Subgroups — C₂×C₄×#14519683 #29600183

Order 15616.

270 conjugacy classes of subgroups.

The subgroup lattice has 114 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519683	C₂	1	2	yes	maximal
2	C₄×#14519683	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519683	C₄	1	8	yes	direct product factor
5	C₂×#14519683	C₂²	1	4	yes	—
6	C₂²×C₁₆×C₆₁	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
9	C₄×C₁₆×C₆₁	C₄	1	2	yes	—
10	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
11	#14519683	C₂×C₄	1	8	yes	direct product factor
12	C₂²×C₈×C₆₁	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²×C₆₁	C₈	1	1	yes	characteristic
15	C₂×C₁₆×C₆₁	C₈	1	4	yes	—
16	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	—
18	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₆₁	C₈	1	2	yes	—
21	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₆₁	C₁₆	1	2	yes	—
23	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
24	C₂×C₈×C₆₁	C₁₆	1	4	yes	—
25	C₂×C₈×C₆₁	C₄²	1	2	yes	—
26	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₆₁	C₁₆	1	2	yes	—
31	C₄²×C₆₁	C₂×C₈	1	2	yes	—
32	C₁₆×C₆₁	C₄²	1	2	yes	—
33	C₁₆×C₆₁	C₂×C₈	1	4	yes	—
34	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
35	C₂³×C₆₁	C₂×C₁₆	1	1	yes	characteristic
36	C₂×C₄×C₆₁	C₃₂	1	8	yes	complemented
37	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
38	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₂×C₁₆	1	2	yes	—
40	C₂×C₄×C₆₁	C₂×C₁₆	1	4	yes	—
41	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
42	C₈×C₆₁	C₄×C₈	1	2	yes	—
43	C₈×C₆₁	C₂×C₁₆	1	4	yes	—
44	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
45	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₄×C₃₂	—	61	1	no	maximal, Sylow
47	C₂²×C₆₁	C₄×C₁₆	1	2	yes	—
48	C₂²×C₆₁	C₂×C₃₂	1	4	yes	—
49	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
50	C₄×C₆₁	C₄×C₁₆	1	2	yes	—
51	C₄×C₆₁	C₂×C₃₂	1	8	yes	complemented
52	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
53	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
54	C₂²×C₃₂	—	61	2	no	—
55	C₂×C₄×C₁₆	D₆₁	1	1	yes	characteristic
56	C₄×C₃₂	—	61	4	no	—
57	C₂×C₆₁	C₄×C₃₂	1	4	yes	complemented
58	C₂×C₆₁	C₂×C₄×C₁₆	1	1	yes	characteristic
59	C₂×C₆₁	C₂²×C₃₂	1	2	yes	—
60	C₂²×C₁₆	Dic₆₁	1	1	yes	characteristic
61	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
62	C₂×C₄×C₈	Dic₆₁	1	1	yes	characteristic
63	C₂×C₃₂	—	61	8	no	—
64	C₂×C₃₂	—	61	4	no	—
65	C₄×C₁₆	Dic₆₁	1	2	yes	—
66	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
67	C₆₁	C₂×C₄×C₃₂	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₈	#92418	1	1	yes	characteristic
69	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
70	C₂×C₄²	#92418	1	1	yes	characteristic
71	C₂×C₁₆	#92418	1	4	yes	—
72	C₂×C₁₆	C₄×D₆₁	1	4	yes	—
73	C₂×C₁₆	C₂×Dic₆₁	1	2	yes	—
74	C₂×C₁₆	C₂×Dic₆₁	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
76	C₄×C₈	#92418	1	2	yes	—
77	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
78	C₃₂	—	61	8	no	—
79	C₂²×C₄	#11758185	1	2	yes	—
80	C₂²×C₄	C₂×#92418	1	1	yes	characteristic
81	C₂×C₈	#11758185	1	4	yes	—
82	C₂×C₈	C₂×#92418	1	1	yes	characteristic
83	C₂×C₈	C₂×#92418	1	2	yes	—
84	C₂×C₈	C₂×#92418	1	2	yes	—
85	C₂×C₈	C₄×Dic₆₁	1	2	yes	—
86	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
87	C₄²	#11758185	1	2	yes	—
88	C₄²	C₂×#92418	1	2	yes	—
89	C₁₆	C₂×#92418	1	4	yes	—
90	C₁₆	C₄×Dic₆₁	1	2	yes	—
91	C₁₆	C₂×C₄×D₆₁	1	2	yes	—
92	C₂³	C₂×#11758185	1	1	yes	characteristic
93	C₂×C₄	#14519683	1	8	yes	direct product factor
94	C₂×C₄	C₄×#92418	1	2	yes	—
95	C₂×C₄	C₂×#11758185	1	1	yes	characteristic
96	C₂×C₄	C₂×#11758185	1	2	yes	—
97	C₂×C₄	C₂×#11758185	1	4	yes	—
98	C₂×C₄	C₂²×#92418	1	1	yes	characteristic
99	C₈	C₄×#92418	1	2	yes	—
100	C₈	C₂×#11758185	1	4	yes	—
101	C₈	C₂×C₄×Dic₆₁	1	1	yes	characteristic
102	C₈	C₂²×#92418	1	1	yes	characteristic
103	C₂²	C₄×#11758185	1	2	yes	—
104	C₂²	C₂×#14519683	1	4	yes	—
105	C₂²	C₂²×#11758185	1	1	yes	characteristic
106	C₄	C₄×#11758185	1	2	yes	—
107	C₄	C₂×#14519683	1	8	yes	direct product factor
108	C₄	C₂×C₄×#92418	1	1	yes	characteristic
109	C₄	C₂²×#11758185	1	1	yes	characteristic
110	C₂	C₄×#14519683	1	4	yes	direct product factor
111	C₂	C₂×C₄×#11758185	1	1	yes	characteristic
112	C₂	C₂²×#14519683	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄×C₁₆	D₆₁
commutator, hypocenter	C₆₁	C₂×C₄×C₃₂
Frattini	C₂×C₁₆	C₂²×D₆₁
Fitting	C₂×C₄×C₁₆×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₁₆
cosocle	C₂×C₁₆×C₆₁	C₂³
LCS	G ▷ C₆₁
UCS	C₁ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#14519683	C₂	direct (N × H)
4. C₂×#14519683	C₄	direct (N × H)
11. #14519683	C₂×C₄	direct (N × H)
36. C₂×C₄×C₆₁	C₃₂	semidirect
51. C₄×C₆₁	C₂×C₃₂	semidirect
57. C₂×C₆₁	C₄×C₃₂	semidirect
67. C₆₁	C₂×C₄×C₃₂	semidirect
93. C₂×C₄	#14519683	direct (N × H)
107. C₄	C₂×#14519683	direct (N × H)
110. C₂	C₄×#14519683	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×#14519683	2
C₄×#14519683	2
C₂×C₄×C₁₆×C₆₁	2
C₂×C₄×C₃₂	61

Subgroups — C2×C4×#14519683 #29600183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14519683 #29600183