Subgroups — Q₈×D₅₃⋊C₁₆ #28328420

Order 13568.

582 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291129	C₂	1	6	yes	maximal, complemented
2	Q₈×#11719910	C₂	1	2	yes	maximal, complemented
3	D₅₃⋊(C₄⋊C₁₆)	C₂	1	3	yes	maximal
4	C₄×D₅₃⋊C₁₆	C₂	1	3	yes	maximal
5	C₈×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×D₅₃⋊C₁₆	C₂²	1	3	yes	—
7	C₂×D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16811544	C₂²	1	6	yes	—
9	C₄⋊#11719910	C₂²	1	6	yes	—
10	#16811509	C₂²	1	6	yes	—
11	C₄×#11719910	C₂²	1	6	yes	—
12	C₄×Q₈×D₅₃	C₄	1	1	yes	characteristic
13	Q₈×#88322	C₂²	1	1	yes	characteristic
14	C₈×Dic₁₀₆	C₄	1	3	yes	—
15	#16810747	C₂²	1	3	yes	—
16	C₄⋊C₈×D₅₃	C₄	1	3	yes	—
17	C₄×C₈×D₅₃	C₂²	1	3	yes	—
18	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
19	D₅₃⋊C₁₆	—	2	6	no	—
20	D₅₃⋊C₁₆	Q₈	1	4	yes	direct product factor
21	C₂×#11719910	C₂³	1	6	yes	—
22	C₂×#11719910	C₂³	1	2	yes	—
23	C₂×Q₈×D₅₃	C₈	1	1	yes	characteristic
24	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
25	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
26	Q₈×Dic₅₃	C₈	1	1	yes	characteristic
27	C₄×Dic₁₀₆	C₂×C₄	1	3	yes	—
28	#13133182	C₈	1	3	yes	—
29	C₄⋊C₄×D₅₃	C₈	1	3	yes	—
30	C₄²×D₅₃	C₂×C₄	1	3	yes	—
31	C₄⋊#88322	C₂×C₄	1	3	yes	—
32	#13133146	C₂×C₄	1	6	yes	—
33	C₄×#88322	C₂³	1	3	yes	—
34	C₈×Dic₅₃	C₂×C₄	1	3	yes	—
35	C₈×Dic₅₃	C₂³	1	3	yes	—
36	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	C₂×C₄	1	3	yes	—
38	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
39	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
40	Q₈×D₅₃	C₁₆	1	8	yes	complemented
41	C₂×C₄×D₅₃	C₂×C₈	1	3	yes	—
42	C₂×C₄×D₅₃	C₂×C₈	1	3	yes	—
43	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₁₀₆	C₂×C₈	1	3	yes	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
46	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₄⋊C₄×C₅₃	C₂×C₈	1	3	yes	—
48	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
49	C₄⋊Dic₅₃	C₂×C₈	1	3	yes	—
50	#11719919	C₂×C₈	1	6	yes	—
51	C₄×Dic₅₃	C₂×C₈	1	3	yes	—
52	C₄×Dic₅₃	C₂×C₈	1	3	yes	—
53	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
54	C₂×#88322	C₂²×C₄	1	3	yes	—
55	C₂×#88322	C₂⁴	1	1	yes	characteristic
56	C₈×D₅₃	—	2	6	no	—
57	C₈×D₅₃	C₂×Q₈	1	2	yes	—
58	C₈×D₅₃	#41	1	2	yes	—
59	#11719910	—	2	6	no	—
60	#11719910	C₂×Q₈	1	4	yes	complemented
61	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
62	Q₈×C₅₃	C₂×C₁₆	1	4	yes	complemented
63	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
64	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
65	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
66	C₂×Dic₅₃	C₂²×C₈	1	3	yes	—
67	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×Dic₅₃	C₂²×C₈	1	3	yes	—
69	C₄×D₅₃	—	2	6	no	—
70	C₄×D₅₃	C₂×C₁₆	1	12	yes	—
71	C₄×D₅₃	C₄×Q₈	1	2	yes	—
72	C₄×D₅₃	#131	1	2	yes	—
73	Dic₁₀₆	C₂×C₁₆	1	12	yes	complemented
74	C₈×C₅₃	—	2	3	no	—
75	C₈×C₅₃	C₄×Q₈	1	2	yes	—
76	#88322	—	2	3	no	—
77	#88322	C₂²×Q₈	1	1	yes	characteristic
78	#88322	C₂×#41	1	1	yes	characteristic
79	C₂×Q₈×C₁₆	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
81	C₂×D₅₃	C₈×Q₈	1	2	yes	—
82	C₂×D₅₃	C₂²×C₁₆	1	2	yes	—
83	C₂×D₅₃	#504	1	2	yes	—
84	C₄×C₅₃	—	2	3	no	—
85	C₄×C₅₃	C₂²×C₁₆	1	6	yes	—
86	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
87	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
88	Dic₅₃	—	2	3	no	—
89	Dic₅₃	C₈×Q₈	1	2	yes	—
90	Dic₅₃	C₂²×C₁₆	1	6	yes	—
91	C₂×C₈×Q₈	—	53	1	no	—
92	Q₈×C₁₆	—	53	2	no	—
93	Q₈×C₁₆	—	53	6	no	—
94	C₂×C₄⋊C₁₆	—	53	3	no	—
95	C₂×C₄×C₁₆	—	53	3	no	—
96	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
97	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
98	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
99	D₅₃	Q₈×C₁₆	1	4	yes	complemented
100	C₂×C₄×Q₈	—	53	1	no	—
101	C₂²×C₁₆	—	53	1	no	—
102	C₂²×C₁₆	—	53	3	no	—
103	C₈×Q₈	—	53	1	no	—
104	C₈×Q₈	—	53	3	no	—
105	C₈×Q₈	—	53	3	no	—
106	C₈×Q₈	#6157	1	1	yes	characteristic
107	C₂×C₄⋊C₈	—	53	3	no	—
108	C₂×C₄×C₈	—	53	3	no	—
109	C₄⋊C₁₆	—	53	6	no	—
110	C₄⋊C₁₆	—	53	6	no	—
111	C₄×C₁₆	—	53	6	no	—
112	C₄×C₁₆	—	53	6	no	—
113	C₅₃	C₂×Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	53	1	no	—
115	C₂²×C₈	—	53	1	no	—
116	C₂²×C₈	—	53	3	no	—
117	C₄×Q₈	—	53	1	no	—
118	C₄×Q₈	—	53	3	no	—
119	C₄×Q₈	—	53	3	no	—
120	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
121	C₂×C₄⋊C₄	—	53	3	no	—
122	C₂×C₄²	—	53	3	no	—
123	C₂×C₁₆	—	53	2	no	—
124	C₂×C₁₆	—	53	4	no	—
125	C₂×C₁₆	—	53	6	no	—
126	C₂×C₁₆	—	106	6	no	—
127	C₄⋊C₈	—	53	6	no	—
128	C₄⋊C₈	—	53	3	no	—
129	C₄⋊C₈	C₂×#6157	1	3	yes	—
130	C₄×C₈	—	53	3	no	—
131	C₄×C₈	—	53	3	no	—
132	C₄×C₈	—	53	3	no	—
133	C₄×C₈	C₂×#6157	1	3	yes	—
134	C₂×Q₈	—	53	8	no	—
135	C₂×Q₈	—	53	3	no	—
136	C₂×Q₈	D₅₃⋊C₈	1	1	yes	characteristic
137	C₂²×C₄	—	53	3	no	—
138	C₂²×C₄	—	53	1	no	—
139	C₂²×C₄	—	53	3	no	—
140	C₂×C₈	—	53	1	no	—
141	C₂×C₈	—	53	2	no	—
142	C₂×C₈	—	53	2	no	—
143	C₂×C₈	—	53	3	no	—
144	C₂×C₈	—	106	6	no	—
145	C₂×C₈	C₂²×#6157	1	3	yes	—
146	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
147	C₄⋊C₄	—	53	6	no	—
148	C₄⋊C₄	—	53	3	no	—
149	C₄⋊C₄	D₅₃⋊C₈	1	3	yes	—
150	C₄²	—	53	3	no	—
151	C₄²	—	53	3	no	—
152	C₄²	—	53	3	no	—
153	C₄²	C₂²×#6157	1	3	yes	—
154	C₁₆	—	53	4	no	—
155	C₁₆	—	106	6	no	—
156	C₂³	—	53	1	no	—
157	Q₈	—	53	12	no	—
158	Q₈	D₅₃⋊C₁₆	1	4	yes	direct product factor
159	C₂×C₄	—	53	1	no	—
160	C₂×C₄	—	53	3	no	—
161	C₂×C₄	—	53	12	no	—
162	C₂×C₄	—	53	2	no	—
163	C₂×C₄	—	53	2	no	—
164	C₂×C₄	—	53	3	no	—
165	C₂×C₄	—	106	6	no	—
166	C₂×C₄	C₂×D₅₃⋊C₈	1	3	yes	—
167	C₂×C₄	C₂×D₅₃⋊C₈	1	3	yes	—
168	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
169	C₈	—	2	3	no	—
170	C₈	—	53	1	no	—
171	C₈	—	53	1	no	—
172	C₈	—	106	3	no	—
173	C₈	Q₈×#6157	1	2	yes	—
174	C₂²	—	53	2	no	—
175	C₂²	—	53	2	no	—
176	C₂²	—	53	2	no	—
177	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	53	6	no	—
180	C₄	—	53	2	no	—
181	C₄	—	106	3	no	—
182	C₄	C₂×D₅₃⋊C₁₆	1	6	yes	—
183	C₄	C₂×Q₈×#6157	1	1	yes	characteristic
184	C₄	D₅₃⋊#131	1	1	yes	characteristic
185	C₂	—	53	4	no	—
186	C₂	Q₈×D₅₃⋊C₈	1	1	yes	characteristic
187	C₂	C₂²×D₅₃⋊C₁₆	1	1	yes	characteristic
188	C₂	D₅₃⋊#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×#6157
commutator	C₂×C₅₃	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₈×Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₂²×D₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₈×Q₈	#6157
hypocenter	C₅₃	C₂×Q₈×C₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. #20291129	C₂	semidirect
2. Q₈×#11719910	C₂	semidirect
20. D₅₃⋊C₁₆	Q₈	direct (N × H)
40. Q₈×D₅₃	C₁₆	semidirect
60. #11719910	C₂×Q₈	semidirect
62. Q₈×C₅₃	C₂×C₁₆	semidirect
73. Dic₁₀₆	C₂×C₁₆	semidirect
99. D₅₃	Q₈×C₁₆	semidirect
113. C₅₃	C₂×Q₈×C₁₆	semidirect
158. Q₈	D₅₃⋊C₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
#20291129	2
Q₈×#11719910	2
D₅₃⋊(C₄⋊C₁₆)	2
C₄×D₅₃⋊C₁₆	2
C₈×Q₈×D₅₃	2
C₂×Q₈×C₁₆	53

Subgroups — Q8×D53⋊C16 #28328420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×D₅₃⋊C₁₆ #28328420