Subgroups — C₂×Q₃₂×Dic₅₃ #27537367

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×Dic₅₃	C₂	1	2	yes	maximal
2	Q₃₂×Dic₅₃	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810449	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810427	C₂	1	2	yes	maximal
5	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₈×Dic₅₃	—	2	2	no	—
8	C₂²×#11719908	C₂²	1	1	yes	characteristic
9	Q₁₆×Dic₅₃	—	2	4	no	—
10	Q₁₆×Dic₅₃	C₂²	1	8	yes	—
11	C₂×#13133151	C₂²	1	2	yes	—
12	C₂×#13133141	—	2	2	no	—
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16810449	C₂²	1	4	yes	—
15	#16810427	C₂²	1	8	yes	—
16	C₁₆×Dic₅₃	C₂²	1	4	yes	—
17	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
18	C₂×Q₃₂×C₅₃	C₄	1	8	yes	complemented
19	C₂×Q₃₂×C₅₃	C₂²	1	4	yes	—
20	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×#88322	D₄	1	1	yes	characteristic
22	Q₈×Dic₅₃	—	2	8	no	—
23	Q₈×Dic₅₃	—	4	4	no	—
24	C₂×C₄⋊Dic₅₃	—	2	2	no	—
25	C₂×C₄⋊Dic₅₃	—	4	2	no	—
26	C₂×C₄×Dic₅₃	—	4	2	no	—
27	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
28	C₂×#11719908	—	2	2	no	—
29	C₂×#11719908	C₂³	1	2	yes	—
30	#13133151	—	2	2	no	—
31	#13133151	C₂³	1	4	yes	—
32	#13133141	—	2	8	no	—
33	C₈×Dic₅₃	D₄	1	2	yes	—
34	C₈×Dic₅₃	C₂³	1	2	yes	—
35	C₂²×Q₈×C₅₃	—	2	2	no	—
36	C₂×Q₁₆×C₅₃	—	2	2	no	—
37	C₂×Q₁₆×C₅₃	—	2	4	no	—
38	C₂×Q₁₆×C₅₃	C₂×C₄	1	8	yes	—
39	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	Q₃₂×C₅₃	C₂×C₄	1	16	yes	complemented
42	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
43	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
44	C₂×Q₈×C₅₃	—	2	8	no	—
45	C₂×Q₈×C₅₃	—	2	4	no	—
46	C₂×Q₈×C₅₃	—	4	2	no	—
47	C₂×Q₈×C₅₃	—	4	4	no	—
48	C₂²×C₄×C₅₃	—	4	2	no	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₅₃	—	4	2	no	—
51	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
52	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
53	Q₁₆×C₅₃	—	2	8	no	—
54	Q₁₆×C₅₃	—	2	4	no	—
55	Q₁₆×C₅₃	C₂²×C₄	1	8	yes	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	#41	1	2	yes	—
59	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
60	C₄⋊Dic₅₃	—	2	4	no	—
61	C₄⋊Dic₅₃	—	4	2	no	—
62	C₄⋊Dic₅₃	—	4	8	no	—
63	C₄×Dic₅₃	—	4	8	no	—
64	C₄×Dic₅₃	D₈	1	2	yes	—
65	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
66	C₂×#88322	—	2	2	no	—
67	C₂×#88322	C₂×D₄	1	2	yes	—
68	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
69	#11719908	—	2	2	no	—
70	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
71	Q₈×C₅₃	—	2	8	no	—
72	Q₈×C₅₃	—	4	4	no	—
73	Q₈×C₅₃	—	4	8	no	—
74	C₂×C₄×C₅₃	—	4	4	no	—
75	C₂×C₄×C₅₃	—	4	8	no	—
76	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#135	1	2	yes	—
79	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
80	C₂×Dic₅₃	—	2	2	no	—
81	C₂×Dic₅₃	—	4	4	no	—
82	C₂×Dic₅₃	—	8	4	no	—
83	C₂×Dic₅₃	Q₃₂	1	4	yes	direct product factor
84	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
85	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
86	C₈×C₅₃	C₄×D₄	1	2	yes	—
87	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
88	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
89	#88322	—	2	2	no	—
90	C₂×C₄×Q₃₂	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	C₄×D₈	1	2	yes	—
92	C₂²×C₅₃	C₂×Q₃₂	1	2	yes	—
93	C₂²×C₅₃	#508	1	2	yes	—
94	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
95	C₄×C₅₃	—	4	8	no	—
96	C₄×C₅₃	C₄×D₈	1	2	yes	—
97	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
98	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
99	Dic₅₃	—	2	2	no	—
100	Dic₅₃	—	8	4	no	—
101	Dic₅₃	C₂×Q₃₂	1	4	yes	direct product factor
102	C₂²×Q₃₂	D₅₃	1	1	yes	characteristic
103	C₂×C₄×Q₁₆	—	53	2	no	—
104	C₄×Q₃₂	—	53	8	no	—
105	C₂×#366	—	53	1	no	—
106	C₂×#358	—	53	2	no	—
107	C₂×C₄×C₁₆	—	53	1	no	—
108	C₂×C₅₃	C₄×Q₃₂	1	4	yes	complemented
109	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
110	C₂×C₅₃	C₂²×Q₃₂	1	1	yes	characteristic
111	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
112	C₂²×Q₁₆	C₂×D₅₃	1	2	yes	—
113	C₂×C₄×Q₈	—	106	2	no	—
114	C₂×Q₃₂	Dic₅₃	1	8	yes	direct product factor
115	C₂×Q₃₂	C₂×D₅₃	1	4	yes	—
116	C₂²×C₁₆	—	53	1	no	—
117	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
118	C₄×Q₁₆	—	53	8	no	—
119	C₄×Q₁₆	—	106	4	no	—
120	C₂×#107	—	53	2	no	—
121	C₂×#103	—	106	2	no	—
122	C₂×C₄×C₈	—	53	1	no	—
123	#366	—	53	4	no	—
124	#358	—	53	8	no	—
125	C₄×C₁₆	—	53	4	no	—
126	C₅₃	C₂×C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₈	—	2	2	no	—
128	C₂×Q₁₆	—	2	2	no	—
129	C₂×Q₁₆	—	2	4	no	—
130	C₂×Q₁₆	C₂×Dic₅₃	1	8	yes	—
131	C₂×Q₁₆	C₂²×D₅₃	1	4	yes	—
132	C₂²×C₈	—	53	1	no	—
133	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
134	C₄×Q₈	—	106	8	no	—
135	C₄×Q₈	—	212	4	no	—
136	C₂×C₄⋊C₄	—	106	2	no	—
137	C₂×C₄⋊C₄	—	212	2	no	—
138	C₂×C₄²	—	53	1	no	—
139	C₂×C₄²	—	212	2	no	—
140	Q₃₂	C₂×Dic₅₃	1	16	yes	direct product factor
141	C₂×C₁₆	—	53	2	no	—
142	C₂×C₁₆	—	106	2	no	—
143	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
144	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
145	#107	—	53	4	no	—
146	#107	—	106	2	no	—
147	#103	—	106	8	no	—
148	C₄×C₈	—	53	2	no	—
149	C₄×C₈	—	53	2	no	—
150	C₂×Q₈	—	2	8	no	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	53	1	no	—
156	C₂²×C₄	—	53	1	no	—
157	C₂²×C₄	—	212	2	no	—
158	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
159	Q₁₆	—	2	8	no	—
160	Q₁₆	—	2	4	no	—
161	Q₁₆	C₂²×Dic₅₃	1	8	yes	—
162	C₂×C₈	—	53	2	no	—
163	C₂×C₈	—	106	2	no	—
164	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
165	C₂×C₈	#11719947	1	2	yes	—
166	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
167	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
168	C₄⋊C₄	—	106	4	no	—
169	C₄⋊C₄	—	212	8	no	—
170	C₄⋊C₄	—	212	2	no	—
171	C₄²	—	53	2	no	—
172	C₄²	—	53	2	no	—
173	C₄²	—	212	8	no	—
174	C₁₆	—	106	2	no	—
175	C₁₆	C₂²×Dic₅₃	1	4	yes	—
176	C₂³	D₈×D₅₃	1	1	yes	characteristic
177	Q₈	—	2	8	no	—
178	Q₈	—	4	4	no	—
179	Q₈	—	4	8	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	8	no	—
182	C₂×C₄	—	53	4	no	—
183	C₂×C₄	—	53	2	no	—
184	C₂×C₄	—	53	2	no	—
185	C₂×C₄	—	106	2	no	—
186	C₂×C₄	—	424	4	no	—
187	C₂×C₄	—	212	4	no	—
188	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
189	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
190	C₂×C₄	#13133236	1	2	yes	—
191	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
192	C₈	—	106	2	no	—
193	C₈	D₄×Dic₅₃	1	2	yes	—
194	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
195	C₈	C₂×#11719947	1	1	yes	characteristic
196	C₂²	D₈×Dic₅₃	1	2	yes	—
197	C₂²	Q₃₂×D₅₃	1	2	yes	—
198	C₂²	#16811056	1	2	yes	—
199	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
200	C₄	—	4	8	no	—
201	C₄	—	53	4	no	—
202	C₄	—	106	2	no	—
203	C₄	—	424	4	no	—
204	C₄	D₈×Dic₅₃	1	2	yes	—
205	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
206	C₄	C₂×#13133236	1	1	yes	characteristic
207	C₂	Q₃₂×Dic₅₃	1	4	yes	direct product factor
208	C₂	C₂×D₈×Dic₅₃	1	1	yes	characteristic
209	C₂	C₂×Q₃₂×D₅₃	1	1	yes	characteristic
210	C₂	C₂×#16811056	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×Q₃₂×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×Q₃₂	D₅₃
hypocenter	C₅₃	C₂×C₄×Q₃₂
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

Complements

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

N	H ≅ G/N	type
2. Q₃₂×Dic₅₃	C₂	direct (N × H)
18. C₂×Q₃₂×C₅₃	C₄	semidirect
41. Q₃₂×C₅₃	C₂×C₄	semidirect
83. C₂×Dic₅₃	Q₃₂	direct (N × H)
101. Dic₅₃	C₂×Q₃₂	direct (N × H)
108. C₂×C₅₃	C₄×Q₃₂	semidirect
114. C₂×Q₃₂	Dic₅₃	direct (N × H)
126. C₅₃	C₂×C₄×Q₃₂	semidirect
140. Q₃₂	C₂×Dic₅₃	direct (N × H)
207. C₂	Q₃₂×Dic₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×Dic₅₃	2
Q₃₂×Dic₅₃	2
C₂×#16810449	2
C₂×#16810427	2
C₂×C₁₆×Dic₅₃	2
C₂²×Q₃₂×C₅₃	2
C₂×C₄×Q₃₂	53

Subgroups — C2×Q32×Dic53 #27537367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₃₂×Dic₅₃ #27537367