Subgroups — #27277829

Order 13568.

462 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	#20274867	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810373	C₂	1	4	yes	maximal
4	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	—	2	2	no	—
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₄	1	2	yes	—
8	C₂×C₄×#88322	C₄	1	1	yes	characteristic
9	C₂×C₄×#88322	C₄	1	1	yes	characteristic
10	C₂×C₄×#88322	C₂²	1	2	yes	—
11	#16810373	—	2	8	no	—
12	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
14	C₄³×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×#88322	—	2	2	no	—
16	C₂²×#88322	C₂×C₄	1	2	yes	—
17	C₂²×#88322	C₂×C₄	1	2	yes	—
18	C₄⋊#88322	—	2	4	no	—
19	C₄⋊#88322	—	4	4	no	—
20	C₄×#88322	—	2	2	no	—
21	C₄×#88322	—	2	4	no	—
22	C₄×#88322	C₈	1	4	yes	complemented
23	C₄×#88322	D₄	1	4	yes	—
24	C₄×#88322	Q₈	1	4	yes	—
25	C₂²×C₈×C₅₃	—	2	2	no	—
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	—	2	4	no	—
32	C₄⋊C₈×C₅₃	—	4	4	no	—
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	—	4	2	no	—
39	C₂×C₈×C₅₃	—	4	4	no	—
40	C₄²×C₅₃	—	2	4	no	—
41	C₄²×C₅₃	—	2	2	no	—
42	C₄²×C₅₃	C₂×C₈	1	2	yes	—
43	C₄²×C₅₃	M₁₆	1	2	yes	—
44	C₄²×C₅₃	SD₁₆	1	4	yes	—
45	C₄²×C₅₃	SD₁₆	1	4	yes	—
46	C₄²×C₅₃	C₂×D₄	1	2	yes	—
47	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
48	C₂×#88322	—	2	2	no	—
49	C₂×#88322	—	2	2	no	—
50	C₂×#88322	—	4	2	no	—
51	C₂×#88322	—	4	4	no	—
52	C₂×#88322	C₄⋊C₄	1	4	yes	—
53	C₂×#88322	C₄⋊C₄	1	4	yes	—
54	C₂×#88322	C₂×C₈	1	4	yes	—
55	C₂×#88322	M₁₆	1	4	yes	—
56	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
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	C₄⋊C₈	1	4	yes	—
61	C₂×C₄×C₅₃	#106	1	4	yes	—
62	C₂×C₄×C₅₃	#106	1	4	yes	—
63	C₂×C₄×C₅₃	#108	1	4	yes	—
64	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
65	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
68	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
70	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
73	C₈×C₅₃	—	8	2	no	—
74	#88322	—	2	4	no	—
75	#88322	—	8	2	no	—
76	#88322	C₄⋊C₈	1	8	yes	complemented
77	#8805	—	53	1	no	maximal, Sylow
78	C₂²×C₅₃	#384	1	1	yes	characteristic
79	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
80	C₂²×C₅₃	#423	1	1	yes	characteristic
81	C₂²×C₅₃	C₂×#106	1	2	yes	—
82	C₂²×C₅₃	C₂×#108	1	2	yes	—
83	C₄×C₅₃	—	2	4	no	—
84	C₄×C₅₃	—	2	2	no	—
85	C₄×C₅₃	#334	1	8	yes	complemented
86	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
87	C₄×C₅₃	#423	1	2	yes	—
88	C₄×C₅₃	#492	1	1	yes	characteristic
89	C₄×C₅₃	#492	1	1	yes	characteristic
90	C₄×C₅₃	#498	1	2	yes	—
91	#1842	—	53	1	no	—
92	#1842	D₅₃	1	1	yes	characteristic
93	C₄²×C₈	—	53	1	no	—
94	C₂×#334	—	53	4	no	—
95	C₂×C₅₃	C₂×#334	1	2	yes	—
96	C₂×C₅₃	#1566	1	2	yes	—
97	C₂×C₅₃	#1842	1	1	yes	characteristic
98	C₂×C₅₃	#1844	1	1	yes	characteristic
99	C₂×C₅₃	#1848	1	1	yes	characteristic
100	C₂×C₄⋊C₈	—	2	2	no	—
101	C₂×C₄⋊C₈	—	53	2	no	—
102	C₂×C₄⋊C₈	—	106	2	no	—
103	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
104	C₂×C₄×C₈	—	53	1	no	—
105	C₂×C₄×C₈	—	53	2	no	—
106	C₂×C₄×C₈	—	53	2	no	—
107	C₂×C₄×C₈	—	53	1	no	—
108	C₄³	C₂×D₅₃	1	1	yes	characteristic
109	#334	—	106	8	no	—
110	C₅₃	#8805	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	53	2	no	—
113	C₂²×C₈	—	53	2	no	—
114	C₂²×C₈	—	106	2	no	—
115	C₂×C₄²	Dic₁₀₆	1	2	yes	—
116	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
117	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
118	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
119	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
120	C₄⋊C₈	—	2	4	no	—
121	C₄⋊C₈	—	4	4	no	—
122	C₄⋊C₈	—	106	4	no	—
123	C₄⋊C₈	—	212	4	no	—
124	C₄×C₈	—	53	4	no	—
125	C₄×C₈	—	53	4	no	—
126	C₄×C₈	—	53	4	no	—
127	C₄×C₈	—	106	4	no	—
128	C₄×C₈	—	106	2	no	—
129	C₂²×C₄	#11719919	1	2	yes	—
130	C₂²×C₄	#11719919	1	2	yes	—
131	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	4	4	no	—
136	C₂×C₈	—	53	4	no	—
137	C₂×C₈	—	53	4	no	—
138	C₂×C₈	—	53	4	no	—
139	C₂×C₈	—	53	4	no	—
140	C₂×C₈	—	106	2	no	—
141	C₂×C₈	—	106	2	no	—
142	C₂×C₈	—	212	2	no	—
143	C₂×C₈	—	212	4	no	—
144	C₄²	—	2	4	no	—
145	C₄²	—	2	2	no	—
146	C₄²	C₈×D₅₃	1	2	yes	—
147	C₄²	#11719912	1	2	yes	—
148	C₄²	#11719923	1	4	yes	—
149	C₄²	#11719924	1	4	yes	—
150	C₄²	D₄×D₅₃	1	2	yes	—
151	C₄²	Q₈×D₅₃	1	2	yes	—
152	C₂³	C₂×#11719919	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	#13133146	1	4	yes	—
157	C₂×C₄	#13133148	1	4	yes	—
158	C₂×C₄	#13133148	1	4	yes	—
159	C₂×C₄	#13133152	1	4	yes	—
160	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
161	C₂×C₄	#13133213	1	1	yes	characteristic
162	C₂×C₄	#13133215	1	1	yes	characteristic
163	C₂×C₄	#13133216	1	2	yes	—
164	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
165	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
166	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
167	C₂×C₄	C₂×#11719924	1	2	yes	—
168	C₂×C₄	C₂×#11719923	1	2	yes	—
169	C₈	—	8	2	no	—
170	C₈	—	53	8	no	—
171	C₈	—	106	4	no	—
172	C₈	—	424	2	no	—
173	C₂²	#16810492	1	1	yes	characteristic
174	C₂²	C₂×#13133146	1	1	yes	characteristic
175	C₂²	#16810646	1	1	yes	characteristic
176	C₂²	C₂×#13133148	1	2	yes	—
177	C₂²	C₂×#13133152	1	2	yes	—
178	C₄	—	2	4	no	—
179	C₄	—	2	2	no	—
180	C₄	#16810373	1	8	yes	complemented
181	C₄	C₄⋊C₈×D₅₃	1	2	yes	—
182	C₄	#16810645	1	2	yes	—
183	C₄	#16810995	1	1	yes	characteristic
184	C₄	#16810997	1	1	yes	characteristic
185	C₄	#16811019	1	2	yes	—
186	C₂	C₂×#16810373	1	2	yes	—
187	C₂	#20273470	1	2	yes	—
188	C₂	#20274866	1	1	yes	characteristic
189	C₂	#20274871	1	1	yes	characteristic
190	C₂	#20274886	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1842	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1842	D₅₃
hypocenter	C₅₃	#8805
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
22. C₄×#88322	C₈	semidirect
76. #88322	C₄⋊C₈	semidirect
85. C₄×C₅₃	#334	semidirect
110. C₅₃	#8805	semidirect
180. C₄	#16810373	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
C₄²×#88322	2
C₂×#16810373	2
C₅₃×#1842	2
#8805	53