Dic₅₃	—	4	4	no	—
51	C₄⋊Dic₅₃	—	4	4	no	—
52	C₄⋊Dic₅₃	—	4	4	no	—
53	C₂×#88322	—	2	4	no	—
54	C₂×#88322	C₄⋊C₄	1	4	yes	—
55	C₂×#88322	C₄⋊C₄	1	4	yes	—
56	C₂×#88322	SD₁₆	1	8	yes	—
57	C₂³×C₅₃	#127	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	8	2	no	—
59	C₂×C₄×C₅₃	—	8	2	no	—
60	C₂×C₄×C₅₃	—	8	2	no	—
61	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
68	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
69	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
70	C₂×C₄×C₅₃	#135	1	4	yes	—
71	C₂×Dic₅₃	—	8	2	no	—
72	C₂×Dic₅₃	—	8	2	no	—
73	C₂×Dic₅₃	—	8	2	no	—
74	C₈×C₅₃	#107	1	8	yes	complemented
75	#88322	—	2	4	no	—
76	#88322	#106	1	8	yes	complemented
77	#8814	—	53	1	no	maximal, Sylow
78	C₂²×C₅₃	#384	1	1	yes	characteristic
79	C₂²×C₅₃	#492	1	2	yes	—
80	C₂²×C₅₃	#493	1	1	yes	characteristic
81	C₂²×C₅₃	#494	1	1	yes	characteristic
82	C₂²×C₅₃	#495	1	2	yes	—
83	C₄×C₅₃	—	16	2	no	—
84	C₄×C₅₃	C₂×#106	1	2	yes	—
85	C₄×C₅₃	C₂×#107	1	2	yes	—
86	C₄×C₅₃	#427	1	2	yes	—
87	C₄×C₅₃	#498	1	2	yes	—
88	C₄×C₅₃	#499	1	2	yes	—
89	C₄×C₅₃	#500	1	2	yes	—
90	Dic₅₃	—	16	2	no	—
91	#1845	—	53	2	no	—
92	#1844	—	53	1	no	—
93	#1844	D₅₃	1	1	yes	characteristic
94	#1834	—	53	2	no	—
95	C₂×C₈²	—	53	1	no	—
96	C₂×C₅₃	#1706	1	2	yes	—
97	C₂×C₅₃	#1707	1	2	yes	—
98	C₂×C₅₃	#1844	1	1	yes	characteristic
99	C₂×C₅₃	#1845	1	1	yes	characteristic
100	C₂×C₅₃	#1846	1	1	yes	characteristic
101	C₂×#107	—	106	4	no	—
102	C₂×#106	—	2	2	no	—
103	C₂×#106	—	106	2	no	—
104	C₂×C₄×C₈	—	53	1	no	—
105	C₂×C₄×C₈	—	53	1	no	—
106	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
107	#384	—	53	2	no	—
108	#384	C₂×D₅₃	1	2	yes	—
109	#340	—	106	4	no	—
110	C₈²	—	53	4	no	—
111	C₅₃	#8814	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₈	—	53	2	no	—
113	C₂²×C₈	—	53	2	no	—
114	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	106	2	no	—
119	C₂×C₄⋊C₄	—	106	2	no	—
120	C₂×C₄⋊C₄	—	106	2	no	—
121	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
122	#107	—	212	8	no	—
123	#106	—	4	4	no	—
124	#106	—	212	4	no	—
125	C₄×C₈	—	53	2	no	—
126	C₄×C₈	—	53	2	no	—
127	C₄×C₈	—	53	2	no	—
128	C₄×C₈	—	53	2	no	—
129	C₄×C₈	Dic₁₀₆	1	2	yes	—
130	C₄×C₈	C₄×D₅₃	1	2	yes	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	212	2	no	—
133	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
134	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
136	C₂×C₈	—	53	4	no	—
137	C₂×C₈	—	53	8	no	—
138	C₂×C₈	—	53	4	no	—
139	C₂×C₈	—	106	4	no	—
140	C₂×C₈	#11719919	1	4	yes	—
141	C₂×C₈	#11719922	1	4	yes	—
142	C₂×C₈	#11719925	1	4	yes	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	212	4	no	—
147	C₄⋊C₄	—	212	4	no	—
148	C₄⋊C₄	—	212	4	no	—
149	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
150	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
151	C₄²	Q₈×D₅₃	1	1	yes	characteristic
152	C₄²	Q₈×D₅₃	1	1	yes	characteristic
153	C₂³	#13133213	1	1	yes	characteristic
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	424	2	no	—
158	C₂×C₄	—	424	2	no	—
159	C₂×C₄	—	424	2	no	—
160	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
161	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
162	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
163	C₂×C₄	#13133215	1	1	yes	characteristic
164	C₂×C₄	#13133216	1	1	yes	characteristic
165	C₂×C₄	#13133216	1	1	yes	characteristic
166	C₂×C₄	C₂×#11719922	1	2	yes	—
167	C₂×C₄	SD₁₆×D₅₃	1	4	yes	—
168	C₂×C₄	C₂×#11719925	1	2	yes	—
169	C₂×C₄	#13133238	1	4	yes	—
170	C₈	—	53	8	no	—
171	C₈	—	106	4	no	—
172	C₈	#13133150	1	8	yes	complemented
173	C₂²	#16810492	1	1	yes	characteristic
174	C₂²	#16810994	1	2	yes	—
175	C₂²	#16810999	1	1	yes	characteristic
176	C₂²	#16811002	1	1	yes	characteristic
177	C₂²	#16811004	1	2	yes	—
178	C₄	—	16	2	no	—
179	C₄	—	848	2	no	—
180	C₄	#106×D₅₃	1	2	yes	—
181	C₄	C₂×#13133150	1	2	yes	—
182	C₄	#16810658	1	2	yes	—
183	C₄	#16811020	1	2	yes	—
184	C₄	#16811023	1	2	yes	—
185	C₄	#16811026	1	2	yes	—
186	C₂	#20274305	1	2	yes	—
187	C₂	#20274310	1	2	yes	—
188	C₂	#20274870	1	1	yes	characteristic
189	C₂	#20274874	1	1	yes	characteristic
190	C₂	#20274876	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1844	C₂
socle, prime radical	C₂³×C₅₃	#127
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1844	D₅₃
hypocenter	C₅₃	#8814
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
12. C₈×#88322	C₄	semidirect
74. C₈×C₅₃	#107	semidirect
76. #88322	#106	semidirect
111. C₅₃	#8814	semidirect
172. C₈	#13133150	semidirect

Maximal subgroups

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

subgroup	index
#20274874	2
#20274872	2
#20274842	2
C₂×C₈×#88322	2
C₅₃×#1844	2
#8814	53