Subgroups — C₂×#20241522 #26136451

Order 12032.

254 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×C₄₇⋊C₃₂	C₂	1	2	yes	maximal
2	#20241522	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₇⋊C₃₂	C₄	1	8	yes	complemented
5	C₂×C₄₇⋊C₃₂	C₂²	1	4	yes	—
6	C₂²×C₁₆×C₄₇	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
9	C₄×C₁₆×C₄₇	C₄	1	2	yes	—
10	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
11	C₄₇⋊C₃₂	C₂×C₄	1	8	yes	complemented
12	C₂²×C₈×C₄₇	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²×C₄₇	C₈	1	1	yes	characteristic
15	C₂×C₁₆×C₄₇	C₈	1	4	yes	—
16	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
18	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₄₇	C₈	1	2	yes	—
21	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₄₇	C₁₆	1	2	yes	—
23	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
24	C₂×C₈×C₄₇	C₁₆	1	4	yes	—
25	C₂×C₈×C₄₇	C₄²	1	2	yes	—
26	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₄₇	C₁₆	1	2	yes	—
31	C₄²×C₄₇	C₂×C₈	1	2	yes	—
32	C₁₆×C₄₇	C₄²	1	2	yes	—
33	C₁₆×C₄₇	C₂×C₈	1	4	yes	—
34	C₁₆×C₄₇	C₂²×C₄	1	2	yes	—
35	C₂³×C₄₇	C₂×C₁₆	1	1	yes	characteristic
36	C₂×C₄×C₄₇	—	2	4	no	—
37	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
38	C₂×C₄×C₄₇	C₂×C₁₆	1	4	yes	—
39	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
40	C₂×C₄×C₄₇	C₂×C₁₆	1	2	yes	—
41	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
42	C₈×C₄₇	C₄×C₈	1	2	yes	—
43	C₈×C₄₇	C₂×C₁₆	1	4	yes	—
44	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
45	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
46	C₂×#1397	—	47	1	no	maximal, Sylow
47	C₂²×C₄₇	C₄×C₁₆	1	2	yes	—
48	C₂²×C₄₇	M₆₄	1	4	yes	—
49	C₂²×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
50	C₄×C₄₇	—	2	4	no	—
51	C₄×C₄₇	C₄×C₁₆	1	2	yes	—
52	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
53	C₄×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
54	C₂²×C₃₂	—	47	2	no	—
55	C₂×C₄×C₁₆	D₄₇	1	1	yes	characteristic
56	#1397	—	47	4	no	—
57	C₂×C₄₇	#1397	1	4	yes	complemented
58	C₂×C₄₇	C₂×C₄×C₁₆	1	1	yes	characteristic
59	C₂×C₄₇	C₂×M₆₄	1	2	yes	—
60	C₂²×C₁₆	Dic₄₇	1	1	yes	characteristic
61	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
62	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
63	C₂×C₃₂	—	47	8	no	—
64	C₂×C₃₂	—	47	4	no	—
65	C₄×C₁₆	Dic₄₇	1	2	yes	—
66	C₄×C₁₆	C₂×D₄₇	1	2	yes	—
67	C₄₇	C₂×#1397	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₈	C₄₇⋊C₈	1	1	yes	characteristic
69	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
70	C₂×C₄²	C₄₇⋊C₈	1	1	yes	characteristic
71	C₂×C₁₆	C₄₇⋊C₈	1	4	yes	—
72	C₂×C₁₆	C₄×D₄₇	1	4	yes	—
73	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
74	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
76	C₄×C₈	C₄₇⋊C₈	1	2	yes	—
77	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
78	C₃₂	—	47	8	no	—
79	C₂²×C₄	C₄₇⋊C₁₆	1	2	yes	—
80	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
81	C₂×C₈	C₄₇⋊C₁₆	1	4	yes	—
82	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
83	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
84	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
85	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
86	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
87	C₄²	C₄₇⋊C₁₆	1	2	yes	—
88	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
89	C₁₆	C₂×C₄₇⋊C₈	1	4	yes	—
90	C₁₆	C₄×Dic₄₇	1	2	yes	—
91	C₁₆	C₂×C₄×D₄₇	1	2	yes	—
92	C₂³	C₂×C₄₇⋊C₁₆	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	C₄×C₄₇⋊C₈	1	2	yes	—
95	C₂×C₄	C₂×C₄₇⋊C₁₆	1	4	yes	—
96	C₂×C₄	C₂×C₄₇⋊C₁₆	1	2	yes	—
97	C₂×C₄	C₂×C₄₇⋊C₁₆	1	1	yes	characteristic
98	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
99	C₈	C₄×C₄₇⋊C₈	1	2	yes	—
100	C₈	C₂×C₄₇⋊C₁₆	1	4	yes	—
101	C₈	C₂×C₄×Dic₄₇	1	1	yes	characteristic
102	C₈	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
103	C₂²	C₄×C₄₇⋊C₁₆	1	2	yes	—
104	C₂²	#15688957	1	4	yes	—
105	C₂²	C₂²×C₄₇⋊C₁₆	1	1	yes	characteristic
106	C₄	—	2	4	no	—
107	C₄	C₄×C₄₇⋊C₁₆	1	2	yes	—
108	C₄	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
109	C₄	C₂²×C₄₇⋊C₁₆	1	1	yes	characteristic
110	C₂	#20241522	1	4	yes	direct product factor
111	C₂	C₂×C₄×C₄₇⋊C₁₆	1	1	yes	characteristic
112	C₂	C₂×#15688957	1	2	yes	—

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₄₇
commutator	C₂×C₄₇	C₂×C₄×C₁₆
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₂×C₄×C₁₆×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₁₆
cosocle	C₂×C₁₆×C₄₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₄₇
hypocenter	C₄₇	C₂×#1397
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20241522	C₂	direct (N × H)
4. C₂×C₄₇⋊C₃₂	C₄	semidirect
11. C₄₇⋊C₃₂	C₂×C₄	semidirect
57. C₂×C₄₇	#1397	semidirect
67. C₄₇	C₂×#1397	semidirect
110. C₂	#20241522	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×C₄₇⋊C₃₂	2
#20241522	2
C₂×C₄×C₁₆×C₄₇	2
C₂×#1397	47

Subgroups — C2×#20241522 #26136451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241522 #26136451