Subgroups — C₂×C₁₆×#88324 #28307743

Order 13568.

398 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811509	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#88324	C₂	1	1	yes	maximal, characteristic
3	C₁₆×#88324	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719910	C₄	1	2	yes	—
6	C₂×C₄×#88324	C₄	1	2	yes	—
7	#16811509	C₄	1	2	yes	—
8	#16811509	C₂²	1	2	yes	—
9	C₈×#88324	C₄	1	2	yes	—
10	C₈×#88324	C₂²	1	2	yes	—
11	C₂²×#11719908	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₁₆×Dic₅₃	C₄	1	2	yes	—
14	C₁₆×Dic₅₃	C₂²	1	2	yes	—
15	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
16	C₂²×#88324	C₈	1	4	yes	—
17	C₂×#11719910	C₈	1	8	yes	complemented
18	C₂×#11719910	C₂×C₄	1	4	yes	—
19	C₄×#88324	C₈	1	4	yes	—
20	C₄×#88324	C₂×C₄	1	4	yes	—
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×#11719908	C₈	1	4	yes	complemented
24	C₂×#11719908	C₂×C₄	1	2	yes	—
25	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	C₈	1	4	yes	complemented
31	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
33	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
34	C₂×#88324	C₁₆	1	16	yes	direct product factor
35	C₂×#88324	C₂×C₈	1	8	yes	—
36	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
41	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
42	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×#88322	C₄²	1	1	yes	characteristic
44	C₂×#88322	C₂×C₈	1	4	yes	—
45	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
46	#11719910	C₂×C₈	1	8	yes	complemented
47	C₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
48	#11719908	C₂×C₈	1	4	yes	complemented
49	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
53	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
54	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
55	C₂×Dic₅₃	C₂×C₁₆	1	8	yes	—
56	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
57	#88324	C₂×C₁₆	1	16	yes	direct product factor
58	C₈×C₅₃	C₄×C₈	1	2	yes	—
59	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
60	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
61	#88322	C₄×C₈	1	2	yes	—
62	#88322	C₂²×C₈	1	2	yes	—
63	C₂×C₈×C₁₆	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	C₈²	1	2	yes	—
65	C₂²×C₅₃	C₄×C₁₆	1	4	yes	—
66	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
67	C₄×C₅₃	C₈²	1	2	yes	—
68	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
69	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
70	Dic₅₃	C₄×C₁₆	1	4	yes	—
71	Dic₅₃	C₂²×C₁₆	1	4	yes	—
72	C₂×C₄×C₁₆	—	53	1	no	—
73	C₂×C₄×C₁₆	—	53	1	no	—
74	C₂×C₈²	—	53	1	no	—
75	C₈×C₁₆	—	53	4	no	—
76	C₂×C₅₃	C₈×C₁₆	1	4	yes	complemented
77	C₂×C₅₃	C₂×C₈²	1	1	yes	characteristic
78	C₂×C₅₃	C₂×C₄×C₁₆	1	2	yes	—
79	C₂²×C₁₆	—	53	2	no	—
80	C₂²×C₁₆	—	53	1	no	—
81	C₂²×C₁₆	#6157	1	1	yes	characteristic
82	C₂×C₄×C₈	—	53	2	no	—
83	C₂×C₄×C₈	—	53	1	no	—
84	C₄×C₁₆	—	53	2	no	—
85	C₄×C₁₆	—	53	2	no	—
86	C₄×C₁₆	—	53	2	no	—
87	C₄×C₁₆	—	53	2	no	—
88	C₈²	—	53	2	no	—
89	C₈²	—	53	2	no	—
90	C₅₃	C₂×C₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₈	—	53	4	no	—
92	C₂²×C₈	—	53	1	no	—
93	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
94	C₂×C₄²	—	53	1	no	—
95	C₂×C₁₆	—	53	4	no	—
96	C₂×C₁₆	—	53	8	no	—
97	C₂×C₁₆	—	53	2	no	—
98	C₂×C₁₆	—	53	4	no	—
99	C₂×C₁₆	#88324	1	4	yes	direct product factor
100	C₂×C₁₆	C₂×#6157	1	2	yes	—
101	C₄×C₈	—	53	4	no	—
102	C₄×C₈	—	53	4	no	—
103	C₄×C₈	—	53	1	no	—
104	C₄×C₈	—	53	1	no	—
105	C₄×C₈	—	53	1	no	—
106	C₄×C₈	—	53	1	no	—
107	C₂²×C₄	—	53	2	no	—
108	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
109	C₂×C₈	—	53	16	no	—
110	C₂×C₈	—	53	8	no	—
111	C₂×C₈	—	53	4	no	—
112	C₂×C₈	—	53	1	no	—
113	C₂×C₈	—	53	1	no	—
114	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
115	C₂×C₈	C₄×#6157	1	1	yes	characteristic
116	C₂×C₈	C₂×#88324	1	2	yes	—
117	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
118	C₄²	—	53	2	no	—
119	C₄²	—	53	1	no	—
120	C₄²	—	53	1	no	—
121	C₁₆	—	53	8	no	—
122	C₁₆	—	53	4	no	—
123	C₁₆	C₂×#88324	1	4	yes	direct product factor
124	C₂³	C₈×#6157	1	1	yes	characteristic
125	C₂×C₄	—	53	2	no	—
126	C₂×C₄	—	53	2	no	—
127	C₂×C₄	—	53	8	no	—
128	C₂×C₄	C₈×#6157	1	2	yes	—
129	C₂×C₄	C₈×#6157	1	1	yes	characteristic
130	C₂×C₄	C₄×#88324	1	2	yes	—
131	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
132	C₈	—	53	16	no	—
133	C₈	—	53	2	no	—
134	C₈	—	53	2	no	—
135	C₈	C₄×#88324	1	2	yes	—
136	C₈	C₂²×#88324	1	1	yes	characteristic
137	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
138	C₂²	C₈×#88324	1	2	yes	—
139	C₂²	C₁₆×#6157	1	4	yes	—
140	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
141	C₄	—	53	4	no	—
142	C₄	—	53	4	no	—
143	C₄	C₈×#88324	1	2	yes	—
144	C₄	C₂×C₈×#6157	1	1	yes	characteristic
145	C₄	C₂×C₄×#88324	1	1	yes	characteristic
146	C₂	C₁₆×#88324	1	4	yes	direct product factor
147	C₂	C₂×C₈×#88324	1	1	yes	characteristic
148	C₂	C₂×C₁₆×#6157	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₁₆	#6157
commutator, hypocenter	C₅₃	C₂×C₈×C₁₆
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₈×Dic₅₃	C₂³
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₁₆×#88324	C₂	direct (N × H)
17. C₂×#11719910	C₈	semidirect
23. C₂×#11719908	C₈	semidirect
30. C₂×C₁₆×C₅₃	C₈	semidirect
34. C₂×#88324	C₁₆	direct (N × H)
46. #11719910	C₂×C₈	semidirect
47. C₁₆×C₅₃	C₂×C₈	semidirect
48. #11719908	C₂×C₈	semidirect
57. #88324	C₂×C₁₆	direct (N × H)
76. C₂×C₅₃	C₈×C₁₆	semidirect
90. C₅₃	C₂×C₈×C₁₆	semidirect
99. C₂×C₁₆	#88324	direct (N × H)
123. C₁₆	C₂×#88324	direct (N × H)
146. C₂	C₁₆×#88324	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×#16811509	2
C₂×C₈×#88324	2
C₁₆×#88324	2
C₂×C₁₆×Dic₅₃	2
C₂×C₈×C₁₆	53

Subgroups — C2×C16×#88324 #28307743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₆×#88324 #28307743