Subgroups — C₂×C₁₆×#88322 #27293063

Order 13568.

398 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
3	C₁₆×#88322	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₄	1	2	yes	—
6	C₂×C₄×#88322	C₄	1	2	yes	—
7	C₄×#11719908	C₄	1	2	yes	—
8	C₄×#11719908	C₂²	1	2	yes	—
9	C₈×#88322	C₄	1	2	yes	—
10	C₈×#88322	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
14	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂²×#88322	C₈	1	4	yes	—
16	C₂×#11719908	C₈	1	8	yes	complemented
17	C₂×#11719908	C₂×C₄	1	4	yes	—
18	C₄×#88322	C₈	1	4	yes	—
19	C₄×#88322	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₅₃	C₈	1	8	yes	complemented
24	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
34	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
35	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₅₃	C₄²	1	1	yes	characteristic
38	C₄²×C₅₃	C₂×C₈	1	2	yes	—
39	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×#88322	C₁₆	1	16	yes	direct product factor
41	C₂×#88322	C₂×C₈	1	8	yes	—
42	C₁₆×C₅₃	C₂×C₈	1	8	yes	complemented
43	#11719908	C₂×C₈	1	8	yes	complemented
44	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
48	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×C₁₆	1	8	yes	—
50	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
52	C₈×C₅₃	C₄×C₈	1	2	yes	—
53	C₈×C₅₃	C₄×C₈	1	2	yes	—
54	C₈×C₅₃	C₂²×C₈	1	2	yes	—
55	C₈×C₅₃	C₂²×C₈	1	2	yes	—
56	#88322	C₂×C₁₆	1	16	yes	direct product factor
57	C₂×C₈×C₁₆	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	C₈²	1	2	yes	—
59	C₂²×C₅₃	C₄×C₁₆	1	4	yes	—
60	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
61	C₄×C₅₃	C₈²	1	2	yes	—
62	C₄×C₅₃	C₄×C₁₆	1	4	yes	—
63	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
64	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
65	C₄×C₅₃	C₂²×C₁₆	1	4	yes	—
66	C₂×C₄×C₁₆	—	53	1	no	—
67	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
68	C₂×C₈²	—	53	1	no	—
69	C₈×C₁₆	—	53	4	no	—
70	C₂×C₅₃	C₈×C₁₆	1	4	yes	complemented
71	C₂×C₅₃	C₂×C₈²	1	1	yes	characteristic
72	C₂×C₅₃	C₂×C₄×C₁₆	1	2	yes	—
73	C₂²×C₁₆	—	53	2	no	—
74	C₂²×C₁₆	Dic₅₃	1	2	yes	—
75	C₂×C₄×C₈	—	53	2	no	—
76	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
77	C₄×C₁₆	—	53	2	no	—
78	C₄×C₁₆	—	53	2	no	—
79	C₄×C₁₆	Dic₅₃	1	2	yes	—
80	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
81	C₈²	—	53	2	no	—
82	C₈²	—	53	2	no	—
83	C₅₃	C₂×C₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	53	4	no	—
85	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
86	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
87	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
88	C₂×C₁₆	—	53	4	no	—
89	C₂×C₁₆	—	53	8	no	—
90	C₂×C₁₆	#88322	1	8	yes	direct product factor
91	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
92	C₄×C₈	—	53	4	no	—
93	C₄×C₈	—	53	4	no	—
94	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
95	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
96	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
98	C₂²×C₄	C₈×D₅₃	1	2	yes	—
99	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
100	C₂×C₈	—	53	16	no	—
101	C₂×C₈	—	53	8	no	—
102	C₂×C₈	C₈×D₅₃	1	4	yes	—
103	C₂×C₈	C₂×#88322	1	4	yes	—
104	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
105	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
106	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
107	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
108	C₄²	C₈×D₅₃	1	2	yes	—
109	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
110	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
111	C₁₆	—	53	8	no	—
112	C₁₆	C₂×#88322	1	8	yes	direct product factor
113	C₂³	C₈×Dic₅₃	1	1	yes	characteristic
114	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
115	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
116	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
117	C₂×C₄	C₄×#88322	1	2	yes	—
118	C₂×C₄	C₁₆×D₅₃	1	8	yes	—
119	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
120	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
121	C₈	—	53	16	no	—
122	C₈	C₈×Dic₅₃	1	2	yes	—
123	C₈	C₄×#88322	1	2	yes	—
124	C₈	C₂×C₈×D₅₃	1	2	yes	—
125	C₈	C₂²×#88322	1	2	yes	—
126	C₂²	C₈×#88322	1	2	yes	—
127	C₂²	C₁₆×Dic₅₃	1	4	yes	—
128	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
129	C₄	C₈×#88322	1	2	yes	—
130	C₄	C₁₆×Dic₅₃	1	4	yes	—
131	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
132	C₄	C₂×C₄×#88322	1	1	yes	characteristic
133	C₄	C₂×C₁₆×D₅₃	1	4	yes	—
134	C₂	C₁₆×#88322	1	4	yes	direct product factor
135	C₂	C₂×C₈×#88322	1	1	yes	characteristic
136	C₂	C₂×C₁₆×Dic₅₃	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₁₆×#88322	C₂	direct (N × H)
16. C₂×#11719908	C₈	semidirect
23. C₂×C₁₆×C₅₃	C₈	semidirect
40. C₂×#88322	C₁₆	direct (N × H)
42. C₁₆×C₅₃	C₂×C₈	semidirect
43. #11719908	C₂×C₈	semidirect
56. #88322	C₂×C₁₆	direct (N × H)
70. C₂×C₅₃	C₈×C₁₆	semidirect
83. C₅₃	C₂×C₈×C₁₆	semidirect
90. C₂×C₁₆	#88322	direct (N × H)
112. C₁₆	C₂×#88322	direct (N × H)
134. C₂	C₁₆×#88322	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×C₄×#11719908	2
C₂×C₈×#88322	2
C₁₆×#88322	2
C₂×C₄×C₁₆×C₅₃	2
C₂×C₈×C₁₆	53

Subgroups — C2×C16×#88322 #27293063

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₆×#88322 #27293063