Subgroups — C₂²×Q₁₆×Dic₅₃ #28248108

Order 13568.

2830 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×Dic₅₃	C₂	1	2	yes	maximal
2	C₂×Q₁₆×Dic₅₃	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13133151	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13133141	C₂	1	2	yes	maximal
5	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88322	C₂²	1	1	yes	characteristic
8	C₂×Q₈×Dic₅₃	—	2	12	no	—
9	C₂×Q₈×Dic₅₃	C₂²	1	24	yes	—
10	C₂²×C₄⋊Dic₅₃	—	2	2	no	—
11	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂²×C₄×Dic₅₃	—	2	2	no	—
13	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	Q₁₆×Dic₅₃	C₂²	1	64	yes	direct product factor
15	C₂×#13133151	C₂²	1	12	yes	—
16	C₂×#13133141	C₂²	1	24	yes	—
17	C₂×C₈×Dic₅₃	C₂²	1	12	yes	—
18	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
19	C₂²×Q₁₆×C₅₃	C₄	1	16	yes	complemented
20	C₂²×Q₁₆×C₅₃	C₂²	1	12	yes	—
21	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂³×Dic₅₃	—	2	2	no	—
23	C₂³×Dic₅₃	D₄	1	1	yes	characteristic
24	C₂³×Dic₅₃	D₄	1	1	yes	characteristic
25	C₂²×#88322	—	2	4	no	—
26	C₂²×#88322	C₂³	1	6	yes	—
27	Q₈×Dic₅₃	—	2	48	no	—
28	Q₈×Dic₅₃	C₂³	1	32	yes	—
29	C₂×C₄⋊Dic₅₃	—	2	6	no	—
30	C₂×C₄⋊Dic₅₃	—	2	24	no	—
31	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
32	C₂×C₄×Dic₅₃	—	2	24	no	—
33	C₂×C₄×Dic₅₃	D₄	1	6	yes	—
34	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
35	#13133151	C₂³	1	16	yes	—
36	#13133141	C₂³	1	32	yes	—
37	C₈×Dic₅₃	C₂³	1	16	yes	—
38	C₂²×Q₈×C₅₃	—	2	6	no	—
39	C₂²×Q₈×C₅₃	—	2	8	no	—
40	C₂²×Q₈×C₅₃	C₂×C₄	1	16	yes	—
41	C₂²×Q₈×C₅₃	C₂³	1	12	yes	—
42	C₂³×C₄×C₅₃	—	2	2	no	—
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×Q₁₆×C₅₃	C₂×C₄	1	96	yes	complemented
45	C₂×Q₁₆×C₅₃	C₂³	1	16	yes	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
48	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×Q₈×C₅₃	—	2	48	no	—
50	C₂×Q₈×C₅₃	—	2	12	no	—
51	C₂×Q₈×C₅₃	—	2	24	no	—
52	C₂×Q₈×C₅₃	C₂²×C₄	1	48	yes	—
53	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
54	C₂²×C₄×C₅₃	—	2	16	no	—
55	C₂²×C₄×C₅₃	—	2	12	no	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₅₃	#41	1	4	yes	—
59	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
60	C₂²×Dic₅₃	—	2	4	no	—
61	C₂²×Dic₅₃	—	2	12	no	—
62	C₂²×Dic₅₃	—	4	8	no	—
63	C₂²×Dic₅₃	Q₁₆	1	8	yes	direct product factor
64	C₂²×Dic₅₃	C₂×D₄	1	6	yes	—
65	C₂²×Dic₅₃	C₂×D₄	1	6	yes	—
66	Q₁₆×C₅₃	C₂²×C₄	1	64	yes	complemented
67	C₂×C₈×C₅₃	C₂²×C₄	1	24	yes	—
68	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
69	C₄⋊Dic₅₃	—	2	12	no	—
70	C₄⋊Dic₅₃	—	2	32	no	—
71	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
72	C₄×Dic₅₃	—	2	32	no	—
73	C₄×Dic₅₃	C₂×D₄	1	12	yes	—
74	C₄×Dic₅₃	C₂⁴	1	4	yes	—
75	C₂×#88322	—	2	12	no	—
76	C₂×#88322	C₂⁴	1	4	yes	—
77	C₂³×C₅₃	C₄×D₄	1	4	yes	—
78	C₂³×C₅₃	C₂×Q₁₆	1	4	yes	—
79	C₂³×C₅₃	#135	1	4	yes	—
80	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
81	Q₈×C₅₃	—	2	48	no	—
82	Q₈×C₅₃	—	2	8	no	—
83	Q₈×C₅₃	C₂³×C₄	1	16	yes	—
84	C₂×C₄×C₅₃	—	2	48	no	—
85	C₂×C₄×C₅₃	—	2	8	no	—
86	C₂×C₄×C₅₃	C₄×D₄	1	12	yes	—
87	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
88	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
89	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
90	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
91	C₂×Dic₅₃	—	2	12	no	—
92	C₂×Dic₅₃	—	2	8	no	—
93	C₂×Dic₅₃	—	4	24	no	—
94	C₂×Dic₅₃	C₂×Q₁₆	1	24	yes	direct product factor
95	C₂×Dic₅₃	C₂²×D₄	1	4	yes	—
96	C₂×Dic₅₃	C₂²×D₄	1	4	yes	—
97	C₈×C₅₃	C₂³×C₄	1	8	yes	—
98	#88322	—	2	4	no	—
99	C₂²×C₄×Q₁₆	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	C₄×Q₁₆	1	16	yes	complemented
101	C₂²×C₅₃	C₂×C₄×D₄	1	6	yes	—
102	C₂²×C₅₃	C₂²×Q₁₆	1	6	yes	—
103	C₂²×C₅₃	C₂×#135	1	6	yes	—
104	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
105	C₄×C₅₃	—	2	16	no	—
106	C₄×C₅₃	C₂×C₄×D₄	1	6	yes	—
107	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
108	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
109	Dic₅₃	—	2	4	no	—
110	Dic₅₃	—	4	8	no	—
111	Dic₅₃	C₂²×Q₁₆	1	8	yes	direct product factor
112	C₂³×Q₁₆	D₅₃	1	1	yes	characteristic
113	C₂²×C₄×Q₈	—	53	2	no	—
114	C₂×C₄×Q₁₆	—	53	24	no	—
115	C₂²×#107	—	53	1	no	—
116	C₂²×#103	—	53	2	no	—
117	C₂²×C₄×C₈	—	53	1	no	—
118	C₂×C₅₃	C₂×C₄×Q₁₆	1	12	yes	complemented
119	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
120	C₂×C₅₃	C₂³×Q₁₆	1	1	yes	characteristic
121	C₂×C₅₃	C₂²×#135	1	1	yes	characteristic
122	C₂³×Q₈	C₂×D₅₃	1	2	yes	—
123	C₂²×Q₁₆	Dic₅₃	1	16	yes	direct product factor
124	C₂²×Q₁₆	C₂×D₅₃	1	12	yes	—
125	C₂³×C₈	—	53	1	no	—
126	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
127	C₂×C₄×Q₈	—	53	24	no	—
128	C₂×C₄×Q₈	—	106	12	no	—
129	C₂²×C₄⋊C₄	—	53	2	no	—
130	C₂²×C₄⋊C₄	—	106	2	no	—
131	C₂²×C₄²	—	53	1	no	—
132	C₂²×C₄²	—	106	2	no	—
133	C₄×Q₁₆	—	53	64	no	—
134	C₂×#107	—	53	12	no	—
135	C₂×#103	—	53	24	no	—
136	C₂×C₄×C₈	—	53	12	no	—
137	C₅₃	C₂²×C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₈	—	2	6	no	—
139	C₂²×Q₈	—	2	8	no	—
140	C₂²×Q₈	C₂×Dic₅₃	1	16	yes	—
141	C₂²×Q₈	C₂²×D₅₃	1	12	yes	—
142	C₂³×C₄	—	2	2	no	—
143	C₂³×C₄	—	53	1	no	—
144	C₂³×C₄	—	53	1	no	—
145	C₂³×C₄	—	106	2	no	—
146	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
147	C₂×Q₁₆	C₂×Dic₅₃	1	96	yes	direct product factor
148	C₂×Q₁₆	C₂²×D₅₃	1	16	yes	—
149	C₂²×C₈	—	53	6	no	—
150	C₂²×C₈	—	106	4	no	—
151	C₂²×C₈	C₂×Dic₅₃	1	8	yes	—
152	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
153	C₄×Q₈	—	53	32	no	—
154	C₄×Q₈	—	106	48	no	—
155	C₂×C₄⋊C₄	—	53	12	no	—
156	C₂×C₄⋊C₄	—	106	24	no	—
157	C₂×C₄⋊C₄	—	106	6	no	—
158	C₂×C₄²	—	53	6	no	—
159	C₂×C₄²	—	53	6	no	—
160	C₂×C₄²	—	106	24	no	—
161	#107	—	53	16	no	—
162	#103	—	53	32	no	—
163	C₄×C₈	—	53	16	no	—
164	C₂⁴	D₄×D₅₃	1	1	yes	characteristic
165	C₂×Q₈	—	2	48	no	—
166	C₂×Q₈	—	2	12	no	—
167	C₂×Q₈	—	2	24	no	—
168	C₂×Q₈	C₂²×Dic₅₃	1	48	yes	—
169	C₂×Q₈	C₂³×D₅₃	1	8	yes	—
170	C₂²×C₄	—	2	16	no	—
171	C₂²×C₄	—	2	12	no	—
172	C₂²×C₄	—	53	8	no	—
173	C₂²×C₄	—	53	6	no	—
174	C₂²×C₄	—	53	6	no	—
175	C₂²×C₄	—	106	4	no	—
176	C₂²×C₄	—	106	12	no	—
177	C₂²×C₄	—	212	8	no	—
178	C₂²×C₄	D₄×D₅₃	1	3	yes	—
179	C₂²×C₄	#11719947	1	4	yes	—
180	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
181	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
182	Q₁₆	C₂²×Dic₅₃	1	64	yes	direct product factor
183	C₂×C₈	—	53	4	no	—
184	C₂×C₈	—	106	12	no	—
185	C₂×C₈	C₂²×Dic₅₃	1	24	yes	—
186	C₂×C₈	C₂³×D₅₃	1	4	yes	—
187	C₄⋊C₄	—	53	8	no	—
188	C₄⋊C₄	—	106	32	no	—
189	C₄⋊C₄	—	106	12	no	—
190	C₄²	—	53	12	no	—
191	C₄²	—	53	4	no	—
192	C₄²	—	106	32	no	—
193	C₂³	D₄×Dic₅₃	1	4	yes	—
194	C₂³	Q₁₆×D₅₃	1	4	yes	—
195	C₂³	#13133240	1	4	yes	—
196	C₂³	C₂×D₄×D₅₃	1	3	yes	—
197	Q₈	—	2	48	no	—
198	Q₈	—	2	8	no	—
199	Q₈	C₂³×Dic₅₃	1	16	yes	—
200	C₂×C₄	—	2	48	no	—
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	53	24	no	—
203	C₂×C₄	—	53	4	no	—
204	C₂×C₄	—	53	4	no	—
205	C₂×C₄	—	106	12	no	—
206	C₂×C₄	—	106	8	no	—
207	C₂×C₄	—	212	24	no	—
208	C₂×C₄	D₄×Dic₅₃	1	12	yes	—
209	C₂×C₄	C₂³×Dic₅₃	1	6	yes	—
210	C₂×C₄	C₂×D₄×D₅₃	1	3	yes	—
211	C₂×C₄	C₂×#11719947	1	6	yes	—
212	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
213	C₈	—	106	4	no	—
214	C₈	C₂³×Dic₅₃	1	8	yes	—
215	C₂²	Q₁₆×Dic₅₃	1	16	yes	direct product factor
216	C₂²	C₂×D₄×Dic₅₃	1	6	yes	—
217	C₂²	C₂×Q₁₆×D₅₃	1	6	yes	—
218	C₂²	C₂×#13133240	1	6	yes	—
219	C₂²	C₂²×D₄×D₅₃	1	1	yes	characteristic
220	C₄	—	2	16	no	—
221	C₄	—	53	8	no	—
222	C₄	—	106	4	no	—
223	C₄	—	212	8	no	—
224	C₄	C₂×D₄×Dic₅₃	1	6	yes	—
225	C₄	C₂⁴×Dic₅₃	1	1	yes	characteristic
226	C₄	C₂²×#11719947	1	1	yes	characteristic
227	C₂	C₂×Q₁₆×Dic₅₃	1	12	yes	direct product factor
228	C₂	C₂²×D₄×Dic₅₃	1	1	yes	characteristic
229	C₂	C₂²×Q₁₆×D₅₃	1	1	yes	characteristic
230	C₂	C₂²×#13133240	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₅₃
commutator	C₄×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂³×Q₁₆×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂³×Q₁₆	D₅₃
hypocenter	C₅₃	C₂²×C₄×Q₁₆
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Q₁₆×Dic₅₃	C₂	direct (N × H)
14. Q₁₆×Dic₅₃	C₂²	direct (N × H)
19. C₂²×Q₁₆×C₅₃	C₄	semidirect
44. C₂×Q₁₆×C₅₃	C₂×C₄	semidirect
63. C₂²×Dic₅₃	Q₁₆	direct (N × H)
66. Q₁₆×C₅₃	C₂²×C₄	semidirect
94. C₂×Dic₅₃	C₂×Q₁₆	direct (N × H)
100. C₂²×C₅₃	C₄×Q₁₆	semidirect
111. Dic₅₃	C₂²×Q₁₆	direct (N × H)
118. C₂×C₅₃	C₂×C₄×Q₁₆	semidirect
123. C₂²×Q₁₆	Dic₅₃	direct (N × H)
137. C₅₃	C₂²×C₄×Q₁₆	semidirect
147. C₂×Q₁₆	C₂×Dic₅₃	direct (N × H)
182. Q₁₆	C₂²×Dic₅₃	direct (N × H)
215. C₂²	Q₁₆×Dic₅₃	direct (N × H)
227. C₂	C₂×Q₁₆×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
C₂²×Q₈×Dic₅₃	2
C₂×Q₁₆×Dic₅₃	2
C₂²×#13133151	2
C₂²×#13133141	2
C₂²×C₈×Dic₅₃	2
C₂³×Q₁₆×C₅₃	2
C₂²×C₄×Q₁₆	53

Subgroups — C22×Q16×Dic53 #28248108

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₁₆×Dic₅₃ #28248108