Subgroups — #27300264

Order 13568.

438 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	C₂×#16810451	C₂	1	2	yes	maximal, complemented
2	#20274744	C₂	1	2	yes	maximal
3	#20272807	C₂	1	2	yes	maximal, complemented
4	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133151	C₂²	1	4	yes	—
6	#16810483	—	2	2	no	—
7	#16810451	—	2	4	no	—
8	#16810386	—	2	2	no	—
9	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₅₃	—	2	4	no	—
11	C₂²×C₁₆×C₅₃	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₅₃	—	2	4	no	—
15	C₂×C₂²⋊Dic₅₃	—	4	2	no	—
16	#13133151	—	2	4	no	—
17	#13133151	—	2	4	no	—
18	#13133124	—	4	2	no	—
19	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	—	2	2	no	—
21	C₂²×C₈×C₅₃	—	2	2	no	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
25	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	—	2	2	no	—
27	C₂×C₁₆×C₅₃	—	2	8	no	—
28	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
31	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	—	2	2	no	—
33	C₂²×C₄×C₅₃	—	2	2	no	—
34	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₅₃	—	4	4	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	2	4	no	—
43	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₅₃	#41	1	2	yes	—
48	C₂²⋊Dic₅₃	—	8	4	no	—
49	C₄⋊Dic₅₃	—	4	4	no	—
50	C₄⋊Dic₅₃	—	4	4	no	—
51	C₁₆×C₅₃	—	2	2	no	—
52	C₁₆×C₅₃	C₂²⋊C₄	1	4	yes	complemented
53	C₂³×C₅₃	—	2	2	no	—
54	C₂³×C₅₃	—	2	2	no	—
55	C₂³×C₅₃	#107	1	2	yes	—
56	C₂³×C₅₃	SD₃₂	1	4	yes	—
57	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
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	—	2	4	no	—
62	C₂×C₄×C₅₃	#107	1	2	yes	—
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#121	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#135	1	2	yes	—
67	C₂×Dic₅₃	—	8	4	no	—
68	C₂×Dic₅₃	—	8	4	no	—
69	C₂×Dic₅₃	—	8	4	no	—
70	C₈×C₅₃	—	2	2	no	—
71	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₈×C₅₃	#117	1	1	yes	characteristic
73	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
74	#12835	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	—	2	2	no	—
76	C₂²×C₅₃	—	2	2	no	—
77	C₂²×C₅₃	—	2	8	no	—
78	C₂²×C₅₃	#367	1	4	yes	complemented
79	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
80	C₂²×C₅₃	#466	1	2	yes	—
81	C₂²×C₅₃	C₂×SD₃₂	1	2	yes	—
82	C₂²×C₅₃	#508	1	2	yes	—
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	#380	1	1	yes	characteristic
85	C₄×C₅₃	#427	1	1	yes	characteristic
86	C₄×C₅₃	#467	1	2	yes	—
87	Dic₅₃	—	16	4	no	—
88	C₂³×C₁₆	D₅₃	1	1	yes	characteristic
89	C₂×#367	—	53	2	no	—
90	#1808	—	53	2	no	—
91	#1380	—	53	2	no	—
92	C₂×C₅₃	—	2	4	no	—
93	C₂×C₅₃	#1808	1	1	yes	characteristic
94	C₂×C₅₃	C₂×#367	1	1	yes	characteristic
95	C₂×C₅₃	#2158	1	1	yes	characteristic
96	C₂×C₅₃	#2217	1	4	yes	—
97	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
98	C₂²×C₁₆	—	2	4	no	—
99	C₂²×C₁₆	Dic₅₃	1	4	yes	complemented
100	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
101	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
102	C₂×#107	—	53	4	no	—
103	#380	—	106	2	no	—
104	#367	—	106	4	no	—
105	#340	—	106	2	no	—
106	C₅₃	#12835	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
108	C₂²×C₈	—	2	2	no	—
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	Dic₁₀₆	1	2	yes	—
111	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
113	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	106	4	no	—
115	C₂×C₂²⋊C₄	—	212	2	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	8	no	—
118	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
119	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
120	C₂×C₁₆	C₂²⋊D₅₃	1	4	yes	—
121	#107	—	106	4	no	—
122	#107	—	106	4	no	—
123	#95	—	212	2	no	—
124	C₂⁴	D₄₂₄	1	1	yes	characteristic
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	212	4	no	—
128	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
129	C₂²×C₄	Dic₂₁₂	1	2	yes	—
130	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
131	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
137	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
138	C₂×C₈	#11719945	1	2	yes	—
139	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
140	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
141	C₄⋊C₄	—	212	4	no	—
142	C₄⋊C₄	—	212	4	no	—
143	C₂²⋊C₄	—	424	4	no	—
144	C₁₆	—	2	2	no	—
145	C₁₆	C₂²⋊Dic₅₃	1	4	yes	complemented
146	C₂³	—	2	2	no	—
147	C₂³	—	2	2	no	—
148	C₂³	#13133151	1	2	yes	—
149	C₂³	#13133162	1	4	yes	—
150	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	424	4	no	—
156	C₂×C₄	—	424	4	no	—
157	C₂×C₄	—	424	4	no	—
158	C₂×C₄	#13133151	1	2	yes	—
159	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
160	C₂×C₄	#13133191	1	2	yes	—
161	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
162	C₂×C₄	#13133239	1	2	yes	—
163	C₈	—	2	2	no	—
164	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
165	C₈	#13133176	1	1	yes	characteristic
166	C₈	#13133196	1	2	yes	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	8	no	—
170	C₂²	#16810451	1	4	yes	complemented
171	C₂²	C₂×#13133151	1	1	yes	characteristic
172	C₂²	#16810859	1	2	yes	—
173	C₂²	C₂×#13133162	1	2	yes	—
174	C₂²	#16811055	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	848	4	no	—
177	C₄	#16810483	1	1	yes	characteristic
178	C₄	#16810661	1	1	yes	characteristic
179	C₄	#16810864	1	2	yes	—
180	C₂	—	2	4	no	—
181	C₂	#20274744	1	1	yes	characteristic
182	C₂	C₂×#16810451	1	1	yes	characteristic
183	C₂	#20276499	1	1	yes	characteristic
184	C₂	#20276811	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂³×C₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂³×C₁₆	D₅₃
hypocenter	C₅₃	#12835
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16810451	C₂	semidirect
3. #20272807	C₂	semidirect
11. C₂²×C₁₆×C₅₃	C₄	semidirect
52. C₁₆×C₅₃	C₂²⋊C₄	semidirect
78. C₂²×C₅₃	#367	semidirect
99. C₂²×C₁₆	Dic₅₃	semidirect
106. C₅₃	#12835	semidirect
145. C₁₆	C₂²⋊Dic₅₃	semidirect
170. C₂²	#16810451	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810451	2
#20274744	2
#20272807	2
C₂³×C₁₆×C₅₃	2
#12835	53