Subgroups — #27300518

Order 13568.

334 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810446	C₂	1	4	yes	maximal
2	#20274887	C₂	1	1	yes	maximal, characteristic
3	#20274872	C₂	1	1	yes	maximal, characteristic
4	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133153	—	2	1	no	—
6	C₂×#13133153	C₂²	1	2	yes	—
7	C₂×#13133149	—	2	1	no	—
8	C₂×#13133149	C₂²	1	2	yes	—
9	#16810647	—	2	1	no	—
10	#16810493	—	2	1	no	—
11	#16810446	—	2	8	no	—
12	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#346	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₅₃	C₄	1	2	yes	complemented
16	C₂×#11719917	—	2	2	no	—
17	C₂×C₄⋊Dic₅₃	—	2	2	no	—
18	C₂×C₄⋊Dic₅₃	—	4	2	no	—
19	#13133153	—	2	4	no	—
20	#13133153	—	4	2	no	—
21	#13133149	—	2	4	no	—
22	#13133149	—	4	2	no	—
23	C₄⋊#88322	—	4	2	no	—
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
27	M₃₂×C₅₃	D₄	1	4	yes	—
28	M₃₂×C₅₃	Q₈	1	4	yes	—
29	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₅₃	—	4	2	no	—
37	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
40	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
41	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
43	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
44	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₅₃	SD₁₆	1	2	yes	—
48	C₄⋊Dic₅₃	—	4	2	no	—
49	C₄⋊Dic₅₃	—	4	2	no	—
50	C₄⋊Dic₅₃	—	8	4	no	—
51	#11719917	—	4	2	no	—
52	#11719917	—	4	2	no	—
53	C₂×#88322	—	4	2	no	—
54	C₁₆×C₅₃	C₄⋊C₄	1	8	yes	complemented
55	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	4	no	—
57	C₂×C₄×C₅₃	#106	1	2	yes	—
58	C₂×C₄×C₅₃	#106	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×Dic₅₃	—	8	4	no	—
66	C₂×Dic₅₃	—	8	2	no	—
67	C₈×C₅₃	#106	1	2	yes	—
68	C₈×C₅₃	#106	1	2	yes	—
69	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
70	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
71	#88322	—	8	2	no	—
72	#12903	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	—	2	2	no	—
74	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
75	C₂²×C₅₃	#492	1	1	yes	characteristic
76	C₂²×C₅₃	#498	1	1	yes	characteristic
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	#365	1	4	yes	complemented
79	C₄×C₅₃	#384	1	1	yes	characteristic
80	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
81	C₄×C₅₃	#492	1	1	yes	characteristic
82	C₄×C₅₃	#498	1	1	yes	characteristic
83	Dic₅₃	—	16	2	no	—
84	C₂×#365	—	53	4	no	—
85	C₄×M₃₂	D₅₃	1	1	yes	characteristic
86	#1848	—	53	1	no	—
87	#1844	—	53	1	no	—
88	C₂×C₅₃	—	2	1	no	—
89	C₂×C₅₃	—	2	1	no	—
90	C₂×C₅₃	#1844	1	1	yes	characteristic
91	C₂×C₅₃	C₂×#365	1	2	yes	—
92	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
93	C₂×#108	—	53	2	no	—
94	C₂×#108	—	106	1	no	—
95	C₂×#106	—	53	2	no	—
96	C₂×#106	—	106	1	no	—
97	#423	—	106	1	no	—
98	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
99	#384	—	106	1	no	—
100	#365	—	106	8	no	—
101	#346	Dic₅₃	1	2	yes	complemented
102	C₄×C₁₆	Dic₅₃	1	2	yes	complemented
103	C₅₃	#12903	1	1	yes	characteristic, complemented, Sylow
104	C₂×M₁₆	—	106	2	no	—
105	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
106	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	106	2	no	—
108	C₂×C₄⋊C₄	—	212	2	no	—
109	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
110	M₃₂	Dic₁₀₆	1	4	yes	—
111	M₃₂	D₂₁₂	1	4	yes	—
112	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
113	#108	—	106	4	no	—
114	#108	—	212	2	no	—
115	#106	—	106	4	no	—
116	#106	—	212	2	no	—
117	C₄⋊C₈	—	212	2	no	—
118	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
119	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
121	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
122	C₂²×C₄	—	212	2	no	—
123	C₂²×C₄	#11719913	1	2	yes	—
124	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
125	M₁₆	—	212	2	no	—
126	M₁₆	—	212	2	no	—
127	C₂×C₈	—	212	2	no	—
128	C₂×C₈	#11719913	1	2	yes	—
129	C₂×C₈	#11719913	1	2	yes	—
130	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
131	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
132	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
133	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
134	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
135	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
136	C₄⋊C₄	—	424	4	no	—
137	C₄⋊C₄	—	212	2	no	—
138	C₄⋊C₄	—	212	2	no	—
139	C₄²	#11719913	1	2	yes	—
140	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
141	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
142	C₁₆	C₄⋊Dic₅₃	1	8	yes	complemented
143	C₂³	C₂×#11719913	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	424	4	no	—
146	C₂×C₄	—	424	2	no	—
147	C₂×C₄	#13133149	1	2	yes	—
148	C₂×C₄	#13133149	1	2	yes	—
149	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
150	C₂×C₄	#13133214	1	1	yes	characteristic
151	C₂×C₄	#13133217	1	1	yes	characteristic
152	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
153	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
154	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
155	C₈	—	424	2	no	—
156	C₈	#13133149	1	2	yes	—
157	C₈	#13133149	1	2	yes	—
158	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
159	C₈	#13133217	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	C₂×#13133149	1	1	yes	characteristic
162	C₂²	#16810996	1	1	yes	characteristic
163	C₂²	#16811021	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	848	2	no	—
166	C₄	#16810446	1	4	yes	complemented
167	C₄	#16810493	1	1	yes	characteristic
168	C₄	C₂×#13133149	1	1	yes	characteristic
169	C₄	#16810996	1	1	yes	characteristic
170	C₄	#16811021	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#20274872	1	1	yes	characteristic
174	C₂	C₂×#16810446	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13133149
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂³×C₅₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#12903
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
14. C₅₃×#346	C₄	semidirect
15. C₄×C₁₆×C₅₃	C₄	semidirect
54. C₁₆×C₅₃	C₄⋊C₄	semidirect
78. C₄×C₅₃	#365	semidirect
101. #346	Dic₅₃	semidirect
102. C₄×C₁₆	Dic₅₃	semidirect
103. C₅₃	#12903	semidirect
142. C₁₆	C₄⋊Dic₅₃	semidirect
166. C₄	#16810446	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810446	2
#20274887	2
#20274872	2
C₄×M₃₂×C₅₃	2
#12903	53