Subgroups — #27277870

Order 13568.

390 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274843	C₂	1	1	yes	maximal, characteristic
2	#20274842	C₂	1	2	yes	maximal
3	#20274842	C₂	1	2	yes	maximal
4	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	#16810493	C₂²	1	2	yes	—
9	#16810386	—	2	2	no	—
10	#16810385	—	2	4	no	—
11	#16810385	—	2	4	no	—
12	C₈×#88322	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#384	C₂²	1	2	yes	—
15	C₅₃×#340	—	2	2	no	—
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	C₄×#88322	C₂×C₄	1	2	yes	—
22	C₄×#88322	C₂×C₄	1	2	yes	—
23	C₄×#88322	Q₈	1	2	yes	—
24	C₄×#88322	Q₈	1	2	yes	—
25	C₂²×C₈×C₅₃	D₄	1	2	yes	—
26	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₅₃	Q₈	1	2	yes	—
32	C₂²×C₄×C₅₃	—	4	2	no	—
33	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₅₃	—	4	2	no	—
37	C₂×C₈×C₅₃	—	2	4	no	—
38	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
39	C₄⋊C₄×C₅₃	—	4	4	no	—
40	C₄⋊C₄×C₅₃	—	4	4	no	—
41	C₄⋊C₄×C₅₃	—	4	4	no	—
42	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₄⋊Dic₅₃	—	4	4	no	—
47	C₄⋊Dic₅₃	—	4	4	no	—
48	C₄⋊Dic₅₃	—	4	4	no	—
49	C₂×#88322	—	2	2	no	—
50	C₂×#88322	—	2	4	no	—
51	C₂×#88322	—	2	2	no	—
52	C₂×#88322	C₄⋊C₄	1	4	yes	—
53	C₂×#88322	C₄⋊C₄	1	4	yes	—
54	C₂³×C₅₃	#127	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	8	2	no	—
56	C₂×C₄×C₅₃	—	8	2	no	—
57	C₂×C₄×C₅₃	—	8	2	no	—
58	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#135	1	2	yes	—
65	C₂×C₄×C₅₃	#135	1	4	yes	—
66	C₂×C₄×C₅₃	#135	1	4	yes	—
67	C₂×C₄×C₅₃	#135	1	2	yes	—
68	C₂×Dic₅₃	—	8	2	no	—
69	C₂×Dic₅₃	—	8	2	no	—
70	C₂×Dic₅₃	—	8	2	no	—
71	C₈×C₅₃	—	2	4	no	—
72	#88322	—	2	4	no	—
73	#88322	—	2	4	no	—
74	#8817	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	#384	1	1	yes	characteristic
76	C₂²×C₅₃	#495	1	1	yes	characteristic
77	C₂²×C₅₃	#495	1	2	yes	—
78	C₂²×C₅₃	#495	1	2	yes	—
79	C₂²×C₅₃	#495	1	1	yes	characteristic
80	C₄×C₅₃	—	16	2	no	—
81	C₄×C₅₃	#427	1	2	yes	—
82	C₄×C₅₃	#427	1	2	yes	—
83	C₄×C₅₃	#427	1	2	yes	—
84	C₄×C₅₃	#499	1	2	yes	—
85	C₄×C₅₃	#499	1	2	yes	—
86	C₄×C₅₃	#499	1	2	yes	—
87	Dic₅₃	—	16	2	no	—
88	#1834	—	53	1	no	—
89	#1834	—	53	2	no	—
90	#1834	—	53	2	no	—
91	#1834	D₅₃	1	1	yes	characteristic
92	C₂×C₈²	—	53	1	no	—
93	C₂×C₅₃	#1711	1	2	yes	—
94	C₂×C₅₃	#1711	1	2	yes	—
95	C₂×C₅₃	#1846	1	1	yes	characteristic
96	C₂×C₅₃	#1846	1	1	yes	characteristic
97	C₂×C₅₃	#1846	1	1	yes	characteristic
98	C₂×C₄×C₈	—	53	1	no	—
99	C₂×C₄×C₈	—	53	1	no	—
100	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
101	#384	—	53	2	no	—
102	#384	C₂×D₅₃	1	2	yes	—
103	#340	—	2	2	no	—
104	#340	—	106	4	no	—
105	#340	—	106	4	no	—
106	#340	—	106	2	no	—
107	C₈²	—	53	4	no	—
108	C₅₃	#8817	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	53	2	no	—
110	C₂²×C₈	—	53	2	no	—
111	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	106	2	no	—
116	C₂×C₄⋊C₄	—	106	2	no	—
117	C₂×C₄⋊C₄	—	106	2	no	—
118	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
119	C₄×C₈	—	53	2	no	—
120	C₄×C₈	—	53	2	no	—
121	C₄×C₈	—	53	2	no	—
122	C₄×C₈	—	53	2	no	—
123	C₄×C₈	Dic₁₀₆	1	2	yes	—
124	C₄×C₈	C₄×D₅₃	1	2	yes	—
125	C₂²×C₄	—	4	2	no	—
126	C₂²×C₄	—	212	2	no	—
127	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
128	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	—	53	4	no	—
132	C₂×C₈	—	53	4	no	—
133	C₂×C₈	—	106	2	no	—
134	C₂×C₈	—	106	4	no	—
135	C₂×C₈	—	106	2	no	—
136	C₂×C₈	#11719919	1	4	yes	—
137	C₄⋊C₄	—	4	4	no	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	212	4	no	—
141	C₄⋊C₄	—	212	4	no	—
142	C₄⋊C₄	—	212	4	no	—
143	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
144	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
145	C₄²	Q₈×D₅₃	1	1	yes	characteristic
146	C₄²	Q₈×D₅₃	1	1	yes	characteristic
147	C₂³	#13133213	1	1	yes	characteristic
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	424	2	no	—
152	C₂×C₄	—	424	2	no	—
153	C₂×C₄	—	424	2	no	—
154	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
155	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
156	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
157	C₂×C₄	#13133215	1	1	yes	characteristic
158	C₂×C₄	#13133216	1	1	yes	characteristic
159	C₂×C₄	#13133216	1	1	yes	characteristic
160	C₂×C₄	#13133236	1	2	yes	—
161	C₂×C₄	#13133237	1	4	yes	—
162	C₂×C₄	#13133238	1	4	yes	—
163	C₂×C₄	#13133240	1	2	yes	—
164	C₈	—	2	4	no	—
165	C₈	—	106	4	no	—
166	C₈	—	106	4	no	—
167	C₂²	#16810492	1	1	yes	characteristic
168	C₂²	#16811004	1	2	yes	—
169	C₂²	#16811005	1	2	yes	—
170	C₂²	#16811006	1	1	yes	characteristic
171	C₂²	#16811008	1	1	yes	characteristic
172	C₄	—	16	2	no	—
173	C₄	—	848	2	no	—
174	C₄	#16810658	1	2	yes	—
175	C₄	#16810659	1	2	yes	—
176	C₄	#16810660	1	2	yes	—
177	C₄	#16811022	1	2	yes	—
178	C₄	#16811023	1	2	yes	—
179	C₄	#16811024	1	2	yes	—
180	C₂	#20274326	1	2	yes	—
181	C₂	#20274328	1	2	yes	—
182	C₂	#20274876	1	1	yes	characteristic
183	C₂	#20274877	1	1	yes	characteristic
184	C₂	#20274878	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1834	C₂
socle, prime radical	C₂³×C₅₃	#127
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1834	D₅₃
hypocenter	C₅₃	#8817
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
12. C₈×#88322	C₄	semidirect
108. C₅₃	#8817	semidirect

Maximal subgroups

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

subgroup	index
#20274843	2
#20274842	2
C₂×C₈×#88322	2
C₅₃×#1834	2
#8817	53