Subgroups — #27300260

Order 13568.

438 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810449	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	#16810449	—	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	D₁₆	1	2	yes	—
57	C₂³×C₅₃	Q₃₂	1	2	yes	—
58	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	4	no	—
63	C₂×C₄×C₅₃	#107	1	2	yes	—
64	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#121	1	2	yes	—
66	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#135	1	2	yes	—
68	C₂×Dic₅₃	—	8	4	no	—
69	C₂×Dic₅₃	—	8	4	no	—
70	C₂×Dic₅₃	—	8	4	no	—
71	C₈×C₅₃	—	2	2	no	—
72	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₈×C₅₃	#117	1	1	yes	characteristic
74	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
75	#12834	—	53	1	no	maximal, Sylow
76	C₂²×C₅₃	—	2	2	no	—
77	C₂²×C₅₃	—	2	2	no	—
78	C₂²×C₅₃	—	2	4	no	—
79	C₂²×C₅₃	—	2	4	no	—
80	C₂²×C₅₃	#366	1	4	yes	complemented
81	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
82	C₂²×C₅₃	#466	1	2	yes	—
83	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
84	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
85	C₂²×C₅₃	#508	1	2	yes	—
86	C₄×C₅₃	—	2	2	no	—
87	C₄×C₅₃	#380	1	1	yes	characteristic
88	C₄×C₅₃	#427	1	1	yes	characteristic
89	C₄×C₅₃	#467	1	2	yes	—
90	Dic₅₃	—	16	4	no	—
91	C₂³×C₁₆	D₅₃	1	1	yes	characteristic
92	C₂×#366	—	53	2	no	—
93	#1808	—	53	2	no	—
94	#1380	—	53	2	no	—
95	C₂×C₅₃	—	2	4	no	—
96	C₂×C₅₃	#1808	1	1	yes	characteristic
97	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
98	C₂×C₅₃	#2158	1	1	yes	characteristic
99	C₂×C₅₃	#2215	1	2	yes	—
100	C₂×C₅₃	#2216	1	2	yes	—
101	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
102	C₂²×C₁₆	—	2	4	no	—
103	C₂²×C₁₆	Dic₅₃	1	4	yes	complemented
104	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
105	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
106	C₂×#107	—	53	4	no	—
107	#380	—	106	2	no	—
108	#366	—	106	4	no	—
109	#340	—	106	2	no	—
110	C₅₃	#12834	1	1	yes	characteristic, complemented, Sylow
111	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	Dic₁₀₆	1	2	yes	—
115	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
116	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
117	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	106	4	no	—
119	C₂×C₂²⋊C₄	—	212	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	2	8	no	—
122	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
123	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
124	C₂×C₁₆	C₂²⋊D₅₃	1	4	yes	—
125	#107	—	106	4	no	—
126	#107	—	106	4	no	—
127	#95	—	212	2	no	—
128	C₂⁴	D₄₂₄	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	212	4	no	—
132	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
133	C₂²×C₄	Dic₂₁₂	1	2	yes	—
134	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
135	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
141	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
142	C₂×C₈	#11719945	1	2	yes	—
143	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
144	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
145	C₄⋊C₄	—	212	4	no	—
146	C₄⋊C₄	—	212	4	no	—
147	C₂²⋊C₄	—	424	4	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	C₂²⋊Dic₅₃	1	4	yes	complemented
150	C₂³	—	2	2	no	—
151	C₂³	—	2	2	no	—
152	C₂³	#13133151	1	2	yes	—
153	C₂³	D₈₄₈	1	2	yes	—
154	C₂³	Dic₄₂₄	1	2	yes	—
155	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	424	4	no	—
161	C₂×C₄	—	424	4	no	—
162	C₂×C₄	—	424	4	no	—
163	C₂×C₄	#13133151	1	2	yes	—
164	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
165	C₂×C₄	#13133191	1	2	yes	—
166	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
167	C₂×C₄	#13133239	1	2	yes	—
168	C₈	—	2	2	no	—
169	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
170	C₈	#13133176	1	1	yes	characteristic
171	C₈	#13133196	1	2	yes	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	4	no	—
175	C₂²	—	2	4	no	—
176	C₂²	#16810449	1	4	yes	complemented
177	C₂²	C₂×#13133151	1	1	yes	characteristic
178	C₂²	#16810859	1	2	yes	—
179	C₂²	C₂×D₈₄₈	1	1	yes	characteristic
180	C₂²	C₂×Dic₄₂₄	1	1	yes	characteristic
181	C₂²	#16811055	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	848	4	no	—
184	C₄	#16810483	1	1	yes	characteristic
185	C₄	#16810661	1	1	yes	characteristic
186	C₄	#16810864	1	2	yes	—
187	C₂	—	2	4	no	—
188	C₂	#20274744	1	1	yes	characteristic
189	C₂	C₂×#16810449	1	1	yes	characteristic
190	C₂	#20276499	1	1	yes	characteristic
191	C₂	#20276801	1	2	yes	—
192	C₂	#20276806	1	2	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	#12834
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×#16810449	C₂	semidirect
3. #20272807	C₂	semidirect
11. C₂²×C₁₆×C₅₃	C₄	semidirect
52. C₁₆×C₅₃	C₂²⋊C₄	semidirect
80. C₂²×C₅₃	#366	semidirect
103. C₂²×C₁₆	Dic₅₃	semidirect
110. C₅₃	#12834	semidirect
149. C₁₆	C₂²⋊Dic₅₃	semidirect
176. C₂²	#16810449	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810449	2
#20274744	2
#20272807	2
C₂³×C₁₆×C₅₃	2
#12834	53